More on Morphisms

Introduction

In this chapter we continue our study of properties of morphisms of schemes. A fundamental reference is [EGA].

Thickenings

The following terminology may not be completely standard, but it is convenient.

Definition

Thickenings.

We say a scheme \(X'\) is a thickening of a scheme \(X\) if \(X\) is a closed subscheme of \(X'\) and the underlying topological spaces are equal.

We say a scheme \(X'\) is a first order thickening of a scheme \(X\) if \(X\) is a closed subscheme of \(X'\) and the quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_{X'}\) defining \(X\) has square zero.

We say a scheme \(X'\) is a finite order thickening of a scheme \(X\) if \(X\) is a closed subscheme of \(X'\) and the quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_{X'}\) defining \(X\) is nilpotent, i.e., there exists an integer \(n \geq 0\) such that \(\mathcal{I}^{n + 1} = 0\).

We say a scheme \(X'\) is an \(n\)th order thickening of a scheme \(X\) if \(X\) is a closed subscheme of \(X'\) and \(\mathcal{I}^{n + 1} = 0\) where \(\mathcal{I} \subset \mathcal{O}_{X'}\) is the quasi-coherent sheaf of ideals defining \(X\).

Given two thickenings \(X \subset X'\) and \(Y \subset Y'\) a morphism of thickenings is a morphism \(f' : X' \to Y'\) such that \(f'(X) \subset Y\), i.e., such that \(f'|_X\) factors through the closed subscheme \(Y\). In this situation we set \(f = f'|_X : X \to Y\) and we say that \((f, f') : (X \subset X') \to (Y \subset Y')\) is a morphism of thickenings.

Let \(S\) be a scheme. We similarly define thickenings over \(S\), and morphisms of thickenings over \(S\). This means that the schemes \(X, X', Y, Y'\) above are schemes over \(S\), and that the morphisms \(X \to X'\), \(Y \to Y'\) and \(f' : X' \to Y'\) are morphisms over \(S\).

Let \(i_X : X \to X'\) be a thickening. Any local section of the kernel \(\mathcal{I} = \Ker(i_X^\sharp)\) is locally nilpotent. This gives some control over the structure sheaf of \(\mathcal{O}_{X'}\), but technically the class of finite order thickenings \(X \subset X'\) is much easier to handle. Namely, if \(X'\) is an \(n\)th order thickening then we have a filtration \[0 = \mathcal{I}^{n + 1} \subset \mathcal{I}^n \subset \mathcal{I}^{n - 1} \subset \ldots \subset \mathcal{I} \subset \mathcal{O}_{X'}\] and we see that \(X'\) is filtered by closed subspaces \[X = X_1 \subset X_2 \subset \ldots \subset X_n \subset X_{n + 1} = X'\] such that each pair \(X_i \subset X_{i + 1}\) is a first order thickening over \(S\). Using simple induction arguments many results proved for first order thickenings can be rephrased as results on finite order thickenings.

First order thickening are described as follows (see Modules, Lemma 01UP).

Lemma

Let \(X\) be a scheme over a base \(S\). Consider a short exact sequence \[0 \to \mathcal{I} \to \mathcal{A} \to \mathcal{O}_X \to 0\] of sheaves on \(X\) where \(\mathcal{A}\) is a sheaf of \(f^{-1}\mathcal{O}_S\)-algebras, \(\mathcal{A} \to \mathcal{O}_X\) is a surjection of sheaves of \(f^{-1}\mathcal{O}_S\)-algebras, and \(\mathcal{I}\) is its kernel. If

\(\mathcal{I}\) is an ideal of square zero in \(\mathcal{A}\), and

\(\mathcal{I}\) is quasi-coherent as an \(\mathcal{O}_X\)-module

then \(X' = (X, \mathcal{A})\) is a scheme and \(X \to X'\) is a first order thickening over \(S\). Moreover, any first order thickening over \(S\) is of this form.

Proof

It is clear that \(X'\) is a locally ringed space. Let \(U = \Spec(B)\) be an affine open of \(X\). Set \(A = \Gamma(U, \mathcal{A})\). Note that since \(H^1(U, \mathcal{I}) = 0\) (see Cohomology of Schemes, Lemma 01XB) the map \(A \to B\) is surjective. By assumption the kernel \(I = \mathcal{I}(U)\) is an ideal of square zero in the ring \(A\). By Schemes, Lemma 01I1 there is a canonical morphism of locally ringed spaces \[(U, \mathcal{A}|_U) \longrightarrow \Spec(A)\] coming from the map \(B \to \Gamma(U, \mathcal{A})\). Since this morphism fits into the commutative diagram \[\xymatrix{ (U, \mathcal{O}_X|_U) \ar[d] \ar[r] & \Spec(B) \ar[d] \\ (U, \mathcal{A}|_U) \ar[r] & \Spec(A) }\] we see that it is a homeomorphism on underlying topological spaces. Thus to see that it is an isomorphism, it suffices to check it induces an isomorphism on the local rings. For \(u \in U\) corresponding to the prime \(\mathfrak p \subset A\) we obtain a commutative diagram of short exact sequences \[\xymatrix{ 0 \ar[r] & I_{\mathfrak p} \ar[r] \ar[d] & A_{\mathfrak p} \ar[r] \ar[d] & B_{\mathfrak p} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{I}_u \ar[r] & \mathcal{A}_u \ar[r] & \mathcal{O}_{X, u} \ar[r] & 0. }\] The left and right vertical arrows are isomorphisms because \(\mathcal{I}\) and \(\mathcal{O}_X\) are quasi-coherent sheaves. Hence also the middle map is an isomorphism. Hence every point of \(X' = (X, \mathcal{A})\) has an affine neighbourhood and \(X'\) is a scheme as desired.

Lemma

Any thickening of an affine scheme is affine.

Proof

This is a special case of Limits, Proposition 05YU.

Proof

Suppose that \(X \subset X'\) is a finite order thickening with \(X\) affine. Then we may use Serre’s criterion to prove \(X'\) is affine. More precisely, we will use Cohomology of Schemes, Lemma 01XF. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_{X'}\)-module. It suffices to show that \(H^1(X', \mathcal{F}) = 0\). Denote \(i : X \to X'\) the given closed immersion and denote \(\mathcal{I} = \Ker(i^\sharp : \mathcal{O}_{X'} \to i_*\mathcal{O}_X)\). By our discussion of finite order thickenings (following Definition 04EX) there exists an \(n \geq 0\) and a filtration \[0 = \mathcal{F}_{n + 1} \subset \mathcal{F}_n \subset \mathcal{F}_{n - 1} \subset \ldots \subset \mathcal{F}_0 = \mathcal{F}\] by quasi-coherent submodules such that \(\mathcal{F}_a/\mathcal{F}_{a + 1}\) is annihilated by \(\mathcal{I}\). Namely, we can take \(\mathcal{F}_a = \mathcal{I}^a\mathcal{F}\). Then \(\mathcal{F}_a/\mathcal{F}_{a + 1} = i_*\mathcal{G}_a\) for some quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{G}_a\), see Morphisms, Lemma 01QY. We obtain \[H^1(X', \mathcal{F}_a/\mathcal{F}_{a + 1}) = H^1(X', i_*\mathcal{G}_a) = H^1(X, \mathcal{G}_a) = 0\] The second equality comes from Cohomology of Schemes, Lemma 089W and the last equality from Cohomology of Schemes, Lemma 01XB. Thus \(\mathcal{F}\) has a finite filtration whose successive quotients have vanishing first cohomology and it follows by a simple induction argument that \(H^1(X', \mathcal{F}) = 0\).

Lemma

Let \(S \subset S'\) be a thickening of schemes. Let \(X' \to S'\) be a morphism and set \(X = S \times_{S'} X'\). Then \((X \subset X') \to (S \subset S')\) is a morphism of thickenings. If \(S \subset S'\) is a first (resp. finite order) thickening, then \(X \subset X'\) is a first (resp. finite order) thickening.

Proof

Omitted.

Lemma

If \(S \subset S'\) and \(S' \subset S''\) are thickenings, then so is \(S \subset S''\).

Proof

Omitted.

Lemma

The property of being a thickening is fpqc local. Similarly for first order thickenings.

Proof

The statement means the following: Let \(X \to X'\) be a morphism of schemes and let \(\{g_i : X'_i \to X'\}\) be an fpqc covering such that the base change \(X_i \to X'_i\) is a thickening for all \(i\). Then \(X \to X'\) is a thickening. Since the morphisms \(g_i\) are jointly surjective we conclude that \(X \to X'\) is surjective. By Descent, Lemma 02L6 we conclude that \(X \to X'\) is a closed immersion. Thus \(X \to X'\) is a thickening. We omit the proof in the case of first order thickenings.

Morphisms of thickenings

If \((f, f') : (X \subset X') \to (Y \subset Y')\) is a morphism of thickenings of schemes, then often properties of the morphism \(f\) are inherited by \(f'\). There are several variants.

Lemma

Let \((f, f') : (X \subset X') \to (S \subset S')\) be a morphism of thickenings. Then

\(f\) is an affine morphism if and only if \(f'\) is an affine morphism,

\(f\) is a surjective morphism if and only if \(f'\) is a surjective morphism,

\(f\) is quasi-compact if and only if \(f'\) quasi-compact,

\(f\) is universally closed if and only if \(f'\) is universally closed,

\(f\) is integral if and only if \(f'\) is integral,

\(f\) is (quasi-)separated if and only if \(f'\) is (quasi-)separated,

\(f\) is universally injective if and only if \(f'\) is universally injective,

\(f\) is universally open if and only if \(f'\) is universally open,

\(f\) is quasi-affine if and only if \(f'\) is quasi-affine, and

add more here.

Proof

Observe that \(S \to S'\) and \(X \to X'\) are universal homeomorphisms (see for example Morphisms, Lemma 054M). This immediately implies parts (2), (3), (4), (7), and (8). Part (1) follows from Lemma 06AD which tells us that there is a 1-to-1 correspondence between affine opens of \(S\) and \(S'\) and between affine opens of \(X\) and \(X'\). Part (9) follows from Limits, Lemma 0B7L and the remark just made about affine opens of \(S\) and \(S'\). Part (5) follows from (1) and (4) by Morphisms, Lemma 01WM. Finally, note that \[S \times_X S = S \times_{X'} S \to S \times_{X'} S' \to S' \times_{X'} S'\] is a thickening (the two arrows are thickenings by Lemma 09ZU). Hence applying (3) and (4) to the morphism \((S \subset S') \to (S \times_X S \to S' \times_{X'} S')\) we obtain (6).

Lemma

Let \((f, f') : (X \subset X') \to (S \subset S')\) be a morphism of thickenings. Let \(\mathcal{L}'\) be an invertible sheaf on \(X'\) and denote \(\mathcal{L}\) the restriction to \(X\). Then \(\mathcal{L}'\) is \(f'\)-ample if and only if \(\mathcal{L}\) is \(f\)-ample.

Proof

Recall that being relatively ample is a condition for each affine open in the base, see Morphisms, Definition 01VH. By Lemma 06AD there is a 1-to-1 correspondence between affine opens of \(S\) and \(S'\). Thus we may assume \(S\) and \(S'\) are affine and we reduce to proving that \(\mathcal{L}'\) is ample if and only if \(\mathcal{L}\) is ample. This is Limits, Lemma 09MW.

Lemma

Let \((f, f') : (X \subset X') \to (S \subset S')\) be a morphism of thickenings such that \(X = S \times_{S'} X'\). If \(S \subset S'\) is a finite order thickening, then

\(f\) is a closed immersion if and only if \(f'\) is a closed immersion,

\(f\) is locally of finite type if and only if \(f'\) is locally of finite type,

\(f\) is locally quasi-finite if and only if \(f'\) is locally quasi-finite,

\(f\) is locally of finite type of relative dimension \(d\) if and only if \(f'\) is locally of finite type of relative dimension \(d\),

\(\Omega_{X/S} = 0\) if and only if \(\Omega_{X'/S'} = 0\),

\(f\) is unramified if and only if \(f'\) is unramified,

\(f\) is proper if and only if \(f'\) is proper,

\(f\) is finite if and only if \(f'\) is finite,

\(f\) is a monomorphism if and only if \(f'\) is a monomorphism,

\(f\) is an immersion if and only if \(f'\) is an immersion, and

add more here.

Proof

The properties \(\mathcal{P}\) listed in the lemma are all stable under base change, hence if \(f'\) has property \(\mathcal{P}\), then so does \(f\). See Schemes, Lemmas 01JY and 02YC and Morphisms, Lemmas 01T4, 01TM, 02NK, 01V0, 02GA, 01W4, and 01WL.

The interesting direction in each case is therefore to assume that \(f\) has the property and deduce that \(f'\) has it too. By induction on the order of the thickening we may assume that \(S \subset S'\) is a first order thickening, see discussion immediately following Definition 04EX.

Most of the proofs will use a reduction to the affine case. Let \(U' \subset S'\) be an affine open and let \(V' \subset X'\) be an affine open lying over \(U'\). Let \(U' = \Spec(A')\) and denote \(I \subset A'\) be the ideal defining the closed subscheme \(U' \cap S\). Say \(V' = \Spec(B')\). Then \(V' \cap X = \Spec(B'/IB')\). Setting \(A = A'/I\) and \(B = B'/IB'\) we get a commutative diagram \[\xymatrix{ 0 \ar[r] & IB' \ar[r] & B' \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & IA' \ar[r] \ar[u] & A' \ar[r] \ar[u] & A \ar[r] \ar[u] & 0 }\] with exact rows and \(I^2 = 0\).

The translation of (1) into algebra: If \(A \to B\) is surjective, then \(A' \to B'\) is surjective. This follows from Nakayama’s lemma (Algebra, Lemma 00DV).

The translation of (2) into algebra: If \(A \to B\) is a finite type ring map, then \(A' \to B'\) is a finite type ring map. This follows from Nakayama’s lemma (Algebra, Lemma 00DV) applied to a map \(A'[x_1, \ldots, x_n] \to B'\) such that \(A[x_1, \ldots, x_n] \to B\) is surjective.

Proof of (3). Follows from (2) and that quasi-finiteness of a morphism which is locally of finite type can be checked on fibres, see Morphisms, Lemma 01TH.

Proof of (4). Follows from (2) and that the additional property of “being of relative dimension \(d\)” can be checked on fibres (by definition, see Morphisms, Definition 02NJ.

The translation of (5) into algebra: If \(\Omega_{B/A} = 0\), then \(\Omega_{B'/A'} = 0\). By Algebra, Lemma 00RV we have \(0 = \Omega_{B/A} = \Omega_{B'/A'}/I\Omega_{B'/A'}\). Hence \(\Omega_{B'/A'} = 0\) by Nakayama’s lemma (Algebra, Lemma 00DV).

The translation of (6) into algebra: If \(A \to B\) is unramified map, then \(A' \to B'\) is unramified. Since \(A \to B\) is of finite type we see that \(A' \to B'\) is of finite type by (2) above. Since \(A \to B\) is unramified we have \(\Omega_{B/A} = 0\). By part (5) we have \(\Omega_{B'/A'} = 0\). Thus \(A' \to B'\) is unramified.

Proof of (7). Follows by combining (2) with results of Lemma 09ZV and the fact that proper equals quasi-compact \(+\) separated \(+\) locally of finite type \(+\) universally closed.

Proof of (8). Follows by combining (2) with results of Lemma 09ZV and using the fact that finite equals integral \(+\) locally of finite type (Morphisms, Lemma 01WJ).

Proof of (9). As \(f\) is a monomorphism we have \(X = X \times_S X\). We may apply the results proved so far to the morphism of thickenings \((X \subset X') \to (X \times_S X \subset X' \times_{S'} X')\). We conclude \(X' \to X' \times_{S'} X'\) is a closed immersion by (1). In fact, it is a first order thickening as the ideal defining the closed immersion \(X' \to X' \times_{S'} X'\) is contained in the pullback of the ideal \(\mathcal{I} \subset \mathcal{O}_{S'}\) cutting out \(S\) in \(S'\). Indeed, \(X = X \times_S X = (X' \times_{S'} X') \times_{S'} S\) is contained in \(X'\). Hence by Morphisms, Lemma 08S2 it suffices to show that \(\Omega_{X'/S'} = 0\) which follows from (5) and the corresponding statement for \(X/S\).

Proof of (10). If \(f : X \to S\) is an immersion, then it factors as \(X \to U \to S\) where \(U \to S\) is an open immersion and \(X \to U\) is a closed immersion. Let \(U' \subset S'\) be the open subscheme whose underlying topological space is the same as \(U\). Then \(X' \to S'\) factors through \(U'\) and we conclude that \(X' \to U'\) is a closed immersion by part (1). This finishes the proof.

The following lemma is a variant on the preceding one. Rather than assume that the thickenings involved are finite order (which allows us to transfer the property of being locally of finite type from \(f\) to \(f'\)), we instead take as given that each of \(f\) and \(f'\) is locally of finite type.

Lemma

Let \((f, f') : (X \subset X') \to (Y \to Y')\) be a morphism of thickenings. Assume \(f\) and \(f'\) are locally of finite type and \(X = Y \times_{Y'} X'\). Then

\(f\) is locally quasi-finite if and only if \(f'\) is locally quasi-finite,

\(f\) is finite if and only if \(f'\) is finite,

\(f\) is a closed immersion if and only if \(f'\) is a closed immersion,

\(\Omega_{X/Y} = 0\) if and only if \(\Omega_{X'/Y'} = 0\),

\(f\) is unramified if and only if \(f'\) is unramified,

\(f\) is a monomorphism if and only if \(f'\) is a monomorphism,

\(f\) is an immersion if and only if \(f'\) is an immersion,

\(f\) is proper if and only if \(f'\) is proper, and

add more here.

Proof

The properties \(\mathcal{P}\) listed in the lemma are all stable under base change, hence if \(f'\) has property \(\mathcal{P}\), then so does \(f\). See Schemes, Lemmas 01JY and 02YC and Morphisms, Lemmas 01TM, 02NK, 01V0, 02GA, 01W4, and 01WL. Hence in each case we need only to prove that if \(f\) has the desired property, so does \(f'\).

A morphism is locally quasi-finite if and only if it is locally of finite type and the scheme theoretic fibres are discrete spaces, see Morphisms, Lemma 06RT. Since the underlying topological space is unchanged by passing to a thickening, we see that \(f'\) is locally quasi-finite if (and only if) \(f\) is. This proves (1).

Case (2) follows from case (5) of Lemma 09ZV and the fact that the finite morphisms are precisely the integral morphisms that are locally of finite type (Morphisms, Lemma 01WJ).

Case (3). This follows immediately from Morphisms, Lemma 0896.

Case (4) follows from the following algebra statement: Let \(A\) be a ring and let \(I \subset A\) be a locally nilpotent ideal. Let \(B\) be a finite type \(A\)-algebra. If \(\Omega_{(B/IB)/(A/I)} = 0\), then \(\Omega_{B/A} = 0\). Namely, the assumption means that \(I\Omega_{B/A} = 0\), see Algebra, Lemma 00RV. On the other hand \(\Omega_{B/A}\) is a finite \(B\)-module, see Algebra, Lemma 00RZ. Hence the vanishing of \(\Omega_{B/A}\) follows from Nakayama’s lemma (Algebra, Lemma 00DV) and the fact that \(IB\) is contained in the Jacobson radical of \(B\).

Case (5) follows immediately from (4) and Morphisms, Lemma 02G5.

Proof of (6). As \(f\) is a monomorphism we have \(X = X \times_Y X\). We may apply the results proved so far to the morphism of thickenings \((X \subset X') \to (X \times_Y X \subset X' \times_{Y'} X')\). We conclude \(\Delta_{X'/Y'} : X' \to X' \times_{Y'} X'\) is a closed immersion by (3). In fact \(\Delta_{X'/Y'}\) is a bijection on underlying sets, hence \(\Delta_{X'/Y'}\) is a thickening. On the other hand \(\Delta_{X'/Y'}\) is locally of finite presentation by Morphisms, Lemma 0818. In other words, \(\Delta_{X'/Y'}(X')\) is cut out by a quasi-coherent sheaf of ideals \(\mathcal{J} \subset \mathcal{O}_{X' \times_{Y'} X'}\) of finite type. Since \(\Omega_{X'/Y'} = 0\) by (5) we see that the conormal sheaf of \(X' \to X' \times_{Y'} X'\) is zero by Morphisms, Lemma 08S2. In other words, \(\mathcal{J}/\mathcal{J}^2 = 0\). This implies \(\Delta_{X'/Y'}\) is an isomorphism, for example by Algebra, Lemma 00EH.

Proof of (7). If \(f : X \to Y\) is an immersion, then it factors as \(X \to V \to Y\) where \(V \to Y\) is an open immersion and \(X \to V\) is a closed immersion. Let \(V' \subset Y'\) be the open subscheme whose underlying topological space is the same as \(V\). Then \(X' \to V'\) factors through \(V'\) and we conclude that \(X' \to V'\) is a closed immersion by part (3).

Case (8) follows from Lemma 09ZV and the definition of proper morphisms as being the quasi-compact, universally closed, and separated morphisms that are locally of finite type.

Picard groups of thickenings

Some material on Picard groups of thickenings.

Lemma

Let \(X \subset X'\) be a first order thickening with ideal sheaf \(\mathcal{I}\). Then there is a canonical exact sequence \[\xymatrix{ 0 \ar[r] & H^0(X, \mathcal{I}) \ar[r] & H^0(X', \mathcal{O}_{X'}^*) \ar[r] & H^0(X, \mathcal{O}^*_X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^1(X, \mathcal{I}) \ar[r] & \Pic(X') \ar[r] & \Pic(X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^2(X, \mathcal{I}) \ar[r] & \ldots \ar[r] & \ldots }\] of abelian groups.

Proof

This is the long exact cohomology sequence associated to the short exact sequence of sheaves of abelian groups \[0 \to \mathcal{I} \to \mathcal{O}_{X'}^* \to \mathcal{O}_X^* \to 0\] where the first map sends a local section \(f\) of \(\mathcal{I}\) to the invertible section \(1 + f\) of \(\mathcal{O}_{X'}\). We also use the identification of the Picard group of a ringed space with the first cohomology group of the sheaf of invertible functions, see Cohomology, Lemma 09NU.

Lemma

Let \(X \subset X'\) be a thickening. Let \(n\) be an integer invertible in \(\mathcal{O}_X\). Then the map \(\Pic(X')[n] \to \Pic(X)[n]\) is bijective.

Proof

By the general principle explained following Definition 04EX this reduces to the case of a first order thickening. Then may use Lemma 0C6R to see that it suffices to show that \(H^1(X, \mathcal{I})[n]\), \(H^1(X, \mathcal{I})/n\), and \(H^2(X, \mathcal{I})[n]\) are zero. This follows as multiplication by \(n\) on \(\mathcal{I}\) is an isomorphism as it is an \(\mathcal{O}_X\)-module.

Proof

Let \(\mathcal{I} \subset \mathcal{O}_{X'}\) be the quasi-coherent ideal sheaf cutting out \(X\). Then we have a short exact sequence of abelian groups \[0 \to (1 + \mathcal{I})^* \to \mathcal{O}_{X'}^* \to \mathcal{O}_X^* \to 0\] We obtain a long exact cohomology sequence as in the statement of Lemma 0C6R with \(H^i(X, \mathcal{I})\) replaced by \(H^i(X, (1 + \mathcal{I})^*)\). Thus it suffices to show that raising to the \(n\)th power is an isomorphism \((1 + \mathcal{I})^* \to (1 + \mathcal{I})^*\). Taking sections over affine opens this follows from Algebra, Lemma 0CAP.

Lemma

Let \(X \subset X'\) be a first order thickening. Let \(n\) be an integer which is invertible on \(X\). Then the diagram \[\xymatrix{ \Pic(X') \ar[r]^{[n]} \ar[d] & \Pic(X') \ar[d] \\ \Pic(X) \ar[r]^{[n]} & \Pic(X) }\] is cartesian.

Proof

Let \(\mathcal{I}\) be the ideal sheaf of \(X\) in \(X'\). We use the exact sequence of Lemma 0C6R. Suppose that \(a \in \Pic(X')\) and \(b \in \Pic(X)\) satisfy \(a|_X = nb\). The obstruction \(\partial(b) \in H^2(X, \mathcal{I})\) to lifting \(b\) satisfies \[n\partial(b) = \partial(nb) = \partial(a|_X) = 0.\] Multiplication by \(n\) is an automorphism of \(\mathcal{I}\) and hence of \(H^2(X, \mathcal{I})\). Thus \(b\) lifts to an element \(b' \in \Pic(X')\). The element \(a - nb'\) is in the kernel of \(\Pic(X') \to \Pic(X)\), so it is the image of some \(\xi \in H^1(X, \mathcal{I})\). Write \(\xi = n\eta\), which is possible because multiplication by \(n\) is also an automorphism of \(H^1(X, \mathcal{I})\). Adding the image of \(\eta\) to \(b'\) produces a lift of \(b\) whose \(n\)-fold multiple is \(a\).

If two such lifts exist, their difference is an element of \(\Pic(X')[n]\) whose restriction to \(X\) is zero. This difference is zero by Lemma 0C6S. Thus the lift is unique.

Picard groups of projective bundles

Lemma

Let \(S\) be a connected scheme. Let \(n \geq 1\) and let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_S\)-module of rank \(n + 1\). If \(\pi : \mathbf{P}(\mathcal{E}) \to S\) is the associated projective bundle, then \[\Pic(S) \oplus \mathbf{Z} \longrightarrow \Pic(\mathbf{P}(\mathcal{E})), \qquad ([\mathcal{N}], d) \longmapsto [\pi^*\mathcal{N} \otimes \mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)]\] is an isomorphism.

Proof

Let \(\mathcal{L}\) be an invertible module on \(\mathbf{P}(\mathcal{E})\) and let \(s \in S\). By Divisors, Lemma 0BXJ, there is a unique integer \(d_s\) such that \[\mathcal{L}|_{\mathbf{P}(\mathcal{E}_s)} \cong \mathcal{O}_{\mathbf{P}(\mathcal{E}_s)}(d_s).\] Set \(\mathcal{F} = \mathcal{L}(-d_s)\). The cohomology of \(\mathcal{F}_s\) is \(\kappa(s)\) in degree \(0\) and zero in all other degrees by Cohomology of Schemes, Lemma 01XT. By Derived Categories of Schemes, Lemma 0B9S, after shrinking \(S\) about \(s\) we have \[R\pi_*\mathcal{F} = \mathcal{N}[0]\] for an invertible \(\mathcal{O}_S\)-module \(\mathcal{N}\), compatibly with base change. The evaluation map \(\pi^*\mathcal{N} \to \mathcal{F}\) is an isomorphism on the fibre over \(s\). Its isomorphism locus is open. Since \(\pi\) is proper, after shrinking \(S\) once more it is an isomorphism everywhere.

We have proved that \(s \mapsto d_s\) is locally constant and that locally on \(S\) every invertible module has the displayed form. Since \(S\) is connected, the integer \(d_s\) has a constant value \(d\). Taking \[\mathcal{N} = \pi_*(\mathcal{L}(-d))\] and using the local result shows globally that \(\mathcal{N}\) is invertible and \(\pi^*\mathcal{N} \to \mathcal{L}(-d)\) is an isomorphism. This proves surjectivity. The integer is unique after restriction to a fibre. Once it is fixed, \(\mathcal{N}\) is recovered by applying \(\pi_*\), the projection formula, and Cohomology of Schemes, Lemma 01XX. This proves injectivity.

Lemma

In the situation of Lemma more-morphisms-lemma-picard-group-projective-bundle, set \[\Gamma_{\mathcal{E}} = \{[\mathcal{N}] \in \Pic(S) \mid \mathcal{E} \otimes \mathcal{N} \cong \mathcal{E}\}.\] This is a subgroup of \(\Pic(S)\), and there is an exact sequence \[1 \longrightarrow \Gamma(S, \mathcal{O}_S^*) \longrightarrow \text{Aut}_{\mathcal{O}_S}(\mathcal{E}) \longrightarrow \text{Aut}_S(\mathbf{P}(\mathcal{E})) \longrightarrow \Gamma_{\mathcal{E}} \longrightarrow 1.\] The first map sends a unit to the corresponding scalar automorphism. Every element of \(\Gamma_{\mathcal{E}}\) is annihilated by \(n + 1\) in \(\Pic(S)\).

Proof

Let \(X = \mathbf{P}(\mathcal{E})\) and let \(u\) be an \(S\)-automorphism of \(X\). By Lemma more-morphisms-lemma-picard-group-projective-bundle, there are a unique integer \(d\) and an invertible module \(\mathcal{N}\) on \(S\) such that \[u^*\mathcal{O}_X(1) \cong \mathcal{O}_X(d) \otimes \pi^*\mathcal{N}.\] On a fibre, pullback by \(u\) is an automorphism of the Picard group \(\mathbf{Z}\) and preserves ample invertible modules. Hence \(d=1\). Applying \(\pi_*\), the projection formula, and Cohomology of Schemes, Lemma 01XX, gives \[\mathcal{E} \cong \mathcal{E} \otimes \mathcal{N}.\] Thus \(u \mapsto [\mathcal{N}]\) defines the fourth arrow in the sequence. It is a group homomorphism because pullback is compatible with composition and \(u\) acts trivially on invertible modules pulled back from \(S\).

If \([\mathcal{N}]\) is trivial, a choice of the displayed isomorphism and the universal quotient \(\pi^*\mathcal{E} \to \mathcal{O}_X(1)\) show that \(u\) is induced by an automorphism of \(\mathcal{E}\). Two such automorphisms induce the same automorphism of \(X\) exactly when they differ by a scalar unit. This can be checked after trivializing \(\mathcal{E}\) by comparing the universal quotient. Conversely, an isomorphism \(\mathcal{E} \cong \mathcal{E} \otimes \mathcal{N}\), together with the canonical identification \(\mathbf{P}(\mathcal{E} \otimes \mathcal{N}) \cong \mathbf{P}(\mathcal{E})\), gives an \(S\)-automorphism of \(X\). This proves exactness and also shows that \(\Gamma_{\mathcal{E}}\) is a subgroup.

Finally, taking determinants in \(\mathcal{E} \cong \mathcal{E} \otimes \mathcal{N}\) gives \(\mathcal{N}^{\otimes(n + 1)} \cong \mathcal{O}_S\).

Infinitesimal neighbourhoods

A natural construction of finite order thickenings is the following. Suppose that \(i : Z \to X\) be an immersion of schemes. Choose an open subscheme \(U \subset X\) such that \(i\) identifies \(Z\) with a closed subscheme \(Z \subset U\). Let \(\mathcal{I} \subset \mathcal{O}_U\) be the quasi-coherent sheaf of ideals defining \(Z\) in \(U\). For \(n \geq 1\) we can consider the closed subscheme \(Z_n \subset U\) defined by the quasi-coherent sheaf of ideals \(\mathcal{I}^{n + 1}\).

Definition

Let \(i : Z \to X\) be an immersion of schemes.

The first order infinitesimal neighbourhood of \(Z\) in \(X\) is the first order thickening \(Z \subset Z_1\) over \(X\) described above.

The \(n\)th order infinitesimal neighbourhood of \(Z\) in \(X\) is the \(n\)th order thickening \(Z \subset Z_n\) over \(X\) described above.

These thickenings have the following universal property (which will assuage any fears that the construction above depends on the choice of the open \(U\)).

Lemma

Let \(i : Z \to X\) be an immersion of schemes.

The first order infinitesimal neighbourhood \(Z'\) of \(Z\) in \(X\) has the following universal property: Given any commutative diagram \[\xymatrix{ Z \ar[d]_i & T \ar[l]^a \ar[d] \\ X & T' \ar[l]_b }\] where \(T \subset T'\) is a first order thickening over \(X\), there exists a unique morphism \((a', a) : (T \subset T') \to (Z \subset Z')\) of thickenings over \(X\).

For \(n \geq 1\) the \(n\)th order infinitesimal neighbourhood \(Z_n\) of \(Z\) in \(X\) has the following universal property: Given any commutative diagram \[\xymatrix{ Z \ar[d]_i & T \ar[l]^a \ar[d] \\ X & T' \ar[l]_b }\] where \(T \subset T'\) is an \(n\)th order thickening over \(X\), there exists a unique morphism \((a', a) : (T \subset T') \to (Z \subset Z_n)\) of thickenings over \(X\).

Proof

We will only prove (1). Let \(U \subset X\) be the open used in the construction of \(Z'\), i.e., an open such that \(Z\) is identified with a closed subscheme of \(U\) cut out by the quasi-coherent sheaf of ideals \(\mathcal{I}\). Since \(|T| = |T'|\) we see that \(b(T') \subset U\). Hence we can think of \(b\) as a morphism into \(U\). Let \(\mathcal{J} \subset \mathcal{O}_{T'}\) be the ideal cutting out \(T\). Since \(b(T) \subset Z\) by the diagram above we see that \(b^\sharp(b^{-1}\mathcal{I}) \subset \mathcal{J}\). As \(T'\) is a first order thickening of \(T\) we see that \(\mathcal{J}^2 = 0\) hence \(b^\sharp(b^{-1}(\mathcal{I}^2)) = 0\). By Schemes, Lemma 01HP this implies that \(b\) factors through \(Z'\). Denote \(a' : T' \to Z'\) this factorization and everything is clear.

Lemma

Let \(i : Z \to X\) be an immersion of schemes. Let \(Z \subset Z'\) be the first order infinitesimal neighbourhood of \(Z\) in \(X\). Then the diagram \[\xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Z \ar[r] & X }\] induces a map of conormal sheaves \(\mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Z'}\) by Morphisms, Lemma 01R4. This map is an isomorphism.

Proof

This is clear from the construction of \(Z'\) above.

Formally unramified morphisms

Recall that a ring map \(R \to A\) is called formally unramified (see Algebra, Definition 00UN) if for every commutative solid diagram \[\xymatrix{ A \ar[r] \ar@{-->}[rd] & B/I \\ R \ar[r] \ar[u] & B \ar[u] }\] where \(I \subset B\) is an ideal of square zero, at most one dotted arrow exists which makes the diagram commute. This motivates the following analogue for morphisms of schemes.

Definition

Let \(f : X \to S\) be a morphism of schemes. We say \(f\) is formally unramified if given any solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] }\] where \(T \subset T'\) is a first order thickening of affine schemes over \(S\) there exists at most one dotted arrow making the diagram commute.

We first prove some formal lemmas, i.e., lemmas which can be proved by drawing the corresponding diagrams.

Lemma

If \(f : X \to S\) is a formally unramified morphism, then given any solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] }\] where \(T \subset T'\) is a first order thickening of schemes over \(S\) there exists at most one dotted arrow making the diagram commute. In other words, in Definition 02H8 the condition that \(T\) be affine may be dropped.

Proof

This is true because a morphism is determined by its restrictions to affine opens.

Lemma

A composition of formally unramified morphisms is formally unramified.

Proof

This is formal.

Lemma

A base change of a formally unramified morphism is formally unramified.

Proof

This is formal.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(U \subset X\) and \(V \subset S\) be open such that \(f(U) \subset V\). If \(f\) is formally unramified, so is \(f|_U : U \to V\).

Proof

Consider a solid diagram \[\xymatrix{ U \ar[d]_{f|_U} & T \ar[d]^i \ar[l]^a \\ V & T' \ar[l] \ar@{-->}[lu] }\] as in Definition 02H8. If \(f\) is formally ramified, then there exists at most one \(S\)-morphism \(a' : T' \to X\) such that \(a'|_T = a\). Hence clearly there exists at most one such morphism into \(U\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume \(X\) and \(S\) are affine. Then \(f\) is formally unramified if and only if \(\mathcal{O}_S(S) \to \mathcal{O}_X(X)\) is a formally unramified ring map.

Proof

This is immediate from the definitions (Definition 02H8 and Algebra, Definition 00UN) by the equivalence of categories of rings and affine schemes, see Schemes, Lemma 01I2.

Here is a characterization in terms of the sheaf of differentials.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Then \(f\) is formally unramified if and only if \(\Omega_{X/S} = 0\).

Proof

We recall some of the arguments of the proof of Morphisms, Lemma 01UT. Let \(W \subset X \times_S X\) be an open such that \(\Delta : X \to X \times_S X\) induces a closed immersion into \(W\). Let \(\mathcal{J} \subset \mathcal{O}_W\) be the ideal sheaf of this closed immersion. Let \(X' \subset W\) be the closed subscheme defined by the quasi-coherent sheaf of ideals \(\mathcal{J}^2\). Consider the two morphisms \(p_1, p_2 : X' \to X\) induced by the two projections \(X \times_S X \to X\). Note that \(p_1\) and \(p_2\) agree when composed with \(\Delta : X \to X'\) and that \(X \to X'\) is a closed immersion defined by a an ideal whose square is zero. Moreover there is a short exact sequence \[0 \to \mathcal{J}/\mathcal{J}^2 \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\] and \(\Omega_{X/S} = \mathcal{J}/\mathcal{J}^2\). Moreover, \(\mathcal{J}/\mathcal{J}^2\) is generated by the local sections \(p_1^\sharp(f) - p_2^\sharp(f)\) for \(f\) a local section of \(\mathcal{O}_X\).

Suppose that \(f : X \to S\) is formally unramified. By assumption this means that \(p_1 = p_2\) when restricted to any affine open \(T' \subset X'\). Hence \(p_1 = p_2\). By what was said above we conclude that \(\Omega_{X/S} = \mathcal{J}/\mathcal{J}^2 = 0\).

Conversely, suppose that \(\Omega_{X/S} = 0\). Then \(X' = X\). Take any pair of morphisms \(f'_1, f'_2 : T' \to X\) fitting as dotted arrows in the diagram of Definition 02H8. This gives a morphism \((f'_1, f'_2) : T' \to X \times_S X\). Since \(f'_1|_T = f'_2|_T\) and \(|T| =|T'|\) we see that the image of \(T'\) under \((f'_1, f'_2)\) is contained in the open \(W\) chosen above. Since \((f'_1, f'_2)(T) \subset \Delta(X)\) and since \(T\) is defined by an ideal of square zero in \(T'\) we see that \((f'_1, f'_2)\) factors through \(X'\). As \(X' = X\) we conclude \(f_1' = f'_2\) as desired.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent:

\(f\) is formally unramified,

for every \(x \in X\) there exist opens \(x \in U \subset X\) and \(f(x) \in V \subset Y\) with \(f(U) \subset V\) such that \(f|_U : U \to V\) is formally unramified,

for every pair of affine opens \(U \subset X\) and \(V \subset Y\) with \(f(U) \subset V\) the ring map \(\mathcal{O}_Y(V) \to \mathcal{O}_X(U)\) is formally unramified, and

there exists an affine open covering \(Y = \bigcup V_j\) and for each \(j\) an affine open covering \(f^{-1}(V_j) = \bigcup U_{ji}\) such that \(\mathcal{O}_Y(V) \to \mathcal{O}_X(U)\) is a formally unramified ring map for all \(j\) and \(i\).

Proof

Combining Lemmas 02HC and 02HD we see that (1) \(\Rightarrow\) (3). It is immediate that (3) \(\Rightarrow\) (4). We have (4) \(\Rightarrow\) (2) by Lemma 02HD. If (2) holds, then \(\Omega_{X/S}\) vanishes in a neighbourhood of every point by Lemma 02H9 (this also uses Morphisms, Lemma 01US) whereupon Lemma 02H9 tells us that \(f\) is formally unramified.

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent:

The morphism \(f\) is unramified (resp. G-unramified), and

the morphism \(f\) is locally of finite type (resp. locally of finite presentation) and formally unramified.

Proof

Use Lemma 02H9 and Morphisms, Lemma 02G5.

Universal first order thickenings

Let \(h : Z \to X\) be a morphism of schemes. A universal first order thickening of \(Z\) over \(X\) is a first order thickening \(Z \subset Z'\) over \(X\) such that given any first order thickening \(T \subset T'\) over \(X\) and a solid commutative diagram \[\xymatrix{ & Z \ar[ld] & & T \ar[rd] \ar[ll]^a \\ Z' \ar[rrd] & & & & T' \ar@{..>}[llll]_{a'} \ar[lld]^b \\ & & X }\] there exists a unique dotted arrow making the diagram commute. Note that in this situation \((a, a') : (T \subset T') \to (Z \subset Z')\) is a morphism of thickenings over \(X\). Thus if a universal first order thickening exists, then it is unique up to unique isomorphism. In general a universal first order thickening does not exist, but if \(h\) is formally unramified then it does.

Lemma

Let \(h : Z \to X\) be a formally unramified morphism of schemes. There exists a universal first order thickening \(Z \subset Z'\) of \(Z\) over \(X\).

Proof

During this proof we will say \(Z \subset Z'\) is a universal first order thickening of \(Z\) over \(X\) if it satisfies the condition of the lemma. We will construct the universal first order thickening \(Z \subset Z'\) over \(X\) by glueing, starting with the affine case which is Algebra, Lemma 04EB. We begin with some general remarks.

If a universal first order thickening of \(Z\) over \(X\) exists, then it is unique up to unique isomorphism. Moreover, suppose that \(V \subset Z\) and \(U \subset X\) are open subschemes such that \(h(V) \subset U\). Let \(Z \subset Z'\) be a universal first order thickening of \(Z\) over \(X\). Let \(V' \subset Z'\) be the open subscheme such that \(V = Z \cap V'\). Then we claim that \(V \subset V'\) is the universal first order thickening of \(V\) over \(U\). Namely, suppose given any diagram \[\xymatrix{ V \ar[d]_h & T \ar[l]^a \ar[d] \\ U & T' \ar[l]_b }\] where \(T \subset T'\) is a first order thickening over \(U\). By the universal property of \(Z'\) we obtain \((a, a') : (T \subset T') \to (Z \subset Z')\). But since we have equality \(|T| = |T'|\) of underlying topological spaces we see that \(a'(T') \subset V'\). Hence we may think of \((a, a')\) as a morphism of thickenings \((a, a') : (T \subset T') \to (V \subset V')\) over \(U\). Uniqueness is clear also. In a completely similar manner one proves that if \(h(Z) \subset U\) and \(Z \subset Z'\) is a universal first order thickening over \(U\), then \(Z \subset Z'\) is a universal first order thickening over \(X\).

Before we glue affine pieces let us show that the lemma holds if \(Z\) and \(X\) are affine. Say \(X = \Spec(R)\) and \(Z = \Spec(S)\). By Algebra, Lemma 04EB there exists a first order thickening \(Z \subset Z'\) over \(X\) which has the universal property of the lemma for diagrams \[\xymatrix{ Z \ar[d]_h & T \ar[l]^a \ar[d] \\ X & T' \ar[l]_b }\] where \(T, T'\) are affine. Given a general diagram we can choose an affine open covering \(T' = \bigcup T'_i\) and we obtain morphisms \(a'_i : T'_i \to Z'\) over \(X\) such that \(a'_i|_{T_i} = a|_{T_i}\). By uniqueness we see that \(a'_i\) and \(a'_j\) agree on any affine open of \(T'_i \cap T'_j\). Hence the morphisms \(a'_i\) glue to a global morphism \(a' : T' \to Z'\) over \(X\) as desired. Thus the lemma holds if \(X\) and \(Z\) are affine.

Choose an affine open covering \(Z = \bigcup Z_i\) such that each \(Z_i\) maps into an affine open \(U_i\) of \(X\). By Lemma 02HC the morphisms \(Z_i \to U_i\) are formally unramified. Hence by the affine case we obtain universal first order thickenings \(Z_i \subset Z_i'\) over \(U_i\). By the general remarks above \(Z_i \subset Z_i'\) is also a universal first order thickening of \(Z_i\) over \(X\). Let \(Z'_{i, j} \subset Z'_i\) be the open subscheme such that \(Z_i \cap Z_j = Z'_{i, j} \cap Z_i\). By the general remarks we see that both \(Z'_{i, j}\) and \(Z'_{j, i}\) are universal first order thickenings of \(Z_i \cap Z_j\) over \(X\). Thus, by the first of our general remarks, we see that there is a canonical isomorphism \(\varphi_{ij} : Z'_{i, j} \to Z'_{j, i}\) inducing the identity on \(Z_i \cap Z_j\). We claim that these morphisms satisfy the cocycle condition of Schemes, Section 01JA. (Verification omitted. Hint: Use that \(Z'_{i, j} \cap Z'_{i, k}\) is the universal first order thickening of \(Z_i \cap Z_j \cap Z_k\) which determines it up to unique isomorphism by what was said above.) Hence we can use the results of Schemes, Section 01JA to get a first order thickening \(Z \subset Z'\) over \(X\) which the property that the open subscheme \(Z'_i \subset Z'\) with \(Z_i = Z'_i \cap Z\) is a universal first order thickening of \(Z_i\) over \(X\).

It turns out that this implies formally that \(Z'\) is a universal first order thickening of \(Z\) over \(X\). Namely, we have the universal property for any diagram \[\xymatrix{ Z \ar[d]_h & T \ar[l]^a \ar[d] \\ X & T' \ar[l]_b }\] where \(a(T)\) is contained in some \(Z_i\). Given a general diagram we can choose an open covering \(T' = \bigcup T'_i\) such that \(a(T_i) \subset Z_i\). We obtain morphisms \(a'_i : T'_i \to Z'\) over \(X\) such that \(a'_i|_{T_i} = a|_{T_i}\). We see that \(a'_i\) and \(a'_j\) necessarily agree on \(T'_i \cap T'_j\) since both \(a'_i|_{T'_i \cap T'_j}\) and \(a'_j|_{T'_i \cap T'_j}\) are solutions of the problem of mapping into the universal first order thickening \(Z'_i \cap Z'_j\) of \(Z_i \cap Z_j\) over \(X\). Hence the morphisms \(a'_i\) glue to a global morphism \(a' : T' \to Z'\) over \(X\) as desired. This finishes the proof.

Definition

Let \(h : Z \to X\) be a formally unramified morphism of schemes.

The universal first order thickening of \(Z\) over \(X\) is the thickening \(Z \subset Z'\) constructed in Lemma 04F3.

The conormal sheaf of \(Z\) over \(X\) is the conormal sheaf of \(Z\) in its universal first order thickening \(Z'\) over \(X\).

We often denote the conormal sheaf \(\mathcal{C}_{Z/X}\) in this situation.

Thus we see that there is a short exact sequence of sheaves \[0 \to \mathcal{C}_{Z/X} \to \mathcal{O}_{Z'} \to \mathcal{O}_Z \to 0\] on \(Z\). The following lemma proves that there is no conflict between this definition and the definition in case \(Z \to X\) is an immersion.

Lemma

Let \(i : Z \to X\) be an immersion of schemes. Then

\(i\) is formally unramified,

the universal first order thickening of \(Z\) over \(X\) is the first order infinitesimal neighbourhood of \(Z\) in \(X\) of Definition 04EY, and

the conormal sheaf of \(i\) in the sense of Morphisms, Definition 01R2 agrees with the conormal sheaf of \(i\) in the sense of Definition 04F4.

Proof

By Morphisms, Lemmas 02GB and 02GC an immersion is unramified, hence formally unramified by Lemma 02HE. The other assertions follow by combining Lemmas 04EZ and 04F0 and the definitions.

Lemma

Let \(Z \to X\) be a formally unramified morphism of schemes. Then the universal first order thickening \(Z'\) is formally unramified over \(X\).

Proof

There are two proofs. The first is to show that \(\Omega_{Z'/X} = 0\) by working affine locally and applying Algebra, Lemma 04EF. Then Lemma 02H9 implies what we want. The second is a direct argument as follows.

Let \(T \subset T'\) be a first order thickening. Let \[\xymatrix{ Z' \ar[d] & T \ar[l]^c \ar[d] \\ X & T' \ar[l] \ar[lu]^{a, b} }\] be a commutative diagram. Consider two morphisms \(a, b : T' \to Z'\) fitting into the diagram. Set \(T_0 = c^{-1}(Z) \subset T\) and \(T'_a = a^{-1}(Z)\) (scheme theoretically). Since \(Z'\) is a first order thickening of \(Z\), we see that \(T'\) is a first order thickening of \(T'_a\). Moreover, since \(c = a|_T\) we see that \(T_0 = T \cap T'_a\) (scheme theoretically). As \(T'\) is a first order thickening of \(T\) it follows that \(T'_a\) is a first order thickening of \(T_0\). Now \(a|_{T'_a}\) and \(b|_{T'_a}\) are morphisms of \(T'_a\) into \(Z'\) over \(X\) which agree on \(T_0\) as morphisms into \(Z\). Hence by the universal property of \(Z'\) we conclude that \(a|_{T'_a} = b|_{T'_a}\). Thus \(a\) and \(b\) are morphism from the first order thickening \(T'\) of \(T'_a\) whose restrictions to \(T'_a\) agree as morphisms into \(Z\). Thus using the universal property of \(Z'\) once more we conclude that \(a = b\). In other words, the defining property of a formally unramified morphism holds for \(Z' \to X\) as desired.

Lemma

Consider a commutative diagram of schemes \[\xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y }\] with \(h\) and \(h'\) formally unramified. Let \(Z \subset Z'\) be the universal first order thickening of \(Z\) over \(X\). Let \(W \subset W'\) be the universal first order thickening of \(W\) over \(Y\). There exists a canonical morphism \((f, f') : (Z, Z') \to (W, W')\) of thickenings over \(Y\) which fits into the following commutative diagram \[\xymatrix{ & & & Z' \ar[ld] \ar[d]^{f'} \\ Z \ar[rr] \ar[d]_f \ar[rrru] & & X \ar[d] & W' \ar[ld] \\ W \ar[rrru]|!{[rr];[rruu]}\hole \ar[rr] & & Y }\] In particular the morphism \((f, f')\) of thickenings induces a morphism of conormal sheaves \(f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}\).

Proof

The first assertion is clear from the universal property of \(W'\). The induced map on conormal sheaves is the map of Morphisms, Lemma 01R4 applied to \((Z \subset Z') \to (W \subset W')\).

Lemma

Let \[\xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y }\] be a fibre product diagram in the category of schemes with \(h'\) formally unramified. Then \(h\) is formally unramified and if \(W \subset W'\) is the universal first order thickening of \(W\) over \(Y\), then \(Z = X \times_Y W \subset X \times_Y W'\) is the universal first order thickening of \(Z\) over \(X\). In particular the canonical map \(f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}\) of Lemma 04F7 is surjective.

Proof

The morphism \(h\) is formally unramified by Lemma 02HB. It is clear that \(X \times_Y W'\) is a first order thickening. It is straightforward to check that it has the universal property because \(W'\) has the universal property (by mapping properties of fibre products). See Morphisms, Lemma 0473 for why this implies that the map of conormal sheaves is surjective.

Lemma

Let \[\xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y }\] be a fibre product diagram in the category of schemes with \(h'\) formally unramified and \(g\) flat. In this case the corresponding map \(Z' \to W'\) of universal first order thickenings is flat, and \(f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}\) is an isomorphism.

Proof

Flatness is preserved under base change, see Morphisms, Lemma 01U9. Hence the first statement follows from the description of \(W'\) in Lemma 04F8. It is clear that \(X \times_Y W'\) is a first order thickening. It is straightforward to check that it has the universal property because \(W'\) has the universal property (by mapping properties of fibre products). See Morphisms, Lemma 0473 for why this implies that the map of conormal sheaves is an isomorphism.

Lemma

Taking the universal first order thickenings commutes with taking opens. More precisely, let \(h : Z \to X\) be a formally unramified morphism of schemes. Let \(V \subset Z\), \(U \subset X\) be opens such that \(h(V) \subset U\). Let \(Z'\) be the universal first order thickening of \(Z\) over \(X\). Then \(h|_V : V \to U\) is formally unramified and the universal first order thickening of \(V\) over \(U\) is the open subscheme \(V' \subset Z'\) such that \(V = Z \cap V'\). In particular, \(\mathcal{C}_{Z/X}|_V = \mathcal{C}_{V/U}\).

Proof

The first statement is Lemma 02HC. The compatibility of universal thickenings can be deduced from the proof of Lemma 04F3, or from Algebra, Lemma 04EE or deduced from Lemma 04F9.

Lemma

Let \(h : Z \to X\) be a formally unramified morphism of schemes over \(S\). Let \(Z \subset Z'\) be the universal first order thickening of \(Z\) over \(X\) with structure morphism \(h' : Z' \to X\). The canonical map \[c_{h'} : (h')^*\Omega_{X/S} \longrightarrow \Omega_{Z'/S}\] induces an isomorphism \(h^*\Omega_{X/S} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z\).

Proof

The map \(c_{h'}\) is the map defined in Morphisms, Lemma 01UV. If \(i : Z \to Z'\) is the given closed immersion, then \(i^*c_{h'}\) is a map \(h^*\Omega_{X/S} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z\). Checking that it is an isomorphism reduces to the affine case by localization, see Lemma 04FA and Morphisms, Lemma 01US. In this case the result is Algebra, Lemma 04EF.

Lemma

Let \(h : Z \to X\) be a formally unramified morphism of schemes over \(S\). There is a canonical exact sequence \[\mathcal{C}_{Z/X} \to h^*\Omega_{X/S} \to \Omega_{Z/S} \to 0.\] The first arrow is induced by \(\text{d}_{Z'/S}\) where \(Z'\) is the universal first order neighbourhood of \(Z\) over \(X\).

Proof

We know that there is a canonical exact sequence \[\mathcal{C}_{Z/Z'} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z \to \Omega_{Z/S} \to 0.\] see Morphisms, Lemma 01UZ. Hence the result follows on applying Lemma 04FB.

Lemma

Let \[\xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d] \\ & Y }\] be a commutative diagram of schemes where \(i\) and \(j\) are formally unramified. Then there is a canonical exact sequence \[\mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/Y} \to 0\] where the first arrow comes from Lemma 04F7 and the second from Lemma 04FC.

Proof

Denote \(Z \to Z'\) the universal first order thickening of \(Z\) over \(X\). Denote \(Z \to Z''\) the universal first order thickening of \(Z\) over \(Y\). By Lemma 04FC here is a canonical morphism \(Z' \to Z''\) so that we have a commutative diagram \[\xymatrix{ Z \ar[r]_{i'} \ar[rd]_{j'} & Z' \ar[r] \ar[d] & X \ar[d] \\ & Z'' \ar[r] & Y }\] Apply Morphisms, Lemma 067L to the left triangle to get an exact sequence \[\mathcal{C}_{Z/Z''} \to \mathcal{C}_{Z/Z'} \to (i')^*\Omega_{Z'/Z''} \to 0\] As \(Z''\) is formally unramified over \(Y\) (see Lemma 04F6) we have \(\Omega_{Z'/Z''} = \Omega_{Z/Y}\) (by combining Lemma 02H9 and Morphisms, Lemma 01UX). Then we have \((i')^*\Omega_{Z'/Y} = i^*\Omega_{X/Y}\) by Lemma 04FB.

Lemma

Let \(Z \to Y \to X\) be formally unramified morphisms of schemes.

If \(Z \subset Z'\) is the universal first order thickening of \(Z\) over \(X\) and \(Y \subset Y'\) is the universal first order thickening of \(Y\) over \(X\), then there is a morphism \(Z' \to Y'\) and \(Y \times_{Y'} Z'\) is the universal first order thickening of \(Z\) over \(Y\).

There is a canonical exact sequence \[i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0\] where the maps come from Lemma 04F7 and \(i : Z \to Y\) is the first morphism.

Proof

The map \(h : Z' \to Y'\) in (1) comes from Lemma 04F7. The assertion that \(Y \times_{Y'} Z'\) is the universal first order thickening of \(Z\) over \(Y\) is clear from the universal properties of \(Z'\) and \(Y'\). By Morphisms, Lemma 062S we have an exact sequence \[(i')^*\mathcal{C}_{Y \times_{Y'} Z'/Z'} \to \mathcal{C}_{Z/Z'} \to \mathcal{C}_{Z/Y \times_{Y'} Z'} \to 0\] where \(i' : Z \to Y \times_{Y'} Z'\) is the given morphism. By Morphisms, Lemma 0473 there exists a surjection \(h^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{Y \times_{Y'} Z'/Z'}\). Combined with the equalities \(\mathcal{C}_{Y/Y'} = \mathcal{C}_{Y/X}\), \(\mathcal{C}_{Z/Z'} = \mathcal{C}_{Z/X}\), and \(\mathcal{C}_{Z/Y \times_{Y'} Z'} = \mathcal{C}_{Z/Y}\) this proves the lemma.

Formally étale morphisms

Recall that a ring map \(R \to A\) is called formally étale (see Algebra, Definition 00UQ) if for every commutative solid diagram \[\xymatrix{ A \ar[r] \ar@{-->}[rd] & B/I \\ R \ar[r] \ar[u] & B \ar[u] }\] where \(I \subset B\) is an ideal of square zero, there exists exactly one dotted arrow which makes the diagram commute. This motivates the following analogue for morphisms of schemes.

Definition

Let \(f : X \to S\) be a morphism of schemes. We say \(f\) is formally étale if given any solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] }\] where \(T \subset T'\) is a first order thickening of affine schemes over \(S\) there exists exactly one dotted arrow making the diagram commute.

It is clear that a formally étale morphism is formally unramified. Hence if \(f : X \to S\) is formally étale, then \(\Omega_{X/S}\) is zero, see Lemma 02H9.

Lemma

If \(f : X \to S\) is a formally étale morphism, then given any solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] }\] where \(T \subset T'\) is a first order thickening of schemes over \(S\) there exists exactly one dotted arrow making the diagram commute. In other words, in Definition 02HG the condition that \(T\) be affine may be dropped.

Proof

Let \(T' = \bigcup T'_i\) be an affine open covering, and let \(T_i = T \cap T'_i\). Then we get morphisms \(a'_i : T'_i \to X\) fitting into the diagram. By uniqueness we see that \(a'_i\) and \(a'_j\) agree on any affine open subscheme of \(T'_i \cap T'_j\). Hence \(a'_i\) and \(a'_j\) agree on \(T'_i \cap T'_j\). Thus we see that the morphisms \(a'_i\) glue to a global morphism \(a' : T' \to X\). The uniqueness of \(a'\) we have seen in Lemma 04F1.

Lemma

A composition of formally étale morphisms is formally étale.

Proof

This is formal.

Lemma

A base change of a formally étale morphism is formally étale.

Proof

This is formal.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(U \subset X\) and \(V \subset S\) be open subschemes such that \(f(U) \subset V\). If \(f\) is formally étale, so is \(f|_U : U \to V\).

Proof

Consider a solid diagram \[\xymatrix{ U \ar[d]_{f|_U} & T \ar[d]^i \ar[l]^a \\ V & T' \ar[l] \ar@{-->}[lu] }\] as in Definition 02HG. If \(f\) is formally ramified, then there exists exactly one \(S\)-morphism \(a' : T' \to X\) such that \(a'|_T = a\). Since \(|T'| = |T|\) we conclude that \(a'(T') \subset U\) which gives our unique morphism from \(T'\) into \(U\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent:

\(f\) is formally étale,

\(f\) is formally unramified and the universal first order thickening of \(X\) over \(S\) is equal to \(X\),

\(f\) is formally unramified and \(\mathcal{C}_{X/S} = 0\), and

\(\Omega_{X/S} = 0\) and \(\mathcal{C}_{X/S} = 0\).

Proof

Actually, the last assertion only make sense because \(\Omega_{X/S} = 0\) implies that \(\mathcal{C}_{X/S}\) is defined via Lemma 02H9 and Definition 04F4. This also makes it clear that (3) and (4) are equivalent.

Either of the assumptions (1), (2), and (3) imply that \(f\) is formally unramified. Hence we may assume \(f\) is formally unramified. The equivalence of (1), (2), and (3) follow from the universal property of the universal first order thickening \(X'\) of \(X\) over \(S\) and the fact that \(X = X' \Leftrightarrow \mathcal{C}_{X/S} = 0\) since after all by definition \(\mathcal{C}_{X/S} = \mathcal{C}_{X/X'}\) is the ideal sheaf of \(X\) in \(X'\).

Lemma

An unramified flat morphism is formally étale.

Proof

Say \(X \to S\) is unramified and flat. Then \(\Delta : X \to X \times_S X\) is an open immersion, see Morphisms, Lemma 02GE. We have to show that \(\mathcal{C}_{X/S}\) is zero. Consider the two projections \(p, q : X \times_S X \to X\). As \(f\) is formally unramified (see Lemma 02HE), \(q\) is formally unramified (see Lemma 02HB). As \(f\) is flat, \(p\) is flat, see Morphisms, Lemma 01U9. Hence \(p^*\mathcal{C}_{X/S} = \mathcal{C}_q\) by Lemma 04F9 where \(\mathcal{C}_q\) denotes the conormal sheaf of the formally unramified morphism \(q : X \times_S X \to X\). But \(\Delta(X) \subset X \times_S X\) is an open subscheme which maps isomorphically to \(X\) via \(q\). Hence by Lemma 04FA we see that \(\mathcal{C}_q|_{\Delta(X)} = \mathcal{C}_{X/X} = 0\). In other words, the pullback of \(\mathcal{C}_{X/S}\) to \(X\) via the identity morphism is zero, i.e., \(\mathcal{C}_{X/S} = 0\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume \(X\) and \(S\) are affine. Then \(f\) is formally étale if and only if \(\mathcal{O}_S(S) \to \mathcal{O}_X(X)\) is a formally étale ring map.

Proof

This is immediate from the definitions (Definition 02HG and Algebra, Definition 00UQ) by the equivalence of categories of rings and affine schemes, see Schemes, Lemma 01I2.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent:

\(f\) is formally étale,

for every \(x \in X\) there exist opens \(x \in U \subset X\) and \(f(x) \in V \subset Y\) with \(f(U) \subset V\) such that \(f|_U : U \to V\) is formally étale,

for every pair of affine opens \(U \subset X\) and \(V \subset Y\) with \(f(U) \subset V\) the ring map \(\mathcal{O}_Y(V) \to \mathcal{O}_X(U)\) is formally étale, and

there exists an affine open covering \(Y = \bigcup V_j\) and for each \(j\) an affine open covering \(f^{-1}(V_j) = \bigcup U_{ji}\) such that \(\mathcal{O}_Y(V) \to \mathcal{O}_X(U)\) is a formally étale ring map for all \(j\) and \(i\).

Proof

Combining Lemmas 02HK and 02HL we see that (1) \(\Rightarrow\) (3). It is immediate that (3) \(\Rightarrow\) (4). We have (4) \(\Rightarrow\) (2) by Lemma 02HL. If (2) holds, then \(\Omega_{X/S}\) and \(\mathcal{C}_{X/S}\) vanish in a neighbourhood of every point by Lemma 04FE (this also uses Morphisms, Lemma 01US and Lemma 04FA) whereupon Lemma 04FE tells us that \(f\) is formally unramified.

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent:

The morphism \(f\) is étale, and

the morphism \(f\) is locally of finite presentation and formally étale.

Proof

Assume \(f\) is étale. An étale morphism is locally of finite presentation, flat and unramified, see Morphisms, Section 02GH. Hence \(f\) is locally of finite presentation and formally étale, see Lemma 04FF.

Conversely, suppose that \(f\) is locally of finite presentation and formally étale. Being étale is local in the Zariski topology on \(X\) and \(S\), see Morphisms, Lemma 02GJ. By Lemma 02HK we can cover \(X\) by affine opens \(U\) which map into affine opens \(V\) such that \(U \to V\) is formally étale (and of finite presentation, see Morphisms, Lemma 01TQ). By Lemma 02HL we see that the ring maps \(\mathcal{O}(V) \to \mathcal{O}(U)\) are formally étale (and of finite presentation). We win by Algebra, Lemma 00UR. (We will give another proof of this implication when we discuss formally smooth morphisms.)

Infinitesimal deformations of maps

In this section we explain how a derivation can be used to infinitesimally move a map. Throughout this section we use that a sheaf on a thickening \(X'\) of \(X\) can be seen as a sheaf on \(X\).

Lemma

Let \(S\) be a scheme. Let \(X \subset X'\) and \(Y \subset Y'\) be two first order thickenings over \(S\). Let \((a, a'), (b, b') : (X \subset X') \to (Y \subset Y')\) be two morphisms of thickenings over \(S\). Assume that

\(a = b\), and

the two maps \(a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) (Morphisms, Lemma 01R4) are equal.

Then the map \((a')^\sharp - (b')^\sharp\) factors as \[\mathcal{O}_{Y'} \to \mathcal{O}_Y \xrightarrow{D} a_*\mathcal{C}_{X/X'} \to a_*\mathcal{O}_{X'}\] where \(D\) is an \(\mathcal{O}_S\)-derivation.

Proof

Instead of working on \(Y\) we work on \(X\). The advantage is that the pullback functor \(a^{-1}\) is exact. Using (1) and (2) we obtain a commutative diagram with exact rows \[\xymatrix{ 0 \ar[r] & \mathcal{C}_{X/X'} \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_X \ar[r] & 0 \\ 0 \ar[r] & a^{-1}\mathcal{C}_{Y/Y'} \ar[r] \ar[u] & a^{-1}\mathcal{O}_{Y'} \ar[r] \ar@<1ex>[u]^{(a')^\sharp} \ar@<-1ex>[u]_{(b')^\sharp} & a^{-1}\mathcal{O}_Y \ar[r] \ar[u] & 0 }\] Now it is a general fact that in such a situation the difference of the \(\mathcal{O}_S\)-algebra maps \((a')^\sharp\) and \((b')^\sharp\) is an \(\mathcal{O}_S\)-derivation from \(a^{-1}\mathcal{O}_Y\) to \(\mathcal{C}_{X/X'}\). By adjointness of the functors \(a^{-1}\) and \(a_*\) this is the same thing as an \(\mathcal{O}_S\)-derivation from \(\mathcal{O}_Y\) into \(a_*\mathcal{C}_{X/X'}\). Some details omitted.

Note that in the situation of the lemma above we may write \(D\) as [04BV]\[\begin{equation} D = \text{d}_{Y/S} \circ \theta \end{equation}\] where \(\theta\) is an \(\mathcal{O}_Y\)-linear map \(\theta : \Omega_{Y/S} \to a_*\mathcal{C}_{X/X'}\). Of course, then by adjunction again we may view \(\theta\) as an \(\mathcal{O}_X\)-linear map \(\theta : a^*\Omega_{Y/S} \to \mathcal{C}_{X/X'}\).

Lemma

Let \(S\) be a scheme. Let \((a, a') : (X \subset X') \to (Y \subset Y')\) be a morphism of first order thickenings over \(S\). Let \[\theta : a^*\Omega_{Y/S} \to \mathcal{C}_{X/X'}\] be an \(\mathcal{O}_X\)-linear map. Then there exists a unique morphism of pairs \((b, b') : (X \subset X') \to (Y \subset Y')\) such that (1) and (2) of Lemma 04FG hold and the derivation \(D\) and \(\theta\) are related by Equation (04BV).

Proof

We simply set \(b = a\) and we define \((b')^\sharp\) to be the map \[(a')^\sharp + D : a^{-1}\mathcal{O}_{Y'} \to \mathcal{O}_{X'}\] where \(D\) is as in Equation (04BV). We omit the verification that \((b')^\sharp\) is a map of sheaves of \(\mathcal{O}_S\)-algebras and that (1) and (2) of Lemma 04FG hold. Equation (04BV) holds by construction.

Remark

Assumptions and notation as in Lemma 02H5. The action of a local section \(\theta\) on \(a'\) is sometimes indicated by \(\theta \cdot a'\). Note that this means nothing else than the fact that \((a')^\sharp\) and \((\theta \cdot a')^\sharp\) differ by a derivation \(D\) which is related to \(\theta\) by Equation (04BV).

Lemma

Let \(S\) be a scheme. Let \(X \subset X'\) and \(Y \subset Y'\) be first order thickenings over \(S\). Assume given a morphism \(a : X \to Y\) and a map \(A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) of \(\mathcal{O}_X\)-modules. For an open subscheme \(U' \subset X'\) consider morphisms \(a' : U' \to Y'\) such that

\(a'\) is a morphism over \(S\),

\(a'|_U = a|_U\), and

the induced map \(a^*\mathcal{C}_{Y/Y'}|_U \to \mathcal{C}_{X/X'}|_U\) is the restriction of \(A\) to \(U\).

Here \(U = X \cap U'\). Then the rule [04FI]\[\begin{equation} U' \mapsto \{a' : U' \to Y'\text{ such that (1), (2), (3) hold.}\} \end{equation}\] defines a sheaf of sets on \(X'\).

Proof

Denote \(\mathcal{F}\) the rule of the lemma. The restriction mapping \(\mathcal{F}(U') \to \mathcal{F}(V')\) for \(V' \subset U' \subset X'\) of \(\mathcal{F}\) is really the restriction map \(a' \mapsto a'|_{V'}\). With this definition in place it is clear that \(\mathcal{F}\) is a sheaf since morphisms are defined locally.

In the following lemma we identify sheaves on \(X\) and any thickening of \(X\).

Lemma

Same notation and assumptions as in Lemma 04FH. There is an action of the sheaf \[\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/S}, \mathcal{C}_{X/X'})\] on the sheaf (04FI). Moreover, the action is simply transitive for any open \(U' \subset X'\) over which the sheaf (04FI) has a section.

Proof

This is a combination of Lemmas 04FG, 02H5, and 04FH.

Remark

A special case of Lemmas 04FG, 02H5, 04FH, and 04FJ is where \(Y = Y'\). In this case the map \(A\) is always zero. The sheaf of Lemma 04FH is just given by the rule \[U' \mapsto \{a' : U' \to Y\text{ over }S\text{ with } a'|_U = a|_U\}\] and we act on this by the sheaf \(\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/S}, \mathcal{C}_{X/X'})\).

Remark

Another special case of Lemmas 04FG, 02H5, 04FH, and 04FJ is where \(S\) itself is a thickening \(Z \subset Z' = S\) and \(Y = Z \times_{Z'} Y'\). Picture \[\xymatrix{ (X \subset X') \ar@{..>}[rr]_{(a, ?)} \ar[rd]_{(g, g')} & & (Y \subset Y') \ar[ld]^{(h, h')} \\ & (Z \subset Z') }\] In this case the map \(A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) is determined by \(a\): the map \(h^*\mathcal{C}_{Z/Z'} \to \mathcal{C}_{Y/Y'}\) is surjective (because we assumed \(Y = Z \times_{Z'} Y'\)), hence the pullback \(g^*\mathcal{C}_{Z/Z'} = a^*h^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'}\) is surjective, and the composition \(g^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) has to be the canonical map induced by \(g'\). Thus the sheaf of Lemma 04FH is just given by the rule \[U' \mapsto \{a' : U' \to Y'\text{ over }Z'\text{ with } a'|_U = a|_U\}\] and we act on this by the sheaf \(\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/Z}, \mathcal{C}_{X/X'})\).

Lemma

Let \(S\) be a scheme. Let \(X \subset X'\) be a first order thickening over \(S\). Let \(Y\) be a scheme over \(S\). Let \(a', b' : X' \to Y\) be two morphisms over \(S\) with \(a = a'|_X = b'|_X\). This gives rise to a commutative diagram \[\xymatrix{ X \ar[r] \ar[d]_a & X' \ar[d]^{(b', a')} \\ Y \ar[r]^-{\Delta_{Y/S}} & Y \times_S Y }\] Since the horizontal arrows are immersions with conormal sheaves \(\mathcal{C}_{X/X'}\) and \(\Omega_{Y/S}\), by Morphisms, Lemma 01R4, we obtain a map \(\theta : a^*\Omega_{Y/S} \to \mathcal{C}_{X/X'}\). Then this \(\theta\) and the derivation \(D\) of Lemma 04FG are related by Equation (04BV).

Proof

Omitted. Hint: The equality may be checked on affine opens where it comes from the following computation. If \(f\) is a local section of \(\mathcal{O}_Y\), then \(1 \otimes f - f \otimes 1\) is a local section of \(\mathcal{C}_{Y/(Y \times_S Y)}\) corresponding to \(\text{d}_{Y/S}(f)\). It is mapped to the local section \((a')^\sharp(f) - (b')^\sharp(f) = D(f)\) of \(\mathcal{C}_{X/X'}\). In other words, \(\theta(\text{d}_{Y/S}(f)) = D(f)\).

For later purposes we need a result that roughly states that the construction of Lemma 02H5 is compatible with étale localization.

Lemma

Let \[\xymatrix{ X_1 \ar[d] & X_2 \ar[l]^f \ar[d] \\ S_1 & S_2 \ar[l] }\] be a commutative diagram of schemes with \(X_2 \to X_1\) and \(S_2 \to S_1\) étale. Then the map \(c_f : f^*\Omega_{X_1/S_1} \to \Omega_{X_2/S_2}\) of Morphisms, Lemma 01UV is an isomorphism.

Proof

We recall that an étale morphism \(U \to V\) is a smooth morphism with \(\Omega_{U/V} = 0\). Using this we see that Morphisms, Lemma 01UX implies \(\Omega_{X_2/S_2} = \Omega_{X_2/S_1}\) and Morphisms, Lemma 02K4 implies that the map \(f^*\Omega_{X_1/S_1} \to \Omega_{X_2/S_1}\) (for the morphism \(f\) seen as a morphism over \(S_1\)) is an isomorphism. Hence the lemma follows.

Lemma

Consider a commutative diagram of first order thickenings \[\vcenter{ \xymatrix{ (T_2 \subset T_2') \ar[d]_{(h, h')} \ar[rr]_{(a_2, a_2')} & & (X_2 \subset X_2') \ar[d]^{(f, f')} \\ (T_1 \subset T_1') \ar[rr]^{(a_1, a_1')} & & (X_1 \subset X_1') } } \quad \begin{matrix} \text{and a commutative} \\ \text{diagram of schemes} \end{matrix} \quad \vcenter{ \xymatrix{ X_2' \ar[r] \ar[d] & S_2 \ar[d] \\ X_1' \ar[r] & S_1 } }\] with \(X_2 \to X_1\) and \(S_2 \to S_1\) étale. For any \(\mathcal{O}_{T_1}\)-linear map \(\theta_1 : a_1^*\Omega_{X_1/S_1} \to \mathcal{C}_{T_1/T'_1}\) let \(\theta_2\) be the composition \[\xymatrix{ a_2^*\Omega_{X_2/S_2} \ar@{=}[r] & h^*a_1^*\Omega_{X_1/S_1} \ar[r]^-{h^*\theta_1} & h^*\mathcal{C}_{T_1/T'_1} \ar[r] & \mathcal{C}_{T_2/T'_2} }\] (equality sign is explained in the proof). Then the diagram \[\xymatrix{ T_2' \ar[rr]_{\theta_2 \cdot a_2'} \ar[d] & & X'_2 \ar[d] \\ T_1' \ar[rr]^{\theta_1 \cdot a_1'} & & X'_1 }\] commutes where the actions \(\theta_2 \cdot a_2'\) and \(\theta_1 \cdot a_1'\) are as in Remark 0CK1.

Proof

The equality sign comes from the identification \(f^*\Omega_{X_1/S_1} = \Omega_{X_2/S_2}\) of Lemma 04BX. Namely, using this we have \(a_2^*\Omega_{X_2/S_2} = a_2^*f^*\Omega_{X_1/S_1} = h^*a_1^*\Omega_{X_1/S_1}\) because \(f \circ a_2 = a_1 \circ h\). Having said this, the commutativity of the diagram may be checked on affine opens. Hence we may assume the schemes in the initial big diagram are affine. Thus we obtain commutative diagrams \[\vcenter{ \xymatrix{ (B'_2, I_2) & & (A'_2, J_2) \ar[ll]^{a_2'} \\ (B'_1, I_1) \ar[u]^{h'} & & (A'_1, J_1) \ar[ll]_{a_1'} \ar[u]_{f'} } } \quad\text{and}\quad \vcenter{ \xymatrix{ A'_2 & & R_2 \ar[ll] \\ A'_1 \ar[u] & & R_1 \ar[ll] \ar[u] } }\] The notation signifies that \(I_1, I_2, J_1, J_2\) are ideals of square zero and maps of pairs are ring maps sending ideals into ideals. Set \(A_1 = A'_1/J_1\), \(A_2 = A'_2/J_2\), \(B_1 = B'_1/I_1\), and \(B_2 = B'_2/I_2\). We are given that \[A_2 \otimes_{A_1} \Omega_{A_1/R_1} \longrightarrow \Omega_{A_2/R_2}\] is an isomorphism. Then \(\theta_1 : B_1 \otimes_{A_1} \Omega_{A_1/R_1} \to I_1\) is \(B_1\)-linear. This gives an \(R_1\)-derivation \(D_1 = \theta_1 \circ \text{d}_{A_1/R_1} : A_1 \to I_1\). In a similar way we see that \(\theta_2 : B_2 \otimes_{A_2} \Omega_{A_2/R_2} \to I_2\) gives rise to a \(R_2\)-derivation \(D_2 = \theta_2 \circ \text{d}_{A_2/R_2} : A_2 \to I_2\). The construction of \(\theta_2\) implies the following compatibility between \(\theta_1\) and \(\theta_2\): for every \(x \in A_1\) we have \[h'(D_1(x)) = D_2(f'(x))\] as elements of \(I_2\). We may view \(D_1\) as a map \(A'_1 \to B'_1\) using \(A'_1 \to A_1 \xrightarrow{D_1} I_1 \to B_1\) similarly we may view \(D_2\) as a map \(A'_2 \to B'_2\). Then the displayed equality holds for \(x \in A'_1\). By the construction of the action in Lemma 02H5 and Remark 0CK1 we know that \(\theta_1 \cdot a_1'\) corresponds to the ring map \(a_1' + D_1 : A'_1 \to B'_1\) and \(\theta_2 \cdot a_2'\) corresponds to the ring map \(a_2' + D_2 : A'_2 \to B'_2\). By the displayed equality we obtain that \(h' \circ (a_1' + D_1) = (a_2' + D_2) \circ f'\) as desired.

Remark

Lemma 04BY can be improved in the following way. Suppose that we have commutative diagrams as in Lemma 04BY but we do not assume that \(X_2 \to X_1\) and \(S_2 \to S_1\) are étale. Next, suppose we have \(\theta_1 : a_1^*\Omega_{X_1/S_1} \to \mathcal{C}_{T_1/T'_1}\) and \(\theta_2 : a_2^*\Omega_{X_2/S_2} \to \mathcal{C}_{T_2/T'_2}\) such that \[\xymatrix{ f_*\mathcal{O}_{X_2} \ar[rr]_{f_*D_2} & & f_*a_{2, *}\mathcal{C}_{T_2/T_2'} \\ \mathcal{O}_{X_1} \ar[rr]^{D_1} \ar[u]^{f^\sharp} & & a_{1, *}\mathcal{C}_{T_1/T_1'} \ar[u]_{\text{induced by }(h')^\sharp} }\] is commutative where \(D_i\) corresponds to \(\theta_i\) as in Equation (04BV). Then we have the conclusion of Lemma 04BY. The importance of the condition that both \(X_2 \to X_1\) and \(S_2 \to S_1\) are étale is that it allows us to construct a \(\theta_2\) from \(\theta_1\).

Infinitesimal deformations of schemes

The following simple lemma is often a convenient tool to check whether an infinitesimal deformation of a map is flat.

Lemma

Let \((f, f') : (X \subset X') \to (S \subset S')\) be a morphism of first order thickenings. Assume that \(f\) is flat. Then the following are equivalent

\(f'\) is flat and \(X = S \times_{S'} X'\), and

the canonical map \(f^*\mathcal{C}_{S/S'} \to \mathcal{C}_{X/X'}\) is an isomorphism.

Proof

As the problem is local on \(X'\) we may assume that \(X, X', S, S'\) are affine schemes. Say \(S' = \Spec(A')\), \(X' = \Spec(B')\), \(S = \Spec(A)\), \(X = \Spec(B)\) with \(A = A'/I\) and \(B = B'/J\) for some square zero ideals. Then we obtain the following commutative diagram \[\xymatrix{ 0 \ar[r] & J \ar[r] & B' \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & I \ar[r] \ar[u] & A' \ar[r] \ar[u] & A \ar[r] \ar[u] & 0 }\] with exact rows. The canonical map of the lemma is the map \[I \otimes_A B = I \otimes_{A'} B' \longrightarrow J.\] The assumption that \(f\) is flat signifies that \(A \to B\) is flat.

Assume (1). Then \(A' \to B'\) is flat and \(J = IB'\). Flatness implies \(\text{Tor}_1^{A'}(B', A) = 0\) (see Algebra, Lemma 00M5). This means \(I \otimes_{A'} B' \to B'\) is injective (see Algebra, Remark 00M6). Hence we see that \(I \otimes_A B \to J\) is an isomorphism.

Assume (2). Then it follows that \(J = IB'\), so that \(X = S \times_{S'} X'\). Moreover, we get \(\text{Tor}_1^{A'}(B', A'/I) = 0\) by reversing the implications in the previous paragraph. Hence \(B'\) is flat over \(A'\) by Algebra, Lemma 051C.

The following lemma is the “nilpotent” version of the “critère de platitude par fibres”, see Section 039A.

Lemma

Consider a commutative diagram \[\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (S \subset S') }\] of thickenings. Assume

\(X'\) is flat over \(S'\),

\(f\) is flat,

\(S \subset S'\) is a finite order thickening, and

\(X = S \times_{S'} X'\) and \(Y = S \times_{S'} Y'\).

Then \(f'\) is flat and \(Y'\) is flat over \(S'\) at all points in the image of \(f'\).

Proof

Immediate consequence of Algebra, Lemma 06A5.

Many properties of morphisms of schemes are preserved under flat deformations.

Lemma

Consider a commutative diagram \[\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (S \subset S') }\] of thickenings. Assume \(S \subset S'\) is a finite order thickening, \(X'\) flat over \(S'\), \(X = S \times_{S'} X'\), and \(Y = S \times_{S'} Y'\). Then

\(f\) is flat if and only if \(f'\) is flat, [06AH]

\(f\) is an isomorphism if and only if \(f'\) is an isomorphism, [06AI]

\(f\) is an open immersion if and only if \(f'\) is an open immersion, [06AJ]

\(f\) is quasi-compact if and only if \(f'\) is quasi-compact, [06AK]

\(f\) is universally closed if and only if \(f'\) is universally closed, [06AL]

\(f\) is (quasi-)separated if and only if \(f'\) is (quasi-)separated, [06AM]

\(f\) is a monomorphism if and only if \(f'\) is a monomorphism, [06AN]

\(f\) is surjective if and only if \(f'\) is surjective, [06AP]

\(f\) is universally injective if and only if \(f'\) is universally injective, [06AQ]

\(f\) is affine if and only if \(f'\) is affine, [06AR]

\(f\) is locally of finite type if and only if \(f'\) is locally of finite type,

\(f\) is locally quasi-finite if and only if \(f'\) is locally quasi-finite, [06AT]

\(f\) is locally of finite presentation if and only if \(f'\) is locally of finite presentation,

\(f\) is locally of finite type of relative dimension \(d\) if and only if \(f'\) is locally of finite type of relative dimension \(d\),

\(f\) is universally open if and only if \(f'\) is universally open, [06AW]

\(f\) is syntomic if and only if \(f'\) is syntomic, [06AX]

\(f\) is smooth if and only if \(f'\) is smooth, [06AY]

\(f\) is unramified if and only if \(f'\) is unramified, [06AZ]

\(f\) is étale if and only if \(f'\) is étale, [06B0]

\(f\) is proper if and only if \(f'\) is proper, [06B1]

\(f\) is integral if and only if \(f'\) is integral, [06B2]

\(f\) is finite if and only if \(f'\) is finite, [06B3]

\(f\) is finite locally free (of rank \(d\)) if and only if \(f'\) is finite locally free (of rank \(d\)), and

add more here.

Proof

The assumptions on \(X\) and \(Y\) mean that \(f\) is the base change of \(f'\) by \(X \to X'\). The properties \(\mathcal{P}\) listed in (1) – (23) above are all stable under base change, hence if \(f'\) has property \(\mathcal{P}\), then so does \(f\). See Schemes, Lemmas 01JY, 01K5, 01KU, and 02YC and Morphisms, Lemmas 01S1, 0472, 01SD, 01T4, 01TM, 01TS, 02NK, 01UI, 01VB, 02GA, 02GO, 01W4, 01WL, and 02KD.

The interesting direction in each case is therefore to assume that \(f\) has the property and deduce that \(f'\) has it too. By induction on the order of the thickening we may assume that \(S \subset S'\) is a first order thickening, see discussion immediately following Definition 04EX. We make a couple of general remarks which we will use without further mention in the arguments below. (I) Let \(W' \subset S'\) be an affine open and let \(U' \subset X'\) and \(V' \subset Y'\) be affine opens lying over \(W'\) with \(f'(U') \subset V'\). Let \(W' = \Spec(R')\) and denote \(I \subset R'\) be the ideal defining the closed subscheme \(W' \cap S\). Say \(U' = \Spec(B')\) and \(V' = \Spec(A')\). Then we get a commutative diagram \[\xymatrix{ 0 \ar[r] & IB' \ar[r] & B' \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & IA' \ar[r] \ar[u] & A' \ar[r] \ar[u] & A \ar[r] \ar[u] & 0 }\] with exact rows. Moreover \(IB' \cong I \otimes_R B\), see proof of Lemma 063Y. (II) The morphisms \(X \to X'\) and \(Y \to Y'\) are universal homeomorphisms. Hence the topology of the maps \(f\) and \(f'\) (after any base change) is identical. (III) If \(f\) is flat, then \(f'\) is flat and \(Y' \to S'\) is flat at every point in the image of \(f'\), see Lemma 06AF.

Ad (06AH). This is general remark (III).

Ad (06AI). Assume \(f\) is an isomorphism. By (III) we see that \(Y' \to S'\) is flat. Choose an affine open \(V' \subset Y'\) and set \(U' = (f')^{-1}(V')\). Then \(V = Y \cap V'\) is affine which implies that \(V \cong f^{-1}(V) = U = Y \times_{Y'} U'\) is affine. By Lemma 06AD we see that \(U'\) is affine. Thus we have a diagram as in the general remark (I) and moreover \(IA \cong I \otimes_R A\) because \(R' \to A'\) is flat. Then \(IB' \cong I \otimes_R B \cong I \otimes_R A \cong IA'\) and \(A \cong B\). By the exactness of the rows in the diagram above we see that \(A' \cong B'\), i.e., \(U' \cong V'\). Thus \(f'\) is an isomorphism.

Ad (06AJ). Assume \(f\) is an open immersion. Then \(f\) is an isomorphism of \(X\) with an open subscheme \(V \subset Y\). Let \(V' \subset Y'\) be the open subscheme whose underlying topological space is \(V\). Then \(f'\) is a map from \(X'\) to \(V'\) which is an isomorphism by (06AI). Hence \(f'\) is an open immersion.

Ad (06AK). Immediate from remark (II). See also Lemma 09ZV for a more general statement.

Ad (06AL). Immediate from remark (II). See also Lemma 09ZV for a more general statement.

Ad (06AM). Note that \(X \times_Y X = Y \times_{Y'} (X' \times_{Y'} X')\) so that \(X' \times_{Y'} X'\) is a thickening of \(X \times_Y X\). Hence the topology of the maps \(\Delta_{X/Y}\) and \(\Delta_{X'/Y'}\) matches and we win. See also Lemma 09ZV for a more general statement.

Ad (06AN). Assume \(f\) is a monomorphism. Consider the diagonal morphism \(\Delta_{X'/Y'} : X' \to X' \times_{Y'} X'\). The base change of \(\Delta_{X'/Y'}\) by \(S \to S'\) is \(\Delta_{X/Y}\) which is an isomorphism by assumption. By (06AI) we conclude that \(\Delta_{X'/Y'}\) is an isomorphism.

Ad (06AP). This is clear. See also Lemma 09ZV for a more general statement.

Ad (06AQ). Immediate from remark (II). See also Lemma 09ZV for a more general statement.

Ad (06AR). Assume \(f\) is affine. Choose an affine open \(V' \subset Y'\) and set \(U' = (f')^{-1}(V')\). Then \(V = Y \cap V'\) is affine which implies that \(U = Y \times_{Y'} U'\) is affine. By Lemma 06AD we see that \(U'\) is affine. Hence \(f'\) is affine. See also Lemma 09ZV for a more general statement.

Ad (06AS). Via remark (I) comes down to proving \(A' \to B'\) is of finite type if \(A \to B\) is of finite type. Suppose that \(x_1, \ldots, x_n \in B'\) are elements whose images in \(B\) generate \(B\) as an \(A\)-algebra. Then \(A'[x_1, \ldots, x_n] \to B\) is surjective as both \(A'[x_1, \ldots, x_n] \to B\) is surjective and \(I \otimes_R A[x_1, \ldots, x_n] \to I \otimes_R B\) is surjective. See also Lemma 09ZW for a more general statement.

Ad (06AT). Follows from (06AS) and that quasi-finiteness of a morphism of finite type can be checked on fibres, see Morphisms, Lemma 01TH. See also Lemma 09ZW for a more general statement.

Ad (06AU). Via remark (I) comes down to proving \(A' \to B'\) is of finite presentation if \(A \to B\) is of finite presentation. We may assume that \(B' = A'[x_1, \ldots, x_n]/K'\) for some ideal \(K'\) by (06AS). We get a short exact sequence \[0 \to K' \to A'[x_1, \ldots, x_n] \to B' \to 0\] As \(B'\) is flat over \(R'\) we see that \(K' \otimes_{R'} R\) is the kernel of the surjection \(A[x_1, \ldots, x_n] \to B\). By assumption on \(A \to B\) there exist finitely many \(f'_1, \ldots, f'_m \in K'\) whose images in \(A[x_1, \ldots, x_n]\) generate this kernel. Since \(I\) is nilpotent we see that \(f'_1, \ldots, f'_m\) generate \(K'\) by Nakayama’s lemma, see Algebra, Lemma 00DV.

Ad (06AV). Follows from (06AS) and general remark (II). See also Lemma 09ZW for a more general statement.

Ad (06AW). Immediate from general remark (II). See also Lemma 09ZV for a more general statement.

Ad (06AX). Assume \(f\) is syntomic. By (06AU) \(f'\) is locally of finite presentation, by general remark (III) \(f'\) is flat and the fibres of \(f'\) are the fibres of \(f\). Hence \(f'\) is syntomic by Morphisms, Lemma 01UF.

Ad (06AY). Assume \(f\) is smooth. By (06AU) \(f'\) is locally of finite presentation, by general remark (III) \(f'\) is flat, and the fibres of \(f'\) are the fibres of \(f\). Hence \(f'\) is smooth by Morphisms, Lemma 01V8.

Ad (06AZ). Assume \(f\) unramified. By (06AS) \(f'\) is locally of finite type and the fibres of \(f'\) are the fibres of \(f\). Hence \(f'\) is unramified by Morphisms, Lemma 02G8. See also Lemma 09ZW for a more general statement.

Ad (06B0). Assume \(f\) étale. By (06AU) \(f'\) is locally of finite presentation, by general remark (III) \(f'\) is flat, and the fibres of \(f'\) are the fibres of \(f\). Hence \(f'\) is étale by Morphisms, Lemma 02GM.

Ad (06B1). This follows from a combination of (06AM), (06AS), (06AK), and (06AL). See also Lemma 09ZW for a more general statement.

Ad (06B2). Combine (06AL) and (06AR) with Morphisms, Lemma 01WM. See also Lemma 09ZV for a more general statement.

Ad (06B3). Combine (06B2), and (06AS) with Morphisms, Lemma 01WJ. See also Lemma 09ZW for a more general statement.

Ad (06B4). Assume \(f\) finite locally free. By (06B3) we see that \(f'\) is finite, by general remark (III) \(f'\) is flat, and by (06AU) \(f'\) is locally of finite presentation. Hence \(f'\) is finite locally free by Morphisms, Lemma 02KB.

The following lemma is the “locally nilpotent” version of the “critère de platitude par fibres”, see Section 039A.

Lemma

Consider a commutative diagram \[\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (S \subset S') }\] of thickenings. Assume

\(Y' \to S'\) is locally of finite type,

\(X' \to S'\) is flat and locally of finite presentation,

\(f\) is flat, and

\(X = S \times_{S'} X'\) and \(Y = S \times_{S'} Y'\).

Then \(f'\) is flat and for all \(y' \in Y'\) in the image of \(f'\) the local ring \(\mathcal{O}_{Y', y'}\) is flat and essentially of finite presentation over \(\mathcal{O}_{S', s'}\).

Proof

Immediate consequence of Algebra, Lemma 0CEL.

Many properties of morphisms of schemes are preserved under flat deformations as in the lemma above.

Lemma

Consider a commutative diagram \[\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (S \subset S') }\] of thickenings. Assume \(Y' \to S'\) locally of finite type, \(X' \to S'\) flat and locally of finite presentation, \(X = S \times_{S'} X'\), and \(Y = S \times_{S'} Y'\). Then

\(f\) is flat if and only if \(f'\) is flat, [0CF5]

\(f\) is an isomorphism if and only if \(f'\) is an isomorphism, [0CF6]

\(f\) is an open immersion if and only if \(f'\) is an open immersion, [0CF7]

\(f\) is quasi-compact if and only if \(f'\) is quasi-compact, [0CF8]

\(f\) is universally closed if and only if \(f'\) is universally closed, [0CF9]

\(f\) is (quasi-)separated if and only if \(f'\) is (quasi-)separated, [0CFA]

\(f\) is a monomorphism if and only if \(f'\) is a monomorphism, [0CFB]

\(f\) is surjective if and only if \(f'\) is surjective, [0CFC]

\(f\) is universally injective if and only if \(f'\) is universally injective, [0CFD]

\(f\) is affine if and only if \(f'\) is affine, [0CFE]

\(f\) is locally quasi-finite if and only if \(f'\) is locally quasi-finite, [0CFF]

\(f\) is locally of finite type of relative dimension \(d\) if and only if \(f'\) is locally of finite type of relative dimension \(d\),

\(f\) is universally open if and only if \(f'\) is universally open, [0CFH]

\(f\) is syntomic if and only if \(f'\) is syntomic, [0CFI]

\(f\) is smooth if and only if \(f'\) is smooth, [0CFJ]

\(f\) is unramified if and only if \(f'\) is unramified, [0CFK]

\(f\) is étale if and only if \(f'\) is étale, [0CFL]

\(f\) is proper if and only if \(f'\) is proper, [0CFM]

\(f\) is finite if and only if \(f'\) is finite, [0CFN]

\(f\) is finite locally free (of rank \(d\)) if and only if \(f'\) is finite locally free (of rank \(d\)), and

add more here.

Proof

The assumptions on \(X\) and \(Y\) mean that \(f\) is the base change of \(f'\) by \(X \to X'\). The properties \(\mathcal{P}\) listed in (1) – (20) above are all stable under base change, hence if \(f'\) has property \(\mathcal{P}\), then so does \(f\). See Schemes, Lemmas 01JY, 01K5, 01KU, and 02YC and Morphisms, Lemmas 01S1, 0472, 01SD, 01TM, 02NK, 01UI, 01VB, 02GA, 02GO, 01W4, 01WL, and 02KD.

The interesting direction in each case is therefore to assume that \(f\) has the property and deduce that \(f'\) has it too. We make a couple of general remarks which we will use without further mention in the arguments below. (I) Let \(W' \subset S'\) be an affine open and let \(U' \subset X'\) and \(V' \subset Y'\) be affine opens lying over \(W'\) with \(f'(U') \subset V'\). Let \(W' = \Spec(R')\) and denote \(I \subset R'\) be the ideal defining the closed subscheme \(W' \cap S\). Say \(U' = \Spec(B')\) and \(V' = \Spec(A')\). Then we get a commutative diagram \[\xymatrix{ 0 \ar[r] & IB' \ar[r] & B' \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & IA' \ar[r] \ar[u] & A' \ar[r] \ar[u] & A \ar[r] \ar[u] & 0 }\] with exact rows. (II) The morphisms \(X \to X'\) and \(Y \to Y'\) are universal homeomorphisms. Hence the topology of the maps \(f\) and \(f'\) (after any base change) is identical. (III) If \(f\) is flat, then \(f'\) is flat and \(Y' \to S'\) is flat at every point in the image of \(f'\), see Lemma 06AF.

Ad (0CF5). This is general remark (III).

Ad (0CF6). Assume \(f\) is an isomorphism. Choose an affine open \(V' \subset Y'\) and set \(U' = (f')^{-1}(V')\). Then \(V = Y \cap V'\) is affine which implies that \(V \cong f^{-1}(V) = U = Y \times_{Y'} U'\) is affine. By Lemma 06AD we see that \(U'\) is affine. Thus we have a diagram as in the general remark (I). By Algebra, Lemma 07RE we see that \(A' \to B'\) is an isomorphism, i.e., \(U' \cong V'\). Thus \(f'\) is an isomorphism.

Ad (0CF7). Assume \(f\) is an open immersion. Then \(f\) is an isomorphism of \(X\) with an open subscheme \(V \subset Y\). Let \(V' \subset Y'\) be the open subscheme whose underlying topological space is \(V\). Then \(f'\) is a map from \(X'\) to \(V'\) which is an isomorphism by (0CF6). Hence \(f'\) is an open immersion.

Ad (0CF8). Immediate from remark (II). See also Lemma 09ZV for a more general statement.

Ad (0CF9). Immediate from remark (II). See also Lemma 09ZV for a more general statement.

Ad (0CFA). Note that \(X \times_Y X = Y \times_{Y'} (X' \times_{Y'} X')\) so that \(X' \times_{Y'} X'\) is a thickening of \(X \times_Y X\). Hence the topology of the maps \(\Delta_{X/Y}\) and \(\Delta_{X'/Y'}\) matches and we win. See also Lemma 09ZV for a more general statement.

Ad (0CFB). Assume \(f\) is a monomorphism. Consider the diagonal morphism \(\Delta_{X'/Y'} : X' \to X' \times_{Y'} X'\). Observe that \(X' \times_{Y'} X' \to S'\) is locally of finite type. The base change of \(\Delta_{X'/Y'}\) by \(S \to S'\) is \(\Delta_{X/Y}\) which is an isomorphism by assumption. By (0CF6) we conclude that \(\Delta_{X'/Y'}\) is an isomorphism.

Ad (0CFC). This is clear. See also Lemma 09ZV for a more general statement.

Ad (0CFD). Immediate from remark (II). See also Lemma 09ZV for a more general statement.

Ad (0CFE). Assume \(f\) is affine. Choose an affine open \(V' \subset Y'\) and set \(U' = (f')^{-1}(V')\). Then \(V = Y \cap V'\) is affine which implies that \(U = Y \times_{Y'} U'\) is affine. By Lemma 06AD we see that \(U'\) is affine. Hence \(f'\) is affine. See also Lemma 09ZV for a more general statement.

Ad (0CFF). Follows from the fact that \(f'\) is locally of finite type (by Morphisms, Lemma 01T8) and that quasi-finiteness of a morphism of finite type can be checked on fibres, see Morphisms, Lemma 01TH.

Ad (0CFG). Follows from general remark (II) and the fact that \(f'\) is locally of finite type (Morphisms, Lemma 01T8).

Ad (0CFH). Immediate from general remark (II). See also Lemma 09ZV for a more general statement.

Ad (0CFI). Assume \(f\) is syntomic. By Morphisms, Lemma 02FV \(f'\) is locally of finite presentation. By general remark (III) \(f'\) is flat. The fibres of \(f'\) are the fibres of \(f\). Hence \(f'\) is syntomic by Morphisms, Lemma 01UF.

Ad (0CFJ). Assume \(f\) is smooth. By Morphisms, Lemma 02FV \(f'\) is locally of finite presentation. By general remark (III) \(f'\) is flat. The fibres of \(f'\) are the fibres of \(f\). Hence \(f'\) is smooth by Morphisms, Lemma 01V8.

Ad (0CFK). Assume \(f\) unramified. By Morphisms, Lemma 01T8 \(f'\) is locally of finite type. The fibres of \(f'\) are the fibres of \(f\). Hence \(f'\) is unramified by Morphisms, Lemma 02G8.

Ad (0CFL). Assume \(f\) étale. By Morphisms, Lemma 02FV \(f'\) is locally of finite presentation. By general remark (III) \(f'\) is flat. The fibres of \(f'\) are the fibres of \(f\). Hence \(f'\) is étale by Morphisms, Lemma 02GM.

Ad (0CFM). This follows from a combination of (0CFA), the fact that \(f\) is locally of finite type (Morphisms, Lemma 01T8), (0CF8), and (0CF9).

Ad (0CFN). Combine (0CF9), (0CFE), Morphisms, Lemma 01WM, the fact that \(f\) is locally of finite type (Morphisms, Lemma 01T8), and Morphisms, Lemma 01WJ.

Ad (0CFP). Assume \(f\) finite locally free. By (0CFN) we see that \(f'\) is finite. By general remark (III) \(f'\) is flat. By Morphisms, Lemma 02FV \(f'\) is locally of finite presentation. Hence \(f'\) is finite locally free by Morphisms, Lemma 02KB.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is proper, flat, and of finite presentation. Let \(\mathcal{L}\) be \(f\)-ample. Assume \(S\) is quasi-compact. There exists a \(d_0 \geq 0\) such that for every cartesian diagram \[\vcenter{ \xymatrix{ X \ar[r]_{i'} \ar[d]_f & X' \ar[d]^{f'} \\ S \ar[r]^i & S' } } \quad\text{and}\quad \begin{matrix} \text{invertible }\mathcal{O}_{X'}\text{-module}\\ \mathcal{L}'\text{ with }\mathcal{L} \cong (i')^*\mathcal{L}' \end{matrix}\] where \(S \subset S'\) is a thickening and \(f'\) is proper, flat, of finite presentation we have

\(R^p(f')_*(\mathcal{L}')^{\otimes d} = 0\) for all \(p > 0\) and \(d \geq d_0\),

\(\mathcal{A}'_d = (f')_*(\mathcal{L}')^{\otimes d}\) is finite locally free for \(d \geq d_0\),

\(\mathcal{A}' = \mathcal{O}_{S'} \oplus \bigoplus_{d \geq d_0} \mathcal{A}'_d\) is a quasi-coherent \(\mathcal{O}_{S'}\)-algebra of finite presentation,

there is a canonical isomorphism \(r' : X' \to \underline{\text{Proj}}_{S'}(\mathcal{A}')\), and

there is a canonical isomorphism \(\theta' : (r')^*\mathcal{O}_{\underline{\text{Proj}}_{S'}(\mathcal{A}')}(1) \to \mathcal{L}'\).

The construction of \(\mathcal{A}'\), \(r'\), \(\theta'\) is functorial in the data \((X', S', i, i', f', \mathcal{L}')\).

Proof

We first describe the maps \(r'\) and \(\theta'\). Observe that \(\mathcal{L}'\) is \(f'\)-ample, see Lemma 0D2R. There is a canonical map of quasi-coherent graded \(\mathcal{O}_{S'}\)-algebras \(\mathcal{A}' \to \bigoplus_{d \geq 0} (f')_*(\mathcal{L}')^{\otimes d}\) which is an isomorphism in degrees \(\geq d_0\). Hence this induces an isomorphism on relative Proj compatible with the Serre twists of the structure sheaf, see Constructions, Lemma 07ZJ. Hence we get the morphism \(r'\) by Morphisms, Lemma 01VJ (which in turn appeals to the construction given in Constructions, Lemma 01O9) and it is an isomorphism by Morphisms, Lemma 0C6J. We get the map \(\theta'\) from Constructions, Lemma 01O9. By Properties, Lemma 01QI we find that \(\theta'\) is an isomorphism (this also uses that the morphism \(r'\) over affine opens of \(S'\) is the same as the morphism from Properties, Lemma 01PZ as is explained in the proof of Morphisms, Lemma 0C6J).

Assuming the vanishing and local freeness stated in parts (1) and (2), the functoriality of the construction can be seen as follows. Suppose that \(h : T \to S'\) is a morphism of schemes, denote \(f_T : X'_T \to T\) the base change of \(f'\) and \(\mathcal{L}_T\) the pullback of \(\mathcal{L}\) to \(X'_T\). By cohomology and base change (as formulated in Derived Categories of Schemes, Lemma 08IB for example) we have the corresponding vanishing over \(T\) and moreover \(h^*\mathcal{A}'_d = f_{T, *}\mathcal{L}_T^{\otimes d}\) (and thus the local freeness of pushforwards as well as the finite generation of the corresponding graded \(\mathcal{O}_T\)-algebra \(\mathcal{A}_T\)). Hence the morphism \(r_T : X_T \to \underline{\text{Proj}}_T(\bigoplus f_{T, *}\mathcal{L}_T^{\otimes d})\) is simply the base change of \(r'\) to \(T\) and the pullback of \(\theta'\) is the map \(\theta_T\).

Having said all of the above, we see that it suffices to prove (1), (2), and (3). Pick \(d_0\) such that \(R^pf_*\mathcal{L}^{\otimes d} = 0\) for all \(d \geq d_0\) and \(p > 0\), see Cohomology of Schemes, Lemma 0B5T. We claim that \(d_0\) works.

By cohomology and base change (Derived Categories of Schemes, Lemma 0B91) we see that \(E'_d = Rf'_*(\mathcal{L}')^{\otimes d}\) is a perfect object of \(D(\mathcal{O}_{S'})\) and its formation commutes with arbitrary base change. In particular, \(E_d = Li^*E'_d = Rf_*\mathcal{L}^{\otimes d}\). By Derived Categories of Schemes, Lemma 0D4E we see that for \(d \geq d_0\) the complex \(E_d\) is isomorphic to the finite locally free \(\mathcal{O}_S\)-module \(f_*\mathcal{L}^{\otimes d}\) placed in cohomological degree \(0\). Then by Derived Categories of Schemes, Lemma 0BDK we conclude that \(E'_d\) is isomorphic to a finite locally free module placed in cohomological degree \(0\). Of course this means that \(E'_d = \mathcal{A}'_d[0]\), that \(R^pf'_*(\mathcal{L}')^{\otimes d} = 0\) for \(p > 0\), and that \(\mathcal{A}'_d\) is finite locally free. This proves (1) and (2).

The last thing we have to show is finite presentation of \(\mathcal{A}'\) as a sheaf of \(\mathcal{O}_{S'}\)-algebras (this notion was introduced in Properties, Section 01PD). Let \(U' = \Spec(R') \subset S'\) be an affine open. Then \(A' = \mathcal{A}'(U')\) is a graded \(R'\)-algebra whose graded parts are finite projective \(R'\)-modules. We have to show that \(A'\) is a finitely presented \(R'\)-algebra. We will prove this by reduction to the Noetherian case. Namely, we can find a finite type \(\mathbf{Z}\)-subalgebra \(R'_0 \subset R'\) and a pair¹ \((X'_0, \mathcal{L}'_0)\) over \(R'_0\) whose base change is \((X'_{U'}, \mathcal{L}'|_{X'_{U'}})\), see Limits, Lemmas 01ZR, 0B8W, 081F, 04AI, and 09MT. Cohomology of Schemes, Lemma 0B5T implies \(A'_0 = \bigoplus_{d \geq 0} H^0(X'_0, (\mathcal{L}'_0)^{\otimes d})\) is a finitely generated graded \(R'_0\)-algebra and implies there exists a \(d'_0\) such that \(H^p(X'_0, (\mathcal{L}'_0)^{\otimes d}) = 0\), \(p > 0\) for \(d \geq d'_0\). By the arguments given above applied to \(X'_0 \to \Spec(R'_0)\) and \(\mathcal{L}'_0\) we see that \((A'_0)_d\) is a finite projective \(R'_0\)-module and that \[A'_d = \mathcal{A}'_d(U') = H^0(X'_{U'}, (\mathcal{L}')^{\otimes d}|_{X'_{U'}}) = H^0(X'_0, (\mathcal{L}'_0)^{\otimes d}) \otimes_{R'_0} R' = (A'_0)_d \otimes_{R'_0} R'\] for \(d \geq d'_0\). Now a small twist in the argument is that we don’t know that we can choose \(d'_0\) equal to \(d_0\)². To get around this we use the following sequence of arguments to finish the proof:

The algebra \(B = R'_0 \oplus \bigoplus_{d \geq \max(d_0, d'_0)} (A'_0)_d\) is an \(R'_0\)-algebra of finite type: apply the Artin-Tate lemma to \(B \subset A'_0\), see Algebra, Lemma 00IS.

As \(R'_0\) is Noetherian we see that \(B\) is an \(R'_0\)-algebra of finite presentation.

By right exactness of tensor product we see that \(B \otimes_{R'_0} R'\) is an \(R'\)-algebra of finite presentation.

By the displayed equalities this exactly says that \(C = R' \oplus \bigoplus_{d \geq \max(d_0, d'_0)} A'_d\) is an \(R'\)-algebra of finite presentation.

The quotient \(A'/C\) is the direct sum of the finite projective \(R'\)-modules \(A'_d\), \(d_0 \leq d \leq \max(d_0, d'_0)\), hence finitely presented as \(R'\)-module.

The quotient \(A'/C\) is finitely presented as a \(C\)-module by Algebra, Lemma 0561.

Thus \(A'\) is finitely presented as a \(C\)-module by Algebra, Lemma 0519.

By Algebra, Lemma 0D46 this implies \(A'\) is finitely presented as a \(C\)-algebra.

Finally, by Algebra, Lemma 00F4 applied to \(R' \to C \to A'\) this implies \(A'\) is finitely presented as an \(R'\)-algebra.

This finishes the proof.

Formally smooth morphisms

Michael Artin’s position on differential criteria of smoothness (e.g., Morphisms, Lemma 01V9) is that they are basically useless (in practice). In this section we introduce the notion of a formally smooth morphism \(X \to S\). Such a morphism is characterized by the property that \(T\)-valued points of \(X\) lift to infinitesimal thickenings of \(T\) provided \(T\) is affine. The main result is that a morphism which is formally smooth and locally of finite presentation is smooth, see Lemma 02H6. It turns out that this criterion is often easier to use than the differential criteria mentioned above.

Recall that a ring map \(R \to A\) is called formally smooth (see Algebra, Definition 00TI) if for every commutative solid diagram \[\xymatrix{ A \ar[r] \ar@{-->}[rd] & B/I \\ R \ar[r] \ar[u] & B \ar[u] }\] where \(I \subset B\) is an ideal of square zero, a dotted arrow exists which makes the diagram commute. This motivates the following analogue for morphisms of schemes.

Definition

Let \(f : X \to S\) be a morphism of schemes. We say \(f\) is formally smooth if given any solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] }\] where \(T \subset T'\) is a first order thickening of affine schemes over \(S\) there exists a dotted arrow making the diagram commute.

In the cases of formally unramified and formally étale morphisms the condition that \(T'\) be affine could be dropped, see Lemmas 04F1 and 04FD. This is no longer true in the case of formally smooth morphisms. In fact, a slightly more natural condition would be that we should be able to fill in the dotted arrow Zariski locally on \(T'\). In fact, analyzing the proof of Lemma 0D0F shows that this would be equivalent to the definition as it currently stands. In particular, being formally smooth is Zariski local on the source (and in fact it is smooth local on the source, insert future reference here).

Lemma

A composition of formally smooth morphisms is formally smooth.

Proof

Omitted.

Lemma

A base change of a formally smooth morphism is formally smooth.

Proof

Omitted, but see Algebra, Lemma 00TJ for the algebraic version.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Then \(f\) is formally étale if and only if \(f\) is formally smooth and formally unramified.

Proof

Omitted.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(U \subset X\) and \(V \subset S\) be open subschemes such that \(f(U) \subset V\). If \(f\) is formally smooth, so is \(f|_U : U \to V\).

Proof

Consider a solid diagram \[\xymatrix{ U \ar[d]_{f|_U} & T \ar[d]^i \ar[l]^a \\ V & T' \ar[l] \ar@{-->}[lu] }\] as in Definition 02H0. If \(f\) is formally smooth, then there exists an \(S\)-morphism \(a' : T' \to X\) such that \(a'|_T = a\). Since the underlying sets of \(T\) and \(T'\) are the same we see that \(a'\) is a morphism into \(U\) (see Schemes, Section 01HD). And it clearly is a \(V\)-morphism as well. Hence the dotted arrow above as desired.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume \(X\) and \(S\) are affine. Then \(f\) is formally smooth if and only if \(\mathcal{O}_S(S) \to \mathcal{O}_X(X)\) is a formally smooth ring map.

Proof

This is immediate from the definitions (Definition 02H0 and Algebra, Definition 00TI) by the equivalence of categories of rings and affine schemes, see Schemes, Lemma 01I2.

The following lemma is the main result of this section. It is a victory of the functorial point of view in that it implies (combined with Limits, Proposition 01ZC) that we can recognize whether a morphism \(f : X \to S\) is smooth in terms of “simple” properties of the functor \(h_X : \Sch/S \to \textit{Sets}\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent:

The morphism \(f\) is smooth, and

the morphism \(f\) is locally of finite presentation and formally smooth.

Proof

Assume \(f : X \to S\) is locally of finite presentation and formally smooth. Consider a pair of affine opens \(\Spec(A) = U \subset X\) and \(\Spec(R) = V \subset S\) such that \(f(U) \subset V\). By Lemma 02H3 we see that \(U \to V\) is formally smooth. By Lemma 02H4 we see that \(R \to A\) is formally smooth. By Morphisms, Lemma 01TQ we see that \(R \to A\) is of finite presentation. By Algebra, Proposition 00TN we see that \(R \to A\) is smooth. Hence by the definition of a smooth morphism we see that \(X \to S\) is smooth.

Conversely, assume that \(f : X \to S\) is smooth. Consider a solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l]^a \\ S & T' \ar[l] \ar@{-->}[lu] }\] as in Definition 02H0. We will show the dotted arrow exists thereby proving that \(f\) is formally smooth.

Let \(\mathcal{F}\) be the sheaf of sets on \(T'\) of Lemma 04FH in the special case discussed in Remark 04FK. Let \[\mathcal{H} = \SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/S}, \mathcal{C}_{T/T'})\] be the sheaf of \(\mathcal{O}_T\)-modules with action \(\mathcal{H} \times \mathcal{F} \to \mathcal{F}\) as in Lemma 04FJ. Our goal is simply to show that \(\mathcal{F}(T) \not = \emptyset\). In other words we are trying to show that \(\mathcal{F}\) is a trivial \(\mathcal{H}\)-torsor on \(T\) (see Cohomology, Section 02FN). There are two steps: (I) To show that \(\mathcal{F}\) is a torsor we have to show that \(\mathcal{F}_t \not = \emptyset\) for all \(t \in T\) (see Cohomology, Definition 02FO). (II) To show that \(\mathcal{F}\) is the trivial torsor it suffices to show that \(H^1(T, \mathcal{H}) = 0\) (see Cohomology, Lemma 02FQ – we may use either cohomology of \(\mathcal{H}\) as an abelian sheaf or as an \(\mathcal{O}_T\)-module, see Cohomology, Lemma 01F1).

First we prove (I). To see this, for every \(t \in T\) we can choose an affine open \(U \subset T\) neighbourhood of \(t\) such that \(a(U)\) is contained in an affine open \(\Spec(A) = W \subset X\) which maps to an affine open \(\Spec(R) = V \subset S\). By Morphisms, Lemma 01V6 the ring map \(R \to A\) is smooth. Hence by Algebra, Proposition 00TN the ring map \(R \to A\) is formally smooth. Lemma 02H4 in turn implies that \(W \to V\) is formally smooth. Hence we can lift \(a|_U : U \to W\) to a \(V\)-morphism \(a' : U' \to W \subset X\) showing that \(\mathcal{F}(U) \not = \emptyset\).

Finally we prove (II). By Morphisms, Lemma 01V3 we see that \(\Omega_{X/S}\) is of finite presentation (it is even finite locally free by Morphisms, Lemma 02G1). Hence \(a^*\Omega_{X/S}\) is of finite presentation (see Modules, Lemma 01BQ). Hence the sheaf \(\mathcal{H} = \SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/S}, \mathcal{C}_{T/T'})\) is quasi-coherent by the discussion in Schemes, Section 01LA. Thus by Cohomology of Schemes, Lemma 01XB we have \(H^1(T, \mathcal{H}) = 0\) as desired.

Locally projective quasi-coherent modules are defined in Properties, Section 05JN.

Lemma

Let \(f : X \to Y\) be a formally smooth morphism of schemes. Then \(\Omega_{X/Y}\) is locally projective on \(X\).

Proof

Choose \(U \subset X\) and \(V \subset Y\) affine open such that \(f(U) \subset V\). By Lemma 02H3 \(f|_U : U \to V\) is formally smooth. Hence \(\Gamma(V, \mathcal{O}_V) \to \Gamma(U, \mathcal{O}_U)\) is a formally smooth ring map, see Lemma 02H4. Hence by Algebra, Lemma 031I the \(\Gamma(U, \mathcal{O}_U)\)-module \(\Omega_{\Gamma(U, \mathcal{O}_U)/\Gamma(V, \mathcal{O}_V)}\) is projective. Hence \(\Omega_{U/V}\) is locally projective, see Properties, Section 05JN.

Lemma

Let \(T\) be an affine scheme. Let \(\mathcal{F}\), \(\mathcal{G}\) be quasi-coherent \(\mathcal{O}_T\)-modules. Consider \(\mathcal{H} = \SheafHom_{\mathcal{O}_T}(\mathcal{F}, \mathcal{G})\). If \(\mathcal{F}\) is locally projective, then \(H^1(T, \mathcal{H}) = 0\).

Proof

By the definition of a locally projective sheaf on a scheme (see Properties, Definition 05JP) we see that \(\mathcal{F}\) is a direct summand of a free \(\mathcal{O}_T\)-module. Hence we may assume that \(\mathcal{F} = \bigoplus_{i \in I} \mathcal{O}_T\) is a free module. In this case \(\mathcal{H} = \prod_{i \in I} \mathcal{G}\) is a product of quasi-coherent modules. By Cohomology, Lemma 0D0A we conclude that \(H^1 = 0\) because the cohomology of a quasi-coherent sheaf on an affine scheme is zero, see Cohomology of Schemes, Lemma 01XB.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent:

\(f\) is formally smooth,

for every \(x \in X\) there exist opens \(x \in U \subset X\) and \(f(x) \in V \subset Y\) with \(f(U) \subset V\) such that \(f|_U : U \to V\) is formally smooth,

for every pair of affine opens \(U \subset X\) and \(V \subset Y\) with \(f(U) \subset V\) the ring map \(\mathcal{O}_Y(V) \to \mathcal{O}_X(U)\) is formally smooth, and

there exists an affine open covering \(Y = \bigcup V_j\) and for each \(j\) an affine open covering \(f^{-1}(V_j) = \bigcup U_{ji}\) such that \(\mathcal{O}_Y(V) \to \mathcal{O}_X(U)\) is a formally smooth ring map for all \(j\) and \(i\).

Proof

The implications (1) \(\Rightarrow\) (2), (1) \(\Rightarrow\) (3), and (2) \(\Rightarrow\) (4) follow from Lemma 02H3. The implication (3) \(\Rightarrow\) (4) is immediate.

Assume (4). The proof that \(f\) is formally smooth is the same as the second part of the proof of Lemma 02H6. Consider a solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l]^a \\ Y & T' \ar[l] \ar@{-->}[lu] }\] as in Definition 02H0. We will show the dotted arrow exists thereby proving that \(f\) is formally smooth. Let \(\mathcal{F}\) be the sheaf of sets on \(T'\) of Lemma 04FH as in the special case discussed in Remark 04FK. Let \[\mathcal{H} = \SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/Y}, \mathcal{C}_{T/T'})\] be the sheaf of \(\mathcal{O}_T\)-modules on \(T\) with action \(\mathcal{H} \times \mathcal{F} \to \mathcal{F}\) as in Lemma 04FJ. The action \(\mathcal{H} \times \mathcal{F} \to \mathcal{F}\) turns \(\mathcal{F}\) into a pseudo \(\mathcal{H}\)-torsor, see Cohomology, Definition 02FO. Our goal is to show that \(\mathcal{F}\) is a trivial \(\mathcal{H}\)-torsor. There are two steps: (I) To show that \(\mathcal{F}\) is a torsor we have to show that \(\mathcal{F}\) locally has a section. (II) To show that \(\mathcal{F}\) is the trivial torsor it suffices to show that \(H^1(T, \mathcal{H}) = 0\), see Cohomology, Lemma 02FQ.

First we prove (I). To see this, for every \(t \in T\) we can choose an affine open \(W \subset T\) neighbourhood of \(t\) such that \(a(W)\) is contained in \(U_{ji}\) for some \(i, j\). Let \(W' \subset T'\) be the corresponding open subscheme. By assumption (4) we can lift \(a|_W : W \to U_{ji}\) to a \(V_j\)-morphism \(a' : W' \to U_{ji}\) showing that \(\mathcal{F}(W')\) is nonempty.

Finally we prove (II). By Lemma 06B5 we see that \(\Omega_{U_{ji}/V_j}\) locally projective. Hence \(\Omega_{X/Y}\) is locally projective, see Properties, Lemma 05JQ. Hence \(a^*\Omega_{X/Y}\) is locally projective, see Properties, Lemma 060M. Hence \[H^1(T, \mathcal{H}) = H^1(T, \SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/Y}, \mathcal{C}_{T/T'})) = 0\] by Lemma 0D0E as desired.

Lemma

Let \(f : X \to Y\), \(g : Y \to S\) be morphisms of schemes. Assume \(f\) is formally smooth. Then \[0 \to f^*\Omega_{Y/S} \to \Omega_{X/S} \to \Omega_{X/Y} \to 0\] (see Morphisms, Lemma 01UX) is short exact.

Proof

The algebraic version of this lemma is the following: Given ring maps \(A \to B \to C\) with \(B \to C\) formally smooth, then the sequence \[0 \to C \otimes_B \Omega_{B/A} \to \Omega_{C/A} \to \Omega_{C/B} \to 0\] of Algebra, Lemma 00RS is exact. This is Algebra, Lemma 031K.

Lemma

Let \(h : Z \to X\) be a formally unramified morphism of schemes over \(S\). Assume that \(Z\) is formally smooth over \(S\). Then the canonical exact sequence \[0 \to \mathcal{C}_{Z/X} \to h^*\Omega_{X/S} \to \Omega_{Z/S} \to 0\] of Lemma 04FC is short exact.

Proof

Let \(Z \to Z'\) be the universal first order thickening of \(Z\) over \(X\). From the proof of Lemma 04FC we see that our sequence is identified with the sequence \[\mathcal{C}_{Z/Z'} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z \to \Omega_{Z/S} \to 0.\] Since \(Z \to S\) is formally smooth we can locally on \(Z'\) find a left inverse \(Z' \to Z\) over \(S\) to the inclusion map \(Z \to Z'\). Thus the sequence is locally split, see Morphisms, Lemma 0474.

Lemma

Let \[\xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d]^f \\ & Y }\] be a commutative diagram of schemes where \(i\) and \(j\) are formally unramified and \(f\) is formally smooth. Then the canonical exact sequence \[0 \to \mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/Y} \to 0\] of Lemma 067V is exact and locally split.

Proof

Denote \(Z \to Z'\) the universal first order thickening of \(Z\) over \(X\). Denote \(Z \to Z''\) the universal first order thickening of \(Z\) over \(Y\). By Lemma 04FC here is a canonical morphism \(Z' \to Z''\) so that we have a commutative diagram \[\xymatrix{ Z \ar[r]_{i'} \ar[rd]_{j'} & Z' \ar[r]_a \ar[d]^k & X \ar[d]^f \\ & Z'' \ar[r]^b & Y }\] In the proof of Lemma 067V we identified the sequence above with the sequence \[\mathcal{C}_{Z/Z''} \to \mathcal{C}_{Z/Z'} \to (i')^*\Omega_{Z'/Z''} \to 0\] Let \(U'' \subset Z''\) be an affine open. Denote \(U \subset Z\) and \(U' \subset Z'\) the corresponding affine open subschemes. As \(f\) is formally smooth there exists a morphism \(h : U'' \to X\) which agrees with \(i\) on \(U\) and such that \(f \circ h\) equals \(b|_{U''}\). Since \(Z'\) is the universal first order thickening we obtain a unique morphism \(g : U'' \to Z'\) such that \(g = a \circ h\). The universal property of \(Z''\) implies that \(k \circ g\) is the inclusion map \(U'' \to Z''\). Hence \(g\) is a left inverse to \(k\). Picture \[\xymatrix{ U \ar[d] \ar[r] & Z' \ar[d]^k \\ U'' \ar[r] \ar[ru]^g & Z'' }\] Thus \(g\) induces a map \(\mathcal{C}_{Z/Z'}|_U \to \mathcal{C}_{Z/Z''}|_U\) which is a left inverse to the map \(\mathcal{C}_{Z/Z''} \to \mathcal{C}_{Z/Z'}\) over \(U\).

Smoothness over a Noetherian base

It turns out that if the base is Noetherian then we can get away with less in the formulation of formal smoothness. In some sense the following lemmas are the beginning of deformation theory.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\). Assume that \(S\) is locally Noetherian and \(f\) locally of finite type. The following are equivalent:

\(f\) is smooth at \(x\),

for every solid commutative diagram \[\xymatrix{ X \ar[d]_f & \Spec(B) \ar[d]^i \ar[l]^-\alpha \\ S & \Spec(B') \ar[l]_-{\beta} \ar@{-->}[lu] }\] where \(B' \to B\) is a surjection of local rings with \(\Ker(B' \to B)\) of square zero, and \(\alpha\) mapping the closed point of \(\Spec(B)\) to \(x\) there exists a dotted arrow making the diagram commute,

same as in (2) but with \(B' \to B\) ranging over small extensions (see Algebra, Definition 02HS), and

same as in (2) but with \(B' \to B\) ranging over small extensions such that \(\alpha\) induces an isomorphism \(\kappa(x) \to \kappa(\mathfrak m)\) where \(\mathfrak m \subset B\) is the maximal ideal.

Proof

Choose an affine neighbourhood \(V \subset S\) of \(f(x)\) and choose an affine neighbourhood \(U \subset X\) of \(x\) such that \(f(U) \subset V\). For any “test” diagram as in (2) the morphism \(\alpha\) will map \(\Spec(B)\) into \(U\) and the morphism \(\beta\) will map \(\Spec(B')\) into \(V\) (see Schemes, Section 01J5). Hence the lemma reduces to the morphism \(f|_U : U \to V\) of affines. (Indeed, \(V\) is Noetherian and \(f|_U\) is of finite type, see Properties, Lemma 01OW and Morphisms, Lemma 01T2.) In this affine case the lemma is identical to Algebra, Lemma 02HT.

Sometimes it is useful to know that one only needs to check the lifting criterion for small extensions “centered” at points of finite type (see Morphisms, Section 01T9).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume that \(S\) is locally Noetherian and \(f\) locally of finite type. The following are equivalent:

\(f\) is smooth,

for every solid commutative diagram \[\xymatrix{ X \ar[d]_f & \Spec(B) \ar[d]^i \ar[l]^-\alpha \\ S & \Spec(B') \ar[l]_-{\beta} \ar@{-->}[lu] }\] where \(B' \to B\) is a small extension of Artinian local rings and \(\beta\) of finite type (!) there exists a dotted arrow making the diagram commute.

Proof

If \(f\) is smooth, then the infinitesimal lifting criterion (Lemma 02H6) says \(f\) is formally smooth and (2) holds.

Assume (2). The set of points \(x \in X\) where \(f\) is not smooth forms a closed subset \(T\) of \(X\). By the discussion in Morphisms, Section 01T9, if \(T \not = \emptyset\) there exists a point \(x \in T \subset X\) such that the morphism \[\Spec(\kappa(x)) \to X \to S\] is of finite type (namely, pick any point \(x\) of \(T\) which is closed in an affine open of \(X\)). By Morphisms, Lemma 02HV given any local Artinian ring \(B'\) with residue field \(\kappa(x)\) then any morphism \(\beta : \Spec(B') \to S\) is of finite type. Thus we see that all the diagrams used in Lemma 02HX (4) correspond to diagrams as in the current lemma (2). Whence \(X \to S\) is smooth a \(x\) a contradiction.

Here is a useful application.

Lemma

Let \(f : X \to S\) be a finite type morphism of locally Noetherian schemes. Let \(Z \subset S\) be a closed subscheme with \(n\)th infinitesimal neighbourhood \(Z_n \subset S\). Set \(X_n = Z_n \times_S X\).

If \(X_n \to Z_n\) is smooth for all \(n\), then \(f\) is smooth at every point of \(f^{-1}(Z)\).

If \(X_n \to Z_n\) is étale for all \(n\), then \(f\) is étale at every point of \(f^{-1}(Z)\).

Proof

Assume \(X_n \to Z_n\) is smooth for all \(n\). Let \(x \in X\) be a point lying over a point of \(Z\). Given a small extension \(B' \to B\) and morphisms \(\alpha\), \(\beta\) as in Lemma 02HX part (3) the maximal ideal of \(B'\) is nilpotent (as \(B'\) is Artinian) and hence the morphism \(\beta\) factors through \(Z_n\) and \(\alpha\) factors through \(X_n\) for a suitable \(n\). Thus the lifting property for \(X_n \to Z_n\) kicks in to get the desired dotted arrow in the diagram. This proves (1). Part (2) follows from (1) and the fact that a morphism is étale if and only if it is smooth of relative dimension \(0\).

Lemma

Let \(f : X \to S\) be a morphism of locally Noetherian schemes. Let \(Z \subset S\) be a closed subscheme with \(n\)th infinitesimal neighbourhood \(Z_n \subset S\). Set \(X_n = Z_n \times_S X\). If \(X_n \to Z_n\) is flat for all \(n\), then \(f\) is flat at every point of \(f^{-1}(Z)\).

Proof

This is a translation of Algebra, Lemma 0523 into the language of schemes.

The naive cotangent complex

This section is the continuation of Modules, Section 08TG which in turn continues the discussion in Algebra, Section 00S0.

Definition

Let \(f : X \to Y\) be a morphism of schemes. The naive cotangent complex of \(f\) is the complex defined in Modules, Definition 08TN. Notation: \(\NL_f\) or \(\NL_{X/Y}\).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(\Spec(A) = U \subset X\) and \(\Spec(R) = V \subset S\) be affine opens with \(f(U) \subset V\). There is a canonical map \[\widetilde{\NL_{A/R}} \longrightarrow \NL_{X/Y}|_U\] of complexes which is an isomorphism in \(D(\mathcal{O}_U)\).

Proof

From the construction of \(\NL_{X/Y}\) in Modules, Section 08TG we see there is a canonical map of complexes \(\NL_{\mathcal{O}_X(U)/f^{-1}\mathcal{O}_Y(U)} \to \NL_{X/Y}(U)\) of \(A = \mathcal{O}_X(U)\)-modules, which is compatible with further restrictions. Using the canonical map \(R \to f^{-1}\mathcal{O}_Y(U)\) we obtain a canonical map \(\NL_{A/R} \to \NL_{\mathcal{O}_X(U)/f^{-1}\mathcal{O}_Y(U)}\) of complexes of \(A\)-modules. Using the universal property of the \(\widetilde{\ }\) functor (see Schemes, Lemma 01I7) we obtain a map as in the statement of the lemma. We may check this map is an isomorphism on cohomology sheaves by checking it induces isomorphisms on stalks. This follows from Algebra, Lemma 07BS and 00S7 and Modules, Lemma 0D09 (and the description of the stalks of \(\mathcal{O}_X\) and \(f^{-1}\mathcal{O}_Y\) at a point \(\mathfrak p \in \Spec(A)\) as \(A_\mathfrak p\) and \(R_\mathfrak q\) where \(\mathfrak q = R \cap \mathfrak p\); references used are Schemes, Lemma 01HV and Sheaves, Lemma 008H).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The cohomology sheaves of the complex \(\NL_{X/Y}\) are quasi-coherent, zero outside degrees \(-1\), \(0\) and equal to \(\Omega_{X/Y}\) in degree \(0\).

Proof

By construction of the naive cotangent complex in Modules, Section 08TG we have that \(\NL_{X/Y}\) is a complex sitting in degrees \(-1\), \(0\) and that its cohomology in degree \(0\) is \(\Omega_{X/Y}\). The sheaf of differentials is quasi-coherent (by Morphisms, Lemma 08S2). To finish the proof it suffices to show that \(H^{-1}(\NL_{X/Y})\) is quasi-coherent. This follows by checking over affines using Lemma 0D0I.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. If \(f\) is locally of finite presentation, then \(\NL_{X/Y}\) is locally on \(X\) quasi-isomorphic to a complex \[\ldots \to 0 \to \mathcal{F}^{-1} \to \mathcal{F}^0 \to 0 \to \ldots\] of quasi-coherent \(\mathcal{O}_X\)-modules with \(\mathcal{F}^0\) of finite presentation and \(\mathcal{F}^{-1}\) of finite type.

Proof

By Lemma 0D0I it suffices to show that \(\NL_{A/R}\) has this shape if \(R \to A\) is a finitely presented ring map. Write \(A = R[x_1, \ldots, x_n]/I\) with \(I\) finitely generated. Then \(I/I^2\) is a finite \(A\)-module and \(\NL_{A/R}\) is quasi-isomorphic to \[\ldots \to 0 \to I/I^2 \to \bigoplus\nolimits_{i = 1, \ldots, n} A\text{d}x_i \to 0 \to \ldots\] by Algebra, Section 00S0 and in particular Algebra, Lemma 00S1.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent

\(f\) is formally smooth,

\(H^{-1}(\NL_{X/Y}) = 0\) and \(H^0(\NL_{X/Y}) = \Omega_{X/Y}\) is locally projective.

Proof

This follows from Algebra, Proposition 031J and Lemma 0D0F.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent

\(f\) is formally étale,

\(H^{-1}(\NL_{X/Y}) = H^0(\NL_{X/Y}) = 0\).

Proof

A formally étale morphism is formally smooth and hence we have \(H^{-1}(\NL_{X/Y}) = 0\) by Lemma 0D0L. On the other hand, we have \(\Omega_{X/Y} = 0\) by Lemma 04FE. Conversely, if (2) holds, then \(f\) is formally smooth by Lemma 0D0L and formally unramified by Lemma 02H9 and hence formally étale by Lemmas 02HH.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent

\(f\) is smooth, and

\(f\) is locally of finite presentation, \(H^{-1}(\NL_{X/Y}) = 0\), and \(H^0(\NL_{X/Y}) = \Omega_{X/Y}\) is finite locally free.

Proof

This follows from the definition of a smooth ring homomorphism (Algebra, Definition 00T2), Lemma 0D0I, and the definition of a smooth morphism of schemes (Morphisms, Definition 01V5). We also use that finite locally free is the same as finite projective for modules over rings (Algebra, Lemma 00NX).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent

\(f\) is étale, and

\(f\) is locally of finite presentation and \(H^{-1}(\NL_{X/Y}) = H^0(\NL_{X/Y}) = 0\).

Proof

This follows from the definition of an étale ring homomorphism (Algebra, Definition 00U1), Lemma 0D0I, and the definition of an étale morphism of schemes (Morphisms, Definition 02GI).

Lemma

Let \(i : Z \to X\) be an immersion of schemes. Then \(\NL_{Z/X}\) is isomorphic to \(\mathcal{C}_{Z/X}[1]\) in \(D(\mathcal{O}_Z)\) where \(\mathcal{C}_{Z/X}\) is the conormal sheaf of \(Z\) in \(X\).

Proof

This follows from Algebra, Lemma 07BP, Morphisms, Lemma 01R3, and Lemma 0D0I.

Lemma

Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of schemes. There is a canonical six term exact sequence \[H^{-1}(f^*\NL_{Y/Z}) \to H^{-1}(\NL_{X/Z}) \to H^{-1}(\NL_{X/Y}) \to f^*\Omega_{Y/Z} \to \Omega_{X/Z} \to \Omega_{X/Y} \to 0\] of cohomology sheaves.

Proof

Special case of Modules, Lemma 0E1Z.

Lemma

Let \(f : X \to Y\) and \(Y \to Z\) be morphisms of schemes. Assume \(X \to Y\) is a complete intersection morphism. Then there is a canonical distinguished triangle \[f^*\NL_{Y/Z} \to \NL_{X/Z} \to \NL_{X/Y} \to f^*\NL_{Y/Z}[1]\] in \(D(\mathcal{O}_X)\) which recovers the \(6\)-term exact sequence of Lemma 0E44.

Proof

It suffices to show the canonical map \[f^*\NL_{Y/Z} \to \text{Cone}(\NL_{X/Z} \to \NL_{X/Y})[-1]\] of Modules, Lemma 0E1Z is an isomorphism in \(D(\mathcal{O}_X)\). In order to show this, it suffices to show that the \(6\)-term sequence has a zero on the left, i.e., that \(H^{-1}(f^*\NL_{Y/Z}) \to H^{-1}(\NL_{X/Z})\) is injective. Affine locally this follows from the corresponding algebra result in More on Algebra, Lemma 07D4. To translate into algebra use Lemma 0D0I.

Lemma

Let \(X \to Y \to Z\) be morphisms of schemes. Assume \(X \to Z\) smooth and \(Y \to Z\) étale. Then \(X \to Y\) is smooth.

Proof

The morphism \(X \to Y\) is locally of finite presentation by Morphisms, Lemma 02FV. By Lemma 0D0N we have \(H^{-1}(\NL_{X/Z}) = 0\) and the module \(\Omega_{X/Z}\) is finite locally free. By Lemma 0G7Z we have \(H^{-1}(\NL_{Y/Z}) = H^0(\NL_{Y/Z}) = 0\). By Lemma 0E44 we get \(H^{-1}(\NL_{X/Y}) = 0\) and \(\Omega_{X/Y} \cong \Omega_{X/Z}\) is finite locally free. By Lemma 0D0N the morphism \(X \to Y\) is smooth.

Lemma

Let \(f : X \to Y\) be a morphism of schemes which factors as \(f = g \circ i\) with \(i\) an immersion and \(g : P \to Y\) formally smooth (for example smooth). Then there is a canonical isomorphism \[\NL_{X/Y} \cong \left(\mathcal{C}_{X/P} \to i^*\Omega_{P/Y}\right)\] in \(D(\mathcal{O}_X)\) where the conormal sheaf \(\mathcal{C}_{X/P}\) is placed in degree \(-1\).

Proof

(For the parenthetical statement see Lemma 02H6.) By Lemmas 0FV2 and 0D0L we have \(\NL_{X/P} = \mathcal{C}_{X/P}[1]\) and \(\NL_{P/Y} = \Omega_{P/Y}\) with \(\Omega_{P/Y}\) locally projective. This implies that \(i^*\NL_{P/Y} \to i^*\Omega_{P/Y}\) is a quasi-isomorphism too (small detail omitted; the reason is that \(i^*\NL_{P/Y}\) is the same thing as \(\tau_{\geq -1}Li^*\NL_{P/Y}\), see More on Algebra, Lemma 0FUY). Thus the canonical map \[i^*\NL_{P/Y} \to \text{Cone}(\NL_{X/Y} \to \NL_{X/P})[-1]\] of Modules, Lemma 0E1Z is an isomorphism in \(D(\mathcal{O}_X)\) because the cohomology group \(H^{-1}(i^*\NL_{P/Y})\) is zero by what we said above. In other words, we have a distinguished triangle \[i^*\NL_{P/Y} \to \NL_{X/Y} \to \NL_{X/P} \to i^*\NL_{P/Y}[1]\] Clearly, this means that \(\NL_{X/Y}\) is the cone on the map \(\NL_{X/P}[-1] \to i^*\NL_{P/Y}\) which is equivalent to the statement of the lemma by our computation of the cohomology sheaves of these objects in the derived category given above.

Lemma

Consider a cartesian diagram of schemes \[\xymatrix{ X' \ar[r]_{g'} \ar[d] & X \ar[d] \\ Y' \ar[r] & Y }\] The canonical map \((g')^*\NL_{X/Y} \to \NL_{X'/Y'}\) induces an isomorphism on \(H^0\) and a surjection on \(H^{-1}\).

Proof

Translated into algebra this is More on Algebra, Lemma 0FUZ. To do the translation use Lemma 0D0I.

Lemma

Consider a cartesian diagram of schemes \[\xymatrix{ X' \ar[d] \ar[r]_{g'} & X \ar[d] \\ Y' \ar[r] & Y }\] If \(Y' \to Y\) is flat, then the canonical map \((g')^*\NL_{X/Y} \to \NL_{X'/Y'}\) is a quasi-isomorphism.

Proof

By Lemma 0D0I this follows from Algebra, Lemma 00S4.

Lemma

Consider a cartesian diagram of schemes \[\xymatrix{ X' \ar[r]_{g'} \ar[d] & X \ar[d] \\ Y' \ar[r] & Y }\] If \(X \to Y\) is flat, then the canonical map \((g')^*\NL_{X/Y} \to \NL_{X'/Y'}\) is a quasi-isomorphism. If in addition \(\NL_{X/Y}\) has tor-amplitude in \([-1, 0]\) then \(L(g')^*\NL_{X/Y} \to \NL_{X'/Y'}\) is a quasi-isomorphism too.

Proof

Translated into algebra this is More on Algebra, Lemma 0FJU. To do the translation use Lemma 0D0I and Derived Categories of Schemes, Lemmas 06Z0 and 08E9.

Pushouts in the category of schemes, I

In this section we construct pushouts of \(Y \leftarrow X \rightarrow X'\) where \(X \to Y\) is affine and \(X \to X'\) is a thickening. This will actually be an important case for us, hence a detailed discussion is merited. In Section 0ECH we discuss a more interesting and more difficult case. See Categories, Section 0025 for a general discussion of pushouts in any category.

Lemma

Let \(A' \to A\) be a surjection of rings and let \(B \to A\) be a ring map. Let \(B' = B \times_A A'\) be the fibre product of rings. Set \(S = \Spec(A)\), \(S' = \Spec(A')\), \(T = \Spec(B)\), and \(T' = \Spec(B')\). Then \[\vcenter{ \xymatrix{ S \ar[r]_i \ar[d]_f & S' \ar[d]^{f'} \\ T \ar[r]^{i'} & T' } } \quad\text{corresponding to}\quad \vcenter{ \xymatrix{ A & A' \ar[l] \\ B \ar[u] & B' \ar[l] \ar[u] } }\] is a pushout of schemes.

Proof

By More on Algebra, Lemma 0B7J we have \(T' = T \amalg_S S'\) as topological spaces, i.e., the diagram is a pushout in the category of topological spaces. Next, consider the map \[((i')^\sharp, (f')^\sharp) : \mathcal{O}_{T'} \longrightarrow i'_*\mathcal{O}_T \times_{g_*\mathcal{O}_S} f'_*\mathcal{O}_{S'}\] where \(g = i' \circ f = f' \circ i\). We claim this map is an isomorphism of sheaves of rings. Namely, we can view both sides as quasi-coherent \(\mathcal{O}_{T'}\)-modules (use Schemes, Lemmas 01LC for the right hand side) and the map is \(\mathcal{O}_{T'}\)-linear. Thus it suffices to show the map is an isomorphism on the level of global sections (Schemes, Lemma 01IB). On global sections we recover the identification \(B' \to B \times_A A'\) from statement of the lemma (this is how we chose \(B'\)).

Let \(X\) be a scheme. Suppose we are given morphisms of schemes \(m' : S' \to X\) and \(n : T \to X\) such that \(m' \circ i = n \circ f\) (call this \(m\)). We get a unique map of topological spaces \(n' : T' \to X\) compatible with \(m'\) and \(n\) as \(T' = T \amalg_S S'\) (see above). By the description of \(\mathcal{O}_{T'}\) in the previous paragraph we obtain a unique homomorphism of sheaves of rings \[(n')^\sharp : \mathcal{O}_X \longrightarrow (n')_*\mathcal{O}_{T'} = m'_*\mathcal{O}_T \times_{m_*\mathcal{O}_T} n_*\mathcal{O}_S\] given by \((m')^\sharp\) and \(n^\sharp\). Thus \((n', (n')^\sharp)\) is the unique morphism of ringed spaces \(T' \to X\) compatible with \(m'\) and \(n\). To finish the proof it suffices to show that \(n'\) is a morphism of schemes, i.e., a morphism of locally ringed spaces.

Let \(t' \in T'\) with image \(x \in X\). We have to show that \(\mathcal{O}_{X, x} \to \mathcal{O}_{T', t'}\) is local. If \(t' \not \in T\), then \(t'\) is the image of a unique point \(s' \in S'\) and \(\mathcal{O}_{T', t'} = \mathcal{O}_{S', s'}\). Namely, \(S' \setminus S \to T' \setminus T\) is an isomorphism of schemes as \(B' \to A'\) induces an isomorphism \(\Ker(B' \to B) = \Ker(A' \to A)\). If \(t'\) is the image of \(t \in T\), then we know that the composition \(\mathcal{O}_{X, x} \to \mathcal{O}_{T', t'} \to \mathcal{O}_{T, t}\) is local and we conclude also.

Lemma

Let \(\mathcal{I} \to (\Sch/S)_{fppf}\), \(i \mapsto X_i\) be a diagram of schemes. Let \((W, X_i \to W)\) be a cocone for the diagram in the category of schemes (Categories, Remark 0G2U). If there exists a fpqc covering \(\{W_a \to W\}_{a \in A}\) of schemes such that

for all \(a \in A\) we have \(W_a = \colim X_i \times_W W_a\) in the category of schemes, and

for all \(a, b \in A\) we have \(W_a \times_W W_b = \colim X_i \times_W W_a \times_W W_b\) in the category of schemes,

then \(W = \colim X_i\) in the category of schemes.

Proof

Namely, for a scheme \(T\) a morphism \(W \to T\) is the same thing as collection of morphism \(W_a \to T\), \(a \in A\) which agree on the overlaps \(W_a \times_W W_b\), see Descent, Lemma 023Q.

Lemma

Let \(X \to X'\) be a thickening of schemes and let \(X \to Y\) be an affine morphism of schemes. Then there exists a pushout \[\xymatrix{ X \ar[r] \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r] & Y' }\] in the category of schemes. Moreover, \(Y \subset Y'\) is a thickening, \(X = Y \times_{Y'} X'\), and \[\mathcal{O}_{Y'} = \mathcal{O}_Y \times_{f_*\mathcal{O}_X} f'_*\mathcal{O}_{X'}\] as sheaves on \(|Y| = |Y'|\).

Proof

We first construct \(Y'\) as a ringed space. Namely, as topological space we take \(Y' = Y\). Denote \(f' : X' \to Y'\) the map of topological spaces which equals \(f\). As structure sheaf \(\mathcal{O}_{Y'}\) we take the right hand side of the equation of the lemma. To see that \(Y'\) is a scheme, we have to show that any point has an affine neighbourhood. Since the formation of the fibre product of sheaves commutes with restricting to opens, we may assume \(Y\) is affine. Then \(X\) is affine (as \(f\) is affine) and \(X'\) is affine as well (see Lemma 06AD). Say \(Y \leftarrow X \rightarrow X'\) corresponds to \(B \rightarrow A \leftarrow A'\). Set \(B' = B \times_A A'\); this is the global sections of \(\mathcal{O}_{Y'}\). As \(A' \to A\) is surjective with locally nilpotent kernel we see that \(B' \to B\) is surjective with locally nilpotent kernel. Hence \(\Spec(B') = \Spec(B)\) (as topological spaces). We claim that \(Y' = \Spec(B')\). To see this we will show for \(g' \in B'\) with image \(g \in B\) that \(\mathcal{O}_{Y'}(D(g)) = B'_{g'}\). Namely, by More on Algebra, Lemma 01Z8 we see that \[(B')_{g'} = B_g \times_{A_h} A'_{h'}\] where \(h \in A\), \(h' \in A'\) are the images of \(g'\). Since \(B_g\), resp. \(A_h\), resp. \(A'_{h'}\) is equal to \(\mathcal{O}_Y(D(g))\), resp. \(f_*\mathcal{O}_X(D(g))\), resp. \(f'_*\mathcal{O}_{X'}(D(g))\) the claim follows.

It remains to show that \(Y'\) is the pushout. The discussion above shows the scheme \(Y'\) has an affine open covering \(Y' = \bigcup W'_i\) such that the corresponding opens \(U'_i \subset X'\), \(W_i \subset Y\), and \(U_i \subset X\) are affine open. Moreover, if \(A'_i\), \(B_i\), \(A_i\) are the rings corresponding to \(U'_i\), \(W_i\), \(U_i\), then \(W'_i\) corresponds to \(B_i \times_{A_i} A'_i\). Thus we can apply Lemmas 0ET0 and 0BMP to conclude our construction is a pushout in the category of schemes.

In the following lemma we use the fibre product of categories as defined in Categories, Example 003R.

Lemma

Let \(X \to X'\) be a thickening of schemes and let \(X \to Y\) be an affine morphism of schemes. Let \(Y' = Y \amalg_X X'\) be the pushout (see Lemma 07RT). Base change gives a functor \[F : (\Sch/Y') \longrightarrow (\Sch/Y) \times_{(\Sch/X)} (\Sch/X')\] given by \(V' \longmapsto (V' \times_{Y'} Y, V' \times_{Y'} X', 1)\) which has a left adjoint \[G : (\Sch/Y) \times_{(\Sch/X)} (\Sch/X') \longrightarrow (\Sch/Y')\] which sends the triple \((V, U', \varphi)\) to the pushout \(V \amalg_{(V \times_Y X)} U'\). Finally, \(F \circ G\) is isomorphic to the identity functor.

Proof

Let \((V, U', \varphi)\) be an object of the fibre product category. Set \(U = U' \times_{X'} X\). Note that \(U \to U'\) is a thickening. Since \(\varphi : V \times_Y X \to U' \times_{X'} X = U\) is an isomorphism we have a morphism \(U \to V\) over \(X \to Y\) which identifies \(U\) with the fibre product \(X \times_Y V\). In particular \(U \to V\) is affine, see Morphisms, Lemma 01SD. Hence we can apply Lemma 07RT to get a pushout \(V' = V \amalg_U U'\). Denote \(V' \to Y'\) the morphism we obtain in virtue of the fact that \(V'\) is a pushout and because we are given morphisms \(V \to Y\) and \(U' \to X'\) agreeing on \(U\) as morphisms into \(Y'\). Setting \(G(V, U', \varphi) = V'\) gives the functor \(G\).

Let us prove that \(G\) is a left adjoint to \(F\). Let \(Z\) be a scheme over \(Y'\). We have to show that \[\Mor(V', Z) = \Mor((V, U', \varphi), F(Z))\] where the morphism sets are taking in their respective categories. Let \(g' : V' \to Z\) be a morphism. Denote \(\tilde g\), resp. \(\tilde f'\) the composition of \(g'\) with the morphism \(V \to V'\), resp. \(U' \to V'\). Base change \(\tilde g\), resp. \(\tilde f'\) by \(Y \to Y'\), resp. \(X' \to Y'\) to get a morphism \(g : V \to Z \times_{Y'} Y\), resp. \(f' : U' \to Z \times_{Y'} X'\). Then \((g, f')\) is an element of the right hand side of the equation above (details omitted). Conversely, suppose that \((g, f') : (V, U', \varphi) \to F(Z)\) is an element of the right hand side. We may consider the composition \(\tilde g : V \to Z\), resp. \(\tilde f' : U' \to Z\) of \(g\), resp. \(f\) by \(Z \times_{Y'} X' \to Z\), resp. \(Z \times_{Y'} Y \to Z\). Then \(\tilde g\) and \(\tilde f'\) agree as morphism from \(U\) to \(Z\). By the universal property of pushout, we obtain a morphism \(g' : V' \to Z\), i.e., an element of the left hand side. We omit the verification that these constructions are mutually inverse.

To prove that \(F \circ G\) is isomorphic to the identity we have to show that the adjunction mapping \((V, U', \varphi) \to F(G(V, U', \varphi))\) is an isomorphism. To do this we may work affine locally. Say \(X = \Spec(A)\), \(X' = \Spec(A')\), and \(Y = \Spec(B)\). Then \(A' \to A\) and \(B \to A\) are ring maps as in More on Algebra, Lemma 07RU and \(Y' = \Spec(B')\) with \(B' = B \times_A A'\). Next, suppose that \(V = \Spec(D)\), \(U' = \Spec(C')\) and \(\varphi\) is given by an \(A\)-algebra isomorphism \(D \otimes_B A \to C' \otimes_{A'} A = C'/IC'\). Set \(D' = D \times_{C'/IC'} C'\). In this case the statement we have to prove is that \(D' \otimes_{B'} B \cong D\) and \(D' \otimes_{B'} A' \cong C'\). This is a special case of More on Algebra, Lemma 07RU.

Lemma

Let \(X \to X'\) be a thickening of schemes and let \(X \to Y\) be an affine morphism of schemes. Let \(Y' = Y \amalg_X X'\) be the pushout (see Lemma 07RT). Let \(V' \to Y'\) be a morphism of schemes. Set \(V = Y \times_{Y'} V'\), \(U' = X' \times_{Y'} V'\), and \(U = X \times_{Y'} V'\). There is an equivalence of categories between

quasi-coherent \(\mathcal{O}_{V'}\)-modules flat over \(Y'\), and

the category of triples \((\mathcal{G}, \mathcal{F}', \varphi)\) where

\(\mathcal{G}\) is a quasi-coherent \(\mathcal{O}_V\)-module flat over \(Y\),

\(\mathcal{F}'\) is a quasi-coherent \(\mathcal{O}_{U'}\)-module flat over \(X'\), and

\(\varphi : (U \to V)^*\mathcal{G} \to (U \to U')^*\mathcal{F}'\) is an isomorphism of \(\mathcal{O}_U\)-modules.

The equivalence maps \(\mathcal{G}'\) to \(((V \to V')^*\mathcal{G}', (U' \to V')^*\mathcal{G}', can)\). Suppose \(\mathcal{G}'\) corresponds to the triple \((\mathcal{G}, \mathcal{F}', \varphi)\). Then

\(\mathcal{G}'\) is a finite type \(\mathcal{O}_{V'}\)-module if and only if \(\mathcal{G}\) and \(\mathcal{F}'\) are finite type \(\mathcal{O}_Y\) and \(\mathcal{O}_{U'}\)-modules.

if \(V' \to Y'\) is locally of finite presentation, then \(\mathcal{G}'\) is an \(\mathcal{O}_{V'}\)-module of finite presentation if and only if \(\mathcal{G}\) and \(\mathcal{F}'\) are \(\mathcal{O}_Y\) and \(\mathcal{O}_{U'}\)-modules of finite presentation.

Proof

A quasi-inverse functor assigns to the triple \((\mathcal{G}, \mathcal{F}', \varphi)\) the fibre product \[(V \to V')_*\mathcal{G} \times_{(U \to V')_*\mathcal{F}} (U' \to V')_*\mathcal{F}'\] where \(\mathcal{F} = (U \to U')^*\mathcal{F}'\). This works, because on affines we recover the equivalence of More on Algebra, Lemma 07RW. Some details omitted.

Parts (a) and (b) follow from More on Algebra, Lemmas 08IH and 08KP.

Lemma

In the situation of Lemma 07RV. If \(V' = G(V, U', \varphi)\) for some triple \((V, U', \varphi)\), then

\(V' \to Y'\) is locally of finite type if and only if \(V \to Y\) and \(U' \to X'\) are locally of finite type,

\(V' \to Y'\) is flat if and only if \(V \to Y\) and \(U' \to X'\) are flat,

\(V' \to Y'\) is flat and locally of finite presentation if and only if \(V \to Y\) and \(U' \to X'\) are flat and locally of finite presentation,

\(V' \to Y'\) is smooth if and only if \(V \to Y\) and \(U' \to X'\) are smooth,

\(V' \to Y'\) is étale if and only if \(V \to Y\) and \(U' \to X'\) are étale, and

add more here as needed.

If \(W'\) is flat over \(Y'\), then the adjunction mapping \(G(F(W')) \to W'\) is an isomorphism. Hence \(F\) and \(G\) define mutually quasi-inverse functors between the category of schemes flat over \(Y'\) and the category of triples \((V, U', \varphi)\) with \(V \to Y\) and \(U' \to X'\) flat.

Proof

Looking over affine pieces the assertions of this lemma are equivalent to the corresponding assertions of More on Algebra, Lemma 08KQ.

Openness of the flat locus

This result takes some work to prove, and (perhaps) deserves its own section. Here it is.

Theorem

Let \(S\) be a scheme. Let \(f : X \to S\) be a morphism which is locally of finite presentation. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module which is locally of finite presentation. Then \[U = \{x \in X \mid \mathcal{F}\text{ is flat over }S\text{ at }x\}\] is open in \(X\).

Proof

We may test for openness locally on \(X\) hence we may assume that \(f\) is a morphism of affine schemes. In this case the theorem is exactly Algebra, Theorem 00RC.

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] be a cartesian diagram of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(x' \in X'\) with images \(x = g'(x')\) and \(s' = f'(x')\).

If \(\mathcal{F}\) is flat over \(S\) at \(x\), then \((g')^*\mathcal{F}\) is flat over \(S'\) at \(x'\).

If \(g\) is flat at \(s'\) and \((g')^*\mathcal{F}\) is flat over \(S'\) at \(x'\), then \(\mathcal{F}\) is flat over \(S\) at \(x\).

In particular, if \(g\) is flat, \(f\) is locally of finite presentation, and \(\mathcal{F}\) is locally of finite presentation, then formation of the open subset of Theorem 0399 commutes with base change.

Proof

Consider the commutative diagram of local rings \[\xymatrix{ \mathcal{O}_{X', x'} & \mathcal{O}_{X, x} \ar[l] \\ \mathcal{O}_{S', s'} \ar[u] & \mathcal{O}_{S, s} \ar[l] \ar[u] }\] Note that \(\mathcal{O}_{X', x'}\) is a localization of \(\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S', s'}\), and that \(((g')^*\mathcal{F})_{x'}\) is equal to \(\mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{X', x'}\). Hence the lemma follows from Algebra, Lemma 00MQ.

Critère de platitude par fibres

Consider a commutative diagram of schemes (left hand diagram) \[\xymatrix{ X \ar[rr]_f \ar[dr] & & Y \ar[dl] \\ & S } \quad \xymatrix{ X_s \ar[rr]_{f_s} \ar[rd] & & Y_s \ar[dl] \\ & \Spec(\kappa(s)) }\] and a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\). Given a point \(x \in X\) lying over \(s \in S\) with image \(y = f(x)\) we consider the question: Is \(\mathcal{F}\) flat over \(Y\) at \(x\)? If \(\mathcal{F}\) is flat over \(S\) at \(x\), then the theorem states this question is intimately related to the question of whether the restriction of \(\mathcal{F}\) to the fibre \[\mathcal{F}_s = (X_s \to X)^*\mathcal{F}\] is flat over \(Y_s\) at \(x\). Below you will find a “Noetherian” version, a “finitely presented” version, and earlier we treated a “nilpotent” version, see Lemma 06AF.

Theorem

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in X\). Set \(y = f(x)\) and \(s \in S\) the image of \(x\) in \(S\). Assume \(S\), \(X\), \(Y\) locally Noetherian, \(\mathcal{F}\) coherent, and \(\mathcal{F}_x \not = 0\). Then the following are equivalent:

\(\mathcal{F}\) is flat over \(S\) at \(x\), and \(\mathcal{F}_s\) is flat over \(Y_s\) at \(x\), and

\(Y\) is flat over \(S\) at \(y\) and \(\mathcal{F}\) is flat over \(Y\) at \(x\).

Proof

Consider the ring maps \[\mathcal{O}_{S, s} \longrightarrow \mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, x}\] and the module \(\mathcal{F}_x\). The stalk of \(\mathcal{F}_s\) at \(x\) is the module \(\mathcal{F}_x/\mathfrak m_s \mathcal{F}_x\) and the local ring of \(Y_s\) at \(y\) is \(\mathcal{O}_{Y, y}/\mathfrak m_s \mathcal{O}_{Y, y}\). Thus the implication (1) \(\Rightarrow\) (2) is Algebra, Lemma 00MP. If (2) holds, then the first ring map is faithfully flat and \(\mathcal{F}_x\) is flat over \(\mathcal{O}_{Y, y}\) so by Algebra, Lemma 00HC we see that \(\mathcal{F}_x\) is flat over \(\mathcal{O}_{S, s}\). Moreover, \(\mathcal{F}_x/\mathfrak m_s \mathcal{F}_x\) is the base change of the flat module \(\mathcal{F}_x\) by \(\mathcal{O}_{Y, y} \to \mathcal{O}_{Y, y}/\mathfrak m_s \mathcal{O}_{Y, y}\), hence flat by Algebra, Lemma 00HI.

Here is the non-Noetherian version.

Theorem

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume

\(X\) is locally of finite presentation over \(S\),

\(\mathcal{F}\) an \(\mathcal{O}_X\)-module of finite presentation, and

\(Y\) is locally of finite type over \(S\).

Let \(x \in X\). Set \(y = f(x)\) and let \(s \in S\) be the image of \(x\) in \(S\). If \(\mathcal{F}_x \not = 0\), then the following are equivalent:

\(\mathcal{F}\) is flat over \(S\) at \(x\), and \(\mathcal{F}_s\) is flat over \(Y_s\) at \(x\), and

\(Y\) is flat over \(S\) at \(y\) and \(\mathcal{F}\) is flat over \(Y\) at \(x\).

Moreover, the set of points \(x\) where (1) and (2) hold is open in \(\text{Supp}(\mathcal{F})\).

Proof

Consider the ring maps \[\mathcal{O}_{S, s} \longrightarrow \mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, x}\] and the module \(\mathcal{F}_x\). The stalk of \(\mathcal{F}_s\) at \(x\) is the module \(\mathcal{F}_x/\mathfrak m_s \mathcal{F}_x\) and the local ring of \(Y_s\) at \(y\) is \(\mathcal{O}_{Y, y}/\mathfrak m_s \mathcal{O}_{Y, y}\). Thus the implication (1) \(\Rightarrow\) (2) is Algebra, Lemma 05UV. If (2) holds, then the first ring map is faithfully flat and \(\mathcal{F}_x\) is flat over \(\mathcal{O}_{Y, y}\) so by Algebra, Lemma 00HC we see that \(\mathcal{F}_x\) is flat over \(\mathcal{O}_{S, s}\). Moreover, \(\mathcal{F}_x/\mathfrak m_s \mathcal{F}_x\) is the base change of the flat module \(\mathcal{F}_x\) by \(\mathcal{O}_{Y, y} \to \mathcal{O}_{Y, y}/\mathfrak m_s \mathcal{O}_{Y, y}\), hence flat by Algebra, Lemma 00HI.

By Morphisms, Lemma 02FV the morphism \(f\) is locally of finite presentation. Consider the set [05VI]\[\begin{equation} U = \{x \in X \mid \mathcal{F} \text{ flat at }x \text{ over both }Y\text{ and }S\}. \end{equation}\] This set is open in \(X\) by Theorem 0399. Note that if \(x \in U\), then \(\mathcal{F}_s\) is flat at \(x\) over \(Y_s\) as a base change of a flat module under the morphism \(Y_s \to Y\), see Morphisms, Lemma 01U8. Hence at every point of \(U \cap \text{Supp}(\mathcal{F})\) condition (1) is satisfied. On the other hand, it is clear that if \(x \in \text{Supp}(\mathcal{F})\) satisfies (1) and (2), then \(x \in U\). Thus the open set we are looking for is \(U \cap \text{Supp}(\mathcal{F})\).

These theorems are often used in the following simplified forms. We give only the global statements – of course there are also pointwise versions.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). Assume

\(S\), \(X\), \(Y\) are locally Noetherian,

\(X\) is flat over \(S\),

for every \(s \in S\) the morphism \(f_s : X_s \to Y_s\) is flat.

Then \(f\) is flat. If \(f\) is also surjective, then \(Y\) is flat over \(S\).

Proof

This is a special case of Theorem 039B.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). Assume

\(X\) is locally of finite presentation over \(S\),

\(X\) is flat over \(S\),

for every \(s \in S\) the morphism \(f_s : X_s \to Y_s\) is flat, and

\(Y\) is locally of finite type over \(S\).

Then \(f\) is flat. If \(f\) is also surjective, then \(Y\) is flat over \(S\).

Proof

This is a special case of Theorem 039C.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume

\(X\) is locally of finite presentation over \(S\),

\(\mathcal{F}\) an \(\mathcal{O}_X\)-module of finite presentation,

\(\mathcal{F}\) is flat over \(S\), and

\(Y\) is locally of finite type over \(S\).

Then the set \[U = \{x \in X \mid \mathcal{F} \text{ flat at }x \text{ over }Y\}.\] is open in \(X\) and its formation commutes with arbitrary base change: If \(S' \to S\) is a morphism of schemes, and \(U'\) is the set of points of \(X' = X \times_S S'\) where \(\mathcal{F}' = \mathcal{F} \times_S S'\) is flat over \(Y' = Y \times_S S'\), then \(U' = U \times_S S'\).

Proof

By Morphisms, Lemma 02FV the morphism \(f\) is locally of finite presentation. Hence \(U\) is open by Theorem 0399. Because we have assumed that \(\mathcal{F}\) is flat over \(S\) we see that Theorem 039C implies \[U = \{x \in X \mid \mathcal{F}_s \text{ flat at }x \text{ over }Y_s\}.\] where \(s\) always denotes the image of \(x\) in \(S\). (This description also works trivially when \(\mathcal{F}_x = 0\).) Moreover, the assumptions of the lemma remain in force for the morphism \(f' : X' \to Y'\) and the sheaf \(\mathcal{F}'\). Hence \(U'\) has a similar description. In other words, it suffices to prove that given \(s' \in S'\) mapping to \(s \in S\) we have \[\{x' \in X'_{s'} \mid \mathcal{F}'_{s'} \text{ flat at }x' \text{ over }Y'_{s'}\}\] is the inverse image of the corresponding locus in \(X_s\). This is true by Lemma 047C because in the cartesian diagram \[\xymatrix{ X'_{s'} \ar[d] \ar[r] & X_s \ar[d] \\ Y'_{s'} \ar[r] & Y_s }\] the horizontal morphisms are flat as they are base changes by the flat morphism \(\Spec(\kappa(s')) \to \Spec(\kappa(s))\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). Assume

\(X\) is locally of finite presentation over \(S\),

\(X\) is flat over \(S\), and

\(Y\) is locally of finite type over \(S\).

Then the set \[U = \{x \in X \mid X\text{ flat at }x \text{ over }Y\}.\] is open in \(X\) and its formation commutes with arbitrary base change.

Proof

This is a special case of Lemma 05VJ.

The following lemma is a variant of Algebra, Lemma 00MH. Note that the hypothesis that \((\mathcal{F}_s)_x\) is a flat \(\mathcal{O}_{X_s, x}\)-module means that \((\mathcal{F}_s)_x\) is a free \(\mathcal{O}_{X_s, x}\)-module which is always the case if \(x \in X_s\) is a generic point of an irreducible component of \(X_s\) and \(X_s\) is reduced (namely, in this case \(\mathcal{O}_{X_s, x}\) is a field, see Algebra, Lemma 00EU).

Lemma

Let \(f : X \to S\) be a morphism of schemes of finite presentation. Let \(\mathcal{F}\) be a finitely presented \(\mathcal{O}_X\)-module. Let \(x \in X\) with image \(s \in S\). If \(\mathcal{F}\) is flat at \(x\) over \(S\) and \((\mathcal{F}_s)_x\) is a flat \(\mathcal{O}_{X_s, x}\)-module, then \(\mathcal{F}\) is finite free in a neighbourhood of \(x\).

Proof

If \(\mathcal{F}_x \otimes \kappa(x)\) is zero, then \(\mathcal{F}_x = 0\) by Nakayama’s lemma (Algebra, Lemma 00DV) and hence \(\mathcal{F}\) is zero in a neighbourhood of \(x\) (Modules, Lemma 01B9) and the lemma holds. Thus we may assume \(\mathcal{F}_x \otimes \kappa(x)\) is not zero and we see that Theorem 039C applies with \(f = \text{id} : X \to X\). We conclude that \(\mathcal{F}_x\) is flat over \(\mathcal{O}_{X, x}\). Hence \(\mathcal{F}_x\) is free, see Algebra, Lemma 00NZ for example. Choose an open neighbourhood \(x \in U \subset X\) and sections \(s_1, \ldots, s_r \in \mathcal{F}(U)\) which map to a basis in \(\mathcal{F}_x\). The corresponding map \(\psi : \mathcal{O}_U^{\oplus r} \to \mathcal{F}|_U\) is surjective after shrinking \(U\) (Modules, Lemma 01B9). Then \(\Ker(\psi)\) is of finite type (see Modules, Lemma 01BP) and \(\Ker(\psi)_x = 0\). Whence after shrinking \(U\) once more \(\psi\) is an isomorphism.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite presentation. Let \(\mathcal{F}\) be a finitely presented \(\mathcal{O}_X\)-module flat over \(S\). Then the set \[\{x \in X : \mathcal{F}\text{ free in a neighbourhood of }x\}\] is open in \(X\) and its formation commutes with arbitrary base change \(S' \to S\).

Proof

Openness holds trivially. Let \(x \in X\) mapping to \(s \in S\). By Lemma 080Q we see that \(x\) is in our set if and only if \(\mathcal{F}|_{X_s}\) is flat at \(x\) over \(X_s\). Clearly this is also equivalent to \(\mathcal{F}\) being flat at \(x\) over \(X\) (because this statement is implied by freeness of \(\mathcal{F}_x\) and implies flatness of \(\mathcal{F}|_{X_s}\) at \(x\) over \(X_s\)). Thus the base change statement follows from Lemma 05VJ applied to \(\text{id} : X \to X\) over \(S\).

Closed immersions between smooth schemes

Some results that do not fit elsewhere very well.

Lemma

Let \(S\) be a scheme. Let \(Y \to X\) be a closed immersion of schemes smooth over \(S\). For every \(y \in Y\) there exist integers \(0 \leq m, n\) and a commutative diagram \[\xymatrix{ Y \ar[d] & V \ar[l] \ar[d] \ar[r] & \mathbf{A}^m_S \ar[d]^{(a_1, \ldots, a_m) \mapsto (a_1, \ldots, a_m, 0 \ldots, 0)} \\ X & U \ar[l] \ar[r]^-\pi & \mathbf{A}^{m + n}_S }\] where \(U \subset X\) is open, \(V = Y \cap U\), \(\pi\) is étale, \(V = \pi^{-1}(\mathbf{A}^m_S)\), and \(y \in V\).

Proof

The question is local on \(X\) hence we may replace \(X\) by an open neighbourhood of \(y\). Since \(Y \to X\) is a regular immersion by Divisors, Lemma 067U we may assume \(X = \Spec(A)\) is affine and there exists a regular sequence \(f_1, \ldots, f_n \in A\) such that \(Y = V(f_1, \ldots, f_n)\). After shrinking \(X\) (and hence \(Y\)) further we may assume there exists an étale morphism \(Y \to \mathbf{A}^m_S\), see Morphisms, Lemma 054L. Let \(\overline{g}_1, \ldots, \overline{g}_m\) in \(\mathcal{O}_Y(Y)\) be the coordinate functions of this étale morphism. Choose lifts \(g_1, \ldots, g_m \in A\) of these functions and consider the morphism \[(g_1, \ldots, g_m, f_1, \ldots, f_n) : X \longrightarrow \mathbf{A}^{m + n}_S\] over \(S\). This is a morphism of schemes locally of finite presentation over \(S\) and hence is locally of finite presentation (Morphisms, Lemma 02FV). The restriction of this morphism to \(\mathbf{A}^m_S \subset \mathbf{A}^{m + n}_S\) is étale by construction. Thus, in order to show that \(X \to \mathbf{A}^{m + n}_S\) is étale at \(y\) it suffices to show that \(X \to \mathbf{A}^{m + n}_S\) is flat at \(y\), see Morphisms, Lemma 02GU. Let \(s \in S\) be the image of \(y\). It suffices to check that \(X_s \to \mathbf{A}^{m + n}_s\) is flat at \(y\), see Theorem 039C. Let \(z \in \mathbf{A}^{m + n}_s\) be the image of \(y\). The local ring map \[\mathcal{O}_{\mathbf{A}^{m + n}_s, z} \longrightarrow \mathcal{O}_{X_s, y}\] is flat by Algebra, Lemma 00R4. Namely, schemes smooth over fields are regular and regular rings are Cohen-Macaulay, see Varieties, Lemma 056S and Algebra, Lemma 00NQ. Thus both source and target are regular local rings (and hence CM). The source and target have the same dimension: namely, we have \(\dim(\mathcal{O}_{Y_s, y}) = \dim(\mathcal{O}_{\mathbf{A}^m_s, z})\) by More on Algebra, Lemma 07QP, we have \(\dim(\mathcal{O}_{\mathbf{A}^{m + n}_s, z}) = n + \dim(\mathcal{O}_{\mathbf{A}^m_s, z})\), and we have \(\dim(\mathcal{O}_{X_s, y}) = n + \dim(\mathcal{O}_{Y_s, y})\) because \(\mathcal{O}_{Y_s, y}\) is the quotient of \(\mathcal{O}_{X_s, y}\) by the regular sequence \(f_1, \ldots, f_n\) of length \(n\) (see Divisors, Remark 0FUD). Finally, the fibre ring of the displayed arrow is finite over \(\kappa(z)\) since \(Y_s \to \mathbf{A}^m_s\) is étale at \(y\). This finishes the proof.

Remark

We fix a ring \(R\) and we set \(S = \Spec(R)\). Fix integers \(0 \leq m\) and \(1 \leq n\). Consider the closed immersion \[Z = \mathbf{A}^m_S \longrightarrow \mathbf{A}^{m + n}_S = X,\quad (a_1, \ldots, a_m) \mapsto (a_1, \ldots, a_m, 0, \ldots 0).\] We are going to consider the blowing up \(X'\) of \(X\) along the closed subscheme \(Z\). Write \[X = \Spec(A) \quad\text{with}\quad A = R[x_1, \ldots, x_m, y_1, \ldots, y_n]\] Then \(X'\) is the Proj of the Rees algebra of \(A\) with respect to the ideal \((y_1, \ldots, y_n)\). This Rees algebra is equal to \(B = A[T_1, \ldots, T_n]/(y_iT_j - y_jT_i)\); details omitted. Hence \(X' = \text{Proj}(B)\) is smooth over \(S\) as it is covered by the affine opens \[\begin{align*} D_+(T_i) & = \Spec(B_{(T_i)}) \\ & = \Spec(A[t_1, \ldots, \hat t_i, \ldots t_n]/(y_j - y_i t_j)) \\ & = \Spec(R[x_1, \ldots, x_m, y_i, t_1, \ldots, \hat t_i, \ldots, t_n]) \end{align*}\] which are isomorphic to \(\mathbf{A}^{n + m}_S\). In this chart the exceptional divisor is cut out by setting \(y_i = 0\) hence the exceptional divisor is smooth over \(S\) as well.

Lemma

Let \(S\) be a scheme. Let \(Z \to X\) be a closed immersion of schemes smooth over \(S\). Let \(b : X' \to X\) be the blowing up of \(Z\) with exceptional divisor \(E \subset X'\). Then \(X'\) and \(E\) are smooth over \(S\). The morphism \(p : E \to Z\) is canonically isomorphic to the projective space bundle \[\mathbf{P}(\mathcal{I}/\mathcal{I}^2) \longrightarrow Z\] where \(\mathcal{I} \subset \mathcal{O}_X\) is the ideal sheaf of \(Z\). The relative \(\mathcal{O}_E(1)\) coming from the projective space bundle structure is isomorphic to the restriction of \(\mathcal{O}_{X'}(-E)\) to \(E\).

Proof

By Divisors, Lemma 067U the immersion \(Z \to X\) is a regular immmersion, hence the ideal sheaf \(\mathcal{I}\) is of finite type, hence \(b\) is a projective morphism with relatively ample invertible sheaf \(\mathcal{O}_{X'}(1) = \mathcal{O}_{X'}(-E)\), see Divisors, Lemmas 02OS and 02NS. The canonical map \(\mathcal{I} \to b_*\mathcal{O}_{X'}(1)\) gives a closed immersion \[X' \longrightarrow \mathbf{P}\left(\bigoplus\nolimits_{n \geq 0} \text{Sym}^n_{\mathcal{O}_X}(\mathcal{I})\right)\] by the very construction of the blowup. The restriction of this morphism to \(E\) gives a canonical map \[E \longrightarrow \mathbf{P}\left(\bigoplus\nolimits_{n \geq 0} \text{Sym}^n_{\mathcal{O}_Z}(\mathcal{I}/\mathcal{I}^2)\right)\] over \(Z\). Since \(\mathcal{I}/\mathcal{I}^2\) is finite locally free if this canonical map is an isomorphism, then the final part of the lemma holds. Having said all of this, now the question is étale local on \(X\). Namely, blowing up commutes with flat base change by Divisors, Lemma 0805 and we can check smoothness after precomposing with a surjective étale morphism. Thus by the étale local structure of a closed immersion of schemes over \(S\) given in Lemma 0FUE this reduces us to the case discussed in Remark 0H1H.

Flat modules and relative assassins

In this section we will prove that the support of a flat module is (in some sense) equidimensional over the base in geometric situations. For the Noetherian case we refer the reader to [EGA, IV Proposition 12.1.1.5]. First, we prove two helper lemmas.

Lemma

Let \(A\) be a valuation ring. Let \(A \to B\) is a local homomorphism of local rings which is essentially of finite type. Let \(u : N \to M\) be a map of finite \(B\)-modules. Assume \(M\) is flat over \(A\) and \(\overline{u} : N/\mathfrak m_A N \to M/\mathfrak m_A M\) is injective. Then \(u\) is injective and \(M/u(N)\) is flat over \(A\).

Proof

We will deduce this lemma from Algebra, Lemma 046Y (please note that we exchanged the roles of \(M\) and \(N\)). To do the reduction we will use More on Algebra, Lemma 0GSE to reduce to the finitely presented case.

By assumption we can write \(B\) as a quotient of the localization of a polynomial algebra \(P = A[x_1, \ldots, x_n]\) at a prime ideal \(\mathfrak q\). Then we can think of \(u : N \to M\) as a map of finite \(P_\mathfrak q\)-modules. Hence we may and do assume that \(B\) is essentially of finite presentation over \(A\).

Next, the \(B\)-module \(N\) is finite but perhaps not of finite presentation. Write \(N = \colim N_\lambda\) as a filtered colimit of finitely presented \(B\)-modules with surjective transition maps. For example choose a presentation \(0 \to K \to B^{\oplus r} \to N \to 0\), write \(K\) as the union of its finite submodules \(K_\lambda\), and set \(N_\lambda = \Coker(K_\lambda \to B^{\oplus r})\). The module \(N/\mathfrak m_A N\) is of finite presentation over the Noetherian ring \(B/\mathfrak m_A B\). Hence for \(\lambda\) large enough we have \(N_\lambda/\mathfrak m_A N_\lambda = N/\mathfrak m_A N\). Now, if we can show the lemma for the composition \(u_\lambda : N_\lambda \to M\), then we conclude that \(N_\lambda = N\) and the result holds for \(u\). Hence we may and do assume \(N\) is of finite presentation over \(B\).

By More on Algebra, Lemma 0GSE the module \(M\) is of finite presentation over \(B\). Thus all the assumptions of Algebra, Lemma 046Y hold and we conclude.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(y \in X\) be a point with image \(t \in S\). Denote \(Y \subset X\) the closure of \(\{y\}\) viewed as an integral closed subscheme of \(X\). Let \(s \in S\) and let \(x \in Y_s\) be a generic point of an irreducible component of \(Y_s\). There exists a cartesian diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] with the following properties:

\(S'\) is the spectrum of a valuation ring with generic point \(t'\) and closed point \(s'\),

\(g(t') = t\) and \(g(s') = s\),

there exists a point \(y' \in X'_{t'}\) which is a generic point of an irreducible component of \((S' \times_S Y)_{t'} = Y_t \times_t t'\) and satisfies \(g'(y') = y\),

denoting \(Y' \subset X'\) the closure of \(\{y'\}\) viewed as an integral closed subscheme of \(X'\) there exists a point \(x' \in Y'_{s'}\) which is a generic point of an irreducible component of \(Y'_{s'}\) with \(g'(x') = x\).

Proof

We choose a valuation ring \(R\), we set \(S' = \Spec(R)\) with generic point \(t'\) and closed point \(s'\), and we choose a morphism \(h : S' \to X\) with \(h(t') = y\) and \(h(s') = x\). See Schemes, Lemma 01J8. Set \(g = f \circ h\) so that \(g(t') = t\) and \(g(s') = s\). Consider the base change \[\xymatrix{ X' \ar[r]_{g'} \ar[d] & X \ar[d] \\ S' \ar@/^1em/[u]^\sigma \ar[r]^-g & S }\] We obtain a section \(\sigma\) of the base change such that \(h = g' \circ \sigma\).

Of course \(\sigma\) factors through the base change \(S' \times_S Y\) of \(Y\) as \(h\) factors through \(Y\). Let \(y' \in X'_{t'} \subset X'\) be the generic point of an irreducible component of the fibre \[(S' \times_S Y)_{t'} = Y_t \times_t t'\] containing the point \(\sigma(t')\), i.e., such that \(y' \leadsto \sigma(t')\). Since \(g'(y') \in Y_t\) and \(g(y') \leadsto g(\sigma(t')) = y\) we find that \(g'(y') = y\) because \(y\) is the generic point of the fibre \(Y_t\). Denote \(Y' \subset X'\) the closure of \(\{y'\}\) in \(X'\) viewed as an integral closed subscheme. Then \(\sigma\) factors through \(Y'\) as \(\sigma(t') \in Y'\). Choose a generic point \(x' \in Y'_{s'}\) of an irreducible component of \(Y'_{s'}\) which contains \(\sigma(s')\), i.e., we get \(x' \leadsto \sigma(s')\) and hence \(g'(x') \leadsto g'(\sigma(s')) = x\). Again as \(x\) is a generic point of an irreducible component of \(Y_s\) by assumption and as \(g'(Y') \subset Y\) we conclude that \(g'(x') = x\).

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a quasi-coherent finite type \(\mathcal{O}_X\)-module. Let \(y \in \text{Ass}_{X/S}(\mathcal{F})\) with image \(t \in S\). Denote \(Y \subset X\) the closure of \(\{y\}\) in \(X\) viewed as an integral closed subscheme. Let \(s \in S\) and let \(x \in Y_s\) be a generic point of an irreducible component of \(Y_s\). If \(\mathcal{F}\) is flat over \(S\) at \(x\), then \(x \in \text{Ass}_{X/S}(\mathcal{F})\) and \(\dim_x(Y_s) = \dim(Y_t)\).

Proof

Choose a diagram as in Lemma 0GSH. Set \(\mathcal{F}' = (g')^*\mathcal{F}\). Divisors, Lemma 05DC implies that \(y' \in \text{Ass}_{X'/S'}(\mathcal{F}')\). By our choice of \(y'\) we also see that \(\dim(Y'_{t'}) = \dim(Y_t)\), see for example Algebra, Lemma 0CWE. By Algebra, Lemma 00QK we see that \(Y'_{s'}\) is equidimensional of dimension equal to \(\dim(Y_t)\). Since \(\mathcal{F}\) is flat at \(x\) over \(S\) we see that \(\mathcal{F}'\) is flat at \(x'\) over \(S'\), see Morphisms, Lemma 01U8.

Suppose that we can show \(x' \in \text{Ass}_{X'/S}(\mathcal{F}')\). Then Divisors, Lemma 05DC implies that \(x \in \text{Ass}_{X/S}(\mathcal{F})\) and that the irreducible component \(C'\) of \(Y'_{s'}\) containing \(x'\) is an irreducible component of \(C \times_s s'\) where \(C \subset Y_s\) is the irreducible component containing \(x\). Whence \(\dim(C) = \dim(C') = \dim(Y_t)\) (see above) and the proof is complete. This reduces us to the case discussed in the next paragraph.

Assume \(S = \Spec(A)\) where \(A\) is a valuation ring and \(t\) and \(s\) are the generic and closed points of \(S\). We will assume \(x \not \in \text{Ass}_{X/S}(\mathcal{F})\) in order to get a contradiction. In other words, we assume \(x \not \in \text{Ass}_{X_s}(\mathcal{F}_s)\) where \(\mathcal{F}_s\) is the pullback of \(\mathcal{F}\) to \(X_s\). Consider the ring map \[A \longrightarrow \mathcal{O}_{X, x} = B\] and the module \(N = \mathcal{F}_x\) over \(B = \mathcal{O}_{X, x}\). Then \(B/\mathfrak m_A B = \mathcal{O}_{X_s, x}\) and \(N/\mathfrak m_A N\) is the stalk of \(\mathcal{F}_s\) at the point \(x\). Denote \(\mathfrak q \subset B\) the prime ideal corresponding to the point \(y\), see Schemes, Lemma 01J7. Since \(x\) is a generic point of \(Y_s\) we see that the radical of \(\mathfrak q + \mathfrak m_A B\) is \(\mathfrak m_B\). Then \(\text{Ass}_{B/\mathfrak m_A B}(N/\mathfrak m_A N)\) is a finite set of prime ideals (Algebra, Lemma 00LC) which doesn’t contain the maximal ideal of \(B/\mathfrak m_A B\) since \(x \not \in \text{Ass}_{X/S}(\mathcal{F})\). Thus the image of of \(\mathfrak q\) in \(B/\mathfrak m_A B\) is not contained in any of those prime ideals. Hence by prime avoidance (Algebra, Lemma 00DS) we can find an element \(g \in \mathfrak q\) whose image in \(B/\mathfrak m_A B\) is a nonzerodivisor on \(N/\mathfrak m_A N\) (this uses the description of zerodivisors in Algebra, Lemma 00LD). Since \(N = \mathcal{F}_x\) is \(A\)-flat by Lemma 0GSG we see that \[g : N \longrightarrow N\] is injective. In particular, if \(K = \text{Frac}(A)\) is the fraction field of \(A\), then we see that \[g : N \otimes_A K \longrightarrow N \otimes_A K\] is injective. Observe that \(\mathfrak q\) corresponds to a prime ideal of \(B \otimes_A K\). Denote \(\mathcal{F}_t\) the restriction of \(\mathcal{F}\) to the generic fibre \(X_t\). We have \((B \otimes_A K)_{\mathfrak q} = \mathcal{O}_{X_t, y}\) and \((N \otimes_A K)_\mathfrak q\) is the stalk at \(y\) of \(\mathcal{F}_t\). Hence we find that \(g \in \mathfrak m_y \subset \mathcal{O}_{X_t, y}\) is a nonzerodivisor on the stalk \((\mathcal{F}_t)_y\) which contradicts our assumption that \(y \in \text{Ass}_{X/S}(\mathcal{F})\).

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module flat over \(S\). Assume \(S\) is irreducible with generic point \(\eta\). If \(\dim(\text{Supp}(\mathcal{F}_\eta)) \leq r\) then for all \(s \in S\) we have \(\dim(\text{Supp}(\mathcal{F}_s)) \leq r\).

Proof

Let \(x \in \text{Supp}(\mathcal{F}_s)\) be a generic point of an irreducible component of \(\text{Supp}(\mathcal{F}_s)\). By Algebra, Lemma 080T we can find a specialization \(y \leadsto x\) in \(\text{Supp}(\mathcal{F})\) with \(f(y) = \eta\). Of course we may assume \(y\) is a generic point of an irreducible component of \(\text{Supp}(\mathcal{F}_\eta)\). We conclude from Lemma 0GSI that the dimension of \(\overline{\{x\}}\) is at most \(r\).

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. Let \(y \in \text{Ass}_{X/S}(\mathcal{F})\). Denote \(Y \subset X\) the closure of \(\{y\}\) in \(X\) viewed as an integral closed subscheme. Denote \(T \subset S\) the closure of \(\{f(y)\}\) viewed as an integral closed subscheme. We obtain a commutative diagram \[\xymatrix{ Y \ar[r] \ar[d] & X \ar[d] \\ T \ar[r] & S }\] where \(Y \to T\) is dominant. Assume \(\mathcal{F}\) is flat over \(S\) at all generic points of irreducible components of fibres of \(Y \to T\) (for example if \(\mathcal{F}\) is flat over \(S\)). Then

if \(s \in S\) and \(x \in Y_s\) is the generic point of an irreducible component of \(Y_s\), then \(x \in \text{Ass}_{X/S}(\mathcal{F})\), and

there is an integer \(d \geq 0\) such that \(Y \to T\) is of relative dimension \(d\), see Morphisms, Definition 02NJ.

Proof

This follows immediately from the pointwise version Lemma 0GSI. Note that to compute the dimension of the locally algebraic schemes \(Y_s\) it suffices to look near the generic points, see Varieties, Section 06LF.

Remark

Here are some cases where the material above, especially Lemma 0GSJ, allows one to conclude that a morphism \(f : X \to S\) of schemes has relative dimension \(d\) as defined in Morphisms, Definition 02NJ. For example, this is true if

\(X\) is integral with generic point \(\xi\),

the transcendence degree of \(\kappa(\xi)\) over \(\kappa(f(\xi))\) is \(d\),

\(f\) is locally of finite type, and

there exists a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) of finite type which is flat over \(S\) with \(\text{Supp}(\mathcal{F}) = X\).

Another set of hypotheses that work are the following:

\(S\) is irreducible with generic point \(\eta\),

\(X_\eta\) is dense in \(X\),

every irreducible component of \(X_\eta\) has dimension \(d\),

\(f\) is locally of finite type, and

there exists a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) of finite type which is flat over \(S\) with \(\text{Supp}(\mathcal{F}) = X\).

Of course, we can relax the flatness condition on \(\mathcal{F}\) and require only that \(\mathcal{F}\) is flat over \(S\) in codimension \(0\), i.e., that \(\mathcal{F}\) is flat over \(S\) at every generic point of every fibre. If we ever need these results, we will carefully state and prove them here.

Normalization revisited

Normalization commutes with smooth base change.

Lemma

Let \(f : Y \to X\) be a smooth morphism of schemes. Let \(\mathcal{A}\) be a quasi-coherent sheaf of \(\mathcal{O}_X\)-algebras. The integral closure of \(\mathcal{O}_Y\) in \(f^*\mathcal{A}\) is equal to \(f^*\mathcal{A}'\) where \(\mathcal{A}' \subset \mathcal{A}\) is the integral closure of \(\mathcal{O}_X\) in \(\mathcal{A}\).

Proof

This is a translation of Algebra, Lemma 03GG into the language of schemes. Details omitted.

Lemma

Let \[\xymatrix{ Y_2 \ar[r] \ar[d]_{f_2} & Y_1 \ar[d]^{f_1} \\ X_2 \ar[r]^\varphi & X_1 }\] be a fibre square in the category of schemes. Assume \(f_1\) is quasi-compact and quasi-separated, and \(\varphi\) is smooth. Let \(Y_i \to X_i' \to X_i\) be the normalization of \(X_i\) in \(Y_i\). Then \(X_2' \cong X_2 \times_{X_1} X_1'\).

Proof

The base change of the factorization \(Y_1 \to X_1' \to X_1\) to \(X_2\) is a factorization \(Y_2 \to X_2 \times_{X_1} X_1' \to X_2\) and \(X_2 \times_{X_1} X_1' \to X_2\) is integral (Morphisms, Lemma 01WL). Hence we get a morphism \(h : X_2' \to X_2 \times_{X_1} X_1'\) by the universal property of Morphisms, Lemma 035I. Observe that \(X_2'\) is the relative spectrum of the integral closure of \(\mathcal{O}_{X_2}\) in \(f_{2, *}\mathcal{O}_{Y_2}\). If \(\mathcal{A}' \subset f_{1, *}\mathcal{O}_{Y_1}\) denotes the integral closure of \(\mathcal{O}_{X_1}\), then \(X_2 \times_{X_1} X_1'\) is the relative spectrum of \(\varphi^*\mathcal{A}'\), see Constructions, Lemma 01LX. By Cohomology of Schemes, Lemma 02KH we know that \(f_{2, *}\mathcal{O}_{Y_2} = \varphi^*f_{1, *}\mathcal{O}_{Y_1}\). Hence the result follows from Lemma 081K.

Lemma

Let \(X \to Y\) be a smooth morphism of schemes. Assume every quasi-compact open of \(Y\) has finitely many irreducible components. Then the same is true for \(X\) and there is a unique isomorphism \(X^\nu = X \times_Y Y^\nu\) over \(X\) where \(X^\nu\), \(Y^\nu\) are the normalizations of \(X\), \(Y\).

Proof

By Descent, Lemma 0BAL every quasi-compact open of \(X\) has finitely many irreducible components. Note that \(X_{red} = X \times_Y Y_{red}\) as a scheme smooth over a reduced scheme is reduced, see Descent, Lemma 034E. Hence we may assume that \(X\) and \(Y\) are reduced (as the normalization of a scheme is equal to the normalization of its reduction by definition). Next, note that \(X' = X \times_Y Y^\nu\) is a normal scheme by Descent, Lemma 034F. The morphism \(X' \to Y^\nu\) is smooth (hence flat) thus the generic points of irreducible components of \(X'\) lie over generic points of irreducible components of \(Y^\nu\). Since \(Y^\nu \to Y\) is birational we conclude that \(X' \to X\) is birational too (because \(X' \to Y^\nu\) induces an isomorphism on fibres over generic points of \(Y\)). We conclude that there exists a factorization \(X^\nu \to X' \to X\), see Morphisms, Lemma 035Q which is an isomorphism as \(X'\) is normal and integral over \(X\).

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\nu : X^\nu \to X\) be the normalization morphism. Then for any point \(x \in X\) the base change \[X^\nu \times_X \Spec(\mathcal{O}_{X, x}^h) \to \Spec(\mathcal{O}_{X, x}^h), \quad\text{resp.}\quad X^\nu \times_X \Spec(\mathcal{O}_{X, x}^{sh}) \to \Spec(\mathcal{O}_{X, x}^{sh})\] is the normalization of \(\Spec(\mathcal{O}_{X, x}^h)\), resp. \(\Spec(\mathcal{O}_{X, x}^{sh})\).

Proof

Let \(\eta_1, \ldots, \eta_r\) be the generic points of the irreducible components of \(X\) passing through \(x\). The base change of the normalization to \(\Spec(\mathcal{O}_{X, x})\) is the spectrum of the integral closure of \(\mathcal{O}_{X, x}\) in \(\prod \kappa(\eta_i)\). This follows from our construction of the normalization of \(X\) in Morphisms, Definition 035N and Morphisms, Lemma 035F; you can also use the description of the normalization in Morphisms, Lemma 035P. Thus we reduce to the following algebra problem. Let \(A\) be a Noetherian local ring; recall that this implies the henselization \(A^h\) and strict henselization \(A^{sh}\) are Noetherian too (More on Algebra, Lemma 06LJ). Let \(\mathfrak p_1, \ldots, \mathfrak p_r\) be its minimal primes. Let \(A'\) be the integral closure of \(A\) in \(\prod \kappa(\mathfrak p_i)\). Problem: show that \(A' \otimes_A A^h\), resp. \(A' \otimes_A A^{sh}\) is constructed from the Noetherian local ring \(A^h\), resp. \(A^{sh}\) in the same manner.

Since \(A^h\), resp. \(A^{sh}\) are colimits of étale \(A\)-algebras, we see that the minimal primes of \(A\) and \(A^{sh}\) are exactly the primes of \(A^h\), resp. \(A^{sh}\) lying over the minimal primes of \(A\) (by going down, see Algebra, Lemmas 00HS and 0CAN). Thus More on Algebra, Lemma 07QQ tells us that \(A^h \otimes_A \prod \kappa(\mathfrak p_i)\), resp. \(A^{sh} \otimes_A \prod \kappa(\mathfrak p_i)\) is the product of the residue fields at the minimal primes of \(A^h\), resp. \(A^{sh}\). We know that taking the integral closure in an overring commutes with étale base change, see Algebra, Lemma 03GE. Writing \(A^h\) and \(A^{sh}\) as a limit of étale \(A\)-algebras we see that the same thing is true for the base change to \(A^h\) and \(A^{sh}\) (you can also use the more general Algebra, Lemma 0CBF).

Normal morphisms

In the article [DM] of Deligne and Mumford the notion of a normal morphism is mentioned. This is just one in a series of types³ of morphisms that can all be defined similarly. Over time we will add these in their own sections as needed.

Definition

Let \(f : X \to Y\) be a morphism of schemes. Assume that all the fibres \(X_y\) are locally Noetherian schemes.

Let \(x \in X\), and \(y = f(x)\). We say that \(f\) is normal at \(x\) if \(f\) is flat at \(x\), and the scheme \(X_y\) is geometrically normal at \(x\) over \(\kappa(y)\) (see Varieties, Definition 038M).

We say \(f\) is a normal morphism if \(f\) is normal at every point of \(X\).

So the condition that the morphism \(X \to Y\) is normal is stronger than just requiring all the fibres to be normal locally Noetherian schemes.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume all fibres of \(f\) are locally Noetherian. The following are equivalent

\(f\) is normal, and

\(f\) is flat and its fibres are geometrically normal schemes.

Proof

This follows directly from the definitions.

Lemma

A smooth morphism is normal.

Proof

Let \(f : X \to Y\) be a smooth morphism. As \(f\) is locally of finite presentation, see Morphisms, Lemma 01VE the fibres \(X_y\) are locally of finite type over a field, hence locally Noetherian. Moreover, \(f\) is flat, see Morphisms, Lemma 01VF. Finally, the fibres \(X_y\) are smooth over a field (by Morphisms, Lemma 01VB) and hence geometrically normal by Varieties, Lemma 056T. Thus \(f\) is normal by Lemma 0391.

We want to show that this notion is local on the source and target for the smooth topology. First we deal with the property of having locally Noetherian fibres.

Lemma

The property \(\mathcal{P}(f)=\)“the fibres of \(f\) are locally Noetherian” is local in the fppf topology on the source and the target.

Proof

Let \(f : X \to Y\) be a morphism of schemes. Let \(\{\varphi_i : Y_i \to Y\}_{i \in I}\) be an fppf covering of \(Y\). Denote \(f_i : X_i \to Y_i\) the base change of \(f\) by \(\varphi_i\). Let \(i \in I\) and let \(y_i \in Y_i\) be a point. Set \(y = \varphi_i(y_i)\). Note that \[X_{i, y_i} = \Spec(\kappa(y_i)) \times_{\Spec(\kappa(y))} X_y.\] Moreover, as \(\varphi_i\) is of finite presentation the field extension \(\kappa(y_i)/\kappa(y)\) is finitely generated. Hence in this situation we have that \(X_y\) is locally Noetherian if and only if \(X_{i, y_i}\) is locally Noetherian, see Varieties, Lemma 038R. This fact implies locality on the target.

Let \(\{X_i \to X\}\) be an fppf covering of \(X\). Let \(y \in Y\). In this case \(\{X_{i, y} \to X_y\}\) is an fppf covering of the fibre. Hence the locality on the source follows from Descent, Lemma 034C.

Lemma

The property \(\mathcal{P}(f)=\)“the fibres of \(f\) are locally Noetherian and \(f\) is normal” is local in the fppf topology on the target and local in the smooth topology on the source.

Proof

We have \(\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f) \wedge \mathcal{P}_3(f)\) where \(\mathcal{P}_1(f)=\)“the fibres of \(f\) are locally Noetherian”, \(\mathcal{P}_2(f)=\)“\(f\) is flat”, and \(\mathcal{P}_3(f)=\)“the fibres of \(f\) are geometrically normal”. We have already seen that \(\mathcal{P}_1\) and \(\mathcal{P}_2\) are local in the fppf topology on the source and the target, see Lemma 0392, and Descent, Lemmas 02L2 and 036K. Thus we have to deal with \(\mathcal{P}_3\).

Let \(f : X \to Y\) be a morphism of schemes. Let \(\{\varphi_i : Y_i \to Y\}_{i \in I}\) be an fpqc covering of \(Y\). Denote \(f_i : X_i \to Y_i\) the base change of \(f\) by \(\varphi_i\). Let \(i \in I\) and let \(y_i \in Y_i\) be a point. Set \(y = \varphi_i(y_i)\). Note that \[X_{i, y_i} = \Spec(\kappa(y_i)) \times_{\Spec(\kappa(y))} X_y.\] Hence in this situation we have that \(X_y\) is geometrically normal if and only if \(X_{i, y_i}\) is geometrically normal, see Varieties, Lemma 038P. This fact implies \(\mathcal{P}_3\) is fpqc local on the target.

Let \(\{X_i \to X\}\) be a smooth covering of \(X\). Let \(y \in Y\). In this case \(\{X_{i, y} \to X_y\}\) is a smooth covering of the fibre. Hence the locality of \(\mathcal{P}_3\) for the smooth topology on the source follows from Descent, Lemma 034F. Combining the above the lemma follows.

Regular morphisms

Compare with Section 038Z. The algebraic version of this notion is discussed in More on Algebra, Section 07BY.

Definition

Let \(f : X \to Y\) be a morphism of schemes. Assume that all the fibres \(X_y\) are locally Noetherian schemes.

Let \(x \in X\), and \(y = f(x)\). We say that \(f\) is regular at \(x\) if \(f\) is flat at \(x\), and the scheme \(X_y\) is geometrically regular at \(x\) over \(\kappa(y)\) (see Varieties, Definition 038T).

We say \(f\) is a regular morphism if \(f\) is regular at every point of \(X\).

The condition that the morphism \(X \to Y\) is regular is stronger than just requiring all the fibres to be regular locally Noetherian schemes.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume all fibres of \(f\) are locally Noetherian. The following are equivalent

\(f\) is regular,

\(f\) is flat and its fibres are geometrically regular schemes,

for every pair of affine opens \(U \subset X\), \(V \subset Y\) with \(f(U) \subset V\) the ring map \(\mathcal{O}_Y(V) \to \mathcal{O}_X(U)\) is regular,

there exists an open covering \(Y = \bigcup_{j \in J} V_j\) and open coverings \(f^{-1}(V_j) = \bigcup_{i \in I_j} U_i\) such that each of the morphisms \(U_i \to V_j\) is regular, and

there exists an affine open covering \(Y = \bigcup_{j \in J} V_j\) and affine open coverings \(f^{-1}(V_j) = \bigcup_{i \in I_j} U_i\) such that the ring maps \(\mathcal{O}_Y(V_j) \to \mathcal{O}_X(U_i)\) are regular.

Proof

The equivalence of (1) and (2) is immediate from the definitions. Let \(x \in X\) with \(y = f(x)\). By definition \(f\) is flat at \(x\) if and only if \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\) is a flat ring map, and \(X_y\) is geometrically regular at \(x\) over \(\kappa(y)\) if and only if \(\mathcal{O}_{X_y, x} = \mathcal{O}_{X, x}/\mathfrak m_y\mathcal{O}_{X, x}\) is a geometrically regular algebra over \(\kappa(y)\). Hence Whether or not \(f\) is regular at \(x\) depends only on the local homomorphism of local rings \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\). Thus the equivalence of (1) and (4) is clear.

Recall (More on Algebra, Definition 07BZ) that a ring map \(A \to B\) is regular if and only if it is flat and the fibre rings \(B \otimes_A \kappa(\mathfrak p)\) are Noetherian and geometrically regular for all primes \(\mathfrak p \subset A\). By Varieties, Lemma 038V this is equivalent to \(\Spec(B \otimes_A \kappa(\mathfrak p))\) being a geometrically regular scheme over \(\kappa(\mathfrak p)\). Thus we see that (2) implies (3). It is clear that (3) implies (5). Finally, assume (5). This implies that \(f\) is flat (see Morphisms, Lemma 01U5). Moreover, if \(y \in Y\), then \(y \in V_j\) for some \(j\) and we see that \(X_y = \bigcup_{i \in I_j} U_{i, y}\) with each \(U_{i, y}\) geometrically regular over \(\kappa(y)\) by Varieties, Lemma 038V. Another application of Varieties, Lemma 038V shows that \(X_y\) is geometrically regular. Hence (2) holds and the proof of the lemma is finished.

Lemma

A smooth morphism is regular.

Proof

Let \(f : X \to Y\) be a smooth morphism. As \(f\) is locally of finite presentation, see Morphisms, Lemma 01VE the fibres \(X_y\) are locally of finite type over a field, hence locally Noetherian. Moreover, \(f\) is flat, see Morphisms, Lemma 01VF. Finally, the fibres \(X_y\) are smooth over a field (by Morphisms, Lemma 01VB) and hence geometrically regular by Varieties, Lemma 056T. Thus \(f\) is regular by Lemma 07R8.

Lemma

The property \(\mathcal{P}(f)=\)“the fibres of \(f\) are locally Noetherian and \(f\) is regular” is local in the fppf topology on the target and local in the smooth topology on the source.

Proof

We have \(\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f) \wedge \mathcal{P}_3(f)\) where \(\mathcal{P}_1(f)=\)“the fibres of \(f\) are locally Noetherian”, \(\mathcal{P}_2(f)=\)“\(f\) is flat”, and \(\mathcal{P}_3(f)=\)“the fibres of \(f\) are geometrically regular”. We have already seen that \(\mathcal{P}_1\) and \(\mathcal{P}_2\) are local in the fppf topology on the source and the target, see Lemma 0392, and Descent, Lemmas 02L2 and 036K. Thus we have to deal with \(\mathcal{P}_3\).

Let \(f : X \to Y\) be a morphism of schemes. Let \(\{\varphi_i : Y_i \to Y\}_{i \in I}\) be an fpqc covering of \(Y\). Denote \(f_i : X_i \to Y_i\) the base change of \(f\) by \(\varphi_i\). Let \(i \in I\) and let \(y_i \in Y_i\) be a point. Set \(y = \varphi_i(y_i)\). Note that \[X_{i, y_i} = \Spec(\kappa(y_i)) \times_{\Spec(\kappa(y))} X_y.\] Hence in this situation we have that \(X_y\) is geometrically regular if and only if \(X_{i, y_i}\) is geometrically regular, see Varieties, Lemma 038W. This fact implies \(\mathcal{P}_3\) is fpqc local on the target.

Let \(\{X_i \to X\}\) be a smooth covering of \(X\). Let \(y \in Y\). In this case \(\{X_{i, y} \to X_y\}\) is a smooth covering of the fibre. Hence the locality of \(\mathcal{P}_3\) for the smooth topology on the source follows from Descent, Lemma 036D. Combining the above the lemma follows.

Cohen-Macaulay morphisms

Compare with Section 038Z. Note that, as pointed out in Algebra, Section 045L and Varieties, Section 045O “geometrically Cohen-Macaulay” is the same as plain Cohen-Macaulay.

Definition

Let \(f : X \to Y\) be a morphism of schemes. Assume that all the fibres \(X_y\) are locally Noetherian schemes.

Let \(x \in X\), and \(y = f(x)\). We say that \(f\) is Cohen-Macaulay at \(x\) if \(f\) is flat at \(x\), and the local ring of the scheme \(X_y\) at \(x\) is Cohen-Macaulay.

We say \(f\) is a Cohen-Macaulay morphism if \(f\) is Cohen-Macaulay at every point of \(X\).

Here is a translation.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume all fibres of \(f\) are locally Noetherian. The following are equivalent

\(f\) is Cohen-Macaulay, and

\(f\) is flat and its fibres are Cohen-Macaulay schemes.

Proof

This follows directly from the definitions.

Lemma

Let \(f : X \to Y\) be a morphism of locally Noetherian schemes which is locally of finite type and Cohen-Macaulay. For every point \(x\) in \(X\) with image \(y\) in \(Y\), \[\dim_x(X) = \dim_y(Y) + \dim_x(X_y),\] where \(X_y\) denotes the fiber over \(y\).

Proof

After replacing \(X\) by an open neighborhood of \(x\), there is a natural number \(d\) such that all fibers of \(X \to Y\) have dimension \(d\) at every point, see Morphisms, Lemma 02NM. Then \(f\) is flat, locally of finite type and of relative dimension \(d\). Hence the result follows from Morphisms, Lemma 0AFE.

Lemma

Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of schemes. Assume that the fibres of \(f\), \(g\), and \(g \circ f\) are locally Noetherian. Let \(x \in X\) with images \(y \in Y\) and \(z \in Z\).

If \(f\) is Cohen-Macaulay at \(x\) and \(g\) is Cohen-Macaulay at \(f(x)\), then \(g \circ f\) is Cohen-Macaulay at \(x\).

If \(f\) and \(g\) are Cohen-Macaulay, then \(g \circ f\) is Cohen-Macaulay.

If \(g \circ f\) is Cohen-Macaulay at \(x\) and \(f\) is flat at \(x\), then \(f\) is Cohen-Macaulay at \(x\) and \(g\) is Cohen-Macaulay at \(f(x)\).

If \(g \circ f\) is Cohen-Macaulay and \(f\) is flat, then \(f\) is Cohen-Macaulay and \(g\) is Cohen-Macaulay at every point in the image of \(f\).

Proof

Consider the map of Noetherian local rings \[\mathcal{O}_{Y_z, y} \to \mathcal{O}_{X_z, x}\] and observe that its fibre is \[\mathcal{O}_{X_z, x}/\mathfrak m_{Y_z, y}\mathcal{O}_{X_z, x} = \mathcal{O}_{X_y, x}\] Thus the lemma this follows from Algebra, Lemma 045J.

Lemma

Let \(f : X \to Y\) be a flat morphism of locally Noetherian schemes. If \(X\) is Cohen-Macaulay, then \(f\) is Cohen-Macaulay and \(\mathcal{O}_{Y, f(x)}\) is Cohen-Macaulay for all \(x \in X\).

Proof

After translating into algebra this follows from Algebra, Lemma 045J.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume that all the fibres \(X_y\) are locally Noetherian schemes. Let \(Y' \to Y\) be locally of finite type. Let \(f' : X' \to Y'\) be the base change of \(f\). Let \(x' \in X'\) be a point with image \(x \in X\).

If \(f\) is Cohen-Macaulay at \(x\), then \(f' : X' \to Y'\) is Cohen-Macaulay at \(x'\).

If \(f\) is flat at \(x\) and \(f'\) is Cohen-Macaulay at \(x'\), then \(f\) is Cohen-Macaulay at \(x\).

If \(Y' \to Y\) is flat at \(f'(x')\) and \(f'\) is Cohen-Macaulay at \(x'\), then \(f\) is Cohen-Macaulay at \(x\).

Proof

Note that the assumption on \(Y' \to Y\) implies that for \(y' \in Y'\) mapping to \(y \in Y\) the field extension \(\kappa(y')/\kappa(y)\) is finitely generated. Hence also all the fibres \(X'_{y'} = (X_y)_{\kappa(y')}\) are locally Noetherian, see Varieties, Lemma 038R. Thus the lemma makes sense. Set \(y' = f'(x')\) and \(y = f(x)\). Hence we get the following commutative diagram of local rings \[\xymatrix{ \mathcal{O}_{X', x'} & \mathcal{O}_{X, x} \ar[l] \\ \mathcal{O}_{Y', y'} \ar[u] & \mathcal{O}_{Y, y} \ar[l] \ar[u] }\] where the upper left corner is a localization of the tensor product of the upper right and lower left corners over the lower right corner.

Assume \(f\) is Cohen-Macaulay at \(x\). The flatness of \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\) implies the flatness of \(\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}\), see Algebra, Lemma 00MQ. The fact that \(\mathcal{O}_{X, x}/\mathfrak m_y\mathcal{O}_{X, x}\) is Cohen-Macaulay implies that \(\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}\) is Cohen-Macaulay, see Varieties, Lemma 045P. Hence we see that \(f'\) is Cohen-Macaulay at \(x'\).

Assume \(f\) is flat at \(x\) and \(f'\) is Cohen-Macaulay at \(x'\). The fact that \(\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}\) is Cohen-Macaulay implies that \(\mathcal{O}_{X, x}/\mathfrak m_y\mathcal{O}_{X, x}\) is Cohen-Macaulay, see Varieties, Lemma 045P. Hence we see that \(f\) is Cohen-Macaulay at \(x\).

Assume \(Y' \to Y\) is flat at \(y'\) and \(f'\) is Cohen-Macaulay at \(x'\). The flatness of \(\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}\) and \(\mathcal{O}_{Y, y} \to \mathcal{O}_{Y', y'}\) implies the flatness of \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\), see Algebra, Lemma 00MQ. The fact that \(\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}\) is Cohen-Macaulay implies that \(\mathcal{O}_{X, x}/\mathfrak m_y\mathcal{O}_{X, x}\) is Cohen-Macaulay, see Varieties, Lemma 045P. Hence we see that \(f\) is Cohen-Macaulay at \(x\).

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat and locally of finite presentation. Let \[W = \{x \in X \mid f\text{ is Cohen-Macaulay at }x\}\] Then

\(W = \{x \in X \mid \mathcal{O}_{X_{f(x)}, x}\text{ is Cohen-Macaulay}\}\),

\(W\) is open in \(X\),

\(W\) is dense in every fibre of \(X \to S\),

the formation of \(W\) commutes with arbitrary base change of \(f\): For any morphism \(g : S' \to S\), consider the base change \(f' : X' \to S'\) of \(f\) and the projection \(g' : X' \to X\). Then the corresponding set \(W'\) for the morphism \(f'\) is equal to \(W' = (g')^{-1}(W)\).

Proof

As \(f\) is flat with locally Noetherian fibres the equality in (1) holds by definition. Parts (2) and (3) follow from Algebra, Lemma 00RI. Part (4) follows either from Algebra, Lemma 00RK or Varieties, Lemma 045P.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat and locally of finite presentation. Let \(x \in X\) with image \(s \in S\). Set \(d = \dim_x(X_s)\). The following are equivalent

\(f\) is Cohen-Macaulay at \(x\),

there exists an open neighbourhood \(U \subset X\) of \(x\) and a locally quasi-finite morphism \(U \to \mathbf{A}^d_S\) over \(S\) which is flat at \(x\),

there exists an open neighbourhood \(U \subset X\) of \(x\) and a locally quasi-finite flat morphism \(U \to \mathbf{A}^d_S\) over \(S\),

for any \(S\)-morphism \(g : U \to \mathbf{A}^d_S\) of an open neighbourhood \(U \subset X\) of \(x\) we have: \(g\) is quasi-finite at \(x\) \(\Rightarrow\) \(g\) is flat at \(x\).

Proof

Openness of flatness shows (2) and (3) are equivalent, see Theorem 0399.

Choose affine open \(U = \Spec(A) \subset X\) with \(x \in U\) and \(V = \Spec(R) \subset S\) with \(f(U) \subset V\). Then \(R \to A\) is a flat ring map of finite presentation. Let \(\mathfrak p \subset A\) be the prime ideal corresponding to \(x\). After replacing \(A\) by a principal localization we may assume there exists a quasi-finite map \(R[x_1, \ldots, x_d] \to A\), see Algebra, Lemma 00QE. Thus there exists at least one pair \((U, g)\) consisting of an open neighbourhood \(U \subset X\) of \(x\) and a locally⁴ quasi-finite morphism \(g : U \to \mathbf{A}^d_S\).

Claim: Given \(R \to A\) flat and of finite presentation, a prime \(\mathfrak p \subset A\) and \(\varphi : R[x_1, \ldots, x_d] \to A\) quasi-finite at \(\mathfrak p\) we have: \(\Spec(\varphi)\) is flat at \(\mathfrak p\) if and only if \(\Spec(A) \to \Spec(R)\) is Cohen-Macaulay at \(\mathfrak p\). Namely, by Theorem 039C flatness may be checked on fibres. The same is true for being Cohen-Macaulay (as \(A\) is already assumed flat over \(R\)). Thus the claim follows from Algebra, Lemma 00RE.

The claim shows that (1) is equivalent to (4) and combined with the fact that we have constructed a suitable \((U, g)\) in the second paragraph, the claim also shows that (1) is equivalent to (2).

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat and locally of finite presentation. For \(d \geq 0\) there exist opens \(U_d \subset X\) with the following properties

\(W = \bigcup_{d \geq 0} U_d\) is dense in every fibre of \(f\), and

\(U_d \to S\) is of relative dimension \(d\) (see Morphisms, Definition 02NJ).

Proof

This follows by combining Lemma 045U with Morphisms, Lemma 02NM.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat and locally of finite presentation. Suppose \(x' \leadsto x\) is a specialization of points of \(X\) with image \(s' \leadsto s\) in \(S\). If \(x\) is a generic point of an irreducible component of \(X_s\) then \(\dim_{x'}(X_{s'}) = \dim_x(X_s)\).

Proof

The point \(x\) is contained in \(U_d\) for some \(d\), where \(U_d\) as in Lemma 054T.

Lemma

The property \(\mathcal{P}(f)=\)“the fibres of \(f\) are locally Noetherian and \(f\) is Cohen-Macaulay” is local in the fppf topology on the target and local in the syntomic topology on the source.

Proof

We have \(\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f)\) where \(\mathcal{P}_1(f)=\)“\(f\) is flat”, and \(\mathcal{P}_2(f)=\)“the fibres of \(f\) are locally Noetherian and Cohen-Macaulay”. We know that \(\mathcal{P}_1\) is local in the fppf topology on the source and the target, see Descent, Lemmas 02L2 and 036K. Thus we have to deal with \(\mathcal{P}_2\).

Let \(f : X \to Y\) be a morphism of schemes. Let \(\{\varphi_i : Y_i \to Y\}_{i \in I}\) be an fppf covering of \(Y\). Denote \(f_i : X_i \to Y_i\) the base change of \(f\) by \(\varphi_i\). Let \(i \in I\) and let \(y_i \in Y_i\) be a point. Set \(y = \varphi_i(y_i)\). Note that \[X_{i, y_i} = \Spec(\kappa(y_i)) \times_{\Spec(\kappa(y))} X_y.\] and that \(\kappa(y_i)/\kappa(y)\) is a finitely generated field extension. Hence if \(X_y\) is locally Noetherian, then \(X_{i, y_i}\) is locally Noetherian, see Varieties, Lemma 038R. And if in addition \(X_y\) is Cohen-Macaulay, then \(X_{i, y_i}\) is Cohen-Macaulay, see Varieties, Lemma 045P. Thus \(\mathcal{P}_2\) is fppf local on the target.

Let \(\{X_i \to X\}\) be a syntomic covering of \(X\). Let \(y \in Y\). In this case \(\{X_{i, y} \to X_y\}\) is a syntomic covering of the fibre. Hence the locality of \(\mathcal{P}_2\) for the syntomic topology on the source follows from Descent, Lemma 036B. Combining the above the lemma follows.

Slicing Cohen-Macaulay morphisms

The results in this section eventually lead to the assertion that the fppf topology is the same as the “finitely presented, flat, quasi-finite” topology. The following lemma is very closely related to Divisors, Lemma 062Y.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) be a point with image \(s \in S\). Let \(h \in \mathfrak m_x \subset \mathcal{O}_{X, x}\). Assume

\(f\) is locally of finite presentation,

\(f\) is flat at \(x\), and

the image \(\overline{h}\) of \(h\) in \(\mathcal{O}_{X_s, x} = \mathcal{O}_{X, x}/\mathfrak m_s\mathcal{O}_{X, x}\) is a nonzerodivisor.

Then there exists an affine open neighbourhood \(U \subset X\) of \(x\) such that \(h\) comes from \(h \in \Gamma(U, \mathcal{O}_U)\) and such that \(D = V(h)\) is an effective Cartier divisor in \(U\) with \(x \in D\) and \(D \to S\) flat and locally of finite presentation.

Proof

We are going to prove this by reducing to the Noetherian case. By openness of flatness (see Theorem 0399) we may assume, after replacing \(X\) by an open neighbourhood of \(x\), that \(X \to S\) is flat. We may also assume that \(X\) and \(S\) are affine. After possible shrinking \(X\) a bit we may assume that there exists an \(h \in \Gamma(X, \mathcal{O}_X)\) which maps to our given \(h\).

We may write \(S = \Spec(A)\) and we may write \(A = \colim_i A_i\) as a directed colimit of finite type \(\mathbf{Z}\) algebras. Then by Algebra, Lemma 02JO or Limits, Lemmas 01ZM, 01ZN, and 01ZM we can find a cartesian diagram \[\xymatrix{ X \ar[r] \ar[d]_f & X_0 \ar[d]^{f_0} \\ S \ar[r] & S_0 }\] with \(f_0\) flat and of finite presentation, \(X_0\) affine, and \(S_0\) affine and Noetherian. Let \(x_0 \in X_0\), resp. \(s_0 \in S_0\) be the image of \(x\), resp. \(s\). We may also assume there exists an element \(h_0 \in \Gamma(X_0, \mathcal{O}_{X_0})\) which restricts to \(h\) on \(X\). (If you used the algebra reference above then this is clear; if you used the references to the chapter on limits then this follows from Limits, Lemma 01ZM by thinking of \(h\) as a morphism \(X \to \mathbf{A}^1_S\).) Note that \(\mathcal{O}_{X_s, x}\) is a localization of \(\mathcal{O}_{(X_0)_{s_0}, x_0} \otimes_{\kappa(s_0)} \kappa(s)\), so that \(\mathcal{O}_{(X_0)_{s_0}, x_0} \to \mathcal{O}_{X_s, x}\) is a flat local ring map, in particular faithfully flat. Hence the image \(\overline{h}_0 \in \mathcal{O}_{(X_0)_{s_0}, x_0}\) is contained in \(\mathfrak m_{(X_0)_{s_0}, x_0}\) and is a nonzerodivisor. We claim that after replacing \(X_0\) by a principal open neighbourhood of \(x_0\) the element \(h_0\) is a nonzerodivisor in \(B_0 = \Gamma(X_0, \mathcal{O}_{X_0})\) such that \(B_0/h_0B_0\) is flat over \(A_0 = \Gamma(S_0, \mathcal{O}_{S_0})\). If so then \[0 \to B_0 \xrightarrow{h_0} B_0 \to B_0/h_0B_0 \to 0\] is a short exact sequence of flat \(A_0\)-modules. Hence this remains exact on tensoring with \(A\) (by Algebra, Lemma 00HL) and the lemma follows.

It remains to prove the claim above. The corresponding algebra statement is the following (we drop the subscript \({}_0\) here): Let \(A \to B\) be a flat, finite type ring map of Noetherian rings. Let \(\mathfrak q \subset B\) be a prime lying over \(\mathfrak p \subset A\). Assume \(h \in \mathfrak q\) maps to a nonzerodivisor in \(B_{\mathfrak q}/\mathfrak p B_{\mathfrak q}\). Goal: show that after possible replacing \(B\) by \(B_g\) for some \(g \in B\), \(g \not \in \mathfrak q\) the element \(h\) becomes a nonzerodivisor and \(B/hB\) becomes flat over \(A\). By Algebra, Lemma 00MF we see that \(h\) is a nonzerodivisor in \(B_{\mathfrak q}\) and that \(B_{\mathfrak q}/hB_{\mathfrak q}\) is flat over \(A\). By openness of flatness, see Algebra, Theorem 00RC or Theorem 0399 we see that \(B/hB\) is flat over \(A\) after replacing \(B\) by \(B_g\) for some \(g \in B\), \(g \not \in \mathfrak q\). Finally, let \(I = \{b \in B \mid hb = 0\}\) be the annihilator of \(h\). Then \(IB_{\mathfrak q} = 0\) as \(h\) is a nonzerodivisor in \(B_{\mathfrak q}\). Also \(I\) is finitely generated as \(B\) is Noetherian. Hence there exists a \(g \in B\), \(g \not \in \mathfrak q\) such that \(IB_g = 0\). After replacing \(B\) by \(B_g\) we see that \(h\) is a nonzerodivisor.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) be a point with image \(s \in S\). Let \(h_1, \ldots, h_r \in \mathcal{O}_{X, x}\). Assume

\(f\) is locally of finite presentation,

\(f\) is flat at \(x\), and

the images of \(h_1, \ldots, h_r\) in \(\mathcal{O}_{X_s, x} = \mathcal{O}_{X, x}/\mathfrak m_s\mathcal{O}_{X, x}\) form a regular sequence.

Then there exists an affine open neighbourhood \(U \subset X\) of \(x\) such that \(h_1, \ldots, h_r\) come from \(h_1, \ldots, h_r \in \Gamma(U, \mathcal{O}_U)\) and such that \(Z = V(h_1, \ldots, h_r) \to U\) is a regular immersion with \(x \in Z\) and \(Z \to S\) flat and locally of finite presentation. Moreover, the base change \(Z_{S'} \to U_{S'}\) is a regular immersion for any scheme \(S'\) over \(S\).

Proof

(Our conventions on regular sequences imply that \(h_i \in \mathfrak m_x\) for each \(i\).) The case \(r = 1\) follows from Lemma 056Y combined with Divisors, Lemma 056Q to see that \(V(h_1)\) remains an effective Cartier divisor after base change. The case \(r > 1\) follows from a straightforward induction on \(r\) (applying the result for \(r = 1\) exactly \(r\) times; details omitted).

Another way to prove the lemma is using the material from Divisors, Section 063P. Namely, first by openness of flatness (see Theorem 0399) we may assume, after replacing \(X\) by an open neighbourhood of \(x\), that \(X \to S\) is flat. We may also assume that \(X\) and \(S\) are affine. After possible shrinking \(X\) a bit we may assume that we have \(h_1, \ldots, h_r \in \Gamma(X, \mathcal{O}_X)\). Set \(Z = V(h_1, \ldots, h_r)\). Note that \(X_s\) is a Noetherian scheme (because it is an algebraic \(\kappa(s)\)-scheme, see Varieties, Section 06LF) and that the topology on \(X_s\) is induced from the topology on \(X\) (see Schemes, Lemma 01K1). Hence after shrinking \(X\) a bit more we may assume that \(Z_s \subset X_s\) is a regular immersion cut out by the \(r\) elements \(h_i|_{X_s}\), see Divisors, Lemma 063I and its proof. It is also clear that \(r = \dim_x(X_s) - \dim_x(Z_s)\) because \[\begin{align*} \dim_x(X_s) & = \dim(\mathcal{O}_{X_s, x}) + \text{trdeg}_{\kappa(s)}(\kappa(x)), \\ \dim_x(Z_s) & = \dim(\mathcal{O}_{Z_s, x}) + \text{trdeg}_{\kappa(s)}(\kappa(x)), \\ \dim(\mathcal{O}_{X_s, x}) & = \dim(\mathcal{O}_{Z_s, x}) + r \end{align*}\] the first two equalities by Algebra, Lemma 00P1 and the second by \(r\) times applying Algebra, Lemma 00KW. Hence Divisors, Lemma 063W part (3) applies to show that (after Zariski shrinking \(X\)) the morphism \(Z \to X\) is a regular immersion to which Divisors, Lemma 063U applies (which gives the flatness and the statement on base change).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) be a point with image \(s \in S\). Assume

\(f\) is locally of finite presentation,

\(f\) is flat at \(x\), and

\(\mathcal{O}_{X_s, x}\) has \(\text{depth} \geq 1\).

Then there exists an affine open neighbourhood \(U \subset X\) of \(x\) and an effective Cartier divisor \(D \subset U\) containing \(x\) such that \(D \to S\) is flat and of finite presentation.

Proof

Pick any \(h \in \mathfrak m_x \subset \mathcal{O}_{X, x}\) which maps to a nonzerodivisor in \(\mathcal{O}_{X_s, x}\) and apply Lemma 056Y.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) be a point with image \(s \in S\). Assume

\(f\) is locally of finite presentation,

\(f\) is Cohen-Macaulay at \(x\), and

\(x\) is a closed point of \(X_s\).

Then there exists a regular immersion \(Z \to X\) containing \(x\) such that

\(Z \to S\) is flat and locally of finite presentation,

\(Z \to S\) is locally quasi-finite, and

\(Z_s = \{x\}\) set theoretically.

Proof

We may and do replace \(S\) by an affine open neighbourhood of \(s\). We will prove the lemma for affine \(S\) by induction on \(d = \dim_x(X_s)\).

The case \(d = 0\). In this case we show that we may take \(Z\) to be an open neighbourhood of \(x\). (Note that an open immersion is a regular immersion.) Namely, if \(d = 0\), then \(X \to S\) is quasi-finite at \(x\), see Morphisms, Lemma 0397. Hence there exists an affine open neighbourhood \(U \subset X\) such that \(U \to S\) is quasi-finite, see Morphisms, Lemma 01TI. Thus after replacing \(X\) by \(U\) we see that the fibre \(X_s\) is a finite discrete set. Hence after replacing \(X\) by a further affine open neighbourhood of \(X\) we see that \(f^{-1}(\{s\}) = \{x\}\) (because the topology on \(X_s\) is induced from the topology on \(X\), see Schemes, Lemma 01K1). This proves the lemma in this case.

Next, assume \(d > 0\). Note that because \(x\) is a closed point of its fibre the extension \(\kappa(x)/\kappa(s)\) is finite (by the Hilbert Nullstellensatz, see Morphisms, Lemma 01TF). Thus we see \[\text{depth}(\mathcal{O}_{X_s, x}) = \dim(\mathcal{O}_{X_s, x}) = d > 0\] the first equality as \(\mathcal{O}_{X_s, x}\) is Cohen-Macaulay and the second by Morphisms, Lemma 02FX. Thus we may apply Lemma 056Z to find a diagram \[\xymatrix{ D \ar[r] \ar[rrd] & U \ar[r] \ar[rd] & X \ar[d] \\ & & S }\] with \(x \in D\). Note that \(\mathcal{O}_{D_s, x} = \mathcal{O}_{X_s, x}/(\overline{h})\) for some nonzerodivisor \(\overline{h}\), see Divisors, Lemma 056Q. Hence \(\mathcal{O}_{D_s, x}\) is Cohen-Macaulay of dimension one less than the dimension of \(\mathcal{O}_{X_s, x}\), see Algebra, Lemma 02JN for example. Thus the morphism \(D \to S\) is flat, locally of finite presentation, and Cohen-Macaulay at \(x\) with \(\dim_x(D_s) = \dim_x(X_s) - 1 = d - 1\). By induction hypothesis we can find a regular immersion \(Z \to D\) having properties (a), (b), (c). As \(Z \to D \to U\) are both regular immersions, we see that also \(Z \to U\) is a regular immersion by Divisors, Lemma 067Q. This finishes the proof.

Lemma

Let \(f : X \to S\) be a flat morphism of schemes which is locally of finite presentation. Let \(s \in S\) be a point in the image of \(f\). Then there exists a commutative diagram \[\xymatrix{ S' \ar[rr] \ar[rd]_g & & X \ar[ld]^f \\ & S }\] where \(g : S' \to S\) is flat, locally of finite presentation, locally quasi-finite, and \(s \in g(S')\).

Proof

The fibre \(X_s\) is not empty by assumption. Hence there exists a closed point \(x \in X_s\) where \(f\) is Cohen-Macaulay, see Lemma 045U. Apply Lemma 0570 and set \(S' = Z\).

The following lemma shows that sheaves for the fppf topology are the same thing as sheaves for the “quasi-finite, flat, finite presentation” topology.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{U} = \{S_i \to S\}_{i \in I}\) be an fppf covering of \(S\), see Topologies, Definition 021M. Then there exists an fppf covering \(\mathcal{V} = \{T_j \to S\}_{j \in J}\) which refines (see Sites, Definition 00VT) \(\mathcal{U}\) such that each \(T_j \to S\) is locally quasi-finite.

Proof

For every \(s \in S\) there exists an \(i \in I\) such that \(s\) is in the image of \(S_i \to S\). By Lemma 0571 we can find a morphism \(g_s : T_s \to S\) such that \(s \in g_s(T_s)\) which is flat, locally of finite presentation and locally quasi-finite and such that \(g_s\) factors through \(S_i \to S\). Hence \(\{T_s \to S\}\) is the desired covering of \(S\) that refines \(\mathcal{U}\).

Generic fibres

Some results on the relationship between generic fibres and nearby fibres.

Lemma

Let \(f : X \to Y\) be a finite type morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\). If \(X_\eta = \emptyset\) then there exists a nonempty open \(V \subset Y\) such that \(X_V = V \times_Y X = \emptyset\).

Proof

Follows immediately from the more general Morphisms, Lemma 02NE.

Lemma

Let \(f : X \to Y\) be a finite type morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\). If \(X_\eta \not = \emptyset\) then there exists a nonempty open \(V \subset Y\) such that \(X_V = V \times_Y X \to V\) is surjective.

Proof

This follows, upon taking affine opens, from Algebra, Lemma 00FH. (Of course it also follows from generic flatness.)

Lemma

Let \(f : X \to Y\) be a finite type morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\). If \(Z \subset X\) is a closed subset with \(Z_\eta\) nowhere dense in \(X_\eta\), then there exists a nonempty open \(V \subset Y\) such that \(Z_y\) is nowhere dense in \(X_y\) for all \(y \in V\).

Proof

Let \(Y' \subset Y\) be the reduction of \(Y\). Set \(X' = Y' \times_Y X\) and \(Z' = Y' \times_Y Z\). As \(Y' \to Y\) is a universal homeomorphism by Morphisms, Lemma 054M we see that it suffices to prove the lemma for \(Z' \subset X' \to Y'\). Thus we may assume that \(Y\) is integral, see Properties, Lemma 01ON. By Morphisms, Proposition 052A there exists a nonempty affine open \(V \subset Y\) such that \(X_V \to V\) and \(Z_V \to V\) are flat and of finite presentation. We claim that \(V\) works. Pick \(y \in V\). If \(Z_y\) has a nonempty interior, then \(Z_y\) contains a generic point \(\xi\) of an irreducible component of \(X_y\). Note that \(\eta \leadsto f(\xi)\). Since \(Z_V \to V\) is flat we can choose a specialization \(\xi' \leadsto \xi\), \(\xi' \in Z\) with \(f(\xi') = \eta\), see Morphisms, Lemma 03HV. By Lemma 054U we see that \[\dim_{\xi'}(Z_\eta) = \dim_{\xi}(Z_y) = \dim_{\xi}(X_y) = \dim_{\xi'}(X_\eta).\] Hence some irreducible component of \(Z_\eta\) passing through \(\xi'\) has dimension \(\dim_{\xi'}(X_\eta)\) which contradicts the assumption that \(Z_\eta\) is nowhere dense in \(X_\eta\) and we win.

Lemma

Let \(f : X \to Y\) be a finite type morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\). Let \(U \subset X\) be an open subscheme such that \(U_\eta\) is scheme theoretically dense in \(X_\eta\). Then there exists a nonempty open \(V \subset Y\) such that \(U_y\) is scheme theoretically dense in \(X_y\) for all \(y \in V\).

Proof

Let \(Y' \subset Y\) be the reduction of \(Y\). Let \(X' = Y' \times_Y X\) and \(U' = Y' \times_Y U\). As \(Y' \to Y\) induces a bijection on points, and as \(U' \to U\) and \(X' \to X\) induce isomorphisms of scheme theoretic fibres, we may replace \(Y\) by \(Y'\) and \(X\) by \(X'\). Thus we may assume that \(Y\) is integral, see Properties, Lemma 01ON. We may also replace \(Y\) by a nonempty affine open. In other words we may assume that \(Y = \Spec(A)\) where \(A\) is a domain with fraction field \(K\).

As \(f\) is of finite type we see that \(X\) is quasi-compact. Write \(X = X_1 \cup \ldots \cup X_n\) for some affine opens \(X_i\). By Morphisms, Definition 01RB we see that \(U_i = X_i \cap U\) is an open subscheme of \(X_i\) such that \(U_{i, \eta}\) is scheme theoretically dense in \(X_{i, \eta}\). Thus it suffices to prove the result for the pairs \((X_i, U_i)\), in other words we may assume that \(X\) is affine.

Write \(X = \Spec(B)\). Note that \(B_K\) is Noetherian as it is a finite type \(K\)-algebra. Hence \(U_\eta\) is quasi-compact. Thus we can find finitely many \(g_1, \ldots, g_m \in B\) such that \(D(g_j) \subset U\) and such that \(U_\eta = D(g_1)_\eta \cup \ldots \cup D(g_m)_\eta\). The fact that \(U_\eta\) is scheme theoretically dense in \(X_\eta\) means that \(B_K \to \bigoplus_j (B_K)_{g_j}\) is injective, see Morphisms, Example 056C. By Algebra, Lemma 0565 this is equivalent to the injectivity of \(B_K \to \bigoplus\nolimits_{j = 1, \ldots, m} B_K\), \(b \mapsto (g_1b, \ldots, g_mb)\). Let \(M\) be the cokernel of this map over \(A\), i.e., such that we have an exact sequence \[0 \to I \to B \xrightarrow{(g_1, \ldots, g_m)} \bigoplus\nolimits_{j = 1, \ldots, m} B \to M \to 0\] After replacing \(A\) by \(A_h\) for some nonzero \(h\) we may assume that \(B\) is a flat, finitely presented \(A\)-algebra, and that \(M\) is flat over \(A\), see Algebra, Lemma 051T. The flatness of \(B\) over \(A\) implies that \(B\) is torsion free as an \(A\)-module, see More on Algebra, Lemma 0538. Hence \(B \subset B_K\). By assumption \(I_K = 0\) which implies that \(I = 0\) (as \(I \subset B \subset B_K\) is a subset of \(I_K\)). Hence now we have a short exact sequence \[0 \to B \xrightarrow{(g_1, \ldots, g_m)} \bigoplus\nolimits_{j = 1, \ldots, m} B \to M \to 0\] with \(M\) flat over \(A\). Hence for every homomorphism \(A \to \kappa\) where \(\kappa\) is a field, we obtain a short exact sequence \[0 \to B \otimes_A \kappa \xrightarrow{(g_1 \otimes 1, \ldots, g_m \otimes 1)} \bigoplus\nolimits_{j = 1, \ldots, m} B \otimes_A \kappa \to M \otimes_A \kappa \to 0\] see Algebra, Lemma 00HL. Reversing the arguments above this means that \(\bigcup D(g_j \otimes 1)\) is scheme theoretically dense in \(\Spec(B \otimes_A \kappa)\). As \(\bigcup D(g_j \otimes 1) = \bigcup D(g_j)_\kappa \subset U_\kappa\) we obtain that \(U_\kappa\) is scheme theoretically dense in \(X_\kappa\) which is what we wanted to prove.

Suppose given a morphism of schemes \(f : X \to Y\) and a point \(y \in Y\). Recall that the fibre \(X_y\) is homeomorphic to the subset \(f^{-1}(\{y\})\) of \(X\) with induced topology, see Schemes, Lemma 01K1. Suppose given a closed subset \(T(y) \subset X_y\). Let \(T\) be the closure of \(T(y)\) in \(X\). Endow \(T\) with the induced reduced scheme structure. Then \(T\) is a closed subscheme of \(X\) with the property that \(T_y = T(y)\) set-theoretically. In fact \(T\) is the smallest closed subscheme of \(X\) with this property. Thus it is “harmless” to denote a closed subset of \(X_y\) by \(T_y\) if we so desire. In the following lemma we apply this to the generic fibre of \(f\).

Lemma

Let \(f : X \to Y\) be a finite type morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\). Let \(X_\eta = Z_{1, \eta} \cup \ldots \cup Z_{n, \eta}\) be a covering of the generic fibre by closed subsets of \(X_\eta\). Let \(Z_i\) be the closure of \(Z_{i, \eta}\) in \(X\) (see discussion above). Then there exists a nonempty open \(V \subset Y\) such that \(X_y = Z_{1, y} \cup \ldots \cup Z_{n, y}\) for all \(y \in V\).

Proof

If \(Y\) is Noetherian then \(U = X \setminus (Z_1 \cup \ldots \cup Z_n)\) is of finite type over \(Y\) and we can directly apply Lemma 054W to get that \(U_V = \emptyset\) for a nonempty open \(V \subset Y\). In general we argue as follows. As the question is topological we may replace \(Y\) by its reduction. Thus \(Y\) is integral, see Properties, Lemma 01ON. After shrinking \(Y\) we may assume that \(X \to Y\) is flat, see Morphisms, Proposition 052A. In this case every point \(x\) in \(X_y\) is a specialization of a point \(x' \in X_\eta\) by Morphisms, Lemma 03HV. As the \(Z_i\) are closed in \(X\) and cover the generic fibre this implies that \(X_y = \bigcup Z_{i, y}\) for \(y \in Y\) as desired.

The following lemma says that generic fibres of morphisms whose source is reduced are reduced.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(\eta \in Y\) be a generic point of an irreducible component of \(Y\). Then \((X_\eta)_{red} = (X_{red})_\eta\).

Proof

Choose an affine neighbourhood \(\Spec(A) \subset Y\) of \(\eta\). Choose an affine open \(\Spec(B) \subset X\) mapping into \(\Spec(A)\) via the morphism \(f\). Let \(\mathfrak p \subset A\) be the minimal prime corresponding to \(\eta\). Let \(B_{red}\) be the quotient of \(B\) by the nilradical \(\sqrt{(0)}\). The algebraic content of the lemma is that \(C = B_{red} \otimes_A \kappa(\mathfrak p)\) is reduced. Denote \(I \subset A\) the nilradical so that \(A_{red} = A/I\). Denote \(\mathfrak p_{red} = \mathfrak p/I\) which is a minimal prime of \(A_{red}\) with \(\kappa(\mathfrak p) = \kappa(\mathfrak p_{red})\). Since \(A \to B_{red}\) and \(A \to \kappa(\mathfrak p)\) both factor through \(A \to A_{red}\) we have \(C = B_{red} \otimes_{A_{red}} \kappa(\mathfrak p_{red})\). Now \(\kappa(\mathfrak p_{red}) = (A_{red})_{\mathfrak p_{red}}\) is a localization by Algebra, Lemma 00EU. Hence \(C\) is a localization of \(B_{red}\) (Algebra, Lemma 00DK) and hence reduced.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume that \(Y\) is irreducible and \(f\) is of finite type. There exists a diagram \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X_V \ar[r] \ar[d] & X \ar[d]^f \\ Y' \ar[r]^g & V \ar[r] & Y }\] where

\(V\) is a nonempty open of \(Y\),

\(X_V = V \times_Y X\),

\(g : Y' \to V\) is a finite universal homeomorphism,

\(X' = (Y' \times_Y X)_{red} = (Y' \times_V X_V)_{red}\),

\(g'\) is a finite universal homeomorphism,

\(Y'\) is an integral affine scheme,

\(f'\) is flat and of finite presentation, and

the generic fibre of \(f'\) is geometrically reduced.

Proof

Let \(V = \Spec(A)\) be a nonempty affine open of \(Y\). By assumption the nilradical \(\mathfrak p = \sqrt{(0)}\) of \(A\) is a prime ideal. Let \(K = \kappa(\mathfrak p)\). Let \(p\) be the characteristic of \(K\) if positive and \(1\) if the characteristic is zero. By Varieties, Lemma 04KT there exists a finite purely inseparable field extension \(K'/K\) such that \((X_{K'})_{red}\) is geometrically reduced over \(K'\). Choose elements \(x_1, \ldots, x_n \in K'\) which generate \(K'\) over \(K\) and such that some \(p\)-power of \(x_i\) is in \(A/\mathfrak p\). Let \(A' \subset K'\) be the finite \(A\)-subalgebra of \(K'\) generated by \(x_1, \ldots, x_n\). Note that \(A'\) is a domain with fraction field \(K'\). By Algebra, Lemma 0BRA we see that \(A \to A'\) induces a universal homeomorphism on spectra. Set \(Y' = \Spec(A')\). Set \(X' = (Y' \times_Y X)_{red}\). The generic fibre of \(X' \to Y'\) is \((X_{K'})_{red}\) by Lemma 054Z which is geometrically reduced by construction. Note that \(X' \to X_V\) is a finite universal homeomorphism as the composition of the reduction morphism \(X' \to Y' \times_Y X\) (see Morphisms, Lemma 054M) and the base change of \(g\). At this point all of the properties of the lemma hold except for possibly (7). This can be achieved by shrinking \(Y'\) and hence \(V\), see Morphisms, Proposition 052A.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume that \(Y\) is irreducible and \(f\) is of finite type. There exists a diagram \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X_V \ar[r] \ar[d] & X \ar[d]^f \\ Y' \ar[r]^g & V \ar[r] & Y }\] where

\(V\) is a nonempty open of \(Y\),

\(X_V = V \times_Y X\),

\(g : Y' \to V\) is surjective finite étale,

\(X' = Y' \times_Y X = Y' \times_V X_V\),

\(g'\) is surjective finite étale,

\(Y'\) is an irreducible affine scheme, and

all irreducible components of the generic fibre of \(f'\) are geometrically irreducible.

Proof

Let \(V = \Spec(A)\) be a nonempty affine open of \(Y\). By assumption the nilradical \(\mathfrak p = \sqrt{(0)}\) of \(A\) is a prime ideal. Let \(K = \kappa(\mathfrak p)\). By Varieties, Lemma 054R there exists a finite separable field extension \(K'/K\) such that all irreducible components of \(X_{K'}\) are geometrically irreducible over \(K'\). Choose an element \(\alpha \in K'\) which generates \(K'\) over \(K\), see Fields, Lemma 030N. Let \(P(T) \in K[T]\) be the minimal polynomial for \(\alpha\) over \(K\). After replacing \(\alpha\) by \(f \alpha\) for some \(f \in A\), \(f \not \in \mathfrak p\) we may assume that there exists a monic polynomial \(T^d + a_1T^{d - 1} + \ldots + a_d \in A[T]\) which maps to \(P(T) \in K[T]\) under the map \(A[T] \to K[T]\). Set \(A' = A[T]/(P)\). Then \(A \to A'\) is a finite free ring map such that there exists a unique prime \(\mathfrak q\) lying over \(\mathfrak p\), such that \(K = \kappa(\mathfrak p) \subset \kappa(\mathfrak q) = K'\) is finite separable, and such that \(\mathfrak pA'_{\mathfrak q}\) is the maximal ideal of \(A'_{\mathfrak q}\). Hence \(g : Y' = \Spec(A') \to V = \Spec(A)\) is étale at \(\mathfrak q\), see Algebra, Lemma 00U6. This means that there exists an open \(W \subset \Spec(A')\) such that \(g|_W : W \to \Spec(A)\) is étale. Since \(g\) is finite and since \(\mathfrak q\) is the only point lying over \(\mathfrak p\) we see that \(Z = g(Y' \setminus W)\) is a closed subset of \(V\) not containing \(\mathfrak p\). Hence after replacing \(V\) by a principal affine open of \(V\) which does not meet \(Z\) we obtain that \(g\) is finite étale.

Relative assassins

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\xi \in \text{Ass}_{X/S}(\mathcal{F})\) and set \(Z = \overline{\{\xi\}} \subset X\). If \(f\) is locally of finite type and \(\mathcal{F}\) is a finite type \(\mathcal{O}_X\)-module, then there exists a nonempty open \(V \subset Z\) such that for every \(s \in f(V)\) the generic points of \(V_s\) are elements of \(\text{Ass}_{X/S}(\mathcal{F})\).

Proof

We may replace \(S\) by an affine open neighbourhood of \(f(\xi)\) and \(X\) by an affine open neighbourhood of \(\xi\). Hence we may assume \(S = \Spec(A)\), \(X = \Spec(B)\) and that \(f\) is given by the finite type ring map \(A \to B\), see Morphisms, Lemma 01T2. Moreover, we may write \(\mathcal{F} = \widetilde{M}\) for some finite \(B\)-module \(M\), see Properties, Lemma 01PB. Let \(\mathfrak q \subset B\) be the prime corresponding to \(\xi\) and let \(\mathfrak p \subset A\) be the corresponding prime of \(A\). By assumption \(\mathfrak q \in \text{Ass}_B(M \otimes_A \kappa(\mathfrak p))\), see Algebra, Remark 05E0 and Divisors, Lemma 02OK. With this notation \(Z = V(\mathfrak q) \subset \Spec(B)\). In particular \(f(Z) \subset V(\mathfrak p)\). Hence clearly it suffices to prove the lemma after replacing \(A\), \(B\), and \(M\) by \(A/\mathfrak pA\), \(B/\mathfrak pB\), and \(M/\mathfrak pM\). In other words we may assume that \(A\) is a domain with fraction field \(K\) and \(\mathfrak q \subset B\) is an associated prime of \(M \otimes_A K\).

At this point we can use generic flatness. Namely, by Algebra, Lemma 051T there exists a nonzero \(g \in A\) such that \(M_g\) is flat as an \(A_g\)-module. After replacing \(A\) by \(A_g\) we may assume that \(M\) is flat as an \(A\)-module.

In this case, by Algebra, Lemma 05C1 we see that \(\mathfrak q\) is also an associated prime of \(M\). Hence we obtain an injective \(B\)-module map \(B/\mathfrak q \to M\). Let \(Q\) be the cokernel so that we obtain a short exact sequence \[0 \to B/\mathfrak q \to M \to Q \to 0\] of finite \(B\)-modules. After applying generic flatness Algebra, Lemma 051T once more, this time to the \(B\)-module \(Q\), we may assume that \(Q\) is a flat \(A\)-module. In particular we may assume the short exact sequence above is universally injective, see Algebra, Lemma 00HL. In this situation \((B/\mathfrak q) \otimes_A \kappa(\mathfrak p') \subset M \otimes_A \kappa(\mathfrak p')\) for any prime \(\mathfrak p'\) of \(A\). The lemma follows as a minimal prime \(\mathfrak q'\) of the support of \((B/\mathfrak q) \otimes_A \kappa(\mathfrak p')\) is an associated prime of \((B/\mathfrak q) \otimes_A \kappa(\mathfrak p')\) by Divisors, Lemma 05AH.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(U \subset X\) be an open subscheme. Assume

\(f\) is of finite type,

\(\mathcal{F}\) is of finite type,

\(Y\) is irreducible with generic point \(\eta\), and

\(\text{Ass}_{X_\eta}(\mathcal{F}_\eta)\) is not contained in \(U_\eta\).

Then there exists a nonempty open subscheme \(V \subset Y\) such that for all \(y \in V\) the set \(\text{Ass}_{X_y}(\mathcal{F}_y)\) is not contained in \(U_y\).

Proof

Let \(Z \subset X\) be the scheme theoretic support of \(\mathcal{F}\), see Morphisms, Definition 05JV. Then \(Z_\eta\) is the scheme theoretic support of \(\mathcal{F}_\eta\) (Morphisms, Lemma 07T9). Hence the generic points of irreducible components of \(Z_\eta\) are contained in \(\text{Ass}_{X_\eta}(\mathcal{F}_\eta)\) by Divisors, Lemma 05AH. Hence we see that \(Z_\eta \cap U_\eta = \emptyset\). Thus \(T = Z \setminus U\) is a closed subset of \(Z\) with \(T_\eta = \emptyset\). If we endow \(T\) with the induced reduced scheme structure then \(T \to Y\) is a morphism of finite type. By Lemma 054W there is a nonempty open \(V \subset Y\) with \(T_V = \emptyset\). Then \(V\) works.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(U \subset X\) be an open subscheme. Assume

\(f\) is of finite type,

\(\mathcal{F}\) is of finite type,

\(Y\) is irreducible with generic point \(\eta\), and

\(\text{Ass}_{X_\eta}(\mathcal{F}_\eta) \subset U_\eta\).

Then there exists a nonempty open subscheme \(V \subset Y\) such that for all \(y \in V\) we have \(\text{Ass}_{X_y}(\mathcal{F}_y) \subset U_y\).

Proof

(This proof is the same as the proof of Lemma 0573. We urge the reader to read that proof first.) Since the statement is about fibres it is clear that we may replace \(Y\) by its reduction. Hence we may assume that \(Y\) is integral, see Properties, Lemma 01ON. We may also assume that \(Y = \Spec(A)\) is affine. Then \(A\) is a domain with fraction field \(K\).

As \(f\) is of finite type we see that \(X\) is quasi-compact. Write \(X = X_1 \cup \ldots \cup X_n\) for some affine opens \(X_i\) and set \(\mathcal{F}_i = \mathcal{F}|_{X_i}\). By assumption the generic fibre of \(U_i = X_i \cap U\) contains \(\text{Ass}_{X_{i, \eta}}(\mathcal{F}_{i, \eta})\). Thus it suffices to prove the result for the triples \((X_i, \mathcal{F}_i, U_i)\), in other words we may assume that \(X\) is affine.

Write \(X = \Spec(B)\). Let \(N\) be a finite \(B\)-module such that \(\mathcal{F} = \widetilde{N}\). Note that \(B_K\) is Noetherian as it is a finite type \(K\)-algebra. Hence \(U_\eta\) is quasi-compact. Thus we can find finitely many \(g_1, \ldots, g_m \in B\) such that \(D(g_j) \subset U\) and such that \(U_\eta = D(g_1)_\eta \cup \ldots \cup D(g_m)_\eta\). Since \(\text{Ass}_{X_\eta}(\mathcal{F}_\eta) \subset U_\eta\) we see that \(N_K \to \bigoplus_j (N_K)_{g_j}\) is injective. By Algebra, Lemma 0565 this is equivalent to the injectivity of \(N_K \to \bigoplus\nolimits_{j = 1, \ldots, m} N_K\), \(n \mapsto (g_1n, \ldots, g_mn)\). Let \(I\) and \(M\) be the kernel and cokernel of this map over \(A\), i.e., such that we have an exact sequence \[0 \to I \to N \xrightarrow{(g_1, \ldots, g_m)} \bigoplus\nolimits_{j = 1, \ldots, m} N \to M \to 0\] After replacing \(A\) by \(A_h\) for some nonzero \(h\) we may assume that \(B\) is a flat, finitely presented \(A\)-algebra and that both \(M\) and \(N\) are flat over \(A\), see Algebra, Lemma 051T. The flatness of \(N\) over \(A\) implies that \(N\) is torsion free as an \(A\)-module, see More on Algebra, Lemma 0538. Hence \(N \subset N_K\). By construction \(I_K = 0\) which implies that \(I = 0\) (as \(I \subset N \subset N_K\) is a subset of \(I_K\)). Hence now we have a short exact sequence \[0 \to N \xrightarrow{(g_1, \ldots, g_m)} \bigoplus\nolimits_{j = 1, \ldots, m} N \to M \to 0\] with \(M\) flat over \(A\). Hence for every homomorphism \(A \to \kappa\) where \(\kappa\) is a field, we obtain a short exact sequence \[0 \to N \otimes_A \kappa \xrightarrow{(g_1 \otimes 1, \ldots, g_m \otimes 1)} \bigoplus\nolimits_{j = 1, \ldots, m} N \otimes_A \kappa \to M \otimes_A \kappa \to 0\] see Algebra, Lemma 00HL. Reversing the arguments above this means that \(\bigcup D(g_j \otimes 1)\) contains \(\text{Ass}_{B \otimes_A \kappa}(N \otimes_A \kappa)\). As \(\bigcup D(g_j \otimes 1) = \bigcup D(g_j)_\kappa \subset U_\kappa\) we obtain that \(U_\kappa\) contains \(\text{Ass}_{X \otimes \kappa}(\mathcal{F} \otimes \kappa)\) which is what we wanted to prove.

Lemma

Let \(f : X \to S\) be a morphism which is locally of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. Let \(U \subset X\) be an open subscheme. Let \(g : S' \to S\) be a morphism of schemes, let \(f' : X' = X_{S'} \to S'\) be the base change of \(f\), let \(g' : X' \to X\) be the projection, set \(\mathcal{F}' = (g')^*\mathcal{F}\), and set \(U' = (g')^{-1}(U)\). Finally, let \(s' \in S'\) with image \(s = g(s')\). In this case \[\text{Ass}_{X_s}(\mathcal{F}_s) \subset U_s \Leftrightarrow \text{Ass}_{X'_{s'}}(\mathcal{F}'_{s'}) \subset U'_{s'}.\]

Proof

This follows immediately from Divisors, Lemma 05DC. See also Divisors, Remark 05KL.

Lemma

Let \(f : X \to Y\) be a morphism of finite presentation. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite presentation. Let \(U \subset X\) be an open subscheme such that \(U \to Y\) is quasi-compact. Then the set \[E = \{y \in Y \mid \text{Ass}_{X_y}(\mathcal{F}_y) \subset U_y\}\] is locally constructible in \(Y\).

Proof

Let \(y \in Y\). We have to show that there exists an open neighbourhood \(V\) of \(y\) in \(Y\) such that \(E \cap V\) is constructible in \(V\). Thus we may assume that \(Y\) is affine. Write \(Y = \Spec(A)\) and \(A = \colim A_i\) as a directed limit of finite type \(\mathbf{Z}\)-algebras. By Limits, Lemma 01ZM we can find an \(i\) and a morphism \(f_i : X_i \to \Spec(A_i)\) of finite presentation whose base change to \(Y\) recovers \(f\). After possibly increasing \(i\) we may assume there exists a quasi-coherent \(\mathcal{O}_{X_i}\)-module \(\mathcal{F}_i\) of finite presentation whose pullback to \(X\) is isomorphic to \(\mathcal{F}\), see Limits, Lemma 01ZR. After possibly increasing \(i\) one more time we may assume there exists an open subscheme \(U_i \subset X_i\) whose inverse image in \(X\) is \(U\), see Limits, Lemma 01Z4. By Lemma 05KQ it suffices to prove the lemma for \(f_i\). Thus we reduce to the case where \(Y\) is the spectrum of a Noetherian ring.

We will use the criterion of Topology, Lemma 053Z to prove that \(E\) is constructible in case \(Y\) is a Noetherian scheme. To see this let \(Z \subset Y\) be an irreducible closed subscheme. We have to show that \(E \cap Z\) either contains a nonempty open subset or is not dense in \(Z\). This follows from Lemmas 05KN and 05KP applied to the base change \((X, \mathcal{F}, U) \times_Y Z\) over \(Z\).

Reduced fibres

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\) and \(f\) of finite type. If \(X_\eta\) is nonreduced, then there exists a nonempty open \(V \subset Y\) such that for all \(y \in V\) the fibre \(X_y\) is nonreduced.

Proof

Let \(Y' \subset Y\) be the reduction of \(Y\). Let \(X' \to Y'\) be the base change of \(f\). Note that \(Y' \to Y\) induces a bijection on points and that \(X' \to X\) identifies fibres. Hence we may assume that \(Y'\) is reduced, i.e., integral, see Properties, Lemma 01ON. We may also replace \(Y\) by an affine open. Hence we may assume that \(Y = \Spec(A)\) with \(A\) a domain. Denote \(K\) the fraction field of \(A\). Pick an affine open \(\Spec(B) = U \subset X\) and a section \(h_\eta \in \Gamma(U_\eta, \mathcal{O}_{U_\eta}) = B_K\) which is nonzero and nilpotent. After shrinking \(Y\) we may assume that \(h\) comes from \(h \in \Gamma(U, \mathcal{O}_U) = B\). After shrinking \(Y\) a bit more we may assume that \(h\) is nilpotent. Let \(I = \{b \in B \mid hb = 0\}\) be the annihilator of \(h\). Then \(C = B/I\) is a finite type \(A\)-algebra whose generic fiber \((B/I)_K\) is nonzero (as \(h_\eta \not = 0\)). We apply generic flatness to \(A \to C\) and \(A \to B/hB\), see Algebra, Lemma 051T, and we obtain a \(g \in A\), \(g \not = 0\) such that \(C_g\) is free as an \(A_g\)-module and \((B/hB)_g\) is flat as an \(A_g\)-module. Replace \(Y\) by \(D(g) \subset Y\). Now we have the short exact sequence \[0 \to C \to B \to B/hB \to 0.\] with \(B/hB\) flat over \(A\) and with \(C\) nonzero free as an \(A\)-module. It follows that for any homomorphism \(A \to \kappa\) to a field the ring \(C \otimes_A \kappa\) is nonzero and the sequence \[0 \to C \otimes_A \kappa \to B \otimes_A \kappa \to B/hB \otimes_A \kappa \to 0\] is exact, see Algebra, Lemma 00HL. Note that \(B/hB \otimes_A \kappa = (B \otimes_A \kappa) / h(B \otimes_A \kappa)\) by right exactness of tensor product. Thus we conclude that multiplication by \(h\) is not zero on \(B \otimes_A \kappa\). This clearly means that for any point \(y \in Y\) the element \(h\) restricts to a nonzero element of \(U_y\), whence \(X_y\) is nonreduced.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(g : Y' \to Y\) be any morphism, and denote \(f' : X' \to Y'\) the base change of \(f\). Then \[\begin{align*} \{y' \in Y' \mid X'_{y'}\text{ is geometrically reduced}\} \\ = g^{-1}(\{y \in Y \mid X_y\text{ is geometrically reduced}\}). \end{align*}\]

Proof

This comes down to the statement that for \(y' \in Y'\) with image \(y \in Y\) the fibre \(X'_{y'} = X_y \times_y y'\) is geometrically reduced over \(\kappa(y')\) if and only if \(X_y\) is geometrically reduced over \(\kappa(y)\). This follows from Varieties, Lemma 0384.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\) and \(f\) of finite type. If \(X_\eta\) is not geometrically reduced, then there exists a nonempty open \(V \subset Y\) such that for all \(y \in V\) the fibre \(X_y\) is not geometrically reduced.

Proof

Apply Lemma 0550 to get \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X_V \ar[r] \ar[d] & X \ar[d]^f \\ Y' \ar[r]^g & V \ar[r] & Y }\] with all the properties mentioned in that lemma. Let \(\eta'\) be the generic point of \(Y'\). Consider the morphism \(X' \to X_{Y'}\) (which is the reduction morphism) and the resulting morphism of generic fibres \(X'_{\eta'} \to X_{\eta'}\). Since \(X'_{\eta'}\) is geometrically reduced, and \(X_\eta\) is not this cannot be an isomorphism, see Varieties, Lemma 0384. Hence \(X_{\eta'}\) is nonreduced. Hence by Lemma 0575 the fibres of \(X_{Y'} \to Y'\) are nonreduced at all points \(y' \in V'\) of a nonempty open \(V' \subset Y'\). Since \(g : Y' \to V\) is a homeomorphism Lemma 0576 proves that \(g(V')\) is the open we are looking for.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume

\(Y\) is irreducible with generic point \(\eta\),

\(X_\eta\) is geometrically reduced, and

\(f\) is of finite type.

Then there exists a nonempty open subscheme \(V \subset Y\) such that \(X_V \to V\) has geometrically reduced fibres.

Proof

Let \(Y' \subset Y\) be the reduction of \(Y\). Let \(X' \to Y'\) be the base change of \(f\). Note that \(Y' \to Y\) induces a bijection on points and that \(X' \to X\) identifies fibres. Hence we may assume that \(Y'\) is reduced, i.e., integral, see Properties, Lemma 01ON. We may also replace \(Y\) by an affine open. Hence we may assume that \(Y = \Spec(A)\) with \(A\) a domain. Denote \(K\) the fraction field of \(A\). After shrinking \(Y\) a bit we may also assume that \(X \to Y\) is flat and of finite presentation, see Morphisms, Proposition 052A.

As \(X_\eta\) is geometrically reduced there exists an open dense subset \(V \subset X_\eta\) such that \(V \to \Spec(K)\) is smooth, see Varieties, Lemma 056V. Let \(U \subset X\) be the set of points where \(f\) is smooth. By Morphisms, Lemma 02V4 we see that \(V \subset U_\eta\). Thus the generic fibre of \(U\) is dense in the generic fibre of \(X\). Since \(X_\eta\) is reduced, it follows that \(U_\eta\) is scheme theoretically dense in \(X_\eta\), see Morphisms, Lemma 056D. We note that as \(U \to Y\) is smooth all the fibres of \(U \to Y\) are geometrically reduced. Thus it suffices to show that, after shrinking \(Y\), for all \(y \in Y\) the scheme \(U_y\) is scheme theoretically dense in \(X_y\), see Morphisms, Lemma 056E. This follows from Lemma 0573.

Lemma

Let \(f : X \to Y\) be a morphism which is quasi-compact and locally of finite presentation. Then the set \[E = \{y \in Y \mid X_y\text{ is geometrically reduced}\}\] is locally constructible in \(Y\).

Proof

Let \(y \in Y\). We have to show that there exists an open neighbourhood \(V\) of \(y\) in \(Y\) such that \(E \cap V\) is constructible in \(V\). Thus we may assume that \(Y\) is affine. Then \(X\) is quasi-compact. Choose a finite affine open covering \(X = U_1 \cup \ldots \cup U_n\). Then the fibres of \(U_i \to Y\) at \(y\) form an affine open covering of the fibre of \(X \to Y\) at \(y\). Hence we may assume \(X\) is affine as well. Write \(Y = \Spec(A)\). Write \(A = \colim A_i\) as a directed limit of finite type \(\mathbf{Z}\)-algebras. By Limits, Lemma 01ZM we can find an \(i\) and a morphism \(f_i : X_i \to \Spec(A_i)\) of finite presentation whose base change to \(Y\) recovers \(f\). By Lemma 0576 it suffices to prove the lemma for \(f_i\). Thus we reduce to the case where \(Y\) is the spectrum of a Noetherian ring.

We will use the criterion of Topology, Lemma 053Z to prove that \(E\) is constructible in case \(Y\) is a Noetherian scheme. To see this let \(Z \subset Y\) be an irreducible closed subscheme with generic point \(\xi\). We have to show that \(E \cap Z\) either contains a nonempty open subset or is not dense in \(Z\). If \(X_\xi\) is geometrically reduced, then Lemma 0578 (applied to the morphism \(X_Z \to Z\)) implies that all fibres \(X_y\) are geometrically reduced for a nonempty open \(V \subset Z\). If \(X_\xi\) is not geometrically reduced, then Lemma 0577 (applied to the morphism \(X_Z \to Z\)) implies that all fibres \(X_y\) are not geometrically reduced for a nonempty open \(V \subset Z\). Thus we win.

Lemma

Let \(f : X \to \Spec(R)\) be a proper morphism where \(R\) is a discrete valuation ring. Assume every irreducible component of \(X\) dominates \(\Spec(R)\) (for example if \(f\) is flat) and assume the special fibre is reduced. Then both \(X\) and the generic fibre \(X_\eta\) are reduced.

Proof

(The fact that a flat morphism satisfies the assumption on irreducible components follows for example from Divisors, Lemma 05DB or from going down for flat ring maps.) Assume the special fibre \(X_s\) is reduced. Let \(x \in X\) be any point, and let us show that \(\mathcal{O}_{X, x}\) is reduced; this will prove that \(X\) and \(X_\eta\) are reduced. Let \(x \leadsto x'\) be a specialization with \(x'\) in the special fibre; such a specialization exists as a proper morphism is closed. Consider the local ring \(A = \mathcal{O}_{X, x'}\). Then \(\mathcal{O}_{X, x}\) is a localization of \(A\), so it suffices to show that \(A\) is reduced. Let \(\pi \in R\) be a uniformizer. If \(a \in A\) is nonzero then there exists an \(n \geq 0\) and an element \(a' \in A\) such that \(a = \pi^n a'\) and \(a' \not \in \pi A\). This follows from Krull intersection theorem (Algebra, Lemma 00IP). If \(a\) is nilpotent the \(a\) is contained in all minimal primes of \(A\), and hence so is \(a'\), because \(\pi\) is not contained in any of the minimal primes of \(A\) by our assumption on the irreducible components of \(X\). Thus \(a'\) is nilpotent. But \(a'\) maps to a nonzero element of the reduced ring \(A/\pi A = \mathcal{O}_{X_s, x'}\). This is a contradiction unless \(A\) is reduced, which is what we wanted to show.

Lemma

Let \(f : X \to Y\) be a flat proper morphism of finite presentation. Then the set \(\{y \in Y \mid X_y\text{ is geometrically reduced}\}\) is open in \(Y\).

Proof

We may assume \(Y\) is affine. Then \(Y\) is a cofiltered limit of affine schemes of finite type over \(\mathbf{Z}\). Hence we can assume \(X \to Y\) is the base change of \(X_0 \to Y_0\) where \(Y_0\) is the spectrum of a finite type \(\mathbf{Z}\)-algebra and \(X_0 \to Y_0\) is flat and proper. See Limits, Lemma 01ZM, 04AI, and 081F. Since the formation of the set of points where the fibres are geometrically reduced commutes with base change (Lemma 0576), we may assume the base is Noetherian.

Assume \(Y\) is Noetherian. The set is constructible by Lemma 0579. Hence it suffices to show the set is stable under generalization (Topology, Lemma 0542). By Properties, Lemma 054F we reduce to the case where \(Y = \Spec(R)\), \(R\) is a discrete valuation ring, and the closed fibre \(X_y\) is geometrically reduced. To show: the generic fibre \(X_\eta\) is geometrically reduced.

If not then there exists a finite extension \(L\) of the fraction field of \(R\) such that \(X_L\) is not reduced, see Varieties, Lemma 035X. There exists a discrete valuation ring \(R' \subset L\) with fraction field \(L\) dominating \(R\), see Algebra, Lemma 09IG. After replacing \(R\) by \(R'\) we reduce to Lemma 0C0D.

Irreducible components of fibres

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\) and \(f\) of finite type. If \(X_\eta\) has \(n\) irreducible components, then there exists a nonempty open \(V \subset Y\) such that for all \(y \in V\) the fibre \(X_y\) has at least \(n\) irreducible components.

Proof

As the question is purely topological we may replace \(X\) and \(Y\) by their reductions. In particular this implies that \(Y\) is integral, see Properties, Lemma 01ON. Let \(X_\eta = X_{1, \eta} \cup \ldots \cup X_{n, \eta}\) be the decomposition of \(X_\eta\) into irreducible components. Let \(X_i \subset X\) be the reduced closed subscheme whose generic fibre is \(X_{i, \eta}\). Note that \(Z_{i, j} = X_i \cap X_j\) is a closed subset of \(X_i\) whose generic fibre \(Z_{i, j, \eta}\) is nowhere dense in \(X_{i, \eta}\). Hence after shrinking \(Y\) we may assume that \(Z_{i, j, y}\) is nowhere dense in \(X_{i, y}\) for every \(y \in Y\), see Lemma 054X. After shrinking \(Y\) some more we may assume that \(X_y = \bigcup X_{i, y}\) for \(y \in Y\), see Lemma 054Y. Moreover, after shrinking \(Y\) we may assume that each \(X_i \to Y\) is flat and of finite presentation, see Morphisms, Proposition 052A. The morphisms \(X_i \to Y\) are open, see Morphisms, Lemma 01UA. Thus there exists an open neighbourhood \(V\) of \(\eta\) which is contained in \(f(X_i)\) for each \(i\). For each \(y \in V\) the schemes \(X_{i, y}\) are nonempty closed subsets of \(X_y\), we have \(X_y = \bigcup X_{i, y}\) and the intersections \(Z_{i, j, y} = X_{i, y} \cap X_{j, y}\) are not dense in \(X_{i, y}\). Clearly this implies that \(X_y\) has at least \(n\) irreducible components.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(g : Y' \to Y\) be any morphism, and denote \(f' : X' \to Y'\) the base change of \(f\). Then \[\begin{align*} \{y' \in Y' \mid X'_{y'}\text{ is geometrically irreducible}\} \\ = g^{-1}(\{y \in Y \mid X_y\text{ is geometrically irreducible}\}). \end{align*}\]

Proof

This comes down to the statement that for \(y' \in Y'\) with image \(y \in Y\) the fibre \(X'_{y'} = X_y \times_y y'\) is geometrically irreducible over \(\kappa(y')\) if and only if \(X_y\) is geometrically irreducible over \(\kappa(y)\). This follows from Varieties, Lemma 054P.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \[n_{X/Y} : Y \to \{0, 1, 2, 3, \ldots, \infty\}\] be the function which associates to \(y \in Y\) the number of irreducible components of \((X_y)_K\) where \(K\) is a separably closed extension of \(\kappa(y)\). This is well defined and if \(g : Y' \to Y\) is a morphism then \[n_{X'/Y'} = n_{X/Y} \circ g\] where \(X' \to Y'\) is the base change of \(f\).

Proof

Suppose that \(y' \in Y'\) has image \(y \in Y\). Suppose \(K \supset \kappa(y)\) and \(K' \supset \kappa(y')\) are separably closed extensions. Then we may choose a commutative diagram \[\xymatrix{ K \ar[r] & K'' & K' \ar[l] \\ \kappa(y) \ar[u] \ar[rr] & & \kappa(y') \ar[u] }\] of fields. The result follows as the morphisms of schemes \[\xymatrix{ (X'_{y'})_{K'} & (X'_{y'})_{K''} = (X_y)_{K''} \ar[l] \ar[r] & (X_y)_K }\] induce bijections between irreducible components, see Varieties, Lemma 038H.

Lemma

Let \(A\) be a domain with fraction field \(K\). Let \(P \in A[x_1, \ldots, x_n]\). Denote \(\overline{K}\) the algebraic closure of \(K\). Assume \(P\) is irreducible in \(\overline{K}[x_1, \ldots, x_n]\). Then there exists a \(f \in A\) such that \(P^\varphi \in \kappa[x_1, \ldots, x_n]\) is irreducible for all homomorphisms \(\varphi : A_f \to \kappa\) into fields.

Proof

There exists an automorphism \(\Psi\) of \(A[x_1, \ldots, x_n]\) over \(A\) such that \(\Psi(P) = ax_n^d +\) lower order terms in \(x_n\) with \(a \not = 0\), see Algebra, Lemma 051N. We may replace \(P\) by \(\Psi(P)\) and we may replace \(A\) by \(A_a\). Thus we may assume that \(P\) is monic in \(x_n\) of degree \(d > 0\). For \(i = 1, \ldots, n - 1\) let \(d_i\) be the degree of \(P\) in \(x_i\). Note that this implies that \(P^\varphi\) is monic of degree \(d\) in \(x_n\) and has degree \(\leq d_i\) in \(x_i\) for every homomorphism \(\varphi : A \to \kappa\) where \(\kappa\) is a field. Thus if \(P^\varphi\) is reducible, then we can write \[P^\varphi = Q_1 Q_2\] with \(Q_1, Q_2\) monic of degree \(e_1, e_2 \geq 0\) in \(x_n\) with \(e_1 + e_2 = d\) and having degree \(\leq d_i\) in \(x_i\) for \(i = 1, \ldots, n - 1\). In other words we can write [0558]\[\begin{equation} Q_j = x_n^{e_j} + \sum\nolimits_{0 \leq l < e_j} \left( \sum\nolimits_{L \in \mathcal{L}} a_{j, l, L} x^L \right) x_n^l \end{equation}\] where the sum is over the set \(\mathcal{L}\) of multi-indices \(L\) of the form \(L = (l_1, \ldots, l_{n - 1})\) with \(0 \leq l_i \leq d_i\). For any \(e_1, e_2 \geq 0\) with \(e_1 + e_2 = d\) we consider the \(A\)-algebra \[B_{e_1, e_2} = A[\{a_{1, l, L}\}_{0 \leq l < e_1, L \in \mathcal{L}}, \{a_{2, l, L}\}_{0 \leq l < e_2, L \in \mathcal{L}}]/(\text{relations})\] where the \((\text{relations})\) is the ideal generated by the coefficients of the polynomial \[P - Q_1Q_2 \in A[\{a_{1, l, L}\}_{0 \leq l < e_1, L \in \mathcal{L}}, \{a_{2, l, L}\}_{0 \leq l < e_2, L \in \mathcal{L}}][x_1, \ldots, x_n]\] with \(Q_1\) and \(Q_2\) defined as in (0558). OK, and the assumption that \(P\) is irreducible over \(\overline{K}\) implies that there does not exist any \(A\)-algebra homomorphism \(B_{e_1, e_2} \to \overline{K}\). By the Hilbert Nullstellensatz, see Algebra, Theorem 00FV this means that \(B_{e_1, e_2} \otimes_A K = 0\). As \(B_{e_1, e_2}\) is a finitely generated \(A\)-algebra this signifies that we can find an \(f_{e_1, e_2} \in A\) such that \((B_{e_1, e_2})_{f_{e_1, e_2}} = 0\). By construction this means that if \(\varphi : A_{f_{e_1, e_2}} \to \kappa\) is a homomorphism to a field, then \(P^\varphi\) does not have a factorization \(P^\varphi = Q_1 Q_2\) with \(Q_1\) of degree \(e_1\) in \(x_n\) and \(Q_2\) of degree \(e_2\) in \(x_n\). Thus taking \(f = \prod_{e_1, e_2 \geq 0, e_1 + e_2 = d} f_{e_1, e_2}\) we win.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume

\(Y\) is irreducible with generic point \(\eta\),

\(X_\eta\) is geometrically irreducible, and

\(f\) is of finite type.

Then there exists a nonempty open subscheme \(V \subset Y\) such that \(X_V \to V\) has geometrically irreducible fibres.

Proof

We give two proofs of the lemma. These are essentially equivalent; the second is more self contained but a bit longer. Choose a diagram \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X_V \ar[r] \ar[d] & X \ar[d]^f \\ Y' \ar[r]^g & V \ar[r] & Y }\] as in Lemma 0550. Note that the generic fibre of \(f'\) is the reduction of the generic fibre of \(f\) (see Lemma 054Z) and hence is geometrically irreducible. Suppose that the lemma holds for the morphism \(f'\). Then after shrinking \(V\) all the fibres of \(f'\) are geometrically irreducible. As \(X' = (Y' \times_V X_V)_{red}\) this implies that all the fibres of \(Y' \times_V X_V\) are geometrically irreducible. Hence by Lemma 0555 all the fibres of \(X_V \to V\) are geometrically irreducible and we win. In this way we see that we may assume that the generic fibre is geometrically reduced as well as geometrically irreducible and we may assume \(Y = \Spec(A)\) with \(A\) a domain.

Let \(x \in X_\eta\) be the generic point. As \(X_\eta\) is geometrically irreducible and reduced we see that \(L = \kappa(x)\) is a finitely generated extension of \(K = \kappa(\eta)\) which is geometrically reduced and geometrically irreducible, see Varieties, Lemmas 035W and 054Q. In particular the field extension \(L/K\) is separable, see Algebra, Lemma 030W. Hence we can find \(x_1, \ldots, x_{r + 1} \in L\) which generate \(L\) over \(K\) and such that \(x_1, \ldots, x_r\) is a transcendence basis for \(L\) over \(K\), see Algebra, Lemma 030Q. Let \(P \in K(x_1, \ldots, x_r)[T]\) be the minimal polynomial for \(x_{r + 1}\). Clearing denominators we may assume that \(P\) has coefficients in \(A[x_1, \ldots, x_r]\). Note that as \(L\) is geometrically reduced and geometrically irreducible over \(K\), the polynomial \(P\) is irreducible in \(\overline{K}[x_1, \ldots, x_r, T]\) where \(\overline{K}\) is the algebraic closure of \(K\). Denote \[B' = A[x_1, \ldots, x_{r + 1}]/(P(x_{r + 1}))\] and set \(X' = \Spec(B')\). By construction the fraction field of \(B'\) is isomorphic to \(L = \kappa(x)\) as \(K\)-extensions. Hence there exists an open \(U \subset X\), and open \(U' \subset X'\) and a \(Y\)-isomorphism \(U \to U'\), see Morphisms, Lemma 0552. Here is a diagram: \[\xymatrix{ X \ar[rd] & U \ar[l] \ar@{=}[r] \ar[d] & U' \ar[r] \ar[d] & X' \ar[ld] \ar@{=}[r] & \Spec(B') \\ & Y \ar@{=}[r] & Y & }\] Note that \(U_\eta \subset X_\eta\) and \(U'_\eta \subset X'_\eta\) are dense opens. Thus after shrinking \(Y\) by applying Lemma 054X we obtain that \(U_y\) is dense in \(X_y\) and \(U'_y\) is dense in \(X'_y\) for all \(y \in Y\). Thus it suffices to prove the lemma for \(X' \to Y\) which is the content of Lemma 0557.

Proof

Let \(Y' \subset Y\) be the reduction of \(Y\). Let \(X' \to X\) be the reduction of \(X\). Note that \(X' \to X \to Y\) factors through \(Y'\), see Schemes, Lemma 0356. As \(Y' \to Y\) and \(X' \to X\) are universal homeomorphisms by Morphisms, Lemma 054M we see that it suffices to prove the lemma for \(X' \to Y'\). Thus we may assume that \(X\) and \(Y\) are reduced. In particular \(Y\) is integral, see Properties, Lemma 01ON. Thus by Morphisms, Proposition 052A there exists a nonempty affine open \(V \subset Y\) such that \(X_V \to V\) is flat and of finite presentation. After replacing \(Y\) by \(V\) we may assume, in addition to (1), (2), (3) that \(Y\) is integral affine, \(X\) is reduced, and \(f\) is flat and of finite presentation. In particular \(f\) is universally open, see Morphisms, Lemma 01UA.

Pick a nonempty affine open \(U \subset X\). Then \(U \to Y\) is flat and of finite presentation with geometrically irreducible generic fibre. The complement \(X_\eta \setminus U_\eta\) is nowhere dense. Thus after shrinking \(Y\) we may assume \(U_y \subset X_y\) is open dense for all \(y \in Y\), see Lemma 054X. Thus we may replace \(X\) by \(U\) and we reduce to the case where \(Y\) is integral affine and \(X\) is reduced affine, flat and of finite presentation over \(Y\) with geometrically irreducible generic fibre \(X_\eta\).

Write \(X = \Spec(B)\) and \(Y = \Spec(A)\). Then \(A\) is a domain, \(B\) is reduced, \(A \to B\) is flat of finite presentation, and \(B_K\) is geometrically irreducible over the fraction field \(K\) of \(A\). In particular we see that \(B_K\) is a domain. Let \(L\) be the fraction field of \(B_K\). Note that \(L\) is a finitely generated field extension of \(K\) as \(B\) is an \(A\)-algebra of finite presentation. Let \(K'/K\) be a finite purely inseparable extension such that \((L \otimes_K K')_{red}\) is a separably generated field extension, see Algebra, Lemma 030R. Choose \(x_1, \ldots, x_n \in K'\) which generate the field extension \(K'\) over \(K\), and such that \(x_i^{q_i} \in A\) for some prime power \(q_i\) (proof existence \(x_i\) omitted). Let \(A'\) be the \(A\)-subalgebra of \(K'\) generated by \(x_1, \ldots, x_n\). Then \(A'\) is a finite \(A\)-subalgebra \(A' \subset K'\) whose fraction field is \(K'\). Note that \(\Spec(A') \to \Spec(A)\) is a universal homeomorphism, see Algebra, Lemma 0BRA. Hence it suffices to prove the result after base changing to \(\Spec(A')\). We are going to replace \(A\) by \(A'\) and \(B\) by \((B \otimes_A A')_{red}\) to arrive at the situation where \(L\) is a separably generated field extension of \(K\). Of course it may happen that \((B \otimes_A A')_{red}\) is no longer flat, or of finite presentation over \(A'\), but this can be remedied by replacing \(A'\) by \(A'_f\) for a suitable \(f \in A'\), see Algebra, Lemma 051T.

At this point we know that \(A\) is a domain, \(B\) is reduced, \(A \to B\) is flat and of finite presentation, \(B_K\) is a domain whose fraction field \(L\) is a separably generated field extension of the fraction field \(K\) of \(A\). By Algebra, Lemma 030Q we may write \(L = K(x_1, \ldots, x_{r + 1})\) where \(x_1, \ldots, x_r\) are algebraically independent over \(K\), and \(x_{r + 1}\) is separable over \(K(x_1, \ldots, x_r)\). After clearing denominators we may assume that the minimal polynomial \(P \in K(x_1, \ldots, x_r)[T]\) of \(x_{r + 1}\) over \(K(x_1, \ldots, x_r)\) has coefficients in \(A[x_1, \ldots, x_r]\). Note that since \(L/K\) is separable and since \(L\) is geometrically irreducible over \(K\), the polynomial \(P\) is irreducible over the algebraic closure \(\overline{K}\) of \(K\). Denote \[B' = A[x_1, \ldots, x_{r + 1}]/(P(x_{r + 1})).\] By construction the fraction fields of \(B\) and \(B'\) are isomorphic as \(K\)-extensions. Hence there exists an isomorphism of \(A\)-algebras \(B_h \cong B'_{h'}\) for suitable \(h \in B\) and \(h' \in B'\), see Morphisms, Lemma 0552. In other words \(X\) and \(X' = \Spec(B')\) have a common affine open \(U\). Here is a diagram: \[\xymatrix{ X = \Spec(B) \ar[rd] & U \ar[l] \ar[r] \ar[d] & \Spec(B') = X' \ar[ld] \\ & Y = \Spec(A) & }\] After shrinking \(Y\) once more (by applying Lemma 054X to \(Z = X \setminus U\) in \(X\) and \(Z' = X' \setminus U\) in \(X'\)) we see that \(U_y\) is dense in \(X_y\) and \(U_y\) is dense in \(X'_y\) for all \(y \in Y\). Thus it suffices to prove the lemma for \(X' \to Y\) which is the content of Lemma 0557.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(n_{X/Y}\) be the function on \(Y\) counting the numbers of geometrically irreducible components of fibres of \(f\) introduced in Lemma 0556. Assume \(f\) of finite type. Let \(y \in Y\) be a point. Then there exists a nonempty open \(V \subset \overline{\{y\}}\) such that \(n_{X/Y}|_V\) is constant.

Proof

Let \(Z\) be the reduced induced scheme structure on \(\overline{\{y\}}\). Let \(f_Z : X_Z \to Z\) be the base change of \(f\). Clearly it suffices to prove the lemma for \(f_Z\) and the generic point of \(Z\). Hence we may assume that \(Y\) is an integral scheme, see Properties, Lemma 01ON. Our goal in this case is to produce a nonempty open \(V \subset Y\) such that \(n_{X/Y}|_V\) is constant.

We apply Lemma 0551 to \(f : X \to Y\) and we get \(g : Y' \to V \subset Y\). As \(g : Y' \to V\) is surjective finite étale, in particular open (see Morphisms, Lemma 03WT), it suffices to prove that there exists an open \(V' \subset Y'\) such that \(n_{X'/Y'}|_{V'}\) is constant, see Lemma 0556. Thus we see that we may assume that all irreducible components of the generic fibre \(X_\eta\) are geometrically irreducible over \(\kappa(\eta)\).

At this point suppose that \(X_\eta = X_{1, \eta} \bigcup \ldots \bigcup X_{n, \eta}\) is the decomposition of the generic fibre into (geometrically) irreducible components. In particular \(n_{X/Y}(\eta) = n\). Let \(X_i\) be the closure of \(X_{i, \eta}\) in \(X\). After shrinking \(Y\) we may assume that \(X = \bigcup X_i\), see Lemma 054Y. After shrinking \(Y\) some more we see that each fibre of \(f\) has at least \(n\) irreducible components, see Lemma 0554. Hence \(n_{X/Y}(y) \geq n\) for all \(y \in Y\). After shrinking \(Y\) some more we obtain that \(X_{i, y}\) is geometrically irreducible for each \(i\) and all \(y \in Y\), see Lemma 0559. Since \(X_y = \bigcup X_{i, y}\) this shows that \(n_{X/Y}(y) \leq n\) and finishes the proof.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(n_{X/Y}\) be the function on \(Y\) counting the numbers of geometrically irreducible components of fibres of \(f\) introduced in Lemma 0556. Assume \(f\) of finite presentation. Then the level sets \[E_n = \{y \in Y \mid n_{X/Y}(y) = n\}\] of \(n_{X/Y}\) are locally constructible in \(Y\).

Proof

Fix \(n\). Let \(y \in Y\). We have to show that there exists an open neighbourhood \(V\) of \(y\) in \(Y\) such that \(E_n \cap V\) is constructible in \(V\). Thus we may assume that \(Y\) is affine. Write \(Y = \Spec(A)\) and \(A = \colim A_i\) as a directed limit of finite type \(\mathbf{Z}\)-algebras. By Limits, Lemma 01ZM we can find an \(i\) and a morphism \(f_i : X_i \to \Spec(A_i)\) of finite presentation whose base change to \(Y\) recovers \(f\). By Lemma 0556 it suffices to prove the lemma for \(f_i\). Thus we reduce to the case where \(Y\) is the spectrum of a Noetherian ring.

We will use the criterion of Topology, Lemma 053Z to prove that \(E_n\) is constructible in case \(Y\) is a Noetherian scheme. To see this let \(Z \subset Y\) be an irreducible closed subscheme. We have to show that \(E_n \cap Z\) either contains a nonempty open subset or is not dense in \(Z\). Let \(\xi \in Z\) be the generic point. Then Lemma 055A shows that \(n_{X/Y}\) is constant in a neighbourhood of \(\xi\) in \(Z\). This clearly implies what we want.

Connected components of fibres

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\) and \(f\) of finite type. If \(X_\eta\) has \(n\) connected components, then there exists a nonempty open \(V \subset Y\) such that for all \(y \in V\) the fibre \(X_y\) has at least \(n\) connected components.

Proof

As the question is purely topological we may replace \(X\) and \(Y\) by their reductions. In particular this implies that \(Y\) is integral, see Properties, Lemma 01ON. Let \(X_\eta = X_{1, \eta} \cup \ldots \cup X_{n, \eta}\) be the decomposition of \(X_\eta\) into connected components. Let \(X_i \subset X\) be the reduced closed subscheme whose generic fibre is \(X_{i, \eta}\). Note that \(Z_{i, j} = X_i \cap X_j\) is a closed subset of \(X\) whose generic fibre \(Z_{i, j, \eta}\) is empty. Hence after shrinking \(Y\) we may assume that \(Z_{i, j} = \emptyset\), see Lemma 054W. After shrinking \(Y\) some more we may assume that \(X_y = \bigcup X_{i, y}\) for \(y \in Y\), see Lemma 054Y. Moreover, after shrinking \(Y\) we may assume that each \(X_i \to Y\) is flat and of finite presentation, see Morphisms, Proposition 052A. The morphisms \(X_i \to Y\) are open, see Morphisms, Lemma 01UA. Thus there exists an open neighbourhood \(V\) of \(\eta\) which is contained in \(f(X_i)\) for each \(i\). For each \(y \in V\) the schemes \(X_{i, y}\) are nonempty closed subsets of \(X_y\), we have \(X_y = \bigcup X_{i, y}\) and the intersections \(Z_{i, j, y} = X_{i, y} \cap X_{j, y}\) are empty! Clearly this implies that \(X_y\) has at least \(n\) connected components.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(g : Y' \to Y\) be any morphism, and denote \(f' : X' \to Y'\) the base change of \(f\). Then \[\begin{align*} \{y' \in Y' \mid X'_{y'}\text{ is geometrically connected}\} \\ = g^{-1}(\{y \in Y \mid X_y\text{ is geometrically connected}\}). \end{align*}\]

Proof

This comes down to the statement that for \(y' \in Y'\) with image \(y \in Y\) the fibre \(X'_{y'} = X_y \times_y y'\) is geometrically connected over \(\kappa(y')\) if and only if \(X_y\) is geometrically connected over \(\kappa(y)\). This follows from Varieties, Lemma 054N.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \[n_{X/Y} : Y \to \{0, 1, 2, 3, \ldots, \infty\}\] be the function which associates to \(y \in Y\) the number of connected components of \((X_y)_K\) where \(K\) is a separably closed extension of \(\kappa(y)\). This is well defined and if \(g : Y' \to Y\) is a morphism then \[n_{X'/Y'} = n_{X/Y} \circ g\] where \(X' \to Y'\) is the base change of \(f\).

Proof

Suppose that \(y' \in Y'\) has image \(y \in Y\). Suppose \(K \supset \kappa(y)\) and \(K' \supset \kappa(y')\) are separably closed extensions. Then we may choose a commutative diagram \[\xymatrix{ K \ar[r] & K'' & K' \ar[l] \\ \kappa(y) \ar[u] \ar[rr] & & \kappa(y') \ar[u] }\] of fields. The result follows as the morphisms of schemes \[\xymatrix{ (X'_{y'})_{K'} & (X'_{y'})_{K''} = (X_y)_{K''} \ar[l] \ar[r] & (X_y)_K }\] induce bijections between connected components, see Varieties, Lemma 0363.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume

\(Y\) is irreducible with generic point \(\eta\),

\(X_\eta\) is geometrically connected, and

\(f\) is of finite type.

Then there exists a nonempty open subscheme \(V \subset Y\) such that \(X_V \to V\) has geometrically connected fibres.

Proof

Choose a diagram \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X_V \ar[r] \ar[d] & X \ar[d]^f \\ Y' \ar[r]^g & V \ar[r] & Y }\] as in Lemma 0551. Note that the generic fibre of \(f'\) is geometrically connected (for example by Lemma 055F). Suppose that the lemma holds for the morphism \(f'\). This means that there exists a nonempty open \(W \subset Y'\) such that every fibre of \(X' \to Y'\) over \(W\) is geometrically connected. Then, as \(g\) is an open morphism by Morphisms, Lemma 03WT all the fibres of \(f\) at points of the nonempty open \(V = g(W)\) are geometrically connected, see Lemma 055F. In this way we see that we may assume that the irreducible components of the generic fibre \(X_\eta\) are geometrically irreducible.

Let \(Y'\) be the reduction of \(Y\), and set \(X' = Y' \times_Y X\). Then it suffices to prove the lemma for the morphism \(X' \to Y'\) (for example by Lemma 055F once again). Since the generic fibre of \(X' \to Y'\) is the same as the generic fibre of \(X \to Y\) we see that we may assume that \(Y\) is irreducible and reduced (i.e., integral, see Properties, Lemma 01ON) and that the irreducible components of the generic fibre \(X_\eta\) are geometrically irreducible.

At this point suppose that \(X_\eta = X_{1, \eta} \bigcup \ldots \bigcup X_{n, \eta}\) is the decomposition of the generic fibre into (geometrically) irreducible components. Let \(X_i\) be the closure of \(X_{i, \eta}\) in \(X\). After shrinking \(Y\) we may assume that \(X = \bigcup X_i\), see Lemma 054Y. Let \(Z_{i, j} = X_i \cap X_j\). Let \[\{1, \ldots, n\} \times \{1, \ldots, n\} = I \amalg J\] where \((i, j) \in I\) if \(Z_{i, j, \eta} = \emptyset\) and \((i, j) \in J\) if \(Z_{i, j, \eta} \not = \emptyset\). After shrinking \(Y\) we may assume that \(Z_{i, j} = \emptyset\) for all \((i, j) \in I\), see Lemma 054W. After shrinking \(Y\) we obtain that \(X_{i, y}\) is geometrically irreducible for each \(i\) and all \(y \in Y\), see Lemma 0559. After shrinking \(Y\) some more we achieve the situation where each \(Z_{i, j} \to Y\) is flat and of finite presentation for all \((i, j) \in J\), see Morphisms, Proposition 052A. This means that \(f(Z_{i, j}) \subset Y\) is open, see Morphisms, Lemma 01UA. We claim that \[V = \bigcap\nolimits_{(i, j) \in J} f(Z_{i, j})\] works, i.e., that \(X_y\) is geometrically connected for each \(y \in V\). Namely, the fact that \(X_\eta\) is connected implies that the equivalence relation generated by the pairs in \(J\) has only one equivalence class. Now if \(y \in V\) and \(K \supset \kappa(y)\) is a separably closed extension, then the irreducible components of \((X_y)_K\) are the fibres \((X_{i, y})_K\). Moreover, we see by construction and \(y \in V\) that \((X_{i, y})_K\) meets \((X_{j, y})_K\) if and only if \((i, j) \in J\). Hence the remark on equivalence classes shows that \((X_y)_K\) is connected and we win.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(n_{X/Y}\) be the function on \(Y\) counting the numbers of geometrically connected components of fibres of \(f\) introduced in Lemma 055F. Assume \(f\) of finite type. Let \(y \in Y\) be a point. Then there exists a nonempty open \(V \subset \overline{\{y\}}\) such that \(n_{X/Y}|_V\) is constant.

Proof

Let \(Z\) be the reduced induced scheme structure on \(\overline{\{y\}}\). Let \(f_Z : X_Z \to Z\) be the base change of \(f\). Clearly it suffices to prove the lemma for \(f_Z\) and the generic point of \(Z\). Hence we may assume that \(Y\) is an integral scheme, see Properties, Lemma 01ON. Our goal in this case is to produce a nonempty open \(V \subset Y\) such that \(n_{X/Y}|_V\) is constant.

We apply Lemma 0551 to \(f : X \to Y\) and we get \(g : Y' \to V \subset Y\). As \(g : Y' \to V\) is surjective finite étale, in particular open (see Morphisms, Lemma 03WT), it suffices to prove that there exists an open \(V' \subset Y'\) such that \(n_{X'/Y'}|_{V'}\) is constant, see Lemma 0556. Thus we see that we may assume that all irreducible components of the generic fibre \(X_\eta\) are geometrically irreducible over \(\kappa(\eta)\). By Varieties, Lemma 054S this implies that also the connected components of \(X_\eta\) are geometrically connected.

At this point suppose that \(X_\eta = X_{1, \eta} \bigcup \ldots \bigcup X_{n, \eta}\) is the decomposition of the generic fibre into (geometrically) connected components. In particular \(n_{X/Y}(\eta) = n\). Let \(X_i\) be the closure of \(X_{i, \eta}\) in \(X\). After shrinking \(Y\) we may assume that \(X = \bigcup X_i\), see Lemma 054Y. After shrinking \(Y\) some more we see that each fibre of \(f\) has at least \(n\) connected components, see Lemma 055D. Hence \(n_{X/Y}(y) \geq n\) for all \(y \in Y\). After shrinking \(Y\) some more we obtain that \(X_{i, y}\) is geometrically connected for each \(i\) and all \(y \in Y\), see Lemma 055G. Since \(X_y = \bigcup X_{i, y}\) this shows that \(n_{X/Y}(y) \leq n\) and finishes the proof.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(n_{X/Y}\) be the function on \(Y\) counting the numbers of geometric connected components of fibres of \(f\) introduced in Lemma 055F. Assume \(f\) of finite presentation. Then the level sets \[E_n = \{y \in Y \mid n_{X/Y}(y) = n\}\] of \(n_{X/Y}\) are locally constructible in \(Y\).

Proof

Fix \(n\). Let \(y \in Y\). We have to show that there exists an open neighbourhood \(V\) of \(y\) in \(Y\) such that \(E_n \cap V\) is constructible in \(V\). Thus we may assume that \(Y\) is affine. Write \(Y = \Spec(A)\) and \(A = \colim A_i\) as a directed limit of finite type \(\mathbf{Z}\)-algebras. By Limits, Lemma 01ZM we can find an \(i\) and a morphism \(f_i : X_i \to \Spec(A_i)\) of finite presentation whose base change to \(Y\) recovers \(f\). By Lemma 055F it suffices to prove the lemma for \(f_i\). Thus we reduce to the case where \(Y\) is the spectrum of a Noetherian ring.

We will use the criterion of Topology, Lemma 053Z to prove that \(E_n\) is constructible in case \(Y\) is a Noetherian scheme. To see this let \(Z \subset Y\) be an irreducible closed subscheme. We have to show that \(E_n \cap Z\) either contains a nonempty open subset or is not dense in \(Z\). Let \(\xi \in Z\) be the generic point. Then Lemma 055H shows that \(n_{X/Y}\) is constant in a neighbourhood of \(\xi\) in \(Z\). This clearly implies what we want.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume that

\(S\) is the spectrum of a discrete valuation ring,

\(f\) is flat,

\(X\) is connected,

the closed fibre \(X_s\) is reduced.

Then the generic fibre \(X_\eta\) is connected.

Proof

Write \(S = \Spec(R)\) and let \(\pi \in R\) be a uniformizer. To get a contradiction assume that \(X_\eta\) is disconnected. This means there exists a nontrivial idempotent \(e \in \Gamma(X_\eta, \mathcal{O}_{X_\eta})\). Let \(U = \Spec(A)\) be any affine open in \(X\). Note that \(\pi\) is a nonzerodivisor on \(A\) as \(A\) is flat over \(R\), see More on Algebra, Lemma 0538 for example. Then \(e|_{U_\eta}\) corresponds to an element \(e \in A[1/\pi]\). Let \(z \in A\) be an element such that \(e = z/\pi^n\) with \(n \geq 0\) minimal. Note that \(z^2 = \pi^nz\). This means that \(z \bmod \pi A\) is nilpotent if \(n > 0\). By assumption \(A/\pi A\) is reduced, and hence minimality of \(n\) implies \(n = 0\). Thus we conclude that \(e \in A\)! In other words \(e \in \Gamma(X, \mathcal{O}_X)\). As \(X\) is connected it follows that \(e\) is a trivial idempotent which is a contradiction.

Connected components meeting a section

The results in this section are in particular applicable to a group scheme \(G \to S\) and its neutral section \(e : S \to G\).

Situation

Here \(f : X \to Y\) be a morphism of schemes, and \(s : Y \to X\) is a section of \(f\). For every \(y \in Y\) we denote \(X^0_y\) the connected component of \(X_y\) containing \(s(y)\). Finally, we set \(X^0 = \bigcup_{y \in Y} X^0_y\).

Lemma

Let \(f : X \to Y\), \(s : Y \to X\) be as in Situation 055L. If \(g : Y' \to Y\) is any morphism, consider the base change diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]^{f'} & X \ar[d]_f \\ Y' \ar@/^1pc/[u]^{s'} \ar[r]^g & Y \ar@/_1pc/[u]_s }\] so that we obtain \((X')^0 \subset X'\). Then \((X')^0 = (g')^{-1}(X^0)\).

Proof

Let \(y' \in Y'\) with image \(y \in Y\). We may think of \(X^0_y\) as a closed subscheme of \(X_y\), see for example Morphisms, Definition 04PX. As \(s(y) \in X^0_y\) we conclude from Varieties, Lemma 04KV that \(X_y^0\) is a geometrically connected scheme over \(\kappa(y)\). Hence \(X_y^0 \times_y y' \to X'_{y'}\) is a connected closed subscheme which contains \(s'(y')\). Thus \(X_y^0 \times_y y' \subset (X'_{y'})^0\). The other inclusion \(X_y^0 \times_y y' \supset (X'_{y'})^0\) is clear as the image of \((X'_{y'})^0\) in \(X_y\) is a connected subset of \(X_y\) which contains \(s(y)\).

Lemma

Let \(f : X \to Y\), \(s : Y \to X\) be as in Situation 055L. Assume \(f\) of finite type. Let \(y \in Y\) be a point. Then there exists a nonempty open \(V \subset \overline{\{y\}}\) such that the inverse image of \(X^0\) in the base change \(X_V\) is open and closed in \(X_V\).

Proof

Let \(Z \subset Y\) be the induced reduced closed subscheme structure on \(\overline{\{y\}}\). Let \(f_Z : X_Z \to Z\) and \(s_Z : Z \to X_Z\) be the base changes of \(f\) and \(s\). By Lemma 055M we have \((X_Z)^0 = (X^0)_Z\). Hence it suffices to prove the lemma for the morphism \(X_Z \to Z\) and the point \(x \in X_Z\) which maps to the generic point of \(Z\). In other words we have reduced the problem to the case where \(Y\) is an integral scheme (see Properties, Lemma 01ON) with generic point \(\eta\). Our goal is to show that after shrinking \(Y\) the subset \(X^0\) becomes an open and closed subset of \(X\).

Note that the scheme \(X_\eta\) is of finite type over a field, hence Noetherian. Thus its connected components are open as well as closed. Hence we may write \(X_\eta = X_\eta^0 \amalg T_\eta\) for some open and closed subset \(T_\eta\) of \(X_\eta\). Next, let \(T \subset X\) be the closure of \(T_\eta\) and let \(X^{00} \subset X\) be the closure of \(X_\eta^0\). Note that \(T_\eta\), resp. \(X^0_\eta\) is the generic fibre of \(T\), resp. \(X^{00}\), see discussion preceding Lemma 054Y. Moreover, that lemma implies that after shrinking \(Y\) we may assume that \(X = X^{00} \cup T\) (set theoretically). Note that \((T \cap X^{00})_\eta = T_\eta \cap X^0_\eta = \emptyset\). Hence after shrinking \(Y\) we may assume that \(T \cap X^{00} = \emptyset\), see Lemma 054W. In particular \(X^{00}\) is open in \(X\). Note that \(X^0_\eta\) is connected and has a rational point, namely \(s(\eta)\), hence it is geometrically connected, see Varieties, Lemma 04KV. Thus after shrinking \(Y\) we may assume that all fibres of \(X^{00} \to Y\) are geometrically connected, see Lemma 055G. At this point it follows that the fibres \(X^{00}_y\) are open, closed, and connected subsets of \(X_y\) containing \(s(y)\). It follows that \(X^0 = X^{00}\) and we win.

Lemma

Let \(f : X \to Y\), \(s : Y \to X\) be as in Situation 055L. If \(f\) is of finite presentation then \(X^0\) is locally constructible in \(X\).

Proof

Let \(x \in X\). We have to show that there exists an open neighbourhood \(U\) of \(x\) such that \(X^0 \cap U\) is constructible in \(U\). This reduces us to the case where \(Y\) is affine. Write \(Y = \Spec(A)\) and \(A = \colim A_i\) as a directed limit of finite type \(\mathbf{Z}\)-algebras. By Limits, Lemma 01ZM we can find an \(i\) and a morphism \(f_i : X_i \to \Spec(A_i)\) of finite presentation, endowed with a section \(s_i : \Spec(A_i) \to X_i\) whose base change to \(Y\) recovers \(f\) and the section \(s\). By Lemma 055M it suffices to prove the lemma for \(f_i, s_i\). Thus we reduce to the case where \(Y\) is the spectrum of a Noetherian ring.

Assume \(Y\) is a Noetherian affine scheme. Since \(f\) is of finite presentation, i.e., of finite type, we see that \(X\) is a Noetherian scheme too, see Morphisms, Lemma 01T6. In order to prove the lemma in this case it suffices to show that for every irreducible closed subset \(Z \subset X\) the intersection \(Z \cap X^0\) either contains a nonempty open of \(Z\) or is not dense in \(Z\), see Topology, Lemma 053Z. Let \(x \in Z\) be the generic point, and let \(y = f(x)\). By Lemma 055N there exists a nonempty open subset \(V \subset \overline{\{y\}}\) such that \(X^0 \cap X_V\) is open and closed in \(X_V\). Since \(f(Z) \subset \overline{\{y\}}\) and \(f(x) = y \in V\) we see that \(W = f^{-1}(V) \cap Z\) is a nonempty open subset of \(Z\). It follows that \(X^0 \cap W\) is open and closed in \(W\). Since \(W\) is irreducible we see that \(X^0 \cap W\) is either empty or equal to \(W\). This proves the lemma.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(G \to S\) be a group scheme locally of finite type. Let \(\eta\) be a generic point of an irreducible component of \(S\). There exist an open neighbourhood \(U \subset S\) of \(\eta\) and an open subgroup scheme \(H \subset G_U\) such that

\(H \to U\) is of finite type and has geometrically connected fibres, and

for every \(u \in U\) the fibre \(H_u\) is the connected component \(G_u^0\) of the identity in \(G_u\).

Proof

After shrinking about \(\eta\), we may assume that \(S\) is affine and that \(\eta\) is its unique generic point. Put \[A = \mathcal{O}_{S, \eta}.\] This is an Artinian local ring. The closed fibre of \(G_A \to \Spec(A)\) has the same underlying topological space as \(G_A\). Its identity component is open and closed and is quasi-compact by Groupoids, Proposition 0B7R. Let \(H_A \subset G_A\) be the corresponding open and closed subscheme. The group operations on \(G_A\) restrict to \(H_A\): multiplication and inversion preserve the connected component containing the identity. Thus \(H_A\) is an open subgroup scheme. It is quasi-compact and locally of finite type over \(A\), and hence of finite type over \(A\).

The scheme \(\Spec(A)\) is the limit of the open neighbourhoods of \(\eta\) in \(S\). Standard limit arguments spread the quasi-compact open immersion \(H_A \to G_A\) to an open immersion \[H \longrightarrow G_U\] over some open neighbourhood \(U\) of \(\eta\); here \(H \to U\) is of finite type. We use Limits, Lemmas 01ZM, 0EUU, and 01ZP. After shrinking \(U\), the identity, inverse, and multiplication of \(G_U\) restrict to \(H\) and satisfy the group axioms, because these are equalities between morphisms of finite presentation which hold after base change to \(\Spec(A)\). Hence \(H\) is an open subgroup scheme of \(G_U\).

The generic fibre of \(H \to U\) is geometrically connected: it is connected and contains the rational identity point. After shrinking \(U\) once more, all fibres of \(H \to U\) are geometrically connected by Lemma 055G. For \(u \in U\), the open subgroup \(H_u \subset G_u\) contains \(G_u^0\). On the other hand, \(H_u\) is connected and contains the identity, so \(H_u \subset G_u^0\). This proves \(H_u = G_u^0\).

Lemma

Let \(S\) be a locally Noetherian scheme and let \(G \to S\) be a group scheme locally of finite type. Then \[G^0 = \bigcup_{s \in S} G_s^0\] is quasi-compact over \(S\). More precisely, every quasi-compact open of \(S\) has a finite stratification by locally closed subschemes \(S_i\) such that \((G_{S_i})^0\) is an open subgroup scheme of finite type over \(S_i\) with geometrically connected fibres.

Proof

The formation of \(G^0\) commutes with arbitrary base change by Lemma 055M. We may therefore work over a quasi-compact open of \(S\), which is Noetherian. Noetherian induction and Lemma more-morphisms-lemma-identity-component-generic-open-subgroup give a finite stratification \[S = \coprod_{i = 1}^r S_i\] by locally closed subschemes with the asserted open subgroups \(H_i\) over \(S_i\). Each \(S_i\) is quasi-compact and each \(H_i \to S_i\) is of finite type. Consequently every \(H_i\) is quasi-compact. Since \(G^0\) is the union of the finitely many locally closed subsets \(|H_i|\), it is quasi-compact.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a proper morphism and let \(e : S \to X\) be a section. Define \(X^0 \subset X\) as in Situation 055L. Then \(X^0\) is closed in \(X\). Endowed with the reduced induced closed subscheme structure, \(X^0\) is proper over \(S\).

Proof

The morphism \(f\) is of finite presentation. Hence \(X^0\) is locally constructible by Lemma 055P. Since \(X\) is locally Noetherian, it suffices to prove that \(X^0\) is stable under specialization, see Topology, Lemma 0542.

Let \(x \leadsto x'\) be a specialization in \(X\) with \(x \in X^0\). By Properties, Lemma 054F, there is a discrete valuation ring \(R\) and a morphism \[b : \Spec(R) \longrightarrow X\] which maps the generic point \(\eta\) to \(x\) and the closed point \(0\) to \(x'\). Think of \(b\) as a section of \(X_R \to \Spec(R)\), and denote by \(e_R\) the base change of \(e\). By Lemma 055M, the points \(b(\eta)\) and \(e_R(\eta)\) lie in the same connected component \(C_\eta\) of \((X_R)_\eta\).

Let \(C \subset X_R\) be the scheme theoretic closure of \(C_\eta\). It is proper and of finite presentation over \(R\). It is flat over \(R\): the definition of scheme theoretic closure makes its structure sheaf torsion free over \(R\), and a torsion free module over a valuation ring is flat by More on Algebra, Lemma 0539. The generic fibre \(C_\eta\) is geometrically connected because it is connected and contains the rational point \(e_R(\eta)\), see Varieties, Lemma 04KV. By Lemma 0BUI, the special fibre \(C_0\) has at most one geometric connected component. It is nonempty because \(C \to \Spec(R)\) is proper and has nonempty generic fibre. Thus \(C_0\) is connected. Both sections \(b\) and \(e_R\) factor through \(C\) because their generic points do. Hence \(b(0)\) and \(e_R(0)\) lie in the same connected component of \((X_R)_0\). It follows that \(x' \in X^0\), as desired.

Thus \(X^0\) is closed. Its reduced induced closed subscheme is proper as a closed subscheme of the proper \(S\)-scheme \(X\).

Lemma

Let \(f : X \to Y\), \(s : Y \to X\) be as in Situation 055L. Assume that \(Y\) is locally Noetherian and that \(f\) is locally of finite type. Let \(y \in Y\). Assume

\(X_y^0\) is geometrically reduced over \(\kappa(y)\), and

\(f\) is universally open along \(X_y^0\).

Then \(X^0\) is a neighbourhood of \(X_y^0\) in \(X\).

Proof

Base change by \(Y_{red} \to Y\). The resulting morphisms on the base and the source are universal homeomorphisms by Morphisms, Lemmas 054M and 0CEU. The formation of \(X^0\) commutes with this base change by Lemma 055M, and universal openness along the indicated locus is preserved by base change. We may therefore assume that \(Y\) is reduced.

It is enough to find, for every \(x \in X_y^0\), an open neighbourhood of \(x\) contained in \(X^0\). The locally Noetherian fibre is locally connected by Topology, Lemma 04MF. Consequently there is a connected quasi-compact open \[V \subset X_y^0\] containing both \(s(y)\) and \(x\). Indeed, the points which can be reached from \(s(y)\) by a finite chain of connected quasi-compact open subsets form an open and closed nonempty subset of the connected space \(X_y^0\).

Choose a quasi-compact open \(W \subset X\) such that \(W_y=V\). After shrinking \(Y\) about \(y\), we may assume that \(Y\) is affine and that the section factors through \(W\). For every \(v \in V\), the fibre \(X_y\) is geometrically reduced at \(v\), and every irreducible component of \(X_y\) through \(v\) is contained in \(X_y^0\). Hence Lemma more-morphisms-lemma-universally-open-geometrically-reduced-flat shows that \(W \to Y\) is flat at every point of \(V\).

The flat locus is open by More on Flatness, Lemma 05IM. Since \(V\) is quasi-compact, it has a quasi-compact open neighbourhood \(W'\) in that locus. Shrinking \(Y\) once more, the section factors through \(W'\). The morphism \(W' \to Y\) is flat and of finite presentation, the latter by Morphisms, Lemma 01TX. Its fibre over \(y\) is \(V\), which is connected and geometrically reduced. Applying Lemma 055Q to \(W' \to Y\) shows that \((W')^0\) is a neighbourhood of \(x\). Since every connected subset of a fibre of \(W'\) containing the section lies in the corresponding component of the fibre of \(X\), we have \((W')^0 \subset X^0\), and the proof is complete.

Lemma

Let \(f : X \to Y\), \(s : Y \to X\) be as in Situation 055L. Let \(y \in Y\) be a point. Assume

\(f\) is of finite presentation and flat, and

the fibre \(X_y\) is geometrically reduced.

Then \(X^0\) is a neighbourhood of \(X^0_y\) in \(X\).

Proof

We may replace \(Y\) with an affine open neighbourhood of \(y\). Write \(Y = \Spec(A)\) and \(A = \colim A_i\) as a directed limit of finite type \(\mathbf{Z}\)-algebras. By Limits, Lemma 01ZM we can find an \(i\) and a morphism \(f_i : X_i \to \Spec(A_i)\) of finite presentation, endowed with a section \(s_i : \Spec(A_i) \to X_i\) whose base change to \(Y\) recovers \(f\) and the section \(s\). After possibly increasing \(i\) we may also assume that \(f_i\) is flat, see Limits, Lemma 04AI. Let \(y_i\) be the image of \(y\) in \(Y_i\). Note that \(X_y = (X_{i, y_i}) \times_{y_i} y\). Hence \(X_{i, y_i}\) is geometrically reduced, see Varieties, Lemma 0384. By Lemma 055M it suffices to prove the lemma for the system \(f_i, s_i, y_i \in Y_i\). Thus we reduce to the case where \(Y\) is the spectrum of a Noetherian ring.

Assume \(Y\) is the spectrum of a Noetherian ring. Since \(f\) is of finite presentation, i.e., of finite type, we see that \(X\) is a Noetherian scheme too, see Morphisms, Lemma 01T6. Let \(x \in X^0\) be a point lying over \(y\). By Topology, Lemma 0540 it suffices to prove that for any irreducible closed \(Z \subset X\) passing through \(x\) the intersection \(X^0 \cap Z\) is dense in \(Z\). In particular it suffices to prove that the generic point \(x' \in Z\) is in \(X^0\). By Properties, Lemma 054F we can find a discrete valuation ring \(R\) and a morphism \(\Spec(R) \to X\) which maps the special point to \(x\) and the generic point to \(x'\). We are going to think of \(\Spec(R)\) as a scheme over \(Y\) via the composition \(\Spec(R) \to X \to Y\). By Lemma 055M we have that \((X_R)^0\) is the inverse image of \(X^0\). By construction we have a second section \(t : \Spec(R) \to X_R\) (besides the base change \(s_R\) of \(s\)) of the structure morphism \(X_R \to \Spec(R)\) such that \(t(\eta_R)\) is a point of \(X_R\) which maps to \(x'\) and \(t(0_R)\) is a point of \(X_R\) which maps to \(x\). Note that \(t(0_R)\) is in \((X_R)^0\) and that \(t(\eta_R) \leadsto t(0_R)\). Thus it suffices to prove that this implies that \(t(\eta_R) \in (X_R)^0\). Hence it suffices to prove the lemma in the case where \(Y\) is the spectrum of a discrete valuation ring and \(y\) its closed point.

Assume \(Y\) is the spectrum of a discrete valuation ring and \(y\) is its closed point. Our goal is to prove that \(X^0\) is a neighbourhood of \(X^0_y\). Note that \(X^0_y\) is open and closed in \(X_y\) as \(X_y\) has finitely many irreducible components. Hence the complement \(C = X_y \setminus X_y^0\) is closed in \(X\). Thus \(U = X \setminus C\) is an open neighbourhood of \(X^0_y\) and \(U^0 = X^0\). Hence it suffices to prove the result for the morphism \(U \to Y\). In other words, we may assume that \(X_y\) is connected. Suppose that \(X\) is disconnected, say \(X = X_1 \amalg \ldots \amalg X_n\) is a decomposition into connected components. Then \(s(Y)\) is completely contained in one of the \(X_i\). Say \(s(Y) \subset X_1\). Then \(X^0 \subset X_1\). Hence we may replace \(X\) by \(X_1\) and assume that \(X\) is connected. At this point Lemma 055J implies that \(X_\eta\) is connected, i.e., \(X^0 = X\) and we win.

Lemma

Let \(f : X \to Y\), \(s : Y \to X\) be as in Situation 055L. Assume

\(f\) is of finite presentation and flat, and

all fibres of \(f\) are geometrically reduced.

Then \(X^0\) is open in \(X\).

Proof

This is an immediate consequence of Lemma 055Q.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : G \to S\) be a commutative group scheme locally of finite type over \(S\). For \(s \in S\), denote \(G_s^0\) the connected component of the identity of \(G_s\). Then \[G^\tau = \{x \in G \mid [n](x) \in G_{f(x)}^0 \text{ for some }n \geq 1\}\] is an open subgroup scheme of \(G\). Formation of this open subscheme commutes with arbitrary base change between locally Noetherian schemes.

Proof

The assertion about base change follows from Lemma 055M. We first show that for every section \(a : S \to G\) the set \[T_a = \{s \in S \mid a(s) \in G_s^\tau\}\] is open. We may work over an affine open of \(S\) and hence assume that \(S\) is Noetherian.

We claim that \(T_a\) is stable under generalization. Let \(s'\) be a specialization of \(s\) and suppose that \(s' \in T_a\). The powers of \(a(s')\) lie in finitely many connected components of \(G_{s'}\). Each of these components is a translate of \(G_{s'}^0\) and is quasi-compact by Groupoids, Proposition 0B7R. Hence there is a quasi-compact open \(W \subset G\) which contains all the powers of \(a(s')\). Since \(a(s)\) is a generalization of \(a(s')\), all the powers of \(a(s)\) lie in \(W_s\). Connected components of the locally Noetherian scheme \(G_s\) are open, and the quasi-compact scheme \(W_s\) meets only finitely many of them. Two powers of \(a(s)\) therefore lie in the same connected component. Translation by the inverse of the smaller power shows that a positive power of \(a(s)\) lies in \(G_s^0\). Thus \(s \in T_a\), proving the claim.

Let \(Z \subset S\) be an irreducible closed subset with generic point \(\eta\). If \(T_a \cap Z\) is nonempty, then \(\eta \in T_a\) by the claim. Choose \(n \geq 1\) such that \([n](a(\eta)) \in G_\eta^0\). The subset \(G^0 = \bigcup G_s^0\) is locally constructible by Lemma 055P. Hence \(([n] \circ a)^{-1}(G^0) \cap Z\) is constructible and contains \(\eta\). It contains a nonempty open of \(Z\) by Topology, Lemma 005K. It follows from Topology, Lemma 053Z that \(T_a\) is constructible. Since it is stable under generalization, it is open by Topology, Lemma 0542.

Apply this result after base change by \(G \to S\) to the section of \(G \times_S G \to G\) which maps \(x\) to \((x, x)\). Its inverse image of the torsion-component locus is \(G^\tau\). Thus \(G^\tau\) is open. It is a subgroup fibrewise, and hence its open subscheme structure makes it an open subgroup scheme of \(G\).

Lemma

In the situation of Lemma more-morphisms-lemma-torsion-component-locus-open, define \[\nu : G^\tau \longrightarrow \{1, 2, 3, \ldots\}, \qquad \nu(x) = \min\{n \geq 1 \mid [n](x) \in G_{f(x)}^0\}.\] Every point of \(G^\tau\) has a neighbourhood on which \(\nu\) has finite image and all its level sets are constructible.

For \(s \in S\), let \(p(s)\) be the characteristic exponent of \(\kappa(s)\). With the convention that every positive integer is prime to \(1\) and that the only power of \(1\) is \(1\), set \[G^\sigma = \{x \in G^\tau \mid (\nu(x),p(f(x)))=1\}.\] The residue-characteristic-primary locus is \[G^\rho = \{x \in G^\tau \mid \nu(x)\text{ is a power of }p(f(x))\}.\] Then \(G^\sigma\) and \(G^\rho\) are locally constructible. If \(G^0 = \bigcup G_s^0\) is closed in \(G\), then \(G^\tau\) is open and closed and \(G^\rho\) is closed. If, in addition, \(p(s)\) is constant on \(S\), then \(G^\sigma\) is closed as well.

Proof

Let \(x \in G^\tau\). By Lemma more-morphisms-lemma-torsion-component-locus-open, we can choose a quasi-compact open neighbourhood \(U \subset G^\tau\) of \(x\). For \(n \geq 1\), set \[E_n = U \cap [n]^{-1}(G^0).\] These are constructible subsets of the Noetherian scheme \(U\) by Lemma 055P. They cover \(U\). Constructible subsets are open in the constructible topology, and this topology on \(U\) is quasi-compact by Topology, Lemma 0901. Thus finitely many \(E_{n_1}, \ldots, E_{n_r}\) cover \(U\). It follows that every value of \(\nu\) on \(U\) divides one of the \(n_i\). In particular \(\nu\) has finite image on \(U\). Moreover, for \(n \geq 1\) we have \[\{u \in U \mid \nu(u)=n\} = E_n \setminus \bigcup_{\substack{d \mid n \\ d < n}} E_d,\] which is constructible. This proves the first assertion.

There are only finitely many values of \(\nu\) on \(U\). On the level set where \(\nu=n\), the condition defining \(G^\sigma\) is the inverse image of \[S\setminus\bigcup_{q\mid n}V(q),\] where \(q\) runs through the prime divisors of \(n\). The condition defining \(G^\rho\) is automatic if \(n=1\), is the inverse image of \(V(q)\) if \(n=q^h\) is a power of a prime, and is empty otherwise. These descriptions prove the local constructibility of both loci without an equal-characteristic hypothesis.

Suppose \(G^0\) is closed. If \(x \in G^\tau\) specializes to \(x'\), then \([n](x) \in G^0\) for some \(n\), and hence \([n](x') \in G^0\). Thus \(G^\tau\) is stable under specialization. Since it is locally constructible, it is closed by Topology, Lemma 0542.

Suppose \(p(s)\) is constant, say equal to \(p\). If \(\nu(x)\) is prime to \(p\), the integer \(n\) can be chosen prime to \(p\). Hence \(G^\sigma\) is stable under specialization and its local constructibility proves that it is closed.

It remains to treat \(G^\rho\) without a restriction on the residue characteristics. Suppose \(x \leadsto x'\) is a specialization in \(G\) with \(x \in G^\rho\). If \(p(f(x))=1\), then \(\nu(x)=1\), so \(x\in G^0\) and closedness of \(G^0\) gives \(x'\in G^0\subset G^\rho\). If \(p(f(x))=p\) is prime, then the specialization \(f(x')\) also has characteristic exponent \(p\). For some \(h\geq 0\) we have \([p^h](x)\in G^0\), and hence \([p^h](x')\in G^0\). Thus \(x'\in G^\rho\). We conclude that \(G^\rho\) is stable under specialization; its local constructibility proves that it is closed.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(G \to S\) be a commutative group scheme locally of finite type. Assume there is a \(p\), equal to \(1\) or to a prime number, which is the characteristic exponent of every residue field of \(S\). If \[[n] : G \longrightarrow G\] is open for every \(n \geq 1\) prime to \(p\), then the \(p\)-primary component locus \[G^\rho = \{x \in G \mid [p^h](x) \in G_{f(x)}^0 \text{ for some }h \geq 0\}\] is open in \(G\).

Proof

The subset \(G^\rho\) is locally constructible by Lemma more-morphisms-lemma-order-in-component-group-constructible. It is therefore enough to prove that it is stable under generalization. Let \(x' \leadsto x\) be a generalization with \(x \in G^\rho\), and write \(y=f(x)\). Replacing \(x\) and \(x'\) by their images under a suitable \([p^h]\), it is enough to consider the case \(x \in G_y^0\).

Choose a quasi-compact open \(U \subset G^\tau\) which contains \(G_y^0\). This is possible because \(G_y^0\) is quasi-compact. By Lemma more-morphisms-lemma-order-in-component-group-constructible the function \(\nu\) has finite image on \(U\). Let \(m\) be a common multiple of the prime-to-\(p\) parts of all the integers \(\nu(z)\) for \(z \in U\).

Groupoids, Lemma groupoids-lemma-connected-group-power-surjective gives \([m](G_y^0)=G_y^0\). As \(G_y^0 \subset U\), it follows that \(x \in [m](U)\). The subset \([m](U)\) is open, and hence it contains the generalization \(x'\). Choose \(z \in U\) with \([m](z)=x'\). Multiplication by \(m\) kills the prime-to-\(p\) part of the order of the image of \(z\) in its fibrewise component group. Thus the image of \(x'\) has \(p\)-power order, so \(x' \in G^\rho\), as desired.

Example

Let \(k\) be a field of characteristic \(p>0\), let \(R=k[[t]]\), and put \(S=\Spec(R)\). The ideal \[(tx,x^p-x)\subset R[x]\] is a Hopf ideal for the additive group law. Hence \[G=\Spec\bigl(R[x]/(tx,x^p-x)\bigr)\] is a commutative finite group scheme over \(S\).

The generic fibre of \(G\) is the trivial group, while the special fibre is the constant group \(\mathbf Z/p\mathbf Z\). Thus \(G^0=e(S)\) and \(G^\rho=G\). Every geometric identity component is trivial and in particular contains no subgroup isomorphic to \(\mathbf G_a\). Moreover, for every \(n\) prime to \(p\), multiplication by \(n\) is an automorphism of \(G\).

Nevertheless, \(G\to S\) is not open. Indeed, \(D(x)\subset G\) is a nonempty open subset contained in the special fibre, and its image is the closed point of \(S\). Thus openness of the primary component locus in \(G\), even together with prime-to-characteristic power-map openness and the absence of additive identity components, does not imply that \(G^\rho\to S\) is open.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(G \to S\) be a commutative group scheme locally of finite type. Let \(n > 1\) and assume that \[[n] : G \longrightarrow G\] is universally open. For \(h \geq 0\), set \[G^{(n^h)} = [n^h]^{-1}(G^0)\] as a subset of \(G\), where \(G^0\) is the identity-component locus. Assume that, for every algebraically closed field \(k\) and every morphism \(\Spec(k) \to S\), the subset \[\bigcup_{h \geq 0} \left|\Ker([n^h] : G_k \to G_k)\right|\] is dense in \(\bigcup_{h \geq 0}G^{(n^h)}_k\). Then \(G \to S\) is universally open along \(\bigcup_{h \geq 0}G^{(n^h)}\).

Proof

The morphism \([n^h]\) is universally open because it is an iterate of \([n]\). Its kernel \(K_h \to S\) is the base change of \([n^h]\) by the identity section \(S \to G\), and hence is universally open. By Lemma more-morphisms-lemma-universally-open-along, the structure morphism \(G \to S\) is universally open along \(|K_h|\), and therefore along \(\bigcup_{h \geq 0}|K_h|\). The density hypothesis and Lemma more-morphisms-lemma-universally-open-along-fibrewise-dense give the result.

Remark

The density hypothesis in Lemma more-morphisms-lemma-power-kernel-density-universally-open cannot be omitted. For example, if \(k\) is algebraically closed of characteristic \(p > 0\) and \(G = \mathbf{G}_{m,k}\), then \([p]\) is finite locally free and hence universally open, and \(G\) has no subgroup isomorphic to \(\mathbf{G}_a\). Nevertheless, the underlying set of every \(\Ker([p^h])\) is just the identity whereas \(G^{(p^h)}=G\). Thus the unrestricted density assertion in the proof of the cited corollary is false when the residue characteristic divides \(n\). When \(n\) is invertible and the geometric identity components have no additive subgroup, the classical structure theorem for connected commutative algebraic groups supplies the required density of \(n\)-primary torsion. This prime-to-characteristic restriction is essential for that argument.

Example

Let \(k\) be a field, put \(S=\Spec(k[t])\), and, for every \(n\in\mathbf Z\), let \(S_n\) be a copy of \(S\). Glue all the \(S_n\) by identifying their open subschemes \(D(t)\) using the identity maps. Denote the resulting scheme by \(G\). The maps \[S_m\times_S S_n\longrightarrow S_{m+n}\] which are the identity on \(S\), together with the maps \(S_n\to S_{-n}\), glue to multiplication and inversion on \(G\). The copy \(S_0\) gives the neutral section. Thus \(G\to S\) is a group scheme locally isomorphic to \(S\to S\).

Every fibre away from the origin of \(S\) is the trivial group. The fibre at the origin is the constant, infinite discrete group \(\mathbf Z\). It follows that, as underlying subsets, \[G^0=G^\tau=S_0.\] This subset is open, but it is not closed. Indeed, the common open \(D(t)\) is dense in every chart \(S_n\), and hence \(S_0\) is dense in \(G\). In particular, without a separatedness hypothesis neither the identity-component locus nor the torsion-component locus has to be closed. This example does not decide whether either locus is always closed for separated group schemes. The cited source also realizes this group as the closure of the neutral section in an open subgroup of the Picard prescheme of a family of conics degenerating to two intersecting lines.

Example

Let \(V\) be a discrete valuation ring of characteristic \(p>0\), let \(t\) be a uniformizer, and set \(S=\Spec(V)\). The additive polynomial \[a\longmapsto a^p-ta\] defines an endomorphism of \(\mathbf G_{a,S}\). Its kernel is the finite locally free group scheme \[G=\Spec\bigl(V[x]/(x^p-tx)\bigr).\] The generic fibre is finite étale of degree \(p\) and becomes the constant group \(\mathbf Z/p\mathbf Z\) after a field extension. The special fibre is \(\Spec(\kappa[x]/(x^p))\) and is connected. Consequently the underlying identity-component and prime-to-\(p\) component loci are \[G^0=G^\sigma=e(S),\] where \(e\) is the neutral section. This closed subset is not open. For example, after adjoining an element \(u\) with \(u^{p-1}=t\), the equation factors as \[x^p-u^{p-1}x=\prod_{a\in\mathbf F_p}(x-au),\] and all \(p\) sections meet in the closed fibre. Thus even for a finite locally free separated group scheme the identity-component locus need not be open.

Example

Let \(V\) be a discrete valuation ring of mixed characteristic \((0,p)\), let \(S=\Spec(V)\), and consider \[H=\mu_{p,S}=\Spec\bigl(V[z]/(z^p-1)\bigr),\qquad C=(\mathbf Z/p\mathbf Z)_S,\] and \(G=H\times_S C\). This is a finite locally free separated group scheme. Its geometric generic fibre is the constant group \((\mathbf Z/p\mathbf Z)^2\). In the special fibre, \(H\) is infinitesimal and connected while \(C\) remains constant.

Define the component loci fibrewise, using the characteristic exponent of the residue field in each fibre. Thus \(G^\sigma\) is the inverse image of the torsion whose order is prime to the residue characteristic, while \(G^\rho\) is the inverse image of the primary part belonging to that characteristic. In characteristic zero the latter is the identity component. If \(\eta\) and \(s\) are the generic and closed points of \(S\), then, as underlying subsets, \[G^\rho=e(S)\cup G_s, \qquad G^\sigma=e(S)\cup G_\eta.\] The first subset is not open: a nonidentity component of the special fibre has generic generalizations which it does not contain. The second subset is not closed: the closure of the generic fibre contains every point of the special fibre, whereas \(G^\sigma\) contains only its identity component there. This shows why a fixed residue characteristic is essential in closedness statements for the two primary component loci.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(G \to S\) be a commutative group scheme locally of finite type. Assume there is a \(p\), equal to \(1\) or to a prime number, which is the characteristic exponent of every residue field of \(S\). Assume

the identity-component locus \(G^0\) is closed in \(G\), and

\(G \to S\) is universally open along \(G^\sigma\).

Then \(G^\sigma\), endowed with its reduced induced closed subscheme structure, is universally open over \(S\).

Proof

The formation of the underlying component loci commutes with locally Noetherian base change, and \(G^\sigma\) is closed by Lemma more-morphisms-lemma-order-in-component-group-constructible. In view of Lemma 0F31, it is enough to prove that \(G^\sigma \to S\) is open after a base change locally of finite presentation. Thus we may prove openness over \(S\) itself.

Since \(G^\sigma \to S\) is locally of finite presentation, it is enough to show that generalizations lift, see Morphisms, Lemma 01U1. The usual trait criterion and quasi-section construction for an open morphism (see [EGA, IV, Proposition 14.3.7]) reduce us, after a further extension of the trait, to the following situation. The scheme \(S\) is the spectrum of a discrete valuation ring, with closed point \(y\) and generic point \(y'\), and there is a section \(g : S \to G\) with \[g(y)=x \in G^\sigma_y.\] Because \(G^\tau\) is open, \(g(y')\) belongs to \(G^\tau_{y'}\). If \(p=1\), then \(G^\sigma_{y'}=G^\tau_{y'}\) and the section \(g\) gives the required lift.

Suppose \(p>1\). Write \(\nu(g(y'))=p^h m\) with \((m,p)=1\), and choose integers \(a,b\) such that \(ap^h+bm=1\). Put \[g_1=[ap^h]\circ g, \qquad g_2=[bm]\circ g.\] Then \(g=g_1g_2\), the point \(g_1(y')\) belongs to \(G^\sigma_{y'}\), and \(g_2(y')\) belongs to \(G^\rho_{y'}\). Closedness of \(G^\sigma\) and \(G^\rho\) gives \(g_1(y)\in G^\sigma_y\) and \(g_2(y)\in G^\rho_y\). Since \(g(y)\in G^\sigma_y\), we also have \(g_2(y)\in G^\sigma_y\), and hence \[g_2(y)\in G^\sigma_y\cap G^\rho_y=G^0_y.\]

The restriction of \(G \to S\) to the open subscheme \[G\setminus(G_y\setminus G^0_y)\] is an open morphism: at its closed-fibre points this follows from universal openness along \(G^\sigma\), and at its generic-fibre points it is automatic. The component quasi-section construction in the cited proof, followed by an extension of the trait, gives a section \(g'_2\) of the identity-component locus with \(g'_2(y)=g_2(y)\). Set \(g'=g_1g'_2\). Then \[g'(y)=g(y)=x \quad\text{and}\quad g'(y')=g_1(y')g'_2(y')\in G^\sigma_{y'}.\] Thus \(g'(y')\) is a generalization of \(x\) in \(G^\sigma\) above \(y'\). This proves that generalizations lift and hence proves the lemma.

Lemma

Let \(k\) be an algebraically closed field of characteristic exponent \(p\). Let \(G\) be a commutative group scheme locally of finite type over \(k\), and let \(n > 1\) be prime to \(p\). Assume that the identity component \(G^0\) contains no subgroup scheme isomorphic to \(\mathbf{G}_a\). For \(h \geq 0\), set \[G^{(n^h)}=[n^h]^{-1}(G^0).\] Then \[\bigcup_{h \geq 0}|\Ker([n^h])|\] is dense in \(\bigcup_{h \geq 0}G^{(n^h)}\).

Proof

The reduction \(H=(G^0)_{red}\) is a smooth connected subgroup scheme, see Groupoids, Lemmas 047R, 047P, and 047N. The classical structure theorem for connected commutative algebraic groups says that the absence of an additive subgroup makes \(H\) an extension of an abelian variety by a torus. The \(n\)-primary torsion is dense in such a group. Indeed, this is clear for a torus, it is the standard degree argument for an abelian variety, and it passes to an extension because multiplication by \(n^r\) is surjective on the torus. Thus \[\bigcup_{h \geq 0}|\Ker([n^h] : H \to H)|\] is dense in \(H\), and hence in \(G^0\).

Let \(C\) be a connected component occurring in \(\bigcup_{h \geq 0}G^{(n^h)}\). Since \(k\) is algebraically closed, choose \(g \in C(k)\). For some \(r\) we have \([n^r](g) \in G^0(k)\). Multiplication by \(n^r\) on \(G^0\) is surjective by Groupoids, Lemma groupoids-lemma-connected-group-power-surjective. Hence, after multiplying \(g\) by a point of \(G^0(k)\), we may assume that \([n^r](g)=e\). Translation by \(g\) now shows that the \(n\)-primary torsion is dense in \(C\). This proves the density assertion component by component.

Lemma

Let \(S\) be a scheme and let \(G \to S\) be a group scheme locally of finite type. Then the function \[S \longrightarrow \{0,1,2,\ldots\}, \qquad s \longmapsto \dim(G_s)\] is upper semicontinuous.

Proof

Let \(e:S\to G\) be the identity section. For every \(n\geq 0\), Morphisms, Lemma 02FZ shows that \[U_n=\{x\in G\mid \dim_x(G_{f(x)})\leq n\}\] is open in \(G\). For every \(s\in S\) we have \[\dim(G_s)=\dim_{e(s)}(G_s).\] Indeed, after extending the residue field to an algebraic closure, every irreducible component is a translate of a component which meets the identity component. Consequently \[\{s\in S\mid \dim(G_s)\leq n\}=e^{-1}(U_n)\] is open in \(S\), which proves the assertion.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(G \to S\) be a group scheme locally of finite type. If \(G \to S\) is universally open along the identity-component locus \(G^0\), then the function \[S \longrightarrow \{0,1,2,\ldots\}, \qquad s \longmapsto \dim(G_s)\] is locally constant.

Proof

The assertion is local on \(S\). After shrinking around a point of \(S\), choose a quasi-compact open \(U \subset G\) containing the identity section. The morphism \(U \to S\) is of finite presentation, and \(\dim(U_s)=\dim(G_s)\) for every \(s\): an open neighbourhood of the identity in a group over a field has the dimension of the group. Thus the level sets of the displayed function are locally constructible by Lemma 05F9.

It remains to show that the dimension is unchanged under specialization. By Properties, Lemma 054F, invariance of fibre dimension under field extension, and the universal hypothesis, we may base change to the spectrum of a discrete valuation ring. Write \(y\) and \(\eta\) for its closed and generic points, and put \(d=\dim(G_y)\). The open \[\{x \in G \mid \dim_x(G_{f(x)}) \leq d\}\] contains the identity of \(G_y\), see Morphisms, Lemma 02FZ. It therefore contains the identity of \(G_\eta\), and hence \(\dim(G_\eta) \leq d\).

For the reverse inequality, choose an irreducible component \(C\) of \(G^0_y\) of dimension \(d\), with generic point \(c\). Universal openness along \(G^0\) implies that \(c\) lies in the closure of the generic fibre: otherwise the complement of that closure would be an open neighbourhood of \(c\) whose image does not contain \(\eta\). Hence \(C\) is contained in an irreducible closed subscheme \(Z \subset G\) which dominates the trait and whose generic fibre is an irreducible component of \(G_\eta\). The scheme \(Z\) is integral and flat over the discrete valuation ring. The dimension formula, Algebra, Lemma 02IJ, gives \[\dim(C)=\dim(Z_\eta) \leq \dim(G_\eta).\] Consequently \(\dim(G_y)=\dim(G_\eta)\). The locally constructible level sets are therefore stable under specialization and generalization, so they are open and closed by Topology, Lemma 0542. This proves local constancy.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(G \to S\) be a commutative group scheme locally of finite type. Assume there is a \(p\), equal to \(1\) or to a prime number, which is the characteristic exponent of every residue field of \(S\). Let \(n > 1\) be prime to \(p\). Assume

\([n] : G \to G\) is universally open, and

for every algebraically closed field \(k\) over a point of \(S\), the identity component of \(G_k\) contains no subgroup isomorphic to \(\mathbf{G}_{a,k}\).

Then \(G \to S\) is universally open along \[\bigcup_{h \geq 0}G^{(n^h)}, \qquad G^{(n^h)}=[n^h]^{-1}(G^0),\] and the function \(s \mapsto \dim(G_s)\) is locally constant.

If condition (1) holds for every \(n > 1\) prime to \(p\), then \(G \to S\) is universally open along \(G^\sigma\). If, in addition, \(G^0\) is closed in \(G\), then \(G^\sigma\) with its reduced induced closed subscheme structure is universally open over \(S\).

Proof

Lemma more-morphisms-lemma-prime-to-characteristic-torsion-dense verifies the geometric fibrewise-density hypothesis of Lemma more-morphisms-lemma-power-kernel-density-universally-open. That lemma proves universal openness along the displayed union. Since \(G^0\) is contained in the union, local constancy of the fibre dimension follows from Lemma more-morphisms-lemma-identity-component-open-fibre-dimension.

If the hypothesis holds for every \(n\) prime to \(p\), then \[G^\sigma= \bigcup_{\substack{n>1 \\ (n,p)=1}}\ \bigcup_{h \geq 0}G^{(n^h)}.\] Thus \(G \to S\) is universally open along \(G^\sigma\). The final assertion is Lemma more-morphisms-lemma-prime-to-characteristic-locus-universally-open.

Dimension of fibres

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(Y\) irreducible with generic point \(\eta\) and \(f\) of finite type. If \(X_\eta\) has dimension \(n\), then there exists a nonempty open \(V \subset Y\) such that for all \(y \in V\) the fibre \(X_y\) has dimension \(n\).

Proof

Let \(Z = \{x \in X \mid \dim_x(X_{f(x)}) > n \}\). By Morphisms, Lemma 02FZ this is a closed subset of \(X\). By assumption \(Z_\eta = \emptyset\). Hence by Lemma 054W we may shrink \(Y\) and assume that \(Z = \emptyset\). Let \(Z' = \{x \in X \mid \dim_x(X_{f(x)}) > n - 1 \} = \{x \in X \mid \dim_x(X_{f(x)}) = n\}\). As before this is a closed subset of \(X\). By assumption we have \(Z'_\eta \not = \emptyset\). Hence after shrinking \(Y\) we may assume that \(Z' \to Y\) is surjective, see Lemma 05F5. Hence we win.

Lemma

Let \(f : X \to Y\) be a morphism of finite type. Let \[n_{X/Y} : Y \to \{0, 1, 2, 3, \ldots, \infty\}\] be the function which associates to \(y \in Y\) the dimension of \(X_y\). If \(g : Y' \to Y\) is a morphism then \[n_{X'/Y'} = n_{X/Y} \circ g\] where \(X' \to Y'\) is the base change of \(f\).

Proof

This follows from Morphisms, Lemma 02FY.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(n_{X/Y}\) be the function on \(Y\) giving the dimension of fibres of \(f\) introduced in Lemma 05F8. Assume \(f\) of finite presentation. Then the level sets \[E_n = \{y \in Y \mid n_{X/Y}(y) = n\}\] of \(n_{X/Y}\) are locally constructible in \(Y\).

Proof

Fix \(n\). Let \(y \in Y\). We have to show that there exists an open neighbourhood \(V\) of \(y\) in \(Y\) such that \(E_n \cap V\) is constructible in \(V\). Thus we may assume that \(Y\) is affine. Write \(Y = \Spec(A)\) and \(A = \colim A_i\) as a directed limit of finite type \(\mathbf{Z}\)-algebras. By Limits, Lemma 01ZM we can find an \(i\) and a morphism \(f_i : X_i \to \Spec(A_i)\) of finite presentation whose base change to \(Y\) recovers \(f\). By Lemma 05F8 it suffices to prove the lemma for \(f_i\). Thus we reduce to the case where \(Y\) is the spectrum of a Noetherian ring.

We will use the criterion of Topology, Lemma 053Z to prove that \(E_n\) is constructible in case \(Y\) is a Noetherian scheme. To see this let \(Z \subset Y\) be an irreducible closed subscheme. We have to show that \(E_n \cap Z\) either contains a nonempty open subset or is not dense in \(Z\). Let \(\xi \in Z\) be the generic point. Then Lemma 05F7 shows that \(n_{X/Y}\) is constant in a neighbourhood of \(\xi\) in \(Z\). This implies what we want.

Lemma

Let \(f : X \to Y\) be a flat morphism of schemes of finite presentation. Let \(n_{X/Y}\) be the function on \(Y\) giving the dimension of fibres of \(f\) introduced in Lemma 05F8. Then \(n_{X/Y}\) is lower semi-continuous.

Proof

Let \(W \subset X\), \(W = \coprod_{d \geq 0} U_d\) be the open constructed in Lemmas 045U and 054T. Let \(y \in Y\) be a point. If \(n_{X/Y}(y) = \dim(X_y) = n\), then \(y\) is in the image of \(U_n \to Y\). By Morphisms, Lemma 01UA we see that \(f(U_n)\) is open in \(Y\). Hence there is an open neighbourhood of \(y\) where \(n_{X/Y}\) is \(\geq n\).

Lemma

Let \(f : X \to Y\) be a proper morphism of schemes. Let \(n_{X/Y}\) be the function on \(Y\) giving the dimension of fibres of \(f\) introduced in Lemma 05F8. Then \(n_{X/Y}\) is upper semi-continuous.

Proof

Let \(Z_d = \{x \in X \mid \dim_x(X_{f(x)}) > d\}\). Then \(Z_d\) is a closed subset of \(X\) by Morphisms, Lemma 02FZ. Since \(f\) is proper \(f(Z_d)\) is closed. Since \(y \in f(Z_d) \Leftrightarrow n_{X/Y}(y) > d\) we see that the lemma is true.

Lemma

Let \(f : X \to Y\) be a proper, flat morphism of schemes of finite presentation. Let \(n_{X/Y}\) be the function on \(Y\) giving the dimension of fibres of \(f\) introduced in Lemma 05F8. Then \(n_{X/Y}\) is locally constant.

Proof

Immediate consequence of Lemmas 0D4H and 0D4I.

Weak relative Noether normalization

The goal of this section is to prove Lemma 0GTG.

Lemma

Let \(R\) be a ring. Let \(\mathfrak p_1, \ldots, \mathfrak p_r\) be prime ideals of \(R\) with \(\mathfrak p_i \not \subset \mathfrak p_j\) if \(i \not = j\). Let \(k_i \subset \kappa(\mathfrak p_i)\) be subfields such that the extensions \(\kappa(\mathfrak p_i)/k_i\) are not algebraic. Let \(J \subset R\) be an ideal not contained in any of the \(\mathfrak p_i\). Then there exists an element \(x \in J\) such that the image of \(x\) in \(\kappa(\mathfrak p_i)\) is transcendental over \(k_i\) for \(i = 1, \ldots, r\).

Proof

The ideal \(J_i = J \mathfrak p_1 \ldots \hat{\mathfrak p}_i \ldots \mathfrak p_r\) is not contained in \(\mathfrak p_i\), see Algebra, Lemma 07K1. It follows that every element \(\xi\) of \(\kappa(\mathfrak p_i) = \text{Frac}(B/\mathfrak p_i)\) is of the form \(\xi = a/b\) with \(a, b \in J_i\) and \(b \not \in \mathfrak p_i\). Choosing \(\xi\) transcendental over \(k_i\) we see that either \(a\) or \(b\) maps to an element of \(\kappa(\mathfrak p_i)\) transcendental over \(k_i\). We conclude that for every \(i = 1, \ldots, r\) we can find an element \(x_i \in J_i = J \mathfrak p_1 \ldots \hat{\mathfrak p}_i \ldots \mathfrak p_r\) which maps to an element of \(\kappa(\mathfrak p_i)\) transcendental over \(k_i\). Then \(x = x_1 + \ldots + x_r\) works.

Lemma

Let \(R \to S\) be a finite type ring map. Let \(d \geq 0\). Let \(a, b \in S\). Assume that the fibres of \[f_a : \Spec(S) \longrightarrow \mathbf{A}^1_R\] given by the \(R\)-algebra map \(R[x] \to S\) sending \(x\) to \(a\) have dimension \(\leq d\). Then there exists an \(n_0\) such that for \(n \geq n_0\) the fibres of \[f_{a^n + b} : \Spec(S) \longrightarrow \mathbf{A}^1_R\] given by the \(R\)-algebra map \(R[x] \to S\) sending \(x\) to \(a^n + b\) have dimension \(\leq d\).

Proof

In this paragraph we reduce to the case where \(R \to S\) is of finite presentation. Namely, write \(S = R[A, B, x_1, \ldots, x_n]/J\) for some ideal \(J \subset R[x_1, \ldots, x_n]\) where \(A\) and \(B\) map to \(a\) and \(b\) in \(S\). Then \(J\) is the union of its finitely generated ideals \(J_\lambda \subset J\). Set \(S_\lambda = R[A, B, x_1, \ldots, x_n]/J_\lambda\) and denote \(a_\lambda, b_\lambda \in S_\lambda\) the images of \(A\) and \(B\). Then for some \(\lambda\) the fibres of \[f_{a_\lambda} : \Spec(S_\lambda) \longrightarrow \mathbf{A}^1_R\] have dimension \(\leq d\), see Limits, Lemma 05M5. Fix such a \(\lambda\). If we can find \(n_0\) which works for \(R \to S_\lambda\), \(a_\lambda\), \(b_\lambda\), then \(n_0\) works for \(R \to S\). Namely, the fibres of \(f_{a_\lambda^n + b_\lambda} : \Spec(S_\lambda) \to \mathbf{A}^1_R\) contain the fibres of \(f_{a^n + b} : \Spec(S) \to \mathbf{A}^1_R\). This reduces us to the case discussed in the next paragraph.

Assume \(R \to S\) is of finite presentation. In this paragraph we reduce to the case where \(R\) is of finite type over \(\mathbf{Z}\). By Algebra, Lemma 00R1 we can find a directed set \(\Lambda\) and a system of ring maps \(R_\lambda \to S_\lambda\) over \(\Lambda\) whose colimit is \(R \to S\) such that \(S_\mu = S_\lambda \otimes_{R_\lambda} R_\mu\) for \(\mu \geq \lambda\) and such that each \(R_\lambda\) and \(S_\lambda\) is of finite type over \(\mathbf{Z}\). Choose \(\lambda_0 \in \Lambda\) and elements \(a_{\lambda_0}, b_{\lambda_0} \in S_{\lambda_0}\) mapping to \(a, b \in S\). For \(\lambda \geq \lambda_0\) denote \(a_\lambda, b_\lambda \in S_\lambda\) the image of \(a_{\lambda_0}, b_{\lambda_0}\). Then for \(\lambda \geq \lambda_0\) large enough the fibres of \[f_{a_\lambda} : \Spec(S_\lambda) \longrightarrow \mathbf{A}^1_{R_\lambda}\] have dimension \(\leq d\), see Limits, Lemma 0EY2. Fix such a \(\lambda\). If we can find \(n_0\) which works for \(R_\lambda \to S_\lambda\), \(a_\lambda\), \(b_\lambda\), then \(n_0\) works for \(R \to S\). Namely, any fibre of \(f_{a^n + b} : \Spec(S) \to \mathbf{A}^1_R\) has the same dimension as a fibre of \(f_{a_\lambda^n + b_\lambda} : \Spec(S_\lambda) \to \mathbf{A}^1_{R_\lambda}\) by Morphisms, Lemma 02FY. This reduces us the the case discussed in the next paragraph.

Assume \(R\) and \(S\) are of finite type over \(\mathbf{Z}\). In particular the dimension of \(R\) is finite, and we may use induction on \(\dim(R)\). Thus we may assume the result holds for all situations with \(R' \to S'\), \(a\), \(b\) as in the lemma with \(R'\) and \(S'\) of finite type over \(\mathbf{Z}\) but with \(\dim(R') < \dim(R)\).

Since the statement is about the topology of the spectrum of \(S\) we may assume \(S\) is reduced. Let \(S^\nu\) be the normalization of \(S\). Then \(S \subset S^\nu\) is a finite extension as \(S\) is excellent, see Algebra, Proposition 0335 and Morphisms, Lemma 035S. Thus \(\Spec(S^\nu) \to \Spec(S)\) is surjective and finite (Algebra, Lemma 00GQ). It follows that if the result holds for \(R \to S^\nu\) and the images of \(a\), \(b\) in \(S^\nu\), then the result holds for \(R \to S\), \(a\), \(b\). (Small detail omitted.) This reduces us to the case discussed in the next paragraph.

Assume \(R\) and \(S\) are of finite type over \(\mathbf{Z}\) and \(S\) normal. Then \(S = S_1 \times \ldots \times S_r\) for some normal domains \(S_i\). If the result holds for each \(R \to S_i\) and the images of \(a\), \(b\) in \(S_i\), then the result holds for \(R \to S\), \(a\), \(b\). (Small detail omitted.) This reduces us to the case discussed in the next paragraph.

Assume \(R\) and \(S\) are of finite type over \(\mathbf{Z}\) and \(S\) a normal domain. We may replace \(R\) by the image of \(R\) in \(S\) (this does not increase the dimension of \(R\)). This reduces us to the case discussed in the next paragraph.

Assume \(R \subset S\) are of finite type over \(\mathbf{Z}\) and \(S\) a normal domain. Consider the morphism \[f_a : \Spec(S) \to \mathbf{A}^1_R\] The assumption tells us that \(f_a\) has fibres of dimension \(\leq d\). Hence the fibres of \(f : \Spec(S) \to \Spec(R)\) have dimension \(\leq d + 1\) (Morphisms, Lemma 02JS). Consider the morphism of integral schemes \[\phi : \Spec(S) \to \mathbf{A}^2_R = \Spec(R[x, y])\] corresponding to the \(R\)-algebra map \(R[x, y] \to S\) sending \(x\) to \(a\) and \(y\) to \(b\). There are two cases to consider

\(\phi\) is dominant, and

\(\phi\) is not dominant.

We claim that in both cases there exists an integer \(n_0\) and a nonempty open \(V \subset \Spec(R)\) such that for \(n \geq n_0\) the fibres of \(f_{a^n + b}\) at points \(q \in \mathbf{A}^1_V\) have dimension \(\leq d\).

Proof of the claim in case (1). We have \(f_{a^n + b} = \pi_n \circ \phi\) where \[\pi_n : \mathbf{A}^2_R \to \mathbf{A}^1_R\] is the flat morphism corresponding to the \(R\)-algebra map \(R[x] \to R[x, y]\) sending \(x\) to \(x^n + y\). Since \(\phi\) is dominant there is a dense open \(U \subset \Spec(S)\) such that \(\phi|_U : U \to \mathbf{A}^2_R\) is flat (this follows for example from generic flatness, see Morphisms, Proposition 052A). Then the composition \[f_{a^n + b}|_U : U \xrightarrow{\phi|_U} \mathbf{A}^2_R \xrightarrow{\pi_n} \mathbf{A}^1_R\] is flat as well. Hence the fibres of this morphism have at least codimension \(1\) in the fibres of \(f|_U : U \to \Spec(R)\) by Morphisms, Lemma 02JS. In other words, the fibres of \(f_{a^n + b}|_U\) have dimension \(\leq d\). On the other hand, since \(U\) is dense in \(\Spec(S)\), we can find a nonempty open \(V \subset \Spec(R)\) such that \(U \cap f^{-1}(p) \subset f^{-1}(p)\) is dense for all \(p \in V\) (see for example Lemma 054X). Thus \(\dim(f^{-1}(p) \setminus U \cap f^{-1}(p)) \leq d\) and we conclude that our claim is true (as any fibres of \(f_{a^n + b} : \Spec(S) \to \mathbf{A}^1_R\) is contained in a fibre of \(f\)).

Case (2). In this case we can find a nonzero \(g = \sum c_{ij} x^i y^j\) in \(R[x, y]\) such that \(\Im(\phi) \subset V(g)\). In fact, we may assume \(g\) is irreducible over \(\text{Frac}(R)\). If \(g \in R[x]\), say with leading coefficient \(c\), then over \(V = D(c) \subset \Spec(R)\) the fibres of \(f\) already have dimension \(\leq d\) (because the image of \(f_a\) is contained in \(V(g) \subset \mathbf{A}^1_R\) which has finite fibres over \(V\)). Hence we may assume \(g\) is not contained in \(R[x]\). Let \(s \geq 1\) be the degree of \(g\) as a polynomial in \(y\) and let \(t\) be the degree of \(\sum c_{is} x^i\) as a polynomial in \(x\). Then \(c_{ts}\) is nonzero and \[g(x, -x^n) = (-1)^s c_{ts} x^{t + sn} + l.o.t.\] provided that \(n\) is bigger than the degree of \(g\) as a polynomial in \(x\) (small detail omitted). For such \(n\) the polynomial \(g(x, -x^n)\) is a nonzero polynomial in \(x\) and maps to a nonzero polynomial in \(\kappa(\mathfrak p)[x]\) for all \(\mathfrak p \subset R\), \(c_{st} \not \in \mathfrak p\). We conclude that our claim is true for \(V\) equal to the principal open \(D(c_{ts})\) of \(\Spec(R)\).

OK, and now we can use induction on \(\dim(R)\). Namely, let \(I \subset R\) be an ideal such that \(V(I) = \Spec(R) \setminus V\). Observe that \(\dim(R/I) < \dim(R)\) as \(R\) is a domain. Let \(n'_0\) be the integer we have by induction on \(\dim(R)\) for \(R/I \to S/IS\) and the images of \(a\) and \(b\) in \(S/IS\). Then \(\max(n_0, n'_0)\) works.

Lemma

Let \(R \to S\) be a finite type ring map. Let \(d\) be the maximum of the dimensions of fibres of \(\Spec(S) \to \Spec(R)\). Then there exists a quasi-finite ring map \(R[t_1, \ldots, t_d] \to S\).

Proof

In this paragraph we reduce to the case where \(R \to S\) is of finite presentation. Namely, write \(S = R[x_1, \ldots, x_n]/J\) for some ideal \(J \subset R[x_1, \ldots, x_n]\). Then \(J\) is the union of its finitely generated ideals \(J_\lambda \subset J\). Set \(S_\lambda = R[x_1, \ldots, x_n]/J_\lambda\). Then for some \(\lambda\) the fibres of \(\Spec(S_\lambda) \to \Spec(R)\) have dimension \(\leq d\), see Limits, Lemma 05M5. Fix such a \(\lambda\). If we can find a quasi-finite \(R[t_1, \ldots, t_d] \to S_\lambda\), then of course the composition \(R[t_1, \ldots, t_d] \to S\) is quasi-finite. This reduces us to the case discussed in the next paragraph.

Assume \(R \to S\) is of finite presentation. In this paragraph we reduce to the case where \(R\) is of finite type over \(\mathbf{Z}\). By Algebra, Lemma 00R1 we can find a directed set \(\Lambda\) and a system of ring maps \(R_\lambda \to S_\lambda\) over \(\Lambda\) whose colimit is \(R \to S\) such that \(S_\mu = S_\lambda \otimes_{R_\lambda} R_\mu\) for \(\mu \geq \lambda\) and such that each \(R_\lambda\) and \(S_\lambda\) is of finite type over \(\mathbf{Z}\). Then for \(\lambda\) large enough the fibres of \(\Spec(S_\lambda) \to \Spec(R_\lambda)\) have dimension \(\leq d\), see Limits, Lemma 0EY2. Fix such a \(\lambda\). If we can find a quasi-finite ring map \(R_\lambda[t_1, \ldots, t_d] \to S_\lambda\), then the base change \(R[t_1, \ldots, t_d] \to S\) is quasi-finite too (Algebra, Lemma 00PP). This reduces us the the case discussed in the next paragraph.

Assume \(R\) and \(S\) are of finite type over \(\mathbf{Z}\). If \(d = 0\), then the ring map is quasi-finite and we are done. Assume \(d > 0\). We will find an element \(a \in S\) such that the fibres of the \(R\)-algebra map \(R[x] \to S\), \(x \mapsto a\) have dimension \(< d\). This will finish the proof by induction.

We will prove the existence of \(a\) by induction on \(\dim(R)\).

Let \(\mathfrak q_1, \ldots, \mathfrak q_r \subset S\) be those among the minimal primes of \(S\) such that \(\dim_{\mathfrak q_i}(S/R) = d\). For notation, see Algebra, Definition 00QD. Say \(\mathfrak q_i\) lies over the prime \(\mathfrak p_i \subset R\). We have \(\text{trdeg}_{\kappa(\mathfrak p_i)}(\kappa(\mathfrak q_i)) = d\) as \(\mathfrak q_i\) is a generic point of its fibre; for example apply Algebra, Lemma 00P1 to \(S \otimes_R \kappa(\mathfrak p_i)\). Hence by Lemma 0GTE we can find an element \(a \in S\) such that the image of \(a\) in \(\kappa(\mathfrak q_i)\) is transcendental over \(\kappa(\mathfrak p_i)\) for \(i = 1, \ldots, r\). Consider the morphism \[f_a : \Spec(S) \longrightarrow \mathbf{A}^1_R\] corresponding the \(R\)-algebra homomorphism \(R[x] \to S\) to mapping \(x\) to \(a\). Let \(U \subset \Spec(S)\) be the open subset where the fibres have dimension \(\leq d - 1\), see Morphisms, Lemma 02FZ. By construction \(U\) contains all the generic points of \(\Spec(S)\). In particular we see that \(U\) contains all generic points of all the generic fibres of \(\Spec(S) \to \Spec(R)\) as such points are necessarily generic points of \(\Spec(S)\). Set \(T = \Spec(S) \setminus U\) viewed as a reduced closed subscheme of \(\Spec(S)\). It follows from what we just said and the assumption that \(\dim(S/R) \leq d\) that the generic fibres of \(T \to \Spec(R)\) have dimension \(\leq d - 1\). Hence by Lemma 05F7, applied several times to produce open neighbourhoods of the generic points of \(\Spec(R)\), we can find a dense open \(V \subset \Spec(R)\) such that \(T_V \to V\) has fibres of dimension \(\leq d - 1\). We conclude that for \(q \in \mathbf{A}^1_V\) the fibre of \(f_a\) over \(q\) has dimension \(\leq d - 1\) (as we have bounded the dimension of the fibre of \(f_a|_U\) and of the fibre of \(f_a|_T\)).

By prime avoidance, we may assume that \(V = D(f)\) for some \(f \in R\). Then we see that the ring map \(R_f[x] \to S_f\), \(x \mapsto a\) has fibres of dimension \(\leq d - 1\). We may replace \(a\) by \(fa\) and assume \(a \in (f)\). By induction on \(\dim(R)\) we can find an element \(\overline{b} \in S/fS\) such that the fibres of \(\Spec(S/fS) \to \Spec(R/fR[x])\), \(x \mapsto \overline{b}\) have dimension \(\leq d - 1\). Let \(b \in S\) be a lift of \(\overline{b}\). By Lemma 0GTF there exists an \(n > 0\) such that \(a^n + b\) still works for \(R_f \to S_f\). On the other hand, the image of \(a^n + b\) in \(S/fS\) is \(\overline{b}\) and the proof is complete.

Bertini theorems

We continue the discussion started in Varieties, Section 0FD4. In this section we prove that general hyperplane sections of geometrically irreducible varieties are geometrically irreducible following the remarkable argument given in [Jou].

Lemma

Let \(K/k\) be a geometrically irreducible and finitely generated field extension. Let \(n \geq 1\). Let \(g_1, \ldots, g_n \in K\) be elements such that there exist \(c_1, \ldots, c_n \in k\) such that the elements \[x_1, \ldots, x_n, \sum g_ix_i, \sum c_ig_i \in K(x_1, \ldots, x_n)\] are algebraically independent over \(k\). Then \(K(x_1, \ldots, x_n)\) is geometrically irreducible over \(k(x_1, \ldots, x_n, \sum g_ix_i)\).

Proof

Let \(c_1, \ldots, c_n \in k\) be as in the statement of the lemma. Write \(\xi = \sum g_ix_i\) and \(\delta = \sum c_ig_i\). For \(a \in k\) consider the automorphism \(\sigma_a\) of \(K(x_1, \ldots, x_n)\) given by the identity on \(K\) and the rules \[\sigma_a(x_i) = x_i + a c_i\] Observe that \(\sigma_a(\xi) = \xi + a \delta\) and \(\sigma_a(\delta) = \delta\). Consider the tower of fields \[K_0 = k(x_1, \ldots, x_n) \subset K_1 = K_0(\xi) \subset K_2 = K_0(\xi, \delta) \subset K(x_1, \ldots, x_n) = \Omega\] Observe that \(\sigma_a(K_0) = K_0\) and \(\sigma_a(K_2) = K_2\). Let \(\theta \in \Omega\) be separable algebraic over \(K_1\). We have to show \(\theta \in K_1\), see Algebra, Lemma 0G33.

Denote \(K'_2\) the separable algebraic closure of \(K_2\) in \(\Omega\). Since \(K'_2/K_2\) is finite (Algebra, Lemma 037Q) and separable there are only a finite number of fields in between \(K'_2\) and \(K_2\) (Fields, Lemma 030N). If \(k\) is infinite⁵, then we can find distinct elements \(a_1, a_2\) of \(k\) such that \[K_2(\sigma_{a_1}(\theta)) = K_2(\sigma_{a_2}(\theta))\] as subfields of \(\Omega\). Write \(\theta_i = \sigma_{a_i}(\theta)\) and \(\xi_i = \sigma_{a_i}(\xi) = \xi + a_i \delta\). Observe that \[K_2 = K_0(\xi_1, \xi_2)\] as we have \(\xi_i = \xi + a_i \delta\), \(\xi = (a_2 \xi_1 - a_1 \xi_2)/(a_2 - a_1)\), and \(\delta = (\xi_1 - \xi_2)/(a_1 - a_2)\). Since \(K_2/K_0\) is purely transcendental of degree \(2\) we conclude that \(\xi_1\) and \(\xi_2\) are algebraically independent over \(K_0\). Since \(\theta_1\) is algebraic over \(K_0(\xi_1)\) we conclude that \(\xi_2\) is transcendental over \(K_0(\xi_1, \theta_1)\).

By assumption \(K/k\) is geometrically irreducible. This implies that \(K(x_1, \ldots, x_n)/K_0\) is geometrically irreducible (Algebra, Lemma 0G31). This in turn implies that \(K_0(\xi_1, \theta_1)/K_0\) is geometrically irreducible as a subextension (Algebra, Lemma 037N). Since \(\xi_2\) is transcendental over \(K_0(\xi_1, \theta_1)\) we conclude that \(K_0(\xi_1, \xi_2, \theta_1)/K_0(\xi_2)\) is geometrically irreducible (Algebra, Lemma 0G32). By our choice of \(a_1, a_2\) above we have \[K_0(\xi_1, \xi_2, \theta_1) = K_2(\sigma_{a_1}(\theta)) = K_2(\sigma_{a_2}(\theta)) = K_0(\xi_1, \xi_2, \theta_2)\] Since \(\theta_2\) is separably algebraic over \(K_0(\xi_2)\) we conclude by Algebra, Lemma 0G33 again that \(\theta_2 \in K_0(\xi_2)\). Taking \(\sigma_{a_2}^{-1}\) of this relation givens \(\theta \in K_0(\xi) = K_1\) as desired.

This finishes the proof in case \(k\) is infinite. If \(k\) is finite, then we can choose a variable \(t\) and consider the extension \(K(t)/k(t)\) which is geometrically irreducible by Algebra, Lemma 0G31. Since it is still be true that \(x_1, \ldots, x_n, \sum g_ix_i, \sum c_ig_i\) in \(K(t, x_1, \ldots, x_n)\) are algebraically independent over \(k(t)\) we conclude that \(K(t, x_1, \ldots, x_n)\) is geometrically irreducible over \(k(t, x_1, \ldots, x_n, \sum g_ix_i)\) by the argument already given. Then using Algebra, Lemma 0G31 once more finishes the job.

Lemma

Let \(A\) be a domain of finite type over a field \(k\). Let \(n \geq 2\). Let \(g_1, \ldots, g_n \in A\) be elements such that \(V(g_1, g_2)\) has an irreducible component of dimension \(\dim(A) - 2\). Then there exist \(c_1, \ldots, c_n \in k\) such that the elements \[x_1, \ldots, x_n, \sum g_ix_i, \sum c_ig_i \in \text{Frac}(A)(x_1, \ldots, x_n)\] are algebraically independent over \(k\).

Proof

The algebraic independence over \(k\) means that the morphism \[T = \Spec(A[x_1, \ldots, x_n]) \longrightarrow \Spec(k[x_1, \ldots, x_n, y, z]) = S\] given by \(y = \sum g_ix_i\) and \(z = \sum c_ig_i\) is dominant. Set \(d = \dim(A)\). If \(T \to S\) is not dominant, then the image has dimension \(< n + 2\) and hence every irreducible component of every fibre has dimension \(> d + n - (n + 2) = d - 2\), see Varieties, Lemma 0B2L. Choose a closed point \(u \in V(g_1, g_2)\) contained in an irreducible component of dimension \(d - 2\) and in no other component of \(V(g_1, g_2)\). Consider the closed point \(t = (u, 1, 0, \ldots 0)\) of \(T\) lying over \(u\). Set \((c_1, \ldots, c_n) = (0, 1, 0, \ldots, 0)\). Then \(t\) maps to the point \(s = (1, 0, \ldots, 0)\) of \(S\). The fibre of \(T \to S\) over \(s\) is cut out by \[x_1 - 1, x_2, \ldots, x_n, \sum x_ig_i, g_2\] and hence equivalently is cut out by \[x_1 - 1, x_2, \ldots, x_n, g_1, g_2\] By our condition on \(g_1, g_2\) this subscheme has an irreducible component of dimension \(d - 2\).

Lemma

In Varieties, Situation 0G47 assume

\(X\) is of finite type over \(k\),

\(X\) is geometrically irreducible over \(k\),

there exist \(v_1, v_2, v_3 \in V\) and an irreducible component \(Z\) of \(H_{v_2} \cap H_{v_3}\) such that \(Z \not \subset H_{v_1}\) and \(\text{codim}(Z, X) = 2\), and

every irreducible component \(Y\) of \(\bigcap_{v \in V} H_v\) has \(\text{codim}(Y, X) \geq 2\).

Then for general \(v \in V \otimes_k k'\) the scheme \(H_v\) is geometrically irreducible over \(k'\).

Proof

In order for assumption (3) to hold, the elements \(v_1, v_2, v_3\) must be \(k\)-linearly independent in \(V\) (small detail omitted). Thus we may choose a basis \(v_1, \ldots, v_r\) of \(V\) incorporating these elements as the first \(3\). Recall that \(H_{univ} \subset \mathbf{A}^r_k \times_k X\) is the “universal divisor”. Consider the projection \(q : H_{univ} \to \mathbf{A}^r_k\) whose scheme theoretic fibres are the divisors \(H_v\). By Lemma 0559 it suffices to show that the generic fibre of \(q\) is geometrically irreducible. To prove this we may replace \(X\) by its reduction, hence we may assume \(X\) is an integral scheme of finite type over \(k\).

Let \(U \subset X\) be a nonempty affine open such that \(\mathcal{L}|_U \cong \mathcal{O}_U\). Write \(U = \Spec(A)\). Denote \(f_i \in A\) the element corresponding to section \(\psi(v_i)|_U\) via the isomorphism \(\mathcal{L}|_U \cong \mathcal{O}_U\). Then \(H_{univ} \cap (\mathbf{A}^r_k \times_k U)\) is given by \[H_U = \Spec(A[x_1, \ldots, x_r]/(x_1f_1 + \ldots + x_rf_r))\] By our choice of basis we see that \(f_1\) cannot be zero because this would mean \(v_1 = 0\) and hence \(H_{v_1} = X\) which contradicts assumption (3). Hence \(\sum x_if_i\) is a nonzerodivisor in \(A[x_1, \ldots, x_r]\). It follows that every irreducible component of \(H_U\) has dimension \(d + r - 1\) where \(d = \dim(X) = \dim(A)\). If \(U' = U \cap D(f_1)\) then we see that \[H_{U'} = \Spec(A_{f_1}[x_1, \ldots, x_r]/(x_1f_1 + \ldots + x_rf_r)) \cong \Spec(A_{f_1}[x_2, \ldots x_r]) = \mathbf{A}^{r - 1}_k \times_k U'\] is irreducible. On the other hand, we have \[H_U \setminus H_{U'} = \Spec(A/(f_1)[x_1, \ldots, x_r]/(x_2f_2 + \ldots + x_rf_r))\] which has dimension at most \(d + r - 2\). Namely, for \(i \not = 1\) the scheme \((H_U \setminus H_{U'}) \times_U D(f_i)\) is either empty (if \(f_i = 0\)) or by the same argument as above isomorphic to an \(r - 1\) dimensional affine space over an open of \(\Spec(A/(f_1))\) and hence has dimension at most \(d + r - 2\). On the other hand, \((H_U \setminus H_{U'}) \times_U V(f_2, \ldots, f_r)\) is an \(r\) dimensional affine space over \(\Spec(A/(f_1, \ldots, f_r))\) and hence assumption (4) tells us this has dimension at most \(d + r - 2\). We conclude that \(H_U\) is irreducible for every \(U\) as above. It follows that \(H_{univ}\) is irreducible.

Thus it suffices to show that the generic point of \(H_{univ}\) is geometrically irreducible over the generic point of \(\mathbf{A}^r_k\), see Varieties, Lemma 054Q. Choose a nonempty affine open \(U = \Spec(A)\) of \(X\) contained in \(X \setminus H_{v_1}\) which meets the irreducible component \(Z\) of \(H_{v_2} \cap H_{v_3}\) whose existence is asserted in assumption (3). With notation as above we have to prove that the field extension \[\text{Frac}(A[x_1, \ldots, x_r]/(x_1f_1 + \ldots + x_rf_r))/ k(x_1, \ldots , x_r)\] is geometrically irreducible. Observe that \(f_1\) is invertible in \(A\) by our choice of \(U\). Set \(K = \text{Frac}(A)\) equal to the fraction field of \(A\). Eliminating the variable \(x_1\) as above, we find that we have to show that the field extension \[K(x_2, \ldots, x_r)/ k(x_2, \ldots, x_r, -\sum\nolimits_{i = 2, \ldots, r} f_1^{-1}f_i x_i)\] is geometrically irreducible. By Lemma 0G4D it suffices to show that for some \(c_2, \ldots, c_r \in k\) the elements \[x_2, \ldots, x_r, \sum\nolimits_{i = 2, \ldots, r} f_1^{-1}f_i x_i, \sum\nolimits_{i = 2, \ldots, r} c_if_1^{-1}f_i\] are algebraically independent over \(k\) in the fraction field of \(A[x_2, \ldots, x_r]\). This follows from Lemma 0G4E and the fact that \(Z \cap U\) is an irreducible component of \(V(f_1^{-1}f_2, f_1^{-1}f_3) \subset U\).

Remark

Let us sketch a “geometric” proof of a special case of Lemma 0G4F. Namely, say \(k\) is an algebraically closed field and \(X \subset \mathbf{P}^n_k\) is smooth and irreducible of dimension \(\geq 2\). Then we claim there is a hyperplane \(H \subset \mathbf{P}^n_k\) such that \(X \cap H\) is smooth and irreducible. Namely, by Varieties, Lemma 0FD6 for a general \(v \in V = kT_0 \oplus \ldots \oplus kT_n\) the corresponding hyperplane section \(X \cap H_v\) is smooth. On the other hand, by Enriques-Severi-Zariski the scheme \(X \cap H_v\) is connected, see Varieties, Lemma 0FD9. Hence \(X \cap H_v\) is smooth and irreducible.

Theorem of the cube

The following lemma tells us that the diagonal of the Picard functor is representable by locally closed immersions under the assumptions made in the lemma.

Lemma

Let \(f : X \to S\) be a flat, proper morphism of finite presentation. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module. For a morphism \(g : T \to S\) consider the base change diagram \[\xymatrix{ X_T \ar[d]_p \ar[r]_q & X \ar[d]^f \\ T \ar[r]^g & S }\] Assume \(\mathcal{O}_T \to p_*\mathcal{O}_{X_T}\) is an isomorphism for all \(g : T \to S\). Then there exists an immersion \(j : Z \to S\) of finite presentation such that a morphism \(g : T \to S\) factors through \(Z\) if and only if there exists a finite locally free \(\mathcal{O}_T\)-module \(\mathcal{N}\) with \(p^*\mathcal{N} \cong q^*\mathcal{E}\).

Proof

Observe that the fibres \(X_s\) of \(f\) are connected by our assumption that \(H^0(X_s, \mathcal{O}_{X_s}) = \kappa(s)\). Thus the rank of \(\mathcal{E}\) is constant on the fibres. Since \(f\) is open (Morphisms, Lemma 01UA) and closed we conclude that there is a decomposition \(S = \coprod S_r\) of \(S\) into open and closed subschemes such that \(\mathcal{E}\) has constant rank \(r\) on the inverse image of \(S_r\). Thus we may assume \(\mathcal{E}\) has constant rank \(r\). We will denote \(\mathcal{E}^\vee = \SheafHom(\mathcal{E}, \mathcal{O}_X)\) the dual rank \(r\) module.

By cohomology and base change (more precisely by Derived Categories of Schemes, Lemma 0B91) we see that \(E = Rf_*\mathcal{E}\) is a perfect object of the derived category of \(S\) and that its formation commutes with arbitrary change of base. Similarly for \(E' = Rf_*\mathcal{E}^\vee\). Since there is never any cohomology in degrees \(< 0\), we see that \(E\) and \(E'\) have (locally) tor-amplitude in \([0, b]\) for some \(b\). Observe that for any \(g : T \to S\) we have \(p_*(q^*\mathcal{E}) = H^0(Lg^*E)\) and \(p_*(q^*\mathcal{E}^\vee) = H^0(Lg^*E')\). Let \(j : Z \to S\) and \(j' : Z' \to S\) be immersions of finite presentation constructed in Derived Categories of Schemes, Lemma 0BDL for \(E\) and \(E'\) with \(a = 0\) and \(r = r\); these are roughly speaking characterized by the property that \(H^0(Lj^*E)\) and \(H^0((j')^*E')\) are finite locally free modules compatible with pullback.

Let \(g : T \to S\) be a morphism. If there exists an \(\mathcal{N}\) as in the lemma, then, using the projection formula Cohomology, Lemma 01E8, we see that the modules \[p_*(q^*\mathcal{E}) \cong p_*(p^*\mathcal{N}) \cong \mathcal{N} \otimes_{\mathcal{O}_T} p_*\mathcal{O}_{X_T} \cong \mathcal{N}\quad\text{and similarly }\quad p_*(q^*\mathcal{E}^\vee) \cong \mathcal{N}^\vee\] are finite locally free modules of rank \(r\) and remain so after any further base change \(T' \to T\). Hence in this case \(T \to S\) factors through \(j\) and through \(j'\). Thus we may replace \(S\) by \(Z \times_S Z'\) and assume that \(f_*\mathcal{E}\) and \(f_*\mathcal{E}^\vee\) are finite locally free \(\mathcal{O}_S\)-modules of rank \(r\) whose formation commutes with arbitrary change of base (small detail omitted).

In this situation if \(g : T \to S\) be a morphism and there exists an \(\mathcal{N}\) as in the lemma, then the map (cup product in degree \(0\)) \[p_*(q^*\mathcal{E}) \otimes_{\mathcal{O}_T} p_*(q^*\mathcal{E}^\vee) \longrightarrow \mathcal{O}_T\] is a perfect pairing. Conversely, if this cup product map is a perfect pairing, then we see that locally on \(T\) we may choose a basis of sections \(\sigma_1, \ldots, \sigma_r\) in \(p_*(q^*\mathcal{E})\) and \(\tau_1, \ldots, \tau_r\) in \(p_*(q^*\mathcal{E}^\vee)\) whose products satisfy \(\sigma_i \tau_j = \delta_{ij}\). Thinking of \(\sigma_i\) as a section of \(q^*\mathcal{E}\) on \(X_T\) and \(\tau_j\) as a section of \(q^*\mathcal{E}^\vee\) on \(X_T\), we conclude that \[\sigma_1, \ldots, \sigma_r : \mathcal{O}_{X_T}^{\oplus r} \longrightarrow q^*\mathcal{E}\] is an isomorphism with inverse given by \[\tau_1, \ldots, \tau_r : q^*\mathcal{E} \longrightarrow \mathcal{O}_{X_T}^{\oplus r}\] In other words, we see that \(p^*p_*q^*\mathcal{E} \cong q^*\mathcal{E}\). But the condition that the cup product is nondegenerate picks out a retrocompact open subscheme (namely, the locus where a suitable determinant is nonzero) and the proof is complete.

The lemma above in particular tells us, that if a vector bundle is trivial on fibres for a proper flat family of proper spaces, then it is the pull back of a vector bundle. Let’s spell this out a bit.

Lemma

Let \(f : X \to S\) be a flat, proper morphism of finite presentation such that \(f_*\mathcal{O}_X = \mathcal{O}_S\) and this remains true after arbitrary base change. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module. Assume

\(\mathcal{E}|_{X_s}\) is isomorphic to \(\mathcal{O}_{X_s}^{\oplus r_s}\) for all \(s \in S\), and

\(S\) is reduced.

Then \(\mathcal{E} = f^*\mathcal{N}\) for some finite locally free \(\mathcal{O}_S\)-module \(\mathcal{N}\).

Proof

Namely, in this case the locally closed immersion \(j : Z \to S\) of Lemma 0BDP is bijective and hence a closed immersion. But since \(S\) is reduced, \(j\) is an isomorphism.

Lemma

Let \(f : X \to S\) be a proper flat morphism of finite presentation. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Assume

\(S\) is the spectrum of a valuation ring,

\(\mathcal{L}\) is trivial on the generic fibre \(X_\eta\) of \(f\),

the closed fibre \(X_0\) of \(f\) is integral,

\(H^0(X_\eta, \mathcal{O}_{X_\eta})\) is equal to the function field of \(S\).

Then \(\mathcal{L}\) is trivial.

Proof

Write \(S = \Spec(A)\). We will first prove the lemma when \(A\) is a discrete valuation ring (as this is the case most often used in practice). Let \(\pi \in A\) be a uniformizer. Take a trivializing section \(s \in \Gamma(X_\eta, \mathcal{L}_\eta)\). After replacing \(s\) by \(\pi^n s\) if necessary we can assume that \(s \in \Gamma(X, \mathcal{L})\). If \(s|_{X_0} = 0\), then we see that \(s\) is divisible by \(\pi\) (because \(X_0\) is the scheme theoretic fibre and \(X\) is flat over \(A\)). Thus we may assume that \(s|_{X_0}\) is nonzero. Then the zero locus \(Z(s)\) of \(s\) is contained in \(X_0\) but does not contain the generic point of \(X_0\) (because \(X_0\) is integral). This means that the \(Z(s)\) has codimension \(\geq 2\) in \(X\) which contradicts Divisors, Lemma 0BCN unless \(Z(s) = \emptyset\) as desired.

Proof in the general case. Since the valuation ring \(A\) is coherent (Algebra, Example 0EWV) we see that \(H^0(X, \mathcal{L})\) is a coherent \(A\)-module, see Derived Categories of Schemes, Lemma 0EX6. Equivalently, \(H^0(X, \mathcal{L})\) is a finitely presented \(A\)-module (Algebra, Lemma 05CX). Since \(H^0(X, \mathcal{L})\) is torsion free (by flatness of \(X\) over \(A\)), we see from More on Algebra, Lemma 0ASP that \(H^0(X, \mathcal{L}) = A^{\oplus n}\) for some \(n\). By flat base change (Cohomology of Schemes, Lemma 02KH) we have \[K = H^0(X_\eta, \mathcal{O}_{X_\eta}) \cong H^0(X_\eta, \mathcal{L}_\eta) = H^0(X, \mathcal{L}) \otimes_A K\] where \(K\) is the fraction field of \(A\). Thus \(n = 1\). Pick a generator \(s \in H^0(X, \mathcal{L})\). Let \(\mathfrak m \subset A\) be the maximal ideal. Then \(\kappa = A/\mathfrak m = \colim A/\pi\) where this is a filtered colimit over nonzero \(\pi \in \mathfrak m\) (here we use that \(A\) is a valuation ring). Thus \(X_0 = \lim X \times_S \Spec(A/\pi)\). If \(s|_{X_0}\) is zero, then for some \(\pi\) we see that \(s\) restricts to zero on \(X \times_S \Spec(A/\pi)\), see Limits, Lemma 01Z0. But if this happens, then \(\pi^{-1} s\) is a global section of \(\mathcal{L}\) which contradicts the fact that \(s\) is a generator of \(H^0(X, \mathcal{L})\). Thus \(s|_{X_0}\) is not zero. Let \(Z(s) \subset X\) be the zero scheme of \(s\). Since \(s|_{X_0}\) is not zero and since \(X_0\) is integral, we see that \(Z(s)_0 \subset X_0\) is an effective Cartier divisor. Since \(f\) is proper and \(S\) is local, every point of \(Z(s)\) specializes to a point of \(Z(s)_0\). Thus by Divisors, Lemma 062Y part (3) we see that \(Z(s)\) is a relative effective Cartier divisor, in particular \(Z(s) \to S\) is flat. Hence if \(Z(s)\) were nonemtpy, then \(Z(s)_\eta\) would be nonempty which contradicts the fact that \(s|_{X_\eta}\) is a trivialization of \(\mathcal{L}_\eta\). Thus \(Z(s) = \emptyset\) as desired.

Lemma

Let \(f : X \to S\) and \(\mathcal{E}\) be as in Lemma 0BDP and in addition assume \(\mathcal{E}\) is an invertible \(\mathcal{O}_X\)-module. If moreover the geometric fibres of \(f\) are integral, then \(Z\) is closed in \(S\).

Proof

Since \(j : Z \to S\) is of finite presentation, it suffices to show: for any morphism \(g : \Spec(A) \to S\) where \(A\) is a valuation ring with fraction field \(K\) such that \(g(\Spec(K)) \in j(Z)\) we have \(g(\Spec(A)) \subset j(Z)\). See Morphisms, Lemma 02JQ. This follows from Lemma 0EX8 and the characterization of \(j : Z \to S\) in Lemma 0BDP.

Lemma

Consider a commutative diagram of schemes \[\xymatrix{ X' \ar[rr] \ar[dr]_{f'} & & X \ar[dl]^f \\ & S }\] with \(f' : X' \to S\) and \(f : X \to S\) satisfying the hypotheses of Lemma 0BDP. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module and let \(\mathcal{L}'\) be the pullback to \(X'\). Let \(Z \subset S\), resp. \(Z' \subset S\) be the locally closed subscheme constructed in Lemma 0BDP for \((f, \mathcal{L})\), resp. \((f', \mathcal{L}')\) so that \(Z \subset Z'\). If \(s \in Z\) and \[H^1(X_s, \mathcal{O}) \longrightarrow H^1(X'_s, \mathcal{O})\] is injective, then \(Z \cap U = Z' \cap U\) for some open neighbourhood \(U\) of \(s\).

Proof

We may replace \(S\) by \(Z'\). After shrinking \(S\) to an affine open neighbourhood of \(s\) we may assume that \(\mathcal{L}' = \mathcal{O}_{X'}\). Let \(E = Rf_*\mathcal{L}\) and \(E' = Rf'_*\mathcal{L}' = Rf'_*\mathcal{O}_{X'}\). These are perfect complexes whose formation commutes with arbitrary change of base (Derived Categories of Schemes, Lemma 0B91). In particular we see that \[E \otimes_{\mathcal{O}_S}^\mathbf{L} \kappa(s) = R\Gamma(X_s, \mathcal{L}_s) = R\Gamma(X_s, \mathcal{O}_{X_s})\] The second equality because \(s \in Z\). Set \(h_i = \dim_{\kappa(s)} H^i(X_s, \mathcal{O}_{X_s})\). After shrinking \(S\) we can represent \(E\) by a complex \[\mathcal{O}_S \to \mathcal{O}_S^{\oplus h_1} \to \mathcal{O}_S^{\oplus h_2} \to \ldots\] see More on Algebra, Lemma 0BCD (strictly speaking this also uses Derived Categories of Schemes, Lemmas 06Z0 and 08EB). Similarly, we may assume \(E'\) is represented by a complex \[\mathcal{O}_S \to \mathcal{O}_S^{\oplus h'_1} \to \mathcal{O}_S^{\oplus h'_2} \to \ldots\] where \(h'_i = \dim_{\kappa(s)} H^i(X'_s, \mathcal{O}_{X'_s})\). By functoriality of cohomology we have a map \[E \longrightarrow E'\] in \(D(\mathcal{O}_S)\) whose formation commutes with change of base. Since the complex representing \(E\) is a finite complex of finite free modules and since \(S\) is affine, we can choose a map of complexes \[\xymatrix{ \mathcal{O}_S \ar[r]_d \ar[d]_a & \mathcal{O}_S^{\oplus h_1} \ar[r] \ar[d]_b & \mathcal{O}_S^{\oplus h_2} \ar[r] \ar[d]_c & \ldots \\ \mathcal{O}_S \ar[r]^{d'} & \mathcal{O}_S^{\oplus h'_1} \ar[r] & \mathcal{O}_S^{\oplus h'_2} \ar[r] & \ldots }\] representing the given map \(E \to E'\). Since \(s \in Z\) we see that the trivializing section of \(\mathcal{L}_s\) pulls back to a trivializing section of \(\mathcal{L}'_s = \mathcal{O}_{X'_s}\). Thus \(a \otimes \kappa(s)\) is an isomorphism, hence after shrinking \(S\) we see that \(a\) is an isomorphism. Finally, we use the hypothesis that \(H^1(X_s, \mathcal{O}) \to H^1(X'_s, \mathcal{O})\) is injective, to see that there exists a \(h_1 \times h_1\) minor of the matrix defining \(b\) which maps to a nonzero element in \(\kappa(s)\). Hence after shrinking \(S\) we may assume that \(b\) is injective. However, since \(\mathcal{L}' = \mathcal{O}_{X'}\) we see that \(d' = 0\). It follows that \(d = 0\). In this way we see that the trivializing section of \(\mathcal{L}_s\) lifts to a section of \(\mathcal{L}\) over \(X\). A straightforward topological argument (omitted) shows that this means that \(\mathcal{L}\) is trivial after possibly shrinking \(S\) a bit further.

Lemma

Consider \(n\) commutative diagrams of schemes \[\xymatrix{ X_i \ar[rr] \ar[dr]_{f_i} & & X \ar[dl]^f \\ & S }\] with \(f_i : X_i \to S\) and \(f : X \to S\) satisfying the hypotheses of Lemma 0BDP. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module and let \(\mathcal{L}_i\) be the pullback to \(X_i\). Let \(Z \subset S\), resp. \(Z_i \subset S\) be the locally closed subscheme constructed in Lemma 0BDP for \((f, \mathcal{L})\), resp. \((f_i, \mathcal{L}_i)\) so that \(Z \subset \bigcap_{i = 1, \ldots, n} Z_i\). If \(s \in Z\) and \[H^1(X_s, \mathcal{O}) \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} H^1(X_{i, s}, \mathcal{O})\] is injective, then \(Z \cap U = (\bigcap_{i = 1, \ldots, n} Z_i) \cap U\) (scheme theoretic intersection) for some open neighbourhood \(U\) of \(s\).

Proof

This lemma is a variant of Lemma 0BF1 and we strongly urge the reader to read that proof first; this proof is basically a copy of that proof with minor modifications. It follows from the description of (scheme valued) points of \(Z\) and the \(Z_i\) that \(Z \subset \bigcap_{i = 1, \ldots, n} Z_i\) where we take the scheme theoretic intersection. Thus we may replace \(S\) by the scheme theoretic intersection \(\bigcap_{i = 1, \ldots, n} Z_i\). After shrinking \(S\) to an affine open neighbourhood of \(s\) we may assume that \(\mathcal{L}_i = \mathcal{O}_{X_i}\) for \(i = 1, \ldots, n\). Let \(E = Rf_*\mathcal{L}\) and \(E_i = Rf_{i, *}\mathcal{L}_i = Rf_{i, *}\mathcal{O}_{X_i}\). These are perfect complexes whose formation commutes with arbitrary change of base (Derived Categories of Schemes, Lemma 0B91). In particular we see that \[E \otimes_{\mathcal{O}_S}^\mathbf{L} \kappa(s) = R\Gamma(X_s, \mathcal{L}_s) = R\Gamma(X_s, \mathcal{O}_{X_s})\] The second equality because \(s \in Z\). Set \(h_j = \dim_{\kappa(s)} H^j(X_s, \mathcal{O}_{X_s})\). After shrinking \(S\) we can represent \(E\) by a complex \[\mathcal{O}_S \to \mathcal{O}_S^{\oplus h_1} \to \mathcal{O}_S^{\oplus h_2} \to \ldots\] see More on Algebra, Lemma 0BCD (strictly speaking this also uses Derived Categories of Schemes, Lemmas 06Z0 and 08EB). Similarly, we may assume \(E_i\) is represented by a complex \[\mathcal{O}_S \to \mathcal{O}_S^{\oplus h_{i, 1}} \to \mathcal{O}_S^{\oplus h_{i, 2}} \to \ldots\] where \(h_{i, j} = \dim_{\kappa(s)} H^j(X_{i, s}, \mathcal{O}_{X_{i, s}})\). By functoriality of cohomology we have a map \[E \longrightarrow E_i\] in \(D(\mathcal{O}_S)\) whose formation commutes with change of base. Since the complex representing \(E\) is a finite complex of finite free modules and since \(S\) is affine, we can choose a map of complexes \[\xymatrix{ \mathcal{O}_S \ar[r]_d \ar[d]_{a_i} & \mathcal{O}_S^{\oplus h_1} \ar[r] \ar[d]_{b_i} & \mathcal{O}_S^{\oplus h_2} \ar[r] \ar[d]_{c_i} & \ldots \\ \mathcal{O}_S \ar[r]^{d_i} & \mathcal{O}_S^{\oplus h_{i, 1}} \ar[r] & \mathcal{O}_S^{\oplus h_{i, 2}} \ar[r] & \ldots }\] representing the given map \(E \to E_i\). Since \(s \in Z\) we see that the trivializing section of \(\mathcal{L}_s\) pulls back to a trivializing section of \(\mathcal{L}_{i, s} = \mathcal{O}_{X_{i, s}}\). Thus \(a_i \otimes \kappa(s)\) is an isomorphism, hence after shrinking \(S\) we see that \(a_i\) is an isomorphism. Finally, we use the hypothesis that \(H^1(X_s, \mathcal{O}) \to \bigoplus_{i = 1, \ldots, n} H^1(X_{i, s}, \mathcal{O})\) is injective, to see that there exists a \(h_1 \times h_1\) minor of the matrix defining \(\oplus b_i\) which maps to a nonzero element in \(\kappa(s)\). Hence after shrinking \(S\) we may assume that \((b_1, \ldots, b_n) : \mathcal{O}_S^{h_1} \to \bigoplus_{i = 1, \ldots, n} \mathcal{O}_S^{h_{i, 1}}\) is injective. However, since \(\mathcal{L}_i = \mathcal{O}_{X_i}\) we see that \(d_i = 0\) for \(i = 1, \ldots n\). It follows that \(d = 0\) because \((b_1, \ldots, b_n) \circ d = (\oplus d_i) \circ (a_1, \ldots, a_n)\). In this way we see that the trivializing section of \(\mathcal{L}_s\) lifts to a section of \(\mathcal{L}\) over \(X\). A straightforward topological argument (omitted) shows that this means that \(\mathcal{L}\) is trivial after possibly shrinking \(S\) a bit further.

Lemma

Let \(f : X \to S\) and \(g : Y \to S\) be morphisms of schemes satisfying the hypotheses of Lemma 0BDP. Let \(\sigma : S \to X\) and \(\tau : S \to Y\) be sections of \(f\) and \(g\). Let \(s \in S\). Let \(\mathcal{L}\) be an invertible sheaf on \(X \times_S Y\). If \((1 \times \tau)^*\mathcal{L}\) on \(X\), \((\sigma \times 1)^*\mathcal{L}\) on \(Y\), and \(\mathcal{L}|_{(X \times_S Y)_s}\) are trivial, then there is an open neighbourhood \(U\) of \(s\) such that \(\mathcal{L}\) is trivial over \((X \times_S Y)_U\).

Proof

By Künneth (Varieties, Lemma 0BED) the map \[H^1(X_s \times_{\Spec(\kappa(s))} Y_s, \mathcal{O}) \to H^1(X_s, \mathcal{O}) \oplus H^1(Y_s, \mathcal{O})\] is injective. Thus we may apply Lemma 0BF2 to the two morphisms \[1 \times \tau : X \to X \times_S Y \quad\text{and}\quad \sigma \times 1 : Y \to X \times_S Y\] to conclude.

Theorem

Let \(S\) be a scheme. Let \(X\), \(Y\), and \(Z\) be schemes over \(S\). Let \(x : S \to X\) and \(y : S \to Y\) be sections of the structure morphisms. Let \(\mathcal{L}\) be an invertible module on \(X \times_S Y \times_S Z\). If

\(X \to S\) and \(Y \to S\) are flat, proper morphisms of finite presentation with geometrically integral fibres,

the pullbacks of \(\mathcal{L}\) by \(x \times \text{id}_Y \times \text{id}_Z\) and \(\text{id}_X \times y \times \text{id}_Z\) are trivial over \(Y \times_S Z\) and \(X \times_S Z\),

there is a point \(z \in Z\) such that \(\mathcal{L}\) restricted to \(X \times_S Y \times_S z\) is trivial, and

\(Z\) is connected,

then \(\mathcal{L}\) is trivial.

An often used special case is the following. Let \(k\) be a field. Let \(X, Y, Z\) be varieties with \(k\)-rational points \(x, y, z\). Let \(\mathcal{L}\) be an invertible module on \(X \times Y \times Z\). If

\(\mathcal{L}\) is trivial over \(x \times Y \times Z\), \(X \times y \times Z\), and \(X \times Y \times z\), and
\(X\) and \(Y\) are geometrically integral and proper over \(k\),

then \(\mathcal{L}\) is trivial.

Proof

Observe that the morphism \(X \times_S Y \to S\) is a flat, proper morphism of finite presentation whose geometrically integral fibres (see Varieties, Lemmas 038K, 038F, and 035Z for the statement about the fibres). By Derived Categories of Schemes, Lemma 0E0L we see that the pushforward of the structure sheaf by \(X \to S\), \(Y \to S\), or \(X \times_S Y \to S\) is the structure sheaf of \(S\) and the same remains true after any base change. Thus we may apply Lemma 0BDP to the morphism \[p : X \times_S Y \times_S Z \longrightarrow Z\] and the invertible module \(\mathcal{L}\) to get a “universal” locally closed subscheme \(Z' \subset Z\) such that \(\mathcal{L}|_{X \times_S Y \times_S Z'}\) is the pullback of an invertible module \(\mathcal{N}\) on \(Z'\). The existence of \(z\) shows that \(Z'\) is nonempty. By Lemma 0BF0 we see that \(Z' \subset Z\) is a closed subscheme. Let \(z' \in Z'\) be a point. Observe that we may write \(p\) as the product morphism \[(X \times_S Z) \times_Z (Y \times_S Z) \longrightarrow Z\] Hence we may apply Lemma 0BF3 to the morphism \(p\), the point \(z'\), and the sections \(\sigma : Z \to X \times_S Z\) and \(\tau : Z \to Y \times_S Z\) given by \(x\) and \(y\). We conclude that \(Z'\) is open. Hence \(Z' = Z\) and \(\mathcal{L} = p^*\mathcal{N}\) for some invertible module \(\mathcal{N}\) on \(Z\). Pulling back via \(x \times y \times \text{id}_Z : Z \to X \times_S Y \times_S Z\) we obtain on the one hand \(\mathcal{N}\) and on the other hand we obtain the trivial invertible module by assumption (2). Thus \(\mathcal{N} = \mathcal{O}_Z\) and the proof is complete.

Limit arguments

Some lemmas involving limits of schemes, and Noetherian approximation. We stick mostly to the affine case. Some of these lemmas are special cases of lemmas in the chapter on limits.

Lemma

Let \(f : X \to S\) be a morphism of affine schemes, which is of finite presentation. Then there exists a cartesian diagram \[\xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^g \ar[d]^f \\ S_0 & S \ar[l] }\] such that

\(X_0\), \(S_0\) are affine schemes,

\(S_0\) is of finite type over \(\mathbf{Z}\),

\(f_0\) is of finite type.

Proof

Write \(S = \Spec(A)\) and \(X = \Spec(B)\). As \(f\) is of finite presentation we see that \(B\) is of finite presentation as an \(A\)-algebra, see Morphisms, Lemma 01TQ. Thus the lemma follows from Algebra, Lemma 00R1.

Lemma

Let \(f : X \to S\) be a morphism of affine schemes, which is of finite presentation. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite presentation. Then there exists a diagram as in Lemma 05FB such that there exists a coherent \(\mathcal{O}_{X_0}\)-module \(\mathcal{F}_0\) with \(g^*\mathcal{F}_0 = \mathcal{F}\).

Proof

Write \(S = \Spec(A)\), \(X = \Spec(B)\), and \(\mathcal{F} = \widetilde{M}\). As \(f\) is of finite presentation we see that \(B\) is of finite presentation as an \(A\)-algebra, see Morphisms, Lemma 01TQ. As \(\mathcal{F}\) is of finite presentation over \(\mathcal{O}_X\) we see that \(M\) is of finite presentation as a \(B\)-module, see Properties, Lemma 01PC. Thus the lemma follows from Algebra, Lemma 00R1.

Lemma

Let \(f : X \to S\) be a morphism of affine schemes, which is of finite presentation. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite presentation and flat over \(S\). Then we may choose a diagram as in Lemma 05FC and sheaf \(\mathcal{F}_0\) such that in addition \(\mathcal{F}_0\) is flat over \(S_0\).

Proof

Write \(S = \Spec(A)\), \(X = \Spec(B)\), and \(\mathcal{F} = \widetilde{M}\). As \(f\) is of finite presentation we see that \(B\) is of finite presentation as an \(A\)-algebra, see Morphisms, Lemma 01TQ. As \(\mathcal{F}\) is of finite presentation over \(\mathcal{O}_X\) we see that \(M\) is of finite presentation as a \(B\)-module, see Properties, Lemma 01PC. As \(\mathcal{F}\) is flat over \(S\) we see that \(M\) is flat over \(A\), see Morphisms, Lemma 01U4. Thus the lemma follows from Algebra, Lemma 02JO.

Lemma

Let \(f : X \to S\) be a morphism of affine schemes, which is of finite presentation and flat. Then there exists a diagram as in Lemma 05FB such that in addition \(f_0\) is flat.

Proof

This is a special case of Lemma 05FD.

Lemma

Let \(f : X \to S\) be a morphism of affine schemes, which is smooth. Then there exists a diagram as in Lemma 05FB such that in addition \(f_0\) is smooth.

Proof

Write \(S = \Spec(A)\), \(X = \Spec(B)\), and as \(f\) is smooth we see that \(B\) is smooth as an \(A\)-algebra, see Morphisms, Lemma 01V6. Hence the lemma follows from Algebra, Lemma 00TP.

Lemma

Let \(f : X \to S\) be a morphism of affine schemes, which is of finite presentation with geometrically reduced fibres. Then there exists a diagram as in Lemma 05FB such that in addition \(f_0\) has geometrically reduced fibres.

Proof

Apply Lemma 05FB to get a cartesian diagram \[\xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^g \ar[d]^f \\ S_0 & S \ar[l]_h }\] of affine schemes with \(X_0 \to S_0\) a finite type morphism of schemes of finite type over \(\mathbf{Z}\). By Lemma 0579 the set \(E \subset S_0\) of points where the fibre of \(f_0\) is geometrically reduced is a constructible subset. By Lemma 0576 we have \(h(S) \subset E\). Write \(S_0 = \Spec(A_0)\) and \(S = \Spec(A)\). Write \(A = \colim_i A_i\) as a direct colimit of finite type \(A_0\)-algebras. By Limits, Lemma 05F4 we see that \(\Spec(A_i) \to S_0\) has image contained in \(E\) for some \(i\). After replacing \(S_0\) by \(\Spec(A_i)\) and \(X_0\) by \(X_0 \times_{S_0} \Spec(A_i)\) we see that all fibres of \(f_0\) are geometrically reduced.

Lemma

Let \(f : X \to S\) be a morphism of affine schemes, which is of finite presentation with geometrically irreducible fibres. Then there exists a diagram as in Lemma 05FB such that in addition \(f_0\) has geometrically irreducible fibres.

Proof

Apply Lemma 05FB to get a cartesian diagram \[\xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^g \ar[d]^f \\ S_0 & S \ar[l]_h }\] of affine schemes with \(X_0 \to S_0\) a finite type morphism of schemes of finite type over \(\mathbf{Z}\). By Lemma 055B the set \(E \subset S_0\) of points where the fibre of \(f_0\) is geometrically irreducible is a constructible subset. By Lemma 0555 we have \(h(S) \subset E\). Write \(S_0 = \Spec(A_0)\) and \(S = \Spec(A)\). Write \(A = \colim_i A_i\) as a direct colimit of finite type \(A_0\)-algebras. By Limits, Lemma 05F4 we see that \(\Spec(A_i) \to S_0\) has image contained in \(E\) for some \(i\). After replacing \(S_0\) by \(\Spec(A_i)\) and \(X_0\) by \(X_0 \times_{S_0} \Spec(A_i)\) we see that all fibres of \(f_0\) are geometrically irreducible.

Lemma

Let \(f : X \to S\) be a morphism of affine schemes, which is of finite presentation with geometrically connected fibres. Then there exists a diagram as in Lemma 05FB such that in addition \(f_0\) has geometrically connected fibres.

Proof

Apply Lemma 05FB to get a cartesian diagram \[\xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^g \ar[d]^f \\ S_0 & S \ar[l]_h }\] of affine schemes with \(X_0 \to S_0\) a finite type morphism of schemes of finite type over \(\mathbf{Z}\). By Lemma 055I the set \(E \subset S_0\) of points where the fibre of \(f_0\) is geometrically connected is a constructible subset. By Lemma 055E we have \(h(S) \subset E\). Write \(S_0 = \Spec(A_0)\) and \(S = \Spec(A)\). Write \(A = \colim_i A_i\) as a direct colimit of finite type \(A_0\)-algebras. By Limits, Lemma 05F4 we see that \(\Spec(A_i) \to S_0\) has image contained in \(E\) for some \(i\). After replacing \(S_0\) by \(\Spec(A_i)\) and \(X_0\) by \(X_0 \times_{S_0} \Spec(A_i)\) we see that all fibres of \(f_0\) are geometrically connected.

Lemma

Let \(d \geq 0\) be an integer. Let \(f : X \to S\) be a morphism of affine schemes, which is of finite presentation all of whose fibres have dimension \(d\). Then there exists a diagram as in Lemma 05FB such that in addition all fibres of \(f_0\) have dimension \(d\).

Proof

Apply Lemma 05FB to get a cartesian diagram \[\xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^g \ar[d]^f \\ S_0 & S \ar[l]_h }\] of affine schemes with \(X_0 \to S_0\) a finite type morphism of schemes of finite type over \(\mathbf{Z}\). By Lemma 05F9 the set \(E \subset S_0\) of points where the fibre of \(f_0\) has dimension \(d\) is a constructible subset. By Lemma 05F8 we have \(h(S) \subset E\). Write \(S_0 = \Spec(A_0)\) and \(S = \Spec(A)\). Write \(A = \colim_i A_i\) as a direct colimit of finite type \(A_0\)-algebras. By Limits, Lemma 05F4 we see that \(\Spec(A_i) \to S_0\) has image contained in \(E\) for some \(i\). After replacing \(S_0\) by \(\Spec(A_i)\) and \(X_0\) by \(X_0 \times_{S_0} \Spec(A_i)\) we see that all fibres of \(f_0\) have dimension \(d\).

Lemma

Let \(f : X \to S\) be a morphism of affine schemes, which is standard syntomic (see Morphisms, Definition 01UC). Then there exists a diagram as in Lemma 05FB such that in addition \(f_0\) is standard syntomic.

Proof

This lemma is a copy of Algebra, Lemma 00SU.

Lemma

(Noetherian approximation and combining properties.) Let \(P\), \(Q\) be properties of morphisms of schemes which are stable under base change. Let \(f : X \to S\) be a morphism of finite presentation of affine schemes. Assume we can find cartesian diagrams \[\vcenter{ \xymatrix{ X_1 \ar[d]_{f_1} & X \ar[l] \ar[d]^f \\ S_1 & S \ar[l] } } \quad\text{and}\quad \vcenter{ \xymatrix{ X_2 \ar[d]_{f_2} & X \ar[l] \ar[d]^f \\ S_2 & S \ar[l] } }\] of affine schemes, with \(S_1\), \(S_2\) of finite type over \(\mathbf{Z}\) and \(f_1\), \(f_2\) of finite type such that \(f_1\) has property \(P\) and \(f_2\) has property \(Q\). Then we can find a cartesian diagram \[\xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l] \ar[d]^f \\ S_0 & S \ar[l] }\] of affine schemes with \(S_0\) of finite type over \(\mathbf{Z}\) and \(f_0\) of finite type such that \(f_0\) has both property \(P\) and property \(Q\).

Proof

The given pair of diagrams correspond to cocartesian diagrams of rings \[\vcenter{ \xymatrix{ B_1 \ar[r] & B \\ A_1 \ar[u] \ar[r] & A \ar[u] } } \quad\text{and}\quad \vcenter{ \xymatrix{ B_2 \ar[r] & B \\ A_2 \ar[u] \ar[r] & A \ar[u] } }\] Let \(A_0 \subset A\) be a finite type \(\mathbf{Z}\)-subalgebra of \(A\) containing the image of both \(A_1 \to A\) and \(A_2 \to A\). Such a subalgebra exists because by assumption both \(A_1\) and \(A_2\) are of finite type over \(\mathbf{Z}\). Note that the rings \(B_{0, 1} = B_1 \otimes_{A_1} A_0\) and \(B_{0, 2} = B_2 \otimes_{A_2} A_0\) are finite type \(A_0\)-algebras with the property that \(B_{0, 1} \otimes_{A_0} A \cong B \cong B_{0, 2} \otimes_{A_0} A\) as \(A\)-algebras. As \(A\) is the directed colimit of its finite type \(A_0\)-subalgebras, by Limits, Lemma 01ZM we may assume after enlarging \(A_0\) that there exists an isomorphism \(B_{0, 1} \cong B_{0, 2}\) as \(A_0\)-algebras. Since properties \(P\) and \(Q\) are assumed stable under base change we conclude that setting \(S_0 = \Spec(A_0)\) and \[X_0 = X_1 \times_{S_1} S_0 = \Spec(B_{0, 1}) \cong \Spec(B_{0, 2}) = X_2 \times_{S_2} S_0\] works.

Étale neighbourhoods

It turns out that some properties of morphisms are easier to study after doing an étale base change. It is convenient to introduce the following terminology.

Definition

Let \(S\) be a scheme. Let \(s \in S\) be a point.

An étale neighbourhood of \((S, s)\) is a pair \((U, u)\) together with an étale morphism of schemes \(\varphi : U \to S\) such that \(\varphi(u) = s\).

A morphism of étale neighbourhoods \(f : (V, v) \to (U, u)\) of \((S, s)\) is simply a morphism of \(S\)-schemes \(f : V \to U\) such that \(f(v) = u\).

An elementary étale neighbourhood is an étale neighbourhood \(\varphi : (U, u) \to (S, s)\) such that \(\kappa(s) = \kappa(u)\).

The notion of an elementary étale neighbourhood has many different names in the literature, for example these are sometimes called “étale neighbourhoods” ([Milne, Page 36] or “strongly étale” ([KPR, Page 108]). Here we follow the convention of the paper [GruRay] by calling them elementary étale neighbourhoods.

If \(f : (V, v) \to (U, u)\) is a morphism of étale neighbourhoods, then \(f\) is automatically étale, see Morphisms, Lemma 02GW. Hence it turns \((V, v)\) into an étale neighbourhood of \((U, u)\). Of course, since the composition of étale morphisms is étale (Morphisms, Lemma 02GN) we see that conversely any étale neighbourhood \((V, v)\) of \((U, u)\) is an étale neighbourhood of \((S, s)\) as well. We also remark that if \(U \subset S\) is an open neighbourhood of \(s\), then \((U, s) \to (S, s)\) is an étale neighbourhood. This follows from the fact that an open immersion is étale (Morphisms, Lemma 02GP). We will use these remarks without further mention throughout this section.

Note that \(\kappa(u)/\kappa(s)\) is a finite separable extension if \((U, u) \to (S, s)\) is an étale neighbourhood, see Morphisms, Lemma 02GU.

Lemma

Let \(S\) be a scheme. Let \(s \in S\). Let \(k/\kappa(s)\) be a finite separable field extension. Then there exists an étale neighbourhood \((U, u) \to (S, s)\) such that the field extension \(\kappa(u)/\kappa(s)\) is isomorphic to \(k/\kappa(s)\).

Proof

We may assume \(S\) is affine. In this case the lemma follows from Algebra, Lemma 00UD.

Lemma

Let \(S\) be a scheme, and let \(s\) be a point of \(S\). The category of étale neighborhoods has the following properties:

Let \((U_i, u_i)_{i=1, 2}\) be two étale neighborhoods of \(s\) in \(S\). Then there exists a third étale neighborhood \((U, u)\) and morphisms \((U, u) \to (U_i, u_i)\), \(i = 1, 2\).

Let \(h_1, h_2: (U, u) \to (U', u')\) be two morphisms between étale neighborhoods of \(s\). Assume \(h_1\), \(h_2\) induce the same map \(\kappa(u') \to \kappa(u)\) of residue fields. Then there exist an étale neighborhood \((U'', u'')\) and a morphism \(h : (U'', u'') \to (U, u)\) which equalizes \(h_1\) and \(h_2\), i.e., such that \(h_1 \circ h = h_2 \circ h\).

Proof

For part (1), consider the fibre product \(U = U_1 \times_S U_2\). It is étale over both \(U_1\) and \(U_2\) because étale morphisms are preserved under base change, see Morphisms, Lemma 02GO. There is a point of \(U\) mapping to both \(u_1\) and \(u_2\) for example by the description of points of a fibre product in Schemes, Lemma 01JT. For part (2), define \(U''\) as the fibre product \[\xymatrix{ U'' \ar[r] \ar[d] & U \ar[d]^{(h_1, h_2)} \\ U' \ar[r]^-\Delta & U' \times_S U'. }\] Since \(h_1\) and \(h_2\) induce the same map of residue fields \(\kappa(u') \to \kappa(u)\) there exists a point \(u'' \in U''\) lying over \(u'\) with \(\kappa(u'') = \kappa(u')\). In particular \(U'' \not = \emptyset\). Moreover, since \(U'\) is étale over \(S\), so is the fibre product \(U'\times_S U'\) (see Morphisms, Lemmas 02GO and 02GN). Hence the vertical arrow \((h_1, h_2)\) is étale by Morphisms, Lemma 02GW. Therefore \(U''\) is étale over \(U'\) by base change, and hence also étale over \(S\) (because compositions of étale morphisms are étale). Thus \((U'', u'')\) is a solution to the problem.

Lemma

Let \(S\) be a scheme, and let \(s\) be a point of \(S\). The category of elementary étale neighborhoods of \((S, s)\) is cofiltered (see Categories, Definition 04AZ).

Proof

This is immediate from the definitions and Lemma 057A.

Lemma

Let \(S\) be a scheme. Let \(s \in S\). Then we have \[\mathcal{O}_{S, s}^h = \colim_{(U, u)} \mathcal{O}(U)\] where the colimit is over the filtered category which is opposite to the category of elementary étale neighbourhoods \((U, u)\) of \((S, s)\).

Proof

Let \(\Spec(A) \subset S\) be an affine neighbourhood of \(s\). Let \(\mathfrak p \subset A\) be the prime ideal corresponding to \(s\). With these choices we have canonical isomorphisms \(\mathcal{O}_{S, s} = A_{\mathfrak p}\) and \(\kappa(s) = \kappa(\mathfrak p)\). A cofinal system of elementary étale neighbourhoods is given by those elementary étale neighbourhoods \((U, u)\) such that \(U\) is affine and \(U \to S\) factors through \(\Spec(A)\). In other words, we see that the right hand side is equal to \(\colim_{(B, \mathfrak q)} B\) where the colimit is over étale \(A\)-algebras \(B\) endowed with a prime \(\mathfrak q\) lying over \(\mathfrak p\) with \(\kappa(\mathfrak p) = \kappa(\mathfrak q)\). Thus the lemma follows from Algebra, Lemma 04GV.

We can lift étale neighbourhoods of points on fibres to the total space.

Lemma

Let \(X \to S\) be a morphism of schemes. Let \(x \in X\) with image \(s \in S\). Let \((V, v) \to (X_s, x)\) be an étale neighbourhood. Then there exists an étale neighbourhood \((U, u) \to (X, x)\) such that there exists a morphism \((U_s, u) \to (V, v)\) of étale neighbourhoods of \((X_s, x)\) which is an open immersion.

Proof

We may assume \(X\), \(V\), and \(S\) affine. Say the morphism \(X \to S\) is given by \(A \to B\) the point \(x\) by a prime \(\mathfrak q \subset B\), the point \(s\) by \(\mathfrak p = A \cap \mathfrak q\), and the morphism \(V \to X_s\) by \(B \otimes_A \kappa(\mathfrak p) \to C\). Since \(\kappa(\mathfrak p)\) is a localization of \(A/\mathfrak p\) there exists an \(f \in A\), \(f \not \in \mathfrak p\) and an étale ring map \(B \otimes_A (A/\mathfrak p)_f \to D\) such that \[C = (B \otimes_A \kappa(\mathfrak p)) \otimes_{B \otimes_A (A/\mathfrak p)_f} D\] See Algebra, Lemma 00U2 part (9). After replacing \(A\) by \(A_f\) and \(B\) by \(B_f\) we may assume \(D\) is étale over \(B \otimes_A A/\mathfrak p = B/\mathfrak p B\). Then we can apply Algebra, Lemma 04D1. This proves the lemma.

Étale neighbourhoods and branches

The number of (geometric) branches of a scheme at a point was defined in Properties, Section 0BQ1. In Varieties, Section 0C3Z we related this to fibres of the normalization morphism. In this section we discuss a characterization of this number in terms of étale neighbourhoods.

Lemma

Let \(R = \colim R_i\) be colimit of a directed system of rings whose transition maps are faithfully flat. Then the number of minimal primes of \(R\) taken as an element of \(\{0, 1, 2, \ldots, \infty\}\) is the supremum of the numbers of minimal primes of the \(R_i\).

Proof

If \(A \to B\) is a flat ring map, then \(\Spec(B) \to \Spec(A)\) maps minimal primes to minimal primes by going down (Algebra, Lemma 00HS). If \(A \to B\) is faithfully flat, then every minimal prime is the image of a minimal prime (by Algebra, Lemma 00HQ and 0CAN). Hence the number of minimal primes of \(R_i\) is \(\geq\) the number of minimal primes of \(R_{i'}\) if \(i \leq i'\). By Algebra, Lemma 090N each of the maps \(R_i \to R\) is faithfully flat and we also see that the number of minimal primes of \(R\) is \(\geq\) the number of minimal primes of \(R_i\). Finally, suppose that \(\mathfrak q_1, \ldots, \mathfrak q_n\) are pairwise distinct minimal primes of \(R\). Then we can find an \(i\) such that \(R_i \cap \mathfrak q_1, \ldots, R_i \cap \mathfrak q_n\) are pairwise distinct (as sets and hence as prime ideals). This implies the lemma.

Lemma

Let \(X\) be a scheme and \(x \in X\) a point. Then

the number of branches of \(X\) at \(x\) is equal to the supremum of the number of irreducible components of \(U\) passing through \(u\) taken over elementary étale neighbourhoods \((U, u) \to (X, x)\),

the number of geometric branches of \(X\) at \(x\) is equal to the supremum of the number of irreducible components of \(U\) passing through \(u\) taken over étale neighbourhoods \((U, u) \to (X, x)\),

\(X\) is unibranch at \(x\) if and only if for every elementary étale neighbourhood \((U, u) \to (X, x)\) there is exactly one irreducible component of \(U\) passing through \(u\), and

\(X\) is geometrically unibranch at \(x\) if and only if for every étale neighbourhood \((U, u) \to (X, x)\) there is exactly one irreducible component of \(U\) passing through \(u\).

Proof

Parts (3) and (4) follow from parts (1) and (2) via Properties, Lemma 0C39.

Proof of (1). Let \(\Spec(A)\) be an affine open neighbourhood of \(x\) and let \(\mathfrak p \subset A\) be the prime ideal corresponding to \(x\). We may replace \(X\) by \(\Spec(A)\) and it suffices to consider affine elementary étale neighbourhoods \((U, u)\) in the supremum as they form a cofinal subsystem. Recall that the henselization \(A_\mathfrak p^h\) is the colimit of the rings \(B_\mathfrak q\) over the category of pairs \((B, \mathfrak q)\) where \(B\) is an étale \(A\)-algebra and \(\mathfrak q\) is a prime lying over \(\mathfrak p\) with \(\kappa(\mathfrak q) = \kappa(\mathfrak p)\), see Algebra, Lemma 04GV. These pairs \((B, \mathfrak q)\) correspond exactly to the affine elementary étale neighbourhoods \((U, u)\) by the correspondence between rings and affine schemes. Observe that irreducible components of \(\Spec(B)\) passing through \(\mathfrak q\) are exactly the minimal prime ideals of \(B_\mathfrak q\). The number of minimal primes of \(A_\mathfrak p^h\) is the number of branches of \(X\) at \(x\) by Properties, Definition 0C38. Observe that the transition maps \(B_\mathfrak q \to B'_{\mathfrak q'}\) in the system are all flat. Since a flat local ring map is faithfully flat (Algebra, Lemma 00HR) we see that the lemma follows from Lemma 0CB3.

Proof of (2). The proof is the same as the proof of (1), except that we use Algebra, Lemma 04GW. There is a tiny difference: given a separable algebraic closure \(\kappa^{sep}\) of \(\kappa(x)\) for every étale neighbourhood \((U, u)\) we can choose a \(\kappa(x)\)-embedding \(\phi : \kappa(u) \to \kappa^{sep}\) because \(\kappa(u)/\kappa(x)\) is finite separable (Morphisms, Lemma 02GU). Hence we can look at the supremum over all triples \((U, u, \phi)\) where \((U, u) \to (X, x)\) is an affine étale neighbourhood and \(\phi : \kappa(u) \to \kappa^{sep}\) is a \(\kappa(x)\)-embedding. These triples correspond exactly to the triples in Algebra, Lemma 04GW and the rest of the proof is exactly the same.

We will need a relative variant of the lemma above.

Lemma

Let \(X \to S\) be a morphism of schemes and \(x \in X\) a point with image \(s\). Then

the number of branches of the fibre \(X_s\) at \(x\) is equal to the supremum of the number of irreducible components of the fibre \(U_s\) passing through \(u\) taken over elementary étale neighbourhoods \((U, u) \to (X, x)\),

the number of geometric branches of the fibre \(X_s\) at \(x\) is equal to the supremum of the number of irreducible components of the fibre \(U_s\) passing through \(u\) taken over étale neighbourhoods \((U, u) \to (X, x)\),

the fibre \(X_s\) is unibranch at \(x\) if and only if for every elementary étale neighbourhood \((U, u) \to (X, x)\) there is exactly one irreducible component of the fibre \(U_s\) passing through \(u\), and

\(X\) is geometrically unibranch at \(x\) if and only if for every étale neighbourhood \((U, u) \to (X, x)\) there is exactly one irreducible component of \(U_s\) passing through \(u\).

Proof

Combine Lemmas 0CB4 and 0CAS.

Lemma

Let \(X \to S\) be a smooth morphism of schemes. Let \(x \in X\) with image \(s \in S\). Then

The number of geometric branches of \(X\) at \(x\) is equal to the number of geometric branches of \(S\) at \(s\).

If \(\kappa(x)/\kappa(s)\) is a purely inseparable⁶ extension of fields, then number of branches of \(X\) at \(x\) is equal to the number of branches of \(S\) at \(s\).

Proof

Follows immediately from More on Algebra, Lemma 0DQ1 and the definitions.

Unramified and étale morphisms

Sometimes unramified morphisms are automatically étale.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(x \in X\) with image \(y \in Y\). Assume

\(Y\) is integral and geometrically unibranch at \(y\),

\(f\) is locally of finite type,

there is a specialization \(x' \leadsto x\) such that \(f(x')\) is the generic point of \(Y\),

\(f\) is unramified at \(x\).

Then \(f\) is étale at \(x\).

Proof

We may replace \(X\) and \(Y\) by suitable affine open neighbourhoods of \(x\) and \(y\). Then \(Y\) is the spectrum of a domain \(A\) and \(X\) is the spectrum of a finite type \(A\)-algebra \(B\). Let \(\mathfrak q \subset B\) be the prime ideal corresponding to \(x\) and \(\mathfrak p \subset A\) the prime ideal corresponding to \(y\). The local ring \(A_\mathfrak p = \mathcal{O}_{Y, y}\) is geometrically unibranch. The ring map \(A \to B\) is unramified at \(\mathfrak q\). Also, the point \(x'\) in (3) corresponds to a prime ideal \(\mathfrak q' \subset \mathfrak q\) such that \(A \cap \mathfrak q' = (0)\). It follows that \(A_\mathfrak p \to B_\mathfrak q\) is injective. We conclude by More on Algebra, Lemma 0GSC.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume

\(Y\) is integral and geometrically unibranch,

at least one irreducible component of \(X\) dominates \(Y\),

\(f\) is unramified, and

\(X\) is connected.

Then \(f\) is étale and \(X\) is irreducible.

Proof

Let \(X' \subset X\) be the irreducible component which dominates \(Y\). This means that the generic point of \(X'\) maps to the generic point of \(Y\) (see for example Morphisms, Lemma 01RM). By Lemma 0GS8 we see that \(f\) is étale at every point of \(X'\). In particular, the open subscheme \(U \subset X\) where \(f\) is étale contains \(X'\). Note that every quasi-compact open of \(U\) has finitely many irreducible components, see Descent, Lemma 0BAL. On the other hand since \(Y\) is geometrically unibranch and \(U\) is étale over \(Y\), the scheme \(U\) is geometrically unibranch. In particular, through every point of \(U\) there passes at most one irreducible component. A simple topological argument now shows that \(X' \subset U\) is both open and closed. Then of course \(X'\) is open and closed in \(X\) and by connectedness we find \(X = U = X'\) as desired.

Lemma

Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of schemes. Let \(x \in X\) with image \(y \in Y\). Assume

\(Y\) is integral and geometrically unibranch at \(y\),

\(g \circ f\) is étale at \(x\),

there is a specialization \(x' \leadsto x\) such that \(f(x')\) is the generic point of \(Y\).

Then \(f\) is étale at \(x\) and \(g\) is étale at \(y\).

Proof

After replacing \(X\) by an open neighbourhood of \(x\) we may assume \(g \circ f\) is étale. Then we find \(f\) is unramified by Morphisms, Lemmas 02GG and 02GK. Hence \(f\) is étale at \(x\) by Lemma 0GS8. Then by étale descent we see that \(g\) is étale at \(y\), see for example Descent, Lemma 02KM.

Lemma

Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of schemes. Assume

\(Y\) is integral and geometrically unibranch,

\(g \circ f\) is étale,

every irreducible component of \(X\) dominates \(Y\).

Then \(f\) is étale and \(g\) is étale at every point in the image of \(f\).

Proof

Immediate from the pointwise version Lemma 0GSA.

Slicing smooth morphisms

In this section we explain a result that roughly states that smooth coverings of a scheme \(S\) can be refined by étale coverings. The technique to prove this relies on a slicing argument.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) be a point with image \(s \in S\). Let \(h \in \mathfrak m_x \subset \mathcal{O}_{X, x}\). Assume

\(f\) is smooth at \(x\), and

the image \(\text{d}\overline{h}\) of \(\text{d}h\) in \[\Omega_{X_s/s, x} \otimes_{\mathcal{O}_{X_s, x}} \kappa(x) = \Omega_{X/S, x} \otimes_{\mathcal{O}_{X, x}} \kappa(x)\] is nonzero.

Then there exists an affine open neighbourhood \(U \subset X\) of \(x\) such that \(h\) comes from \(h \in \Gamma(U, \mathcal{O}_U)\) and such that \(D = V(h)\) is an effective Cartier divisor in \(U\) with \(x \in D\) and \(D \to S\) smooth.

Proof

As \(f\) is smooth at \(x\) we may assume, after replacing \(X\) by an open neighbourhood of \(x\) that \(f\) is smooth. In particular we see that \(f\) is flat and locally of finite presentation. By Lemma 056Y we already know there exists an open neighbourhood \(U \subset X\) of \(x\) such that \(h\) comes from \(h \in \Gamma(U, \mathcal{O}_U)\) and such that \(D = V(h)\) is an effective Cartier divisor in \(U\) with \(x \in D\) and \(D \to S\) flat and of finite presentation. By Morphisms, Lemma 01UZ we have a short exact sequence \[\mathcal{C}_{D/U} \to i^*\Omega_{U/S} \to \Omega_{D/S} \to 0\] where \(i : D \to U\) is the closed immersion and \(\mathcal{C}_{D/U}\) is the conormal sheaf of \(D\) in \(U\). As \(D\) is an effective Cartier divisor cut out by \(h \in \Gamma(U, \mathcal{O}_U)\) we see that \(\mathcal{C}_{D/U} = h \cdot \mathcal{O}_S\). Since \(U \to S\) is smooth the sheaf \(\Omega_{U/S}\) is finite locally free, hence its pullback \(i^*\Omega_{U/S}\) is finite locally free also. The first arrow of the sequence maps the free generator \(h\) to the section \(\text{d}h|_D\) of \(i^*\Omega_{U/S}\) which has nonzero value in the fibre \(\Omega_{U/S, x} \otimes \kappa(x)\) by assumption. By right exactness of \(\otimes \kappa(x)\) we conclude that \[\dim_{\kappa(x)} \left( \Omega_{D/S, x} \otimes \kappa(x) \right) = \dim_{\kappa(x)} \left( \Omega_{U/S, x} \otimes \kappa(x) \right) - 1.\] By Morphisms, Lemma 01V9 we see that \(\Omega_{U/S, x} \otimes \kappa(x)\) can be generated by at most \(\dim_x(U_s)\) elements. By the displayed formula we see that \(\Omega_{D/S, x} \otimes \kappa(x)\) can be generated by at most \(\dim_x(U_s) - 1\) elements. Note that \(\dim_x(D_s) = \dim_x(U_s) - 1\) for example because \(\dim(\mathcal{O}_{D_s, x}) = \dim(\mathcal{O}_{U_s, x}) - 1\) by Algebra, Lemma 00KW (also \(D_s \subset U_s\) is effective Cartier, see Divisors, Lemma 056Q) and then using Morphisms, Lemma 02FX. Thus we conclude that \(\Omega_{D/S, x} \otimes \kappa(x)\) can be generated by at most \(\dim_x(D_s)\) elements and we conclude that \(D \to S\) is smooth at \(x\) by Morphisms, Lemma 01V9 again. After shrinking \(U\) we get that \(D \to S\) is smooth and we win.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) be a point with image \(s \in S\). Assume

\(f\) is smooth at \(x\), and

the map \[\Omega_{X_s/s, x} \otimes_{\mathcal{O}_{X_s, x}} \kappa(x) \longrightarrow \Omega_{\kappa(x)/\kappa(s)}\] has a nonzero kernel.

Then there exists an affine open neighbourhood \(U \subset X\) of \(x\) and an effective Cartier divisor \(D \subset U\) containing \(x\) such that \(D \to S\) is smooth.

Proof

Write \(k = \kappa(s)\) and \(R = \mathcal{O}_{X_s, x}\). Denote \(\mathfrak m\) the maximal ideal of \(R\) and \(\kappa = R/\mathfrak m\) so that \(\kappa = \kappa(x)\). As formation of modules of differentials commutes with localization (see Algebra, Lemma 00RT) we have \(\Omega_{X_s/s, x} = \Omega_{R/k}\). By Algebra, Lemma 00RU there is an exact sequence \[\mathfrak m/\mathfrak m^2 \xrightarrow{\text{d}} \Omega_{R/k} \otimes_R \kappa \to \Omega_{\kappa/k} \to 0.\] Hence if (2) holds, there exists an element \(\overline{h} \in \mathfrak m\) such that \(\text{d}\overline{h}\) is nonzero. Choose a lift \(h \in \mathcal{O}_{X, x}\) of \(\overline{h}\) and apply Lemma 057C.

Remark

The second condition in Lemma 057D is necessary even if \(x\) is a closed point of a positive dimensional fibre. An example is the following: Let \(k\) be a field of characteristic \(p > 0\) which is imperfect. Let \(a \in k\) be an element which is not a \(p\)th power. Let \(\mathfrak m = (x, y^p - a) \subset k[x, y]\). This corresponds to a closed point \(w\) of \(X = \mathbf{A}^2_k\). Set \(S = \mathbf{A}^1_k\) and let \(f : X \to S\) be the morphism corresponding to \(k[x] \to k[x, y]\). Then there does not exist any commutative diagram \[\xymatrix{ S' \ar[rr]_h \ar[rd]_g & & X \ar[ld]^f \\ & S }\] with \(g\) étale and \(w\) in the image of \(h\). This is clear as the residue field extension \(\kappa(w)/\kappa(f(w))\) is purely inseparable, but for any \(s' \in S'\) with \(g(s') = f(w)\) the extension \(\kappa(s')/\kappa(f(w))\) would be separable.

If you assume the residue field extension is separable then the phenomenon of Remark 057E does not happen. Here is the precise result.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) be a point with image \(s \in S\). Assume

\(f\) is smooth at \(x\),

the residue field extension \(\kappa(x)/\kappa(s)\) is separable, and

\(x\) is not a generic point of \(X_s\).

Then there exists an affine open neighbourhood \(U \subset X\) of \(x\) and an effective Cartier divisor \(D \subset U\) containing \(x\) such that \(D \to S\) is smooth.

Proof

Write \(k = \kappa(s)\) and \(R = \mathcal{O}_{X_s, x}\). Denote \(\mathfrak m\) the maximal ideal of \(R\) and \(\kappa = R/\mathfrak m\) so that \(\kappa = \kappa(x)\). As formation of modules of differentials commutes with localization (see Algebra, Lemma 00RT) we have \(\Omega_{X_s/s, x} = \Omega_{R/k}\). By assumption (2) and Algebra, Lemma 00TU the map \[\text{d} : \mathfrak m/\mathfrak m^2 \longrightarrow \Omega_{R/k} \otimes_R \kappa(\mathfrak m)\] is injective. Assumption (3) implies that \(\mathfrak m/\mathfrak m^2 \not = 0\). Thus there exists an element \(\overline{h} \in \mathfrak m\) such that \(\text{d}\overline{h}\) is nonzero. Choose a lift \(h \in \mathcal{O}_{X, x}\) of \(\overline{h}\) and apply Lemma 057C.

The subscheme \(Z\) constructed in the following lemma is really a complete intersection in an affine open neighbourhood of \(x\). If we ever need this we will explicitly formulate a separate lemma stating this fact.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) be a point with image \(s \in S\). Assume

\(f\) is smooth at \(x\), and

\(x\) is a closed point of \(X_s\) and \(\kappa(s) \subset \kappa(x)\) is separable.

Then there exists an immersion \(Z \to X\) containing \(x\) such that

\(Z \to S\) is étale, and

\(Z_s = \{x\}\) set theoretically.

Proof

We may and do replace \(S\) by an affine open neighbourhood of \(s\). We may and do replace \(X\) by an affine open neighbourhood of \(x\) such that \(X \to S\) is smooth. We will prove the lemma for smooth morphisms of affines by induction on \(d = \dim_x(X_s)\).

The case \(d = 0\). In this case we show that we may take \(Z\) to be an open neighbourhood of \(x\). Namely, if \(d = 0\), then \(X \to S\) is quasi-finite at \(x\), see Morphisms, Lemma 0397. Hence there exists an affine open neighbourhood \(U \subset X\) such that \(U \to S\) is quasi-finite, see Morphisms, Lemma 01TI. Thus after replacing \(X\) by \(U\) we see that \(X\) is quasi-finite and smooth over \(S\), hence smooth of relative dimension \(0\) over \(S\), hence étale over \(S\). Moreover, the fibre \(X_s\) is a finite discrete set. Hence after replacing \(X\) by a further affine open neighbourhood of \(X\) we see that \(f^{-1}(\{s\}) = \{x\}\) (because the topology on \(X_s\) is induced from the topology on \(X\), see Schemes, Lemma 01K1). This proves the lemma in this case.

Next, assume \(d > 0\). Note that because \(x\) is a closed point of its fibre the extension \(\kappa(x)/\kappa(s)\) is finite (by the Hilbert Nullstellensatz, see Morphisms, Lemma 01TF). Thus we see \(\Omega_{\kappa(x)/\kappa(s)} = 0\) as this holds for algebraic separable field extensions. Thus we may apply Lemma 057D to find a diagram \[\xymatrix{ D \ar[r] \ar[rrd] & U \ar[r] \ar[rd] & X \ar[d] \\ & & S }\] with \(x \in D\). Note that \(\dim_x(D_s) = \dim_x(X_s) - 1\) for example because \(\dim(\mathcal{O}_{D_s, x}) = \dim(\mathcal{O}_{X_s, x}) - 1\) by Algebra, Lemma 00KW (also \(D_s \subset X_s\) is effective Cartier, see Divisors, Lemma 056Q) and then using Morphisms, Lemma 02FX. Thus the morphism \(D \to S\) is smooth with \(\dim_x(D_s) = \dim_x(X_s) - 1 = d - 1\). By induction hypothesis we can find an immersion \(Z \to D\) as desired, which finishes the proof.

Lemma

Let \(f : X \to S\) be a smooth morphism of schemes. Let \(s \in S\) be a point in the image of \(f\). Then there exists an étale neighbourhood \((S', s') \to (S, s)\) and a \(S\)-morphism \(S' \to X\).

Proof

By assumption \(X_s \not = \emptyset\). By Varieties, Lemma 056U there exists a closed point \(x \in X_s\) such that \(\kappa(x)\) is a finite separable field extension of \(\kappa(s)\). Hence by Lemma 057G there exists an immersion \(Z \to X\) such that \(Z \to S\) is étale and such that \(x \in Z\). Take \((S' , s') = (Z, x)\).

Proof

Pick a point \(x \in X\) with \(f(x) = s\). Choose a diagram \[\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-\pi & \mathbf{A}^d_V \ar[ld] \\ S & V \ar[l] }\] with \(\pi\) étale, \(x \in U\) and \(V = \Spec(R)\) affine, see Morphisms, Lemma 054L. In particular \(s \in V\). The morphism \(\pi : U \to \mathbf{A}^d_V\) is open, see Morphisms, Lemma 03WT. Thus \(W = \pi(U) \cap \mathbf{A}^d_s\) is a nonempty open subset of \(\mathbf{A}^d_s\). Let \(w \in W\) be a point with \(\kappa(s) \subset \kappa(w)\) finite separable, see Varieties, Lemma 055T. By Algebra, Lemma 00OP there exist \(d\) elements \(\overline{f}_1, \ldots, \overline{f}_d \in \kappa(s)[x_1, \ldots, x_d]\) which generate the maximal ideal corresponding to \(w\) in \(\kappa(s)[x_1, \ldots, x_d]\). After replacing \(R\) by a principal localization we may assume there are \(f_1, \ldots, f_d \in R[x_1, \ldots, x_d]\) which map to \(\overline{f}_1, \ldots, \overline{f}_d \in \kappa(s)[x_1, \ldots, x_d]\). Consider the \(R\)-algebra \[R' = R[x_1, \ldots, x_d]/(f_1, \ldots, f_d)\] and set \(S' = \Spec(R')\). By construction we have a closed immersion \(j : S' \to \mathbf{A}^d_V\) over \(V\). By construction the fibre of \(S' \to V\) over \(s\) is a single point \(s'\) whose residue field is finite separable over \(\kappa(s)\). Let \(\mathfrak q' \subset R'\) be the corresponding prime. By Algebra, Lemma 00ST we see that \((R')_g\) is a relative global complete intersection over \(R\) for some \(g \in R'\), \(g \not \in \mathfrak q\). Thus \(S' \to V\) is flat and of finite presentation in a neighbourhood of \(s'\), see Algebra, Lemma 00SW. By construction the scheme theoretic fibre of \(S' \to V\) over \(s\) is \(\Spec(\kappa(s'))\). Hence it follows from Morphisms, Lemma 02GU that \(S' \to S\) is étale at \(s'\). Set \[S'' = U \times_{\pi, \mathbf{A}^d_V, j} S'.\] By construction there exists a point \(s'' \in S''\) which maps to \(s'\) via the projection \(p : S'' \to S'\). Note that \(p\) is étale as the base change of the étale morphism \(\pi\), see Morphisms, Lemma 02GO. Choose a small affine neighbourhood \(S''' \subset S''\) of \(s''\) which maps into the nonempty open neighbourhood of \(s' \in S'\) where the morphism \(S' \to S\) is étale. Then the étale neighbourhood \((S''', s'') \to (S, s)\) is a solution to the problem posed by the lemma.

The following lemma shows that sheaves for the smooth topology are the same thing as sheaves for the étale topology.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{U} = \{S_i \to S\}_{i \in I}\) be a smooth covering of \(S\), see Topologies, Definition 021Z. Then there exists an étale covering \(\mathcal{V} = \{T_j \to S\}_{j \in J}\) (see Topologies, Definition 0215) which refines (see Sites, Definition 00VT) \(\mathcal{U}\).

Proof

For every \(s \in S\) there exists an \(i \in I\) such that \(s\) is in the image of \(S_i \to S\). By Lemma 055U we can find an étale morphism \(g_s : T_s \to S\) such that \(s \in g_s(T_s)\) and such that \(g_s\) factors through \(S_i \to S\). Hence \(\{T_s \to S\}\) is an étale covering of \(S\) that refines \(\mathcal{U}\).

Lemma

Let \(f : X \to S\) be a smooth morphism of schemes. Then there exists an étale covering \(\{U_i \to X\}_{i \in I}\) such that \(U_i \to S\) factors as \(U_i \to V_i \to S\) where \(V_i \to S\) is étale and \(U_i \to V_i\) is a smooth morphism of affine schemes, which has a section, and has geometrically connected fibres.

Proof

Let \(s \in S\). By Varieties, Lemma 056U the set of closed points \(x \in X_s\) such that \(\kappa(x)/\kappa(s)\) is separable is dense in \(X_s\). Thus it suffices to construct an étale morphism \(U \to X\) with \(x\) in the image such that \(U \to S\) factors in the manner described in the lemma. To do this, choose an immersion \(Z \to X\) passing through \(x\) such that \(Z \to S\) is étale (Lemma 057G). After replacing \(S\) by \(Z\) and \(X\) by \(Z \times_S X\) we see that we may assume \(X \to S\) has a section \(\sigma : S \to X\) with \(\sigma(s) = x\). Then we can first replace \(S\) by an affine open neighbourhood of \(s\) and next replace \(X\) by an affine open neighbourhood of \(x\). Then finally, we consider the subset \(X^0 \subset X\) of Section 055K. By Lemmas 055R and 055P this is a retrocompact open subscheme containing \(\sigma\) such that the fibres \(X^0 \to S\) are geometrically connected. If \(X^0\) is not affine, then we choose an affine open \(U \subset X^0\) containing \(x\). Since \(X^0 \to S\) is smooth, the image of \(U\) is open. Choose an affine open neighbourhood \(V \subset S\) of \(s\) contained in \(\sigma^{-1}(U)\) and in the image of \(U \to S\). Finally, the reader sees that \(U \cap f^{-1}(V) \to V\) has all the desired properties. For example \(U \cap f^{-1}(V)\) is equal to \(U \times_S V\) is affine as a fibre product of affine schemes. Also, the geometric fibres of \(U \cap f^{-1}(V) \to V\) are nonempty open subschemes of the irreducible fibres of \(X^0 \to S\) and hence connected. Some details omitted.

Étale neighbourhoods and Artin approximation

In this section we prove results of the form: if two pointed schemes have isomorphic complete local rings, then they have isomorphic étale neighbourhoods. We will rely on Popescu’s theorem, see Smoothing Ring Maps, Theorem 07GC.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(X\), \(Y\) be schemes locally of finite type over \(S\). Let \(x \in X\) and \(y \in Y\) be points lying over the same point \(s \in S\). Assume \(\mathcal{O}_{S, s}\) is a G-ring. Assume further we are given a local \(\mathcal{O}_{S, s}\)-algebra map \[\varphi : \mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, x}^\wedge\] For every \(N \geq 1\) there exists an elementary étale neighbourhood \((U, u) \to (X, x)\) and an \(S\)-morphism \(f : U \to Y\) mapping \(u\) to \(y\) such that the diagram \[\xymatrix{ \mathcal{O}_{X, x}^\wedge \ar[r] & \mathcal{O}_{U, u}^\wedge \\ \mathcal{O}_{Y, y} \ar[r]^{f^\sharp_u} \ar[u]^\varphi & \mathcal{O}_{U, u} \ar[u] }\] commutes modulo \(\mathfrak m_u^N\).

Proof

The question is local on \(X\) hence we may assume \(X\), \(Y\), \(S\) are affine. Say \(S = \Spec(R)\), \(X = \Spec(A)\), \(Y = \Spec(B)\). Write \(B = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)\). Let \(\mathfrak p \subset A\) be the prime ideal corresponding to \(x\). The local ring \(\mathcal{O}_{X, x} = A_\mathfrak p\) is a G-ring by More on Algebra, Proposition 07PV. The map \(\varphi\) is a map \[B_\mathfrak q^\wedge \longrightarrow A_\mathfrak p^\wedge\] where \(\mathfrak q \subset B\) is the prime corresponding to \(y\). Let \(a_1, \ldots, a_n \in A_\mathfrak p^\wedge\) be the images of \(x_1, \ldots, x_n\) via \(R[x_1, \ldots, x_n] \to B \to B_\mathfrak q^\wedge \to A_\mathfrak p^\wedge\). Then we can apply Smoothing Ring Maps, Lemma 0CAR to get an étale ring map \(A \to A'\) and a prime ideal \(\mathfrak p' \subset A'\) and \(b_1, \ldots, b_n \in A'\) such that \(\kappa(\mathfrak p) = \kappa(\mathfrak p')\), \(a_i - b_i \in (\mathfrak p')^N(A'_{\mathfrak p'})^\wedge\), and \(f_j(b_1, \ldots, b_n) = 0\) for \(j = 1, \ldots, n\). This determines an \(R\)-algebra map \(B \to A'\) by sending the class of \(x_i\) to \(b_i \in A'\). This finishes the proof by taking \(U = \Spec(A') \to \Spec(B)\) as the morphism \(f\) and \(u = \mathfrak p'\).

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(X\), \(Y\) be schemes locally of finite type over \(S\). Let \(x \in X\) and \(y \in Y\) be points lying over the same point \(s \in S\). Assume \(\mathcal{O}_{S, s}\) is a G-ring. Assume we have an \(\mathcal{O}_{S, s}\)-algebra isomorphism \[\varphi : \mathcal{O}_{Y, y}^\wedge \longrightarrow \mathcal{O}_{X, x}^\wedge\] between the complete local rings. Then for every \(N \geq 1\) there exists morphisms \[(X, x) \leftarrow (U, u) \rightarrow (Y, y)\] of pointed schemes over \(S\) such that both arrows define elementary étale neighbourhoods and such that the diagram \[\xymatrix{ & \mathcal{O}_{U, u}^\wedge \\ \mathcal{O}_{Y, y}^\wedge \ar[rr]^\varphi \ar[ru] & & \mathcal{O}_{X, x}^\wedge \ar[lu] }\] commutes modulo \(\mathfrak m_u^N\).

Proof

We may assume \(N \geq 2\). Apply Lemma 0CAU to get \((U, u) \to (X, x)\) and \(f : (U, u) \to (Y, y)\). We claim that \(f\) is étale at \(u\) which will finish the proof. In fact, we will show that the induced map \(\mathcal{O}_{Y, y}^\wedge \to \mathcal{O}_{U, u}^\wedge\) is an isomorphism. Having proved this, Lemma 02HX will show that \(f\) is smooth at \(u\) and of course \(f\) is unramified at \(u\) as well, so Morphisms, Lemma 02GK tells us \(f\) is étale at \(u\). For a local ring \((R, \mathfrak m)\) we set \(\text{Gr}_\mathfrak m(R) = \bigoplus_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1}\). To prove the claim we look at the induced diagram of graded rings \[\xymatrix{ & \text{Gr}_{\mathfrak m_u}(\mathcal{O}_{U, u}) \\ \text{Gr}_{\mathfrak m_y}(\mathcal{O}_{Y, y}) \ar[rr]^\varphi \ar[ru] & & \text{Gr}_{\mathfrak m_x}(\mathcal{O}_{X, x}) \ar[lu] }\] Since \(N \geq 2\) this diagram is actually commutative as the displayed graded algebras are generated in degree \(1\)! By assumption the lower arrow is an isomorphism. By More on Algebra, Lemma 0AGX (for example) the map \(\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{U, u}^\wedge\) is an isomorphism and hence the north-west arrow in the diagram is an isomorphism. We conclude that \(f\) induces an isomorphism \(\text{Gr}_{\mathfrak m_x}(\mathcal{O}_{X, x}) \to \text{Gr}_{\mathfrak m_y}(\mathcal{O}_{U, u})\). Using induction and the short exact sequences \[0 \to \text{Gr}^n_{\mathfrak m}(R) \to R/\mathfrak m^{n + 1} \to R/\mathfrak m^n \to 0\] for both local rings we conclude (from the snake lemma) that \(f\) induces isomorphisms \(\mathcal{O}_{Y, y}/\mathfrak m_y^n \to \mathcal{O}_{U, u}/\mathfrak m_u^n\) for all \(n\) which is what we wanted to show.

Lemma

Let \(X \to S\), \(Y \to T\), \(x\), \(s\), \(y\), \(t\), \(\sigma\), \(y_\sigma\), and \(\varphi\) be given as follows: we have morphisms of schemes \[\vcenter{ \xymatrix{ X \ar[d] & Y \ar[d] \\ S & T } } \quad\text{with points}\quad \vcenter{ \xymatrix{ x \ar[d] & y \ar[d] \\ s & t } }\] Here \(S\) is locally Noetherian and \(T\) is of finite type over \(\mathbf{Z}\). The morphisms \(X \to S\) and \(Y \to T\) are locally of finite type. The local ring \(\mathcal{O}_{S, s}\) is a G-ring. The map \[\sigma : \mathcal{O}_{T, t} \longrightarrow \mathcal{O}_{S, s}^\wedge\] is a local homomorphism. Set \(Y_\sigma = Y \times_{T, \sigma} \Spec(\mathcal{O}_{S, s}^\wedge)\). Next, \(y_\sigma\) is a point of \(Y_\sigma\) mapping to \(y\) and the closed point of \(\Spec(\mathcal{O}_{S, s}^\wedge)\). Finally \[\varphi : \mathcal{O}_{X, x}^\wedge \longrightarrow \mathcal{O}_{Y_\sigma, y_\sigma}^\wedge\] is an isomorphism of \(\mathcal{O}_{S, s}^\wedge\)-algebras. In this situation there exists a commutative diagram \[\xymatrix{ X \ar[d] & W \ar[l] \ar[rd] \ar[rr] & & Y \times_{T, \tau} V \ar[r] \ar[ld] & Y \ar[d] \\ S & & V \ar[ll] \ar[rr]^\tau & & T }\] of schemes and points \(w \in W\), \(v \in V\) such that

\((V, v) \to (S, s)\) is an elementary étale neighbourhood,

\((W, w) \to (X, x)\) is an elementary étale neighbourhood, and

\(\tau(v) = t\).

Let \(y_\tau \in Y \times_T V\) correspond to \(y_\sigma\) via the identification \((Y_\sigma)_s = (Y \times_T V)_v\). Then

\((W, w) \to (Y \times_{T, \tau} V, y_\tau)\) is an elementary étale neighbourhood.

Proof

Denote \(X_\sigma = X \times_S \Spec(\mathcal{O}_{S, s}^\wedge)\) and \(x_\sigma \in X_\sigma\) the unique point lying over \(x\). Observe that \(\mathcal{O}_{S, s}^\wedge\) is a G-ring by More on Algebra, Proposition 07PS. By Lemma 0CAV we can choose \[(X_\sigma, x_\sigma) \leftarrow (U, u) \rightarrow (Y_\sigma, y_\sigma)\] where both arrows are elementary étale neighbourhoods.

After replacing \(S\) by an open neighbourhood of \(s\), we may assume \(S = \Spec(R)\) is affine. Since \(\mathcal{O}_{S, s}\) is a G-ring by Smoothing Ring Maps, Theorem 07GC the ring \(\mathcal{O}_{S, s}^\wedge\) is a filtered colimit of smooth \(R\)-algebras. Thus we can write \[\Spec(\mathcal{O}_{S, s}^\wedge) = \lim S_i\] as a directed limit of affine schemes \(S_i\) smooth over \(S\). Denote \(s_i \in S_i\) the image of the closed point of \(\Spec(\mathcal{O}_{S, s}^\wedge)\). Observe that \(\kappa(s) = \kappa(s_i)\). Set \(X_i = X \times_S S_i\) and denote \(x_i \in X_i\) the unique point mapping to \(x\). Note that \(\kappa(x) = \kappa(x_i)\). Since \(T\) is of finite type over \(\mathbf{Z}\) by Limits, Proposition 01ZC we can choose an \(i\) and a morphism \(\sigma_i : (S_i, s_i) \to (T, t)\) of pointed schemes whose composition with \(\Spec(\mathcal{O}_{S, s}^\wedge) \to S_i\) is equal to \(\sigma\). Set \(Y_i = Y \times_T S_i\) and denote \(y_i\) the image of \(y_\sigma\). Note that \(\kappa(y_i) = \kappa(y_\sigma)\). By Limits, Lemma 01ZM we can choose an \(i\) and a diagram \[\xymatrix{ X_i \ar[rd] & U_i \ar[l] \ar[d] \ar[r] & Y_i \ar[ld] \\ & S_i }\] whose base change to \(\Spec(\mathcal{O}_{S, s}^\wedge)\) recovers \(X_\sigma \leftarrow U \rightarrow Y_\sigma\). By Limits, Lemma 07RP after increasing \(i\) we may assume the morphisms \(X_i \leftarrow U_i \rightarrow Y_i\) are étale. Let \(u_i \in U_i\) be the image of \(u\). Then \(u_i \mapsto x_i\) hence \(\kappa(x) = \kappa(x_\sigma) = \kappa(u) \supset \kappa(u_i) \supset \kappa(x_i) = \kappa(x)\) and we see that \(\kappa(u_i) = \kappa(x_i)\). Hence \((X_i, x_i) \leftarrow (U_i, u_i)\) is an elementary étale neighbourhood. Since also \(\kappa(y_i) = \kappa(y_\sigma) = \kappa(u)\) we see that also \((U_i, u_i) \to (Y_i, y_i)\) is an elementary étale neighbourhood.

At this point we have constructed a diagram \[\xymatrix{ X \ar[d] & X \times_S S_i \ar[l] \ar[rd] & U_i \ar[l] \ar[r] \ar[d] & Y \times_T S_i \ar[r] \ar[ld] & Y \ar[d] \\ S & & S_i \ar[ll] \ar[rr] & & T }\] as in the statement of the lemma, except that \(S_i \to S\) is smooth. By Lemma 057G and after shrinking \(S_i\) we can assume there exists a closed subscheme \(V \subset S_i\) passing through \(s_i\) such that \(V \to S\) is étale. Setting \(W\) equal to the scheme theoretic inverse image of \(V\) in \(U_i\) we conclude.

We strongly encourage the reader to skip the rest of this section.

Lemma

Consider a diagram \[\vcenter{ \xymatrix{ X \ar[d] & Y \ar[d] \\ S & T \ar[l] } } \quad\text{with points}\quad \vcenter{ \xymatrix{ x \ar[d] & y \ar[d] \\ s & t \ar[l] } }\] where \(S\) be a locally Noetherian scheme and the morphisms are locally of finite type. Assume \(\mathcal{O}_{S, s}\) is a G-ring. Assume further we are given a local \(\mathcal{O}_{S, s}\)-algebra map \[\sigma : \mathcal{O}_{T, t} \longrightarrow \mathcal{O}_{S, s}^\wedge\] and a local \(\mathcal{O}_{S, s}\)-algebra map \[\varphi : \mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{Y_\sigma, y_\sigma}^\wedge\] where \(Y_\sigma = Y \times_{T, \sigma} \Spec(\mathcal{O}_{S, s}^\wedge)\) and \(y_\sigma\) is the unique point of \(Y_\sigma\) lying over \(y\). For every \(N \geq 1\) there exists a commutative diagram \[\xymatrix{ X \ar[d] & X \times_S V \ar[l] \ar[rd] & W \ar[l]^-f \ar[r] \ar[d] & Y \times_{T, \tau} V \ar[r] \ar[ld] & Y \ar[d] \\ S & & V \ar[ll] \ar[rr]^\tau & & T }\] of schemes over \(S\) and points \(w \in W\), \(v \in V\) such that

\(v \mapsto s\), \(\tau(v) = t\), \(f(w) = (x, v)\), and \(w \mapsto (y, v)\),

\((V, v) \to (S, s)\) is an elementary étale neighbourhood,

the diagram \[\xymatrix{ \mathcal{O}_{S, s}^\wedge \ar[r] & \mathcal{O}_{V, v}^\wedge \\ \mathcal{O}_{T, t} \ar[r]^{\tau^\sharp_v} \ar[u]_\sigma & \mathcal{O}_{V, v} \ar[u] }\] commutes module \(\mathfrak m_v^N\),

\((W, w) \to (Y \times_{T, \tau} V, (y, v))\) is an elementary étale neighbourhood,

the diagram \[\xymatrix{ \mathcal{O}_{X, x} \ar[r]_\varphi & \mathcal{O}_{Y_\sigma, y_\sigma}^\wedge \ar[r] & \mathcal{O}_{Y_\sigma, y_\sigma}/\mathfrak m_{y_\sigma}^N \ar@{=}[r] & \mathcal{O}_{Y \times_{T, \tau} V, (y, v)}/\mathfrak m_{(y, v)}^N \ar[d]_{\cong} \\ \mathcal{O}_{X, x} \ar[r] \ar@{=}[u] & \mathcal{O}_{X \times_S V, (x, v)} \ar[r]^{f^\sharp_w} & \mathcal{O}_{W, w} \ar[r] & \mathcal{O}_{W, w}/\mathfrak m_w^N }\] commutes. The equality comes from the fact that \(Y_\sigma\) and \(Y \times_{T, \tau} V\) are canonically isomorphic over \(\mathcal{O}_{V, v}/\mathfrak m_v^N = \mathcal{O}_{S, s}/\mathfrak m_s^N\) by parts (2) and (3).

Proof

After replacing \(X\), \(S\), \(T\), \(Y\) by affine open subschemes we may assume the diagram in the statement of the lemma comes from applying \(\Spec\) to a diagram \[\vcenter{ \xymatrix{ A & B \\ R \ar[u] \ar[r] & C \ar[u] } } \quad\text{with primes}\quad \vcenter{ \xymatrix{ \mathfrak p_A & \mathfrak p_B \\ \mathfrak p_R \ar@{-}[u] \ar@{-}[r] & \mathfrak p_C \ar@{-}[u] } }\] of Noetherian rings and finite type ring maps. In this proof every ring \(E\) will be a Noetherian \(R\)-algebra endowed with a prime ideal \(\mathfrak p_E\) lying over \(\mathfrak p_R\) and all ring maps will be \(R\)-algebra maps compatible with the given primes. Moreover, if we write \(E^\wedge\) we mean the completion of the localization of \(E\) at \(\mathfrak p_E\). We will also use without further mention that an étale ring map \(E_1 \to E_2\) such that \(\kappa(\mathfrak p_{E_1}) = \kappa(\mathfrak p_{E_2})\) induces an isomorphism \(E_1^\wedge = E_2^\wedge\) by More on Algebra, Lemma 0AGX.

With this notation \(\sigma\) and \(\varphi\) correspond to ring maps \[\sigma : C \to R^\wedge \quad\text{and}\quad \varphi : A \longrightarrow (B \otimes_{C, \sigma} R^\wedge)^\wedge\] Here is a picture \[\xymatrix{ A \ar@/^1em/[rrr]^\varphi & B \ar[r] & B \otimes_{C, \sigma} R^\wedge \ar[r] & (B \otimes_{C, \sigma} R^\wedge)^\wedge \\ R \ar[r] \ar[u] & C \ar[r]^\sigma \ar[u] & R^\wedge \ar[u] \ar[ru] }\] Observe that \(R^\wedge\) is a G-ring by More on Algebra, Proposition 07PS. Thus \(B \otimes_{C, \sigma} R^\wedge\) is a G-ring by More on Algebra, Proposition 07PV. By Lemma 0CAU (translated into algebra) there exists an étale ring map \(B \otimes_{C, \sigma} R^\wedge \to B'\) inducing an isomorphism \(\kappa(\mathfrak p_{B \otimes_{C, \sigma} R^\wedge}) \to \kappa(\mathfrak p_{B'})\) and an \(R\)-algebra map \(A \to B'\) such that the composition \[A \to B' \to (B')^\wedge = (B \otimes_{C, \sigma} R^\wedge)^\wedge\] is the same as \(\varphi\) modulo \((\mathfrak p_{(B \otimes_{C, \sigma} R^\wedge)^\wedge})^N\). Thus we may replace \(\varphi\) by this composition because the only way \(\varphi\) enters the conclusion is via the commutativity requirement in part (5) of the statement of the lemma. Picture: \[\xymatrix{ & & B' \ar[r] & (B')^\wedge \ar@{=}[d] \\ A \ar[rru] & B \ar[r] & B \otimes_{C, \sigma} R^\wedge \ar[r] \ar[u] & (B \otimes_{C, \sigma} R^\wedge)^\wedge \\ R \ar[r] \ar[u] & C \ar[r]^\sigma \ar[u] & R^\wedge \ar[u] \ar[ru] }\] Next, we use that \(R^\wedge\) is a filtered colimit of smooth \(R\)-algebras (Smoothing Ring Maps, Theorem 07GC) because \(R_{\mathfrak p_R}\) is a G-ring by assumption. Since \(C\) is of finite presentation over \(R\) we get a factorization \[C \to R' \to R^\wedge\] for some \(R \to R'\) smooth, see Algebra, Lemma 00QO. After increasing \(R'\) we may assume there exists an étale \(B \otimes_C R'\)-algebra \(B''\) whose base change to \(B \otimes_{C, \sigma} R^\wedge\) is \(B'\), see Algebra, Lemma 00U2. Then \(B'\) is the filtered colimit of these \(B''\) and we conclude that after increasing \(R'\) we may assume there is an \(R\)-algebra map \(A \to B''\) such that \(A \to B'' \to B'\) is the previously constructed map (same reference as above). Picture \[\xymatrix{ & & B'' \ar[r] & B' \ar[r] & (B')^\wedge \ar@{=}[d] \\ A \ar[rru] & B \ar[r] & B \otimes_C R' \ar[r] \ar[u] & B \otimes_{C, \sigma} R^\wedge \ar[r] \ar[u] & (B \otimes_{C, \sigma} R^\wedge)^\wedge \\ R \ar[r] \ar[u] & C \ar[r] \ar[u] & R' \ar[r] \ar[u] & R^\wedge \ar[u] \ar[ru] }\] and \[B' = B'' \otimes_{(B \otimes_C R')} (B \otimes_{C, \sigma} R^\wedge)\] This means that we may replace \(C\) by \(R'\), \(\sigma : C \to R^\wedge\) by \(R' \to R^\wedge\), and \(B\) by \(B''\) so that we simplify to the diagram \[\xymatrix{ A \ar[r] & B \ar[r] & B \otimes_{C, \sigma} R^\wedge \\ R \ar[r] \ar[u] & C \ar[r]^\sigma \ar[u] & R^\wedge \ar[u] }\] with \(\varphi\) equal to the composition of the horizontal arrows followed by the canonical map from \(B \otimes_{C, \sigma} R^\wedge\) to its completion. The final step in the proof is to apply Lemma 0CAU (or its proof) one more time to \(\Spec(C)\) and \(\Spec(R)\) over \(\Spec(R)\) and the map \(C \to R^\wedge\). The lemma produces a ring map \(C \to D\) such that \(R \to D\) is étale, such that \(\kappa(\mathfrak p_R) = \kappa(\mathfrak p_D)\), and such that \[C \to D \to D^\wedge = R^\wedge\] is equal to \(\sigma : C \to R^\wedge\) modulo \((\mathfrak p_{R^\wedge})^N\). Then we can take \[V = \Spec(D) \quad\text{and}\quad W = \Spec(B \otimes_C D)\] as our solution to the problem posed by the lemma. Namely the diagram \[\xymatrix{ A \ar[r] & B \otimes_{C, \sigma} R^\wedge \ar[r] & B \otimes_{C, \sigma} R^\wedge/(\mathfrak p_{R^\wedge})^N \ar@{=}[r] & B \otimes_C D/(\mathfrak p_D)^N \\ A \ar@{=}[u] \ar[r] & A \otimes_R D \ar[r] & B \otimes_R D \ar[r] & B \otimes_C D/(\mathfrak p_D)^N \ar@{=}[u] }\] commutes because \(C \to D \to D^\wedge = R^\wedge\) is equal to \(\sigma\) modulo \((\mathfrak p_{R^\wedge})^N\). This proves part (5) and the other properties are immediate from the construction.

Lemma

Let \(T \to S\) be finite type morphisms of Noetherian schemes. Let \(t \in T\) map to \(s \in S\) and let \(\sigma : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge\) be a local \(\mathcal{O}_{S, s}\)-algebra map. For every \(N \geq 1\) there exists a finite type morphism \((T', t') \to (T, t)\) such that \(\sigma\) factors through \(\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}\) and such that for every local \(\mathcal{O}_{S, s}\)-algebra map \(\sigma' : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge\) which factors through \(\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}\) the maps \(\sigma\) and \(\sigma'\) agree modulo \(\mathfrak m_s^N\).

Proof

We may assume \(S\) and \(T\) are affine. Say \(S = \Spec(R)\) and \(T = \Spec(C)\). Let \(c_1, \ldots, c_n \in C\) be generators of \(C\) as an \(R\)-algebra. Let \(\mathfrak p \subset R\) be the prime ideal corresponding to \(s\). Say \(\mathfrak p = (f_1, \ldots, f_m)\). After replacing \(R\) by a principal localization (to clear denominators in \(R_\mathfrak p\)) we may assume there exist \(r_1, \ldots, r_n \in R\) and \(a_{i, I} \in \mathcal{O}_{S, s}^\wedge\) where \(I = (i_1, \ldots, i_m)\) with \(\sum i_j = N\) such that \[\sigma(c_i) = r_i + \sum\nolimits_I a_{i, I} f_1^{i_1} \ldots f_m^{i_m}\] in \(\mathcal{O}_{S, s}^\wedge\). Then we consider \[C' = C[t_{i, I}]/ \left(c_i - r_i - \sum\nolimits_I t_{i, I} f_1^{i_1} \ldots f_m^{i_m}\right)\] with \(\mathfrak p' = \mathfrak pC' + (t_{i, I})\) and factorization of \(\sigma : C \to \mathcal{O}_{S, s}^\wedge\) through \(C'\) given by sending \(t_{i, I}\) to \(a_{i, I}\). Taking \(T' = \Spec(C')\) works because any \(\sigma'\) as in the statement of the lemma will send \(c_i\) to \(r_i\) modulo the maximal ideal to the power \(N\).

Lemma

Let \(Y \to T \to S\) be finite type morphisms of Noetherian schemes. Let \(t \in T\) map to \(s \in S\) and let \(\sigma : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge\) be a local \(\mathcal{O}_{S, s}\)-algebra map. There exists a finite type morphism \((T', t') \to (T, t)\) such that \(\sigma\) factors through \(\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}\) and such that for every local \(\mathcal{O}_{S, s}\)-algebra map \(\sigma' : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge\) which factors through \(\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}\) the closed immersions \[Y \times_{T, \sigma} \Spec(\mathcal{O}_{S, s}^\wedge) = Y_\sigma \longleftarrow Y_t \longrightarrow Y_{\sigma'} = Y \times_{T, \sigma'} \Spec(\mathcal{O}_{S, s}^\wedge)\] have isomorphic conormal algebras.

Proof

A useful observation is that \(\kappa(s) = \kappa(t)\) by the existence of \(\sigma\). Observe that the statement makes sense as the fibres of \(Y_\sigma\) and \(Y_{\sigma'}\) over \(s \in \Spec(\mathcal{O}_{S, s}^\wedge)\) are both canonically isomorphic to \(Y_t\). We will think of the property “\(\sigma'\) factors through \(\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}\)” as a constraint on \(\sigma'\). If we have several such constraints, say given by \((T'_i, t'_i) \to (T, t)\), \(i = 1, \ldots, n\) then we can combined them by considering \((T'_1 \times_T \ldots \times_T T'_n, (t'_1, \ldots, t'_n)) \to (T, t)\). We will use this without further mention in the following.

By Lemma 0CAX we can assume that any \(\sigma'\) as in the statement of the lemma is the same as \(\sigma\) modulo \(\mathfrak m_s^2\). Note that the conormal algebra of \(Y_t\) in \(Y_\sigma\) is just the quasi-coherent graded \(\mathcal{O}_{Y_t}\)-algebra \[\bigoplus\nolimits_{n \geq 0} \mathfrak m_s^n\mathcal{O}_{Y_\sigma}/ \mathfrak m_s^{n + 1}\mathcal{O}_{Y_\sigma}\] and similarly for \(Y_{\sigma'}\). Since \(\sigma\) and \(\sigma'\) agree modulo \(\mathfrak m_s^2\) we see that these two algebras are the same in degrees \(0\) and \(1\). On the other hand, these conormal algebras are generated in degree \(1\) over degree \(0\). Hence if there is an isomorphism extending the isomorphism just constructed in degrees \(0\) and \(1\), then it is unique.

We may assume \(S\) and \(T\) are affine. Let \(Y = Y_1 \cup \ldots \cup Y_n\) be an affine open covering. If we can construct \((T_i', t'_i) \to (T, t)\) as in the lemma such that the desired isomorphism (see previous paragraph) exists for \(Y_i \to T \to S\) and \(\sigma\), then these glue by uniqueness to prove the result for \(Y \to T\). Thus we may assume \(Y\) is affine.

Write \(S = \Spec(R)\), \(T = \Spec(C)\), and \(Y = \Spec(B)\). Choose a presentation \(B = C[x_1, \ldots, x_n]/(f_1, \ldots, f_m)\). Denote \(R^\wedge = \mathcal{O}_{S, s}^\wedge\). Let \(a_{kj} \in R^\wedge[x_1, \ldots, x_n]\) be polynomials such that \[\sum\nolimits_{j = 1, \ldots, m} a_{kj}\sigma(f_j) = 0,\quad \text{for }k = 1, \ldots, K\] is a set of generators for the module of relations among the \(\sigma(f_j) \in R^\wedge[x_1, \ldots, x_n]\). Thus we have an exact sequence [0CAZ]\[\begin{equation} R^\wedge[x_1, \ldots, x_n]^{\oplus K} \to R^\wedge[x_1, \ldots, x_n]^{\oplus m} \to R^\wedge[x_1, \ldots, x_n] \to B \otimes_{C, \sigma} R^\wedge \to 0 \end{equation}\] Let \(c\) be an integer which works in the Artin-Rees lemma for both the first and the second map in this sequence and the ideal \(\mathfrak m_{R^\wedge}R^\wedge[x_1, \ldots, x_n]\) as defined in More on Algebra, Section 07VD. Write \[a_{kj} = \sum\nolimits_{I \in \Omega} a_{kj, I} x^I \quad\text{and}\quad f_j = \sum\nolimits_{I \in \Omega} f_{j, I} x^I\] in multiindex notation where \(a_{kj, I} \in R^\wedge\), \(f_{j, I} \in C\), and \(\Omega\) a finite set of multiindices. Then we see that \[\sum\nolimits_{j = 1, \ldots, m,\ I, I' \in \Omega,\ I + I' = I''} a_{kj, I} \sigma(f_{j, I'}) = 0,\quad I''\text{ a multiindex}\] in \(R^\wedge\). Thus we take \[C' = C[t_{jk, I}]/ \left( \sum\nolimits_{j = 1, \ldots, m,\ I, I' \in \Omega,\ I + I' = I''} t_{kj, I} f_{j, I'},\ I''\text{ a multiindex}\right)\] Then \(\sigma\) factors through a map \(\tilde\sigma : C' \to R^\wedge\) sending \(t_{kj, I}\) to \(a_{jk, I}\). Thus \(T' = \Spec(C')\) comes with a point \(t' \in T'\) such that \(\sigma\) factors through \(\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}\). Let \(t_{kj} = \sum t_{kj, I} x^I\) in \(C'[x_1, \ldots, x_n]\). Then we see that we have a complex [0CB0]\[\begin{equation} C'[x_1, \ldots, x_n]^{\oplus K} \to C'[x_1, \ldots, x_n]^{\oplus m} \to C'[x_1, \ldots, x_n] \to B \otimes_C C' \to 0 \end{equation}\] which is exact at \(C'[x_1, \ldots, x_n]\) and whose base change by \(\tilde\sigma\) gives (0CAZ).

By Lemma 0CAX we can find a further morphism \((T'', t'') \to (T', t')\) such that \(\tilde\sigma\) factors through \(\mathcal{O}_{T', t'} \to \mathcal{O}_{T'', t''}\) and such that if \(\sigma' : C \to R^\wedge\) factors through \(\mathcal{O}_{T'', t''}\), then the induced map \(\tilde \sigma' : C' \to R^\wedge\) agrees modulo \(\mathfrak m_s^{c + 1}\) with \(\tilde \sigma\). Thus if \(\sigma'\) is such a map, then we obtain a complex \[R^\wedge[x_1, \ldots, x_n]^{\oplus K} \to R^\wedge[x_1, \ldots, x_n]^{\oplus m} \to R^\wedge[x_1, \ldots, x_n] \to B \otimes_{C, \sigma'} R^\wedge \to 0\] over \(R^\wedge[x_1, \ldots, x_n]\) by applying \(\tilde\sigma'\) to the polynomials \(t_{kj}\) and \(f_j\). In other words, this is the base change of the complex (0CB0) by \(\tilde\sigma'\). The matrices defining this complex are congruent modulo \(\mathfrak m_s^{c + 1}\) to the matrices defining the complex (0CAZ) because \(\tilde \sigma\) and \(\tilde \sigma'\) are congruent modulo \(\mathfrak m_s^{c + 1}\). Since (0CAZ) is exact, we can apply More on Algebra, Lemma 07VF to conclude that \[\text{Gr}_{\mathfrak m_s}(B \otimes_{C, \sigma'} R^\wedge) \cong \text{Gr}_{\mathfrak m_s}(B \otimes_{C, \sigma} R^\wedge)\] as desired.

Lemma

With notation an assumptions as in Lemma 0CAW assume that \(\varphi\) induces an isomorphism on completions. Then we can choose our diagram such that \(f\) is étale.

Proof

We may assume \(N \geq 2\) and we may replace \((T, t)\) with \((T', t')\) as in Lemma 0CAY. Since \((V, v) \to (S, s)\) is an elementary étale neighbourhood, so is \((X \times_S V, (x, v)) \to (X, x)\). Thus \(\mathcal{O}_{X, x} \to \mathcal{O}_{X \times_S V, (x, v)}\) induces an isomorphism on completions by More on Algebra, Lemma 0AGX. We claim \(\mathcal{O}_{X, x} \to \mathcal{O}_{W, w}\) induces an isomorphism on completions. Having proved this, Lemma 02HX will show that \(f\) is smooth at \(w\) and of course \(f\) is unramified at \(u\) as well, so Morphisms, Lemma 02GK tells us \(f\) is étale at \(w\).

First we use the commutativity in part (5) of Lemma 0CAW to see that for \(i \leq N\) there is a commutative diagram \[\xymatrix{ \text{Gr}^i_{\mathfrak m_x}(\mathcal{O}_{X, x}) \ar[r]_-\varphi & \text{Gr}^i_{\mathfrak m_{y_\sigma}}(\mathcal{O}_{Y_\sigma, y_\sigma}^\wedge) \ar@{=}[r] & \text{Gr}^i_{\mathfrak m_{(y, v)}}(\mathcal{O}_{Y \times_{T, \tau} V, (y, v)}) \ar[d]_{\cong} \\ \text{Gr}^i_{\mathfrak m_x}(\mathcal{O}_{X, x}) \ar[r]^-{\cong} \ar@{=}[u] & \text{Gr}^i_{\mathfrak m_{(x, v)}}(\mathcal{O}_{X \times_S V, (x, v)}) \ar[r]^{f^\sharp_w} & \text{Gr}^i_{\mathfrak m_w}(\mathcal{O}_{W, w}) }\] This implies that \(f^\sharp_w\) defines an isomorphism \(\kappa(x) \to \kappa(w)\) on residue fields and an isomorphism \(\mathfrak m_x/\mathfrak m_x^2 \to \mathfrak m_w/\mathfrak m_w^2\) on cotangent spaces. Hence \(f^\sharp_w\) defines a surjection \(\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{W, w}^\wedge\) on complete local rings.

By Lemma 0CAY there is an isomorphism of \(\text{Gr}_{\mathfrak m_s}(\mathcal{O}_{(Y \times_{T, \tau} V, (y, v)})\) with \(\text{Gr}_{\mathfrak m_s}(\mathcal{O}_{Y_\sigma, y_\sigma})\). This follows by taking stalks of the isomorphism of conormal sheaves at the point \(y\). Since our local rings are Noetherian taking associated graded with respect to \(\mathfrak m_s\) commutes with completion because completion with respect to an ideal is an exact functor on finite modules over Noetherian rings. This produces the right vertical isomorphism in the diagram of graded rings \[\xymatrix{ \text{Gr}_{\mathfrak m_s}(\mathcal{O}_{W, w}^\wedge) & \text{Gr}_{\mathfrak m_s} (\mathcal{O}_{(Y \times_{T, \tau} V, (y, v)}^\wedge) \ar[l] \\ \text{Gr}_{\mathfrak m_s}(\mathcal{O}_{X, x}^\wedge) \ar[r]^\varphi \ar[u] & \text{Gr}_{\mathfrak m_s}(\mathcal{O}_{Y_\sigma, y_\sigma}^\wedge) \ar[u]_{\cong} }\] We do not claim the diagram commutes. By the result of the previous paragraph the left arrow is surjective. The other three arrows are isomorphisms. It follows that the left arrow is a surjective map between isomorphic Noetherian rings. Hence it is an isomorphism by Algebra, Lemma 06RN (you can argue this directly using Hilbert functions as well). In particular \(\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{W, w}^\wedge\) must be injective as well as surjective which finishes the proof.

Finite free locally dominates étale

In this section we explain a result that roughly states that étale coverings of a scheme \(S\) can be refined by Zariski coverings of finite locally free covers of \(S\).

Lemma

Let \(S\) be a scheme. Let \(s \in S\). Let \(f : (U, u) \to (S, s)\) be an étale neighbourhood. There exists an affine open neighbourhood \(s \in V \subset S\) and a surjective, finite locally free morphism \(\pi : T \to V\) such that for every \(t \in \pi^{-1}(s)\) there exists an open neighbourhood \(t \in W_t \subset T\) and a commutative diagram \[\xymatrix{ T \ar[d]^\pi & W_t \ar[l] \ar[rr]_{h_t} \ar[rd] & & U \ar[dl] \\ V \ar[rr] & & S }\] with \(h_t(t) = u\).

Proof

The problem is local on \(S\) hence we may replace \(S\) by any open neighbourhood of \(s\). We may also replace \(U\) by an open neighbourhood of \(u\). Hence, by Morphisms, Lemma 02GT we may assume that \(U \to S\) is a standard étale morphism of affine schemes. In this case the lemma (with \(V = S\)) follows from Algebra, Lemma 00UF.

Lemma

Let \(f : U \to S\) be a surjective étale morphism of affine schemes. There exists a surjective, finite locally free morphism \(\pi : T \to S\) and a finite open covering \(T = T_1 \cup \ldots \cup T_n\) such that each \(T_i \to S\) factors through \(U \to S\). Diagram: \[\xymatrix{ & \coprod T_i \ar[rd] \ar[ld] & \\ T \ar[rd]^\pi & & U \ar[ld]_f \\ & S & }\] where the south-west arrow is a Zariski-covering.

Proof

This is a restatement of Algebra, Lemma 00UG.

Remark

In terms of topologies Lemmas 02LG and 02LH mean the following. Let \(S\) be any scheme. Let \(\{f_i : U_i \to S\}\) be an étale covering of \(S\). There exists a Zariski open covering \(S = \bigcup V_j\), for each \(j\) a finite locally free, surjective morphism \(W_j \to V_j\), and for each \(j\) a Zariski open covering \(\{W_{j, k} \to W_j\}\) such that the family \(\{W_{j, k} \to S\}\) refines the given étale covering \(\{f_i : U_i \to S\}\). What does this mean in practice? Well, for example, suppose we have a descent problem which we know how to solve for Zariski coverings and for fppf coverings of the form \(\{\pi : T \to S\}\) with \(\pi\) finite locally free and surjective. Then this descent problem has an affirmative answer for étale coverings as well. This trick was used by Gabber in his proof that \(\text{Br}(X) = \text{Br}'(X)\) for an affine scheme \(X\), see [Hoobler].

Étale localization of quasi-finite morphisms

Now we come to a series of lemmas around the theme “quasi-finite morphisms become finite after étale localization”. The general idea is the following. Suppose given a morphism of schemes \(f : X \to S\) and a point \(s \in S\). Let \(\varphi : (U, u) \to (S, s)\) be an étale neighbourhood of \(s\) in \(S\). Consider the fibre product \(X_U = U \times_S X\) and the basic diagram [02LJ]\[\begin{equation} \vcenter{ \xymatrix{ V \ar[r] \ar[dr] & X_U \ar[d] \ar[r] & X \ar[d]^f \\ & U \ar[r]^\varphi & S } } \end{equation}\] where \(V \subset X_U\) is open. Is there some standard model for the morphism \(f_U : X_U \to U\), or for the morphism \(V \to U\) for suitable opens \(V\)? Of course the answer is no in general. But for quasi-finite morphisms we can say something.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\). Set \(s = f(x)\). Assume that

\(f\) is locally of finite type, and

\(x \in X_s\) is isolated⁷.

Then there exist

an elementary étale neighbourhood \((U, u) \to (S, s)\),

an open subscheme \(V \subset X_U\) (see 02LJ)

such that

\(V \to U\) is a finite morphism,

there is a unique point \(v\) of \(V\) mapping to \(u\) in \(U\), and

the point \(v\) maps to \(x\) under the morphism \(X_U \to X\), inducing \(\kappa(x) = \kappa(v)\).

Moreover, for any elementary étale neighbourhood \((U', u') \to (U, u)\) setting \(V' = U' \times_U V \subset X_{U'}\) the triple \((U', u', V')\) satisfies the properties (i), (ii), and (iii) as well.

Proof

Let \(Y \subset X\), \(W \subset S\) be affine opens such that \(f(Y) \subset W\) and such that \(x \in Y\). Note that \(x\) is also an isolated point of the fibre of the morphism \(f|_Y : Y \to W\). If we can prove the theorem for \(f|_Y : Y \to W\), then the theorem follows for \(f\). Hence we reduce to the case where \(f\) is a morphism of affine schemes. This case is Algebra, Lemma 00UJ.

In the preceding and following lemma we do not assume that the morphism \(f\) is separated. This means that the opens \(V\), \(V_i\) created in them are not necessarily closed in \(X_U\). Moreover, if we choose the neighbourhood \(U\) to be affine, then each \(V_i\) is affine, but the intersections \(V_i \cap V_j\) need not be affine (in the nonseparated case).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x_1, \ldots, x_n \in X\) be points having the same image \(s\) in \(S\). Assume that

\(f\) is locally of finite type, and

\(x_i \in X_s\) is isolated for \(i = 1, \ldots, n\).

Then there exist

an elementary étale neighbourhood \((U, u) \to (S, s)\),

for each \(i\) an open subscheme \(V_i \subset X_U\),

such that for each \(i\) we have

\(V_i \to U\) is a finite morphism,

there is a unique point \(v_i\) of \(V_i\) mapping to \(u\) in \(U\), and

the point \(v_i\) maps to \(x_i\) in \(X\) and \(\kappa(x_i) = \kappa(v_i)\).

Proof

We will use induction on \(n\). Namely, suppose \((U, u) \to (S, s)\) and \(V_i \subset X_U\), \(i = 1, \ldots, n - 1\) work for \(x_1, \ldots, x_{n - 1}\). Since \(\kappa(s) = \kappa(u)\) the fibre \((X_U)_u = X_s\). Hence there exists a unique point \(x'_n \in X_u \subset X_U\) corresponding to \(x_n \in X_s\). Also \(x'_n\) is isolated in \(X_u\). Hence by Lemma 02LK there exists an elementary étale neighbourhood \((U', u') \to (U, u)\) and an open \(V_n \subset X_{U'}\) which works for \(x'_n\) and hence for \(x_n\). By the final assertion of Lemma 02LK the open subschemes \(V'_i = U'\times_U V_i\) for \(i = 1, \ldots, n - 1\) still work with respect to \(x_1, \ldots, x_{n - 1}\). Hence we win.

If we allow a nontrivial field extension \(\kappa(u)/\kappa(s)\), i.e., general étale neighbourhoods, then we can split the points as follows.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x_1, \ldots, x_n \in X\) be points having the same image \(s\) in \(S\). Assume that

\(f\) is locally of finite type, and

\(x_i \in X_s\) is isolated for \(i = 1, \ldots, n\).

Then there exist

an étale neighbourhood \((U, u) \to (S, s)\),

for each \(i\) an integer \(m_i\) and open subschemes \(V_{i, j} \subset X_U\), \(j = 1, \ldots, m_i\)

such that we have

each \(V_{i, j} \to U\) is a finite morphism,

there is a unique point \(v_{i, j}\) of \(V_{i, j}\) mapping to \(u\) in \(U\) with \(\kappa(u) \subset \kappa(v_{i, j})\) finite purely inseparable,

if \(v_{i, j} = v_{i', j'}\), then \(i = i'\) and \(j = j'\), and

the points \(v_{i, j}\) map to \(x_i\) in \(X\) and no other points of \((X_U)_u\) map to \(x_i\).

Proof

This proof is a variant of the proof of Algebra, Lemma 00UL in the language of schemes. By Morphisms, Lemma 01TH the morphism \(f\) is quasi-finite at each of the points \(x_i\). Hence \(\kappa(s) \subset \kappa(x_i)\) is finite for each \(i\) (Morphisms, Lemma 01TG). For each \(i\), let \(\kappa(s) \subset L_i \subset \kappa(x_i)\) be the subfield such that \(L_i/\kappa(s)\) is separable, and \(\kappa(x_i)/L_i\) is purely inseparable. Choose a finite Galois extension \(L/\kappa(s)\) such that there exist \(\kappa(s)\)-embeddings \(L_i \to L\) for \(i = 1, \ldots, n\). Choose an étale neighbourhood \((U, u) \to (S, s)\) such that \(L \cong \kappa(u)\) as \(\kappa(s)\)-extensions (Lemma 02LF).

Let \(y_{i, j}\), \(j = 1, \ldots, m_i\) be the points of \(X_U\) lying over \(x_i \in X\) and \(u \in U\). By Schemes, Lemma 01JT these points \(y_{i, j}\) correspond exactly to the primes in the rings \(\kappa(u) \otimes_{\kappa(s)} \kappa(x_i)\). This also explains why there are finitely many; in fact \(m_i = [L_i : \kappa(s)]\) but we do not need this. By our choice of \(L\) (and elementary field theory) we see that \(\kappa(u) \subset \kappa(y_{i, j})\) is finite purely inseparable for each pair \(i, j\). Also, by Morphisms, Lemma 01TM for example, the morphism \(X_U \to U\) is quasi-finite at the points \(y_{i, j}\) for all \(i, j\).

Apply Lemma 02LL to the morphism \(X_U \to U\), the point \(u \in U\) and the points \(y_{i, j} \in (X_U)_u\). This gives an étale neighbourhood \((U', u') \to (U, u)\) with \(\kappa(u) = \kappa(u')\) and opens \(V_{i, j} \subset X_{U'}\) with the properties (i), (ii), and (iii) of that lemma. We claim that the étale neighbourhood \((U', u') \to (S, s)\) and the opens \(V_{i, j} \subset X_{U'}\) are a solution to the problem posed by the lemma. We omit the verifications.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(s \in S\). Let \(x_1, \ldots, x_n \in X_s\). Assume that

\(f\) is locally of finite type,

\(f\) is separated, and

\(x_1, \ldots, x_n\) are pairwise distinct isolated points of \(X_s\).

Then there exists an elementary étale neighbourhood \((U, u) \to (S, s)\) and a decomposition \[U \times_S X = W \amalg V_1 \amalg \ldots \amalg V_n\] into open and closed subschemes such that the morphisms \(V_i \to U\) are finite, the fibres of \(V_i \to U\) over \(u\) are singletons \(\{v_i\}\), each \(v_i\) maps to \(x_i\) with \(\kappa(x_i) = \kappa(v_i)\), and the fibre of \(W \to U\) over \(u\) contains no points mapping to any of the \(x_i\).

Proof

Choose \((U, u) \to (S, s)\) and \(V_i \subset X_U\) as in Lemma 02LL. Since \(X_U \to U\) is separated (Schemes, Lemma 01KU) and \(V_i \to U\) is finite hence proper (Morphisms, Lemma 01WN) we see that \(V_i \subset X_U\) is closed by Morphisms, Lemma 01W6. Hence \(V_i \cap V_j\) is a closed subset of \(V_i\) which does not contain \(v_i\). Hence the image of \(V_i \cap V_j\) in \(U\) is a closed set (because \(V_i \to U\) proper) not containing \(u\). After shrinking \(U\) we may therefore assume that \(V_i \cap V_j = \emptyset\) for all \(i, j\). This gives the decomposition as in the lemma.

Here is the variant where we reduce to purely inseparable field extensions.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(s \in S\). Let \(x_1, \ldots, x_n \in X_s\). Assume that

\(f\) is locally of finite type,

\(f\) is separated, and

\(x_1, \ldots, x_n\) are pairwise distinct isolated points of \(X_s\).

Then there exists an étale neighbourhood \((U, u) \to (S, s)\) and a decomposition \[U \times_S X = W \amalg \ \coprod\nolimits_{i = 1, \ldots, n} \ \coprod\nolimits_{j = 1, \ldots, m_i} V_{i, j}\] into open and closed subschemes such that the morphisms \(V_{i, j} \to U\) are finite, the fibres of \(V_{i, j} \to U\) over \(u\) are singletons \(\{v_{i, j}\}\), each \(v_{i, j}\) maps to \(x_i\), \(\kappa(u) \subset \kappa(v_{i, j})\) is purely inseparable, and the fibre of \(W \to U\) over \(u\) contains no points mapping to any of the \(x_i\).

Proof

This is proved in exactly the same way as the proof of Lemma 02LN except that it uses Lemma 02LM instead of Lemma 02LL.

The following version may be a little easier to parse.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(s \in S\). Assume that

\(f\) is locally of finite type,

\(f\) is separated, and

\(X_s\) has at most finitely many isolated points.

Then there exists an elementary étale neighbourhood \((U, u) \to (S, s)\) and a decomposition \[U \times_S X = W \amalg V\] into open and closed subschemes such that the morphism \(V \to U\) is finite, and the fibre \(W_u\) of the morphism \(W \to U\) contains no isolated points. In particular, if \(f^{-1}(s)\) is a finite set, then \(W_u = \emptyset\).

Proof

This is clear from Lemma 02LN by choosing \(x_1, \ldots, x_n\) the complete set of isolated points of \(X_s\) and setting \(V = \bigcup V_i\).

Étale localization of integral morphisms

Some variants of the results of Section 04HF for the case of integral morphisms.

Lemma

Let \(R \to S\) be an integral ring map. Let \(\mathfrak p \subset R\) be a prime ideal. Assume

there are finitely many primes \(\mathfrak q_1, \ldots, \mathfrak q_n\) lying over \(\mathfrak p\), and

for each \(i\) the maximal separable subextension \(\kappa(\mathfrak q)/\kappa(\mathfrak q_i)_{sep}/\kappa(\mathfrak p)\) (Fields, Lemma 030K) is finite over \(\kappa(\mathfrak p)\).

Then there exists an étale ring map \(R \to R'\) and a prime \(\mathfrak p'\) lying over \(\mathfrak p\) such that \[S \otimes_R R' = A_1 \times \ldots \times A_m\] with \(R' \to A_j\) integral having a unique prime \(\mathfrak r_j\) over \(\mathfrak p'\) such that \(\kappa(\mathfrak r_j)/\kappa(\mathfrak p')\) is purely inseparable.

Proof

This proof uses Algebra, Lemma 00UL. Namely, choose a generator \(\theta_i \in \kappa(\mathfrak q_i)_{sep}\) of this field over \(\kappa(\mathfrak p)\) (Fields, Lemma 030N). The spectrum of the fibre ring \(S \otimes_R \kappa(\mathfrak p)\) is finite discrete with points corresponding to \(\mathfrak q_1, \ldots, \mathfrak q_n\). By the Chinese remainder theorem (Algebra, Lemma 00DT) we see that \(S \otimes_R \kappa(\mathfrak p) \to \prod \kappa(\mathfrak q_i)\) is surjective. Hence after replacing \(R\) by \(R_g\) for some \(g \in R\), \(g \not \in \mathfrak p\) we may assume that \((0, \ldots, 0, \theta_i, 0, \ldots, 0) \in \prod \kappa(\mathfrak q_i)\) is the image of some \(x_i \in S\). Let \(S' \subset S\) be the \(R\)-subalgebra generated by our \(x_i\). Since \(\Spec(S) \to \Spec(S')\) is surjective (Algebra, Lemma 00GQ) we conclude that \(\mathfrak q_i' = S' \cap \mathfrak q_i\) are the primes of \(S'\) over \(\mathfrak p\). By our choice of \(x_i\) we conclude these primes are distinct that and \(\kappa(\mathfrak q'_i)_{sep} = \kappa(\mathfrak q_i)_{sep}\). In particular the field extensions \(\kappa(\mathfrak q_i)/\kappa(\mathfrak q'_i)\) are purely inseparable. Since \(R \to S'\) is finite we may apply Algebra, Lemma 00UL. and we get \(R \to R'\) and \(\mathfrak p'\) and a decomposition \[S' \otimes_R R' = A'_1 \times \ldots \times A'_m \times B'\] with \(R' \to A'_j\) integral having a unique prime \(\mathfrak r'_j\) over \(\mathfrak p'\) such that \(\kappa(\mathfrak r'_j)/\kappa(\mathfrak p')\) is purely inseparable and such that \(B'\) does not have a prime lying over \(\mathfrak p'\). Since \(R' \to B'\) is finite (as \(R \to S'\) is finite) we can after localizing \(R'\) at some \(g' \in R'\), \(g' \not \in \mathfrak p'\) assume that \(B' = 0\). Via the map \(S' \otimes_R R' \to S \otimes_R R'\) we get the corresponding decomposition for \(S\).

Proof

This proof uses strict henselization. First, assume \(R\) is strictly henselization with maximal ideal \(\mathfrak p\). Then \(S/\mathfrak p S\) has finitely many primes corresponding to \(\mathfrak q_1, \ldots, \mathfrak q_n\), each maximal, each with purely inseparable residue field over \(\kappa(\mathfrak p)\). Hence \(S/\mathfrak p S\) is equal to \(\prod (S/\mathfrak p S)_{\mathfrak p_i}\). By More on Algebra, Lemma 09XI we can lift this product decomposition to a product composition of \(S\) as in the statement.

In the general case, let \(R^{sh}\) be the strict henselization of \(R_\mathfrak p\). Then we can apply the result of the first paragraph to \(R^{sh} \to S \otimes_R R^{sh}\). Consider the \(m\) mutually orthogonal idempotents in \(S \otimes_R R^{sh}\) corresponding to the product decomposition. Since \(R^{sh}\) is a filtered colimit of étale ring maps \((R, \mathfrak p) \to (R', \mathfrak p')\) by Algebra, Lemma 04GW we see that these idempotents descend to some \(R'\) as desired.

Zariski’s Main Theorem

In this section we prove Zariski’s main theorem as reformulated by Grothendieck. Often when we say “Zariski’s main theorem” in this content we mean either of Lemma 03GW, Lemma 02LR, or Lemma 05K0. In most texts people refer to the last of these as Zariski’s main theorem.

We have already proved the algebraic version in Algebra, Theorem 00Q9 and we have already restated this algebraic version in the language of schemes, see Morphisms, Theorem 03GT. The version in this section is more subtle; to get the full result we use the étale localization techniques of Section 04HF to reduce to the algebraic case.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume \(f\) is of finite type and separated. Let \(S'\) be the normalization of \(S\) in \(X\), see Morphisms, Definition 035H. Picture: \[\xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & S' \ar[ld]^\nu \\ & S & }\] Then there exists an open subscheme \(U' \subset S'\) such that

\((f')^{-1}(U') \to U'\) is an isomorphism, and

\((f')^{-1}(U') \subset X\) is the set of points at which \(f\) is quasi-finite.

Proof

By Morphisms, Lemma 01TI the subset \(U \subset X\) of points where \(f\) is quasi-finite is open. The lemma is equivalent to

\(U' = f'(U) \subset S'\) is open,

\(U = (f')^{-1}(U')\), and

\(U \to U'\) is an isomorphism.

Let \(x \in U\) be arbitrary. We claim there exists an open neighbourhood \(f'(x) \in V \subset S'\) such that \((f')^{-1}V \to V\) is an isomorphism. We first prove the claim implies the lemma. Namely, then \((f')^{-1}V \cong V\) is both locally of finite type over \(S\) (as an open subscheme of \(X\)) and for \(v \in V\) the residue field extension \(\kappa(v)/\kappa(\nu(v))\) is algebraic (as \(V \subset S'\) and \(S'\) is integral over \(S\)). Hence the fibres of \(V \to S\) are discrete (Morphisms, Lemma 01TE) and \((f')^{-1}V \to S\) is locally quasi-finite (Morphisms, Lemma 06RT). This implies \((f')^{-1}V \subset U\) and \(V \subset U'\). Since \(x\) was arbitrary we see that (a), (b), and (c) are true.

Let \(s = f(x)\). Let \((T, t) \to (S, s)\) be an elementary étale neighbourhood. Denote by a subscript \({}_T\) the base change to \(T\). Let \(y = (x, t) \in X_T\) be the unique point in the fibre \(X_t\) lying over \(x\). Note that \(U_T \subset X_T\) is the set of points where \(f_T\) is quasi-finite, see Morphisms, Lemma 01TM. Note that \[X_T \xrightarrow{f'_T} S'_T \xrightarrow{\nu_T} T\] is the normalization of \(T\) in \(X_T\), see Lemma 03GV. Suppose that the claim holds for \(y \in U_T \subset X_T \to S'_T \to T\), i.e., suppose that we can find an open neighbourhood \(f'_T(y) \in V' \subset S'_T\) such that \((f'_T)^{-1}V' \to V'\) is an isomorphism. The morphism \(S'_T \to S'\) is étale hence the image \(V \subset S'\) of \(V'\) is open. Observe that \(f'(x) \in V\) as \(f'_T(y) \in V'\). Observe that \[\xymatrix{ (f'_T)^{-1}V' \ar[r] \ar[d] & (f')^{-1}(V) \ar[d] \\ V' \ar[r] & V }\] is a fibre square (as \(S'_T \times_{S'} X = X_T\)). Since the left vertical arrow is an isomorphism and \(\{V' \to V\}\) is a étale covering, we conclude that the right vertical arrow is an isomorphism by Descent, Lemma 02L4. In other words, the claim holds for \(x \in U \subset X \to S' \to S\).

By the result of the previous paragraph we may replace \(S\) by an elementary étale neighbourhood of \(s = f(x)\) in order to prove the claim. Thus we may assume there is a decomposition \[X = V \amalg W\] into open and closed subschemes where \(V \to S\) is finite and \(x \in V\), see Lemma 02LN. Since \(X\) is a disjoint union of \(V\) and \(W\) over \(S\) and since \(V \to S\) is finite we see that the normalization of \(S\) in \(X\) is the morphism \[X = V \amalg W \longrightarrow V \amalg W' \longrightarrow S\] where \(W'\) is the normalization of \(S\) in \(W\), see Morphisms, Lemmas 03GO, 01WJ, and 03GP. The claim follows and we win.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume \(f\) is quasi-finite and separated. Let \(S'\) be the normalization of \(S\) in \(X\), see Morphisms, Definition 035H. Picture: \[\xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & S' \ar[ld]^\nu \\ & S & }\] Then \(f'\) is a quasi-compact open immersion and \(\nu\) is integral. In particular \(f\) is quasi-affine.

Proof

This follows from Lemma 03GW. Namely, by that lemma there exists an open subscheme \(U' \subset S'\) such that \((f')^{-1}(U') = X\) and \(X \to U'\) is an isomorphism. In other words, \(f'\) is an open immersion. Note that \(f'\) is quasi-compact as \(f\) is quasi-compact and \(\nu : S' \to S\) is separated (Schemes, Lemma 03GI). It follows that \(f\) is quasi-affine by Morphisms, Lemma 01SM.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume \(f\) is quasi-finite and separated and assume that \(S\) is quasi-compact and quasi-separated. Then there exists a factorization \[\xymatrix{ X \ar[rd]_f \ar[rr]_j & & T \ar[ld]^\pi \\ & S & }\] where \(j\) is a quasi-compact open immersion and \(\pi\) is finite.

Proof

Let \(X \to S' \to S\) be as in the conclusion of Lemma 02LR. By Properties, Lemma 0817 we can write \(\nu_*\mathcal{O}_{S'} = \colim_{i \in I} \mathcal{A}_i\) as a directed colimit of finite quasi-coherent \(\mathcal{O}_S\)-algebras \(\mathcal{A}_i \subset \nu_*\mathcal{O}_{S'}\). Then \(\pi_i : T_i = \underline{\Spec}_S(\mathcal{A}_i) \to S\) is a finite morphism for each \(i\). Note that the transition morphisms \(T_{i'} \to T_i\) are affine and that \(S' = \lim T_i\).

By Limits, Lemma 01Z4 there exists an \(i\) and a quasi-compact open \(U_i \subset T_i\) whose inverse image in \(S'\) equals \(f'(X)\). For \(i' \geq i\) let \(U_{i'}\) be the inverse image of \(U_i\) in \(T_{i'}\). Then \(X \cong f'(X) = \lim_{i' \geq i} U_{i'}\), see Limits, Lemma 01YX. By Limits, Lemma 081B we see that \(X \to U_{i'}\) is a closed immersion for some \(i' \geq i\). (In fact \(X \cong U_{i'}\) for sufficiently large \(i'\) but we don’t need this.) Hence \(X \to T_{i'}\) is an immersion. By Morphisms, Lemma 01QV we can factor this as \(X \to T \to T_{i'}\) where the first arrow is an open immersion and the second a closed immersion. Thus we win.

Lemma

With notation and hypotheses as in Lemma 05K0. Assume moreover that \(f\) is locally of finite presentation. Then we can choose the factorization such that \(T\) is finite and of finite presentation over \(S\).

Proof

By Limits, Lemma 09YY we can write \(T = \lim T_i\) where all \(T_i\) are finite and of finite presentation over \(S\) and the transition morphisms \(T_{i'} \to T_i\) are closed immersions. By Limits, Lemma 01Z4 there exists an \(i\) and an open subscheme \(U_i \subset T_i\) whose inverse image in \(T\) is \(X\). By Limits, Lemma 081B we see that \(X \cong U_i\) for large enough \(i\). Replacing \(T\) by \(T_i\) finishes the proof.

Applications of Zariski’s Main Theorem, I

A first application is the characterization of finite morphisms as proper morphisms with finite fibres.

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent:

\(f\) is finite,

\(f\) is proper with finite fibres,

\(f\) is proper and locally quasi-finite,

\(f\) is universally closed, separated, locally of finite type and has finite fibres.

Proof

We have (1) implies (2) by Morphisms, Lemmas 01WN, 02NH, and 02NU. We have (2) implies (3) by Morphisms, Lemma 02NG. We have (3) implies (4) by the definition of proper morphisms and Morphisms, Lemmas 01TJ and 02NH.

Assume (4). Pick \(s \in S\). By Morphisms, Lemma 02NG we see that all the finitely many points of \(X_s\) are isolated in \(X_s\). Choose an elementary étale neighbourhood \((U, u) \to (S, s)\) and decomposition \(X_U = V \amalg W\) as in Lemma 02LP. Note that \(W_u = \emptyset\) because all points of \(X_s\) are isolated. Since \(f\) is universally closed we see that the image of \(W\) in \(U\) is a closed set not containing \(u\). After shrinking \(U\) we may assume that \(W = \emptyset\). In other words we see that \(X_U = V\) is finite over \(U\). Since \(s \in S\) was arbitrary this means there exists a family \(\{U_i \to S\}\) of étale morphisms whose images cover \(S\) such that the base changes \(X_{U_i} \to U_i\) are finite. Note that \(\{U_i \to S\}\) is an étale covering, see Topologies, Definition 0215. Hence it is an fpqc covering, see Topologies, Lemma 022C. Hence we conclude \(f\) is finite by Descent, Lemma 02LA.

As a consequence we have the following useful results.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(s \in S\). Assume that \(f\) is proper and \(f^{-1}(\{s\})\) is a finite set. Then there exists an open neighbourhood \(V \subset S\) of \(s\) such that \(f|_{f^{-1}(V)} : f^{-1}(V) \to V\) is finite.

Proof

The morphism \(f\) is quasi-finite at all the points of \(f^{-1}(\{s\})\) by Morphisms, Lemma 02NG. By Morphisms, Lemma 01TI the set of points at which \(f\) is quasi-finite is an open \(U \subset X\). Let \(Z = X \setminus U\). Then \(s \not \in f(Z)\). Since \(f\) is proper the set \(f(Z) \subset S\) is closed. Choose any open neighbourhood \(V \subset S\) of \(s\) with \(f(Z) \cap V = \emptyset\). Then \(f^{-1}(V) \to V\) is locally quasi-finite and proper. Hence it is quasi-finite (Morphisms, Lemma 01TJ), hence has finite fibres (Morphisms, Lemma 02NH), hence is finite by Lemma 02LS.

Lemma

Consider a commutative diagram of schemes \[\xymatrix{ X \ar[rr]_h \ar[rd]_f & & Y \ar[ld]^g \\ & S }\] Let \(s \in S\). Assume

\(X \to S\) is a proper morphism,

\(Y \to S\) is separated and locally of finite type, and

the image of \(X_s \to Y_s\) is finite.

Then there is an open subspace \(U \subset S\) containing \(s\) such that \(X_U \to Y_U\) factors through a closed subscheme \(Z \subset Y_U\) finite over \(U\).

Proof

Let \(Z \subset Y\) be the scheme theoretic image of \(h\), see Morphisms, Section 01R5. By Morphisms, Lemma 0AH6 the morphism \(X \to Z\) is surjective and \(Z \to S\) is proper. Thus \(X_s \to Z_s\) is surjective. We see that either (3) implies \(Z_s\) is finite. Hence \(Z \to S\) is finite in an open neighbourhood of \(s\) by Lemma 02UP.

Applications of Zariski’s Main Theorem, II

In this section we give a few more consequences of Zariski’s main theorem to the structure of quasi-finite morphisms.

Lemma

Let \(f : X \to Y\) be a separated, locally quasi-finite morphism with \(Y\) affine. Then every finite set of points of \(X\) is contained in an open affine of \(X\).

Proof

Let \(x_1, \ldots, x_n \in X\). Choose a quasi-compact open \(U \subset X\) with \(x_i \in U\). Then \(U \to Y\) is quasi-affine by Lemma 02LR. Hence there exists an affine open \(V \subset U\) containing \(x_1, \ldots, x_n\) by Properties, Lemma 01ZY.

Lemma

Let \(f : Y \to X\) be a quasi-finite morphism. There exists a dense open \(U \subset X\) such that \(f|_{f^{-1}(U)} : f^{-1}(U) \to U\) is finite.

Proof

If \(U_i \subset X\), \(i \in I\) is a collection of opens such that the restrictions \(f|_{f^{-1}(U_i)} : f^{-1}(U_i) \to U_i\) are finite, then with \(U = \bigcup U_i\) the restriction \(f|_{f^{-1}(U)} : f^{-1}(U) \to U\) is finite, see Morphisms, Lemma 01WI. Thus the problem is local on \(X\) and we may assume that \(X\) is affine.

Assume \(X\) is affine. Write \(Y = \bigcup_{j = 1, \ldots, m} V_j\) with \(V_j\) affine. This is possible since \(f\) is quasi-finite and hence in particular quasi-compact. Each \(V_j \to X\) is quasi-finite and separated. Let \(\eta \in X\) be a generic point of an irreducible component of \(X\). We see from Morphisms, Lemmas 02NH and 02NW that there exists an open neighbourhood \(\eta \in U_\eta\) such that \(f^{-1}(U_\eta) \cap V_j \to U_\eta\) is finite. We may choose \(U_\eta\) such that it works for each \(j = 1, \ldots, m\). Note that the collection of generic points of \(X\) is dense in \(X\). Thus we see there exists a dense open \(W = \bigcup_\eta U_\eta\) such that each \(f^{-1}(W) \cap V_j \to W\) is finite. It suffices to show that there exists a dense open \(U \subset W\) such that \(f|_{f^{-1}(U)} : f^{-1}(U) \to U\) is finite. Thus we may replace \(X\) by an affine open subscheme of \(W\) and assume that each \(V_j \to X\) is finite.

Assume \(X\) is affine, \(Y = \bigcup_{j = 1, \ldots, m} V_j\) with \(V_j\) affine, and the restrictions \(f|_{V_j} : V_j \to X\) are finite. Set \[\Delta_{ij} = \Big(\overline{V_i \cap V_j} \setminus V_i \cap V_j\Big) \cap V_j.\] This is a nowhere dense closed subset of \(V_j\) because it is the boundary of the open subset \(V_i \cap V_j\) in \(V_j\). By Morphisms, Lemma 03HX the image \(f(\Delta_{ij})\) is a nowhere dense closed subset of \(X\). By Topology, Lemma 03HO the union \(T = \bigcup f(\Delta_{ij})\) is a nowhere dense closed subset of \(X\). Thus \(U = X \setminus T\) is a dense open subset of \(X\). We claim that \(f|_{f^{-1}(U)} : f^{-1}(U) \to U\) is finite. To see this let \(U' \subset U\) be an affine open. Set \(Y' = f^{-1}(U') = U' \times_X Y\), \(V_j' = Y' \cap V_j = U' \times_X V_j\). Consider the restriction \[f' = f|_{Y'} : Y' \longrightarrow U'\] of \(f\). This morphism now has the property that \(Y' = \bigcup_{j = 1, \ldots, m} V'_j\) is an affine open covering, each \(V'_j \to U'\) is finite, and \(V_i' \cap V_j'\) is (open and) closed both in \(V'_i\) and \(V'_j\). Hence \(V_i' \cap V_j'\) is affine, and the map \[\mathcal{O}(V'_i) \otimes_{\mathbf{Z}} \mathcal{O}(V'_j) \longrightarrow \mathcal{O}(V'_i \cap V'_j)\] is surjective. This implies that \(Y'\) is separated, see Schemes, Lemma 01KP. Finally, consider the commutative diagram \[\xymatrix{ \coprod_{j = 1, \ldots, m} V'_j \ar[rd] \ar[rr] & & Y' \ar[ld] \\ & U' & }\] The south-east arrow is finite, hence proper, the horizontal arrow is surjective, and the south-west arrow is separated. Hence by Morphisms, Lemma 03GN we conclude that \(Y' \to U'\) is proper. Since it is also quasi-finite, we see that it is finite by Lemma 02LS, and we win.

Lemma

Let \(f : X \to S\) be flat, locally of finite presentation, separated, locally quasi-finite with universally bounded fibres. Then there exist closed subsets \[\emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset Z_n = S\] such that with \(S_r = Z_r \setminus Z_{r - 1}\) the stratification \(S = \coprod_{r = 0, \ldots, n} S_r\) is characterized by the following universal property: Given \(g : T \to S\) the projection \(X \times_S T \to T\) is finite locally free of degree \(r\) if and only if \(g(T) \subset S_r\) (set theoretically).

Proof

Let \(n\) be an integer bounding the degree of the fibres of \(X \to S\). By Morphisms, Lemma 03J7 we see that any base change has degrees of fibres bounded by \(n\) also. In particular, all the integers \(r\) that occur in the statement of the lemma will be \(\leq n\). We will prove the lemma by induction on \(n\). The base case is \(n = 0\) which is obvious.

We claim the set of points \(s \in S\) with \(\deg_{\kappa(s)}(X_s) = n\) is an open subset \(S_n \subset S\) and that \(X \times_S S_n \to S_n\) is finite locally free of degree \(n\). Namely, suppose that \(s \in S\) is such a point. Choose an elementary étale morphism \((U, u) \to (S, s)\) and a decomposition \(U \times_S X = W \amalg V\) as in Lemma 02LP. Since \(V \to U\) is finite, flat, and locally of finite presentation, we see that \(V \to U\) is finite locally free, see Morphisms, Lemma 02KB. After shrinking \(U\) to a smaller neighbourhood of \(u\) we may assume \(V \to U\) is finite locally free of some degree \(d\), see Morphisms, Lemma 04MH. As \(u \mapsto s\) and \(W_u = \emptyset\) we see that \(d = n\). Since \(n\) is the maximum degree of a fibre we see that \(W = \emptyset\)! Thus \(U \times_S X \to U\) is finite locally free of degree \(n\). By Descent, Lemma 02VO we conclude that \(X \to S\) is finite locally free of degree \(n\) over \(\Im(U \to S)\) which is an open neighbourhood of \(s\) (Morphisms, Lemma 03WT). This proves the claim.

Let \(S' = S \setminus S_n\) endowed with the reduced induced scheme structure and set \(X' = X \times_S S'\). Note that the degrees of fibres of \(X' \to S'\) are universally bounded by \(n - 1\). By induction we find a stratification \(S' = S_0 \amalg \ldots \amalg S_{n - 1}\) adapted to the morphism \(X' \to S'\). We claim that \(S = \coprod_{r = 0, \ldots, n} S_r\) works for the morphism \(X \to S\). Let \(g : T \to S\) be a morphism of schemes and assume that \(X \times_S T \to T\) is finite locally free of degree \(r\). As remarked above this implies that \(r \leq n\). If \(r = n\), then it is clear that \(T \to S\) factors through \(S_n\). If \(r < n\), then \(g(T) \subset S' = S \setminus S_d\) (set theoretically) hence \(T_{red} \to S\) factors through \(S'\), see Schemes, Lemma 0356. Note that \(X \times_S T_{red} \to T_{red}\) is also finite locally free of degree \(r\) as a base change. By the universal property of the stratification \(S' = \coprod_{r = 0, \ldots, n - 1} S_r\) we see that \(g(T) = g(T_{red})\) is contained in \(S_r\). Conversely, suppose that we have \(g : T \to S\) such that \(g(T) \subset S_r\) (set theoretically). If \(r = n\), then \(g\) factors through \(S_n\) and it is clear that \(X \times_S T \to T\) is finite locally free of degree \(n\) as a base change. If \(r < n\), then \(X \times_S T \to T\) is a morphism which is separated, flat, and locally of finite presentation, such that the restriction to \(T_{red}\) is finite locally free of degree \(r\). Since \(T_{red} \to T\) is a universal homeomorphism, we conclude that \(X \times_S T_{red} \to X \times_S T\) is a universal homeomorphism too and hence \(X \times_S T \to T\) is universally closed (as this is true for the finite morphism \(X \times_S T_{red} \to T_{red}\)). It follows that \(X \times_S T \to T\) is finite, for example by Lemma 02LS. Then we can use Morphisms, Lemma 02KB to see that \(X \times_S T \to T\) is finite locally free. Finally, the degree is \(r\) as all the fibres have degree \(r\).

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat, locally of finite presentation, separated, and quasi-finite. Then there exist closed subsets \[\emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset S\] such that with \(S_r = Z_r \setminus Z_{r - 1}\) the stratification \(S = \coprod S_r\) is characterized by the following universal property: Given a morphism \(g : T \to S\) the projection \(X \times_S T \to T\) is finite locally free of degree \(r\) if and only if \(g(T) \subset S_r\) (set theoretically). Moreover, the inclusion maps \(S_r \to S\) are quasi-compact.

Proof

The question is local on \(S\), hence we may assume that \(S\) is affine. By Morphisms, Lemma 03JA the fibres of \(f\) are universally bounded in this case. Hence the existence of the stratification follows from Lemma 07RY.

We will show that \(U_r = S \setminus Z_r \to S\) is quasi-compact for each \(r \geq 0\). This will prove the final statement by elementary topology. Since a composition of quasi-compact maps is quasi-compact it suffices to prove that \(U_r \to U_{r - 1}\) is quasi-compact. Choose an affine open \(W \subset U_{r - 1}\). Write \(W = \Spec(A)\). Then \(Z_r \cap W = V(I)\) for some ideal \(I \subset A\) and \(X \times_S \Spec(A/I) \to \Spec(A/I)\) is finite locally free of degree \(r\). Note that \(A/I = \colim A/I_i\) where \(I_i \subset I\) runs through the finitely generated ideals. By Limits, Lemma 06AC we see that \(X \times_S \Spec(A/I_i) \to \Spec(A/I_i)\) is finite locally free of degree \(r\) for some \(i\). (This uses that \(X \to S\) is of finite presentation, as it is locally of finite presentation, separated, and quasi-compact.) Hence \(\Spec(A/I_i) \to \Spec(A) = W\) factors (set theoretically) through \(Z_r \cap W\). It follows that \(Z_r \cap W = V(I_i)\) is the zero set of a finite subset of elements of \(A\). This means that \(W \setminus Z_r\) is a finite union of standard opens, hence quasi-compact, as desired.

Lemma

Let \(f : X \to S\) be a flat, locally of finite presentation, separated, and locally quasi-finite morphism of schemes. Then there exist open subschemes \[S = U_0 \supset U_1 \supset U_2 \supset \ldots\] such that a morphism \(\Spec(k) \to S\) where \(k\) is a field factors through \(U_d\) if and only if \(X \times_S \Spec(k)\) has degree \(\geq d\) over \(k\).

Proof

The statement simply means that the collection of points where the degree of the fibre is \(\geq d\) is open. Thus we can work locally on \(S\) and assume \(S\) is affine. In this case, for every \(W \subset X\) quasi-compact open, the set of points \(U_d(W)\) where the fibres of \(W \to S\) have degree \(\geq d\) is open by Lemma 07RZ. Since \(U_d = \bigcup_W U_d(W)\) the result follows.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat, locally of finite presentation, and locally quasi-finite. Let \(g \in \Gamma(X, \mathcal{O}_X)\) nonzero. Then there exist an open \(V \subset X\) such that \(g|_V \not = 0\), an open \(U \subset S\) fitting into a commutative diagram \[\xymatrix{ V \ar[r] \ar[d]_\pi & X \ar[d]^f \\ U \ar[r] & S, }\] a quasi-coherent subsheaf \(\mathcal{F} \subset \mathcal{O}_U\), an integer \(r > 0\), and an injective \(\mathcal{O}_U\)-module map \(\mathcal{F}^{\oplus r} \to \pi_*\mathcal{O}_V\) whose image contains \(g|_V\).

Proof

We may assume \(X\) and \(S\) affine. We obtain a filtration \(\emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset Z_n = S\) as in Lemmas 07RY and 07RZ. Let \(T \subset X\) be the scheme theoretic support of the finite \(\mathcal{O}_X\)-module \(\Im(g : \mathcal{O}_X \to \mathcal{O}_X)\). Note that \(T\) is the support of \(g\) as a section of \(\mathcal{O}_X\) (Modules, Definition 01AT) and for any open \(V \subset X\) we have \(g|_V \not = 0\) if and only if \(V \cap T \not = \emptyset\). Let \(r\) be the smallest integer such that \(f(T) \subset Z_r\) set theoretically. Let \(\xi \in T\) be a generic point of an irreducible component of \(T\) such that \(f(\xi) \not \in Z_{r - 1}\) (and hence \(f(\xi) \in Z_r\)). We may replace \(S\) by an affine neighbourhood of \(f(\xi)\) contained in \(S \setminus Z_{r - 1}\). Write \(S = \Spec(A)\) and let \(I = (a_1, \ldots, a_m) \subset A\) be a finitely generated ideal such that \(V(I) = Z_r\) (set theoretically, see Algebra, Lemma 00F6). Since the support of \(g\) is contained in \(f^{-1}V(I)\) by our choice of \(r\) we see that there exists an integer \(N\) such that \(a_j^N g = 0\) for \(j = 1, \ldots, m\). Replacing \(a_j\) by \(a_j^r\) we may assume that \(Ig = 0\). For any \(A\)-module \(M\) write \(M[I]\) for the \(I\)-torsion of \(M\), i.e., \(M[I] = \{m \in M \mid Im = 0\}\). Write \(X = \Spec(B)\), so \(g \in B[I]\). Since \(A \to B\) is flat we see that \[B[I] = A[I] \otimes_A B \cong A[I] \otimes_{A/I} B/IB\] By our choice of \(Z_r\), the \(A/I\)-module \(B/IB\) is finite locally free of rank \(r\). Hence after replacing \(S\) by a smaller affine open neighbourhood of \(f(\xi)\) we may assume that \(B/IB \cong (A/IA)^{\oplus r}\) as \(A/I\)-modules. Choose a map \(\psi : A^{\oplus r} \to B\) which reduces modulo \(I\) to the isomorphism of the previous sentence. Then we see that the induced map \[A[I]^{\oplus r} \longrightarrow B[I]\] is an isomorphism. The lemma follows by taking \(\mathcal{F}\) the quasi-coherent sheaf associated to the \(A\)-module \(A[I]\) and the map \(\mathcal{F}^{\oplus r} \to \pi_*\mathcal{O}_V\) the one corresponding to \(A[I]^{\oplus r} \subset A^{\oplus r} \to B\).

Lemma

Let \(U \to X\) be a surjective étale morphism of schemes. Assume \(X\) is quasi-compact and quasi-separated. Then there exists a surjective integral morphism \(Y \to X\), such that for every \(y \in Y\) there is an open neighbourhood \(V \subset Y\) such that \(V \to X\) factors through \(U\). In fact, we may assume \(Y \to X\) is finite and of finite presentation.

Proof

Since \(X\) is quasi-compact, there exist finitely many affine opens \(U_i \subset U\) such that \(U' = \coprod U_i \to X\) is surjective. After replacing \(U\) by \(U'\), we see that we may assume \(U\) is affine. In particular \(U \to X\) is separated (Schemes, Lemma 01KN). Then there exists an integer \(d\) bounding the degree of the geometric fibres of \(U \to X\) (see Morphisms, Lemma 03JA). We will prove the lemma by induction on \(d\) for all quasi-compact and separated schemes \(U\) mapping surjective and étale onto \(X\). If \(d = 1\), then \(U = X\) and the result holds with \(Y = U\). Assume \(d > 1\).

We apply Lemma 02LR and we obtain a factorization \[\xymatrix{ U \ar[rr]_j \ar[rd] & & Y \ar[ld]^\pi \\ & X }\] with \(\pi\) integral and \(j\) a quasi-compact open immersion. We may and do assume that \(j(U)\) is scheme theoretically dense in \(Y\). Note that \[U \times_X Y = U \amalg W\] where the first summand is the image of \(U \to U \times_X Y\) (which is closed by Schemes, Lemma 01KS and open because it is étale as a morphism between schemes étale over \(Y\)) and the second summand is the (open and closed) complement. The image \(V \subset Y\) of \(W\) is an open subscheme containing \(Y \setminus U\).

The étale morphism \(W \to Y\) has geometric fibres of cardinality \(< d\). Namely, this is clear for geometric points of \(U \subset Y\) by inspection. Since \(U \subset Y\) is dense, it holds for all geometric points of \(Y\) for example by Lemma 07RY (the degree of the fibres of a quasi-compact separated étale morphism does not go up under specialization). Thus we may apply the induction hypothesis to \(W \to V\) and find a surjective integral morphism \(Z \to V\) with \(Z\) a scheme, which Zariski locally factors through \(W\). Choose a factorization \(Z \to Z' \to Y\) with \(Z' \to Y\) integral and \(Z \to Z'\) open immersion (Lemma 02LR). After replacing \(Z'\) by the scheme theoretic closure of \(Z\) in \(Z'\) we may assume that \(Z\) is scheme theoretically dense in \(Z'\). After doing this we have \(Z' \times_Y V = Z\). Finally, let \(T \subset Y\) be the induced reduced closed subscheme structure on \(Y \setminus V\). Consider the morphism \[Z' \amalg T \longrightarrow X\] This is a surjective integral morphism by construction. Since \(T \subset U\) it is clear that the morphism \(T \to X\) factors through \(U\). On the other hand, let \(z \in Z'\) be a point. If \(z \not \in Z\), then \(z\) maps to a point of \(Y \setminus V \subset U\) and we find a neighbourhood of \(z\) on which the morphism factors through \(U\). If \(z \in Z\), then we have a neighbourhood \(\Omega \subset Z\) which factors through \(W \subset U \times_X Y\) and hence through \(U\). This proves existence.

Assume we have found \(Y \to X\) integral and surjective which Zariski locally factors through \(U\). Choose a finite affine open covering \(Y = \bigcup V_j\) such that \(V_j \to X\) factors through \(U\). We can write \(Y = \lim Y_i\) with \(Y_i \to X\) finite and of finite presentation, see Limits, Lemma 09YZ. For large enough \(i\) we can find affine opens \(V_{i, j} \subset Y_i\) whose inverse image in \(Y\) recovers \(V_j\), see Limits, Lemma 01Z4. For even larger \(i\) the morphisms \(V_j \to U\) over \(X\) come from morphisms \(V_{i, j} \to U\) over \(X\), see Limits, Proposition 01ZC. This finishes the proof.

Application to morphisms with connected fibres

In this section we prove some lemmas that produce morphisms all of whose fibres are geometrically connected or geometrically integral. This will be useful in our study of the local structure of morphisms of finite type later.

Lemma

Consider a diagram of morphisms of schemes \[\xymatrix{ Z \ar[r]_{\sigma} \ar[rd] & X \ar[d] \\ & Y }\] an a point \(y \in Y\). Assume

\(X \to Y\) is of finite presentation and flat,

\(Z \to Y\) is finite locally free,

\(Z_y \not = \emptyset\),

all fibres of \(X \to Y\) are geometrically reduced, and

\(X_y\) is geometrically connected over \(\kappa(y)\).

Then there exists a quasi-compact open \(X^0 \subset X\) such that \(X^0_y = X_y\) and such that all nonempty fibres of \(X^0 \to Y\) are geometrically connected.

Proof

In this proof we will use that flat, finite presentation, finite locally free are properties that are preserved under base change and composition. We will also use that a finite locally free morphism is both open and closed. You can find these facts as Morphisms, Lemmas 01U9, 01TS, 02KD, 01U7, 01TR, 02KC, 01UA, and 01WN.

Note that \(X_Z \to Z\) is flat morphism of finite presentation which has a section \(s\) coming from \(\sigma\). Let \(X_Z^0\) denote the subset of \(X_Z\) defined in Situation 055L. By Lemma 055R it is an open subset of \(X_Z\).

The pullback \(X_{Z \times_Y Z}\) of \(X\) to \(Z \times_Y Z\) comes equipped with two sections \(s_0, s_1\), namely the base changes of \(s\) by \(\text{pr}_0, \text{pr}_1 : Z \times_Y Z \to Z\). The construction of Situation 055L gives two subsets \((X_{Z \times_Y Z})_{s_0}^0\) and \((X_{Z \times_Y Z})_{s_1}^0\). By Lemma 055M these are the inverse images of \(X_Z^0\) under the morphisms \(1_X \times \text{pr}_0, 1_X \times \text{pr}_1 : X_{Z \times_Y Z} \to X_Z\). In particular these subsets are open.

Let \((Z \times_Y Z)_y = \{z_1, \ldots, z_n\}\). As \(X_y\) is geometrically connected, we see that the fibres of \((X_{Z \times_Y Z})_{s_0}^0\) and \((X_{Z \times_Y Z})_{s_1}^0\) over each \(z_i\) agree (being equal to the whole fibre). Another way to say this is that \[s_0(z_i) \in (X_{Z \times_Y Z})_{s_1}^0 \quad\text{and}\quad s_1(z_i) \in (X_{Z \times_Y Z})_{s_0}^0.\] Since the sets \((X_{Z \times_Y Z})_{s_0}^0\) and \((X_{Z \times_Y Z})_{s_1}^0\) are open in \(X_{Z \times_Y Z}\) there exists an open neighbourhood \(W \subset Z \times_Y Z\) of \((Z \times_Y Z)_y\) such that \[s_0(W) \subset (X_{Z \times_Y Z})_{s_1}^0 \quad\text{and}\quad s_1(W) \subset (X_{Z \times_Y Z})_{s_0}^0.\] Then it follows directly from the construction in Situation 055L that \[p^{-1}(W) \cap (X_{Z \times_Y Z})_{s_0}^0 = p^{-1}(W) \cap (X_{Z \times_Y Z})_{s_1}^0\] where \(p : X_{Z \times_Y Z} \to Z \times_W Z\) is the projection. Because \(Z \times_Y Z \to Y\) is finite locally free, hence open and closed, there exists an affine open neighbourhood \(V \subset Y\) of \(y\) such that \(q^{-1}(V) \subset W\), where \(q : Z \times_Y Z \to Y\) is the structure morphism. To prove the lemma we may replace \(Y\) by \(V\). After we do this we see that \(X_Z^0 \subset Y_Z\) is an open such that \[(1_X \times \text{pr}_0)^{-1}(X_Z^0) = (1_X \times \text{pr}_1)^{-1}(X_Z^0).\] This means that the image \(X^0 \subset X\) of \(X_Z^0\) is an open such that \((X_Z \to X)^{-1}(X^0) = X_Z^0\), see Descent, Lemma 03N0. Finally, \(X^0\) is quasi-compact because \(X_Z^0\) is quasi-compact by Lemma 055P (use that at this point \(Y\) is affine, hence \(X\) is quasi-compact and quasi-separated, hence locally constructible is the same as constructible and in particular quasi-compact; details omitted). In this way we see that \(X^0\) has all the desired properties.

Lemma

Let \(h : Y \to S\) be a morphism of schemes. Let \(s \in S\) be a point. Let \(T \subset Y_s\) be an open subscheme. Assume

\(h\) is flat and of finite presentation,

all fibres of \(h\) are geometrically reduced, and

\(T\) is geometrically connected over \(\kappa(s)\).

Then we can find an affine elementary étale neighbourhood \((S', s') \to (S, s)\) and a quasi-compact open \(V \subset Y_{S'}\) such that

all fibres of \(V \to S'\) are geometrically connected,

\(V_{s'} = T \times_s s'\).

Proof

The problem is clearly local on \(S\), hence we may replace \(S\) by an affine open neighbourhood of \(s\). The topology on \(Y_s\) is induced from the topology on \(Y\), see Schemes, Lemma 01K1. Hence we can find a quasi-compact open \(V \subset Y\) such that \(V_s = T\). The restriction of \(h\) to \(V\) is quasi-compact (as \(S\) affine and \(V\) quasi-compact), quasi-separated, locally of finite presentation, and flat hence flat of finite presentation. Thus after replacing \(Y\) by \(V\) we may assume, in addition to (1) and (2) that \(Y_s = T\) and \(S\) affine.

Pick a closed point \(y \in Y_s\) such that \(h\) is Cohen-Macaulay at \(y\), see Lemma 045U. By Lemma 0570 there exists a diagram \[\xymatrix{ Z \ar[r] \ar[rd] & Y \ar[d] \\ & S }\] such that \(Z \to S\) is flat, locally of finite presentation, locally quasi-finite with \(Z_s = \{y\}\). Apply Lemma 02LK to find an elementary neighbourhood \((S', s') \to (S, s)\) and an open \(Z' \subset Z_{S'} = S' \times_S Z\) with \(Z' \to S'\) finite with a unique point \(z' \in Z'\) lying over \(s\). Note that \(Z' \to S'\) is also locally of finite presentation and flat (as an open of the base change of \(Z \to S\)), hence \(Z' \to S'\) is finite locally free, see Morphisms, Lemma 02KB. Note that \(Y_{S'} \to S'\) is flat and of finite presentation with geometrically reduced fibres as a base change of \(h\). Also \(Y_{s'} = Y_s\) is geometrically connected. Apply Lemma 057I to \(Z' \to Y_{S'}\) over \(S'\) to get \(V \subset Y_{S'}\) quasi-compact open satisfying (2) whose fibres over \(S'\) are either empty or geometrically connected. As \(V \to S'\) is open (Morphisms, Lemma 01UA), after replacing \(S'\) by an affine open neighbourhood of \(s'\) we may assume \(V \to S'\) is surjective, whence (1) holds.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite presentation and flat with geometrically reduced fibres. Then there exists an étale covering \(\{X_i \to X\}_{i \in I}\) such that \(X_i \to S\) factors as \(X_i \to S_i \to S\) where \(S_i \to S\) is étale and \(X_i \to S_i\) is flat of finite presentation with geometrically connected and geometrically reduced fibres.

Proof

Pick a point \(x \in X\) with image \(s \in S\). We will produce a diagram \[\xymatrix{ X' \ar[r] \ar[rd] & S' \times_S X \ar[r] \ar[d] & X \ar[d] \\ & S' \ar[r] & S }\] and points \(s' \in S'\), \(x' \in X'\), \(y \in S' \times_S X\) such that \(x'\) maps to \(x\), \((S', s') \to (S, s)\) is an étale neighbourhood, \((X', x') \to (S' \times_S X, y)\) is an étale neighbourhood⁸, and \(X' \to S'\) has geometrically connected fibres. If we can do this for every \(x \in X\), then the lemma follows (with members of the covering given by the collection of étale morphisms \(X' \to X\) so produced). The first step is the replace \(X\) and \(S\) by affine open neighbourhoods of \(x\) and \(s\) which reduces us to the case that \(X\) and \(S\) are affine (and hence \(f\) of finite presentation).

Choose a separable algebraic extension \(\overline{k}\) of \(\kappa(s)\). Denote \(X_{\overline{k}}\) the base change of \(X_s\). Choose a point \(\overline{x}\) in \(X_{\overline{k}}\) mapping to \(x \in X_s\). Choose a connected quasi-compact open neighbourhood \(\overline{V} \subset X_{\overline{k}}\) of \(\overline{x}\). (This is possible because any scheme locally of finite type over a field is locally connected as a locally Noetherian topological space.) By Varieties, Lemma 04KU we can find a finite separable extension \(k'/\kappa(s)\) and a quasi-compact open \(V' \subset X_{k'}\) whose base change is \(\overline{V}\). In particular \(V'\) is geometrically connected over \(k'\), see Varieties, Lemma 0387. By Lemma 02LF we can find an étale neighbourhood \((S', s') \to (S, s)\) such that \(\kappa(s')\) is isomorphic to \(k'\) as an extension of \(\kappa(s)\). Denote \(x' \in (S' \times_S X)_{s'} = X_{k'}\) the image of \(\overline{x}\). Thus after replacing \((S, s)\) by \((S', s')\) and \((X, x)\) by \((S' \times_S X, x')\) we reduce to the case handled in the next paragraph.

Assume there is a quasi-compact open \(V \subset X_s\) which contains \(x\) and is geometrically connected. Then we can apply Lemma 055W to find an affine étale neighbourhood \((S', s') \to (S, s)\) and a quasi-compact open \(X' \subset S' \times_S X\) such that \(X' \to S'\) has geometrically connected fibres and such that \(X'\) contains a point mapping to \(x\). This finishes the proof.

Lemma

Let \(h : Y \to S\) be a morphism of schemes. Let \(s \in S\) be a point. Let \(T \subset Y_s\) be an open subscheme. Assume

\(h\) is of finite presentation,

\(h\) is normal, and

\(T\) is geometrically irreducible over \(\kappa(s)\).

Then we can find an affine elementary étale neighbourhood \((S', s') \to (S, s)\) and a quasi-compact open \(V \subset Y_{S'}\) such that

all fibres of \(V \to S'\) are geometrically integral,

\(V_{s'} = T \times_s s'\).

Proof

Apply Lemma 055W to find an affine elementary étale neighbourhood \((S', s') \to (S, s)\) and a quasi-compact open \(V \subset Y_{S'}\) such that all fibres of \(V \to S'\) are geometrically connected and \(V_{s'} = T \times_s s'\). As \(V\) is an open of the base change of \(h\) all fibres of \(V \to S'\) are geometrically normal, see Lemma 0391. In particular, they are geometrically reduced. To finish the proof we have to show they are geometrically irreducible. But, if \(t \in S'\) then \(V_t\) is of finite type over \(\kappa(t)\) and hence \(V_t \times_{\kappa(t)} \overline{\kappa(t)}\) is of finite type over \(\overline{\kappa(t)}\) hence Noetherian. By choice of \(S' \to S\) the scheme \(V_t \times_{\kappa(t)} \overline{\kappa(t)}\) is connected. Hence \(V_t \times_{\kappa(t)} \overline{\kappa(t)}\) is irreducible by Properties, Lemma 033M and we win.

Application to the structure of finite type morphisms

The result in this section can be found in [GruRay]. Loosely stated it says that a finite type morphism is étale locally on the source and target the composition of a finite morphism by a smooth morphism with geometrically connected fibres of relative dimension equal to the fibre dimension of the original morphism.

Lemma

Let \(f : X \to S\) be a morphism. Let \(x \in X\) and set \(s = f(x)\). Assume that \(f\) is locally of finite type and that \(n = \dim_x(X_s)\). Then there exists a commutative diagram \[\xymatrix{ X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] \\ & Y \ar[d]^h & & y \ar@{|->}[d] \\ S \ar@{=}[r] & S & s & s \ar@{=}[l] }\] and a point \(x' \in X'\) with \(g(x') = x\) such that with \(y = \pi(x')\) we have

\(h : Y \to S\) is smooth of relative dimension \(n\),

\(g : (X', x') \to (X, x)\) is an elementary étale neighbourhood,

\(\pi\) is finite, and \(\pi^{-1}(\{y\}) = \{x'\}\), and

\(\kappa(y)\) is a purely transcendental extension of \(\kappa(s)\).

Moreover, if \(f\) is locally of finite presentation then \(\pi\) is of finite presentation.

Proof

The problem is local on \(X\) and \(S\), hence we may assume that \(X\) and \(S\) are affine. By Algebra, Lemma 0520 after replacing \(X\) by a standard open neighbourhood of \(x\) in \(X\) we may assume there is a factorization \[\xymatrix{ X \ar[r]^\pi & \mathbf{A}^n_S \ar[r] & S }\] such that \(\pi\) is quasi-finite and such that \(\kappa(\pi(x))\) is purely transcendental over \(\kappa(s)\). By Lemma 02LK there exists an elementary étale neighbourhood \[(Y, y) \to (\mathbf{A}^n_S, \pi(x))\] and an open \(X' \subset X \times_{\mathbf{A}^n_S} Y\) which contains a unique point \(x'\) lying over \(y\) such that \(X' \to Y\) is finite. This proves (1) – (4) hold. For the final assertion, use Morphisms, Lemma 02FV.

Lemma

Let \(f : X \to S\) be a morphism. Let \(x \in X\) and set \(s = f(x)\). Assume that \(f\) is locally of finite type and that \(n = \dim_x(X_s)\). Then there exists a commutative diagram \[\xymatrix{ X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] \\ & Y' \ar[d]^h & & y' \ar@{|->}[d] \\ S & S' \ar[l]_e & s & s' \ar@{|->}[l] }\] and a point \(x' \in X'\) with \(g(x') = x\) such that with \(y' = \pi(x')\), \(s' = h(y')\) we have

\(h : Y' \to S'\) is smooth of relative dimension \(n\),

all fibres of \(Y' \to S'\) are geometrically integral,

\(g : (X', x') \to (X, x)\) is an elementary étale neighbourhood,

\(\pi\) is finite, and \(\pi^{-1}(\{y'\}) = \{x'\}\),

\(\kappa(y')\) is a purely transcendental extension of \(\kappa(s')\), and

\(e : (S', s') \to (S, s)\) is an elementary étale neighbourhood.

Moreover, if \(f\) is locally of finite presentation, then \(\pi\) is of finite presentation.

Proof

The question is local on \(S\), hence we may replace \(S\) by an affine open neighbourhood of \(s\). Next, we apply Lemma 052E to get a commutative diagram \[\xymatrix{ X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] \\ & Y \ar[d]^h & & y \ar@{|->}[d] \\ S \ar@{=}[r] & S & s & s \ar@{=}[l] }\] where \(h\) is smooth of relative dimension \(n\) and \(\kappa(y)\) is a purely transcendental extension of \(\kappa(s)\). Since the question is local on \(X\) also, we may replace \(Y\) by an affine neighbourhood of \(y\) (and \(X'\) by the inverse image of this under \(\pi\)). As \(S\) is affine this guarantees that \(Y \to S\) is quasi-compact, separated and smooth, in particular of finite presentation. Let \(T\) be the connected component of \(Y_s\) containing \(y\). As \(Y_s\) is Noetherian we see that \(T\) is open. We also see that \(T\) is geometrically connected over \(\kappa(s)\) by Varieties, Lemma 04KV. Since \(T\) is also smooth over \(\kappa(s)\) it is geometrically normal, see Varieties, Lemma 056T. We conclude that \(T\) is geometrically irreducible over \(\kappa(s)\) (as a connected Noetherian normal scheme is irreducible, see Properties, Lemma 033M). Finally, note that the smooth morphism \(h\) is normal by Lemma 056W. At this point we have verified all assumption of Lemma 057J hold for the morphism \(h : Y \to S\) and open \(T \subset Y_s\). As a result of applying Lemma 057J we obtain \(e : S' \to S\), \(s' \in S'\), \(Y'\) as in the commutative diagram \[\xymatrix{ X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & X' \times_Y Y' \ar[l] \ar[d] & x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] & (x', s') \ar@{|->}[l] \ar@{|->}[d] \\ & Y \ar[d]^h & Y' \ar[d] \ar[l] & & y \ar@{|->}[d] & (y, s') \ar@{|->}[l] \ar@{|->}[d] \\ S \ar@{=}[r] & S & S' \ar[l]_e & s & s \ar@{=}[l] & s' \ar@{|->}[l] }\] where \(e : (S', s') \to (S, s)\) is an elementary étale neighbourhood, and where \(Y' \subset Y_{S'}\) is an open neighbourhood all of whose fibres over \(S'\) are geometrically irreducible, such that \(Y'_{s'} = T\) via the identification \(Y_s = Y_{S', s'}\). Let \((y, s') \in Y'\) be the point corresponding to \(y \in T\); this is also the unique point of \(Y \times_S S'\) lying over \(y\) with residue field equal to \(\kappa(y)\) which maps to \(s'\) in \(S'\). Similarly, let \((x', s') \in X' \times_Y Y' \subset X' \times_S S'\) be the unique point over \(x'\) with residue field equal to \(\kappa(x')\) lying over \(s'\). Then the outer part of this diagram is a solution to the problem posed in the lemma. Some minor details omitted.

Lemma

Assumption and notation as in Lemma 057K. In addition to properties (1) – (6) we may also arrange it so that

\(S'\), \(Y'\), \(X'\) are affine.

Proof

Note that if \(Y'\) is affine, then \(X'\) is affine as \(\pi\) is finite. Choose an affine open neighbourhood \(U' \subset S'\) of \(s'\). Choose an affine open neighbourhood \(V' \subset h^{-1}(U')\) of \(y'\). Let \(W' = h(V')\). This is an open neighbourhood of \(s'\) in \(S'\), see Morphisms, Lemma 056G, contained in \(U'\). Choose an affine open neighbourhood \(U'' \subset W'\) of \(s'\). Then \(h^{-1}(U'') \cap V'\) is affine because it is equal to \(U'' \times_{U'} V'\). By construction \(h^{-1}(U'') \cap V' \to U''\) is a surjective smooth morphism whose fibres are (nonempty) open subschemes of geometrically integral fibres of \(Y' \to S'\), and hence geometrically integral. Thus we may replace \(S'\) by \(U''\) and \(Y'\) by \(h^{-1}(U'') \cap V'\).

The significance of the property \(\pi^{-1}(\{y'\}) = \{x'\}\) is partially explained by the following lemma.

Lemma

Let \(\pi : X \to Y\) be a finite morphism. Let \(x \in X\) with \(y = \pi(x)\) such that \(\pi^{-1}(\{y\}) = \{x\}\). Then

For every neighbourhood \(U \subset X\) of \(x\) in \(X\), there exists a neighbourhood \(V \subset Y\) of \(y\) such that \(\pi^{-1}(V) \subset U\).

The ring map \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\) is finite.

If \(\pi\) is of finite presentation, then \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\) is of finite presentation.

For any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \(\mathcal{F}_x = \pi_*\mathcal{F}_y\) as \(\mathcal{O}_{Y, y}\)-modules.

Proof

The first assertion is purely topological; use that \(\pi\) is a continuous and closed map such that \(\pi^{-1}(\{y\}) = \{x\}\). To prove the second and third parts we may assume \(X = \Spec(B)\) and \(Y = \Spec(A)\). Then \(A \to B\) is a finite ring map and \(y\) corresponds to a prime \(\mathfrak p\) of \(A\) such that there exists a unique prime \(\mathfrak q\) of \(B\) lying over \(\mathfrak p\). Then \(B_{\mathfrak q} = B_{\mathfrak p}\), see Algebra, Lemma 00EA. In other words, the map \(A_{\mathfrak p} \to B_{\mathfrak q}\) is equal to the map \(A_{\mathfrak p} \to B_{\mathfrak p}\) you get from localizing \(A \to B\) at \(\mathfrak p\). Thus (2) and (3) follow from simple properties of localization (some details omitted). For the final statement, suppose that \(\mathcal{F} = \widetilde M\) for some \(B\)-module \(M\). Then \(\mathcal{F} = M_{\mathfrak q}\) and \(\pi_*\mathcal{F}_y = M_{\mathfrak p}\). By the above these localizations agree. Alternatively you can use part (1) and the definition of stalks to see that \(\mathcal{F}_x = \pi_*\mathcal{F}_y\) directly.

Application to the fppf topology

We can use the above étale localization techniques to prove the following result describing the fppf topology as being equal to the topology “generated by” Zariski coverings and by coverings of the form \(\{f : T \to S\}\) where \(f\) is surjective finite locally free.

Lemma

Let \(S\) be a scheme. Let \(\{S_i \to S\}_{i \in I}\) be an fppf covering. Then there exist

an étale covering \(\{S'_a \to S\}\),

surjective finite locally free morphisms \(V_a \to S'_a\),

such that the fppf covering \(\{V_a \to S\}\) refines the given covering \(\{S_i \to S\}\).

Proof

We may assume that each \(S_i \to S\) is locally quasi-finite, see Lemma 0572.

Fix a point \(s \in S\). Pick an \(i \in I\) and a point \(s_i \in S_i\) mapping to \(s\). Choose an elementary étale neighbourhood \((S', s) \to (S, s)\) such that there exists an open \[S_i \times_S S' \supset V\] which contains a unique point \(v \in V\) mapping to \(s \in S'\) and such that \(V \to S'\) is finite, see Lemma 02LK. Then \(V \to S'\) is finite locally free, because it is finite and because \(S_i \times_S S' \to S'\) is flat and locally of finite presentation as a base change of the morphism \(S_i \to S\), see Morphisms, Lemmas 01TS, 01U9, and 02KB. Hence \(V \to S'\) is open, and after shrinking \(S'\) we may assume that \(V \to S'\) is surjective finite locally free. Since we can do this for every point of \(S\) we conclude that \(\{S_i \to S\}\) can be refined by a covering of the form \(\{V_a \to S\}_{a \in A}\) where each \(V_a \to S\) factors as \(V_a \to S'_a \to S\) with \(S'_a \to S\) étale and \(V_a \to S'_a\) surjective finite locally free.

Lemma

Let \(S\) be a scheme. Let \(\{S_i \to S\}_{i \in I}\) be an fppf covering. Then there exist

a Zariski open covering \(S = \bigcup U_j\),

surjective finite locally free morphisms \(W_j \to U_j\),

Zariski open coverings \(W_j = \bigcup_k W_{j, k}\),

surjective finite locally free morphisms \(T_{j, k} \to W_{j, k}\)

such that the fppf covering \(\{T_{j, k} \to S\}\) refines the given covering \(\{S_i \to S\}\).

Proof

Let \(\{V_a \to S\}_{a \in A}\) be the fppf covering found in Lemma 0DET. In other words, this covering refines \(\{S_i \to S\}\) and each \(V_a \to S\) factors as \(V_a \to S'_a \to S\) with \(S'_a \to S\) étale and \(V_a \to S'_a\) surjective finite locally free.

By Remark 02LI there exists a Zariski open covering \(S = \bigcup U_j\), for each \(j\) a finite locally free, surjective morphism \(W_j \to U_j\), and for each \(j\) a Zariski open covering \(\{W_{j, k} \to W_j\}\) such that the family \(\{W_{j, k} \to S\}\) refines the étale covering \(\{S'_a \to S\}\), i.e., for each pair \(j, k\) there exists an \(a(j, k)\) and a factorization \(W_{j, k} \to S'_a \to S\) of the morphism \(W_{j, k} \to S\). Set \(T_{j, k} = W_{j, k} \times_{S'_a} V_a\) and everything is clear.

Lemma

Let \(S\) be a scheme. If \(U \subset S\) is open and \(V \to U\) is a surjective integral morphism, then there exists a surjective integral morphism \(\overline{V} \to S\) with \(\overline{V} \times_S U\) isomorphic to \(V\) as schemes over \(U\).

Proof

Let \(V' \to S\) be the normalization of \(S\) in \(U\), see Morphisms, Section 0BAK. By construction \(V' \to S\) is integral. By Morphisms, Lemmas 035K and 03GP we see that the inverse image of \(U\) in \(V'\) is \(V\). Let \(Z\) be the reduced induced scheme structure on \(S \setminus U\). Then \(\overline{V} = V' \amalg Z\) works.

Lemma

Let \(S\) be a quasi-compact and quasi-separated scheme. If \(U \subset S\) is a quasi-compact open and \(V \to U\) is a surjective finite morphism, then there exists a surjective finite morphism \(\overline{V} \to S\) with \(\overline{V} \times_S U\) isomorphic to \(V\) as schemes over \(U\).

Proof

By Zariski’s Main Theorem (Lemma 05K0) we can assume \(V\) is a quasi-compact open in a scheme \(V'\) finite over \(S\). After replacing \(V'\) by the scheme theoretic image of \(V\) we may assume that \(V\) is dense in \(V'\). It follows that \(V' \times_S U = V\) because \(V \to V' \times_S U\) is closed as \(V\) is finite over \(U\). Let \(Z\) be the reduced induced scheme structure on \(S \setminus U\). Then \(\overline{V} = V' \amalg Z\) works.

Lemma

Let \(S\) be a scheme. Let \(\{S_i \to S\}_{i \in I}\) be an fppf covering. Then there exists a surjective integral morphism \(S' \to S\) and an open covering \(S' = \bigcup U'_\alpha\) such that for each \(\alpha\) the morphism \(U'_\alpha \to S\) factors through \(S_i \to S\) for some \(i\).

Proof

Choose \(S = \bigcup U_j\), \(W_j \to U_j\), \(W_j = \bigcup W_{j, k}\), and \(T_{j, k} \to W_{j, k}\) as in Lemma 05WN. By Lemma 0CNX we can extend \(W_j \to U_j\) to a surjective integral morphism \(\overline{W}_j \to S\). After this we can extend \(T_{j, k} \to W_{j, k}\) to a surjective integral morphism \(\overline{T}_{j, k} \to \overline{W}_j\). We set \(\overline{T}_j\) equal to the product of all the schemes \(\overline{T}_{j, k}\) over \(\overline{W}_j\) (Limits, Lemma 0CNI). Then we set \(S'\) equal to the product of all the schemes \(\overline{T}_j\) over \(S\). If \(x \in S'\), then there is a \(j\) such that the image of \(x\) in \(S\) lies in \(U_j\). Hence there is a \(k\) such that the image of \(x\) under the projection \(S' \to \overline{W}_j\) lies in \(W_{j, k}\). Hence under the projection \(S' \to \overline{T}_j \to \overline{T}_{j, k}\) the point \(x\) ends up in \(T_{j, k}\). And \(T_{j, k} \to S\) factors through \(S_i\) for some \(i\). Finally, the morphism \(S' \to S\) is integral and surjective by Limits, Lemmas 0CNK and 0CNJ.

Lemma

Let \(S\) be a quasi-compact and quasi-separated scheme. Let \(\{S_i \to S\}_{i \in I}\) be an fppf covering. Then there exists a surjective finite morphism \(S' \to S\) of finite presentation and an open covering \(S' = \bigcup U'_\alpha\) such that for each \(\alpha\) the morphism \(U'_\alpha \to S\) factors through \(S_i \to S\) for some \(i\).

Proof

Let \(Y \to X\) be the integral surjective morphism found in Lemma 0CNZ. Choose a finite affine open covering \(Y = \bigcup V_j\) such that \(V_j \to X\) factors through \(S_{i(j)}\). We can write \(Y = \lim Y_\lambda\) with \(Y_\lambda \to X\) finite and of finite presentation, see Limits, Lemma 09YZ. For large enough \(\lambda\) we can find affine opens \(V_{\lambda, j} \subset Y_\lambda\) whose inverse image in \(Y\) recovers \(V_j\), see Limits, Lemma 01Z4. For even larger \(\lambda\) the morphisms \(V_j \to S_{i(j)}\) over \(X\) come from morphisms \(V_{\lambda, j} \to S_{i(j)}\) over \(X\), see Limits, Proposition 01ZC. Setting \(S' = Y_\lambda\) for this \(\lambda\) finishes the proof.

Lemma

An fppf covering of schemes is a ph covering.

Proof

Let \(\{T_i \to T\}\) be an fppf covering of schemes, see Topologies, Definition 021M. Observe that \(T_i \to T\) is locally of finite type. Let \(U \subset T\) be an affine open. It suffices to show that \(\{T_i \times_T U \to U\}\) can be refined by a standard ph covering, see Topologies, Definition 0DBG. This follows immediately from Lemma 0CP0 and the fact that a finite morphism is proper (Morphisms, Lemma 01WN).

Remark

As a consequence of Lemma 0DBT we obtain a comparison morphism \[\epsilon : (\Sch/S)_{ph} \longrightarrow (\Sch/S)_{fppf}\] This is the morphism of sites given by the identity functor on underlying categories (with suitable choices of sites as in Topologies, Remark 03FF). The functor \(\epsilon_*\) is the identity on underlying presheaves and the functor \(\epsilon^{-1}\) associated to an fppf sheaf its ph sheafification. By composition we can in addition compare the ph topology with the syntomic, smooth, étale, and Zariski topologies.

Quasi-projective schemes

The term “quasi-projective scheme” has not yet been defined. A possible definition could be a scheme which has an ample invertible sheaf. However, if \(X\) is a scheme over a base scheme \(S\), then we say that \(X\) is quasi-projective over \(S\) if the morphism \(X \to S\) is quasi-projective (Morphisms, Definition 01VW). Since the identity morphism of any scheme is quasi-projective, we see that a scheme quasi-projective over \(S\) doesn’t necessarily have an ample invertible sheaf. For this reason it seems better to leave the term “quasi-projective scheme” undefined.

Lemma

Let \(S\) be a scheme which has an ample invertible sheaf. Let \(f : X \to S\) be a morphism of schemes. The following are equivalent

\(X \to S\) is quasi-projective,

\(X \to S\) is H-quasi-projective,

there exists a quasi-compact open immersion \(X \to X'\) of schemes over \(S\) with \(X' \to S\) projective,

\(X \to S\) is of finite type and \(X\) has an ample invertible sheaf, and

\(X \to S\) is of finite type and there exists an \(f\)-very ample invertible sheaf.

Proof

The implication (2) \(\Rightarrow\) (1) is Morphisms, Lemma 01VY. The implication (1) \(\Rightarrow\) (2) is Morphisms, Lemma 087S. The implication (2) \(\Rightarrow\) (3) is Morphisms, Lemma 01WA

Assume \(X \subset X'\) is as in (3). In particular \(X \to S\) is of finite type. By Morphisms, Lemma 01WA the morphism \(X \to S\) is H-projective. Thus there exists a quasi-compact immersion \(i : X \to \mathbf{P}^n_S\). Hence \(\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_S}(1)\) is \(f\)-very ample. As \(X \to S\) is quasi-compact we conclude from Morphisms, Lemma 01VN that \(\mathcal{L}\) is \(f\)-ample. Thus \(X \to S\) is quasi-projective by definition.

The implication (4) \(\Rightarrow\) (2) is Morphisms, Lemma 01VS.

The implication (5) \(\Rightarrow\) (4) follows from Morphisms, Lemma 0892.

The equivalence of (1) and (5) follows from Morphisms, Lemmas 01VN and 01VU.

Lemma

Let \(S\) be a scheme which has an ample invertible sheaf. Let \(\text{QP}_S\) be the full subcategory of the category of schemes over \(S\) satisfying the equivalent conditions of Lemma 0B42.

if \(S' \to S\) is a morphism of schemes and \(S'\) has an ample invertible sheaf, then base change determines a functor \(\text{QP}_S \to \text{QP}_{S'}\),

if \(X \in \text{QP}_S\) and \(Y \in \text{QP}_X\), then \(Y \in \text{QP}_S\),

the category \(\text{QP}_S\) is closed under fibre products,

the category \(\text{QP}_S\) is closed under finite disjoint unions,

if \(X \to S\) is projective, then \(X \in \text{QP}_S\),

if \(X \to S\) is quasi-affine of finite type, then \(X\) is in \(\text{QP}_S\),

if \(X \to S\) is quasi-finite and separated, then \(X \in \text{QP}_S\),

if \(X \to S\) is a quasi-compact immersion, then \(X \in \text{QP}_S\),

add more here.

Proof

Part (1) follows from Morphisms, Lemma 0B3G.

Part (2) follows from the fourth characterization of Lemma 0B42.

If \(X \to S\) and \(Y \to S\) are quasi-projective, then \(X \times_S Y \to Y\) is quasi-projective by Morphisms, Lemma 0B3G. Hence (3) follows from (2).

If \(X = Y \amalg Z\) is a disjoint union of schemes and \(\mathcal{L}\) is an invertible \(\mathcal{O}_X\)-module such that \(\mathcal{L}|_Y\) and \(\mathcal{L}|_Z\) are ample, then \(\mathcal{L}\) is ample (details omitted). Thus part (4) follows from the fourth characterization of Lemma 0B42.

Part (5) follows from Morphisms, Lemma 07RL.

Part (6) follows from Morphisms, Lemma 0B3H.

Part (7) follows from part (6) and Lemma 02LR.

Part (8) follows from part (7) and Morphisms, Lemma 01TN.

The following lemma doesn’t really belong in this section, but there does not seem to be a good spot for it anywhere else.

Lemma

Let \(X\) be a quasi-affine scheme. Let \(f : U \to X\) be an integral morphism. Then \(U\) is quasi-affine and the diagram \[\xymatrix{ U \ar[r] \ar[d] & \Spec(\Gamma(U, \mathcal{O}_U)) \ar[d] \\ X \ar[r] & \Spec(\Gamma(X, \mathcal{O}_X)) }\] is cartesian.

Proof

The scheme \(U\) is quasi-affine because integral morphisms are affine, affine morphisms are quasi-affine, a scheme is quasi-affine if and only if the structure morphism to \(\Spec(\mathbf{Z})\) is quasi-affine, and compositions of quasi-affine morphisms are quasi-affine. The first two statements follow immediately from the definition and the third is Morphisms, Lemma 01SN. Set \(U' = X \times_{\Spec(\Gamma(X, \mathcal{O}_X))} \Spec(\Gamma(U, \mathcal{O}_U))\) and consider the extended diagram \[\xymatrix{ U \ar[r]_j \ar[rd] & U' \ar[d] \ar[r] & \Spec(\Gamma(U, \mathcal{O}_U)) \ar[d] \\ & X \ar[r] & \Spec(\Gamma(X, \mathcal{O}_X)) }\] The morphism \(j\) is closed by Morphisms, Lemma 01W6 combined with the fact that an integral morphism is universally closed (Morphisms, Lemma 01WM) and the fact that the vertical arrows are in the diagram are separated. On the other hand, \(j\) is open because the horizontal arrows in the diagram of the lemma are open by Properties, Lemma 01P9. Thus \(j\) identifies \(U\) with an open and closed subscheme of \(U'\). If \(U \not = U'\) then \(U\) isn’t dense in \(U'\) and a fortiori not dense in the spectrum of \(\Gamma(U, \mathcal{O}_U)\). However, the scheme theoretic image of \(U\) in \(\Spec(\Gamma(U, \mathcal{O}_U))\) is \(\Spec(\Gamma(U, \mathcal{O}_U))\) because any ideal in \(\Gamma(U, \mathcal{O}_U)\) cutting out a closed subscheme through which \(U\) factors would have to be zero. Hence \(U\) is dense in \(\Spec(\Gamma(U, \mathcal{O}_U))\) for example by Morphisms, Lemma 01R8. Thus \(U = U'\) and we win.

Projective schemes

This section is the analogue of Section 0B41 for projective morphisms.

Lemma

Let \(S\) be a scheme which has an ample invertible sheaf. Let \(f : X \to S\) be a morphism of schemes. The following are equivalent

\(X \to S\) is projective,

\(X \to S\) is H-projective,

\(X \to S\) is quasi-projective and proper,

\(X \to S\) is H-quasi-projective and proper,

\(X \to S\) is proper and \(X\) has an ample invertible sheaf,

\(X \to S\) is proper and there exists an \(f\)-ample invertible sheaf,

\(X \to S\) is proper and there exists an \(f\)-very ample invertible sheaf,

there is a quasi-coherent graded \(\mathcal{O}_S\)-algebra \(\mathcal{A}\) generated by \(\mathcal{A}_1\) over \(\mathcal{A}_0\) with \(\mathcal{A}_1\) a finite type \(\mathcal{O}_S\)-module such that \(X = \underline{\text{Proj}}_S(\mathcal{A})\).

Proof

Observe first that in each case the morphism \(f\) is proper, see Morphisms, Lemmas 01W9 and 01WC. Hence it suffices to prove the equivalence of the notions in case \(f\) is a proper morphism. We will use this without further mention in the following.

The equivalences (1) \(\Leftrightarrow\) (3) and (2) \(\Leftrightarrow\) (4) are Morphisms, Lemma 0BCL.

The implication (2) \(\Rightarrow\) (1) is Morphisms, Lemma 01W9.

The implications (1) \(\Rightarrow\) (2) and (3) \(\Rightarrow\) (4) are Morphisms, Lemma 087S.

The implication (1) \(\Rightarrow\) (7) is immediate from Morphisms, Definitions 01W8 and 01VM.

The conditions (3) and (6) are equivalent by Morphisms, Definition 01VW.

Thus (1) – (4), (6) are equivalent and imply (7). By Lemma 0B42 conditions (3), (5), and (7) are equivalent. Thus we see that (1) – (7) are equivalent.

By Divisors, Lemma 0B3U we see that (8) implies (1). Conversely, if (2) holds, then we can choose a closed immersion \[i : X \longrightarrow \mathbf{P}^n_S = \underline{\text{Proj}}_S(\mathcal{O}_S[T_0, \ldots, T_n]).\] See Constructions, Lemma 01OE for the equality. By Divisors, Lemma 0801 we see that \(X\) is the relative Proj of a quasi-coherent graded quotient algebra \(\mathcal{A}\) of \(\mathcal{O}_S[T_0, \ldots, T_n]\). Then \(\mathcal{A}\) satisfies the conditions of (8).

Lemma

Let \(S\) be a scheme which has an ample invertible sheaf. Let \(\text{P}_S\) be the full subcategory of the category of schemes over \(S\) satisfying the equivalent conditions of Lemma 0B45.

if \(S' \to S\) is a morphism of schemes and \(S'\) has an ample invertible sheaf, then base change determines a functor \(\text{P}_S \to \text{P}_{S'}\),

if \(X \in \text{P}_S\) and \(Y \in \text{P}_X\), then \(Y \in \text{P}_S\),

the category \(\text{P}_S\) is closed under fibre products,

the category \(\text{P}_S\) is closed under finite disjoint unions,

if \(X \to S\) is finite, then \(X\) is in \(\text{P}_S\),

add more here.

Proof

Part (1) follows from Morphisms, Lemma 02V6.

Part (2) follows from the fifth characterization of Lemma 0B45 and the fact that compositions of proper morphisms are proper (Morphisms, Lemma 01W3).

If \(X \to S\) and \(Y \to S\) are projective, then \(X \times_S Y \to Y\) is projective by Morphisms, Lemma 02V6. Hence (3) follows from (2).

If \(X = Y \amalg Z\) is a disjoint union of schemes and \(\mathcal{L}\) is an invertible \(\mathcal{O}_X\)-module such that \(\mathcal{L}|_Y\) and \(\mathcal{L}|_Z\) are ample, then \(\mathcal{L}\) is ample (details omitted). Thus part (4) follows from the fifth characterization of Lemma 0B45.

Part (5) follows from Morphisms, Lemma 0B3I.

Here is a slightly different type of result.

Lemma

Let \(f : X \to Y\) be a proper morphism of schemes. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(y \in Y\) be a point such that \(\mathcal{L}_y\) is ample on \(X_y\). Then there is an open neighbourhood \(V \subset Y\) of \(y\) such that \(\mathcal{L}|_{f^{-1}(V)}\) is ample on \(f^{-1}(V)/V\).

Proof

We may assume \(Y\) is affine. Then we find a directed set \(I\) and an inverse system of morphisms \(X_i \to Y_i\) of schemes with \(Y_i\) of finite type over \(\mathbf{Z}\), with affine transition morphisms \(X_i \to X_{i'}\) and \(Y_i \to Y_{i'}\), with \(X_i \to Y_i\) proper, such that \(X \to Y = \lim (X_i \to Y_i)\). See Limits, Lemma 0A0P. After shrinking \(I\) we can assume we have a compatible system of invertible \(\mathcal{O}_{X_i}\)-modules \(\mathcal{L}_i\) pulling back to \(\mathcal{L}\), see Limits, Lemma 0B8W. Let \(y_i \in Y_i\) be the image of \(y\). Then \(\kappa(y) = \colim \kappa(y_i)\). Hence for some \(i\) we have \(\mathcal{L}_{i, y_i}\) is ample on \(X_{i, y_i}\) by Limits, Lemma 09MT. By Cohomology of Schemes, Lemma 0D2N we find an open neighbourhood \(V_i \subset Y_i\) of \(y_i\) such that \(\mathcal{L}_i\) restricted to \(f_i^{-1}(V_i)\) is ample relative to \(V_i\). Letting \(V \subset Y\) be the inverse image of \(V_i\) finishes the proof (hints: use Morphisms, Lemma 0893 and the fact that \(X \to Y \times_{Y_i} X_i\) is affine and the fact that the pullback of an ample invertible sheaf by an affine morphism is ample by Morphisms, Lemma 0892).

Proj and Spec

In this section we clarify the relationship between the Proj and the spectrum of a graded ring.

Let \(R\) be a ring. Let \(A\) be a graded \(R\)-algebra, see Algebra, Section 00JL. For \(m \geq 0\) we denote \(A_{\geq m} = \bigoplus_{d \geq m} A_d\). Consider the graded ring \[B = \bigoplus\nolimits_{d \geq 0} A_{\geq d}\] For \(d' \geq d\) and \(a \in A_{d'}\) let us denote \(a^{(d)} \in B\) the element in \(B_d\) corresponding to \(a\). Let us denote \(\sigma : A \to B\) and \(\psi : A \to B\) the two obvious ring maps: if \(a \in A_d\), then \(\sigma(a) = a^{(0)}\) and \(\psi(a) = a^{(d)}\). Then \(\psi\) is a graded ring map and \(\sigma\) turns \(B\) into a graded algebra over \(A\). There is also a surjective graded ring map \(\tau : B \to A\) which for \(d' \geq d\) and \(a \in A_{d'}\) sends \(a^{(d)}\) to \(0\) if \(d' > d\) and to \(a\) if \(d' = d\).

Affine schemes and spectra. We set \(X = \Spec(A)\). The irrelevant ideal \(A_+\) cuts out a closed subscheme \(Z = V(A_+) = \Spec(A/A_+) = \Spec(A_0)\). Set \(U = X \setminus Z\). \[U \longrightarrow X \longrightarrow Z\] Projective schemes and Proj. Set \(P = \text{Proj}(A)\). We may and do view \(P\) as a scheme over \(\Spec(A_0) = Z\). Set \(L = \text{Proj}(B)\). We may and do view \(L\) as a scheme over \(\Spec(B_0) = \Spec(A) = X\); observe that the identification of \(B_0\) with \(A\) is given by \(\sigma\). The surjection \(\tau\) defines a closed immersion \(0 : P \to L\). Since \(A \xrightarrow{\sigma} B \to A\) is equal to the map \(A \to A_0 \to A\) we conclude that \[\xymatrix{ P \ar[d] \ar[r]_0 & L \ar[d] \\ Z \ar[r] & X }\] is commutative.

We claim that \(\psi\) defines a morphism \(L \to P\). To see this, by Constructions, Lemma 01MY, it suffices to check \(\psi(A_+) \not \subset \mathfrak p\) for every homogeneous prime ideal \(\mathfrak p \subset B\) with \(B_+ \not \subset \mathfrak p\). First, pick \(g \in B_+\) homogeneous \(g \not \in \mathfrak p\). Then we can write \(g\) as a finite sum \(g = \sum a_i^{(d)}\) with \(a_i \in A_{d_i}\) for some \(d_i \geq d\). We conclude that there exist \(d' \geq d\) and \(a \in A_{d'}\) such that \(a^{(d)} \not \in \mathfrak p\). Then \[(a^{(d)})^{d'} = (a^{d'})^{(d'd)} = a^{(d)} (a^{d' - 1})^{(d(d' - 1))} = \psi(a) (a^{d' - 1})^{(d(d' - 1))}\] (the notation leaves something to be desired) is not in \(\mathfrak p\). Hence \(\psi(a) \not \in \mathfrak p\), proving the claim. Thus we can extend our diagram above to a commutative diagram \[\xymatrix{ P \ar[d] \ar[r]_0 & L \ar[d] \ar[r]_\pi & P \ar[d] \\ Z \ar[r] & X \ar[r] & Z }\] where \(X \to Z\) is given by \(A_0 \to A\). Since \(\tau \circ \psi = \text{id}_A\) we see \(\pi \circ 0 = \text{id}_P\).

Observe that \(\pi\) is an affine morphism. This is clear from the construction in Constructions, Lemma 01MY. In fact, if \(f \in A_d\) for some \(d > 0\), then setting \(g = \psi(f)\) we have \(\pi^{-1}(D_+(f)) = D_+(g)\). In this case we have the following equality of homogeneous parts \[(B[1/g])_{m'} = \bigoplus\nolimits_{m \geq m'} (A[1/f])_m\] This isomorphism is compatible with further localization. Taking \(m' = 0\) we see that \(\pi_*\mathcal{O}_L\) is the direct sum of \(\mathcal{O}_P(m)\) for \(m \geq 0\)⁹. We conclude \(L\) is identified with the relative spectrum: \[L = \underline{\Spec}_P \left( \bigoplus\nolimits_{m \geq 0} \mathcal{O}_P(m) \right)\] In particular \(L \to P\) is a cone¹⁰, see Constructions, Section 062P. Moreover, it is clear that \(0 : P \to L\) is the vertex of the cone.

Let \(f \in A_d\) for some \(d > 0\) and \(g = \psi(f) \in B_d\) as in the previous paragraph. Looking at the structure of the ring maps \[\xymatrix{ A_0 \ar[r] \ar[d] & A \ar[d]^\sigma \ar[r] & A_0 \ar[d] \\ (A[1/f])_0 \ar[r]^-\psi & (B[1/g])_0 = \bigoplus\nolimits_{m \geq 0} (A[1/f])_m \ar[r]^-\tau & (A[1/f])_0 }\] some compuations¹¹ in graded rings will show that

\(\sigma(A_+)(B[1/g])_0 \subset \Ker(\tau : (B[1/g])_0 \to (A[1/f])_0)\),
\(\sigma(f) \in (B[1/g])_0\) is a nonzerodivisor,
\(\sigma(f) (B[1/g])_0 = \sigma(A_d) (B[1/g])_0\) as ideals,
\(\sigma(f) (B[1/g])_0\) and \(\Ker(\tau : (B[1/g])_0 \to (A[1/f])_0)\) have the same radical,
if \(d = 1\), then \(\sigma(f) (B[1/g])_0 = \Ker(\tau : (B[1/g])_0 \to (A[1/f])_0)\).

We see in particular that \[0(D_+(f)) = V(\sigma(f)) \subset D_+(g) = \Spec((B[1/g])_0)\] set theoretically. In other words, the ideal generated by \(\sigma(A_d)\) cuts out an effective Cartier divisor on \(D_+(g)\) which is set theoretically equal to the image of the closed immersion \(0 : P \to L\).

We claim that \(L \to X\) is an isomorphism over \(U\). Namely, if \(f \in A_d\) for some \(d > 0\), then \[\Spec(A_f) \times_X L = \text{Proj}(A_f \otimes_A B) = \text{Proj}(B_{\sigma(f)})\] For each \(e\) we have \((B_{\sigma(f)})_e = A_f \otimes_B B_e = A_f \otimes_A A_{\geq e} = A_f\), the final equality induced by the injection \(A_{\geq e} \subset A\). Hence \(B_{\sigma(f)} \cong A_f[T]\) with \(T\) in degree \(1\). This proves the claim as \(\text{Proj}(A_f[T]) \to \Spec(A_f)\) is an isomorphism. From now on we identify \(U\) with the corresponding open of \(L\).

The identification made in the previous paragraph lets us consider the restriction \(\pi|_U : U \to P\). Pick \(f \in A_d\) for some \(d > 0\) and \(g = \psi(f) \in B_d\) as we have done above several times. Then \[U \cap \pi^{-1}(D_+(f)) = U \cap D_+(g)\] is the complement of the zero locus of \(\sigma(f) \in (B[1/g])_0\) via the identification of \(D_+(g)\) with the spectrum of \((B[1/g])_0\). This is assertion (4) above. Therefore \(U \cap D_+(g)\) is affine and \[\mathcal{O}_L(U \cap D_+(g)) = (B[1/g])_0[1/\sigma(f)] = \bigoplus\nolimits_{m \in \mathbf{Z}} (A[1/f])_m\] where the last equal sign is the natural extension of the identification \((B[1/g])_0 = \bigoplus_{m \geq 0} (A[1/f])_m\) made above. Exactly as we did before with \(\pi : L \to P\) we conclude that \(\pi|_U : U \to P\) is affine and \[U = \underline{\Spec}_P \left( \bigoplus\nolimits_{m \in \mathbf{Z}} \mathcal{O}_P(m) \right)\] as schemes over \(P\).

Summarising the above, our constructions produce a commutative diagram [0EKG]\[\begin{equation} \vcenter{ \xymatrix{ \underline{\Spec}_P \left( \bigoplus\nolimits_{m \in \mathbf{Z}} \mathcal{O}_P(m) \right) \ar[r] \ar@{=}[d] & L = \underline{\Spec}_P \left( \bigoplus\nolimits_{m \geq 0} \mathcal{O}_P(m) \right) \ar[d]^\sigma \ar[r]_-\pi & P \ar[d] \\ U \ar[r] & X \ar[r] & Z } } \end{equation}\] of schemes where \(\pi\) is a cone whose zero section \(0 : P \to L\) maps set theoretically onto the inverse image of \(Z\) in \(L\).

Let \(W \subset P\) be the largest open such that \(\mathcal{O}_P(1)|_W\) is invertible and the natural maps induce isomorphisms \(\mathcal{O}_P(m)|_W \cong \mathcal{O}_P(1)^{\otimes m}|_W\) for all \(m \in \mathbf{Z}\), i.e., the open of Constructions, Lemma 01MU for \(d = 1\). Then we see that \(L|_W = \pi^{-1}(W) \to W\) is a vector bundle (Constructions, Section 01M1) of rank \(1\), namely, \[L|_W = \mathbf{V}(\mathcal{O}_P(1)|_W)\] in Grothendieckian notation. This is immediate from the above showing that \(L|_W\) is equal to the relative spectrum of the symmetric algebra over \(\mathcal{O}_W\) on \(\mathcal{O}_P(1)|_W\). Then clearly the morphism \(0|_W : W \to L|_W\) is the zero section of this vector bundle. In particular \(0(W)\) is an effective Cartier divisor on \(L|_W\). Moreover, the open \(U|_W = (\pi|_U)^{-1}(W)\) is the complement of the zero section.

If \(A\) is generated by \(f_1, \ldots, f_r \in A_1\) over \(A_0\), then \((f_1, \ldots, f_r)^m = A_{\geq m}\) for all \(m \geq 0\) and hence our \(B\) above is the Rees algebra for \(A_+ = (f_1, \ldots, f_r)\). Thus in this case \(L \to X\) is the blowup of \(Z\) and \(W = P\) where \(W\) is as in the preceding paragraph.

If \(P\) is quasi-compact, then for \(d\) sufficiently divisible, the closed subscheme \(D \subset L\) cut out by \(\sigma(A_d)\mathcal{O}_L\) is an effective Cartier divisor, \(0 : P \to L\) factors through \(D\), and \(0(P) = D\) set theoretically. This follows from Constructions, Lemma 01MD and (1), (2), (3), and (4) proved above. (Take any \(d\) divisible by the lcm of the degrees of the elements found in the lemma.)

We continue to assume \(P\) is quasi-compact. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_P\)-module. Let us set \(\mathcal{F}_U = \pi^*\mathcal{F}|_U\). Then we have [0EKH]\[\begin{equation} R\Gamma(U, \mathcal{F}_U) = \bigoplus\nolimits_{m \in \mathbf{Z}} R\Gamma(P, \mathcal{F} \otimes_{\mathcal{O}_P} \mathcal{O}_P(m)) \end{equation}\] Moreover, this direct sum decomposition is functorial in \(\mathcal{F}\) and the induced \(A\)-module structure on the right is the same as the \(A\)-module structure on the left coming from \(U \subset X\). To prove the formula, since \(\pi|_U\) is affine and \((\pi|_U)_*\mathcal{O}_U = \bigoplus_{m \in \mathbf{Z}} \mathcal{O}_P(m)\) we get \[\begin{align*} R(\pi|_U)_*\mathcal{F}_U & = (\pi|_U)_*\mathcal{F}_U \\ & = (\pi|_U)_*(\pi|_U)^*\mathcal{F} \\ & = \mathcal{F} \otimes_{\mathcal{O}_P} \bigoplus\nolimits_{m \in \mathbf{Z}} \mathcal{O}_P(m) \\ & = \bigoplus\nolimits_{m \in \mathbf{Z}} \mathcal{F} \otimes_{\mathcal{O}_P} \mathcal{O}_P(m) \end{align*}\] By Leray we find that \(R\Gamma(U, \mathcal{F}_U) = R\Gamma(P, R(\pi|_U)_*\mathcal{F}_U)\), see Cohomology, Lemma 01F4. The proof is finished because taking cohomology commutes with direct sums in this case, see Derived Categories of Schemes, Lemma 08DZ. This is where we use that \(P\) is quasi-compact; \(P\) is separated by Constructions, Lemma 01MC.

Lemma

Let \(R\) be a ring. Let \(P\) be a proper scheme over \(R\) and let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_P\)-module. Set \(A = \bigoplus_{m \geq 0} \Gamma(P, \mathcal{L}^{\otimes m})\). Then \(P = \text{Proj}(A)\) and diagram (0EKG) becomes the diagram \[\xymatrix{ \underline{\Spec}_P \left( \bigoplus\nolimits_{m \in \mathbf{Z}} \mathcal{L}^{\otimes m} \right) \ar[r] \ar@{=}[d] & L = \underline{\Spec}_P \left( \bigoplus\nolimits_{m \geq 0} \mathcal{L}^{\otimes m} \right) \ar[d]^\sigma \ar[r]_-\pi & P \ar[d] \\ U \ar[r] & X \ar[r] & Z }\] having the properties explained above.

Proof

We have \(P = \text{Proj}(A)\) by Morphisms, Lemma 0C6J. Moreover, by Properties, Lemma 01QI via this identification we have \(\mathcal{O}_P(m) = \mathcal{L}^{\otimes m}\) for all \(m \in \mathbf{Z}\).

Closed points in fibres

Some of the material in this section is taken from the preprint [Osserman-Payne].

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(Z \subset X\) be a closed subscheme. Let \(s \in S\). Assume

\(S\) is irreducible with generic point \(\eta\),

\(X\) is irreducible,

\(f\) is dominant,

\(f\) is locally of finite type,

\(\dim(X_s) \leq \dim(X_\eta)\),

\(Z\) is locally principal in \(X\), and

\(Z_\eta = \emptyset\).

Then the fibre \(Z_s\) is (set theoretically) a union of irreducible components of \(X_s\).

Proof

Let \(X_{red}\) denote the reduction of \(X\). Then \(Z \cap X_{red}\) is a locally principal closed subscheme of \(X_{red}\), see Divisors, Lemma 053P. Hence we may assume that \(X\) is reduced. In other words \(X\) is integral, see Properties, Lemma 01ON. In this case the morphism \(X \to S\) factors through \(S_{red}\), see Schemes, Lemma 0356. Thus we may replace \(S\) by \(S_{red}\) and assume that \(S\) is integral too.

The assertion that \(f\) is dominant signifies that the generic point of \(X\) is mapped to \(\eta\), see Morphisms, Lemma 01RM. Moreover, the scheme \(X_\eta\) is an integral scheme which is locally of finite type over the field \(\kappa(\eta)\). Hence \(d = \dim(X_\eta) \geq 0\) is equal to \(\dim_\xi(X_\eta)\) for every point \(\xi\) of \(X_\eta\), see Algebra, Lemmas 00OS and 00OT. In view of Morphisms, Lemma 02FZ and condition (5) we conclude that \(\dim_x(X_s) = d\) for every \(x \in X_s\).

In the Noetherian case the assertion can be proved as follows. If the lemma does not holds there exists \(x \in Z_s\) which is a generic point of an irreducible component of \(Z_s\) but not a generic point of any irreducible component of \(X_s\). Then we see that \(\dim_x(Z_s) \leq d - 1\), because \(\dim_x(X_s) = d\) and in a neighbourhood of \(x\) in \(X_s\) the closed subscheme \(Z_s\) does not contain any of the irreducible components of \(X_s\). Hence after replacing \(X\) by an open neighbourhood of \(x\) we may assume that \(\dim_z(Z_{f(z)}) \leq d - 1\) for all \(z \in Z\), see Morphisms, Lemma 02FZ. Let \(\xi' \in Z\) be a generic point of an irreducible component of \(Z\) and set \(s' = f(\xi)\). As \(Z \not = X\) is locally principal we see that \(\dim(\mathcal{O}_{X, \xi}) = 1\), see Algebra, Lemma 00KV (this is where we use \(X\) is Noetherian). Let \(\xi \in X\) be the generic point of \(X\) and let \(\xi_1\) be a generic point of any irreducible component of \(X_{s'}\) which contains \(\xi'\). Then we see that we have the specializations \[\xi \leadsto \xi_1 \leadsto \xi'.\] As \(\dim(\mathcal{O}_{X, \xi}) = 1\) one of the two specializations has to be an equality. By assumption \(s' \not = \eta\), hence the first specialization is not an equality. Hence \(\xi' = \xi_1\) is a generic point of an irreducible component of \(X_{s'}\). Applying Morphisms, Lemma 02FZ one more time this implies \(\dim_{\xi'}(Z_{s'}) = \dim_{\xi'}(X_{s'}) \geq \dim(X_\eta) = d\) which gives the desired contradiction.

In the general case we reduce to the Noetherian case as follows. If the lemma is false then there exists a point \(x \in X\) lying over \(s\) such that \(x\) is a generic point of an irreducible component of \(Z_s\), but not a generic point of any of the irreducible components of \(X_s\). Let \(U \subset S\) be an affine neighbourhood of \(s\) and let \(V \subset X\) be an affine neighbourhood of \(x\) with \(f(V) \subset U\). Write \(U = \Spec(A)\) and \(V = \Spec(B)\) so that \(f|_V\) is given by a ring map \(A \to B\). Let \(\mathfrak q \subset B\), resp. \(\mathfrak p \subset A\) be the prime corresponding to \(x\), resp. \(s\). After possibly shrinking \(V\) we may assume \(Z \cap V\) is cut out by some element \(g \in B\). Denote \(K\) the fraction field of \(A\). What we know at this point is the following:

\(A \subset B\) is a finitely generated extension of domains,

the element \(g \otimes 1\) is invertible in \(B \otimes_A K\),

\(d = \dim(B \otimes_A K) = \dim(B \otimes_A \kappa(\mathfrak p))\),

\(g \otimes 1\) is not a unit of \(B \otimes_A \kappa(\mathfrak p)\), and

\(g \otimes 1\) is not in any of the minimal primes of \(B \otimes_A \kappa(\mathfrak p)\).

We are seeking a contradiction.

Pick elements \(x_1, \ldots, x_n \in B\) which generate \(B\) over \(A\). For a finitely generated \(\mathbf{Z}\)-algebra \(A_0 \subset A\) let \(B_0 \subset B\) be the \(A_0\)-subalgebra generated by \(x_1, \ldots, x_n\), denote \(K_0\) the fraction field of \(A_0\), and set \(\mathfrak p_0 = A_0 \cap \mathfrak p\). We claim that when \(A_0\) is large enough then (1) – (5) also hold for the system \((A_0 \subset B_0, g, \mathfrak p_0)\).

We prove each of the conditions in turn. Part (1) holds by construction. For part (2) write \((g \otimes 1) h = 1\) for some \(h \otimes 1/a \in B \otimes_A K\). Write \(g = \sum a_I x^I\), \(h = \sum a'_I x^I\) (multi-index notation) for some coefficients \(a_I, a'_I \in A\). As soon as \(A_0\) contains \(a\) and the \(a_I, a'_I\) then (2) holds because \(B_0 \otimes_{A_0} K_0 \subset B \otimes_A K\) (as localizations of the injective map \(B_0 \to B\)). To achieve (3) consider the exact sequence \[0 \to I \to A[X_1, \ldots, X_n] \to B \to 0\] which defines \(I\) where the second map sends \(X_i\) to \(x_i\). Since \(\otimes\) is right exact we see that \(I \otimes_A K\), respectively \(I \otimes_A \kappa(\mathfrak p)\) is the kernel of the surjection \(K[X_1, \ldots, X_n] \to B \otimes_A K\), respectively \(\kappa(\mathfrak p)[X_1, \ldots, X_n] \to B \otimes_A \kappa(\mathfrak p)\). As a polynomial ring over a field is Noetherian there exist finitely many elements \(h_j \in I\), \(j = 1, \ldots, m\) which generate \(I \otimes_A K\) and \(I \otimes_A \kappa(\mathfrak p)\). Write \(h_j = \sum a_{j, I}X^I\). As soon as \(A_0\) contains all \(a_{j, I}\) we get to the situation where \[B_0 \otimes_{A_0} K_0 \otimes_{K_0} K = B \otimes_A K \quad\text{and}\quad B_0 \otimes_{A_0} \kappa(\mathfrak p_0) \otimes_{\kappa(\mathfrak p_0)} \kappa(\mathfrak p) = B \otimes_A \kappa(\mathfrak p).\] By either Morphisms, Lemma 02FY or Algebra, Lemma 00P3 we see that the dimension equalities of (3) are satisfied. Part (4) is immediate. As \(B_0 \otimes_{A_0} \kappa(\mathfrak p_0) \subset B \otimes_A \kappa(\mathfrak p)\) each minimal prime of \(B_0 \otimes_{A_0} \kappa(\mathfrak p_0)\) lies under a minimal prime of \(B \otimes_A \kappa(\mathfrak p)\) by Algebra, Lemma 00FL. This implies that (5) holds. In this way we reduce the problem to the Noetherian case which we have dealt with above.

Here is an algebraic application of the lemma above. The fourth assumption of the lemma holds if \(A \to B\) is flat, see Lemma 053T.

Lemma

Let \(A \to B\) be a local homomorphism of local rings, and \(g \in \mathfrak m_B\). Assume

\(A\) and \(B\) are domains and \(A \subset B\),

\(B\) is essentially of finite type over \(A\),

\(g\) is not contained in any minimal prime over \(\mathfrak m_AB\), and

\(\dim(B/\mathfrak m_AB) + \text{trdeg}_{\kappa(\mathfrak m_A)}(\kappa(\mathfrak m_B)) = \text{trdeg}_A(B)\).

Then \(A \subset B/gB\), i.e., the generic point of \(\Spec(A)\) is in the image of the morphism \(\Spec(B/gB) \to \Spec(A)\).

Proof

Note that the two assertions are equivalent by Algebra, Lemma 00FL. To start the proof let \(C\) be an \(A\)-algebra of finite type and \(\mathfrak q\) a prime of \(C\) such that \(B = C_{\mathfrak q}\). Of course we may assume that \(C\) is a domain and that \(g \in C\). After replacing \(C\) by a localization we see that \(\dim(C/\mathfrak m_AC) = \dim(B/\mathfrak m_AB) + \text{trdeg}_{\kappa(\mathfrak m_A)}(\kappa(\mathfrak m_B))\), see Morphisms, Lemma 02FX. Setting \(K\) equal to the fraction field of \(A\) we see by the same reference that \(\dim(C \otimes_A K) = \text{trdeg}_A(B)\). Hence assumption (4) means that the generic and closed fibres of the morphism \(\Spec(C) \to \Spec(A)\) have the same dimension.

Suppose that the lemma is false. Then \((B/gB) \otimes_A K = 0\). This means that \(g \otimes 1\) is invertible in \(B \otimes_A K = C_{\mathfrak q} \otimes_A K\). As \(C_{\mathfrak q}\) is a limit of principal localizations we conclude that \(g \otimes 1\) is invertible in \(C_h \otimes_A K\) for some \(h \in C\), \(h \not \in \mathfrak q\). Thus after replacing \(C\) by \(C_h\) we may assume that \((C/gC) \otimes_A K = 0\). We do one more replacement of \(C\) to make sure that the minimal primes of \(C/\mathfrak m_AC\) correspond one-to-one with the minimal primes of \(B/\mathfrak m_AB\). At this point we apply Lemma 053R to \(X = \Spec(C) \to \Spec(A) = S\) and the locally closed subscheme \(Z = \Spec(C/gC)\). Since \(Z_K = \emptyset\) we see that \(Z \otimes \kappa(\mathfrak m_A)\) has to contain an irreducible component of \(X \otimes \kappa(\mathfrak m_A) = \Spec(C/\mathfrak m_AC)\). But this contradicts the assumption that \(g\) is not contained in any prime minimal over \(\mathfrak m_AB\). The lemma follows.

Lemma

Let \(A \to B\) be a local homomorphism of local rings. Assume

\(A\) and \(B\) are domains and \(A \subset B\),

\(B\) is essentially of finite type over \(A\), and

\(B\) is flat over \(A\).

Then we have \[\dim(B/\mathfrak m_AB) + \text{trdeg}_{\kappa(\mathfrak m_A)}(\kappa(\mathfrak m_B)) = \text{trdeg}_A(B).\]

Proof

Let \(C\) be an \(A\)-algebra of finite type and \(\mathfrak q\) a prime of \(C\) such that \(B = C_{\mathfrak q}\). We may assume \(C\) is a domain. We have \(\dim_{\mathfrak q}(C/\mathfrak m_AC) = \dim(B/\mathfrak m_AB) + \text{trdeg}_{\kappa(\mathfrak m_A)}(\kappa(\mathfrak m_B))\), see Morphisms, Lemma 02FX. Setting \(K\) equal to the fraction field of \(A\) we see by the same reference that \(\dim(C \otimes_A K) = \text{trdeg}_A(B)\). Thus we are really trying to prove that \(\dim_{\mathfrak q}(C/\mathfrak m_AC) = \dim(C \otimes_A K)\). Choose a valuation ring \(A'\) in \(K\) dominating \(A\), see Algebra, Lemma 00IA. Set \(C' = C \otimes_A A'\). Choose a prime \(\mathfrak q'\) of \(C'\) lying over \(\mathfrak q\); such a prime exists because \[C'/\mathfrak m_{A'}C' = C/\mathfrak m_AC \otimes_{\kappa(\mathfrak m_A)} \kappa(\mathfrak m_{A'})\] which proves that \(C/\mathfrak m_AC \to C'/\mathfrak m_{A'}C'\) is faithfully flat. This also proves that \(\dim_{\mathfrak q}(C/\mathfrak m_AC) = \dim_{\mathfrak q'}(C'/\mathfrak m_{A'}C')\), see Algebra, Lemma 00P4. Note that \(B' = C'_{\mathfrak q'}\) is a localization of \(B \otimes_A A'\). Hence \(B'\) is flat over \(A'\). The generic fibre \(B' \otimes_{A'} K\) is a localization of \(B \otimes_A K\). Hence \(B'\) is a domain. If we prove the lemma for \(A' \subset B'\), then we get the equality \(\dim_{\mathfrak q'}(C'/\mathfrak m_{A'}C') = \dim(C' \otimes_{A'} K)\) which implies the desired equality \(\dim_{\mathfrak q}(C/\mathfrak m_AC) = \dim(C \otimes_A K)\) by what was said above. This reduces the lemma to the case where \(A\) is a valuation ring.

Let \(A \subset B\) be as in the lemma with \(A\) a valuation ring. As before write \(B = C_{\mathfrak q}\) for some domain \(C\) of finite type over \(A\). By Algebra, Lemma 00QK we obtain \(\dim(C/\mathfrak m_AC) = \dim(C \otimes_A K)\) and we win.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \leadsto x'\) be a specialization of points in \(X\). Set \(s = f(x)\) and \(s' = f(x')\). Assume

\(x'\) is a closed point of \(X_{s'}\), and

\(f\) is locally of finite type.

Then the set \[\{x_1 \in X \text{ such that } f(x_1) = s \text{ and } x_1\text{ is closed in }X_s \text{ and } x \leadsto x_1 \leadsto x' \}\] is dense in the closure of \(x\) in \(X_s\).

Proof

We apply Schemes, Lemma 01J8 to the specialization \(x \leadsto x'\). This produces a morphism \(\varphi : \Spec(B) \to X\) where \(B\) is a valuation ring such that \(\varphi\) maps the generic point to \(x\) and the closed point to \(x'\). We may also assume that \(\kappa(x)\) is the fraction field of \(B\). Let \(A = B \cap \kappa(s)\). Note that this is a valuation ring (see Algebra, Lemma 052L) which dominates the image of \(\mathcal{O}_{S, s'} \to \kappa(s)\). Consider the commutative diagram \[\xymatrix{ \Spec(B) \ar[rd] \ar[r] & X_A \ar[d] \ar[r] & X \ar[d] \\ & \Spec(A) \ar[r] & S }\] The generic (resp. closed) point of \(B\) maps to a point \(x_A\) (resp. \(x'_A\)) of \(X_A\) lying over the generic (resp. closed) point of \(\Spec(A)\). Note that \(x'_A\) is a closed point of the special fibre of \(X_A\) by Morphisms, Lemma 053M. Note that the generic fibre of \(X_A \to \Spec(A)\) is isomorphic to \(X_s\). Thus we have reduced the lemma to the case where \(S\) is the spectrum of a valuation ring, \(s = \eta \in S\) is the generic point, and \(s' \in S\) is the closed point.

We will prove the lemma by induction on \(\dim_x(X_\eta)\). If \(\dim_x(X_\eta) = 0\), then there are no other points of \(X_\eta\) specializing to \(x\) and \(x\) is closed in its fibre, see Morphisms, Lemma 01TH, and the result holds. Assume \(\dim_x(X_\eta) > 0\).

Let \(X' \subset X\) be the reduced induced scheme structure on the irreducible closed subscheme \(\overline{\{x\}}\) of \(X\), see Schemes, Definition 01J4. To prove the lemma we may replace \(X\) by \(X'\) as this only decreases \(\dim_x(X_\eta)\). Hence we may also assume that \(X\) is an integral scheme and that \(x\) is its generic point. In addition, we may replace \(X\) by an affine neighbourhood of \(x'\). Thus we have \(X = \Spec(B)\) where \(A \subset B\) is a finite type extension of domains. Note that in this case \(\dim_x(X_\eta) = \dim(X_\eta) = \dim(X_{s'})\), and that in fact \(X_{s'}\) is equidimensional, see Algebra, Lemma 00QK.

Let \(W \subset X_\eta\) be a proper closed subset (this is the subset we want to “avoid”). As \(X_s\) is of finite type over a field we see that \(W\) has finitely many irreducible components \(W = W_1 \cup \ldots \cup W_n\). Let \(\mathfrak q_j \subset B\), \(j = 1, \ldots, r\) be the corresponding prime ideals. Let \(\mathfrak q \subset B\) be the maximal ideal corresponding to the point \(x'\). Let \(\mathfrak p_1, \ldots, \mathfrak p_s \subset B\) be the minimal primes lying over \(\mathfrak m_AB\). There are finitely many as these correspond to the irreducible components of the Noetherian scheme \(X_{s'}\). Moreover, each of these irreducible components has dimension \(> 0\) (see above) hence we see that \(\mathfrak p_i \not = \mathfrak q\) for all \(i\). Now, pick an element \(g \in \mathfrak q\) such that \(g \not \in \mathfrak q_j\) for all \(j\) and \(g \not \in \mathfrak p_i\) for all \(i\), see Algebra, Lemma 00DS. Denote \(Z \subset X\) the locally principal closed subscheme defined by \(g\). Let \(Z_\eta = Z_{1, \eta} \cup \ldots \cup Z_{n, \eta}\), \(n \geq 0\) be the decomposition of the generic fibre of \(Z\) into irreducible components (finitely many as the generic fibre is Noetherian). Denote \(Z_i \subset X\) the closure of \(Z_{i, \eta}\). After replacing \(X\) by a smaller affine neighbourhood we may assume that \(x' \in Z_i\) for each \(i = 1, \ldots, n\). By construction \(Z \cap X_{s'}\) does not contain any irreducible component of \(X_{s'}\). Hence by Lemma 053R we conclude that \(Z_\eta \not = \emptyset\)! In other words \(n \geq 1\). Letting \(x_1 \in Z_1\) be the generic point we see that \(x_1 \leadsto x'\) and \(f(x_1) = \eta\). Also, by construction \(Z_{1, \eta} \cap W_j \subset W_j\) is a proper closed subset. Hence every irreducible component of \(Z_{1, \eta} \cap W_j\) has codimension \(\geq 2\) in \(X_\eta\) whereas \(\text{codim}(Z_{1, \eta}, X_\eta) = 1\) by Algebra, Lemma 00KV. Thus \(W \cap Z_{1, \eta}\) is a proper closed subset. At this point we see that the induction hypothesis applies to \(Z_1 \to S\) and the specialization \(x_1 \leadsto x'\). This produces a closed point \(x_2\) of \(Z_{1, \eta}\) not contained in \(W\) which specializes to \(x'\). Thus we obtain \(x \leadsto x_2 \leadsto x'\), the point \(x_2\) is closed in \(X_\eta\), and \(x_2 \not \in W\) as desired.

Remark

The proof of Lemma 053U actually shows that there exists a sequence of specializations \[x \leadsto x_1 \leadsto x_2 \leadsto \ldots \leadsto x_d \leadsto x'\] where all \(x_i\) are in the fibre \(X_s\), each specialization is immediate, and \(x_d\) is a closed point of \(X_s\). The integer \(d = \text{trdeg}_{\kappa(s)}(\kappa(x)) = \dim(\overline{\{x\}})\) where the closure is taken in \(X_s\). Moreover, the points \(x_i\) can be chosen to avoid any closed subset of \(X_s\) which does not contain the point \(x\).

Examples, Section 05JH shows that the following lemma is false if \(A\) is not assumed Noetherian.

Lemma

Let \(\varphi : A \to B\) be a local ring map of local rings. Let \(V \subset \Spec(B)\) be an open subscheme which contains at least one prime not lying over \(\mathfrak m_A\). Assume \(A\) is Noetherian, \(\varphi\) essentially of finite type, and \(A/\mathfrak m_A \subset B/\mathfrak m_B\) is finite. Then there exists a \(\mathfrak q \in V\), \(\mathfrak m_A \not = \mathfrak q \cap A\) such that \(A \to B/\mathfrak q\) is the localization of a quasi-finite ring map.

Proof

Since \(A\) is Noetherian and \(A \to B\) is essentially of finite type, we know that \(B\) is Noetherian too. By Properties, Lemma 02IM the topological space \(\Spec(B) \setminus \{\mathfrak m_B\}\) is Jacobson. Hence we can choose a closed point \(\mathfrak q\) which is contained in the nonempty open \[V \setminus \{\mathfrak q \subset B \mid \mathfrak m_A = \mathfrak q \cap A\}.\] (Nonempty by assumption, open because \(\{\mathfrak m_A\}\) is a closed subset of \(\Spec(A)\).) Then \(\Spec(B/\mathfrak q)\) has two points, namely \(\mathfrak m_B\) and \(\mathfrak q\) and \(\mathfrak q\) does not lie over \(\mathfrak m_A\). Write \(B/\mathfrak q = C_{\mathfrak m}\) for some finite type \(A\)-algebra \(C\) and prime ideal \(\mathfrak m\). Then \(A \to C\) is quasi-finite at \(\mathfrak m\) by Algebra, Lemma 00PK (2). Hence by Algebra, Lemma 00QA we see that after replacing \(C\) by a principal localization the ring map \(A \to C\) is quasi-finite.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) with image \(s \in S\). Let \(U \subset X\) be an open subscheme. Assume \(f\) locally of finite type, \(S\) locally Noetherian, \(x\) a closed point of \(X_s\), and assume there exists a point \(x' \in U\) with \(x' \leadsto x\) and \(f(x') \not = s\). Then there exists a closed subscheme \(Z \subset X\) such that (a) \(x \in Z\), (b) \(f|_Z : Z \to S\) is quasi-finite at \(x\), and (c) there exists a \(z \in Z\), \(z \in U\), \(z \leadsto x\) and \(f(z) \not = s\).

Proof

This is a reformulation of Lemma 05GT. Namely, set \(A = \mathcal{O}_{S, s}\) and \(B = \mathcal{O}_{X, x}\). Denote \(V \subset \Spec(B)\) the inverse image of \(U\). The ring map \(f^\sharp : A \to B\) is essentially of finite type. By assumption there exists at least one point of \(V\) which does not map to the closed point of \(\Spec(A)\). Hence all the assumptions of Lemma 05GT hold and we obtain a prime \(\mathfrak q \subset B\) which does not lie over \(\mathfrak m_A\) and such that \(A \to B/\mathfrak q\) is the localization of a quasi-finite ring map. Let \(z \in X\) be the image of the point \(\mathfrak q\) under the canonical morphism \(\Spec(B) \to X\). Set \(Z = \overline{\{z\}}\) with the induced reduced scheme structure. As \(z \leadsto x\) we see that \(x \in Z\) and \(\mathcal{O}_{Z, x} = B/\mathfrak q\). By construction \(Z \to S\) is quasi-finite at \(x\).

Remark

We can use Lemma 05GT or its variant Lemma 05GU to give an alternative proof of Lemma 053U in case \(S\) is locally Noetherian. Here is a rough sketch. Namely, first replace \(S\) by the spectrum of the local ring at \(s'\). Then we may use induction on \(\dim(S)\). The case \(\dim(S) = 0\) is trivial because then \(s' = s\). Replace \(X\) by the reduced induced scheme structure on \(\overline{\{x\}}\). Apply Lemma 05GU to \(X \to S\) and \(x' \mapsto s'\) and any nonempty open \(U \subset X\) containing \(x\). This gives us a closed subscheme \(x' \in Z \subset X\) a point \(z \in Z\) such that \(Z \to S\) is quasi-finite at \(x'\) and such that \(f(z) \not = s'\). Then \(z\) is a closed point of \(X_{f(z)}\), and \(z \leadsto x'\). As \(f(z) \not = s'\) we see \(\dim(\mathcal{O}_{S, f(z)}) < \dim(S)\). Since \(x\) is the generic point of \(X\) we see \(x \leadsto z\), hence \(s = f(x) \leadsto f(z)\). Apply the induction hypothesis to \(s \leadsto f(z)\) and \(z \mapsto f(z)\) to win.

Lemma

Suppose that \(f : X \to S\) is locally of finite type, \(S\) locally Noetherian, \(x \in X\) a closed point of its fibre \(X_s\), and \(U \subset X\) an open subscheme such that \(U \cap X_s = \emptyset\) and \(x \in \overline{U}\), then the conclusions of Lemma 05GU hold.

Proof

Namely, we can reduce this to the cited lemma as follows: First we replace \(X\) and \(S\) by affine neighbourhoods of \(x\) and \(s\). Then \(X\) is Noetherian, in particular \(U\) is quasi-compact (see Morphisms, Lemma 01T6 and Topology, Lemmas 0052 and 04ZA). Hence there exists a specialization \(x' \leadsto x\) with \(x' \in U\) (see Morphisms, Lemma 02JQ). Note that \(f(x') \not = s\). Thus we see all hypotheses of the lemma are satisfied and we win.

Stein factorization

Stein factorization is the statement that a proper morphism \(f : X \to S\) with \(f_*\mathcal{O}_X = \mathcal{O}_S\) has connected fibres.

Lemma

Let \(S\) be a scheme. Let \(f : X \to S\) be a universally closed and quasi-separated morphism. There exists a factorization \[\xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & S' \ar[dl]^\pi \\ & S & }\] with the following properties:

the morphism \(f'\) is universally closed, quasi-compact, quasi-separated, and surjective,

the morphism \(\pi : S' \to S\) is integral,

we have \(f'_*\mathcal{O}_X = \mathcal{O}_{S'}\),

we have \(S' = \underline{\Spec}_S(f_*\mathcal{O}_X)\), and

\(S'\) is the normalization of \(S\) in \(X\), see Morphisms, Definition 035H.

Formation of the factorization \(f = \pi \circ f'\) commutes with flat base change.

Proof

By Morphisms, Lemma 04XU the morphism \(f\) is quasi-compact. Hence the normalization \(S'\) of \(S\) in \(X\) is defined (Morphisms, Definition 035H) and we have the factorization \(X \to S' \to S\). By Morphisms, Lemma 03GQ we have (2), (4), and (5). The morphism \(f'\) is universally closed by Morphisms, Lemma 01W6. It is quasi-compact by Schemes, Lemma 03GI and quasi-separated by Schemes, Lemma 01KV.

To show the remaining statements we may assume the base scheme \(S\) is affine, say \(S = \Spec(R)\). Then \(S' = \Spec(A)\) with \(A = \Gamma(X, \mathcal{O}_X)\) an integral \(R\)-algebra. Thus it is clear that \(f'_*\mathcal{O}_X\) is \(\mathcal{O}_{S'}\) (because \(f'_*\mathcal{O}_X\) is quasi-coherent, by Schemes, Lemma 01LC, and hence equal to \(\widetilde{A}\)). This proves (3).

Let us show that \(f'\) is surjective. As \(f'\) is universally closed (see above) the image of \(f'\) is a closed subset \(V(I) \subset S' = \Spec(A)\). Pick \(h \in I\). Then \(h|_X = f^\sharp(h)\) is a global section of the structure sheaf of \(X\) which vanishes at every point. As \(X\) is quasi-compact this means that \(h|_X\) is a nilpotent section, i.e., \(h^n|X = 0\) for some \(n > 0\). But \(A = \Gamma(X, \mathcal{O}_X)\), hence \(h^n = 0\). As every element of \(I\) is nilpotent, we conclude that \(V(I) = S'\) as desired.

By Cohomology of Schemes, Lemma 02KH we see that formation of \(f_*\mathcal{O}_X\) commutes with flat base change. Formation of the relative spectrum commutes with any base change by Constructions, Lemma 01LX. Thus formation of the factorization commutes with flat base change.

Lemma

In Lemma 03GY assume in addition that \(f\) is locally of finite type. Then for \(s \in S\) the fibre \(\pi^{-1}(\{s\}) = \{s_1, \ldots, s_n\}\) is finite and the field extensions \(\kappa(s_i)/\kappa(s)\) are finite.

Proof

Recall that there are no specializations among the points of \(\pi^{-1}(\{s\})\), see Algebra, Lemma 00GT. As \(f'\) is surjective, we find that \(|X_s| \to \pi^{-1}(\{s\})\) is surjective. Observe that \(X_s\) is a quasi-separated scheme of finite type over a field (quasi-compactness was shown in the proof of the referenced lemma). Thus \(X_s\) is Noetherian (Morphisms, Lemma 01T6). A topological argument (omitted) now shows that \(\pi^{-1}(\{s\})\) is finite. For each \(i\) we can pick a finite type point \(x_i \in X_s\) mapping to \(s_i\) (Morphisms, Lemma 02J4). We conclude that \(\kappa(s_i)/\kappa(s)\) is finite: \(x_i\) can be represented by a morphism \(\Spec(k_i) \to X_s\) of finite type (by our definition of finite type points) and hence \(\Spec(k_i) \to s = \Spec(\kappa(s))\) is of finite type (as a composition of finite type morphisms), hence \(k_i/\kappa(s)\) is finite (Morphisms, Lemma 01TA).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(s \in S\). Then \(X_s\) is geometrically connected, if and only if for every étale neighbourhood \((U, u) \to (S, s)\) the base change \(X_U \to U\) has connected fibre \(X_u\).

Proof

If \(X_s\) is geometrically connected, then any base change of it is connected. On the other hand, suppose that \(X_s\) is not geometrically connected. Then by Varieties, Lemma 0389 we see that \(X_s \times_{\Spec(\kappa(s))} \Spec(k)\) is disconnected for some finite separable field extension \(k/\kappa(s)\). By Lemma 02LF there exists an affine étale neighbourhood \((U, u) \to (S, s)\) such that \(\kappa(u)/\kappa(s)\) is identified with \(k/\kappa(s)\). In this case \(X_u\) is disconnected.

Theorem

Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a proper morphism. There exists a factorization \[\xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & S' \ar[dl]^\pi \\ & S & }\] with the following properties:

the morphism \(f'\) is proper with geometrically connected fibres,

the morphism \(\pi : S' \to S\) is finite,

we have \(f'_*\mathcal{O}_X = \mathcal{O}_{S'}\),

we have \(S' = \underline{\Spec}_S(f_*\mathcal{O}_X)\), and

\(S'\) is the normalization of \(S\) in \(X\), see Morphisms, Definition 035H.

Proof

Let \(f = \pi \circ f'\) be the factorization of Lemma 03GY. Note that besides the conclusions of Lemma 03GY we also have that \(f'\) is separated (Schemes, Lemma 01KV) and finite type (Morphisms, Lemma 01T8). Hence \(f'\) is proper. By Cohomology of Schemes, Proposition 02O5 we see that \(f_*\mathcal{O}_X\) is a coherent \(\mathcal{O}_S\)-module. Hence we see that \(\pi\) is finite, i.e., (2) holds.

This proves all but the most interesting assertion, namely that all the fibres of \(f'\) are geometrically connected. It is clear from the discussion above that we may replace \(S\) by \(S'\), and we may therefore assume that \(S\) is Noetherian, affine, \(f : X \to S\) is proper, and \(f_*\mathcal{O}_X = \mathcal{O}_S\). Let \(s \in S\) be a point of \(S\). We have to show that \(X_s\) is geometrically connected. By Lemma 03GZ we see that it suffices to show \(X_u\) is connected for every étale neighbourhood \((U, u) \to (S, s)\). We may assume \(U\) is affine. Thus \(U\) is Noetherian (Morphisms, Lemma 01T6), the base change \(f_U : X_U \to U\) is proper (Morphisms, Lemma 01W4), and that also \((f_U)_*\mathcal{O}_{X_U} = \mathcal{O}_U\) (Cohomology of Schemes, Lemma 02KH). Hence after replacing \((f : X \to S, s)\) by the base change \((f_U : X_U \to U, u)\) it suffices to prove that the fibre \(X_s\) is connected when \(f_*\mathcal{O}_X = \mathcal{O}_S\). We can deduce this from Derived Categories of Schemes, Lemma 0G7X (by looking at idempotents in the structure sheaf of \(X_s\)) but we will also give a direct argument below.

Namely, we apply the theorem on formal functions, more precisely Cohomology of Schemes, Lemma 02OD. It tells us that \[\mathcal{O}^\wedge_{S, s} = (f_*\mathcal{O}_X)_s^\wedge = \lim_n H^0(X_n, \mathcal{O}_{X_n})\] where \(X_n\) is the \(n\)th infinitesimal neighbourhood of \(X_s\). Since the underlying topological space of \(X_n\) is equal to that of \(X_s\) we see that if \(X_s = T_1 \amalg T_2\) is a disjoint union of nonempty open and closed subschemes, then similarly \(X_n = T_{1, n} \amalg T_{2, n}\) for all \(n\). And this in turn means \(H^0(X_n, \mathcal{O}_{X_n})\) contains a nontrivial idempotent \(e_{1, n}\), namely the function which is identically \(1\) on \(T_{1, n}\) and identically \(0\) on \(T_{2, n}\). It is clear that \(e_{1, n + 1}\) restricts to \(e_{1, n}\) on \(X_n\). Hence \(e_1 = \lim e_{1, n}\) is a nontrivial idempotent of the limit. This contradicts the fact that \(\mathcal{O}^\wedge_{S, s}\) is a local ring. Thus the assumption was wrong, i.e., \(X_s\) is connected, and we win.

Theorem

Let \(S\) be a scheme. Let \(f : X \to S\) be a proper morphism. There exists a factorization \[\xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & S' \ar[dl]^\pi \\ & S & }\] with the following properties:

the morphism \(f'\) is proper with geometrically connected fibres,

the morphism \(\pi : S' \to S\) is integral,

we have \(f'_*\mathcal{O}_X = \mathcal{O}_{S'}\),

we have \(S' = \underline{\Spec}_S(f_*\mathcal{O}_X)\), and

\(S'\) is the normalization of \(S\) in \(X\), see Morphisms, Definition 035H.

Proof

We may apply Lemma 03GY to get the morphism \(f' : X \to S'\). Note that besides the conclusions of Lemma 03GY we also have that \(f'\) is separated (Schemes, Lemma 01KV) and finite type (Morphisms, Lemma 01T8). Hence \(f'\) is proper. At this point we have proved all of the statements except for the statement that \(f'\) has geometrically connected fibres.

We may assume that \(S = \Spec(R)\) is affine. Set \(R' = \Gamma(X, \mathcal{O}_X)\). Then \(S' = \Spec(R')\). Thus we may replace \(S\) by \(S'\) and assume that \(S = \Spec(R)\) is affine \(R = \Gamma(X, \mathcal{O}_X)\). Next, let \(s \in S\) be a point. Let \(U \to S\) be an étale morphism of affine schemes and let \(u \in U\) be a point mapping to \(s\). Let \(X_U \to U\) be the base change of \(X\). By Lemma 03GZ it suffices to show that the fibre of \(X_U \to U\) over \(u\) is connected. By Cohomology of Schemes, Lemma 02KH we see that \(\Gamma(X_U, \mathcal{O}_{X_U}) = \Gamma(U, \mathcal{O}_U)\). Hence we have to show: Given \(S = \Spec(R)\) affine, \(X \to S\) proper with \(\Gamma(X, \mathcal{O}_X) = R\) and \(s \in S\) is a point, the fibre \(X_s\) is connected.

To do this it suffices to show that the only idempotents \(e \in H^0(X_s, \mathcal{O}_{X_s})\) are \(0\) and \(1\) (we already know that \(X_s\) is nonempty by Lemma 03GY). By Derived Categories of Schemes, Lemma 0G7X after replacing \(R\) by a principal localization we may assume \(e\) is the image of an element of \(R\). Since \(R \to H^0(X_s, \mathcal{O}_{X_s})\) factors through \(\kappa(s)\) we conclude.

Here is an application.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume

\(f\) is proper,

\(S\) is integral with generic point \(\xi\),

\(S\) is normal,

\(X\) is reduced,

every generic point of an irreducible component of \(X\) maps to \(\xi\),

we have \(H^0(X_\xi, \mathcal{O}) = \kappa(\xi)\).

Then \(f_*\mathcal{O}_X = \mathcal{O}_S\) and \(f\) has geometrically connected fibres.

Proof

Apply Theorem 03H2 to get a factorization \(X \to S' \to S\). It is enough to show that \(S' = S\). This will follow from Morphisms, Lemma 0AB1. Namely, \(S'\) is reduced because \(X\) is reduced (Morphisms, Lemma 0AXN). The morphism \(S' \to S\) is integral by the theorem cited above. Every generic point of \(S'\) lies over \(\xi\) by Morphisms, Lemma 0AXP and assumption (5). On the other hand, since \(S'\) is the relative spectrum of \(f_*\mathcal{O}_X\) we see that the scheme theoretic fibre \(S'_\xi\) is the spectrum of \(H^0(X_\xi, \mathcal{O})\) which is equal to \(\kappa(\xi)\) by assumption. Hence \(S'\) is an integral scheme with function field equal to the function field of \(S\). This finishes the proof.

Here is another application.

Lemma

Let \(X \to S\) be a flat proper morphism of finite presentation. Let \(n_{X/S}\) be the function on \(S\) counting the numbers of geometric connected components of fibres of \(f\) introduced in Lemma 055F. Then \(n_{X/S}\) is lower semi-continuous.

Proof

Let \(s \in S\). Set \(n = n_{X/S}(s)\). Note that \(n < \infty\) as the geometric fibre of \(X \to S\) at \(s\) is a proper scheme over a field, hence Noetherian, hence has a finite number of connected components. We have to find an open neighbourhood \(V\) of \(s\) such that \(n_{X/S}|_V \geq n\). Let \(X \to S' \to S\) be the Stein factorization as in Theorem 03H2. By Lemma 0E0M there are finitely many points \(s'_1, \ldots, s'_m \in S'\) lying over \(s\) and the extensions \(\kappa(s'_i)/\kappa(s)\) are finite. Then Lemma 0BSR tells us that after replacing \(S\) by an étale neighbourhood of \(s\) we may assume \(S' = V_1 \amalg \ldots \amalg V_m\) as a scheme with \(s'_i \in V_i\) and \(\kappa(s'_i)/\kappa(s)\) purely inseparable. Then the schemes \(X_{s_i'}\) are geometrically connected over \(\kappa(s)\), hence \(m = n\). The schemes \(X_i = (f')^{-1}(V_i)\), \(i = 1, \ldots, n\) are flat and of finite presentation over \(S\). Hence the image of \(X_i \to S\) is open (Morphisms, Lemma 01UA). Thus in a neighbourhood of \(s\) we see that \(n_{X/S}\) is at least \(n\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume

\(f\) is proper, flat, and of finite presentation, and

the geometric fibres of \(f\) are reduced.

Then the function \(n_{X/S} : S \to \mathbf{Z}\) counting the numbers of geometric connected components of fibres of \(f\) is locally constant.

Proof

By Lemma 0BUI the function \(n_{X/S}\) is lower semincontinuous. For \(s \in S\) consider the \(\kappa(s)\)-algebra \[A_s = H^0(X_s, \mathcal{O}_{X_s})\] Define a function \(\beta_0 : X \to \mathbf{Z}\) sending \(s\) to \(\dim_{\kappa(s)} A_s\). By Varieties, Lemma 0BUG and the fact that \(X_s\) is geometrically reduced \(A_s\) is finite product of finite separable extensions of \(\kappa(s)\). Hence \(A_s \otimes_{\kappa(s)} \kappa(\overline{s})\) is a product of \(\beta_0(s)\) copies of \(\kappa(\overline{s})\). Thus \(X_{\overline{s}}\) has \(\beta_0(s)\) connected components. In other words, we have \(n_{X/S} = \beta_0\) as functions on \(S\). Thus \(n_{X/S}\) is upper semi-continuous by Derived Categories of Schemes, Lemma 0BDN. This finishes the proof.

A final application.

Lemma

Let \((A, I)\) be a henselian pair. Let \(X \to \Spec(A)\) be separated and of finite type. Set \(X_0 = X \times_{\Spec(A)} \Spec(A/I)\). Let \(Y \subset X_0\) be an open and closed subscheme such that \(Y \to \Spec(A/I)\) is proper. Then there exists an open and closed subscheme \(W \subset X\) which is proper over \(A\) with \(W \times_{\Spec(A)} \Spec(A/I) = Y\).

Proof

We will denote \(T \mapsto T_0\) the base change by \(\Spec(A/I) \to \Spec(A)\). By Chow’s lemma (in the form of Limits, Lemma 0202) there exists a surjective proper morphism \(\varphi : X' \to X\) such that \(X'\) admits an immersion into \(\mathbf{P}^n_A\). Set \(Y' = \varphi^{-1}(Y)\). This is an open and closed subscheme of \(X'_0\). Suppose the lemma holds for \((X', Y')\). Let \(W' \subset X'\) be the open and closed subscheme proper over \(A\) such that \(Y' = W'_0\). By Morphisms, Lemma 01W6 \(W = \varphi(W') \subset X\) and \(Q = \varphi(X' \setminus W') \subset X\) are closed subsets and by Morphisms, Lemma 03GN \(W\) is proper over \(A\). The image of \(W \cap Q\) in \(\Spec(A)\) is closed. Since \((A, I)\) is henselian, if \(W \cap Q\) is nonempty, then we find that \(W \cap Q\) has a point lying over \(\Spec(A/I)\). This is impossible as \(W'_0 = Y' = \varphi^{-1}(Y)\). We conclude that \(W\) is an open and closed subscheme of \(X\) proper over \(A\) with \(W_0 = Y\). Thus we reduce to the case described in the next paragraph.

Assume there exists an immersion \(j : X \to \mathbf{P}^n_A\) over \(A\). Let \(\overline{X}\) be the scheme theoretic image of \(j\). Since \(j\) is a quasi-compact morphism (Schemes, Lemma 03GI) we see that \(j : X \to \overline{X}\) is an open immersion (Morphisms, Lemma 01RG). Hence the base change \(j_0 : X_0 \to \overline{X}_0\) is an open immersion as well. Thus \(j_0(Y) \subset \overline{X}_0\) is open. It is also closed by Morphisms, Lemma 01W6. Suppose that the lemma holds for \((\overline{X}, j_0(Y))\). Let \(\overline{W} \subset \overline{X}\) be the corresponding open and closed subscheme proper over \(A\) such that \(j_0(Y) = \overline{W}_0\). Then \(T = \overline{W} \setminus j(X)\) is closed in \(\overline{W}\), hence has closed image in \(\Spec(A)\) by properness of \(\overline{W}\) over \(A\). Since \((A, I)\) is henselian, we find that if \(T\) is nonempty, then there is a point of \(T\) mapping into \(\Spec(A/I)\). This is impossible because \(j_0(Y) = \overline{W}_0\) is contained in \(j(X)\). Hence \(\overline{W}\) is contained in \(j(X)\) and we can set \(W \subset X\) equal to the unique open and closed subscheme mapping isomorphically to \(\overline{W}\) via \(j\). Thus we reduce to the case described in the next paragraph.

Assume \(X \subset \mathbf{P}^n_A\) is a closed subscheme. Then \(X \to \Spec(A)\) is a proper morphism. Let \(Z = X_0 \setminus Y\). This is an open and closed subscheme of \(X_0\) and \(X_0 = Y \amalg Z\). Let \(X \to X' \to \Spec(A)\) be the Stein factorization as in Theorem 03H2. Let \(Y' \subset X'_0\) and \(Z' \subset X'_0\) be the images of \(Y\) and \(Z\). Since the fibres of \(X \to Z\) are geometrically connected, we see that \(Y' \cap Z' = \emptyset\). Hence \(X'_0 = Y' \amalg Z'\) as \(X \to X'\) is surjective. Since \(X' \to \Spec(A)\) is integral, we see that \(X'\) is the spectrum of an \(A\)-algebra integral over \(A\). Recall that open and closed subsets of spectra correspond \(1\)-to-\(1\) with idempotents in the corresponding ring, see Algebra, Lemma 00EE. Hence by More on Algebra, Lemma 09XI we see that we may write \(X' = W' \amalg V'\) with \(W'\) and \(V'\) open and closed and with \(Y' = W'_0\) and \(Z' = V'_0\). Let \(W\) be the inverse image in \(X\) to finish the proof.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a separated morphism locally of finite type and let \(e : S \to X\) be a section. Let \(s \in S\). If the connected component \(X_s^0\) containing \(e(s)\) is proper over \(\kappa(s)\), then there is an open neighbourhood \(U \subset S\) of \(s\) such that \(X_U^0\) is closed in \(X_U\). With the reduced induced closed subscheme structure, \(X_U^0\) is proper over \(U\).

Proof

We may first replace \(S\) by an affine open neighbourhood of \(s\). The proper scheme \(X_s^0\) is quasi-compact. Choose a quasi-compact open \(W\subset X\) which contains \(X_s^0\). After shrinking \(S\) inside the open neighbourhood \(e^{-1}(W)\) of \(s\), the section factors through \(W\). The scheme \(W\) is Noetherian: it is quasi-compact and locally Noetherian, and it is quasi-separated because \(X\to S\) is separated. Hence \(W\to S\) is of finite type. Replacing \(X\) by \(W\), which does not change the connected component of the fibre containing \(e(s)\), reduces us to the case where \(f\) is of finite type.

Let \(A\) be the henselization of \(\mathcal{O}_{S, s}\). Apply Lemma 0CT9 to \(X_A\) and the open and closed proper subscheme \(X_s^0\) of its closed fibre. We obtain an open and closed subscheme \(W_A \subset X_A\) which is proper over \(A\) and whose closed fibre is \(X_s^0\).

By the construction of the henselization in Algebra, Lemma 04GN, and the limit results in Limits, Lemmas 01ZM, 0EUU, 01ZP, and 081F, there is an affine étale neighbourhood \((S', s') \to (S, s)\) and an open and closed subscheme \[W' \subset X_{S'}\] which is proper over \(S'\) and whose fibre at \(s'\) is \(X_s^0\). After shrinking \(S'\) about \(s'\), we may assume that the base change \(e'\) of the section factors through \(W'\).

Let \(U\) be the image of \(S' \to S\). After replacing \(S\) by \(U\), the morphism \(S' \to S\) is an fpqc covering. In every fibre the connected component containing \(e'\) is contained in the open and closed subscheme \(W'\). Hence \((X_{S'})^0\) is the connected-component locus of the proper \(S'\)-scheme \(W'\) along \(e'\). By Lemma more-morphisms-lemma-connected-along-section-proper, this locus is closed and, with its reduced induced structure, proper over \(S'\).

Formation of the connected-component locus commutes with base change by Lemma 055M. Since \(X_{S'} \to X\) is a quasi-compact surjective flat morphism, Morphisms, Lemma 02JY shows that \(X^0\) is closed in \(X\). Let \(Z \subset X\) be the reduced induced closed subscheme. The base change \(Z_{S'}\) has underlying set \((X_{S'})^0\) and is a closed subscheme of the proper scheme \(W'\). Hence \(Z_{S'}\) is proper over \(S'\). Properness is fpqc local on the base by Descent, Lemma 02L1, so \(Z\) is proper over \(S\).

Lemma

Let \(S\) be a locally Noetherian scheme all of whose residue fields have characteristic zero. Let \(G\) be a commutative group scheme locally of finite type over \(S\). Then \[G^\tau = G^\sigma \quad\text{is open in }G, \qquad G^0 = G^\rho.\] If \(G \to S\) is separated and every \(G_s^0\) is proper over \(\kappa(s)\), then \(G^0\) is closed in \(G\) and its reduced induced closed subscheme is proper over \(S\). In this case \(G^\tau\) is open and closed in \(G\).

Assume that \([n] : G\to G\) is universally open for every \(n>1\). Then \(G^0\) is open in \(G\). Assume, in addition, that for every algebraically closed field \(k\) over a point of \(S\), the identity component of \(G_k\) contains no subgroup scheme isomorphic to \(\mathbf G_{a,k}\). Then \(G^\tau\to S\) is universally open. If moreover \(S\) is reduced, then \(G^\tau\to S\) is smooth.

Proof

The characteristic exponent is \(p=1\). Thus the definitions in Lemma more-morphisms-lemma-order-in-component-group-constructible give \(G^\sigma=G^\tau\) and \(G^\rho=G^0\). The openness of \(G^\tau\) follows from Lemma more-morphisms-lemma-torsion-component-locus-open.

Assume the additional hypotheses. For every \(s \in S\), Lemma more-morphisms-lemma-proper-connected-component-neighbourhood gives an open neighbourhood \(U\) of \(s\) such that \(G_U^0\) is closed and its reduced induced structure is proper over \(U\). Thus \(G^0\) is closed and its reduced induced structure is proper over \(S\), as both assertions are local on the base. Finally, Lemma more-morphisms-lemma-order-in-component-group-constructible shows that \(G^\tau\) is closed when \(G^0\) is closed.

Suppose every power map in the statement is universally open. Since \(p=1\), Lemma more-morphisms-lemma-primary-component-locus-open shows that \(G^\rho\) is open. As \(G^\rho=G^0\), this proves the first additional assertion.

Under the hypothesis excluding additive subgroups, Lemma more-morphisms-corollary-prime-to-characteristic-component-openness shows that \(G\to S\) is universally open along \(G^\sigma=G^\tau\). The latter is an open subscheme of \(G\), so its structure morphism to \(S\) is universally open. Finally, every fibre of \(G\to S\) is smooth by Groupoids, Lemma 047N. If \(S\) is reduced, apply Lemma more-morphisms-corollary-universally-open-smooth-fibre-flat to \(G^\tau\to S\), with the whole fibre as the indicated open subset, at every point of \(S\). This proves that \(G^\tau\to S\) is smooth.

Generic flatness stratification

We can use generic flatness to construct a stratification of the base such that a given module becomes flat over the strata.

Lemma

Let \(f : X \to S\) be a morphism of finite presentation between quasi-compact and quasi-separated schemes. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation. Then there exists a \(t \geq 0\) and closed subschemes \[S \supset S_0 \supset S_1 \supset \ldots \supset S_t = \emptyset\] such that \(S_i \to S\) is defined by a finite type ideal sheaf, \(S_0 \subset S\) is a thickening, and \(\mathcal{F}\) pulled back to \(X \times_S (S_i \setminus S_{i + 1})\) is flat over \(S_i \setminus S_{i + 1}\).

Proof

We can find a cartesian diagram \[\xymatrix{ X \ar[d] \ar[r] & X_0 \ar[d] \\ S \ar[r] & S_0 }\] and a finitely presented \(\mathcal{O}_{X_0}\)-module \(\mathcal{F}_0\) which pulls back to \(\mathcal{F}\) such that \(X_0\) and \(S_0\) are of finite type over \(\mathbf{Z}\). See Limits, Proposition 01ZA and Lemmas 01ZM and 01ZR. Thus we may assume \(X\) and \(S\) are of finite type over \(\mathbf{Z}\) and \(\mathcal{F}\) is a coherent \(\mathcal{O}_X\)-module.

Assume \(X\) and \(S\) are of finite type over \(\mathbf{Z}\) and \(\mathcal{F}\) is a coherent \(\mathcal{O}_X\)-module. In this case every quasi-coherent ideal is of finite type, hence we do not have to check the condition that \(S_i\) is cut out by a finite type ideal. Set \(S_0 = S_{red}\) equal to the reduction of \(S\). By generic flatness as stated in Morphisms, Proposition 052B there is a dense open \(U_0 \subset S_0\) such that \(\mathcal{F}\) pulled back to \(X \times_S U_0\) is flat over \(U_0\). Let \(S_1 \subset S_0\) be the reduced closed subscheme whose underlying closed subset is \(S \setminus U_0\). We continue in this way, provided \(S_1 \not = \emptyset\), to find \(S_0 \supset S_1 \supset \ldots\). Because \(S\) is Noetherian any descending chain of closed subsets stabilizes hence we see that \(S_t = \emptyset\) for some \(t \geq 0\).

Lemma

Let \(f : X \to S\) be a morphism of finite presentation between quasi-compact and quasi-separated schemes. Then there exists a \(t \geq 0\) and closed subschemes \[S \supset S_0 \supset S_1 \supset \ldots \supset S_t = \emptyset\] such that \(S_i \to S\) is defined by a finite type ideal sheaf, \(S_0 \subset S\) is a thickening, and \(X \times_S (S_i \setminus S_{i + 1})\) is flat over \(S_i \setminus S_{i + 1}\).

Proof

Apply Lemma 0ASY with \(\mathcal{F} = \mathcal{O}_X\).

Lemma

Let \(f : X \to S\) be a surjective morphism of finite presentation. Let \(M\) be an object of \(D_\QCoh(\mathcal{O}_S)\) such that \(H^i(M) = 0\) for \(i > 0\). The following are equivalent

\(M\) is isomorphic to a flat \(\mathcal{O}_S\)-module placed in degree \(0\),

\(Lf^*M\) is isomorphic to a flat \(\mathcal{O}_X\)-module placed in degree \(0\).

Proof

We omit the proof that (1) implies (2). Assume (2). To prove (1) is a local question, hence we may assume that \(S\) is affine. Choose a stratification \(S \supset S_0 \supset S_1 \supset \ldots \supset S_t = \emptyset\) as in Lemma 0H3Z. Set \(T_i = S_i \setminus S_{i - 1}\) and \(Y_i = X \times_S T_i\). Denote \(g_i : T_i \to S\) and \(h_i : Y_i \to X\) the inclusion morphisms. Since \(f\) is surjective, the morphisms \(f_i : X_i \to T_i\) are surjective and flat. Since \(Lh_i^* \circ Lf^* = Lf_i^* \circ Lg_i^*\) we conclude from (2) that \(Lf_i^* Lg_i^*M\) is isomorphic to a flat \(\mathcal{O}_{Y_i}\)-module placed in degree \(0\). Since \(f_i\) is flat and surjective, we see that \(Lg_i^*M\) is isomorphic to a flat \(\mathcal{O}_{T_i}\)-module placed in degree \(0\).

We will prove this implies the result by induction on \(t\). We will use the following notation: \(S = \Spec(A)\) and \(M\) is the object of \(D_\QCoh(\mathcal{O}_S)\) corresponding to \(N\) in \(D(A)\), see Derived Categories of Schemes, Lemma 06Z0.

Base case: \(t = 1\). Write \(S_0 = \Spec(A/I)\) where \(I\) is a finite generated ideal. Then \(I\) is nilpotent as \(S_0 = S\) set theoretically. The assumption is that \(N \otimes_A^\mathbf{L} A/I\) has tor amplitude in \([0, 0]\). By More on Algebra, Lemma 0H75 the same is true for \(N\).

Induction step. Assume \(t > 1\). Write \(S_{t - 1} = \Spec(A/I)\) where \(I = (f_1, \ldots, f_r)\) is a finitely generated ideal. We will argue by induction on \(r\). Observe that \(N_{f_r} = N \otimes_A^\mathbf{L} A_{f_r}\) in \(D(A_{f_r})\) has tor amplitude in \([0, 0]\) by induction hypothesis (because the stratification on the principal open \(D(f_r)\) has length at most \(t - 1\)). On the other hand, consider \(N' = N \otimes_A^\mathbf{L} A/f_rA\) in \(D(A/f_rA)\). Setting \(S' = \Spec(A/f_rA)\) we obtain a stratification \[S' \supset S' \cap S_0 \supset S' \cap S_1 \supset \ldots \supset S' \cap S_{t - 1} \supset S' \cap S_t = \emptyset\] where \(S' \cap S_{t - 1}\) is cut out by \(r - 1\) equations in \(S'\). Similarly to before, the derived pullbacks of \(M\) to the parts \(S' \cap S_i \setminus S' \cap S_{i + 1}\) are of tor amplitude in \([0, 0]\). Whence by induction on \(r\) we conclude that \(N'\) has tor amplitude in \([0, 0]\). Then we finally conclude that \(N\) has tor amplitude in \([0, 0]\) by More on Algebra, Lemma 0H85.

Lemma

Let \(R\) be a Noetherian domain. Let \(R \to A \to B\) be finite type ring maps. Let \(M\) be a finite \(A\)-module and let \(N\) a finite \(B\)-module. Let \(M \to N\) be an \(A\)-linear map. There exists an nonzero \(f \in R\) such that the cokernel of \(M_f \to N_f\) is a flat \(R_f\)-module.

Proof

By replacing \(M\) by the image of \(M \to N\), we may assume \(M \subset N\). Choose a filtration \(0 = N_0 \subset N_1 \subset \ldots \subset N_t = N\) such that \(N_i/N_{i - 1} = B/\mathfrak q_i\) for some prime ideal \(\mathfrak q_i \subset B\), see Algebra, Lemma 00L0. Set \(M_i = M \cap N_i\). Then \(Q = N/M\) has a filtration by the submodules \(Q_i = N_i/M_i\). It suffices to prove \(Q_i/Q_{i - 1}\) becomes flat after localizing at a nonzero element of \(f\) (since extensions of flat modules are flat by Algebra, Lemma 00HM). Since \(Q_i/Q_{i - 1}\) is isomorphic to the cokernel of the map \(M_i/M_{i - 1} \to N_i/N_{i - 1}\), we reduce to the case discussed in the next paragraph.

Assume \(B\) is a domain and \(M \subset N = B\). After replacing \(A\) by the image of \(A\) in \(B\) we may assume \(A \subset B\). By generic flatness, we may assume \(A\) and \(B\) are flat over \(R\) (Algebra, Lemma 051R). It now suffices to show \(M \to B\) becomes \(R\)-universally injective after replacing \(R\) by a principal localization (Algebra, Lemma 058P). By generic freeness, we can find a nonzero \(g \in A\) such that \(B_g\) is a free \(A_g\)-module (Algebra, Lemma 051R). Thus we may choose a direct summand \(M' \subset B_g\) as an \(A_g\)-module, which is finite free as an \(A_g\)-module, and such that \(M \to B \to B_g\) factors through \(M'\). Clearly, it suffices to show that \(M \to M'\) becomes \(R\)-universally injective after replacing \(R\) by a principal localization.

Say \(M' = A_g^{\oplus n}\). Since \(M \subset M'\) is a finite \(A\)-module, we see that \(M\) is contained in \((1/g^m)A^{\oplus n}\) for some \(m \geq 0\). After changing our basis for \(M'\) we may assume \(M \subset A^{\oplus n}\). Then it suffices to show that \(A^{\oplus n}/M\) and \(A_g/A\) become \(R\)-flat after replacing \(R\) by a principal localization. Namely, then \(M \to A^{\oplus n}\) and \(A^{\oplus n} \to A_g^{\oplus n}\) are universally injective by Algebra, Lemma 00HL and consequently so is the composition \(M \to M' = A_g^{\oplus n}\).

By generic flatness (see reference above), we may assume the module \(A^{\oplus n}/M\) is \(R\)-flat. For the quotient \(A_g/A\) we use the fact that \[A_g/A = \colim (1/g^m)A/A \cong \colim A/g^mA\] and the module \(A/g^mA\) has a filtration of length \(m\) whose successive quotients are isomorphic to \(A/gA\). Again by generic flatness we may assume \(A/gA\) is \(R\)-flat and hence each \(A/g^mA\) is \(R\)-flat, and hence so is \(A_g/A\).

Let \(f : X \to Y\) be a morphism of schemes over a base scheme \(S\). Let \(Z \subset Y\) be the scheme theoretic image of \(f\), see Morphisms, Section 01R5. Let \(g : S' \to S\) be a morphism of schemes and let \(f' : X \times_S S' \to Y \times_S S'\) be the base change of \(f\) by \(g\). It is not always true that \(Z \times_S S' \subset Y \times_S S'\) is the scheme theoretic image of \(f'\). Let us say that formation of the scheme theoretic image of \(f/S\) commutes with arbitrary base change if for every \(g\) as above the scheme theoretic image of \(f'\) is equal to \(Z \times_S S'\).

Lemma

Let \(S\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\) with both \(X\) and \(Y\) of finite presentation over \(S\). Then there exists a \(t \geq 0\) and closed subschemes \[S \supset S_0 \supset S_1 \supset \ldots \supset S_t = \emptyset\] with the following properties:

\(S_i \to S\) is defined by a finite type ideal sheaf,

\(S_0 \subset S\) is a thickening, and

with \(T_i = S_i \setminus S_{i + 1}\) and \(f_i\) the base change of \(f\) to \(T_i\) we have: formation of the scheme theoretic image of \(f_i/T_i\) commutes with arbitrary base change (see discussion above the lemma).

Proof

We can find a commutative diagram \[\xymatrix{ X \ar[d] \ar[r] & Y \ar[d] \ar[r] & S \ar[d] \\ U \ar[r] & V \ar[r] & W }\] with cartesian squares such that \(U\), \(V\), \(W\) are of finite type over \(\mathbf{Z}\). Namely, first write \(S\) as a cofiltered limit of finite type schemes over \(\mathbf{Z}\) with affine transition morphisms using Limits, Proposition 01ZA and then descend the morphism \(X \to Y\) using Limits, Lemma 01ZM. This reduces us to the case discussed in the next paragraph.

Assume \(S\) is Noetherian. In this case every quasi-coherent ideal is of finite type, hence we do not have to check the condition that \(S_i\) is cut out by a finite type ideal. Set \(S_0 = S_{red}\) equal to the reduction of \(S\). Let \(\eta \in S_0\) be a generic point of an irreducible component of \(S_0\). By Noetherian induction on the underlying topological space of \(S_0\), we may assume the result holds for any closed subscheme of \(S_0\) not containing \(\eta\). Thus it suffices to show that there exists an open neighbourhood \(U_0 \subset S_0\) such that the base change \(f_0\) of \(f\) to \(U_0\) has property (3).

Let \(R\) be a Noetherian domain. Let \(f : X \to Y\) be a morphism of finite type schemes over \(R\). By the discussion in the previous paragraph it suffices to show that after replacing \(R\) by \(R_g\) for some \(g \in R\) nonzero and \(X\), \(Y\) by their base changes to \(R_g\), formation of the scheme theoretic image of \(f/R\) commutes with arbitrary base change.

Let \(Y = V_1 \cup \ldots V_n\) be an affine open covering. Let \(U_i = f^{-1}(V_i)\). If the statement is true for each of the morphisms \(U_i \to V_i\) over \(R\), then it holds for \(f\). Namely, the scheme theoretic image of \(U_i \to V_i\) is the intersection of \(V_i\) with the scheme theoretic image of \(f : X \to Y\) by Morphisms, Lemma 01R8. Thus we may assume \(Y\) is affine.

Let \(X = U_1 \cup \ldots U_n\) be an affine open covering. Then the scheme theoretic image of \(X \to Y\) is the same as the scheme theoretic imge of \(\coprod U_i \to Y\). Thus we may assume \(X\) is affine.

Say \(X = \Spec(A)\) and \(Y = \Spec(B)\) and \(f\) corresponds to the \(R\)-algebra map \(\varphi : A \to B\). Then the scheme theoretic image of \(f\) is \(\Spec(A/\Ker(\varphi))\) and similarly after base change (by an affine morphism, but it is enough to check for those). Thus formation of the scheme theoretic image commutes with base change if \(\Ker(\varphi \otimes_R R') = \Ker(\varphi) \otimes_R R'\) for all ring maps \(R \to R'\).

After replacing \(R\), \(A\), \(B\) by \(R_g\), \(A_g\), \(B_g\) for a suitable nonzero \(g\) in \(R\), we may assume \(A\) and \(B\) are flat over \(R\). By Lemma 0H40 we may also assume \(B/A\) is a flat \(R\)-module. Then \(0 \to \Ker(\varphi) \to A \to B \to B/A \to 0\) is an exact sequence of flat \(R\)-modules, which implies the desired base change statement.

Stratifying a morphism

Let \(f : X \to S\) be a finitely presented morphism of quasi-compact and quasi-separated schemes. In Section 0H3Y we have seen that we can stratify \(S\) such that \(X\) is flat over the strata. In this section look for stratifications of both \(S\) and \(X\) such that we obtain smooth strata; this won’t quite work and we’ll need a base change by finite locally free morphisms as well.

Lemma

Let \(f : X \to S\) be a morphism of schemes of finite presentation. Let \(\eta \in S\) be a generic point of an irreducible component of \(S\). Assume \(S\) is reduced. Then there exist

an open subscheme \(U \subset S\) containing \(\eta\),

a surjective, universally injective, finite locally free morphism \(V \to U\),

a \(t \geq 0\) and closed subschemes \[X \times_S V \supset Z_0 \supset Z_1 \supset \ldots \supset Z_t = \emptyset\] such that \(Z_i \to X \times_S V\) is defined by a finite type ideal sheaf, \(Z_0 \subset X \times_S V\) is a thickening, and such that the morphism \(Z_i \setminus Z_{i + 1} \to V\) is smooth.

Proof

It is clear that we may replace \(S\) by an open neighbourhood of \(\eta\) and \(X\) by the restriction to this open. Thus we may assume \(S = \Spec(A)\) where \(A\) is a reduced ring and \(\eta\) corresponds to a minimal prime ideal \(\mathfrak p\). Recall that the local ring \(\mathcal{O}_{S, \eta} = A_\mathfrak p\) is equal to \(\kappa(\mathfrak p)\) in this case, see Algebra, Lemma 00EU.

Apply Varieties, Lemma 0H3W to the scheme \(X_\eta\) over \(k = \kappa(\eta)\). Denote \(k'/k\) the purely inseparable field extension this produces. In the next paragraph we reduce to the case \(k' = k\). (This step corresponds to finding the morphism \(V \to U\) in the statement of the lemma; in particular we can take \(V = U\) if the characteristic of \(\kappa(\mathfrak p)\) is zero.)

If the characteristic of \(k = \kappa(\mathfrak p)\) is zero, then \(k' = k\). If the characteristic of \(k = \kappa(\mathfrak p)\) is \(p > 0\), then \(p\) maps to zero in \(A_\mathfrak p = \kappa(\mathfrak p)\). Hence after replacing \(A\) by a principal localization (i.e., shrinking \(S\)) we may assume \(p = 0\) in \(A\). If \(k' \not = k\), then there exists an \(\beta \in k'\), \(\beta \not \in k\) such that \(\beta^p \in k\). After replacing \(A\) by a principal localization we may assume there exists an \(a \in A\) such that \(\beta^p = a\). Set \(A' = A[x]/(x^p - a)\). Then \(S' = \Spec(A') \to \Spec(A) = S\) is finite locally free, surjective, and universally injective. Moreover, if \(\mathfrak p' \subset A'\) denotes the unique prime ideal lying over \(\mathfrak p\), then \(A'_{\mathfrak p'} = k(\beta)\) and \(k'/k(\beta)\) has smaller degree. Thus after replacing \(S\) by \(S'\) and \(\eta\) by the point \(\eta'\) corresponding to \(\mathfrak p'\) we see that the degree of \(k'\) over the residue field of \(\eta\) has decreased. Continuing like this, by induction we reduce to the case \(k' = \kappa(\mathfrak p) = \kappa(\eta)\).

Thus we may assume \(S\) is affine, reduced, and that we have a \(t \geq 0\) and closed subschemes \[X_\eta \supset Z_{\eta, 0} \supset Z_{\eta, 1} \supset \ldots \supset Z_{\eta, t} = \emptyset\] such that \(Z_{\eta, 0} = (X_\eta)_{red}\) and \(Z_{\eta, i} \setminus Z_{\eta, i + 1}\) is smooth over \(\eta\) for all \(i\). Recall that \(\kappa(\eta) = \kappa(\mathfrak p) = A_\mathfrak p\) is the filtered colimit of \(A_a\) for \(a \in A\), \(a \not \in \mathfrak p\). See Algebra, Lemma 00CR. Thus we can descend the diagram above to a corresponding diagram over \(\Spec(A_a)\) for some \(a \in A\), \(a \not \in \mathfrak p\). More precisely, after replacing \(S\) by \(\Spec(A_a)\) we may assume we have a \(t \geq 0\) and closed subschemes \[X \supset Z_0 \supset Z_1 \supset \ldots \supset Z_t = \emptyset\] such that \(Z_i \to X\) is a closed immersion of finite presentation, such that \(Z_0 \to X\) is a thickening, and such that \(Z_i \setminus Z_{i + 1}\) is smooth over \(S\). In other words, the lemma holds. More precisely, we first use Limits, Lemma 01ZM to obtain morphisms \[Z_t \to Z_{t - 1} \to \ldots \to Z_0 \to X\] over \(S\), each of finite presentation, and whose base change to \(\eta\) produces the inclusions between the given closed subschemes above. After shrinking \(S\) further we may assume each of the morphisms is a closed immersion, see Limits, Lemma 01ZP. After shrinking \(S\) we may assume \(Z_0 \to X\) is surjective and hence a thickening, see Limits, Lemma 07RR. After shrinking \(S\) once more we may assume \(Z_i \setminus Z_{i + 1} \to S\) is smooth, see Limits, Lemma 0C0C. This finishes the proof.

Lemma

Let \(f : X \to S\) be a morphism of finite presentation between quasi-compact and quasi-separated schemes. Then there exists a \(t \geq 0\) and closed subschemes \[S \supset S_0 \supset S_1 \supset \ldots \supset S_t = \emptyset\] such that

\(S_i \to S\) is defined by a finite type ideal sheaf,

\(S_0 \subset S\) is a thickening,

for each \(i\) there exists a surjective finite locally free morphism \(T_i \to S_i \setminus S_{i + 1}\),

for each \(i\) there exists a \(t_i \geq 0\) and closed subschemes \[X_i = X \times_S T_i \supset Z_{i, 0} \supset Z_{i, 1} \supset \ldots \supset Z_{i, t_i} = \emptyset\] such that \(Z_{i, j} \to X_i\) is defined by a finite type ideal sheaf, \(Z_{i, 0} \subset X_i\) is a thickening, and such that the morphism \(Z_{i, j} \setminus Z_{i, j + 1} \to T_i\) is smooth.

Proof

We can find a cartesian diagram \[\xymatrix{ X \ar[d] \ar[r] & X_0 \ar[d] \\ S \ar[r] & S_0 }\] such that \(X_0\) and \(S_0\) are of finite type over \(\mathbf{Z}\). See Limits, Proposition 01ZA and Lemma 01ZM. Thus we may assume \(X\) and \(S\) are of finite type over \(\mathbf{Z}\). Namely, a solution of the problem posed by the lemma for \(X_0 \to S_0\) will base change to a solution over \(S\); details omitted.

Assume \(X\) and \(S\) are of finite type over \(\mathbf{Z}\). In this case every quasi-coherent ideal is of finite type, hence we do not have to check the condition that \(S_i\) is cut out by a finite type ideal. Set \(S_0 = S_{red}\) equal to the reduction of \(S\). Let \(\eta \in S_0\) be a generic point of an irreducible component. By Lemma 0H43 we can find an open subscheme \(U \subset S_0\), a surjective, universally injective, finite locally free morphism \(V \to U\), a \(t_0 \geq 0\) and closed subschemes \[X \times_S V \supset Z_{0, 0} \supset Z_{0, 1} \supset \ldots \supset Z_{0, t_0} = \emptyset\] such that \(Z_{0, i} \to X \times_S V\) is defined by a finite type ideal sheaf, \(Z_{0, 0} \subset X \times_S V\) is a thickening, and such that the morphism \(Z_{0, i} \setminus Z_{0, i + 1} \to V\) is smooth. Then we let \(S_1 \subset S_0\) be the reduced induced subscheme structure on \(S_0 \setminus U\). By Noetherian induction on the underlying topological space of \(S\), we may assume that the lemma holds for \(X \times_S S_1 \to S_1\). This produces \(t \geq 1\) and \[S_1 = S_1 \supset S_2 \supset \ldots \supset S_t = \emptyset\] and \(t_i\) and \(Z_{i, j}\) as in the statement of the lemma. This proves the lemma.

Improving morphisms of relative dimension one

We can make any curve be smooth and projective after extending the ground field, compactifying, and normalizing. This also implies results about finite type morphisms whose generic fibres have dimension \(1\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\eta \in S\) be a generic point of an irreducible component of \(S\). Assume \(f\) is separated, of finite presentation, and \(\dim(X_\eta) \leq 1\). Then there exists a commutative diagram \[\xymatrix{ \overline{Y}_1 \amalg \ldots \amalg \overline{Y}_n \ar[rd] & Y_1 \amalg \ldots \amalg Y_n \ar[r]_-\nu \ar[d] \ar[l]^j & X_V \ar[r] \ar[d] & X_U \ar[r] \ar[d] & X \ar[d]^f \\ & T_1 \amalg \ldots \amalg T_n \ar[r] & V \ar[r] & U \ar[r] & S }\] of schemes with the following properties:

\(U \subset S\) is an open neighbourhood of \(\eta\),

\(V \to U\) is a finite, surjective, universally injective morphism,

\(X_U = U \times_S X\) and \(X_V = V \times_S X\) are the base changes,

\(\nu\) is finite, surjective, and there is an open \(W \subset X_V\) such that

\(W\) is dense in all fibres of \(X_V \to V\),

\(\nu^{-1}(W) \cap Y_i\) is dense in all fibres of \(Y_i \to T_i\), and

\(\nu^{-1}(W) \to W\) is a thickening,

\(j\) is an open immersion,

\(T_i \to V\) is finite étale,

\(Y_i \to T_i\) is surjective and smooth,

\(\overline{Y}_i \to T_i\) is smooth, proper, with geometrically connected fibres of dimension \(\leq 1\).

Proof

It is clear that we may replace \(S\) by an open neighbourhood of \(\eta\) and \(X\) by the restriction to this open. Moreover, we may replace \(S\) by its reduction and \(X\) by the base change to this reduction. Thus we may assume \(S = \Spec(A)\) where \(A\) is a reduced ring and \(\eta\) corresponds to a minimal prime ideal \(\mathfrak p\). Recall that the local ring \(\mathcal{O}_{S, \eta} = A_\mathfrak p\) is equal to \(\kappa(\mathfrak p)\) in this case, see Algebra, Lemma 00EU.

Apply Varieties, Lemma 0GK5 to the scheme \(X_\eta\) over \(k = \kappa(\eta)\). Denote \(k'/k\) the purely inseparable field extension this produces. In the next paragraph we reduce to the case \(k' = k\). (This step corresponds to finding the morphism \(V \to U\) in the statement of the lemma; in particular we can take \(V = U\) if the characteristic of \(\kappa(\mathfrak p)\) is zero.)

If the characteristic of \(k = \kappa(\mathfrak p)\) is zero, then \(k' = k\). If the characteristic of \(k = \kappa(\mathfrak p)\) is \(p > 0\), then \(p\) maps to zero in \(A_\mathfrak p = \kappa(\mathfrak p)\). Hence after replacing \(A\) by a principal localization (i.e., shrinking \(S\)) we may assume \(p = 0\) in \(A\). If \(k' \not = k\), then there exists an \(\beta \in k'\), \(\beta \not \in k\) such that \(\beta^p \in k\). After replacing \(A\) by a principal localization we may assume there exists an \(a \in A\) such that \(\beta^p = a\). Set \(A' = A[x]/(x^p - a)\). Then \(S' = \Spec(A') \to \Spec(A) = S\) is finite, surjective, and universally injective. Moreover, if \(\mathfrak p' \subset A'\) denotes the unique prime ideal lying over \(\mathfrak p\), then \(A'_{\mathfrak p'} = k(\beta)\) and \(k'/k(\beta)\) has smaller degree. Thus after replacing \(S\) by \(S'\) and \(\eta\) by the point \(\eta'\) corresponding to \(\mathfrak p'\) we see that the degree of \(k'\) over the residue field of \(\eta\) has decreased. Continuing like this, by induction we reduce to the case \(k' = \kappa(\mathfrak p) = \kappa(\eta)\).

Thus we may assume \(S\) is affine, reduced, and that we have a diagram \[\xymatrix{ \overline{Y}_{1, \eta} \amalg \ldots \amalg \overline{Y}_{n, \eta} \ar[rd] & Y_{1, \eta} \amalg \ldots \amalg Y_{n, \eta} \ar[r]_-\nu \ar[d] \ar[l]^j & X_\eta \ar[d] \\ & \Spec(k_1) \amalg \ldots \amalg \Spec(k_n) \ar[r] & \eta }\] of schemes with the following properties:

\(\nu\) is the normalization of \(X_\eta\),

\(j\) is an open immersion with dense image,

\(k_i/\kappa(\eta)\) is a finite separable extension for \(i = 1, \ldots, n\),

\(\overline{Y}_{i, \eta}\) is smooth, projective, and geometrically irreducible of dimension \(\leq 1\) over \(k_i\).

Recall that \(\kappa(\eta) = \kappa(\mathfrak p) = A_\mathfrak p\) is the filtered colimit of \(A_a\) for \(a \in A\), \(a \not \in \mathfrak p\). See Algebra, Lemma 00CR. Thus we can descend the diagram above to a corresponding diagram over \(\Spec(A_a)\) for some \(a \in A\), \(a \not \in \mathfrak p\). More precisely, after replacing \(S\) by \(\Spec(A_a)\) we may assume we have a commutative diagram \[\xymatrix{ \overline{Y}_1 \amalg \ldots \amalg \overline{Y}_n \ar[rd] & Y_1 \amalg \ldots \amalg Y_n \ar[r]_-\nu \ar[d] \ar[l]^j & X \ar[d] \\ & T_1 \amalg \ldots \amalg T_n \ar[r] & S }\] of schemes whose base change to \(\eta\) is the diagram above with the following properties

\(\nu\) is a finite, surjective morphism,

\(j\) is an open immersion,

\(T_i \to S\) is finite étale for \(i = 1, \ldots, n\),

\(Y_i \to T_i\) is smooth and surjective,

\(\overline{Y}_i \to T_i\) is smooth and proper and has geometrically connected fibres of dimension \(\leq 1\).

For this we first use Limits, Lemma 01ZM to obtain the diagram base changing to the previous diagram. Then we use Limits, Lemmas 07RP, 0C0C, 01ZO, 01Z6, 0EUU, 081F, and 07RR to obtain \(\nu\) finite, surjective, \(j\) open immersion, \(T_i \to S\) finite étale, \(Y_i \to T\) smooth, \(\overline{Y}_i \to T_i\) proper and smooth. Since \(Y_i\) cannot be empty, since smooth morphisms are open, and since \(T_i \to S\) is finite étale, after shrinking \(S\) we may assume \(Y_i \to T_i\) is surjective. Finally, the fibre of \(\overline{Y}_i \to T_i\) over the unique point \(\eta_i = \Spec(k_i)\) of \(T_i\) lying over \(\eta\) is geometrically connected. Hence by another shrinking we may assume the same thing is true for all fibres, see Lemma 0E0N.

It remains to prove the existence of an open \(W \subset X\) satisfying (a), (b), and (c). Since \(\nu_\eta : \coprod Y_{i, \eta} \to X_\eta\) is the normalization morphism, we know by Varieties, Lemma 0BXR there exists a dense open \(W_\eta \subset X_\eta\) such that \(\nu^{-1}(W_\eta) \to W_\eta\) is equal to the inclusion of the reduction of \(W_\eta\) into \(W_\eta\). Let \(W \subset X\) be a quasi-compact open whose fibre over \(\eta\) is the open \(W_\eta\) we just found. After replacing \(A = \Gamma(S, \mathcal{O}_S)\) by another localization we may assume \(\nu^{-1}(W) \to W\) is a closed immersion, see Limits, Lemma 01ZP. Since \(\nu\) is also surjective we conclude \(\nu^{-1}(W) \to W\) is a thickening. Set \(W_i = \nu^{-1}(W) \cap Y_i\). Shrinking \(S\) once more we can assume \(W_i \to T_i\) is surjective for all \(i\) (same argument as above). Then we find that \(W_i \subset Y_i\) is dense in all fibres of \(Y_i \to T_i\) as \(Y_i \to T_i\) has geometrically irreducible fibres. Since \(\nu\) is finite and surjective, it then follows that \(W = \nu(\nu^{-1}(W))\) is dense in all fibres of \(X \to S\) too.

Descending separated locally quasi-finite morphisms

In this section we show that “separated locally quasi-finite morphisms satisfy descent for fppf-coverings”. See Descent, Definition 02W2 for terminology. This is in the marvellous (for many reasons) paper by Raynaud and Gruson hidden in the proof of [GruRay, Lemma 5.7.1]. It can also be found in [Murre-representation], and [SGA3, Exposé X, Lemma 5.4] under the additional hypothesis that the morphism is locally of finite presentation. Here is the formal statement.

Lemma

Let \(S\) be a scheme. Let \(\{X_i \to S\}_{i\in I}\) be an fppf covering, see Topologies, Definition 021M. Let \((V_i/X_i, \varphi_{ij})\) be a descent datum relative to \(\{X_i \to S\}\). If each morphism \(V_i \to X_i\) is separated and locally quasi-finite, then the descent datum is effective.

Proof

Being separated and being locally quasi-finite are properties of morphisms of schemes which are preserved under any base change, see Schemes, Lemma 01KU and Morphisms, Lemma 01TM. Hence Descent, Lemma 02W3 applies and it suffices to prove the statement of the lemma in case the fppf-covering is given by a single \(\{X \to S\}\) flat surjective morphism of finite presentation of affines. Say \(X = \Spec(A)\) and \(S = \Spec(R)\) so that \(R \to A\) is a faithfully flat ring map. Let \((V, \varphi)\) be a descent datum relative to \(X\) over \(S\) and assume that \(\pi : V \to X\) is separated and locally quasi-finite.

Let \(W^1 \subset V\) be any affine open. Consider \(W = \text{pr}_1(\varphi(W^1 \times_S X)) \subset V\). Here is a picture \[\xymatrix{ W^1 \times_S X \ar[rrrrr] \ar[ddd] \ar[rd] & & & & & \varphi(W^1 \times_S X) \ar[ddd] \ar[ld] \\ & V \times_S X \ar[rrr]^\varphi \ar[rd] \ar[dd] & & & X \times_S V \ar[ld] \ar[dd] & \\ & & X \times_S X \ar[r]^1 \ar[d]_{\text{pr}_0} & X \times_S X \ar[d]^{\text{pr}_1} & & \\ W^1 \ar[r] & V \ar[r] & X & X & V \ar[l] & W \ar[l] }\] Ok, and now since \(X \to S\) is flat and of finite presentation it is universally open (Morphisms, Lemma 01UA). Hence we conclude that \(W\) is open. Moreover, it is also clearly the case that \(W\) is quasi-compact, and \(W^1 \subset W\). Moreover, we note that \(\varphi(W \times_S X) = X \times_S W\) by the cocycle condition for \(\varphi\). Hence we obtain a new descent datum \((W, \varphi')\) by restricting \(\varphi\) to \(W \times_S X\). Note that the morphism \(W \to X\) is quasi-compact, separated and locally quasi-finite. This implies that it is separated and quasi-finite by definition. Hence it is quasi-affine by Lemma 02LR. Thus by Descent, Lemma 0247 we see that the descent datum \((W, \varphi')\) is effective.

In other words, we find that there exists an open covering \(V = \bigcup W_i\) by quasi-compact opens \(W_i\) which are stable for the descent morphism \(\varphi\). Moreover, for each such quasi-compact open \(W \subset V\) the corresponding descent data \((W, \varphi')\) is effective. This means the original descent datum is effective by glueing the schemes obtained from descending the opens \(W_i\), see Descent, Lemma 0AP4.

Relative finite presentation

Let \(R \to A\) be a finite type ring map. Let \(M\) be an \(A\)-module. In More on Algebra, Section 0659 we defined what it means for \(M\) to be finitely presented relative to \(R\). We also proved this notion has good localization properties and glues. Hence we can define the corresponding global notion as follows.

Definition

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. We say \(\mathcal{F}\) is finitely presented relative to \(S\) or of finite presentation relative to \(S\) if there exists an affine open covering \(S = \bigcup V_i\) and for every \(i\) an affine open covering \(f^{-1}(V_i) = \bigcup_j U_{ij}\) such that \(\mathcal{F}(U_{ij})\) is a \(\mathcal{O}_X(U_{ij})\)-module of finite presentation relative to \(\mathcal{O}_S(V_i)\).

Note that this implies that \(\mathcal{F}\) is a finite type \(\mathcal{O}_X\)-module. If \(X \to S\) is just locally of finite type, then \(\mathcal{F}\) may be of finite presentation relative to \(S\), without \(X \to S\) being locally of finite presentation. We will see that \(X \to S\) is locally of finite presentation if and only if \(\mathcal{O}_X\) is of finite presentation relative to \(S\).

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The following are equivalent

\(\mathcal{F}\) is of finite presentation relative to \(S\),

for every affine opens \(U \subset X\), \(V \subset S\) with \(f(U) \subset V\) the \(\mathcal{O}_X(U)\)-module \(\mathcal{F}(U)\) is finitely presented relative to \(\mathcal{O}_S(V)\).

Moreover, if this is true, then for every open subschemes \(U \subset X\) and \(V \subset S\) with \(f(U) \subset V\) the restriction \(\mathcal{F}|_U\) is of finite presentation relative to \(V\).

Proof

The final statement is clear from the equivalence of (1) and (2). It is also clear that (2) implies (1). Assume (1) holds. Let \(S = \bigcup V_i\) and \(f^{-1}(V_i) = \bigcup U_{ij}\) be affine open coverings as in Definition 05H1. Let \(U \subset X\) and \(V \subset S\) be as in (2). By More on Algebra, Lemma 065D it suffices to find a standard open covering \(U = \bigcup U_k\) of \(U\) such that \(\mathcal{F}(U_k)\) is finitely presented relative to \(\mathcal{O}_S(V)\). In other words, for every \(u \in U\) it suffices to find a standard affine open \(u \in U' \subset U\) such that \(\mathcal{F}(U')\) is finitely presented relative to \(\mathcal{O}_S(V)\). Pick \(i\) such that \(f(u) \in V_i\) and then pick \(j\) such that \(u \in U_{ij}\). By Schemes, Lemma 01IW we can find \(v \in V' \subset V \cap V_i\) which is standard affine open in \(V'\) and \(V_i\). Then \(f^{-1}V' \cap U\), resp. \(f^{-1}V' \cap U_{ij}\) are standard affine opens of \(U\), resp. \(U_{ij}\). Applying the lemma again we can find \(u \in U' \subset f^{-1}V' \cap U \cap U_{ij}\) which is standard affine open in both \(f^{-1}V' \cap U\) and \(f^{-1}V' \cap U_{ij}\). Thus \(U'\) is also a standard affine open of \(U\) and \(U_{ij}\). By More on Algebra, Lemma 065A the assumption that \(\mathcal{F}(U_{ij})\) is finitely presented relative to \(\mathcal{O}_S(V_i)\) implies that \(\mathcal{F}(U')\) is finitely presented relative to \(\mathcal{O}_S(V_i)\). Since \(\mathcal{O}_X(U') = \mathcal{O}_X(U') \otimes_{\mathcal{O}_S(V_i)} \mathcal{O}_S(V')\) we see from More on Algebra, Lemma 065B that \(\mathcal{F}(U')\) is finitely presented relative to \(\mathcal{O}_S(V')\). Applying More on Algebra, Lemma 065A again we conclude that \(\mathcal{F}(U')\) is finitely presented relative to \(\mathcal{O}_S(V)\). This finishes the proof.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module.

If \(f\) is locally of finite presentation, then \(\mathcal{F}\) is of finite presentation relative to \(S\) if and only if \(\mathcal{F}\) is of finite presentation.

The morphism \(f\) is locally of finite presentation if and only if \(\mathcal{O}_X\) is of finite presentation relative to \(S\).

Proof

Follows immediately from the definitions, see discussion following More on Algebra, Definition 05GZ.

Lemma

Let \(\pi : X \to Y\) be a finite morphism of schemes locally of finite type over a base scheme \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then \(\mathcal{F}\) is of finite presentation relative to \(S\) if and only if \(\pi_*\mathcal{F}\) is of finite presentation relative to \(S\).

Proof

Translation of the result of More on Algebra, Lemma 05H0 into the language of schemes.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(S' \to S\) be a morphism of schemes, set \(X' = X \times_S S'\) and denote \(\mathcal{F}'\) the pullback of \(\mathcal{F}\) to \(X'\). If \(\mathcal{F}\) is of finite presentation relative to \(S\), then \(\mathcal{F}'\) is of finite presentation relative to \(S'\).

Proof

Translation of the result of More on Algebra, Lemma 065B into the language of schemes.

Lemma

Let \(X \to Y \to S\) be morphisms of schemes which are locally of finite type. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module. If \(f : X \to Y\) is locally of finite presentation and \(\mathcal{G}\) of finite presentation relative to \(S\), then \(f^*\mathcal{G}\) is of finite presentation relative to \(S\).

Proof

Translation of the result of More on Algebra, Lemma 0670 into the language of schemes.

Lemma

Let \(X \to Y \to S\) be morphisms of schemes which are locally of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. If \(Y \to S\) is locally of finite presentation and \(\mathcal{F}\) is of finite presentation relative to \(Y\), then \(\mathcal{F}\) is of finite presentation relative to \(S\).

Proof

Translation of the result of More on Algebra, Lemma 065C into the language of schemes.

Lemma

Let \(X \to S\) be a morphism of schemes which is locally of finite type. Let \(0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0\) be a short exact sequence of quasi-coherent \(\mathcal{O}_X\)-modules.

If \(\mathcal{F}', \mathcal{F}''\) are finitely presented relative to \(S\), then so is \(\mathcal{F}\).

If \(\mathcal{F}'\) is a finite type \(\mathcal{O}_X\)-module and \(\mathcal{F}\) is finitely presented relative to \(S\), then \(\mathcal{F}''\) is finitely presented relative to \(S\).

Proof

Translation of the result of More on Algebra, Lemma 0671 into the language of schemes.

Lemma

Let \(X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}, \mathcal{F}'\) be quasi-coherent \(\mathcal{O}_X\)-modules. If \(\mathcal{F} \oplus \mathcal{F}'\) is finitely presented relative to \(S\), then so are \(\mathcal{F}\) and \(\mathcal{F}'\).

Proof

Translation of the result of More on Algebra, Lemma 0672 into the language of schemes.

Relative pseudo-coherence

This section is the analogue of More on Algebra, Section 065E for schemes. We strongly urge the reader to take a look at that section first. Although we have developed the material in this section and the material on pseudo-coherent complexes in Cohomology, Sections 08C3, 08CA, 08CF, and 08CL for arbitrary complexes of \(\mathcal{O}_X\)-modules, if \(X\) is a scheme then working exclusively with objects in \(D_\QCoh(\mathcal{O}_X)\) greatly simplifies many of the lemmmas and arguments, often reducing the problem at hand immediately to the algebraic counterpart. Moreover, one of the first thing we do is to show that being relatively pseudo-coherent implies the cohomology sheaves are quasi-coherent, see Lemma 0CSU. Hence, on a first reading we suggest the reader work exclusively with objects in \(D_\QCoh(\mathcal{O}_X)\).

Lemma

Let \(X \to S\) be a finite type morphism of affine schemes. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Let \(m \in \mathbf{Z}\). The following are equivalent

for some closed immersion \(i : X \to \mathbf{A}^n_S\) the object \(Ri_*E\) of \(D(\mathcal{O}_{\mathbf{A}^n_S})\) is \(m\)-pseudo-coherent, and

for all closed immersions \(i : X \to \mathbf{A}^n_S\) the object \(Ri_*E\) of \(D(\mathcal{O}_{\mathbf{A}^n_S})\) is \(m\)-pseudo-coherent.

Proof

Say \(S = \Spec(R)\) and \(X = \Spec(A)\). Let \(i\) correspond to the surjection \(\alpha : R[x_1, \ldots, x_n] \to A\) and let \(X \to \mathbf{A}^m_S\) correspond to \(\beta : R[y_1, \ldots, y_m] \to A\). Choose \(f_j \in R[x_1, \ldots, x_n]\) with \(\alpha(f_j) = \beta(y_j)\) and \(g_i \in R[y_1, \ldots, y_m]\) with \(\beta(g_i) = \alpha(x_i)\). Then we get a commutative diagram \[\xymatrix{ R[x_1, \ldots, x_n, y_1, \ldots, y_m] \ar[d]^{x_i \mapsto g_i} \ar[rr]_-{y_j \mapsto f_j} & & R[x_1, \ldots, x_n] \ar[d] \\ R[y_1, \ldots, y_m] \ar[rr] & & A }\] corresponding to the commutative diagram of closed immersions \[\xymatrix{ \mathbf{A}^{n + m}_S & \mathbf{A}^n_S \ar[l] \\ \mathbf{A}^m_S \ar[u] & X \ar[u] \ar[l] }\] Thus it suffices to show that under a closed immersion \[f : \mathbf{A}^m_S \to \mathbf{A}^{n + m}_S\] an object \(E\) of \(D(\mathcal{O}_{\mathbf{A}^m_S})\) is \(m\)-pseudo-coherent if and only if \(Rf_*E\) is \(m\)-pseudo-coherent. This follows from Derived Categories of Schemes, Lemma 09VA and the fact that \(f_*\mathcal{O}_{\mathbf{A}^m_S}\) is a pseudo-coherent \(\mathcal{O}_{\mathbf{A}^{n + m}_S}\)-module. The pseudo-coherence of \(f_*\mathcal{O}_{\mathbf{A}^m_S}\) is straightforward to prove directly, but it also follows from Derived Categories of Schemes, Lemma 08E7 and More on Algebra, Lemma 065H.

Recall that if \(f : X \to S\) is a morphism of scheme which is locally of finite type, then for every pair of affine opens \(U \subset X\) and \(V \subset S\) such that \(f(U) \subset V\), the ring map \(\mathcal{O}_S(V) \to \mathcal{O}_X(U)\) is of finite type (Morphisms, Lemma 01T2). Hence there always exist closed immersions \(U \to \mathbf{A}^n_V\) and the following definition makes sense.

Definition

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. Fix \(m \in \mathbf{Z}\).

We say \(E\) is \(m\)-pseudo-coherent relative to \(S\) if there exists an affine open covering \(S = \bigcup V_i\) and for each \(i\) an affine open covering \(f^{-1}(V_i) = \bigcup U_{ij}\) such that the equivalent conditions of Lemma 09VC are satisfied for each of the pairs \((U_{ij} \to V_i, E|_{U_{ij}})\).

We say \(E\) is pseudo-coherent relative to \(S\) if \(E\) is \(m\)-pseudo-coherent relative to \(S\) for all \(m \in \mathbf{Z}\).

We say \(\mathcal{F}\) is \(m\)-pseudo-coherent relative to \(S\) if \(\mathcal{F}\) viewed as an object of \(D(\mathcal{O}_X)\) is \(m\)-pseudo-coherent relative to \(S\).

We say \(\mathcal{F}\) is pseudo-coherent relative to \(S\) if \(\mathcal{F}\) viewed as an object of \(D(\mathcal{O}_X)\) is pseudo-coherent relative to \(S\).

If \(X\) is quasi-compact and \(E\) is \(m\)-pseudo-coherent relative to \(S\) for some \(m\), then \(E\) is bounded above. If \(E\) is pseudo-coherent relative to \(S\), then \(E\) has quasi-coherent cohomology sheaves.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. If \(E\) in \(D(\mathcal{O}_X)\) is \(m\)-pseudo-coherent relative to \(S\), then \(H^i(E)\) is a quasi-coherent \(\mathcal{O}_X\)-module for \(i > m\). If \(E\) is pseudo-coherent relative to \(S\), then \(E\) is an object of \(D_\QCoh(\mathcal{O}_X)\).

Proof

Choose an affine open covering \(S = \bigcup V_i\) and for each \(i\) an affine open covering \(f^{-1}(V_i) = \bigcup U_{ij}\) such that the equivalent conditions of Lemma 09VC are satisfied for each of the pairs \((U_{ij} \to V_i, E|_{U_{ij}})\). Since being quasi-coherent is local on \(X\), we may assume that there exists an closed immersion \(i : X \to \mathbf{A}^n_S\) such that \(Ri_*E\) is \(m\)-pseudo-coherent on \(\mathbf{A}^n_S\). By Derived Categories of Schemes, Lemma 08E5 this means that \(H^q(Ri_*E)\) is quasi-coherent for \(q > m\). Since \(i_*\) is an exact functor, we have \(i_*H^q(E) = H^q(Ri_*E)\) is quasi-coherent on \(\mathbf{A}^n_S\). By Morphisms, Lemma 01QY this implies that \(H^q(E)\) is quasi-coherent as desired (strictly speaking it implies there exists some quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) such that \(i_*\mathcal{F} = i_*H^q(E)\) and then Modules, Lemma 08KS tells us that \(\mathcal{F} \cong H^q(E)\) hence the result).

Next, we prove the condition of relative pseudo-coherence localizes well.

Lemma

Let \(S\) be an affine scheme. Let \(V \subset S\) be a standard open. Let \(X \to V\) be a finite type morphism of affine schemes. Let \(U \subset X\) be an affine open. Let \(E\) be an object of \(D(\mathcal{O}_X)\). If the equivalent conditions of Lemma 09VC are satisfied for the pair \((X \to V, E)\), then the equivalent conditions of Lemma 09VC are satisfied for the pair \((U \to S, E|_U)\).

Proof

Write \(S = \Spec(R)\), \(V = D(f)\), \(X = \Spec(A)\), and \(U = D(g)\). Assume the equivalent conditions of Lemma 09VC are satisfied for the pair \((X \to V, E)\).

Choose \(R_f[x_1, \ldots, x_n] \to A\) surjective. Write \(R_f = R[x_0]/(fx_0 - 1)\). Then \(R[x_0, x_1, \ldots, x_n] \to A\) is surjective, and \(R_f[x_1, \ldots, x_n]\) is pseudo-coherent as an \(R[x_0, \ldots, x_n]\)-module. Thus we have \[X \to \mathbf{A}^n_V \to \mathbf{A}^{n + 1}_S\] and we can apply Derived Categories of Schemes, Lemma 09VA to conclude that the pushforward \(E'\) of \(E\) to \(\mathbf{A}^{n + 1}_S\) is \(m\)-pseudo-coherent.

Choose an element \(g' \in R[x_0, x_1, \ldots, x_n]\) which maps to \(g \in A\). Consider the surjection \(R[x_0, \ldots, x_{n + 1}] \to R[x_0, \ldots, x_n, 1/g']\). We obtain \[\xymatrix{ X \ar[d] & U \ar[d] \ar[l] \ar[dr] \\ \mathbf{A}^{n + 1}_S & D(g')\ar[l] \ar[r] & \mathbf{A}^{n + 2}_S }\] where the lower left arrow is an open immersion and the lower right arrow is a closed immersion. We conclude as before that the pushforward of \(E'|_{D(g')}\) to \(\mathbf{A}^{n + 2}_S\) is \(m\)-pseudo-coherent. Since this is also the pushforward of \(E|_U\) to \(\mathbf{A}^{n + 2}_S\) we conclude the lemma is true.

Lemma

Let \(X \to S\) be a finite type morphism of affine schemes. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Let \(m \in \mathbf{Z}\). Let \(X = \bigcup U_i\) be a standard affine open covering. The following are equivalent

the equivalent conditions of Lemma 09VC hold for the pairs \((U_i \to S, E|_{U_i})\),

the equivalent conditions of Lemma 09VC hold for the pair \((X \to S, E)\).

Proof

The implication (2) \(\Rightarrow\) (1) is Lemma 09VD. Assume (1). Say \(S = \Spec(R)\) and \(X = \Spec(A)\) and \(U_i = D(f_i)\). Write \(1 = \sum f_ig_i\) in \(A\). Consider the surjections \[R[x_i, y_i, z_i] \to R[x_i, y_i, z_i]/(\sum y_iz_i - 1) \to A.\] which sends \(y_i\) to \(f_i\) and \(z_i\) to \(g_i\). Note that \(R[x_i, y_i, z_i]/(\sum y_iz_i - 1)\) is pseudo-coherent as an \(R[x_i, y_i, z_i]\)-module. Thus it suffices to prove that the pushforward of \(E\) to \(T = \Spec(R[x_i, y_i, z_i]/(\sum y_iz_i - 1))\) is \(m\)-pseudo-coherent, see Derived Categories of Schemes, Lemma 09VA. For each \(i_0\) it suffices to prove the restriction of this pushforward to \(W_{i_0} = \Spec(R[x_i, y_i, z_i, 1/y_{i_0}]/(\sum y_iz_i - 1))\) is \(m\)-pseudo-coherent. Note that there is a commutative diagram \[\xymatrix{ X \ar[d] & U_{i_0} \ar[l] \ar[d] \\ T & W_{i_0} \ar[l] }\] which implies that the pushforward of \(E\) to \(T\) restricted to \(W_{i_0}\) is the pushforward of \(E|_{U_{i_0}}\) to \(W_{i_0}\). Since \(R[x_i, y_i, z_i, 1/y_{i_0}]/(\sum y_iz_i - 1)\) is isomorphic to a polynomial ring over \(R\) this proves what we want.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Fix \(m \in \mathbf{Z}\). The following are equivalent

\(E\) is \(m\)-pseudo-coherent relative to \(S\),

for every affine opens \(U \subset X\) and \(V \subset S\) with \(f(U) \subset V\) the equivalent conditions of Lemma 09VC are satisfied for the pair \((U \to V, E|_U)\).

Moreover, if this is true, then for every open subschemes \(U \subset X\) and \(V \subset S\) with \(f(U) \subset V\) the restriction \(E|_U\) is \(m\)-pseudo-coherent relative to \(V\).

Proof

The final statement is clear from the equivalence of (1) and (2). It is also clear that (2) implies (1). Assume (1) holds. Let \(S = \bigcup V_i\) and \(f^{-1}(V_i) = \bigcup U_{ij}\) be affine open coverings as in Definition 09UI. Let \(U \subset X\) and \(V \subset S\) be as in (2). By Lemma 09VE it suffices to find a standard open covering \(U = \bigcup U_k\) of \(U\) such that the equivalent conditions of Lemma 09VC are satisfied for the pairs \((U_k \to V, E|_{U_k})\). In other words, for every \(u \in U\) it suffices to find a standard affine open \(u \in U' \subset U\) such that the equivalent conditions of Lemma 09VC are satisfied for the pair \((U' \to V, E|_{U'})\). Pick \(i\) such that \(f(u) \in V_i\) and then pick \(j\) such that \(u \in U_{ij}\). By Schemes, Lemma 01IW we can find \(v \in V' \subset V \cap V_i\) which is standard affine open in \(V'\) and \(V_i\). Then \(f^{-1}V' \cap U\), resp. \(f^{-1}V' \cap U_{ij}\) are standard affine opens of \(U\), resp. \(U_{ij}\). Applying the lemma again we can find \(u \in U' \subset f^{-1}V' \cap U \cap U_{ij}\) which is standard affine open in both \(f^{-1}V' \cap U\) and \(f^{-1}V' \cap U_{ij}\). Thus \(U'\) is also a standard affine open of \(U\) and \(U_{ij}\). By Lemma 09VD the assumption that the equivalent conditions of Lemma 09VC are satisfied for the pair \((U_{ij} \to V_i, E|_{U_{ij}})\) implies that the equivalent conditions of Lemma 09VC are satisfied for the pair \((U' \to V, E|_{U'})\).

For objects of the derived category whose cohomology sheaves are quasi-coherent, we can relate relative \(m\)-pseudo-coherence to the notion defined in More on Algebra, Definition 065I. We will use the fact that for an affine scheme \(U = \Spec(A)\) the functor \(R\Gamma(U, -)\) induces an equivalence between \(D_\QCoh(\mathcal{O}_U)\) and \(D(A)\), see Derived Categories of Schemes, Lemma 06Z0. This functor is compatible with pullbacks: if \(E\) is an object of \(D_\QCoh(\mathcal{O}_U)\) and \(A \to B\) is a ring map corresponding to a morphism of affine schemes \(g : V = \Spec(B) \to \Spec(A) = U\), then \(R\Gamma(V, Lg^*E) = R\Gamma(U, E) \otimes_A^\mathbf{L} B\). See Derived Categories of Schemes, Lemma 08DW.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Fix \(m \in \mathbf{Z}\). The following are equivalent

\(E\) is \(m\)-pseudo-coherent relative to \(S\),

there exists an affine open covering \(S = \bigcup V_i\) and for each \(i\) an affine open covering \(f^{-1}(V_i) = \bigcup U_{ij}\) such that the complex of \(\mathcal{O}_X(U_{ij})\)-modules \(R\Gamma(U_{ij}, E)\) is \(m\)-pseudo-coherent relative to \(\mathcal{O}_S(V_i)\), and

for every affine opens \(U \subset X\) and \(V \subset S\) with \(f(U) \subset V\) the complex of \(\mathcal{O}_X(U)\)-modules \(R\Gamma(U, E)\) is \(m\)-pseudo-coherent relative to \(\mathcal{O}_S(V)\).

Proof

Let \(U\) and \(V\) be as in (2) and choose a closed immersion \(i : U \to \mathbf{A}^n_V\). A formal argument, using Lemma 09UJ, shows it suffices to prove that \(Ri_*(E|_U)\) is \(m\)-pseudo-coherent if and only if \(R\Gamma(U, E)\) is \(m\)-pseudo-coherent relative to \(\mathcal{O}_S(V)\). Say \(U = \Spec(A)\), \(V = \Spec(R)\), and \(\mathbf{A}^n_V = \Spec(R[x_1, \ldots, x_n]\). By the remarks preceding the lemma, \(E|_U\) is quasi-isomorphic to the complex of quasi-coherent sheaves on \(U\) associated to the object \(R\Gamma(U, E)\) of \(D(A)\). Note that \(R\Gamma(U, E) = R\Gamma(\mathbf{A}^n_V, Ri_*(E|_U))\) as \(i\) is a closed immersion (and hence \(i_*\) is exact). Thus \(Ri_*E\) is associated to \(R\Gamma(U, E)\) viewed as an object of \(D(R[x_1, \ldots, x_n])\). We conclude as \(m\)-pseudo-coherence of \(Ri_*(E|_U)\) is equivalent to \(m\)-pseudo-coherence of \(R\Gamma(U, E)\) in \(D(R[x_1, \ldots, x_n])\) by Derived Categories of Schemes, Lemma 08E7 which is equivalent to \(R\Gamma(U, E)\) is \(m\)-pseudo-coherent relative to \(R = \mathcal{O}_S(V)\) by definition.

Lemma

Let \(i : X \to Y\) morphism of schemes locally of finite type over a base scheme \(S\). Assume that \(i\) induces a homeomorphism of \(X\) with a closed subset of \(Y\). Let \(E\) be an object of \(D(\mathcal{O}_X)\). Then \(E\) is \(m\)-pseudo-coherent relative to \(S\) if and only if \(Ri_*E\) is \(m\)-pseudo-coherent relative to \(S\).

Proof

By Morphisms, Lemma 04DE the morphism \(i\) is affine. Thus we may assume \(S\), \(Y\), and \(X\) are affine. Say \(S = \Spec(R)\), \(Y = \Spec(A)\), and \(X = \Spec(B)\). The condition means that \(A/\text{rad}(A) \to B/\text{rad}(B)\) is surjective; here \(\text{rad}(A)\) and \(\text{rad}(B)\) denote the Jacobson radical of \(A\) and \(B\). As \(B\) is of finite type over \(A\), we can find \(b_1, \ldots, b_m \in \text{rad}(B)\) which generate \(B\) as an \(A\)-algebra. Say \(b_j^N = 0\) for all \(j\). Consider the diagram of rings \[\xymatrix{ B & R[x_i, y_j]/(y_j^N) \ar[l] & R[x_i, y_j] \ar[l] \\ A \ar[u] & R[x_i] \ar[l] \ar[u] \ar[ru] }\] which translates into a diagram \[\xymatrix{ X \ar[d] \ar[r] & T \ar[d] \ar[r] & \mathbf{A}^{n + m}_S \ar[ld] \\ Y \ar[r] & \mathbf{A}^n_S }\] of affine schemes. By Lemma 09UJ we see that \(E\) is \(m\)-pseudo-coherent relative to \(S\) if and only if its pushforward to \(\mathbf{A}^{n + m}_S\) is \(m\)-pseudo-coherent. By Derived Categories of Schemes, Lemma 09VA we see that this is true if and only if its pushforward to \(T\) is \(m\)-pseudo-coherent. The same lemma shows that this holds if and only if the pushforward to \(\mathbf{A}^n_S\) is \(m\)-pseudo-coherent. Again by Lemma 09UJ this holds if and only if \(Ri_*E\) is \(m\)-pseudo-coherent relative to \(S\).

Lemma

Let \(\pi : X \to Y\) be a finite morphism of schemes locally of finite type over a base scheme \(S\). Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Then \(E\) is \(m\)-pseudo-coherent relative to \(S\) if and only if \(R\pi_*E\) is \(m\)-pseudo-coherent relative to \(S\).

Proof

Translation of the result of More on Algebra, Lemma 0673 into the language of schemes. Observe that \(R\pi_*\) indeed maps \(D_\QCoh(\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_Y)\) by Derived Categories of Schemes, Lemma 08D5. To do the translation use Lemma 09UJ.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \((E, E', E'')\) be a distinguished triangle of \(D(\mathcal{O}_X)\). Let \(m \in \mathbf{Z}\).

If \(E\) is \((m + 1)\)-pseudo-coherent relative to \(S\) and \(E'\) is \(m\)-pseudo-coherent relative to \(S\) then \(E''\) is \(m\)-pseudo-coherent relative to \(S\).

If \(E, E''\) are \(m\)-pseudo-coherent relative to \(S\), then \(E'\) is \(m\)-pseudo-coherent relative to \(S\).

If \(E'\) is \((m + 1)\)-pseudo-coherent relative to \(S\) and \(E''\) is \(m\)-pseudo-coherent relative to \(S\), then \(E\) is \((m + 1)\)-pseudo-coherent relative to \(S\).

Moreover, if two out of three of \(E, E', E''\) are pseudo-coherent relative to \(S\), the so is the third.

Proof

Immediate from Lemma 09UJ and Cohomology, Lemma 08CD.

Lemma

Let \(X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. Then

\(\mathcal{F}\) is \(m\)-pseudo-coherent relative to \(S\) for all \(m > 0\),

\(\mathcal{F}\) is \(0\)-pseudo-coherent relative to \(S\) if and only if \(\mathcal{F}\) is a finite type \(\mathcal{O}_X\)-module,

\(\mathcal{F}\) is \((-1)\)-pseudo-coherent relative to \(S\) if and only if \(\mathcal{F}\) is quasi-coherent and finitely presented relative to \(S\).

Proof

Part (1) is immediate from the definition. To see part (3) we may work locally on \(X\) (both properties are local). Thus we may assume \(X\) and \(S\) are affine. Choose a closed immersion \(i : X \to \mathbf{A}^n_S\). Then we see that \(\mathcal{F}\) is \((-1)\)-pseudo-coherent relative to \(S\) if and only if \(i_*\mathcal{F}\) is \((-1)\)-pseudo-coherent, which is true if and only if \(i_*\mathcal{F}\) is an \(\mathcal{O}_{\mathbf{A}^n_S}\)-module of finite presentation, see Cohomology, Lemma 08DN. A module of finite presentation is quasi-coherent, see Modules, Lemma 01BO. By Morphisms, Lemma 01QY we see that \(\mathcal{F}\) is quasi-coherent if and only if \(i_*\mathcal{F}\) is quasi-coherent. Having said this part (3) follows. The proof of (2) is similar but less involved.

Lemma

Let \(X \to S\) be a morphism of schemes which is locally of finite type. Let \(m \in \mathbf{Z}\). Let \(E, K\) be objects of \(D(\mathcal{O}_X)\). If \(E \oplus K\) is \(m\)-pseudo-coherent relative to \(S\) so are \(E\) and \(K\).

Proof

Follows from Cohomology, Lemma 08CE and the definitions.

Lemma

Let \(X \to S\) be a morphism of schemes which is locally of finite type. Let \(m \in \mathbf{Z}\). Let \(\mathcal{F}^\bullet\) be a (locally) bounded above complex of \(\mathcal{O}_X\)-modules such that \(\mathcal{F}^i\) is \((m - i)\)-pseudo-coherent relative to \(S\) for all \(i\). Then \(\mathcal{F}^\bullet\) is \(m\)-pseudo-coherent relative to \(S\).

Proof

Follows from Cohomology, Lemma 09V7 and the definitions.

Lemma

Let \(X \to S\) be a morphism of schemes which is locally of finite type. Let \(m \in \mathbf{Z}\). Let \(E\) be an object of \(D(\mathcal{O}_X)\). If \(E\) is (locally) bounded above and \(H^i(E)\) is \((m - i)\)-pseudo-coherent relative to \(S\) for all \(i\), then \(E\) is \(m\)-pseudo-coherent relative to \(S\).

Proof

Follows from Cohomology, Lemma 09V8 and the definitions.

Lemma

Let \(X \to S\) be a morphism of schemes which is locally of finite type. Let \(m \in \mathbf{Z}\). Let \(E\) be an object of \(D(\mathcal{O}_X)\) which is \(m\)-pseudo-coherent relative to \(S\). Let \(S' \to S\) be a morphism of schemes. Set \(X' = X \times_S S'\) and denote \(E'\) the derived pullback of \(E\) to \(X'\). If \(S'\) and \(X\) are Tor independent over \(S\), then \(E'\) is \(m\)-pseudo-coherent relative to \(S'\).

Proof

The problem is local on \(X\) and \(X'\) hence we may assume \(X\), \(S\), \(S'\), and \(X'\) are affine. Choose a closed immersion \(i : X \to \mathbf{A}^n_S\) and denote \(i' : X' \to \mathbf{A}^n_{S'}\) the base change to \(S'\). Denote \(g : X' \to X\) and \(g' : \mathbf{A}^n_{S'} \to \mathbf{A}^n_S\) the projections, so \(E' = Lg^*E\). Since \(X\) and \(S'\) are tor-independent over \(S\), the base change map (Cohomology, Remark 08HY) induces an isomorphism \[Ri'_*(Lg^*E) = L(g')^*Ri_*E\] Namely, for a point \(x' \in X'\) lying over \(x \in X\) the base change map on stalks at \(x'\) is the map \[E_x \otimes_{\mathcal{O}_{\mathbf{A}^n_S, x}}^\mathbf{L} \mathcal{O}_{\mathbf{A}^n_{S'}, x'} \longrightarrow E_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X', x'}\] coming from the closed immersions \(i\) and \(i'\). Note that the source is quasi-isomorphic to a localization of \(E_x \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S', s'}\) which is isomorphic to the target as \(\mathcal{O}_{X', x'}\) is isomorphic to (the same) localization of \(\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S', s'}\) by assumption. We conclude the lemma holds by an application of Cohomology, Lemma 09U7.

Lemma

Let \(f : X \to Y\) be a morphism of schemes locally of finite type over a base \(S\). Let \(m \in \mathbf{Z}\). Let \(E\) be an object of \(D(\mathcal{O}_Y)\). Assume

\(\mathcal{O}_X\) is pseudo-coherent relative to \(Y\)¹², and

\(E\) is \(m\)-pseudo-coherent relative to \(S\).

Then \(Lf^*E\) is \(m\)-pseudo-coherent relative to \(S\).

Proof

The problem is local on \(X\). Thus we may assume \(X\), \(Y\), and \(S\) are affine. Arguing as in the proof of More on Algebra, Lemma 067B we can find a commutative diagram \[\xymatrix{ X \ar[r]_i \ar[d]_f & \mathbf{A}^d_Y \ar[r]_j \ar[ld]^p & \mathbf{A}^{n + d}_S \ar[ld] \\ Y \ar[r] & \mathbf{A}^n_S }\] Observe that \[Ri_* Lf^*E = Ri_* Li^* Lp^*E = Lp^*E \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}}^\mathbf{L} Ri_*\mathcal{O}_X\] by Cohomology, Lemma 0B55. By assumption and the fact that \(Y\) is affine, we can represent \(Ri_*\mathcal{O}_X = i_*\mathcal{O}_X\) by a complexes of finite free \(\mathcal{O}_{\mathbf{A}_Y^n}\)-modules \(\mathcal{F}^\bullet\), with \(\mathcal{F}^q = 0\) for \(q > 0\) (details omitted; use Derived Categories of Schemes, Lemma 08E7 and More on Algebra, Lemma 0675). By assumption \(E\) is bounded above, say \(H^q(E) = 0\) for \(q > a\). Represent \(E\) by a complex \(\mathcal{E}^\bullet\) of \(\mathcal{O}_Y\)-modules with \(\mathcal{E}^q = 0\) for \(q > a\). Then the derived tensor product above is represented by \(\text{Tot}(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^\bullet)\).

Since \(j\) is a closed immersion, the functor \(j_*\) is exact and \(Rj_*\) is computed by applying \(j_*\) to any representing complex of sheaves. Thus we have to show that \(j_*\text{Tot}(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^\bullet)\) is \(m\)-pseudo-coherent as a complex of \(\mathcal{O}_{\mathbf{A}^{n + m}_S}\)-modules. Note that \(\text{Tot}(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^\bullet)\) has a filtration by subcomplexes with successive quotients the complexes \(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^q[-q]\). Note that for \(q \ll 0\) the complexes \(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^q[-q]\) have zero cohomology in degrees \(\leq m\) and hence are \(m\)-pseudo-coherent. Hence, applying Lemma 09UL and induction, it suffices to show that \(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^q[-q]\) is pseudo-coherent relative to \(S\) for all \(q\). Note that \(\mathcal{F}^q = 0\) for \(q > 0\). Since also \(\mathcal{F}^q\) is finite free this reduces to proving that \(p^*\mathcal{E}^\bullet\) is \(m\)-pseudo-coherent relative to \(S\) which follows from Lemma 09UR for instance.

Lemma

Let \(f : X \to Y\) be a morphism of schemes locally of finite type over a base \(S\). Let \(m \in \mathbf{Z}\). Let \(E\) be an object of \(D(\mathcal{O}_X)\). Assume \(\mathcal{O}_Y\) is pseudo-coherent relative to \(S\)¹³. Then the following are equivalent

\(E\) is \(m\)-pseudo-coherent relative to \(Y\), and

\(E\) is \(m\)-pseudo-coherent relative to \(S\).

Proof

The question is local on \(X\), hence we may assume \(X\), \(Y\), and \(S\) are affine. Arguing as in the proof of More on Algebra, Lemma 067B we can find a commutative diagram \[\xymatrix{ X \ar[r]_i \ar[d]_f & \mathbf{A}^m_Y \ar[r]_j \ar[ld]^p & \mathbf{A}^{n + m}_S \ar[ld] \\ Y \ar[r] & \mathbf{A}^n_S }\] The assumption that \(\mathcal{O}_Y\) is pseudo-coherent relative to \(S\) implies that \(\mathcal{O}_{\mathbf{A}^m_Y}\) is pseudo-coherent relative to \(\mathbf{A}^m_S\) (by flat base change; this can be seen by using for example Lemma 09UR). This in turn implies that \(j_*\mathcal{O}_{\mathbf{A}^n_Y}\) is pseudo-coherent as an \(\mathcal{O}_{\mathbf{A}^{n + m}_S}\)-module. Then the equivalence of the lemma follows from Derived Categories of Schemes, Lemma 09VA.

Lemma

Let \[\xymatrix{ X \ar[rd] \ar[rr]_i & & P \ar[ld] \\ & S }\] be a commutative diagram of schemes. Assume \(i\) is a closed immersion and \(P \to S\) flat and locally of finite presentation. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Then the following are equivalent

\(E\) is \(m\)-pseudo-coherent relative to \(S\),

\(Ri_*E\) is \(m\)-pseudo-coherent relative to \(S\), and

\(Ri_*E\) is \(m\)-pseudo-coherent on \(P\).

Proof

The equivalence of (1) and (2) is Lemma 09UK. The equivalence of (2) and (3) follows from Lemma 09UT applied to \(\text{id} : P \to P\) provided we can show that \(\mathcal{O}_P\) is pseudo-coherent relative to \(S\). This follows from More on Algebra, Lemma 067J and the definitions.

Pseudo-coherent morphisms

Avoid reading this section at all cost. If you need some of this material, first take a look at the corresponding algebra sections, see More on Algebra, Sections 064N, 065E, and 067G. For now the only thing you need to know is that a ring map \(A \to B\) is pseudo-coherent if and only if \(B = A[x_1, \ldots, x_n]/I\) and \(B\) as an \(A[x_1, \ldots, x_n]\)-module has a resolution by finite free \(A[x_1, \ldots, x_n]\)-modules.

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent

there exist an affine open covering \(S = \bigcup V_j\) and for each \(j\) an affine open covering \(f^{-1}(V_j) = \bigcup U_{ji}\) such that \(\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_{ij})\) is a pseudo-coherent ring map,

for every pair of affine opens \(U \subset X\), \(V \subset S\) such that \(f(U) \subset V\) the ring map \(\mathcal{O}_S(V) \to \mathcal{O}_X(U)\) is pseudo-coherent, and

\(f\) is locally of finite type and \(\mathcal{O}_X\) is pseudo-coherent relative to \(S\).

Proof

To see the equivalence of (1) and (2) it suffices to check conditions (1)(a), (b), (c) of Morphisms, Definition 01SR for the property of being a pseudo-coherent ring map. These properties follow (using localization is flat) from More on Algebra, Lemmas 067A, 0679, and 067E.

If (1) holds, then \(f\) is locally of finite type as a pseudo-coherent ring map is of finite type by definition. Moreover, (1) implies via Lemma 09VF and the definitions that \(\mathcal{O}_X\) is pseudo-coherent relative to \(S\). Conversely, if (3) holds, then we see that for every \(U\) and \(V\) as in (2) the ring \(\mathcal{O}_X(U)\) is of finite type over \(\mathcal{O}_S(V)\) and \(\mathcal{O}_X(U)\) is as a module pseudo-coherent relative to \(\mathcal{O}_S(V)\), see Lemmas 09UJ and 09VF. This is the definition of a pseudo-coherent ring map, hence (2) and (1) hold.

Definition

A morphism of schemes \(f : X \to S\) is called pseudo-coherent if the equivalent conditions of Lemma 067Y are satisfied. In this case we also say that \(X\) is pseudo-coherent over \(S\).

Beware that a base change of a pseudo-coherent morphism is not pseudo-coherent in general.

Lemma

A flat base change of a pseudo-coherent morphism is pseudo-coherent.

Proof

This translates into the following algebra result: Let \(A \to B\) be a pseudo-coherent ring map. Let \(A \to A'\) be flat. Then \(A' \to B \otimes_A A'\) is pseudo-coherent. This follows from the more general More on Algebra, Lemma 067A.

Lemma

A composition of pseudo-coherent morphisms of schemes is pseudo-coherent.

Proof

This translates into the following algebra result: If \(A \to B \to C\) are composable pseudo-coherent ring maps then \(A \to C\) is pseudo-coherent. This follows from either More on Algebra, Lemma 067B or More on Algebra, Lemma 067D.

Lemma

A pseudo-coherent morphism is locally of finite presentation.

Proof

Immediate from the definitions.

Lemma

A flat morphism which is locally of finite presentation is pseudo-coherent.

Proof

This follows from the fact that a flat ring map of finite presentation is pseudo-coherent (and even perfect), see More on Algebra, Lemma 067J.

Lemma

Let \(f : X \to Y\) be a morphism of schemes pseudo-coherent over a base scheme \(S\). Then \(f\) is pseudo-coherent.

Proof

This translates into the following algebra result: If \(R \to A \to B\) are composable ring maps and \(R \to A\), \(R \to B\) pseudo-coherent, then \(R \to B\) is pseudo-coherent. This follows from More on Algebra, Lemma 067D.

Lemma

Let \(f : X \to S\) be a finite morphism of schemes. Then \(f\) is pseudo-coherent if and only if \(f_*\mathcal{O}_X\) is pseudo-coherent as an \(\mathcal{O}_S\)-module.

Proof

Translated into algebra this lemma says the following: If \(R \to A\) is a finite ring map, then \(R \to A\) is pseudo-coherent as a ring map (which means by definition that \(A\) as an \(A\)-module is pseudo-coherent relative to \(R\)) if and only if \(A\) is pseudo-coherent as an \(R\)-module. This follows from the more general More on Algebra, Lemma 0673.

Lemma

Let \(f : X \to S\) be a morphism of schemes. If \(S\) is locally Noetherian, then \(f\) is pseudo-coherent if and only if \(f\) is locally of finite type.

Proof

This translates into the following algebra result: If \(R \to A\) is a finite type ring map with \(R\) Noetherian, then \(R \to A\) is pseudo-coherent if and only if \(R \to A\) is of finite type. To see this, note that a pseudo-coherent ring map is of finite type by definition. Conversely, if \(R \to A\) is of finite type, then we can write \(A = R[x_1, \ldots, x_n]/I\) and it follows from More on Algebra, Lemma 066E that \(A\) is pseudo-coherent as an \(R[x_1, \ldots, x_n]\)-module, i.e., \(R \to A\) is a pseudo-coherent ring map.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is pseudo-coherent” is fpqc local on the base.

Proof

We will use the criterion of Descent, Lemma 02KP to prove this. By Definition 067Z being pseudo-coherent is Zariski local on the base. By Lemma 0680 being pseudo-coherent is preserved under flat base change. The final hypothesis (3) of Descent, Lemma 02KP translates into the following algebra statement: Let \(A \to B\) be a faithfully flat ring map. Let \(C = A[x_1, \ldots, x_n]/I\) be an \(A\)-algebra. If \(C \otimes_A B\) is pseudo-coherent as an \(B[x_1, \ldots, x_n]\)-module, then \(C\) is pseudo-coherent as a \(A[x_1, \ldots, x_n]\)-module. This is More on Algebra, Lemma 068R.

Lemma

Let \(A \to B\) be a flat ring map of finite presentation. Let \(I \subset B\) be an ideal. Then \(A \to B/I\) is pseudo-coherent if and only if \(I\) is pseudo-coherent as a \(B\)-module.

Proof

Choose a presentation \(B = A[x_1, \ldots, x_n]/J\). Note that \(B\) is pseudo-coherent as an \(A[x_1, \ldots, x_n]\)-module because \(A \to B\) is a pseudo-coherent ring map by Lemma 0695. Note that \(A \to B/I\) is pseudo-coherent if and only if \(B/I\) is pseudo-coherent as an \(A[x_1, \ldots, x_n]\)-module. By More on Algebra, Lemma 064Z we see this is equivalent to the condition that \(B/I\) is pseudo-coherent as an \(B\)-module. This proves the lemma as the short exact sequence \(0 \to I \to B \to B/I \to 0\) shows that \(I\) is pseudo-coherent if and only if \(B/I\) is (see More on Algebra, Lemma 064V).

The following lemma will be obsoleted by the stronger Lemma 0699.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is pseudo-coherent” is syntomic local on the source.

Proof

We will use the criterion of Descent, Lemma 036H to prove this. It follows from Lemmas 0695 and 0681 that being pseudo-coherent is preserved under precomposing with flat morphisms locally of finite presentation, in particular under precomposing with syntomic morphisms (see Morphisms, Lemmas 01UL and 01UK). It is clear from Definition 067Z that being pseudo-coherent is Zariski local on the source and target. Hence, according to the aforementioned Descent, Lemma 036H it suffices to prove the following: Suppose \(X' \to X \to Y\) are morphisms of affine schemes with \(X' \to X\) syntomic and \(X' \to Y\) pseudo-coherent. Then \(X \to Y\) is pseudo-coherent. To see this, note that in any case \(X \to Y\) is of finite presentation by Descent, Lemma 02KK. Choose a closed immersion \(X \to \mathbf{A}^n_Y\). By Algebra, Lemma 00T0 we can find an affine open covering \(X' = \bigcup_{i = 1, \ldots, n} X'_i\) and syntomic morphisms \(W_i \to \mathbf{A}^n_Y\) lifting the morphisms \(X'_i \to X\), i.e., such that there are fibre product diagrams \[\xymatrix{ X'_i \ar[d] \ar[r] & W_i \ar[d] \\ X \ar[r] & \mathbf{A}^n_Y }\] After replacing \(X'\) by \(\coprod X'_i\) and setting \(W = \coprod W_i\) we obtain a fibre product diagram \[\xymatrix{ X' \ar[d] \ar[r] & W \ar[d]^h \\ X \ar[r] & \mathbf{A}^n_Y }\] with \(W \to \mathbf{A}^n_Y\) flat and of finite presentation and \(X' \to Y\) still pseudo-coherent. Since \(W \to \mathbf{A}^n_Y\) is open (see Morphisms, Lemma 01UA) and \(X' \to X\) is surjective we can find \(f \in \Gamma(\mathbf{A}^n_Y, \mathcal{O})\) such that \(X \subset D(f) \subset \Im(h)\). Write \(Y = \Spec(R)\), \(X = \Spec(A)\), \(X' = \Spec(A')\) and \(W = \Spec(B)\), \(A = R[x_1, \ldots, x_n]/I\) and \(A' = B/IB\). Then \(R \to A'\) is pseudo-coherent. Picture \[\xymatrix{ A' = B/IB & B \ar[l] \\ A = R[x_1, \ldots, x_n]/I \ar[u] & R[x_1, \ldots, x_n] \ar[l] \ar[u] }\] By Lemma 0697 we see that \(IB\) is pseudo-coherent as a \(B\)-module. The ring map \(R[x_1, \ldots, x_n]_f \to B_f\) is faithfully flat by our choice of \(f\) above. This implies that \(I_f \subset R[x_1, \ldots, x_n]_f\) is pseudo-coherent, see More on Algebra, Lemma 068R. Applying Lemma 0697 one more time we see that \(R \to A\) is pseudo-coherent.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is pseudo-coherent” is fppf local on the source.

Proof

Let \(f : X \to S\) be a morphism of schemes. Let \(\{g_i : X_i \to X\}\) be an fppf covering such that each composition \(f \circ g_i\) is pseudo-coherent. According to Lemma 05WN there exist

a Zariski open covering \(X = \bigcup U_j\),

surjective finite locally free morphisms \(W_j \to U_j\),

Zariski open coverings \(W_j = \bigcup_k W_{j, k}\),

surjective finite locally free morphisms \(T_{j, k} \to W_{j, k}\)

such that the fppf covering \(\{h_{j, k} : T_{j, k} \to X\}\) refines the given covering \(\{X_i \to X\}\). Denote \(\psi_{j, k} : T_{j, k} \to X_{\alpha(j, k)}\) the morphisms that witness the fact that \(\{T_{j, k} \to X\}\) refines the given covering \(\{X_i \to X\}\). Note that \(T_{j, k} \to X\) is a flat, locally finitely presented morphism, so both \(X_i\) and \(T_{j, k}\) are pseudo-coherent over \(X\) by Lemma 0695. Hence \(\psi_{j, k} : T_{j, k} \to X_i\) is pseudo-coherent, see Lemma 0683. Hence \(T_{j, k} \to S\) is pseudo coherent as the composition of \(\psi_{j, k}\) and \(f \circ g_{\alpha(j, k)}\), see Lemma 0681. Thus we see we have reduced the lemma to the case of a Zariski open covering (which is OK) and the case of a covering given by a single surjective finite locally free morphism which we deal with in the following paragraph.

Assume that \(X' \to X \to S\) is a sequence of morphisms of schemes with \(X' \to X\) surjective finite locally free and \(X' \to Y\) pseudo-coherent. Our goal is to show that \(X \to S\) is pseudo-coherent. Note that by Descent, Lemma 02KL the morphism \(X \to S\) is locally of finite presentation. It is clear that the problem reduces to the case that \(X'\), \(X\) and \(S\) are affine and \(X' \to X\) is free of some rank \(r > 0\). The corresponding algebra problem is the following: Suppose \(R \to A \to A'\) are ring maps such that \(R \to A'\) is pseudo-coherent, \(R \to A\) is of finite presentation, and \(A' \cong A^{\oplus r}\) as an \(A\)-module. Goal: Show \(R \to A\) is pseudo-coherent. The assumption that \(R \to A'\) is pseudo-coherent means that \(A'\) as an \(A'\)-module is pseudo-coherent relative to \(R\). By More on Algebra, Lemma 0673 this implies that \(A'\) as an \(A\)-module is pseudo-coherent relative to \(R\). Since \(A' \cong A^{\oplus r}\) as an \(A\)-module we see that \(A\) as an \(A\)-module is pseudo-coherent relative to \(R\), see More on Algebra, Lemma 0676. This by definition means that \(R \to A\) is pseudo-coherent and we win.

Perfect morphisms

In order to understand the material in this section you have to understand the material of the section on pseudo-coherent morphisms just a little bit. For now the only thing you need to know is that a ring map \(A \to B\) is perfect if and only if it is pseudo-coherent and \(B\) has finite tor dimension as an \(A\)-module.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. The following are equivalent

there exist an affine open covering \(S = \bigcup V_j\) and for each \(j\) an affine open covering \(f^{-1}(V_j) = \bigcup U_{ji}\) such that \(\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_{ij})\) is a perfect ring map, and

for every pair of affine opens \(U \subset X\), \(V \subset S\) such that \(f(U) \subset V\) the ring map \(\mathcal{O}_S(V) \to \mathcal{O}_X(U)\) is perfect.

Proof

Assume (1) and let \(U, V\) be as in (2). It follows from Lemma 067Y that \(\mathcal{O}_S(V) \to \mathcal{O}_X(U)\) is pseudo-coherent. Hence it suffices to prove that the property of a ring map being "of finite tor dimension" satisfies conditions (1)(a), (b), (c) of Morphisms, Definition 01SR. These properties follow from More on Algebra, Lemmas 066J, 066M, and 066N. Some details omitted.

Definition

A morphism of schemes \(f : X \to S\) is called perfect if the equivalent conditions of Lemma 0686 are satisfied. In this case we also say that \(X\) is perfect over \(S\).

Note that a perfect morphism is in particular pseudo-coherent, hence locally of finite presentation. Beware that a base change of a perfect morphism is not perfect in general.

Lemma

A flat base change of a perfect morphism is perfect.

Proof

This translates into the following algebra result: Let \(A \to B\) be a perfect ring map. Let \(A \to A'\) be flat. Then \(A' \to B \otimes_A A'\) is perfect. This result for pseudo-coherent ring maps we have seen in Lemma 0680. The corresponding fact for finite tor dimension follows from More on Algebra, Lemma 066M.

Lemma

A composition of perfect morphisms of schemes is perfect.

Proof

This translates into the following algebra result: If \(A \to B \to C\) are composable perfect ring maps then \(A \to C\) is perfect. We have seen this is the case for pseudo-coherent in Lemma 0681 and its proof. By assumption there exist integers \(n\), \(m\) such that \(B\) has tor dimension \(\leq n\) over \(A\) and \(C\) has tor dimension \(\leq m\) over \(B\). Then for any \(A\)-module \(M\) we have \[M \otimes_A^{\mathbf{L}} C = (M \otimes_A^{\mathbf{L}} B) \otimes_B^{\mathbf{L}} C\] and the spectral sequence of More on Algebra, Example 0662 shows that \(\text{Tor}^A_p(M, C) = 0\) for \(p > n + m\) as desired.

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent

\(f\) is flat and perfect, and

\(f\) is flat and locally of finite presentation.

Proof

The implication (2) \(\Rightarrow\) (1) is More on Algebra, Lemma 067J. The converse follows from the fact that a pseudo-coherent morphism is locally of finite presentation, see Lemma 0682.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is regular and \(f\) is locally of finite type. Then \(f\) is perfect.

Proof

See More on Algebra, Lemma 067K.

Lemma

A regular immersion of schemes is perfect. A Koszul-regular immersion of schemes is perfect.

Proof

Since a regular immersion is a Koszul-regular immersion, see Divisors, Lemma 063K, it suffices to prove the second statement. This translates into the following algebraic statement: Suppose that \(I \subset A\) is an ideal generated by a Koszul-regular sequence \(f_1, \ldots, f_r\) of \(A\). Then \(A \to A/I\) is a perfect ring map. Since \(A \to A/I\) is surjective this is a presentation of \(A/I\) by a polynomial algebra over \(A\). Hence it suffices to see that \(A/I\) is pseudo-coherent as an \(A\)-module and has finite tor dimension. By definition of a Koszul sequence the Koszul complex \(K(A, f_1, \ldots, f_r)\) is a finite free resolution of \(A/I\). Hence \(A/I\) is a perfect complex of \(A\)-modules and we win.

Lemma

Let \[\xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S }\] be a commutative diagram of morphisms of schemes. Assume \(Y \to S\) smooth and \(X \to S\) perfect. Then \(f : X \to Y\) is perfect.

Proof

We can factor \(f\) as the composition \[X \longrightarrow X \times_S Y \longrightarrow Y\] where the first morphism is the map \(i = (1, f)\) and the second morphism is the projection. Since \(Y \to S\) is flat, see Morphisms, Lemma 01VF, we see that \(X \times_S Y \to Y\) is perfect by Lemma 0688. As \(Y \to S\) is smooth, also \(X \times_S Y \to X\) is smooth, see Morphisms, Lemma 01VB. Hence \(i\) is a section of a smooth morphism, therefore \(i\) is a regular immersion, see Divisors, Lemma 067R. This implies that \(i\) is perfect, see Lemma 068C. We conclude that \(f\) is perfect because the composition of perfect morphisms is perfect, see Lemma 0689.

Remark

It is not true that a morphism between schemes \(X, Y\) perfect over a base \(S\) is perfect. An example is \(S = \Spec(k)\), \(X = \Spec(k)\), \(Y = \Spec(k[x]/(x^2)\) and \(X \to Y\) the unique \(S\)-morphism.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is perfect” is fpqc local on the base.

Proof

We will use the criterion of Descent, Lemma 02KP to prove this. By Definition 0687 being perfect is Zariski local on the base. By Lemma 0688 being perfect is preserved under flat base change. The final hypothesis (3) of Descent, Lemma 02KP translates into the following algebra statement: Let \(A \to B\) be a faithfully flat ring map. Let \(C = A[x_1, \ldots, x_n]/I\) be an \(A\)-algebra. If \(C \otimes_A B\) is perfect as an \(B[x_1, \ldots, x_n]\)-module, then \(C\) is perfect as a \(A[x_1, \ldots, x_n]\)-module. This is More on Algebra, Lemma 068T.

Lemma

Let \(f : X \to S\) be a pseudo-coherent morphism of schemes. The following are equivalent

\(f\) is perfect,

\(\mathcal{O}_X\) locally has finite tor dimension as a sheaf of \(f^{-1}\mathcal{O}_S\)-modules, and

for all \(x \in X\) the ring \(\mathcal{O}_{X, x}\) has finite tor dimension as an \(\mathcal{O}_{S, f(x)}\)-module.

Proof

The problem is local on \(X\) and \(S\). Hence we may assume that \(X = \Spec(B)\), \(S = \Spec(A)\) and \(f\) corresponds to a pseudo-coherent ring map \(A \to B\).

If (1) holds, then \(B\) has finite tor dimension \(d\) as \(A\)-module. Then \(B_\mathfrak q\) has tor dimension \(d\) as an \(A_\mathfrak p\)-module for all primes \(\mathfrak q \subset B\) with \(\mathfrak p = A \cap \mathfrak q\), see More on Algebra, Lemma 0B67. Then \(\mathcal{O}_X\) has tor dimension \(d\) as a sheaf of \(f^{-1}\mathcal{O}_S\)-modules by Cohomology, Lemma 09U9. Thus (1) implies (2).

By Cohomology, Lemma 09U9 (2) implies (3).

Assume (3). We cannot use More on Algebra, Lemma 0B67 to conclude as we are not given that the tor dimension of \(B_\mathfrak q\) over \(A_\mathfrak p\) is bounded independent of \(\mathfrak q\). Choose a presentation \(A[x_1, \ldots, x_n] \to B\). Then \(B\) is pseudo-coherent as a \(A[x_1, \ldots, x_n]\)-module. Let \(\mathfrak q \subset A[x_1, \ldots, x_n]\) be a prime ideal lying over \(\mathfrak p \subset A\). Then either \(B_\mathfrak q\) is zero or by assumption it has finite tor dimension as an \(A_\mathfrak p\)-module. Since the fibres of \(A \to A[x_1, \ldots, x_n]\) have finite global dimension, we can apply More on Algebra, Lemma 068X to \(A_\mathfrak p \to A[x_1, \ldots, x_n]_\mathfrak q\) to see that \(B_\mathfrak q\) is a perfect \(A[x_1, \ldots, x_n]_\mathfrak q\)-module. Hence \(B\) is a perfect \(A[x_1, \ldots, x_n]\)-module by More on Algebra, Lemma 068W. Thus \(A \to B\) is a perfect ring map by definition.

Lemma

Let \(i : Z \to X\) be a perfect closed immersion of schemes. Then \(i_*\mathcal{O}_Z\) is a perfect \(\mathcal{O}_X\)-module, i.e., it is a perfect object of \(D(\mathcal{O}_X)\).

Proof

This is more or less immediate from the definition. Namely, let \(U = \Spec(A)\) be an affine open of \(X\). Then \(i^{-1}(U) = \Spec(A/I)\) for some ideal \(I \subset A\) and \(A/I\) has a finite resolution by finite projective \(A\)-modules by More on Algebra, Lemma 068Y. Hence \(i_*\mathcal{O}_Z|_U\) can be represented by a finite length complex of finite locally free \(\mathcal{O}_U\)-modules. This is what we had to show, see Cohomology, Section 08CL.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a perfect proper morphism of schemes. Let \(E \in D(\mathcal{O}_X)\) be perfect. Then \(Rf_*E\) is a perfect object of \(D(\mathcal{O}_S)\).

Proof

We claim that Derived Categories of Schemes, Lemma 08EV applies. Conditions (1) and (2) are immediate. Condition (3) is local on \(X\). Thus we may assume \(X\) and \(S\) affine and \(E\) represented by a strictly perfect complex of \(\mathcal{O}_X\)-modules. Thus it suffices to show that \(\mathcal{O}_X\) has finite tor dimension as a sheaf of \(f^{-1}\mathcal{O}_S\)-modules. This is equivalent to being perfect by Lemma 069C.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is perfect” is fppf local on the source.

Proof

Let \(\{g_i : X_i \to X\}_{i \in I}\) be an fppf covering of schemes and let \(f : X \to S\) be a morphism such that each \(f \circ g_i\) is perfect. By Lemma 0699 we conclude that \(f\) is pseudo-coherent. Hence by Lemma 069C it suffices to check that \(\mathcal{O}_{X, x}\) is an \(\mathcal{O}_{S, f(x)}\)-module of finite tor dimension for all \(x \in X\). Pick \(i \in I\) and \(x_i \in X_i\) mapping to \(x\). Then we see that \(\mathcal{O}_{X_i, x_i}\) has finite tor dimension over \(\mathcal{O}_{S, f(x)}\) and that \(\mathcal{O}_{X, x} \to \mathcal{O}_{X_i, x_i}\) is faithfully flat. The desired conclusion follows from More on Algebra, Lemma 068S.

Lemma

Let \(i : Z \to Y\) and \(j : Y \to X\) be immersions of schemes. Assume

\(X\) is locally Noetherian,

\(j \circ i\) is a regular immersion, and

\(i\) is perfect.

Then \(i\) and \(j\) are regular immersions.

Proof

Since \(X\) (and hence \(Y\)) is locally Noetherian all 4 types of regular immersions agree, and moreover we may check whether a morphism is a regular immersion on the level of local rings, see Divisors, Lemma 063I. Thus the result follows from Divided Power Algebra, Lemma 09PW.

Local complete intersection morphisms

In Divisors, Section 0638 we have defined 4 different types of regular immersions: regular, Koszul-regular, \(H_1\)-regular, and quasi-regular. In this section we consider morphisms \(f : X \to S\) which locally on \(X\) factor as \[\xymatrix{ X \ar[rr]_i \ar[rd] & & \mathbf{A}^n_S \ar[ld] \\ & S }\] where \(i\) is a \(*\)-regular immersion for \(* \in \{\emptyset, Koszul, H_1, quasi\}\). However, we don’t know how to prove that this condition is independent of the factorization if \(* = \emptyset\), i.e., when we require \(i\) to be a regular immersion. On the other hand, we want a local complete intersection morphism to be perfect, which is only going to be true if \(* = Koszul\) or \(* = \emptyset\). Hence we will define a local complete intersection morphism or Koszul morphism to be a morphism of schemes \(f : X \to S\) that locally on \(X\) has a factorization as above with \(i\) a Koszul-regular immersion. To see that this works we first prove this is independent of the chosen factorizations.

Lemma

Let \(S\) be a scheme. Let \(U\), \(P\), \(P'\) be schemes over \(S\). Let \(u \in U\). Let \(i : U \to P\), \(i' : U \to P'\) be immersions over \(S\). Assume \(P\) and \(P'\) smooth over \(S\). Then the following are equivalent

\(i\) is a Koszul-regular immersion in a neighbourhood of \(u\), and

\(i'\) is a Koszul-regular immersion in a neighbourhood of \(u\).

Proof

Assume \(i\) is a Koszul-regular immersion in a neighbourhood of \(u\). Consider the morphism \(j = (i, i') : U \to P \times_S P' = P''\). Since \(P'' = P \times_S P' \to P\) is smooth, it follows from Divisors, Lemma 067S that \(j\) is a Koszul-regular immersion, whereupon it follows from Divisors, Lemma 0693 that \(i'\) is a Koszul-regular immersion.

Before we state the definition, let us make the following simple remark. Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(x \in X\). Then there exist an open neighbourhood \(U \subset X\) and a factorization of \(f|_U\) as the composition of an immersion \(i : U \to \mathbf{A}^n_S\) followed by the projection \(\mathbf{A}^n_S \to S\) which is smooth. Picture \[\xymatrix{ X \ar[rd] & U \ar[l] \ar[d] \ar[r]_-i & \mathbf{A}^n_S = P \ar[ld]^\pi \\ & S }\] In fact you can do this with any affine open neighbourhood \(U\) of \(x\) in \(X\), see Morphisms, Lemma 04II.

Definition

Let \(f : X \to S\) be a morphism of schemes.

Let \(x \in X\). We say that \(f\) is Koszul at \(x\) if \(f\) is of finite type at \(x\) and there exists an open neighbourhood and a factorization of \(f|_U\) as \(\pi \circ i\) where \(i : U \to P\) is a Koszul-regular immersion and \(\pi : P \to S\) is smooth.

We say \(f\) is a Koszul morphism, or that \(f\) is a local complete intersection morphism if \(f\) is Koszul at every point.

We have seen above that the choice of the factorization \(f|_U = \pi \circ i\) is irrelevant, i.e., given a factorization of \(f|_U\) as an immersion \(i\) followed by a smooth morphism \(\pi\), whether or not \(i\) is Koszul regular in a neighbourhood of \(x\) is an intrinsic property of \(f\) at \(x\). Let us record this here explicitly as a lemma so that we can refer to it

Lemma

Let \(f : X \to S\) be a local complete intersection morphism. Let \(P\) be a scheme smooth over \(S\). Let \(U \subset X\) be an open subscheme and \(i : U \to P\) an immersion of schemes over \(S\). Then \(i\) is a Koszul-regular immersion.

Proof

This is the defining property of a local complete intersection morphism. See discussion above.

It seems like a good idea to collect here some properties in common with all Koszul morphisms.

Lemma

Let \(f : X \to S\) be a local complete intersection morphism. Then

\(f\) is locally of finite presentation,

\(f\) is pseudo-coherent, and

\(f\) is perfect.

Proof

Since a perfect morphism is pseudo-coherent (because a perfect ring map is pseudo-coherent) and a pseudo-coherent morphism is locally of finite presentation (because a pseudo-coherent ring map is of finite presentation) it suffices to prove the last statement. Being perfect is a local property, hence we may assume that \(f\) factors as \(\pi \circ i\) where \(\pi\) is smooth and \(i\) is a Koszul-regular immersion. A Koszul-regular immersion is perfect, see Lemma 068C. A smooth morphism is perfect as it is flat and locally of finite presentation, see Lemma 068A. Finally a composition of perfect morphisms is perfect, see Lemma 0689.

Lemma

Let \(f : X = \Spec(B) \to S = \Spec(A)\) be a morphism of affine schemes. Then \(f\) is a local complete intersection morphism if and only if \(A \to B\) is a local complete intersection homomorphism, see More on Algebra, Definition 07D0.

Proof

Follows immediately from the definitions.

Beware that a base change of a Koszul morphism is not Koszul in general.

Lemma

A flat base change of a local complete intersection morphism is a local complete intersection morphism.

Proof

Omitted. Hint: This is true because a base change of a smooth morphism is smooth and a flat base change of a Koszul-regular immersion is a Koszul-regular immersion, see Divisors, Lemma 063L.

Lemma

A composition of local complete intersection morphisms is a local complete intersection morphism.

Proof

Let \(g : Y \to S\) and \(f : X \to Y\) be local complete intersection morphisms. Let \(x \in X\) and set \(y = f(x)\). Choose an open neighbourhood \(V \subset Y\) of \(y\) and a factorization \(g|_V = \pi \circ i\) for some Koszul-regular immersion \(i : V \to P\) and smooth morphism \(\pi : P \to S\). Next choose an open neighbourhood \(U\) of \(x \in X\) and a factorization \(f|_U = \pi' \circ i'\) for some Koszul-regular immersion \(i' : U \to P'\) and smooth morphism \(\pi' : P' \to Y\). In fact, we may assume that \(P' = \mathbf{A}^n_V\), see discussion preceding and following Definition 069F. Picture: \[\xymatrix{ X \ar[d] & U \ar[l] \ar[r]_-{i'} & P' = \mathbf{A}^n_V \ar[d] \\ Y \ar[d] & & V \ar[ll] \ar[r]_i & P \ar[d] \\ S & & & S \ar[lll] }\] Set \(P'' = \mathbf{A}^n_P\). Then \(U \to P' \to P''\) is a Koszul-regular immersion as a composition of Koszul-regular immersions, namely \(i'\) and the flat base change of \(i\) via \(P'' \to P\), see Divisors, Lemma 063L and Divisors, Lemma 067Q. Also \(P'' \to P \to S\) is smooth as a composition of smooth morphisms, see Morphisms, Lemma 01VA. Hence we conclude that \(X \to S\) is Koszul at \(x\) as desired.

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent

\(f\) is flat and a local complete intersection morphism, and

\(f\) is syntomic.

Proof

Working affine locally this is More on Algebra, Lemma 07D3. We also give a more geometric proof.

Assume (2). By Morphisms, Lemma 01UE for every point \(x\) of \(X\) there exist affine open neighbourhoods \(U\) of \(x\) and \(V\) of \(f(x)\) such that \(f|_U : U \to V\) is standard syntomic. This means that \(U = \Spec(R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)) \to V = \Spec(R)\) where \(R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)\) is a relative global complete intersection over \(R\). By Algebra, Lemma 00SV the sequence \(f_1, \ldots, f_c\) is a regular sequence in each local ring \(R[x_1, \ldots, x_n]_{\mathfrak q}\) for every prime \(\mathfrak q \supset (f_1, \ldots, f_c)\). Consider the Koszul complex \(K_\bullet = K_\bullet(R[x_1, \ldots, x_n], f_1, \ldots, f_c)\) with homology groups \(H_i = H_i(K_\bullet)\). By More on Algebra, Lemma 062F we see that \((H_i)_{\mathfrak q} = 0\), \(i > 0\) for every \(\mathfrak q\) as above. On the other hand, by More on Algebra, Lemma 0663 we see that \(H_i\) is annihilated by \((f_1, \ldots, f_c)\). Hence we see that \(H_i = 0\), \(i > 0\) and \(f_1, \ldots, f_c\) is a Koszul-regular sequence. This proves that \(U \to V\) factors as a Koszul-regular immersion \(U \to \mathbf{A}^n_V\) followed by a smooth morphism as desired.

Assume (1). Then \(f\) is a flat and locally of finite presentation (Lemma 069H). Hence, according to Morphisms, Lemma 01UE it suffices to show that the local rings \(\mathcal{O}_{X_s, x}\) are local complete intersection rings. Choose, locally on \(X\), a factorization \(f = \pi \circ i\) for some Koszul-regular immersion \(i : X \to P\) and smooth morphism \(\pi : P \to S\). Note that \(X \to P\) is a relative quasi-regular immersion over \(S\), see Divisors, Definition 063S. Hence according to Divisors, Lemma 063U we see that \(X \to P\) is a regular immersion and the same remains true after any base change. Thus each fibre is a regular immersion, whence all the local rings of all the fibres of \(X\) are local complete intersections.

Lemma

A regular immersion of schemes is a local complete intersection morphism. A Koszul-regular immersion of schemes is a local complete intersection morphism.

Proof

Since a regular immersion is a Koszul-regular immersion, see Divisors, Lemma 063K, it suffices to prove the second statement. The second statement follows immediately from the definition.

Lemma

Let \[\xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S }\] be a commutative diagram of morphisms of schemes. Assume \(Y \to S\) smooth and \(X \to S\) is a local complete intersection morphism. Then \(f : X \to Y\) is a local complete intersection morphism.

Proof

Immediate from the definitions.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. If \(f\) is locally of finite type and \(X\) and \(Y\) are regular, then \(f\) is a local complete intersection morphism.

Proof

We may assume there is a factorization \(X \to \mathbf{A}^n_Y \to Y\) where the first arrow is an immersion. As \(Y\) is regular also \(\mathbf{A}^n_Y\) is regular by Algebra, Lemma 07NF. Hence \(X \to \mathbf{A}^n_Y\) is a regular immersion by Divisors, Lemma 0E9J.

The following lemma is of a different nature.

Lemma

Let \[\xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S }\] be a commutative diagram of morphisms of schemes. Assume

\(S\) is locally Noetherian,

\(Y \to S\) is locally of finite type,

\(f : X \to Y\) is perfect,

\(X \to S\) is a local complete intersection morphism.

Then \(X \to Y\) is a local complete intersection morphism and \(Y \to S\) is Koszul at \(f(x)\) for all \(x \in X\).

Proof

In the course of this proof all schemes will be locally Noetherian and all rings will be Noetherian. We will use without further mention that regular sequences and Koszul regular sequences agree in this setting, see More on Algebra, Lemma 09CC. Moreover, whether an ideal (resp. ideal sheaf) is regular may be checked on local rings (resp. stalks), see Algebra, Lemma 061L (resp. Divisors, Lemma 063I)

The question is local. Hence we may assume \(S\), \(X\), \(Y\) are affine. In this situation we may choose a commutative diagram \[\xymatrix{ \mathbf{A}^{n + m}_S \ar[d] & X \ar[l] \ar[d] \\ \mathbf{A}^n_S \ar[d] & Y \ar[l] \ar[ld] \\ S }\] whose horizontal arrows are closed immersions. Let \(x \in X\) be a point and consider the corresponding commutative diagram of local rings \[\xymatrix{ J \ar[r] & \mathcal{O}_{\mathbf{A}^{n + m}_S, x} \ar[r] & \mathcal{O}_{X, x} \\ I \ar[r] \ar[u] & \mathcal{O}_{\mathbf{A}^n_S, f(x)} \ar[r] \ar[u] & \mathcal{O}_{Y, f(x)} \ar[u] }\] where \(J\) and \(I\) are the kernels of the horizontal arrows. Since \(X \to S\) is a local complete intersection morphism, the ideal \(J\) is generated by a regular sequence. Since \(X \to Y\) is perfect the ring \(\mathcal{O}_{X, x}\) has finite tor dimension over \(\mathcal{O}_{Y, f(x)}\). Hence we may apply Divided Power Algebra, Lemma 09PX to conclude that \(I\) and \(J/I\) are generated by regular sequences. By our initial remarks, this finishes the proof.

Lemma

Let \[\xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S }\] be a commutative diagram of morphisms of schemes. Assume \(S\) is locally Noetherian, \(Y \to S\) is locally of finite type, \(Y\) is regular, and \(X \to S\) is a local complete intersection morphism. Then \(f : X \to Y\) is a local complete intersection morphism and \(Y \to S\) is Koszul at \(f(x)\) for all \(x \in X\).

Proof

This is a special case of Lemma 09RL in view of Lemma 068B (and Morphisms, Lemma 01T8).

Lemma

Let \(i : X \to Y\) be an immersion. If

\(i\) is perfect,

\(Y\) is locally Noetherian, and

the conormal sheaf \(\mathcal{C}_{X/Y}\) is finite locally free,

then \(i\) is a regular immersion.

Proof

Translated into algebra, this is Divided Power Algebra, Proposition 0FJS.

Lemma

Let \(f : X \to Y\) be a local complete intersection homomorphism. Then the naive cotangent complex \(\NL_{X/Y}\) is a perfect object of \(D(\mathcal{O}_X)\) of tor-amplitude in \([-1, 0]\).

Proof

Translated into algebra this is More on Algebra, Lemma 0FV0. To do the translation use Lemmas 07DB and 0D0I as well as Derived Categories of Schemes, Lemmas 06Z0, 08E9 and 08EB.

Lemma

Let \(f : X \to Y\) be a perfect morphism of locally Noetherian schemes. The following are equivalent

\(f\) is a local complete intersection morphism,

\(\NL_{X/Y}\) has tor-amplitude in \([-1, 0]\), and

\(\NL_{X/Y}\) is perfect with tor-amplitude in \([-1, 0]\).

Proof

Translated into algebra this is Divided Power Algebra, Lemma 0FJT. To do the translation use Lemmas 07DB and 0D0I as well as Derived Categories of Schemes, Lemmas 06Z0, 08E9 and 08EB.

Lemma

Let \(f : X \to Y\) be a flat morphism of finite presentation. The following are equivalent

\(f\) is a local complete intersection morphism,

\(f\) is syntomic,

\(\NL_{X/Y}\) has tor-amplitude in \([-1, 0]\), and

\(\NL_{X/Y}\) is perfect with tor-amplitude in \([-1, 0]\).

Proof

Translated into algebra this is Divided Power Algebra, Lemma 0FJV. To do the translation use Lemmas 07DB and 0D0I as well as Derived Categories of Schemes, Lemmas 06Z0, 08E9 and 08EB.

The following lemma gives a characterization of smooth morphisms as flat morphisms whose diagonal is perfect.

Lemma

Let \(f : X \to Y\) be a finite type morphism of locally Noetherian schemes. Denote \(\Delta : X \to X \times_Y X\) the diagonal morphism. The following are equivalent

\(f\) is smooth,

\(f\) is flat and \(\Delta : X \to X \times_Y X\) is a regular immersion,

\(f\) is flat and \(\Delta : X \to X \times_Y X\) is a local complete intersection morphism,

\(f\) is flat and \(\Delta : X \to X \times_Y X\) is perfect.

Proof

Assume (1). Then \(f\) is flat by Morphisms, Lemma 01VF. The projections \(X \times_Y X \to X\) are smooth by Morphisms, Lemma 01VB. Hence the diagonal is a section to a smooth morphism and hence a regular immersion, see Divisors, Lemma 067R. Hence (1) \(\Rightarrow\) (2). The implication (2) \(\Rightarrow\) (3) is Lemma 069L. The implication (3) \(\Rightarrow\) (4) is Lemma 069H. The interesting implication (4) \(\Rightarrow\) (1) follows immediately from Divided Power Algebra, Lemma 0FCX.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a local complete intersection morphism” is fpqc local on the base.

Proof

Let \(f : X \to S\) be a morphism of schemes. Let \(\{S_i \to S\}\) be an fpqc covering of \(S\). Assume that each base change \(f_i : X_i \to S_i\) of \(f\) is a local complete intersection morphism. Note that this implies in particular that \(f\) is locally of finite type, see Lemma 069H and Descent, Lemma 02KX. Let \(x \in X\). Choose an open neighbourhood \(U\) of \(x\) and an immersion \(j : U \to \mathbf{A}^n_S\) over \(S\) (see discussion preceding Definition 069F). We have to show that \(j\) is a Koszul-regular immersion. Since \(f_i\) is a local complete intersection morphism, we see that the base change \(j_i : U \times_S S_i \to \mathbf{A}^n_{S_i}\) is a Koszul-regular immersion, see Lemma 069G. Because \(\{\mathbf{A}^n_{S_i} \to \mathbf{A}^n_S\}\) is a fpqc covering we see from Descent, Lemma 0694 that \(j\) is a Koszul-regular immersion as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a local complete intersection morphism” is syntomic local on the source.

Proof

We will use the criterion of Descent, Lemma 036H to prove this. It follows from Lemmas 069K and 069J that being a local complete intersection morphism is preserved under precomposing with syntomic morphisms. It is clear from Definition 069F that being a local complete intersection morphism is Zariski local on the source and target. Hence, according to the aforementioned Descent, Lemma 036H it suffices to prove the following: Suppose \(X' \to X \to Y\) are morphisms of affine schemes with \(X' \to X\) syntomic and \(X' \to Y\) a local complete intersection morphism. Then \(X \to Y\) is a local complete intersection morphism. To see this, note that in any case \(X \to Y\) is of finite presentation by Descent, Lemma 02KK. Choose a closed immersion \(X \to \mathbf{A}^n_Y\). By Algebra, Lemma 00T0 we can find an affine open covering \(X' = \bigcup_{i = 1, \ldots, n} X'_i\) and syntomic morphisms \(W_i \to \mathbf{A}^n_Y\) lifting the morphisms \(X'_i \to X\), i.e., such that there are fibre product diagrams \[\xymatrix{ X'_i \ar[d] \ar[r] & W_i \ar[d] \\ X \ar[r] & \mathbf{A}^n_Y }\] After replacing \(X'\) by \(\coprod X'_i\) and setting \(W = \coprod W_i\) we obtain a fibre product diagram of affine schemes \[\xymatrix{ X' \ar[d] \ar[r] & W \ar[d]^h \\ X \ar[r] & \mathbf{A}^n_Y }\] with \(h : W \to \mathbf{A}^n_Y\) syntomic and \(X' \to Y\) still a local complete intersection morphism. Since \(W \to \mathbf{A}^n_Y\) is open (see Morphisms, Lemma 01UA) and \(X' \to X\) is surjective we see that \(X\) is contained in the image of \(W \to \mathbf{A}^n_Y\). Choose a closed immersion \(W \to \mathbf{A}^{n + m}_Y\) over \(\mathbf{A}^n_Y\). Now the diagram looks like \[\xymatrix{ X' \ar[d] \ar[r] & W \ar[d]^h \ar[r] & \mathbf{A}^{n + m}_Y \ar[ld] \\ X \ar[r] & \mathbf{A}^n_Y }\] Because \(h\) is syntomic and hence a local complete intersection morphism (see above) the morphism \(W \to \mathbf{A}^{n + m}_Y\) is a Koszul-regular immersion. Because \(X' \to Y\) is a local complete intersection morphism the morphism \(X' \to \mathbf{A}^{n + m}_Y\) is a Koszul-regular immersion. We conclude from Divisors, Lemma 068Z that \(X' \to W\) is a Koszul-regular immersion. Hence, since being a Koszul-regular immersion is fpqc local on the target (see Descent, Lemma 0694) we conclude that \(X \to \mathbf{A}^n_Y\) is a Koszul-regular immersion which is what we had to show.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). Assume both \(X\) and \(Y\) are flat and locally of finite presentation over \(S\). Then the set \[\{x \in X \mid f\text{ Koszul at }x\}.\] is open in \(X\) and its formation commutes with arbitrary base change \(S' \to S\).

Proof

The set is open by definition (see Definition 069F). Let \(S' \to S\) be a morphism of schemes. Set \(X' = S' \times_S X\), \(Y' = S' \times_S Y\), and denote \(f' : X' \to Y'\) the base change of \(f\). Let \(x' \in X'\) be a point such that \(f'\) is Koszul at \(x'\). Denote \(s' \in S'\), \(x \in X\), \(y' \in Y'\) , \(y \in Y\), \(s \in S\) the image of \(x'\). Note that \(f\) is locally of finite presentation, see Morphisms, Lemma 02FV. Hence we may choose an affine neighbourhood \(U \subset X\) of \(x\) and an immersion \(i : U \to \mathbf{A}^n_Y\). Denote \(U' = S' \times_S U\) and \(i' : U' \to \mathbf{A}^n_{Y'}\) the base change of \(i\). The assumption that \(f'\) is Koszul at \(x'\) implies that \(i'\) is a Koszul-regular immersion in a neighbourhood of \(x'\), see Lemma 069G. The scheme \(X'\) is flat and locally of finite presentation over \(S'\) as a base change of \(X\) (see Morphisms, Lemmas 01U9 and 01TS). Hence \(i'\) is a relative \(H_1\)-regular immersion over \(S'\) in a neighbourhood of \(x'\) (see Divisors, Definition 063S). Thus the base change \(i'_{s'} : U'_{s'} \to \mathbf{A}^n_{Y'_{s'}}\) is a \(H_1\)-regular immersion in an open neighbourhood of \(x'\), see Divisors, Lemma 063R and the discussion following Divisors, Definition 063S. Since \(s' = \Spec(\kappa(s')) \to \Spec(\kappa(s)) = s\) is a surjective flat universally open morphism (see Morphisms, Lemma 0383) we conclude that the base change \(i_s : U_s \to \mathbf{A}^n_{Y_s}\) is an \(H_1\)-regular immersion in a neighbourhood of \(x\), see Descent, Lemma 0694. Finally, note that \(\mathbf{A}^n_Y\) is flat and locally of finite presentation over \(S\), hence Divisors, Lemma 063W implies that \(i\) is a (Koszul-)regular immersion in a neighbourhood of \(x\) as desired.

Lemma

Let \(f : X \to Y\) be a local complete intersection morphism of schemes. Then \(f\) is unramified if and only if \(f\) is formally unramified and in this case the conormal sheaf \(\mathcal{C}_{X/Y}\) is finite locally free on \(X\).

Proof

The first assertion follows immediately from Lemma 02HE and the fact that a local complete intersection morphism is locally of finite type. To compute the conormal sheaf of \(f\) we choose, locally on \(X\), a factorization of \(f\) as \(f = p \circ i\) where \(i : X \to V\) is a Koszul-regular immersion and \(V \to Y\) is smooth. By Lemma 067W we see that \(\mathcal{C}_{X/Y}\) is a locally direct summand of \(\mathcal{C}_{X/V}\) which is finite locally free as \(i\) is a Koszul-regular (hence quasi-regular) immersion, see Divisors, Lemma 063M.

Lemma

Let \(Z \to Y \to X\) be formally unramified morphisms of schemes. Assume that \(Z \to Y\) is a local complete intersection morphism. The exact sequence \[0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0\] of Lemma 06AE is short exact.

Proof

The question is local on \(Z\) hence we may assume there exists a factorization \(Z \to \mathbf{A}^n_Y \to Y\) of the morphism \(Z \to Y\). Then we get a commutative diagram \[\xymatrix{ Z \ar[r]_{i'} \ar@{=}[d] & \mathbf{A}^n_Y \ar[r] \ar[d] & \mathbf{A}^n_X \ar[d] \\ Z \ar[r]^i & Y \ar[r] & X }\] As \(Z \to Y\) is a local complete intersection morphism, we see that \(Z \to \mathbf{A}^n_Y\) is a Koszul-regular immersion. Hence by Divisors, Lemma 063N the sequence \[0 \to (i')^*\mathcal{C}_{\mathbf{A}^n_Y/\mathbf{A}^n_X} \to \mathcal{C}_{Z/\mathbf{A}^n_X} \to \mathcal{C}_{Z/\mathbf{A}^n_Y} \to 0\] is exact and locally split. Note that \(i^*\mathcal{C}_{Y/X} = (i')^*\mathcal{C}_{\mathbf{A}^n_Y/\mathbf{A}^n_X}\) by Lemma 04F9 and note that the diagram \[\xymatrix{ (i')^*\mathcal{C}_{\mathbf{A}^n_Y/\mathbf{A}^n_X} \ar[r] & \mathcal{C}_{Z/\mathbf{A}^n_X} \\ i^*\mathcal{C}_{Y/X} \ar[u]^{\cong} \ar[r] & \mathcal{C}_{Z/X} \ar[u] }\] is commutative. Hence the lower horizontal arrow is a locally split injection. This proves the lemma.

Exact sequences of differentials and conormal sheaves

In this section we collect some results on exact sequences of conormal sheaves and sheaves of differentials. In some sense these are all realizations of the triangle of cotangent complexes associated to a pair of composable morphisms of schemes.

Let \(g : Z \to Y\) and \(f : Y \to X\) be morphisms of schemes.

There is a canonical exact sequence \[g^*\Omega_{Y/X} \to \Omega_{Z/X} \to \Omega_{Z/Y} \to 0,\] see Morphisms, Lemma 01UX. If \(g : Z \to Y\) is smooth or more generally formally smooth, then this sequence is a short exact sequence, see Morphisms, Lemma 02K4 or see Lemma 06B6.
If \(g\) is an immersion or more generally formally unramified, then there is a canonical exact sequence \[\mathcal{C}_{Z/Y} \to g^*\Omega_{Y/X} \to \Omega_{Z/X} \to 0,\] see Morphisms, Lemma 01UZ or see Lemma 04FC. If \(f \circ g : Z \to X\) is smooth or more generally formally smooth, then this sequence is a short exact sequence, see Morphisms, Lemma 06AA or see Lemma 06B7.
If \(g\) and \(f \circ g\) are immersions or more generally formally unramified, then there is a canonical exact sequence \[\mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to g^*\Omega_{Y/X} \to 0,\] see Morphisms, Lemma 067L or see Lemma 067V. If \(f : Y \to X\) is smooth or more generally formally smooth, then this sequence is a short exact sequence, see Morphisms, Lemma 06AB or see Lemma 067W.
If \(g\) and \(f\) are immersions or more generally formally unramified, then there is a canonical exact sequence \[g^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0.\] see Morphisms, Lemma 062S or see Lemma 06AE. If \(g : Z \to Y\) is a regular immersion¹⁴ or more generally a local complete intersection morphism, then this sequence is a short exact sequence, see Divisors, Lemma 063N or see Lemma 06BA.

Weakly étale morphisms

A ring homomorphism \(A \to B\) is weakly étale if both \(A \to B\) and \(B \otimes_A B \to B\) are flat, see More on Algebra, Definition 092B. The analogous notion for morphisms of schemes is the following.

Definition

A morphism of schemes \(X \to Y\) is weakly étale or absolutely flat if both \(X \to Y\) and the diagonal morphism \(X \to X \times_Y X\) are flat.

An étale morphism is weakly étale and conversely it turns out that a weakly étale morphism is indeed somewhat like an étale morphism. For example, if \(X \to Y\) is weakly étale, then \(L_{X/Y} = 0\), as follows from Cotangent, Lemma 08R2. We will prove a very precise result relating weakly étale morphisms to étale morphisms later (see Pro-étale Cohomology, Section 097Y). In this section we stick with the basics.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent

\(X \to Y\) is weakly étale, and

for every \(x \in X\) the ring map \(\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}\) is weakly étale.

Proof

Observe that under both assumptions (1) and (2) the morphism \(f\) is flat. Thus we may assume \(f\) is flat. Let \(x \in X\) with image \(y = f(x)\) in \(Y\). There are canonical maps of rings \[\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{X \times_Y X, \Delta_{X/Y}(x)} \longrightarrow \mathcal{O}_{X, x}\] where the first map is a localization (hence flat) and the second map is a surjection. Condition (1) means that the second arrow is flat for all \(x\). Condition (2) is that the composition is flat for all \(x\). Thus the equivalence by Algebra, Lemma 00HT part (2).

Lemma

Let \(X \to Y\) be a morphism of schemes such that \(X \to X \times_Y X\) is flat. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. If \(\mathcal{F}\) is flat over \(Y\), then \(\mathcal{F}\) is flat over \(X\).

Proof

Let \(x \in X\) with image \(y = f(x)\) in \(Y\). Since \(X \to X \times_Y X\) is flat, we see that \(\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \to \mathcal{O}_{X, x}\) is flat. Hence the result follows from More on Algebra, Lemma 092C and the definitions.

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent

The morphism \(f\) is weakly étale.

For every affine opens \(U \subset X\), \(V \subset S\) with \(f(U) \subset V\) the ring map \(\mathcal{O}_S(V) \to \mathcal{O}_X(U)\) is weakly étale.

There exists an open covering \(S = \bigcup_{j \in J} V_j\) and open coverings \(f^{-1}(V_j) = \bigcup_{i \in I_j} U_i\) such that each of the morphisms \(U_i \to V_j\), \(j\in J, i\in I_j\) is weakly étale.

There exists an affine open covering \(S = \bigcup_{j \in J} V_j\) and affine open coverings \(f^{-1}(V_j) = \bigcup_{i \in I_j} U_i\) such that the ring map \(\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)\) is of weakly étale, for all \(j\in J, i\in I_j\).

Moreover, if \(f\) is weakly étale then for any open subschemes \(U \subset X\), \(V \subset S\) with \(f(U) \subset V\) the restriction \(f|_U : U \to V\) is weakly-étale.

Proof

Suppose given open subschemes \(U \subset X\), \(V \subset S\) with \(f(U) \subset V\). Then \(U \times_V U \subset X \times_Y X\) is open (Schemes, Lemma 01JR) and the diagonal \(\Delta_{U/V}\) of \(f|_U : U \to V\) is the restriction \(\Delta_{X/Y}|_U : U \to U \times_V U\). Since flatness is a local property of morphisms of schemes (Morphisms, Lemma 01U5) the final statement of the lemma is follows as well as the equivalence of (1) and (3). If \(X\) and \(Y\) are affine, then \(X \to Y\) is weakly étale if and only if \(\mathcal{O}_Y(Y) \to \mathcal{O}_X(X)\) is weakly étale (use again Morphisms, Lemma 01U5). Thus (1) and (3) are also equivalent to (2) and (4).

Lemma

Let \(X \to Y \to Z\) be morphisms of schemes.

If \(X \to X \times_Y X\) and \(Y \to Y \times_Z Y\) are flat, then \(X \to X \times_Z X\) is flat.

If \(X \to Y\) and \(Y \to Z\) are weakly étale, then \(X \to Z\) is weakly étale.

Proof

Part (1) follows from the factorization \[X \to X \times_Y X \to X \times_Z X\] of the diagonal of \(X\) over \(Z\), the fact that \[X \times_Y X = (X \times_Z X) \times_{(Y \times_Z Y)} Y,\] the fact that a base change of a flat morphism is flat, and the fact that the composition of flat morphisms is flat (Morphisms, Lemmas 01U9 and 01U7). Part (2) follows from part (1) and the fact (just used) that the composition of flat morphisms is flat.

Lemma

Let \(X \to Y\) and \(Y' \to Y\) be morphisms of schemes and let \(X' = Y' \times_Y X\) be the base change of \(X\).

If \(X \to X \times_Y X\) is flat, then \(X' \to X' \times_{Y'} X'\) is flat.

If \(X \to Y\) is weakly étale, then \(X' \to Y'\) is weakly étale.

Proof

Assume \(X \to X \times_Y X\) is flat. The morphism \(X' \to X' \times_{Y'} X'\) is the base change of \(X \to X \times_Y X\) by \(Y' \to Y\). Hence it is flat by Morphisms, Lemmas 01U9. This proves (1). Part (2) follows from (1) and the fact (just used) that the base change of a flat morphism is flat.

Lemma

Let \(X \to Y \to Z\) be morphisms of schemes. Assume that \(X \to Y\) is flat and surjective and that \(X \to X \times_Z X\) is flat. Then \(Y \to Y \times_Z Y\) is flat.

Proof

Consider the commutative diagram \[\xymatrix{ X \ar[r] \ar[d] & X \times_Z X \ar[d] \\ Y \ar[r] & Y \times_Z Y }\] The top horizontal arrow is flat and the vertical arrows are flat. Hence \(X\) is flat over \(Y \times_Z Y\). By Morphisms, Lemma 02JZ we see that \(Y\) is flat over \(Y \times_Z Y\).

Lemma

Let \(f : X \to Y\) be a weakly étale morphism of schemes. Then \(f\) is formally unramified, i.e., \(\Omega_{X/Y} = 0\).

Proof

Recall that \(f\) is formally unramified if and only if \(\Omega_{X/Y} = 0\) by Lemma 02H9. Via Lemma 094S and Morphisms, Lemma 01UT this follows from the case of rings which is More on Algebra, Lemma 092M.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Then \(X \to Y\) is weakly étale in each of the following cases

\(X \to Y\) is a flat monomorphism,

\(X \to Y\) is an open immersion,

\(X \to Y\) is flat and unramified,

\(X \to Y\) is étale.

Proof

If (1) holds, then \(\Delta_{X/Y}\) is an isomorphism (Schemes, Lemma 01L3), hence certainly \(f\) is weakly étale. Case (2) is a special case of (1). The diagonal of an unramified morphism is an open immersion (Morphisms, Lemma 02GE), hence flat. Thus a flat unramified morphism is weakly étale. An étale morphism is flat and unramified (Morphisms, Lemma 02GK), hence (4) follows from (3).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. If \(Y\) is reduced and \(f\) weakly étale, then \(X\) is reduced.

Proof

Via Lemma 094S this follows from the case of rings which is More on Algebra, Lemma 092I.

The following lemma uses a nontrivial result about weakly étale ring maps.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent

\(f\) is weakly étale, and

for \(x \in X\) the local ring map \(\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}\) induces an isomorphism on strict henselizations.

Proof

Let \(x \in X\) be a point with image \(y = f(x)\) in \(Y\). Choose a separable algebraic closure \(\kappa^{sep}\) of \(\kappa(x)\). Let \(\mathcal{O}_{X, x}^{sh}\) be the strict henselization corresponding to \(\kappa^{sep}\) and \(\mathcal{O}_{Y, y}^{sh}\) the strict henselization relative to the separable algebraic closure of \(\kappa(y)\) in \(\kappa^{sep}\). Consider the commutative diagram \[\xymatrix{ \mathcal{O}_{X, x} \ar[r] & \mathcal{O}_{X, x}^{sh} \\ \mathcal{O}_{Y, y} \ar[u] \ar[r] & \mathcal{O}_{Y, y}^{sh} \ar[u] }\] local homomorphisms of local rings, see Algebra, Lemma 04GU. Since the strict henselization is a filtered colimit of étale ring maps, More on Algebra, Lemma 092N shows the horizontal maps are weakly étale. Moreover, the horizontal maps are faithfully flat by More on Algebra, Lemma 07QM.

Assume \(f\) weakly étale. By Lemma 094Q the left vertical arrow is weakly étale. By More on Algebra, Lemmas 092J and 092L the right vertical arrow is weakly étale. By More on Algebra, Theorem 092Z we conclude the right vertical map is an isomorphism.

Assume \(\mathcal{O}_{Y, y}^{sh} \to \mathcal{O}_{X, x}^{sh}\) is an isomorphism. Then \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}^{sh}\) is weakly étale. Since \(\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^{sh}\) is faithfully flat we conclude that \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\) is weakly étale by More on Algebra, Lemma 092K. Thus (2) implies (1) by Lemma 094Q.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. If \(Y\) is a normal scheme and \(f\) weakly étale, then \(X\) is a normal scheme.

Proof

By More on Algebra, Lemma 06DI a scheme \(S\) is normal if and only if for all \(s \in S\) the strict henselization of \(\mathcal{O}_{S, s}\) is a normal domain. Hence the lemma follows from Lemma 094Z.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes over \(S\). If \(X\), \(Y\) are weakly étale over \(S\), then \(f\) is weakly étale.

Proof

We will use Morphisms, Lemmas 01U9 and 01U7 without further mention. Write \(X \to Y\) as the composition \(X \to X \times_S Y \to Y\). The second morphism is flat as the base change of the flat morphism \(X \to S\). The first is the base change of the flat morphism \(Y \to Y \times_S Y\) by the morphism \(X \times_S Y \to Y \times_S Y\), hence flat. Thus \(X \to Y\) is flat. The morphism \(X \times_Y X \to X \times_S X\) is an immersion. Thus Lemma 094R implies, that since \(X\) is flat over \(X \times_S X\) it follows that \(X\) is flat over \(X \times_Y X\).

The following is a scheme theoretic generalization of the observation that a field extension that is simultaneously separable and purely inseparable must be an isomorphism.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. If \(f\) is weakly étale and a universal homeomorphism, it is an isomorphism.

Proof

Since \(f\) is a universal homeomorphism, the diagonal \(\Delta : X \to X \times_Y X\) is a surjective closed immersion by Morphisms, Lemmas 04DE and 01S4. Since \(\Delta\) is also flat, we see that \(\Delta\) must be an isomorphism by Morphisms, Lemma 04PW. In other words, \(f\) is a monomorphism (Schemes, Lemma 01L3). Since \(f\) is a universal homeomorphism it is certainly quasi-compact. Hence by Descent, Lemma 06NC we find that \(f\) is an isomorphism.

The following is a weakly étale generalization of Étale Morphisms, Lemma 0EBS.

Lemma

Let \(U \to X\) be a weakly étale morphism of schemes where \(X\) is a scheme in characteristic \(p\). Then the relative Frobenius \(F_{U/X} : U \to U \times_{X, F_X} X\) is an isomorphism.

Proof

The morphism \(F_{U/X}\) is a universal homeomorphism by Varieties, Lemma 0CCB. The morphism \(F_{U/X}\) is weakly étale as a morphism between schemes weakly étale over \(X\) by Lemma 0951. Hence \(F_{U/X}\) is an isomorphism by Lemma 0F6V.

Reduced fibre theorem

In this section we discuss the simplest kind of theorem of the kind advertised by the title. Although the proof of the result is kind of laborious, in essence it follows in a straightforward manner from Epp’s result on eliminating ramification, see More on Algebra, Theorem 09F9.

Let \(A\) be a Dedekind domain with fraction field \(K\). Let \(X\) be a scheme flat and of finite type over \(A\). Let \(L\) be a finite extension of \(K\). Let \(B\) be the integral closure of \(A\) in \(L\). Then \(B\) is a Dedekind domain (Algebra, Lemma 09IG). Let \(X_B = X \times_{\Spec(A)} \Spec(B)\) be the base change. Then \(X_B \to \Spec(B)\) is of finite type (Morphisms, Lemma 01T4). Hence \(X_B\) is Noetherian (Morphisms, Lemma 01T6). Thus the normalization \(\nu : Y \to X_B\) exists (see Morphisms, Definition 035N and the discussion following). Picture [09IK]\[\begin{equation} \xymatrix{ Y \ar[rd] \ar[r]_\nu & X_B \ar[r] \ar[d] & X \ar[d] \\ & \Spec(B) \ar[r] & \Spec(A) } \end{equation}\] We sometimes call \(Y\) the normalized base change of \(X\). In general the morphism \(\nu\) may not be finite. But if \(A\) is a Nagata ring (a condition that is virtually always satisfied in practice) then \(\nu\) is finite and \(Y\) is of finite type over \(B\), see Morphisms, Lemmas 035S and 035A.

Taking the normalized base change commutes with composition. More precisely, if \(M/L/K\) are finite extensions of fields with integral closures \(A \subset B \subset C\) then the normalized base change \(Z\) of \(Y \to \Spec(B)\) relative to \(M/L\) is equal to the normalized base change of \(X \to \Spec(A)\) relative to \(M/K\).

Theorem

Let \(A\) be a Dedekind ring with fraction field \(K\). Let \(X\) be a scheme flat and of finite type over \(A\). Assume \(A\) is a Nagata ring. There exists a finite extension \(L/K\) such that the normalized base change \(Y\) is smooth over \(\Spec(B)\) at all generic points of all fibres.

Proof

During the proof we will repeatedly use that formation of the set of points where a (flat, finitely presented) morphism like \(X \to \Spec(A)\) is smooth commutes with base change, see Morphisms, Lemma 02V4.

We first choose a finite extension \(L/K\) such that \((X_L)_{red}\) is geometrically reduced over \(L\), see Varieties, Lemma 04KT. Since \(Y \to (X_B)_{red}\) is birational we see applying Varieties, Lemma 04KS that \(Y_L\) is geometrically reduced over \(L\) as well. Hence \(Y_L \to \Spec(L)\) is smooth on a dense open \(V \subset Y_L\) by Varieties, Lemma 056V. Thus the smooth locus \(U \subset Y\) of the morphism \(Y \to \Spec(B)\) is open (by Morphisms, Definition 01V5) and is dense in the generic fibre. Replacing \(A\) by \(B\) and \(X\) by \(Y\) we reduce to the case treated in the next paragraph.

Assume \(X\) is normal and the smooth locus \(U \subset X\) of \(X \to \Spec(A)\) is dense in the generic fibre. This implies that \(U\) is dense in all but finitely many fibres, see Lemma 054X. Let \(x_1, \ldots, x_r \in X \setminus U\) be the finitely many generic points of irreducible components of \(X \setminus U\) which are moreover generic points of irreducible components of fibres of \(X \to \Spec(A)\). Set \(\mathcal{O}_i = \mathcal{O}_{X, x_i}\). Let \(A_i\) be the localization of \(A\) at the maximal ideal corresponding to the image of \(x_i\) in \(\Spec(A)\). By More on Algebra, Proposition 09II there exist finite extensions \(K_i/K\) which are solutions for the extension of discrete valuation rings \(A_i \to \mathcal{O}_i\). Let \(L/K\) be a finite extension dominating all of the extensions \(K_i/K\). Then \(L/K\) is still a solution for \(A_i \to \mathcal{O}_i\) by More on Algebra, Lemma 0GLR.

Consider the diagram (09IK) with the extension \(L/K\) we just produced. Note that \(U_B \subset X_B\) is smooth over \(B\), hence normal (for example use Algebra, Lemma 033C). Thus \(Y \to X_B\) is an isomorphism over \(U_B\). Let \(y \in Y\) be a generic point of an irreducible component of a fibre of \(Y \to \Spec(B)\) lying over the maximal ideal \(\mathfrak m \subset B\). Assume that \(y \not \in U_B\). Then \(y\) maps to one of the points \(x_i\). It follows that \(\mathcal{O}_{Y, y}\) is a local ring of the integral closure of \(\mathcal{O}_i\) in \(R(X) \otimes_K L\) (details omitted). Hence because \(L/K\) is a solution for \(A_i \to \mathcal{O}_i\) we see that \(B_\mathfrak m \to \mathcal{O}_{Y, y}\) is formally smooth in the \(\mathfrak m_y\)-adic topology (this is the definition of being a "solution"). In other words, \(\mathfrak m\mathcal{O}_{Y, y} = \mathfrak m_y\) and the residue field extension is separable, see More on Algebra, Lemma 09E7. Hence the local ring of the fibre at \(y\) is \(\kappa(y)\). This implies the fibre is smooth over \(\kappa(\mathfrak m)\) at \(y\) for example by Algebra, Lemma 00TV. This finishes the proof.

Lemma

Let \(f : X \to S\) be a flat, finite type morphism of schemes. Assume \(S\) is Nagata, integral with function field \(K\), and regular of dimension \(1\). Then there exists a finite extension \(L/K\) such that in the diagram \[\xymatrix{ Y \ar[rd]_g \ar[r]_-\nu & X \times_S T \ar[d] \ar[r] & X \ar[d]_f \\ & T \ar[r] & S }\] the morphism \(g\) is smooth at all generic points of fibres. Here \(T\) is the normalization of \(S\) in \(\Spec(L)\) and \(\nu : Y \to X \times_S T\) is the normalization.

Proof

Choose a finite affine open covering \(S = \bigcup \Spec(A_i)\). Then \(K\) is equal to the fraction field of \(A_i\) for all \(i\). Let \(X_i = X \times_S \Spec(A_i)\). Choose \(L_i/K\) as in Theorem 09IL for the morphism \(X_i \to \Spec(A_i)\). Let \(B_i \subset L_i\) be the integral closure of \(A_i\) and let \(Y_i\) be the normalized base change of \(X\) to \(B_i\). Let \(L/K\) be a finite extension dominating each \(L_i\). Let \(T_i \subset T\) be the inverse image of \(\Spec(A_i)\). For each \(i\) we get a commutative diagram \[\xymatrix{ g^{-1}(T_i) \ar[r] \ar[d] & Y_i \ar[r] \ar[d] & X \times_S \Spec(A_i) \ar[d] \\ T_i \ar[r] & \Spec(B_i) \ar[r] & \Spec(A_i) }\] and in fact the left hand square is a normalized base change as discussed at the beginning of the section. In the proof of Theorem 09IL we have seen that the smooth locus of \(Y \to T\) contains the inverse image in \(g^{-1}(T_i)\) of the set of points where \(Y_i\) is smooth over \(B_i\). This proves the lemma.

Lemma

Let \(A\) be a Dedekind ring with fraction field \(K\). Let \(X\) be a scheme flat and of finite type over \(A\). Assume \(A\) is a Nagata ring and that for every generic point \(\eta\) of an irreducible component of \(X\) the field extension \(\kappa(\eta)/K\) is separable. Then there exists a finite separable extension \(L/K\) such that the normalized base change \(Y\) is smooth over \(\Spec(B)\) at all generic points of all fibres.

Proof

This is proved in exactly the same manner as Theorem 09IL with a few minor modifications. The most important change is to use More on Algebra, Lemma 0BRP instead of More on Algebra, Proposition 09II. During the proof we will repeatedly use that formation of the set of points where a (flat, finitely presented) morphism like \(X \to \Spec(A)\) is smooth commutes with base change, see Morphisms, Lemma 02V4.

Since \(X\) is flat over \(A\) every generic point \(\eta\) of \(X\) maps to the generic point of \(\Spec(A)\). After replacing \(X\) by its reduction we may assume \(X\) is reduced. In this case \(X_K\) is geometrically reduced over \(K\) by Varieties, Lemma 04KS. Hence \(X_K \to \Spec(K)\) is smooth on a dense open by Varieties, Lemma 056V. Thus the smooth locus \(U \subset X\) of the morphism \(X \to \Spec(A)\) is open (by Morphisms, Definition 01V5) and is dense in the generic fibre. This reduces us to the situation of the following paragraph.

Assume \(X\) is normal and the smooth locus \(U \subset X\) of \(X \to \Spec(A)\) is dense in the generic fibre. This implies that \(U\) is dense in all but finitely many fibres, see Lemma 054X. Let \(x_1, \ldots, x_r \in X \setminus U\) be the finitely many generic points of irreducible components of \(X \setminus U\) which are moreover generic points of irreducible components of fibres of \(X \to \Spec(A)\). Set \(\mathcal{O}_i = \mathcal{O}_{X, x_i}\). Observe that the fraction field of \(\mathcal{O}_i\) is the residue field of a generic point of \(X\). Let \(A_i\) be the localization of \(A\) at the maximal ideal corresponding to the image of \(x_i\) in \(\Spec(A)\). We may apply More on Algebra, Lemma 0BRP and we find finite separable extensions \(K_i/K\) which are solutions for \(A_i \to \mathcal{O}_i\). Let \(L/K\) be a finite separable extension dominating all of the extensions \(K_i/K\). Then \(L/K\) is still a solution for \(A_i \to \mathcal{O}_i\) by More on Algebra, Lemma 0GLR.

Consider the diagram (09IK) with the extension \(L/K\) we just produced. Note that \(U_B \subset X_B\) is smooth over \(B\), hence normal (for example use Algebra, Lemma 033C). Thus \(Y \to X_B\) is an isomorphism over \(U_B\). Let \(y \in Y\) be a generic point of an irreducible component of a fibre of \(Y \to \Spec(B)\) lying over the maximal ideal \(\mathfrak m \subset B\). Assume that \(y \not \in U_B\). Then \(y\) maps to one of the points \(x_i\). It follows that \(\mathcal{O}_{Y, y}\) is a local ring of the integral closure of \(\mathcal{O}_i\) in \(R(X) \otimes_K L\) (details omitted). Hence because \(L/K\) is a solution for \(A_i \to \mathcal{O}_i\) we see that \(B_\mathfrak m \to \mathcal{O}_{Y, y}\) is formally smooth (this is the definition of being a "solution"). In other words, \(\mathfrak m\mathcal{O}_{Y, y} = \mathfrak m_y\) and the residue field extension is separable. Hence the local ring of the fibre at \(y\) is \(\kappa(y)\). This implies the fibre is smooth over \(\kappa(\mathfrak m)\) at \(y\) for example by Algebra, Lemma 00TV. This finishes the proof.

Lemma

Let \(f : X \to S\) be a flat, finite type morphism of schemes. Assume \(S\) is Nagata, integral with function field \(K\), and regular of dimension \(1\). Assume the field extensions \(\kappa(\eta)/K\) are separable for every generic point \(\eta\) of an irreducible component of \(X\). Then there exists a finite separable extension \(L/K\) such that in the diagram \[\xymatrix{ Y \ar[rd]_g \ar[r]_-\nu & X \times_S T \ar[d] \ar[r] & X \ar[d]_f \\ & T \ar[r] & S }\] the morphism \(g\) is smooth at all generic points of fibres. Here \(T\) is the normalization of \(S\) in \(\Spec(L)\) and \(\nu : Y \to X \times_S T\) is the normalization.

Proof

This follows from Lemma 0BRR in exactly the same manner that Lemma 0BRQ follows from Theorem 09IL.

Ind-quasi-affine morphisms

A bit of theory to be used later.

Definition

A scheme \(X\) is ind-quasi-affine if every quasi-compact open of \(X\) is quasi-affine. Similarly, a morphism of schemes \(X \to Y\) is ind-quasi-affine if \(f^{-1}(V)\) is ind-quasi-affine for each affine open \(V\) in \(Y\).

An example of an ind-quasi-affine scheme is an open of an affine scheme. If \(X = \bigcup_{i \in I} U_i\) is a union of quasi-affine opens such that any two \(U_i\) are contained in a third, then \(X\) is ind-quasi-affine. An ind-quasi-affine scheme \(X\) is separated because any two affine opens \(U, V\) are contained in a separated open subscheme of \(X\), namely \(U \cup V\). Similarly an ind-quasi-affine morphism is separated.

Lemma

For a morphism of schemes \(f : X \to Y\), the following are equivalent:

\(f\) is ind-quasi-affine,

for every affine open subscheme \(V \subset Y\) and every quasi-compact open subscheme \(U \subset f^{-1}(V)\), the induced morphism \(U \to V\) is quasi-affine.

for some cover \(\{ V_j \}_{j \in J}\) of \(Y\) by quasi-compact and quasi-separated open subschemes \(V_j \subset Y\), every \(j \in J\), and every quasi-compact open subscheme \(U \subset f^{-1}(V_j)\), the induced morphism \(U \to V_j\) is quasi-affine.

for every quasi-compact and quasi-separated open subscheme \(V \subset Y\) and every quasi-compact open subscheme \(U \subset f^{-1}(V)\), the induced morphism \(U \to V\) is quasi-affine.

In particular, the property of being an ind-quasi-affine morphism is Zariski local on the base.

Proof

The equivalence (1) \(\Leftrightarrow\) (2) follows from the definitions and Morphisms, Lemma 01SM. For (2) \(\Rightarrow\) (4), let \(U\) and \(V\) be as in (4). By Schemes, Lemma 03GI, the induced morphism \(U \to V\) is quasi-compact. Thus, for every affine open \(V' \subset V\), the fiber product \(V' \times_V U\) is quasi-compact, so, by (2), the induced map \(V' \times_V U \to V'\) is quasi-affine. Thus, \(U \to V\) is also quasi-affine by Morphisms, Lemma 01SM. This argument also gives (3) \(\Rightarrow\) (4): indeed, keeping the same notation, those affine opens \(V' \subset V\) that lie in one of the \(V_j\) cover \(V\), so one needs to argue that the quasi-compact map \(V' \times_V U \to V'\) is quasi-affine. However, by (3), the composition \(V' \times_V U \to V' \to V_j\) is quasi-affine and, by Schemes, Lemma 01KV, the map \(V' \to V_j\) is quasi-separated. Thus, \(V' \times_V U \to V'\) is quasi-affine by Morphisms, Lemma 054G. The final implications (4) \(\Rightarrow\) (2) and (4) \(\Rightarrow\) (3) are evident.

Lemma

The property of being an ind-quasi-affine morphism is stable under composition.

Proof

Let \(f : X \to Y\) and \(g : Y \to Z\) be ind-quasi-affine morphisms. Let \(V \subset Z\) and \(U \subset f^{-1}(g^{-1}(V))\) be quasi-compact opens such that \(V\) is also quasi-separated. The image \(f(U)\) is a quasi-compact subset of \(g^{-1}(V)\), so it is contained in some quasi-compact open \(W \subset g^{-1}(V)\) (a union of finitely many affines). We obtain a factorization \(U \to W \to V\). The map \(W \to V\) is quasi-affine by Lemma 0F1U, so, in particular, \(W\) is quasi-separated. Then, by Lemma 0F1U again, \(U \to W\) is quasi-affine as well. Consequently, by Morphisms, Lemma 01SN, the composition \(U \to V\) is also quasi-affine, and it remains to apply Lemma 0F1U once more.

Lemma

Any quasi-affine morphism is ind-quasi-affine. Any immersion is ind-quasi-affine.

Proof

The first assertion is immediate from the definitions. In particular, affine morphisms, such as closed immersions, are ind-quasi-affine. Thus, by Lemma 0F1V, it remains to show that an open immersion is ind-quasi-affine. This, however, is immediate from the definitions.

Lemma

If \(f : X \to Y\) and \(g : Y \to Z\) are morphisms of schemes such that \(g \circ f\) is ind-quasi-affine, then \(f\) is ind-quasi-affine.

Proof

By Lemma 0F1U, we may work Zariski locally on \(Z\) and then on \(Y\), so we lose no generality by assuming that \(Z\), and then also \(Y\), is affine. Then any quasi-compact open of \(X\) is quasi-affine, so Lemma 0F1U gives the claim.

Lemma

The property of being ind-quasi-affine is stable under base change.

Proof

Let \(f : X \to Y\) be an ind-quasi-affine morphism. For checking that every base change of \(f\) is ind-quasi-affine, by Lemma 0F1U, we may work Zariski locally on \(Y\), so we assume that \(Y\) is affine. Furthermore, we may also assume that in the base change morphism \(Z \to Y\) the scheme \(Z\) is affine, too. The base change \(X \times_Y Z \to X\) is an affine morphism, so, by Lemmas 0F1V and 0F1W, the map \(X \times_Y Z \to Y\) is ind-quasi-affine. Then, by Lemma 0F1X, the base change \(X \times_Y Z \to Z\) is ind-quasi-affine, as desired.

Lemma

The property of being ind-quasi-affine is fpqc local on the base.

Proof

The stability of ind-quasi-affineness under base change supplied by Lemma 0AP7 gives one direction. For the other, let \(f : X \to Y\) be a morphism of schemes and let \(\{g_i : Y_i \to Y\}\) be an fpqc covering such that the base change \(f_i : X_i \to Y_i\) is ind-quasi-affine for all \(i\). We need to show \(f\) is ind-quasi-affine.

By Lemma 0F1U, we may work Zariski locally on \(Y\), so we assume that \(Y\) is affine. Then we use stability under base change ensured by Lemma 0AP7 to refine the cover and assume that it is given by a single affine, faithfully flat morphism \(g : Y' \to Y\). For any quasi-compact open \(U \subset X\), its \(Y'\)-base change \(U \times_Y Y' \subset X \times_Y Y'\) is also quasi-compact. It remains to observe that, by Descent, Lemma 02L7, the map \(U \to Y\) is quasi-affine if and only if so is \(U \times_Y Y' \to Y'\).

Lemma

A separated locally quasi-finite morphism of schemes is ind-quasi-affine.

Proof

Let \(f : X \to Y\) be a separated locally quasi-finite morphism of schemes. Let \(V \subset Y\) be affine and \(U \subset f^{-1}(V)\) quasi-compact open. We have to show \(U\) is quasi-affine. Since \(U \to V\) is a separated quasi-finite morphism of schemes, this follows from Zariski’s Main Theorem. See Lemma 02LR.

Pushouts in the category of schemes, II

This section is a continuation of Section 07RS. In this section we construct pushouts of \(Y \leftarrow Z \rightarrow X\) where \(Z \to X\) is a closed immersion and \(Z \to Y\) is integral and an additional condition is satisfied. Please see the detailed discussion in [Ferrand-Conducteur].

Situation

Here \(S\) is a scheme and \(i : Z \to X\) and \(j : Z \to Y\) are morphisms of schemes over \(S\). We assume

\(i\) is a closed immersion,

\(j\) is an integral morphism of schemes,

for \(y \in Y\) there exists an affine open \(U \subset X\) with \(j^{-1}(\{y\}) \subset i^{-1}(U)\).

Lemma

In Situation 0ECI then for \(y \in Y\) there exist affine opens \(U \subset X\) and \(V \subset Y\) with \(i^{-1}(U) = j^{-1}(V)\) and \(y \in V\).

Proof

Let \(y \in Y\). Choose an affine open \(U \subset X\) such that \(j^{-1}(\{y\}) \subset i^{-1}(U)\) (possible by assumption). Choose an affine open \(V \subset Y\) neighbourhood of \(y\) such that \(j^{-1}(V) \subset i^{-1}(U)\). This is possible because \(j : Z \to Y\) is a closed morphism (Morphisms, Lemma 01WM) and \(i^{-1}(U)\) contains the fibre over \(y\). Since \(j\) is integral, the scheme theoretic fibre \(Z_y\) is the spectrum of an algebra integral over a field. By Limits, Lemma 0E21 we can find an \(\overline{f} \in \Gamma(i^{-1}(U), \mathcal{O}_{i^{-1}(U)})\) such that \(Z_y \subset D(\overline{f}) \subset j^{-1}(V)\). Since \(i|_{i^{-1}(U)} : i^{-1}(U) \to U\) is a closed immersion of affines, we can choose an \(f \in \Gamma(U, \mathcal{O}_U)\) whose restriction to \(i^{-1}(U)\) is \(\overline{f}\). After replacing \(U\) by the principal open \(D(f) \subset U\) we find affine opens \(y \in V \subset Y\) and \(U \subset X\) with \[j^{-1}(\{y\}) \subset i^{-1}(U) \subset j^{-1}(V)\] Now we (in some sense) repeat the argument. Namely, we choose \(g \in \Gamma(V, \mathcal{O}_V)\) such that \(y \in D(g)\) and \(j^{-1}(D(g)) \subset i^{-1}(U)\) (possible by the same argument as above). Then we can pick \(f \in \Gamma(U, \mathcal{O}_U)\) whose restriction to \(i^{-1}(U)\) is the pullback of \(g\) by \(i^{-1}(U) \to V\) (again possible by the same reason as above). Then we finally have affine opens \(y \in V' = D(g) \subset V \subset Y\) and \(U' = D(f) \subset U \subset X\) with \(j^{-1}(V') = i^{-1}(V')\).

Proposition

In Situation 0ECI the pushout \(Y \amalg_Z X\) exists in the category of schemes. Picture \[\xymatrix{ Z \ar[r]_i \ar[d]_j & X \ar[d]^a \\ Y \ar[r]^-b & Y \amalg_Z X }\] The diagram is a fibre square, the morphism \(a\) is integral, the morphism \(b\) is a closed immersion, and \[\mathcal{O}_{Y \amalg_Z X} = b_*\mathcal{O}_Y \times_{c_*\mathcal{O}_Z} a_*\mathcal{O}_X\] as sheaves of rings where \(c = a \circ i = b \circ j\).

Proof

As a topological space we set \(Y \amalg_Z X\) equal to the pushout of the diagram in the category of topological spaces (Topology, Section 0B1W). This is just the pushout of the underlying sets (Topology, Lemma 0B1X) endowed with the quotient topology. On \(Y \amalg_Z X\) we have the maps of sheaves of rings \[b_*\mathcal{O}_Y \longrightarrow c_*\mathcal{O}_Z \longleftarrow a_*\mathcal{O}_X\] and we can define \[\mathcal{O}_{Y \amalg_Z X} = b_*\mathcal{O}_Y \times_{c_*\mathcal{O}_Z} a_*\mathcal{O}_X\] as the fibre product in the category of sheaves of rings. To prove that we obtain a scheme we have to show that every point has an affine open neighbourhood. This is clear for points not in the image of \(c\) as the image of \(c\) is a closed subset whose complement is isomorphic as a ringed space to \((Y \setminus j(Z)) \amalg (X \setminus i(Z))\).

A point in the image of \(c\) corresponds to a unique \(y \in Y\) in the image of \(j\). By Lemma 0ECJ we find affine opens \(U \subset X\) and \(V \subset Y\) with \(y \in V\) and \(i^{-1}(U) = j^{-1}(V)\). Since the construction of the first paragraph is clearly compatible with restriction to compatible open subschemes, to prove that it produces a scheme we may assume \(X\), \(Y\), and \(Z\) are affine.

If \(X = \Spec(A)\), \(Y = \Spec(B)\), and \(Z = \Spec(C)\) are affine, then More on Algebra, Lemma 0B7J shows that \(Y \amalg_Z X = \Spec(B \times_C A)\) as topological spaces. To finish the proof that \(Y \times_Z X\) is a scheme, it suffices to show that on \(\Spec(B \times_C A)\) the structure sheaf is the fibre product of the pushforwards. This follows by applying More on Algebra, Lemma 01Z8 to principal affine opens of \(\Spec(B \times_C A)\).

The discussion above shows the scheme \(Y \amalg_Z X\) has an affine open covering \(Y \amalg_Z X = \bigcup W_i\) such that \(U_i = a^{-1}(W_i)\), \(V_i = b^{-1}(W_i)\), and \(\Omega_i = c^{-1}(W_i)\) are affine open in \(X\), \(Y\), and \(Z\). Thus \(a\) and \(b\) are affine. Moreover, if \(A_i\), \(B_i\), \(C_i\) are the rings corresponding to \(U_i\), \(V_i\), \(\Omega_i\), then \(A_i \to C_i\) is surjective and \(W_i\) corresponds to \(A_i \times_{C_i} B_i\) which surjects onto \(B_i\). Hence \(b\) is a closed immersion. The ring map \(A_i \times_{C_i} B_i \to A_i\) is integral by More on Algebra, Lemma 0E1S hence \(a\) is integral. The diagram is cartesian because \[C_i \cong B_i \otimes_{B_i \times_{C_i} A_i} A_i\] This follows as \(B_i \times_{C_i} A_i \to B_i\) and \(A_i \to C_i\) are surjective maps whose kernels are the same.

Finally, we can apply Lemmas 0ET0 and 0BMP to conclude our construction is a pushout in the category of schemes.

Lemma

In Situation 0ECI. If \(X\) and \(Y\) are separated, then the pushout \(Y \amalg_Z X\) (Proposition 0E25) is separated. Same with “separated over \(S\)”, “quasi-separated”, and “quasi-separated over \(S\)”.

Proof

The morphism \(Y \amalg X \to Y \amalg_Z X\) is surjective and universally closed. Thus we may apply Morphisms, Lemma 09MQ.

Lemma

In Situation 0ECI assume \(S\) is a locally Noetherian scheme and \(X\), \(Y\), and \(Z\) are locally of finite type over \(S\). Then the pushout \(Y \amalg_Z X\) (Proposition 0E25) is locally of finite type over \(S\).

Proof

Looking on affine opens we recover the result of More on Algebra, Lemma 00IT.

Lemma

In Situation 0ECI suppose given a commutative diagram \[\xymatrix{ Y' \ar[d]^g & Z' \ar[l]^{j'} \ar[r]_{i'} \ar[d]^h & X' \ar[d]^f \\ Y & Z \ar[l] \ar[r] & X }\] with cartesian squares and \(f, g, h\) separated and locally quasi-finite. Then

the pushouts \(Y \amalg_Z X\) and \(Y' \amalg_{Z'} X'\) exist,

\(Y' \amalg_{Z'} X' \to Y \amalg_Z X\) is separated and locally quasi-finite, and

the squares \[\xymatrix{ Y' \ar[r] \ar[d] & Y' \amalg_{Z'} X' \ar[d] & X' \ar[l] \ar[d] \\ Y \ar[r] & Y \amalg_Z X & X \ar[l] }\] are cartesian.

Proof

The pushout \(Y \amalg_Z X\) exists by Proposition 0E25. To see that the pushout \(Y' \amalg_{Z'} X'\) exists, we check condition (3) of Situation 0ECI holds for \((X', Y', Z', i', j')\). Namely, let \(y' \in Y'\) and denote \(y \in Y\) the image. Choose \(U \subset X\) affine open with \(i(j^{-1}(y)) \subset U\). Choose a quasi-compact open \(U' \subset X'\) contained in \(f^{-1}(U)\) containing the quasi-compact subset \(i'((j')^{-1}(\{y'\}))\). By Lemma 0AP9 we see that \(U'\) is quasi-affine. Since \(Z'_{y'}\) is the spectrum of an algebra integral over a field, we can apply Limits, Lemma 0E21 and we find there exists an affine open subscheme of \(U'\) containing \(i'((j')^{-1}(\{y'\}))\) as desired.

Having verified existence we check the other assertions. Affine locally we are exactly in the situation of More on Algebra, Lemma 08KQ with \(B \to D\) and \(A' \to C'\) locally quasi-finite¹⁵. In particular, the morphism \(Y' \amalg_{Z'} X' \to Y \amalg_Z X\) is locally of finite type. The squares in of the diagram are cartesian by More on Algebra, Lemma 07RU. Since being locally quasi-finite can be checked on fibres (Morphisms, Lemma 01TH) we conclude that \(Y' \amalg_{Z'} X' \to Y \amalg_Z X\) is locally quasi-finite.

We still have to check \(Y' \amalg_{Z'} X' \to Y \amalg_Z X\) is separated. Observe that \(Y' \amalg X' \to Y' \amalg_{Z'} X'\) is universally closed and surjective by Proposition 0E25. Since also the morphism \(Y' \amalg X' \to Y \amalg_Z X\) is separated (as it factors as \(Y' \amalg X' \to Y \amalg X \to Y \amalg_Z X\)) we conclude by Morphisms, Lemma 09MQ.

Lemma

In Situation 0ECI the category of schemes flat, separated, and locally quasi-finite over the pushout \(Y \amalg_Z X\) is equivalent to the category of \((X', Y', Z', i', j', f, g, h)\) as in Lemma 0ECK with \(f, g, h\) flat. Similarly with “flat” replaced with “étale”.

Proof

If we start with \((X', Y', Z', i', j', f, g, h)\) as in Lemma 0ECK with \(f, g, h\) flat or étale, then \(Y' \amalg_{Z'} X' \to Y \amalg_Z X\) is flat or étale by More on Algebra, Lemma 08KQ.

For the converse, let \(W \to Y \amalg_Z X\) be a separated and locally quasi-finite morphism. Set \(X' = W \times_{Y \amalg_Z X} X\), \(Y' = W \times_{Y \amalg_Z X} Y\), and \(Z' = W \times_{Y \amalg_Z X} Z\) with obvious morphisms \(i', j', f, g, h\). Form the pushout \(Y' \amalg_{Z'} X'\). We obtain a morphism \[Y' \amalg_{Z'} X' \longrightarrow W\] of schemes over \(Y \amalg_X Z\) by the universal property of the pushout. If we do not assume that \(W \to Y \amalg_Z X\) is flat, then in general this morphism won’t be an isomorphism. (In fact, More on Algebra, Lemma 08IG shows the displayed arrow is a closed immersion but not an isomorphism in general.) However, if \(W \to Y \times_Z X\) is flat, then it is an isomorphism by More on Algebra, Lemma 08KQ.

Next, we discuss existence in the case where both morphisms are closed immersions.

Lemma

Let \(i : Z \to X\) and \(j : Z \to Y\) be closed immersions of schemes. Then the pushout \(Y \amalg_Z X\) exists in the category of schemes. Picture \[\xymatrix{ Z \ar[r]_i \ar[d]_j & X \ar[d]^a \\ Y \ar[r]^-b & Y \amalg_Z X }\] The diagram is a fibre square, the morphisms \(a\) and \(b\) are closed immersions, and there is a short exact sequence \[0 \to \mathcal{O}_{Y \amalg_Z X} \to a_*\mathcal{O}_X \oplus b_*\mathcal{O}_Y \to c_*\mathcal{O}_Z \to 0\] where \(c = a \circ i = b \circ j\).

Proof

This is a special case of Proposition 0E25. Observe that hypothesis (3) in Situation 0ECI is immediate because the fibres of \(j\) are singletons. Finally, reverse the roles of the arrows to conclude that both \(a\) and \(b\) are closed immersions.

Lemma

Let \(i : Z \to X\) and \(j : Z \to Y\) be closed immersions of schemes. Let \(f : X' \to X\) and \(g : Y' \to Y\) be morphisms of schemes and let \(\varphi : X' \times_{X, i} Z \to Y' \times_{Y, j} Z\) be an isomorphism of schemes over \(Z\). Consider the morphism \[h : X' \amalg_{X' \times_{X, i} Z, \varphi} Y' \longrightarrow X \amalg_Z Y\] Then we have

\(h\) is locally of finite type if and only if \(f\) and \(g\) are locally of finite type,

\(h\) is flat if and only if \(f\) and \(g\) are flat,

\(h\) is flat and locally of finite presentation if and only if \(f\) and \(g\) are flat and locally of finite presentation,

\(h\) is smooth if and only if \(f\) and \(g\) are smooth,

\(h\) is étale if and only if \(f\) and \(g\) are étale, and

add more here as needed.

Proof

We know that the pushouts exist by Lemma 0B7M. In particular we get the morphism \(h\). Hence we may replace all schemes in sight by affine schemes. In this case the assertions of the lemma are equivalent to the corresponding assertions of More on Algebra, Lemma 08KQ.

Relative morphisms

In this section we prove a representability result which we will use in Fundamental Groups, Section 0BL6 to prove a result on the category of finite étale coverings of a scheme. The material in this section is discussed in the correct generality in Criteria for Representability, Section 05Y0.

Let \(S\) be a scheme. Let \(Z\) and \(X\) be schemes over \(S\). Given a scheme \(T\) over \(S\) we can consider morphisms \(b : T \times_S Z \to T \times_S X\) over \(S\). Picture [0BL1]\[\begin{equation} \vcenter{ \xymatrix{ T \times_S Z \ar[rd] \ar[rr]_b & & T \times_S X \ar[ld] & Z \ar[rd] & & X \ar[ld] \\ & T \ar[rrr] & & & S } } \end{equation}\] Of course, we can also think of \(b\) as a morphism \(b : T \times_S Z \to X\) such that \[\xymatrix{ T \times_S Z \ar[r] \ar[d] \ar@/^1pc/[rrr]_-b & Z \ar[rd] & & X \ar[ld] \\ T \ar[rr] & & S }\] commutes. In this situation we can define a functor [0BL2]\[\begin{equation} \mathit{Mor}_S(Z, X) : (\Sch/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \{b\text{ as above}\} \end{equation}\] Here is a basic representability result.

Lemma

Let \(Z \to S\) and \(X \to S\) be morphisms of affine schemes. Assume \(\Gamma(Z, \mathcal{O}_Z)\) is a finite free \(\Gamma(S, \mathcal{O}_S)\)-module. Then \(\mathit{Mor}_S(Z, X)\) is representable by an affine scheme over \(S\).

Proof

Write \(S = \Spec(R)\). Choose a basis \(\{e_1, \ldots, e_m\}\) for \(\Gamma(Z, \mathcal{O}_Z)\) over \(R\). Choose a presentation \[\Gamma(X, \mathcal{O}_X) = R[\{x_i\}_{i \in I}]/(\{f_k\}_{k \in K}).\] We will denote \(\overline{x}_i\) the image of \(x_i\) in this quotient. Write \[P = R[\{a_{ij}\}_{i \in I, 1 \leq j \leq m}].\] Consider the \(R\)-algebra map \[\Psi : R[\{x_i\}_{i \in I}] \longrightarrow P \otimes_R \Gamma(Z, \mathcal{O}_Z), \quad x_i \longmapsto \sum\nolimits_j a_{ij} \otimes e_j.\] Write \(\Psi(f_k) = \sum c_{kj} \otimes e_j\) with \(c_{kj} \in P\). Finally, denote \(J \subset P\) the ideal generated by the elements \(c_{kj}\), \(k \in K\), \(1 \leq j \leq m\). We claim that \(W = \Spec(P/J)\) represents the functor \(\mathit{Mor}_S(Z, X)\).

First, note that by construction \(P/J\) is an \(R\)-algebra, hence a morphism \(W \to S\). Second, by construction the map \(\Psi\) factors through \(\Gamma(X, \mathcal{O}_X)\), hence we obtain an \(P/J\)-algebra homomorphism \[P/J \otimes_R \Gamma(X, \mathcal{O}_X) \longrightarrow P/J \otimes_R \Gamma(Z, \mathcal{O}_Z)\] which determines a morphism \(b_{univ} : W \times_S Z \to W \times_S X\). By the Yoneda lemma \(b_{univ}\) determines a transformation of functors \(W \to \mathit{Mor}_S(Z, X)\) which we claim is an isomorphism. To show that it is an isomorphism it suffices to show that it induces a bijection of sets \(W(T) \to \mathit{Mor}_S(Z, X)(T)\) over any affine scheme \(T\).

Suppose \(T = \Spec(R')\) is an affine scheme over \(S\) and \(b \in \mathit{Mor}_S(Z, X)(T)\). The structure morphism \(T \to S\) defines an \(R\)-algebra structure on \(R'\) and \(b\) defines an \(R'\)-algebra map \[b^\sharp : R' \otimes_R \Gamma(X, \mathcal{O}_X) \longrightarrow R' \otimes_R \Gamma(Z, \mathcal{O}_Z).\] In particular we can write \(b^\sharp(1 \otimes \overline{x}_i) = \sum \alpha_{ij} \otimes e_j\) for some \(\alpha_{ij} \in R'\). This corresponds to an \(R\)-algebra map \(P \to R'\) determined by the rule \(a_{ij} \mapsto \alpha_{ij}\). This map factors through the quotient \(P/J\) by the construction of the ideal \(J\) to give a map \(P/J \to R'\). This in turn corresponds to a morphism \(T \to W\) such that \(b\) is the pullback of \(b_{univ}\). Some details omitted.

Lemma

Let \(Z \to S\) and \(X \to S\) be morphisms of schemes. If \(Z \to S\) is finite locally free and \(X \to S\) is affine, then \(\mathit{Mor}_S(Z, X)\) is representable by a scheme affine over \(S\).

Proof

Choose an affine open covering \(S = \bigcup U_i\) such that \(\Gamma(Z \times_S U_i, \mathcal{O}_{Z \times_S U_i})\) is finite free over \(\mathcal{O}_S(U_i)\). Let \(F_i \subset \mathit{Mor}_S(Z, X)\) be the subfunctor which assigns to \(T/S\) the empty set if \(T \to S\) does not factor through \(U_i\) and \(\mathit{Mor}_S(Z, X)(T)\) otherwise. Then the collection of these subfunctors satisfy the conditions (2)(a), (2)(b), (2)(c) of Schemes, Lemma 01JJ which proves the lemma. Condition (2)(a) follows from Lemma 05Y6 and the other two follow from straightforward arguments.

The condition on the morphism \(f : X \to S\) in the lemma below is very useful to prove statements like it. It holds if one of the following is true: \(X\) is quasi-affine, \(f\) is quasi-affine, \(f\) is quasi-projective, \(f\) is locally projective, there exists an ample invertible sheaf on \(X\), there exists an \(f\)-ample invertible sheaf on \(X\), or there exists an \(f\)-very ample invertible sheaf on \(X\).

Lemma

Let \(Z \to S\) and \(X \to S\) be morphisms of schemes. Assume

\(Z \to S\) is finite locally free, and

for all \((s, x_1, \ldots, x_d)\) where \(s \in S\) and \(x_1, \ldots, x_d \in X_s\) there exists an affine open \(U \subset X\) with \(x_1, \ldots, x_d \in U\).

Then \(\mathit{Mor}_S(Z, X)\) is representable by a scheme.

Proof

Consider the set \(I\) of pairs \((U, V)\) where \(U \subset X\) and \(V \subset S\) are affine open and \(U \to S\) factors through \(V\). For \(i \in I\) denote \((U_i, V_i)\) the corresponding pair. Set \(F_i = \mathit{Mor}_{V_i}(Z_{V_i}, U_i)\). It is immediate that \(F_i\) is a subfunctor of \(\mathit{Mor}_S(Z, X)\). Then we claim that conditions (2)(a), (2)(b), (2)(c) of Schemes, Lemma 01JJ which proves the lemma.

Condition (2)(a) follows from Lemma 0BL3.

To check condition (2)(b) consider \(T/S\) and \(b \in \mathit{Mor}_S(Z, X)(T)\). Thinking of \(b\) as a morphism \(T \times_S Z \to X\) we find an open \(b^{-1}(U_i) \subset T \times_S Z\). Clearly, \(b \in F_i(T)\) if and only if \(b^{-1}(U_i) = T \times_S Z\). Since the projection \(p : T \times_S Z \to T\) is finite hence closed, the set \(U_{i, b} \subset T\) of points \(t \in T\) with \(p^{-1}(\{t\}) \subset b^{-1}(U_i)\) is open. Then \(f : T' \to T\) factors through \(U_{i, b}\) if and only if \(b \circ f \in F_i(T')\) and we are done checking (2)(b).

Finally, we check condition (2)(c) and this is where our condition on \(X \to S\) is used. Namely, consider \(T/S\) and \(b \in \mathit{Mor}_S(Z, X)(T)\). It suffices to prove that every \(t \in T\) is contained in one of the opens \(U_{i, b}\) defined in the previous paragraph. This is equivalent to the condition that \(b(p^{-1}(\{t\})) \subset U_i\) for some \(i\) where \(p : T \times_S Z \to T\) is the projection and \(b : T \times_S Z \to X\) is the given morphism. Since \(p\) is finite, the set \(b(p^{-1}(\{t\})) \subset X\) is finite and contained in the fibre of \(X \to S\) over the image \(s\) of \(t\) in \(S\). Thus our condition on \(X \to S\) exactly shows a suitable pair exists.

Lemma

Let \(Z \to S\) and \(X \to S\) be morphisms of schemes. Assume \(Z \to S\) is finite locally free and \(X \to S\) is separated and locally quasi-finite. Then \(\mathit{Mor}_S(Z, X)\) is representable by a scheme.

Proof

This follows from Lemmas 0BL4 and 07S0.

Characterizing pseudo-coherent complexes, III

In this section we discuss characterizations of pseudo-coherent complexes in terms of cohomology. This is a continuation of Derived Categories of Schemes, Section 0CSE. A basic tool will be to reduce to the case of projective space using a derived version of Chow’s lemma, see Lemma 0CSJ.

Lemma

Consider a commutative diagram of schemes \[\xymatrix{ Z' \ar[d] \ar[r] & Y' \ar[d] \\ X' \ar[r] & S' }\] Let \(S \to S'\) be a morphism. Denote by \(X\) and \(Y\) the base changes of \(X'\) and \(Y'\) to \(S\). Assume \(Y' \to S'\) and \(Z' \to X'\) are flat. Then \(X \times_S Y\) and \(Z'\) are Tor independent over \(X' \times_{S'} Y'\).

Proof

The question is local, hence we may assume all schemes are affine (some details omitted). Observe that \[\xymatrix{ X \times_S Y \ar[r] \ar[d] & X' \times_{S'} Y' \ar[d] \\ X \ar[r] & X' }\] is cartesian with flat vertical arrows. Write \(X = \Spec(A)\), \(X' = \Spec(A')\), \(X' \times_{S'} Y' = \Spec(B')\). Then \(X \times_S Y = \Spec(A \otimes_{A'} B')\). Write \(Z' = \Spec(C')\). We have to show \[\text{Tor}_p^{B'}(A \otimes_{A'} B', C') = 0, \quad\text{for } p > 0\] Since \(A' \to B'\) is flat we have \(A \otimes_{A'} B' = A \otimes_{A'}^\mathbf{L} B'\). Hence \[(A \otimes_{A'} B') \otimes_{B'}^\mathbf{L} C' = (A \otimes_{A'}^\mathbf{L} B') \otimes_{B'}^\mathbf{L} C' = A \otimes_{A'}^\mathbf{L} C' = A \otimes_{A'} C'\] The second equality by More on Algebra, Lemma 08YU. The last equality because \(A' \to C'\) is flat. This proves the lemma.

Lemma

Let \(A\) be a ring. Let \(X\) be a separated scheme of finite presentation over \(A\). Let \(x \in X\). Then there exist an open neighbourhood \(U \subset X\) of \(x\), an \(n \geq 0\), an open \(V \subset \mathbf{P}^n_A\), a closed subscheme \(Z \subset X \times_A \mathbf{P}^n_A\), a point \(z \in Z\), and an object \(E\) in \(D(\mathcal{O}_{X \times_A \mathbf{P}^n_A})\) such that

\(Z \to X \times_A \mathbf{P}^n_A\) is of finite presentation,

\(b : Z \to X\) is an isomorphism over \(U\) and \(b(z) = x\),

\(c : Z \to \mathbf{P}^n_A\) is a closed immersion over \(V\),

\(b^{-1}(U) = c^{-1}(V)\), in particular \(c(z) \in V\),

\(E|_{X \times_A V} \cong (b, c)_*\mathcal{O}_Z|_{X \times_A V}\),

\(E\) is pseudo-coherent and supported on \(Z\).

Proof

We can find a finite type \(\mathbf{Z}\)-subalgebra \(A' \subset A\) and a scheme \(X'\) separated and of finite presentation over \(A'\) whose base change to \(A\) is \(X\). See Limits, Lemmas 01ZM and 01ZQ. Let \(x' \in X'\) be the image of \(x\). If we can prove the lemma for \(x' \in X'/A'\), then the lemma follows for \(x \in X/A\). Namely, if \(U', n', V', Z', z', E'\) provide the solution for \(x' \in X'/A'\), then we can let \(U \subset X\) be the inverse image of \(U'\), let \(n = n'\), let \(V \subset \mathbf{P}^n_A\) be the inverse image of \(V'\), let \(Z \subset X \times \mathbf{P}^n\) be the scheme theoretic inverse image of \(Z'\), let \(z \in Z\) be the unique point mapping to \(x\), and let \(E\) be the derived pullback of \(E'\). Observe that \(E\) is pseudo-coherent by Cohomology, Lemma 09U7. It only remains to check (5). To see this set \(W = b^{-1}(U) = c^{-1}(V)\) and \(W' = (b')^{-1}(U) = (c')^{-1}(V')\) and consider the cartesian square \[\xymatrix{ W \ar[d]_{(b, c)} \ar[r] & W' \ar[d]^{(b', c')} \\ X \times_A V \ar[r] & X' \times_{A'} V' }\] By Lemma 0CTA the schemes \(X \times_A V\) and \(W'\) are Tor independent over \(X' \times_{A'} V'\). Hence the derived pullback of \((b', c')_*\mathcal{O}_{W'}\) to \(X \times_A V\) is \((b, c)_*\mathcal{O}_W\) by Derived Categories of Schemes, Lemma 08IB. This also uses that \(R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'}\) because \((b', c')\) is a closed immersion and similarly for \((b, c)_*\mathcal{O}_Z\). Since \(E'|_{U' \times_{A'} V'} = (b', c')_*\mathcal{O}_{W'}\) we obtain \(E|_{U \times_A V} = (b, c)_*\mathcal{O}_W\) and (5) holds. This reduces us to the situation described in the next paragraph.

Assume \(A\) is of finite type over \(\mathbf{Z}\). Choose an affine open neighbourhood \(U \subset X\) of \(x\). Then \(U\) is of finite type over \(A\). Choose a closed immersion \(U \to \mathbf{A}^n_A\) and denote \(j : U \to \mathbf{P}^n_A\) the immersion we get by composing with the open immersion \(\mathbf{A}^n_A \to \mathbf{P}^n_A\). Let \(Z\) be the scheme theoretic closure of \[(\text{id}_U, j) : U \longrightarrow X \times_A \mathbf{P}^n_A\] Since the projection \(X \times \mathbf{P}^n \to X\) is separated, we conclude from Morphisms, Lemma 0CNG that \(b : Z \to X\) is an isomorphism over \(U\). Let \(z \in Z\) be the unique point lying over \(x\).

Let \(Y \subset \mathbf{P}^n_A\) be the scheme theoretic closure of \(j\). Then it is clear that \(Z \subset X \times_A Y\) is the scheme theoretic closure of \((\text{id}_U, j) : U \to X \times_A Y\). As \(X\) is separated, the morphism \(X \times_A Y \to Y\) is separated as well. Hence we see that \(Z \to Y\) is an isomorphism over the open subscheme \(j(U) \subset Y\) by the same lemma we used above. Choose \(V \subset \mathbf{P}^n_A\) open with \(V \cap Y = j(U)\). Then we see that (3) and (4) hold.

Because \(A\) is Noetherian we see that \(X\) and \(X \times_A \mathbf{P}^n_A\) are Noetherian schemes. Hence we can take \(E = (b, c)_*\mathcal{O}_Z\) in this case, see Derived Categories of Schemes, Lemma 08E8. This finishes the proof.

Lemma

Let \(A\), \(x \in X\), and \(U, n, V, Z, z, E\) be as in Lemma 0CSJ. For any \(K \in D_\QCoh(\mathcal{O}_X)\) we have \[Rq_*(Lp^*K \otimes^\mathbf{L} E)|_V = R(U \to V)_*K|_U\] where \(p : X \times_A \mathbf{P}^n_A \to X\) and \(q : X \times_A \mathbf{P}^n_A \to \mathbf{P}^n_A\) are the projections and where the morphism \(U \to V\) is the finitely presented closed immersion \(c \circ (b|_U)^{-1}\).

Proof

Since \(b^{-1}(U) = c^{-1}(V)\) and since \(c\) is a closed immersion over \(V\), we see that \(c \circ (b|_U)^{-1}\) is a closed immersion. It is of finite presentation because \(U\) and \(V\) are of finite presentation over \(A\), see Morphisms, Lemma 02FV. First we have \[Rq_*(Lp^*K \otimes^\mathbf{L} E)|_V = Rq'_*\left((Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V}\right)\] where \(q' : X \times_A V \to V\) is the projection because formation of total direct image commutes with localization. Set \(W = b^{-1}(U) = c^{-1}(V)\) and denote \(i : W \to X \times_A V\) the closed immersion \(i = (b, c)|_W\). Then \[Rq'_*\left((Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V}\right) = Rq'_*(Lp^*K|_{X \times_A V} \otimes^\mathbf{L} i_*\mathcal{O}_W)\] by property (5). Since \(i\) is a closed immersion we have \(i_*\mathcal{O}_W = Ri_*\mathcal{O}_W\). Using Derived Categories of Schemes, Lemma 08EU we can rewrite this as \[Rq'_* Ri_* Li^* Lp^*K|_{X \times_A V} = R(q' \circ i)_* Lb^*K|_W = R(U \to V)_* K|_U\] which is what we want.

Lemma

Let \(A\) be a ring. Let \(X\) be a scheme separated and of finite presentation over \(A\). Let \(K \in D_\QCoh(\mathcal{O}_X)\). If \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is pseudo-coherent in \(D(A)\) for every pseudo-coherent \(E\) in \(D(\mathcal{O}_X)\), then \(K\) is pseudo-coherent relative to \(A\).

Proof

Assume \(K \in D_\QCoh(\mathcal{O}_X)\) and \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is pseudo-coherent in \(D(A)\) for every pseudo-coherent \(E\) in \(D(\mathcal{O}_X)\). Let \(x \in X\). We will show that \(K\) is pseudo-coherent relative to \(A\) in a neighbourhood of \(x\) and this will prove the lemma.

Choose \(U, n, V, Z, z, E\) as in Lemma 0CSJ. Denote \(p : X \times \mathbf{P}^n \to X\) and \(q : X \times \mathbf{P}^n \to \mathbf{P}^n_A\) the projections. Then for any \(i \in \mathbf{Z}\) we have \[\begin{align*} & R\Gamma(\mathbf{P}^n_A, Rq_*(Lp^*K \otimes^\mathbf{L} E) \otimes^\mathbf{L} \mathcal{O}_{\mathbf{P}^n_A}(i)) \\ & = R\Gamma(X \times \mathbf{P}^n, Lp^*K \otimes^\mathbf{L} E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i)) \\ & = R\Gamma(X, K \otimes^\mathbf{L} Rp_*(E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i))) \end{align*}\] by Derived Categories of Schemes, Lemma 08EU. By Derived Categories of Schemes, Lemma 0CSD the complex \(Rp_*(E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i))\) is pseudo-coherent on \(X\). Hence the assumption tells us the expression in the displayed formula is a pseudo-coherent object of \(D(A)\). By Derived Categories of Schemes, Lemma 0CSG we conclude that \(Rq_*(Lp^*K \otimes^\mathbf{L} E)\) is pseudo-coherent on \(\mathbf{P}^n_A\). By Lemma 0CSK we have \[Rq_*(Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V} = R(U \to V)_*K|_U\] Since \(U \to V\) is a closed immersion into an open subscheme of \(\mathbf{P}^n_A\) this means \(K|_U\) is pseudo-coherent relative to \(A\) by Lemma 09UU.

Lemma

Let \(A\) be a ring. Let \(X\) be a scheme separated and of finite presentation over \(A\). Let \(K \in D_\QCoh(\mathcal{O}_X).\) If \(R \Gamma (X, E \otimes ^{\mathbf{L}} K)\) is pseudo-coherent in \(D(A)\) for every perfect \(E \in D(\mathcal{O}_X)\), then \(K\) is pseudo-coherent relative to \(A\).

Proof

In view of Lemma 0CSL, it suffices to show \(R \Gamma (X, E \otimes ^{\mathbf{L}} K)\) is pseudo-coherent in \(D(A)\) for every pseudo-coherent \(E \in D(\mathcal{O}_X)\). By Derived Categories of Schemes, Proposition 0GEN it follows that \(K \in D^-_\QCoh (\mathcal{O}_X)\). Now the result follows by Derived Categories of Schemes, Lemma 0CSH.

Lemma

Let \(A\) be a ring. Let \(X\) be a scheme separated, of finite presentation, and flat over \(A\). Let \(K \in D_\QCoh(\mathcal{O}_X).\) If \(R \Gamma (X, E \otimes^\mathbf{L} K)\) is perfect in \(D(A)\) for every perfect \(E \in D(\mathcal{O}_X)\), then \(K\) is \(\Spec(A)\)-perfect.

Proof

By Lemma 0GES, \(K\) is pseudo-coherent relative to \(A\). By Lemma 09UU, \(K\) is pseudo-coherent in \(D( \mathcal{O}_X)\). By Derived Categories of Schemes, Proposition 0GEQ we see that \(K\) is in \(D^-(\mathcal{O}_X)\). Let \(\mathfrak{p}\) be a prime ideal of \(A\) and denote \(i : Y \to X\) the inclusion of the scheme theoretic fibre over \(\mathfrak{p}\), i.e., \(Y\) is a scheme over \(\kappa(\mathfrak p)\). By Derived Categories of Schemes, Lemma 0GEH, we will be done if we can show \(Li^*(K)\) is bounded below. Let \(G \in D_{perf} (\mathcal{O}_X)\) be a perfect complex which generates \(D_\QCoh (\mathcal{O}_X)\), see Derived Categories of Schemes, Theorem 09IS. We have \[\begin{align*} R\Hom _{\mathcal{O}_Y}(Li^*(G), Li^*(K)) & = R\Gamma(Y, Li^*(G ^\vee \otimes ^\mathbf{L} K)) \\ & = R\Gamma(X, G^\vee \otimes ^{\mathbf{L}} K) \otimes^\mathbf{L}_A \kappa(\mathfrak{p}) \end{align*}\] The first equality uses that \(Li^*\) preserves perfect objects and duals and Cohomology, Lemma 08DQ; we omit some details. The second equality follows from Derived Categories of Schemes, Lemma 08IB as \(X\) is flat over \(A\). It follows from our hypothesis that this is a perfect object of \(D(\kappa(\mathfrak{p}))\). The object \(Li^*(G) \in D_{perf}(\mathcal{O}_Y)\) generates \(D_\QCoh(\mathcal{O}_Y)\) by Derived Categories of Schemes, Remark 0BQT. Hence Derived Categories of Schemes, Proposition 0GEQ now implies that \(Li^*(K)\) is bounded below and we win.

Descent finiteness properties of complexes

This section is the continuation of Derived Categories of Schemes, Section 09UC.

Lemma

Let \(X \to S\) be locally of finite type. Let \(\{f_i : X_i \to X\}\) be an fppf covering of schemes. Let \(E \in D_\QCoh(\mathcal{O}_X)\). Let \(m \in \mathbf{Z}\). Then \(E\) is \(m\)-pseudo-coherent relative to \(S\) if and only if each \(Lf_i^*E\) is \(m\)-pseudo-coherent relative to \(S\).

Proof

Assume \(E\) is \(m\)-pseudo-coherent relative to \(S\). The morphisms \(f_i\) are pseudo-coherent by Lemma 0695. Hence \(Lf_i^*E\) is \(m\)-pseudo-coherent relative to \(S\) by Lemma 09US.

Conversely, assume that \(Lf_i^*E\) is \(m\)-pseudo-coherent relative to \(S\) for each \(i\). Pick \(S = \bigcup U_j\), \(W_j \to U_j\), \(W_j = \bigcup W_{j, k}\), \(T_{j, k} \to W_{j, k}\), and morphisms \(\alpha_{j, k} : T_{j, k} \to X_{i(j, k)}\) over \(S\) as in Lemma 05WN. Since the morphism \(T_{j, K} \to S\) is flat and of finite presentation, we see that \(\alpha_{j, k}\) is pseudo-coherent by Lemma 0683. Hence \[L\alpha_{j, k}^*Lf_{i(j, k)}^*E = L(T_{i, k} \to S)^*E\] is \(m\)-pseudo-coherent relative to \(S\) by Lemma 09US. Now we want to descend this property through the coverings \(\{T_{j, k} \to W_{j, k}\}\), \(W_j = \bigcup W_{j, k}\), \(\{W_j \to U_j\}\), and \(S = \bigcup U_j\). Since for Zariski coverings the result is true (by the definition of \(m\)-pseudo-coherence relative to \(S\)), this means we may assume we have a single surjective finite locally free morphism \(\pi : Y \to X\) such that \(L\pi^*E\) is pseudo-coherent relative to \(S\). In this case \(R\pi_*L\pi^*E\) is pseudo-coherent relative to \(S\) by Lemma 09UK (this is the first time we use that \(E\) has quasi-coherent cohomology sheaves). We have \(R\pi_*L\pi^*E = E \otimes^\mathbf{L}_{\mathcal{O}_X} \pi_*\mathcal{O}_Y\) for example by Derived Categories of Schemes, Lemma 08EU and locally on \(X\) the map \(\mathcal{O}_X \to \pi_*\mathcal{O}_Y\) is the inclusion of a direct summand. Hence we conclude by Lemma 09UN.

Lemma

Let \(X \to T \to S\) be morphisms of schemes. Assume \(T \to S\) is flat and locally of finite presentation and \(X \to T\) locally of finite type. Let \(E \in D(\mathcal{O}_X)\). Let \(m \in \mathbf{Z}\). Then \(E\) is \(m\)-pseudo-coherent relative to \(S\) if and only if \(E\) is \(m\)-pseudo-coherent relative to \(T\).

Proof

Locally on \(X\) we can choose a closed immersion \(i : X \to \mathbf{A}^n_T\). Then \(\mathbf{A}^n_T \to S\) is flat and locally of finite presentation. Thus we may apply Lemma 09UT to see the equivalence holds.

Lemma

Let \(f : X \to S\) be locally of finite type. Let \(\{S_i \to S\}\) be an fppf covering of schemes. Denote \(f_i : X_i \to S_i\) the base change of \(f\) and \(g_i : X_i \to X\) the projection. Let \(E \in D_\QCoh(\mathcal{O}_X)\). Let \(m \in \mathbf{Z}\). Then \(E\) is \(m\)-pseudo-coherent relative to \(S\) if and only if each \(Lg_i^*E\) is \(m\)-pseudo-coherent relative to \(S_i\).

Proof

This follows formally from Lemmas 0CSN and 0CSP. Namely, if \(E\) is \(m\)-pseudo-coherent relative to \(S\), then \(Lg_i^*E\) is \(m\)-pseudo-coherent relative to \(S\) (by the first lemma), hence \(Lg_i^*E\) is \(m\)-pseudo-coherent relative to \(S_i\) (by the second). Conversely, if \(Lg_i^*E\) is \(m\)-pseudo-coherent relative to \(S_i\), then \(Lg_i^*E\) is \(m\)-pseudo-coherent relative to \(S\) (by the second lemma), hence \(E\) is \(m\)-pseudo-coherent relative to \(S\) (by the first lemma).

Relatively perfect objects

This section is a continuation of the discussion in Derived Categories of Schemes, Section 0DHZ.

Lemma

Let \(i : X \to X'\) be a finite order thickening of schemes. Let \(K' \in D(\mathcal{O}_{X'})\) be an object such that \(K = Li^*K'\) is pseudo-coherent. Then \(K'\) is pseudo-coherent.

Proof

We first prove \(K'\) has quasi-coherent cohomology sheaves. To do this, we may reduce to the case of a first order thickening, see Section 04EW. Let \(\mathcal{I} \subset \mathcal{O}_{X'}\) be the quasi-coherent sheaf of ideals cutting out \(X\). Tensoring the short exact sequence \[0 \to \mathcal{I} \to \mathcal{O}_{X'} \to i_*\mathcal{O}_X \to 0\] with \(K'\) we obtain a distinguished triangle \[K' \otimes_{\mathcal{O}_{X'}}^\mathbf{L} \mathcal{I} \to K' \to K' \otimes_{\mathcal{O}_{X'}}^\mathbf{L} i_*\mathcal{O}_X \to (K' \otimes_{\mathcal{O}_{X'}}^\mathbf{L} \mathcal{I})[1]\] Since \(i_* = Ri_*\) and since we may view \(\mathcal{I}\) as a quasi-coherent \(\mathcal{O}_X\)-module (as we have a first order thickening) we may rewrite this as \[i_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I}) \to K' \to i_*K \to i_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I})[1]\] Please use Cohomology, Lemma 0B55 to identify the terms. Since \(K\) is in \(D_\QCoh(\mathcal{O}_X)\) we conclude that \(K'\) is in \(D_\QCoh(\mathcal{O}_{X'})\); this uses Derived Categories of Schemes, Lemmas 08E5, 08DX, and 08D5.

Assume \(K'\) is in \(D_\QCoh(\mathcal{O}_{X'})\). The question is local on \(X'\) hence we may assume \(X'\) is affine. Say \(X' = \Spec(A')\) and \(X = \Spec(A)\) with \(A = A'/I\) and \(I\) nilpotent. Then \(K'\) comes from an object \(M' \in D(A')\), see Derived Categories of Schemes, Lemma 06Z0. Thus \(M = M' \otimes_{A'}^\mathbf{L} A\) is a pseudo-coherent object of \(D(A)\) by Derived Categories of Schemes, Lemma 08E7 and our assumption on \(K\). It follows that \(M'\) is pseudo-coherent by More on Algebra, Lemma 0H76. This finishes the proof.

Lemma

Consider a cartesian diagram \[\xymatrix{ X \ar[r]_i \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r]^j & Y' }\] of schemes. Assume \(X' \to Y'\) is flat and locally of finite presentation and \(Y \to Y'\) is a finite order thickening. Let \(E' \in D(\mathcal{O}_{X'})\). If \(E = Li^*(E')\) is \(Y\)-perfect, then \(E'\) is \(Y'\)-perfect.

Proof

Recall that being \(Y\)-perfect for \(E\) means \(E\) is pseudo-coherent and locally has finite tor dimension as a complex of \(f^{-1}\mathcal{O}_Y\)-modules (Derived Categories of Schemes, Definition 0DI0). By Lemma 0DJX we find that \(E'\) is pseudo-coherent. In particular, \(E'\) is in \(D_\QCoh(\mathcal{O}_{X'})\), see Derived Categories of Schemes, Lemma 08E5. To prove that \(E'\) locally has finite tor dimension we may work locally on \(X'\). Hence we may assume \(X'\), \(S'\), \(X\), \(S\) are affine, say given by rings \(A'\), \(R'\), \(A\), \(R\). Then we reduce to the commutative algebra version by Derived Categories of Schemes, Lemma 0DI2. The commutative algebra version in More on Algebra, Lemma 0DJG.

Lemma

Let \((R, I)\) be a pair consisting of a ring and an ideal \(I\) contained in the Jacobson radical. Set \(S = \Spec(R)\) and \(S_0 = \Spec(R/I)\). Let \(f : X \to S\) be proper, flat, and of finite presentation. Denote \(X_0 = S_0 \times_S X\). Let \(E \in D(\mathcal{O}_X)\) be pseudo-coherent. If the derived restriction \(E_0\) of \(E\) to \(X_0\) is \(S_0\)-perfect, then \(E\) is \(S\)-perfect.

Proof

Choose a finite affine open covering \(X = U_1 \cup \ldots \cup U_n\). For each \(i\) we can choose a closed immersion \(U_i \to \mathbf{A}^{d_i}_S\). Set \(U_{i, 0} = S_0 \times_S U_i\). For each \(i\) the complex \(E_0|_{U_{i, 0}}\) has tor amplitude in \([a_i, b_i]\) for some \(a_i, b_i \in \mathbf{Z}\). Let \(x \in X\) be a point. We will show that the tor amplitude of \(E_x\) over \(R\) is in \([a_i - d_i, b_i]\) for some \(i\). This will finish the proof as the tor amplitude can be read off from the stalks by Cohomology, Lemma 09U9.

Since \(f\) is proper \(f(\overline{\{x\}})\) is a closed subset of \(S\). Since \(I\) is contained in the Jacobson radical, we see that \(f(\overline{\{x\}})\) meeting the closed subset \(S_0 \subset S\). Hence there is a specialization \(x \leadsto x_0\) with \(x_0 \in X_0\). Pick an \(i\) with \(x_0 \in U_i\), so \(x_0 \in U_{i, 0}\). We will fix \(i\) for the rest of the proof. Write \(U_i = \Spec(A)\). Then \(A\) is a flat, finitely presented \(R\)-algebra which is a quotient of a polynomial \(R\)-algebra in \(d_i\)-variables. The restriction \(E|_{U_i}\) corresponds (by Derived Categories of Schemes, Lemma 06Z0 and 08E7) to a pseudo-coherent object \(K\) of \(D(A)\). Observe that \(E_0\) corresponds to \(K \otimes_A^\mathbf{L} A/IA\). Let \(\mathfrak q \subset \mathfrak q_0 \subset A\) be the prime ideals corresponding to \(x \leadsto x_0\). Then \(E_x = K_{\mathfrak q}\) and \(K_{\mathfrak q}\) is a localization of \(K_{\mathfrak q_0}\). Hence it suffices to show that \(K_{\mathfrak q_0}\) has tor amplitude in \([a_i - d_i, b_i]\) as a complex of \(R\)-modules. Let \(I \subset \mathfrak p_0 \subset R\) be the prime ideal corresponding to \(f(x_0)\). Then we have \[\begin{align*} K \otimes_R^\mathbf{L} \kappa(\mathfrak p_0) & = (K \otimes_R^\mathbf{L} R/I) \otimes_{R/I}^\mathbf{L} \kappa(\mathfrak p_0) \\ & = (K \otimes_A^\mathbf{L} A/IA) \otimes_{R/I}^\mathbf{L} \kappa(\mathfrak p_0) \end{align*}\] the second equality because \(R \to A\) is flat. By our choice of \(a_i, b_i\) this complex has cohomology only in degrees in the interval \([a_i, b_i]\). Thus we may finally apply More on Algebra, Lemma 0DJH to \(R \to A\), \(\mathfrak q_0\), \(\mathfrak p_0\) and \(K\) to conclude.

Contracting rational curves

In this section we study proper morphisms \(f : X \to Y\) whose fibres have dimension \(\leq 1\) having \(R^1f_*\mathcal{O}_X = 0\). To understand the title of this section, please take a look at Algebraic Curves, Sections 0E3G, 0E7M, and 0E7N.

Lemma

Let \(f : X \to Y\) be a proper morphism of schemes. Let \(y \in Y\) be a point with \(\dim(X_y) \leq 1\). If

\(R^1f_*\mathcal{O}_X = 0\), or more generally

there is a morphism \(g : Y' \to Y\) such that \(y\) is in the image of \(g\) and such that \(R^1f'_*\mathcal{O}_{X'} = 0\) where \(f' : X' \to Y'\) is the base change of \(f\) by \(g\).

Then \(H^1(X_y, \mathcal{O}_{X_y}) = 0\).

Proof

To prove the lemma we may replace \(Y\) by an open neighbourhood of \(y\). Thus we may assume \(Y\) is affine and that all fibres of \(f\) have dimension \(\leq 1\), see Morphisms, Lemma 02FZ. In this case \(R^1f_*\mathcal{O}_X\) is a quasi-coherent \(\mathcal{O}_Y\)-module of finite type and its formation commutes with arbitrary base change, see Limits, Lemmas 0EX4 and 0E7D. The lemma follows immediately.

Lemma

Let \(f : X \to Y\) be a proper morphism of schemes. Let \(y \in Y\) be a point with \(\dim(X_y) \leq 1\) and \(H^1(X_y, \mathcal{O}_{X_y}) = 0\). Then there is an open neighbourhood \(V \subset Y\) of \(y\) such that \(R^1f_*\mathcal{O}_X|_V = 0\) and the same is true after base change by any \(Y' \to V\).

Proof

To prove the lemma we may replace \(Y\) by an open neighbourhood of \(y\). Thus we may assume \(Y\) is affine and that all fibres of \(f\) have dimension \(\leq 1\), see Morphisms, Lemma 02FZ. In this case \(R^1f_*\mathcal{O}_X\) is a quasi-coherent \(\mathcal{O}_Y\)-module of finite type and its formation commutes with arbitrary base change, see Limits, Lemmas 0EX4 and 0E7D. Say \(Y = \Spec(A)\), \(y\) corresponds to the prime \(\mathfrak p \subset A\), and \(R^1f_*\mathcal{O}_X\) corresponds to the finite \(A\)-module \(M\). Then \(H^1(X_y, \mathcal{O}_{X_y}) = 0\) means that \(\mathfrak pM_\mathfrak p = M_\mathfrak p\) by the statement on base change. By Nakayama’s lemma we conclude \(M_\mathfrak p = 0\). Since \(M\) is finite, we find an \(f \in A\), \(f \not \in \mathfrak p\) such that \(M_f = 0\). Thus taking \(V\) the principal open \(D(f)\) we obtain the desired result.

Lemma

Let \(f : X \to Y\) be a proper morphism of schemes such that \(\dim(X_y) \leq 1\) and \(H^1(X_y, \mathcal{O}_{X_y}) = 0\) for all \(y \in Y\). Let \(\mathcal{F}\) be quasi-coherent on \(X\). Then

\(R^pf_*\mathcal{F} = 0\) for \(p > 1\), and

\(R^1f_*\mathcal{F} = 0\) if there is a surjection \(f^*\mathcal{G} \to \mathcal{F}\) with \(\mathcal{G}\) quasi-coherent on \(Y\).

If \(Y\) is affine, then we also have

\(H^p(X, \mathcal{F}) = 0\) for \(p \not \in \{0, 1\}\), and

\(H^1(X, \mathcal{F}) = 0\) if \(\mathcal{F}\) is globally generated.

Proof

The vanishing in (1) is Limits, Lemma 0E7D. To prove (2) we may work locally on \(Y\) and assume \(Y\) is affine. Then \(R^1f_*\mathcal{F}\) is the quasi-coherent module on \(Y\) associated to the module \(H^1(X, \mathcal{F})\). Here we use that \(Y\) is affine, quasi-coherence of higher direct images (Cohomology of Schemes, Lemma 01XJ), and Cohomology of Schemes, Lemma 01XK. Since \(Y\) is affine, the quasi-coherent module \(\mathcal{G}\) is globally generated, and hence so is \(f^*\mathcal{G}\) and \(\mathcal{F}\). In this way we see that (4) implies (2). Part (3) follows from (1) as well as the remarks on quasi-coherence of direct images just made. Thus all that remains is the prove (4). If \(\mathcal{F}\) is globally generated, then there is a surjection \(\bigoplus_{i \in I} \mathcal{O}_X \to \mathcal{F}\). By part (1) and the long exact sequence of cohomology this induces a surjection on \(H^1\). Since \(H^1(X, \mathcal{O}_X) = 0\) because \(R^1f_*\mathcal{O}_X = 0\) by Lemma 0E7G, and since \(H^1(X, -)\) commutes with direct sums (Cohomology, Lemma 01FF) we conclude.

Lemma

Let \(f : X \to Y\) be a proper morphism of schemes. Assume

for all \(y \in Y\) we have \(\dim(X_y) \leq 1\) and \(H^1(X_y, \mathcal{O}_{X_y}) = 0\), and

\(\mathcal{O}_Y \to f_*\mathcal{O}_X\) is surjective.

Then \(\mathcal{O}_{Y'} \to f'_*\mathcal{O}_{X'}\) is surjective for any base change \(f' : X' \to Y'\) of \(f\).

Proof

We may assume \(Y\) and \(Y'\) affine. Then we can choose a closed immersion \(Y' \to Y''\) with \(Y'' \to Y\) a flat morphism of affines. By flat base change (Cohomology of Schemes, Lemma 02KH) we see that the result holds for \(X'' \to Y''\). Thus we may assume \(Y'\) is a closed subscheme of \(Y\). Let \(\mathcal{I} \subset \mathcal{O}_Y\) be the ideal cutting out \(Y'\). Then there is a short exact sequence \[0 \to \mathcal{I}\mathcal{O}_X \to \mathcal{O}_X \to \mathcal{O}_{X'} \to 0\] where we view \(\mathcal{O}_{X'}\) as a quasi-coherent module on \(X\). By Lemma 0E7H we have \(H^1(X, \mathcal{I}\mathcal{O}_X) = 0\). It follows that \[H^0(Y, \mathcal{O}_Y) \to H^0(Y, f_*\mathcal{O}_X) = H^0(X, \mathcal{O}_X) \to H^0(X, \mathcal{O}_{X'})\] is surjective as desired. The first arrow is surjective as \(Y\) is affine and since we assumed \(\mathcal{O}_Y \to f_*\mathcal{O}_X\) is surjective and the second by the long exact sequence of cohomology associated to the short exact sequence above and the vanishing just proved.

Lemma

Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S }\] of morphisms of schemes. Let \(s \in S\) be a point. Assume

\(X \to S\) is locally of finite presentation and flat at points of \(X_s\),

\(f\) is proper,

the fibres of \(f_s : X_s \to Y_s\) have dimension \(\leq 1\) and \(R^1f_{s, *}\mathcal{O}_{X_s} = 0\),

\(\mathcal{O}_{Y_s} \to f_{s, *}\mathcal{O}_{X_s}\) is surjective.

Then there is an open \(Y_s \subset V \subset Y\) such that (a) \(f^{-1}(V)\) is flat over \(S\), (b) \(\dim(X_y) \leq 1\) for \(y \in V\), (c) \(R^1f_*\mathcal{O}_X|_V = 0\), (d) \(\mathcal{O}_V \to f_*\mathcal{O}_X|_V\) is surjective, and (b), (c), and (d) remain true after base change by any \(Y' \to V\).

Proof

Let \(y \in Y\) be a point over \(s\). It suffices to find an open neighbourhood of \(y\) with the desired properties. As a first step, we replace \(Y\) by the open \(V\) found in Lemma 0E7G so that \(R^1f_*\mathcal{O}_X\) is zero universally (the hypothesis of the lemma holds by Lemma 0E7F). We also shrink \(Y\) so that all fibres of \(f\) have dimension \(\leq 1\) (use Morphisms, Lemma 02FZ and properness of \(f\)). Thus we may assume we have (b) and (c) with \(V = Y\) and after any base change \(Y' \to Y\). Thus by Lemma 0E7I it now suffices to show (d) over \(Y\). We may still shrink \(Y\) further; for example, we may and do assume \(Y\) and \(S\) are affine.

By Theorem 0399 there is an open subset \(U \subset X\) where \(X \to S\) is flat which contains \(X_s\) by hypothesis. Then \(f(X \setminus U)\) is a closed subset not containing \(y\). Thus after shrinking \(Y\) we may assume \(X\) is flat over \(S\).

Say \(S = \Spec(R)\). Choose a closed immersion \(Y \to Y'\) where \(Y'\) is the spectrum of a polynomial ring \(R[x_e; e \in E]\) on a set \(E\). Denote \(f' : X \to Y'\) the composition of \(f\) with \(Y \to Y'\). Then the hypotheses (1) – (4) as well as (b) and (c) hold for \(f'\) and \(s\). If we we show \(\mathcal{O}_{Y'} \to f'_*\mathcal{O}_X\) is surjective in an open neighbourhood of \(y\), then the same is true for \(\mathcal{O}_Y \to f_*\mathcal{O}_X\). Thus we may assume \(Y\) is the spectrum of \(R[x_e; e \in E]\).

At this point \(X\) and \(Y\) are flat over \(S\). Then \(Y_s\) and \(X\) are tor independent over \(Y\). We urge the reader to find their own proof, but it also follows from Lemma 0CTA applied to the square with corners \(X, Y, S, S\) and its base change by \(s \to S\). Hence \[Rf_{s, *}\mathcal{O}_{X_s} = L(Y_s \to Y)^*Rf_*\mathcal{O}_X\] by Derived Categories of Schemes, Lemma 08IB. Because of the vanishing already established this implies \(f_{s, *}\mathcal{O}_{X_s} = (Y_s \to Y)^*f_*\mathcal{O}_X\). We conclude that \(\mathcal{O}_Y \to f_*\mathcal{O}_X\) is a map of quasi-coherent \(\mathcal{O}_Y\)-modules whose pullback to \(Y_s\) is surjective. We claim \(f_*\mathcal{O}_X\) is a finite type \(\mathcal{O}_Y\)-module. If true, then the cokernel \(\mathcal{F}\) of \(\mathcal{O}_Y \to f_*\mathcal{O}_X\) is a finite type quasi-coherent \(\mathcal{O}_Y\)-module such that \(\mathcal{F}_y \otimes \kappa(y) = 0\). By Nakayama’s lemma (Algebra, Lemma 00DV) we have \(\mathcal{F}_y = 0\). Thus \(\mathcal{F}\) is zero in an open neighbourhood of \(y\) (Modules, Lemma 01B9) and the proof is complete.

Proof of the claim. For a finite subset \(E' \subset E\) set \(Y' = \Spec(R[x_e; e \in E'])\). For large enough \(E'\) the morphism \(f' : X \to Y \to Y'\) is proper, see Limits, Lemma 0EX1. We fix \(E'\) and \(Y'\) in the following. Write \(R = \colim R_i\) as the colimit of its finite type \(\mathbf{Z}\)-subalgebras. Set \(S_i = \Spec(R_i)\) and \(Y'_i = \Spec(R_i[x_e; e \in E'])\). For \(i\) large enough we can find a diagram \[\xymatrix{ X \ar[d] \ar[r]_{f'} & Y' \ar[d] \ar[r] & S \ar[d] \\ X_i \ar[r]^{f'_i} & Y'_i \ar[r] & S_i }\] with cartesian squares such that \(X_i\) is flat over \(S_i\) and \(X_i \to Y'_i\) is proper. See Limits, Lemmas 01ZM, 04AI, and 081F. The same argument as above shows \(Y'\) and \(X_i\) are tor independent over \(Y'_i\) and hence \[R\Gamma(X, \mathcal{O}_X) = R\Gamma(X_i, \mathcal{O}_{X_i}) \otimes^\mathbf{L}_{R_i[x_e; e \in E']} R[x_e; e \in E']\] by the same reference as above. By Cohomology of Schemes, Lemma 02O6 the complex \(R\Gamma(X_i, \mathcal{O}_{X_i})\) is pseudo-coherent in the derived category of the Noetherian ring \(R_i[x_e; e \in E']\) (see More on Algebra, Lemma 066E). Hence \(R\Gamma(X, \mathcal{O}_X)\) is pseudo-coherent in the derived category of \(R[x_e; e \in E']\), see More on Algebra, Lemma 0650. Since the only nonvanishing cohomology module is \(H^0(X, \mathcal{O}_X)\) we conclude it is a finite \(R[x_e; e \in E']\)-module, see More on Algebra, Lemma 064T. This concludes the proof.

Lemma

Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S }\] of morphisms of schemes. Assume \(X \to S\) is flat, \(f\) is proper, \(\dim(X_y) \leq 1\) for \(y \in Y\), and \(R^1f_*\mathcal{O}_X = 0\). Then \(f_*\mathcal{O}_X\) is \(S\)-flat and formation of \(f_*\mathcal{O}_X\) commutes with arbitrary base change \(S' \to S\).

Proof

We may assume \(Y\) and \(S\) are affine, say \(S = \Spec(A)\). To show the quasi-coherent \(\mathcal{O}_Y\)-module \(f_*\mathcal{O}_X\) is flat relative to \(S\) it suffices to show that \(H^0(X, \mathcal{O}_X)\) is flat over \(A\) (some details omitted). By Lemma 0E7H we have \(H^1(X, \mathcal{O}_X \otimes_A M) = 0\) for every \(A\)-module \(M\). Since also \(\mathcal{O}_X\) is flat over \(A\) we deduce the functor \(M \mapsto H^0(X, \mathcal{O}_X \otimes_A M)\) is exact. Moreover, this functor commutes with direct sums by Cohomology, Lemma 01FF. Then it is an exercise to see that \(H^0(X, \mathcal{O}_X \otimes_A M) = M \otimes_A H^0(X, \mathcal{O}_X)\) functorially in \(M\) and this gives the desired flatness. Finally, if \(S' \to S\) is a morphism of affines given by the ring map \(A \to A'\), then in the affine case just discussed we see that \[H^0(X \times_S S', \mathcal{O}_{X \times_S S'}) = H^0(X, \mathcal{O}_X \otimes_A A') = H^0(X, \mathcal{O}_X) \otimes_A A'\] This shows that formation of \(f_*\mathcal{O}_X\) commutes with any base change \(S' \to S\). Some details omitted.

Lemma

Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S }\] of morphisms of schemes. Let \(s \in S\) be a point. Assume

\(X \to S\) is locally of finite presentation and flat at points of \(X_s\),

\(Y \to S\) is locally of finite presentation,

\(f\) is proper,

the fibres of \(f_s : X_s \to Y_s\) have dimension \(\leq 1\) and \(R^1f_{s, *}\mathcal{O}_{X_s} = 0\),

\(\mathcal{O}_{Y_s} \to f_{s, *}\mathcal{O}_{X_s}\) is an isomorphism.

Then there is an open \(Y_s \subset V \subset Y\) such that (a) \(V\) is flat over \(S\), (b) \(f^{-1}(V)\) is flat over \(S\), (c) \(\dim(X_y) \leq 1\) for \(y \in V\), (d) \(R^1f_*\mathcal{O}_X|_V = 0\), (e) \(\mathcal{O}_V \to f_*\mathcal{O}_X|_V\) is an isomorphism, and (a) – (e) remain true after base change of \(f^{-1}(V) \to V\) by any \(S' \to S\).

Proof

Let \(y \in Y_s\). We may always replace \(Y\) by an open neighbourhood of \(y\). Thus we may assume \(Y\) and \(S\) affine. We may also assume that \(X\) is flat over \(S\), \(\dim(X_y) \leq 1\) for \(y \in Y\), \(R^1f_*\mathcal{O}_X = 0\) universally, and that \(\mathcal{O}_Y \to f_*\mathcal{O}_X\) is surjective, see Lemma 0E7J. (We won’t use all of this.)

Assume \(S\) and \(Y\) affine. Write \(S = \lim S_i\) as a cofiltered of affine Noetherian schemes \(S_i\). By Limits, Lemma 01ZM there exists an element \(0 \in I\) and a diagram \[\xymatrix{ X_0 \ar[rr]_{f_0} \ar[rd] & & Y_0 \ar[ld] \\ & S_0 }\] of finite type morphisms of schemes whose base change to \(S\) is the diagram of the lemma. After increasing \(0\) we may assume \(Y_0\) is affine and \(X_0 \to S_0\) proper, see Limits, Lemmas 081F and 01Z6. Let \(s_0 \in S_0\) be the image of \(s\). As \(Y_s\) is affine, we see that \(R^1f_{s, *}\mathcal{O}_{X_s} = 0\) is equivalent to \(H^1(X_s, \mathcal{O}_{X_s}) = 0\). Since \(X_s\) is the base change of \(X_{0, s_0}\) by the faithfully flat map \(\kappa(s_0) \to \kappa(s)\) we see that \(H^1(X_{0, s_0}, \mathcal{O}_{X_{0, s_0}}) = 0\) and hence \(R^1f_{0, *}\mathcal{O}_{X_{0, s_0}} = 0\). Similarly, as \(\mathcal{O}_{Y_s} \to f_{s, *}\mathcal{O}_{X_s}\) is an isomorphism, so is \(\mathcal{O}_{Y_{0, s_0}} \to f_{0, *}\mathcal{O}_{X_{0, s_0}}\). Since the dimensions of the fibres of \(X_s \to Y_s\) are at most \(1\), the same is true for the morphism \(X_{0, s_0} \to Y_{0, s_0}\). Finally, since \(X \to S\) is flat, after increasing \(0\) we may assume \(X_0\) is flat over \(S_0\), see Limits, Lemma 04AI. Thus it suffices to prove the lemma for \(X_0 \to Y_0 \to S_0\) and the point \(s_0\).

Combining the reduction arguments above we reduce to the case where \(S\) and \(Y\) affine, \(S\) Noetherian, the fibres of \(f\) have dimension \(\leq 1\), and \(R^1f_*\mathcal{O}_X = 0\) universally. Let \(y \in Y_s\) be a point. Claim: \[\mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_X)_y\] is an isomorphism. The claim implies the lemma. Namely, since \(f_*\mathcal{O}_X\) is coherent (Cohomology of Schemes, Proposition 02O5) the claim means we can replace \(Y\) by an open neighbourhood of \(y\) and obtain an isomorphism \(\mathcal{O}_Y \to f_*\mathcal{O}_X\). Then we conclude that \(Y\) is flat over \(S\) by Lemma 0E7K. Finally, the isomorphism \(\mathcal{O}_Y \to f_*\mathcal{O}_X\) remains an isomorphism after any base change \(S' \to S\) by the final statement of Lemma 0E7K.

Proof of the claim. We already know that \(\mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_X)_y\) is surjective (Lemma 0E7J) and that \((f_*\mathcal{O}_X)_y\) is \(\mathcal{O}_{S, s}\)-flat (Lemma 0E7K) and that the induced map \[\mathcal{O}_{Y_s, y} = \mathcal{O}_{Y, y}/\mathfrak m_s\mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_X)_y/\mathfrak m_s (f_*\mathcal{O}_X)_y \to (f_{s, *}\mathcal{O}_{X_s})_y\] is injective by the assumption in the lemma. Then it follows from Algebra, Lemma 00ME that \(\mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_X)_y\) is injective as desired.

Lemma

Let \(f : X \to Y\) be a proper morphism of Noetherian schemes such that \(f_*\mathcal{O}_X = \mathcal{O}_Y\), such that the fibres of \(f\) have dimension \(\leq 1\), and such that \(H^1(X_y, \mathcal{O}_{X_y}) = 0\) for \(y \in Y\). Then \(f^* : \Pic(Y) \to \Pic(X)\) is a bijection onto the subgroup of \(\mathcal{L} \in \Pic(X)\) with \(\mathcal{L}|_{X_y} \cong \mathcal{O}_{X_y}\) for all \(y \in Y\).

Proof

By the projection formula (Cohomology, Lemma 01E8) we see that \(f_*f^*\mathcal{N} \cong \mathcal{N}\) for \(\mathcal{N} \in \Pic(Y)\). We claim that for \(\mathcal{L} \in \Pic(X)\) with \(\mathcal{L}|_{X_y} \cong \mathcal{O}_{X_y}\) for all \(y \in Y\) we have \(\mathcal{N} = f_*\mathcal{L}\) is invertible and \(\mathcal{L} \cong f^*\mathcal{N}\). This will finish the proof.

The \(\mathcal{O}_Y\)-module \(\mathcal{N} = f_*\mathcal{L}\) is coherent by Cohomology of Schemes, Proposition 02O5. Thus to see that it is an invertible \(\mathcal{O}_Y\)-module, it suffices to check on stalks (Algebra, Lemma 00NX). Since the map from a Noetherian local ring to its completion is faithfully flat, it suffices to check the completion \((f_*\mathcal{L})_y^\wedge\) is free (see Algebra, Section 0BNH and Lemma 00O1). For this we will use the theorem of formal functions as formulated in Cohomology of Schemes, Lemma 02OD. Since \(f_*\mathcal{O}_X = \mathcal{O}_Y\) and hence \((f_*\mathcal{O}_X)_y^\wedge \cong \mathcal{O}_{Y, y}^\wedge\), it suffices to show that \(\mathcal{L}|_{X_n} \cong \mathcal{O}_{X_n}\) for each \(n\) (compatibly for varying \(n\). By Lemma 0C6R we have an exact sequence \[H^1(X_y, \mathfrak m_y^n\mathcal{O}_X/\mathfrak m_y^{n + 1}\mathcal{O}_X) \to \Pic(X_{n + 1}) \to \Pic(X_n)\] with notation as in the theorem on formal functions. Observe that we have a surjection \[\mathcal{O}_{X_y}^{\oplus r_n} \cong \mathfrak m_y^n/\mathfrak m_y^{n + 1} \otimes_{\kappa(y)} \mathcal{O}_{X_y} \longrightarrow \mathfrak m_y^n\mathcal{O}_X/\mathfrak m_y^{n + 1}\mathcal{O}_X\] for some integers \(r_n \geq 0\). Since \(\dim(X_y) \leq 1\) this surjection induces a surjection on first cohomology groups (by the vanishing of cohomology in degrees \(\geq 2\) coming from Cohomology, Proposition 02UZ). Hence the \(H^1\) in the sequence is zero and the transition maps \(\Pic(X_{n + 1}) \to \Pic(X_n)\) are injective as desired.

We still have to show that \(f^*\mathcal{N} \cong \mathcal{L}\). This is proved by the same method and we omit the details.

Affine stratifications

This material is taken from [RV]. Please read a little bit about stratifications in Topology, Section 09XY before reading this section.

If \(X\) is a scheme, then a stratification of \(X\) usually means a stratification of the underlying topological space of \(X\). The strata are locally closed subsets. We will view these strata as reduced locally closed subschemes of \(X\) using Schemes, Remark 0F2L.

Definition

Let \(X\) be a scheme. An affine stratification is a locally finite stratification \(X = \coprod_{i \in I} X_i\) whose strata \(X_i\) are affine and such that the inclusion morphisms \(X_i \to X\) are affine.

The condition that a stratification \(X = \coprod X_i\) is locally finite is, in the presence of the condition that the inclusion morphisms \(X_i \to X\) are quasi-compact, equivalent to the condition that the strata are locally constructible subsets of \(X\), see Properties, Lemma 0F2M.

The condition that \(X_i \to X\) is an affine morphism is independent on the scheme structure we put on the locally closed subset \(X_i\), see Lemma 09ZV. Moreover, if \(X\) is separated (or more generally has affine diagonal) and \(X = \coprod X_i\) is a locally finite stratification with affine strata, then the morphisms \(X_i \to X\) are affine. See Morphisms, Lemma 01SG. This allows us to disregard the condition of affineness of the inclusion morphisms \(X_i \to X\) in most cases of interest.

We are often interested in the case where the partially ordered index set \(I\) of the stratification is finite. Recall that the length of a partially ordered set \(I\) is the supremum of the lengths \(p\) of chains \(i_0 < i_1 < \ldots < i_p\) of elements of \(I\).

Lemma

Let \(X\) be a scheme. Let \(X = \coprod_{i \in I} X_i\) be a finite affine stratification. There exists an affine stratification with index set \(\{0, \ldots, n\}\) where \(n\) is the length of \(I\).

Proof

Recall that we have a partial ordering on \(I\) such that the closure of \(X_i\) is contained in \(\bigcup_{j \leq i} X_j\) for all \(i \in I\). Let \(I' \subset I\) be the set of maximal indices of \(I\). If \(i \in I'\), then \(X_i\) is open in \(X\) because the union of the closures of the other strata is the complement of \(X_i\). Let \(U = \bigcup_{i \in I'} X_i\) viewed as an open subscheme of \(X\) so that \(U_{red} = \coprod_{i \in I'} X_i\) as schemes. Then \(U\) is an affine scheme by Schemes, Lemma 01I5 and Lemma 06AD. The morphism \(U \to X\) is affine as each \(X_i \to X\), \(i \in I'\) is affine by the same reasoning using Lemma 09ZV. The complement \(Z = X \setminus U\) endowed with the reduced induced scheme structure has the affine stratification \(Z = \bigcup_{i \in I \setminus I'} X_i\). Here we use that a morphism of schemes \(T \to Z\) is affine if and only if the composition \(T \to X\) is affine; this follows from Morphisms, Lemmas 01SE, 01SC, and 01SG. Observe that the partially ordered set \(I \setminus I'\) has length exactly one less than the length of \(I\). Hence by induction we find that \(Z\) has an affine stratification \(Z = Z_0 \amalg \ldots \amalg Z_{n - 1}\) with index set \(\{1, \ldots, n\}\). Setting \(Z_n = U\) we obtain the desired stratification of \(X\).

If a scheme \(X\) has a finite affine stratification, then of course \(X\) is quasi-compact. A bit less obvious is the fact that it forces \(X\) to be quasi-separated as well.

Lemma

Let \(X\) be a scheme. The following are equivalent

\(X\) has a finite affine stratification, and

\(X\) is quasi-compact and quasi-separated.

Proof

Let \(X = \bigcup X_i\) be a finite affine stratification. Since each \(X_i\) is affine hence quasi-compact, we conclude that \(X\) is quasi-compact. Let \(U, V \subset X\) be affine open. Then \(U \cap X_i\) and \(V \cap X_i\) are affine open in \(X_i\) since \(X_i \to X\) is an affine morphism. Hence \(U \cap V \cap X_i\) is an affine open of the affine scheme \(X_i\) (see Schemes, Lemma 01KP for example). Therefore \(U \cap V = \coprod U \cap V \cap X_i\) is quasi-compact as a finite union of affine strata. We conclude that \(X\) is quasi-separated by Schemes, Lemma 01KO.

Assume \(X\) is quasi-compact and quasi-separated. We may use the induction principle of Cohomology of Schemes, Lemma 08DR to prove the assertion that \(X\) has a finite affine stratification. If \(X\) is empty, then it has an empty affine stratification. If \(X\) is nonempty affine then it has an affine stratification with one stratum. Next, assume \(X = U \cup V\) where \(U\) is quasi-compact open, \(V\) is affine open, and we have a finite affine stratifications \(U = \bigcup_{i \in I} U_i\) and \(U \cap V = \coprod_{j \in J} W_j\). Denote \(Z = X \setminus V\) and \(Z' = X \setminus U\). Note that \(Z\) is closed in \(U\) and \(Z'\) is closed in \(V\). Observe that \(U_i \cap Z\) and \(U_i \cap W_j = U_i \times_U W_j\) are affine schemes affine over \(U\). (Hints: use that \(U_i \times_U W_j \to W_j\) is affine as a base change of \(U_i \to U\), hence \(U_i \cap W_j\) is affine, hence \(U_i \cap W_j \to U_i\) is affine, hence \(U_i \cap W_j \to U\) is affine.) It follows that \[U = \coprod\nolimits_{i \in I} (U_i \cap Z) \amalg \coprod\nolimits_{(i, j) \in I \times J} (U_i \cap W_j)\] is a finite affine stratification with partial ordering on \(I \amalg I \times J\) given by \(i' \leq (i, j) \Leftrightarrow i' \leq i\) and \((i', j') \leq (i, j) \Leftrightarrow i' \leq i\) and \(j' \leq j\). Observe that \((U_i \cap Z) \times_X V = \emptyset\) and \((U_i \cap W_j) \times_X V = U_i \cap W_j\) are affine. Hence the morphisms \(U_i \cap Z \to X\) and \(U_i \cap W_j \to X\) are affine because we can check affineness of a morphism locally on the target (Morphisms, Lemma 01S8) and we have affineness over both \(U\) and \(V\). To finish the proof we take the stratification above and we add one additional stratum, namely \(Z'\), whose index we add as a minimal element to the partially ordered set.

Definition

Let \(X\) be a nonempty quasi-compact and quasi-separated scheme. The affine stratification number is the smallest integer \(n \geq 0\) such that the following equivalent conditions are satisfied

there exists a finite affine stratification \(X = \coprod_{i \in I} X_i\) where \(I\) has length \(n\),

there exists an affine stratification \(X = X_0 \amalg X_1 \amalg \ldots \amalg X_n\) with index set \(\{0, \ldots, n\}\).

The equivalence of the conditions holds by Lemma 0F2T. The existence of a finite affine stratification is proven in Lemma 0F2U.

Lemma

Let \(X\) be a separated scheme which has an open covering by \(n + 1\) affines. Then the affine stratification number of \(X\) is at most \(n\).

Proof

Say \(X = U_0 \cup \ldots \cup U_n\) is an affine open covering. Set \[X_i = (U_i \cup \ldots \cup U_n) \setminus (U_{i + 1} \cup \ldots \cup U_n)\] Then \(X_i\) is affine as a closed subscheme of \(U_i\). The morphism \(X_i \to X\) is affine by Morphisms, Lemma 01SG. Finally, we have \(\overline{X_i} \subset X_i \cup X_{i - 1} \cup \ldots X_0\).

Lemma

Let \(X\) be a Noetherian scheme of dimension \(\infty > d \geq 0\). Then the affine stratification number of \(X\) is at most \(d\).

Proof

By induction on \(d\). If \(d = 0\), then \(X\) is affine, see Properties, Lemma 0AAX. Assume \(d > 0\). Let \(\eta_1, \ldots, \eta_n\) be the generic points of the irreducible components of \(X\) (Properties, Lemma 0BA8). We can cover \(X\) by affine opens containing \(\eta_1, \ldots, \eta_n\), see Properties, Lemma 01ZX. Since \(X\) is quasi-compact we can find a finite affine open covering \(X = \bigcup_{j = 1, \ldots, m} U_j\) with \(\eta_1, \ldots, \eta_n \in U_j\) for all \(j = 1, \ldots, m\). Choose an affine open \(U \subset U_1 \cap \ldots \cap U_m\) containing \(\eta_1, \ldots, \eta_n\) (possible by the lemma already quoted). Then the morphism \(U \to X\) is affine because \(U \to U_j\) is affine for all \(j\), see Morphisms, Lemma 01S8. Let \(Z = X \setminus U\). By construction \(\dim(Z) < \dim(X)\). By induction hypothesis we can find an affine stratification \(Z = \bigcup_{i \in \{0, \ldots, n\}} Z_i\) of \(Z\) with \(n \leq \dim(Z)\). Setting \(U = X_{n + 1}\) and \(X_i = Z_i\) for \(i \leq n\) we conclude.

Proposition

Let \(X\) be a nonempty quasi-compact and quasi-separated scheme with affine stratification number \(n\). Then \(H^p(X, \mathcal{F}) = 0\), \(p > n\) for every quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\).

Proof

We will prove this by induction on the affine stratification number \(n\). If \(n = 0\), then \(X\) is affine and the result is Cohomology of Schemes, Lemma 01XB. Assume \(n > 0\). By Definition 0F2V there is an affine scheme \(U\) and an affine open immersion \(j : U \to X\) such that the complement \(Z\) has affine stratification number \(n - 1\). As \(U\) and \(j\) are affine we have \(H^p(X, j_*(\mathcal{F}|_U)) = 0\) for \(p > 0\), see Cohomology of Schemes, Lemmas 089W and 01XC. Denote \(\mathcal{K}\) and \(\mathcal{Q}\) the kernel and cokernel of the map \(\mathcal{F} \to j_*(\mathcal{F}|_U)\). Thus we obtain an exact sequence \[0 \to \mathcal{K} \to \mathcal{F} \to j_*(\mathcal{F}|_U) \to \mathcal{Q} \to 0\] of quasi-coherent \(\mathcal{O}_X\)-modules (see Schemes, Section 01LA). A standard argument, breaking our exact sequence into short exact sequences and using the long exact cohomology sequence, shows it suffices to prove \(H^p(X, \mathcal{K}) = 0\) and \(H^p(X, \mathcal{Q}) = 0\) for \(p \geq n\). Since \(\mathcal{F} \to j_*(\mathcal{F}|_U)\) restricts to an isomorphism over \(U\), we see that \(\mathcal{K}\) and \(\mathcal{Q}\) are supported on \(Z\). By Properties, Lemma 01PG we can write these modules as the filtered colimits of their finite type quasi-coherent submodules. Using the fact that cohomology of sheaves on \(X\) commutes with filtered colimits, see Cohomology, Lemma 01FF, we conclude it suffices to show that if \(\mathcal{G}\) is a finite type quasi-coherent module whose support is contained in \(Z\), then \(H^p(X, \mathcal{G}) = 0\) for \(p \geq n\). Let \(Z' \subset X\) be the scheme theoretic support of \(\mathcal{G} \oplus \mathcal{O}_Z\); we may and do think of \(\mathcal{G}\) as a quasi-coherent module on \(Z'\), see Morphisms, Section 056H. Then \(Z'\) and \(Z\) have the same underlying topological space and hence the same affine stratification number, namely \(n - 1\). Hence \(H^p(X, \mathcal{G}) = H^p(Z', \mathcal{G})\) (equality by Cohomology of Schemes, Lemma 089W) vanishes for \(p \geq n\) by induction hypothesis.

Example

Let \(k\) be a field and let \(X = \mathbf{P}^n_k\) be \(n\)-dimensional projective space over \(k\). Lemma 0F2W applies to this by Constructions, Lemma 01NG. Hence the affine stratification number of \(\mathbf{P}^n_k\) is at most \(n\). On the other hand, we have nonzero cohomology in degree \(n\) for some quasi-coherent modules on \(\mathbf{P}^n_k\), see Cohomology of Schemes, Lemma 01XT. Using Proposition 0F2Y we conclude that the affine stratification number of \(\mathbf{P}^n_k\) is equal to \(n\).

Universally open morphisms

Some material on universally open morphisms.

Definition

Let \(f : X \to S\) be a morphism of schemes and let \(E \subset |X|\). For a morphism \(S' \to S\), denote \(E_{S'} \subset |X_{S'}|\) the inverse image of \(E\).

We say that \(f\) is open along \(E\) if, for every \(x \in E\) and every open neighbourhood \(U \subset X\) of \(x\), the image \(f(U)\) is a neighbourhood of \(f(x)\).

We say that \(f\) is universally open along \(E\) if, for every morphism \(S' \to S\), the base change \(X_{S'} \to S'\) is open along \(E_{S'}\).

We also say that \(f\) is (universally) open at the points of \(E\).

Remark

For a morphism \(f : X \to S\) which is locally of finite presentation, universal openness is equivalent to the lifting of generalizations after every base change. This follows by combining Morphisms, Lemmas 01U1 and 040F. Over a locally Noetherian base, the specialization lemma for discrete valuation rings, Properties, Lemma 054F, explains the usual valuative test: after making the relevant base change, a failure to lift a generalization can be witnessed on a trait.

This also explains the irreducible-component formulation used in the cited source. If an irreducible component of a locally Noetherian base change of \(X\) is contained over a proper closed subset of the corresponding component of the base, then an open neighbourhood of its generic point can be chosen whose image is not a neighbourhood. Such a vertical component obstructs universal openness. The general lifting criterion is the precise statement and remains valid when a simple component formulation would require extra hypotheses.

For the group schemes considered in the cited source, Lemma more-morphisms-lemma-prime-to-characteristic-locus-universally-open and Corollary more-morphisms-corollary-prime-to-characteristic-component-openness give the proved universal-openness statement for the prime-to-characteristic component locus. The source’s parenthetical suggestion that the full locus of torsion components is “most often” universally open is explicitly tentative and is not used as a theorem here.

Lemma

Let \(f : X \to S\) be a morphism of schemes and let \(E \subset |X|\).

Universal openness along \(E\) is preserved by arbitrary base change.

The morphism \(f\) is universally open if and only if it is universally open along \(|X|\).

Let \(g : Z \to X\) be an \(S\)-morphism. If \(Z \to S\) is universally open, then \(f\) is universally open along the image of \(|g|\).

Proof

The first assertion follows from the definition. For the second, only the converse requires a comment. After any base change, if \(U \subset X\) is open, then every point of \(f(U)\) has a neighbourhood contained in \(f(U)\); hence \(f(U)\) is open. For (3), after any base change, an open neighbourhood of a point in the image of \(|g|\) pulls back to an open neighbourhood of a point of \(Z\). Its image in the base is a neighbourhood and is contained in the image of the original open.

Lemma

Let \(f : X \to S\) be a morphism of schemes and let \(E \subset Z \subset |X|\). Assume

\(f\) is universally open along \(E\), and

for every algebraically closed field \(k\) and every morphism \(\Spec(k) \to S\), the inverse image \(E_k\) is dense in \(Z_k\).

Then \(f\) is universally open along \(Z\).

Proof

The hypotheses have the same form after arbitrary base change, so it is enough to prove openness along \(Z\). Let \(z \in Z\), let \(s=f(z)\), and let \(U \subset X\) be an open neighbourhood of \(z\). Choose an algebraic closure \(k\) of \(\kappa(s)\) and a point \(\overline{z} \in X_k\) above \(z\). The open subset \(U_k \cap Z_k\) contains \(\overline{z}\) and hence meets \(E_k\). The image of a point of this intersection is a point \(e \in U \cap E\) lying over \(s\). Since \(f\) is open at \(e\), the set \(f(U)\) is a neighbourhood of \(s\).

Lemma

Let \(f : X \to S\) be a morphism which is locally of finite type, where \(S\) is locally Noetherian. Let \(x \in X\) and set \(s=f(x)\). Assume

\(\mathcal{O}_{S,s}\) is reduced,

\(X_s\) is geometrically reduced at \(x\), and

\(f\) is universally open at the generic points of every irreducible component of \(X_s\) which contains \(x\).

Then \(f\) is flat at \(x\).

Proof

We use the Noetherian valuative criterion for flatness in [EGA, IV, 11.8.1]. Thus it is enough to consider a local map \[\mathcal{O}_{S,s} \longrightarrow R\] to a discrete valuation ring and a point \(x'\) of \(X_R\) over the closed point of \(\Spec(R)\) which maps to \(x\). Universal openness is preserved by base change. Moreover, the generic point of every irreducible component of the closed fibre of \(X_R\) through \(x'\) maps to the generic point of an irreducible component of \(X_s\) through \(x\). Finally, the closed fibre is reduced at \(x'\) by the geometric reducedness hypothesis.

Let \(C=\mathcal{O}_{X_R,x'}\) and let \(\pi \in R\) be a uniformizer. We claim that \(\pi\) is not contained in a minimal prime of \(C\). Otherwise, since \(C\) has only finitely many minimal primes, there would be an open neighbourhood of such a minimal prime which meets no other irreducible component of \(\Spec(C)\). This neighbourhood would be contained in the closed fibre. The minimal prime in question is then the generic point of an irreducible component of the closed fibre through \(x'\), contradicting openness there.

We next show that \(\pi\) is a nonzerodivisor on \(C\). Suppose that \(\pi^n c=0\). Since \(\pi\) avoids every minimal prime of \(C\), the element \(c\) belongs to every minimal prime of \(C\). Hence its image in \(C/\pi C\) belongs to every minimal prime of that ring. The ring \(C/\pi C\) is reduced, so \(c \in \pi C\). Starting with an element \(c\) such that \(\pi c=0\) and applying this observation repeatedly, we obtain \(c \in \pi^rC\) for every \(r\). Krull’s intersection theorem, Algebra, Lemma 00IP, gives \(c=0\). Consequently \(C\) is torsion free over \(R\), and hence flat by More on Algebra, Lemma 0539. The valuative criterion now proves that \(f\) is flat at \(x\).

Lemma

Let \(f : X \to S\) be a morphism which is locally of finite type, where \(S\) is locally Noetherian. Let \(s \in S\) and let \(U \subset X_s\) be open. Assume

\(f\) is universally open along \(U\),

\(X_s\) is smooth over \(\kappa(s)\), and

\(\mathcal{O}_{S,s}\) is reduced.

Then \(f\) is flat and smooth at every point of \(U\).

Proof

Let \(x \in U\). The generic point of every irreducible component of \(X_s\) through \(x\) belongs to \(U\). Thus Lemma more-morphisms-lemma-universally-open-geometrically-reduced-flat shows that \(f\) is flat at \(x\). Since finite type is finite presentation over a locally Noetherian base by Morphisms, Lemma 01TX, smoothness follows from Morphisms, Lemma 01V8.

Lemma

Let \(S\) be a locally Noetherian scheme and let \(G \to S\) be a group scheme locally of finite type. Assume

\(G \to S\) is universally open along \(G^0\), and

every fibre \(G_s\) is smooth over \(\kappa(s)\).

Then \(G^0\) is open in \(G\). If \(S\) is reduced, then \(G^0 \to S\) is smooth.

Proof

For every \(s \in S\), apply Lemma more-morphisms-lemma-connected-along-section-universally-open-neighbourhood to the neutral section. It shows that \(G^0\) is a neighbourhood of \(G_s^0\) in \(G\). Hence \(G^0\) is open. If \(S\) is reduced, apply Lemma more-morphisms-corollary-universally-open-smooth-fibre-flat with \(U=G_s^0\) for every \(s\). This shows that the restriction \(G^0 \to S\) is smooth. The fibre-smoothness hypothesis is automatic over points of residue characteristic zero by Groupoids, Lemma 047N.

Lemma

Let \(f : X \to S\) be a morphism of schemes. The following are equivalent

\(f\) is universally open,

for every morphism \(S' \to S\) which is locally of finite presentation the base change \(X_{S'} \to S'\) is open, and

for every \(n\) the morphism \(\mathbf{A}^n \times X \to \mathbf{A}^n \times S\) is open.

Proof

It is clear that (1) implies (2) and (2) implies (3). Let us prove that (3) implies (1). Suppose that the base change \(X_T \to T\) is not open for some morphism of schemes \(g : T \to S\). Then we can find some affine opens \(V \subset S\), \(U \subset X\), \(W \subset T\) with \(f(U) \subset V\) and \(g(W) \subset V\) such that \(U \times_V W \to W\) is not open. If we can show that this implies \(\mathbf{A}^n \times U \to \mathbf{A}^n \times V\) is not open, then \(\mathbf{A}^n \times X \to \mathbf{A}^n \times S\) is not open and the proof is complete. This reduces us to the result proved in the next paragraph.

Let \(A \to B\) be a ring map such that \(A' \to B' = A' \otimes_A B\) does not induce an open map of spectra for some \(A\)-algebra \(A'\). As the principal opens give a basis for the topology of \(\Spec(B')\) we conclude that the image of \(D(g)\) in \(\Spec(A')\) is not open for some \(g \in B'\). Write \(g = \sum_{i = 1, \ldots, n} a'_i \otimes b_i\) for some \(n\), \(a'_i \in A'\), and \(b_i \in B\). Consider the element \(h = \sum_{i = 1, \ldots, n} x_i b_i\) in \(B[x_1, \ldots, x_n]\). Assume that \(D(h)\) maps to an open subset under the morphism \[\Spec(B[x_1, \ldots, x_n]) \longrightarrow \Spec(A[x_1, \ldots, x_n])\] in order to get a contradiction. Then \(D(h)\) would map surjectively onto a quasi-compact open \(U \subset \Spec(A[x_1, \ldots, x_n])\). Let \(A[x_1, \ldots, x_n] \to A'\) be the \(A\)-algebra homomorphism sending \(x_i\) to \(a'_i\). This also induces a \(B\)-algebra homomorphism \(B[x_1, \ldots, x_n] \to B'\) sending \(h\) to \(g\). Since \[\xymatrix{ \Spec(B[x_1, \ldots, x_n]) \ar[d] & \Spec(B') \ar[l] \ar[d] \\ \Spec(A[x_1, \ldots, x_n]) & \Spec(A') \ar[l] }\] is cartesian the image of \(D(g)\) in \(\Spec(A')\) is equal to the inverse image of \(U\) in \(\Spec(A')\) and hence open which is the desired contradiction.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. If

\(f\) is locally quasi-finite,

\(Y\) is geometrically unibranch and locally Noetherian, and

every irreducible component of \(X\) dominates an irreducible component of \(Y\),

then \(f\) is universally open.

Proof

For any \(n\) the scheme \(\mathbf{A}^n \times Y\) is geometrically unibranch by Lemma 0DQ2 and Properties, Lemma 0C39. Hence the hypotheses of the lemma hold for the morphisms \(\mathbf{A}^n \times X \to \mathbf{A}^n \times Y\) for all \(n\). By Lemma 0F31 it suffices to prove \(f\) is open. By Morphisms, Lemma 01U1 it suffices to show that generalizations lift along \(f\). Suppose that \(y' \leadsto y\) is a specialization of points in \(Y\) and \(x \in X\) is a point mapping to \(y\). As in Lemma 02LK choose a diagram \[\xymatrix{ u \ar[d] & U \ar[d] \ar[r] & X \ar[d] \\ v & V \ar[r] & Y }\] where \((V, v) \to (Y, y)\) is an elementary étale neighbourhood, \(U \to V\) is finite, \(u\) is the unique point of \(U\) mapping to \(v\), \(U \subset V \times_Y X\) is open, and \(v \mapsto y\) and \(u \mapsto x\). Let \(E\) be an irreducible component of \(U\) passing through \(u\) (there is at least one of these). Since \(U \to X\) is étale, \(E\) maps to an irreducible component of \(X\), which in turn dominates an irreducible component of \(Y\) (by assumption). Since \(U \to V\) is finite hence closed, we conclude that the image \(E' \subset V\) of \(E\) is an irreducible closed subset passing through \(v\) which dominates an irreducible component of \(Y\). Since \(V \to Y\) is étale \(E'\) must be an irreducible component of \(V\) passing through \(v\). Since \(Y\) is geometrically unibranch we see that \(E'\) is the unique irreducible component of \(V\) passing through \(v\) (Lemma 0CB4). Since \(V\) is locally Noetherian we may after shrinking \(V\) assume that \(E' = V\) (equality of sets).

Since \(V \to Y\) is étale we can find a specialization \(v' \leadsto v\) whose image is \(y' \leadsto y\). By the above we can find \(u' \in U\) mapping to \(v'\). Then \(u' \leadsto u\) because \(u\) is the only point of \(U\) mapping to \(v\) and \(U \to V\) is closed. Then finally the image \(x' \in X\) of \(u'\) is a point specializing to \(x\) and mapping to \(y'\) and the proof is complete.

Lemma

Let \(A \to B\) be a ring map. Say \(B\) is generated as an \(A\)-module by \(b_1, \ldots, b_d \in B\). Set \(h = \sum x_ib_i \in B[x_1, \ldots, x_d]\). Then \(\Spec(B) \to \Spec(A)\) is universally open if and only if the image of \(D(h)\) in \(\Spec(A[x_1, \ldots, x_d])\) is open.

Proof

If \(\Spec(B) \to \Spec(A)\) is universally open, then of course the image of \(D(h)\) is open. Conversely, assume the image \(U\) of \(D(h)\) is open. Let \(A \to A'\) be a ring map. It suffices to show that the image of any principal open \(D(g) \subset \Spec(A' \otimes_A B)\) in \(\Spec(A')\) is open. We may write \(g = \sum_{i = 1, \ldots, d} a'_i \otimes b_i\) for some \(a'_i \in A'\). Let \(A[x_1, \ldots, x_n] \to A'\) be the \(A\)-algebra homomorphism sending \(x_i\) to \(a'_i\). This also induces a \(B\)-algebra homomorphism \(B[x_1, \ldots, x_n] \to A' \otimes_A B\) sending \(h\) to \(g\). Since \[\xymatrix{ \Spec(B[x_1, \ldots, x_n]) \ar[d] & \Spec(B') \ar[l] \ar[d] \\ \Spec(A[x_1, \ldots, x_n]) & \Spec(A') \ar[l] }\] is cartesian the image of \(D(g)\) in \(\Spec(A')\) is equal to the inverse image of \(U\) in \(\Spec(A')\) and hence open.

Lemma

Let \(S = \lim S_i\) be a limit of a directed system of schemes with affine transition morphisms. Let \(0 \in I\) and let \(f_0 : X_0 \to Y_0\) be a morphism of schemes over \(S_0\). Assume \(S_0\), \(X_0\), \(Y_0\) are quasi-compact and quasi-separated. Let \(f_i : X_i \to Y_i\) be the base change of \(f_0\) to \(S_i\) and let \(f : X \to Y\) be the base change of \(f_0\) to \(S\). If

\(f\) is locally quasi-finite and universally open, and

\(f_0\) is locally of finite presentation,

then there exists an \(i \geq 0\) such that \(f_i\) is locally quasi-finite and universally open.

Proof

By Limits, Lemma 094M after increasing \(0\) we may assume \(f_0\) is locally quasi-finite. Let \(x \in X\). By étale localization of quasi-finite morphisms we can find a diagram \[\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \\ Y & V \ar[l] }\] where \(V \to Y\) is étale, \(U \subset X_V\) is open, \(U \to V\) is finite, and \(x\) is in the image of \(U \to X\), see Lemma 02LK. After shrinking \(V\) we may assume \(V\) and \(U\) are affine. Since \(X\) is quasi-compact, it follows, by taking a finite disjoint union of such \(V\) and \(U\), that we can make a diagram as above such that \(U \to X\) is surjective. By Limits, Lemmas 01ZM, 01Z4, 07RR, 01ZO, 07RP, and 01Z6 after possibly increasing \(0\) we may assume we have a diagram \[\xymatrix{ X_0 \ar[d] & U_0 \ar[l] \ar[d] \\ Y_0 & V_0 \ar[l] }\] where \(V_0\) is affine, \(V_0 \to Y_0\) is étale, \(U_0 \subset (X_0)_{V_0}\) is open, \(U_0 \to V_0\) is finite, and \(U_0 \to X_0\) is surjective. Since \(V_i \to Y_i\) is étale and hence universally open, follows that it suffices to prove that \(U_i \to V_i\) is universally open for large enough \(i\). This reduces us to the case discussed in the next paragraph.

Let \(A = \colim A_i\) be a filtered colimit of rings. Let \(A_0 \to B_0\) be a ring map. Set \(B = A \otimes_{A_0} B_0\) and \(B_i = A_i \otimes_{A_0} B_0\). Assume \(A_0 \to B_0\) is finite, of finite presentation, and \(A \to B\) is universally open. We have to show that \(A_i \to B_i\) is universally open for \(i\) large enough. Pick \(b_{0, 1}, \ldots, b_{0, d} \in B_0\) which generate \(B_0\) as an \(A_0\)-module. Set \(h_0 = \sum_{j = 1, \ldots, d} x_jb_{0, j}\) in \(B_0[x_1, \ldots, x_d]\). Denote \(h\), resp. \(h_i\) the image of \(h_0\) in \(B[x_1, \ldots, x_d]\), resp. \(B_i[x_1, \ldots, x_d]\). The image \(U\) of \(D(h)\) in \(\Spec(A[x_1, \ldots, x_d])\) is open as \(A \to B\) is universally open. Of course \(U\) is quasi-compact as the image of an affine scheme. For \(i\) large enough there is a quasi-compact open \(U_i \subset \Spec(A_i[x_1, \ldots, x_d])\) whose inverse image in \(\Spec(A[x_1, \ldots, x_d])\) is \(U\), see Limits, Lemma 01Z4. After increasing \(i\) we may assume that \(D(h_i)\) maps into \(U_i\); this follows from the same lemma by considering the pullback of \(U_i\) in \(D(h_i)\). Finally, for \(i\) even larger the morphism of schemes \(D(h_i) \to U_i\) will be surjective by an application of the already used Limits, Lemma 07RR. We conclude \(A_i \to B_i\) is universally open by Lemma 0F33.

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism. Then

the functions \(n_{X/Y}\) of Lemmas 0556 and 055F agree,

if \(X\) is quasi-compact, then \(n_{X/Y}\) attains a maximum \(d < \infty\).

Proof

Agreement of the functions is immediate from the fact that the (geometric) fibres of a locally quasi-finite morphism are discrete, see Morphisms, Lemma 06RT. Boundedness follows from Morphisms, Lemmas 03J5 and 03JA.

Lemma

Let \(f : X \to Y\) be a separated, locally quasi-finite, and universally open morphism of schemes. Let \(n_{X/Y}\) be as in Lemma 0F35. If \(n_{X/Y}(y) \geq d\) for some \(y \in Y\) and \(d \geq 0\), then \(n_{X/Y} \geq d\) in an open neighbourhood of \(y\).

Proof

The question is local on \(Y\) hence we may assume \(Y\) affine. Let \(K\) be an algebraic closure of the residue field \(\kappa(y)\). Our assumption is that \((X_y)_K\) has \(\geq d\) connected components. Then for a suitable quasi-compact open \(X' \subset X\) the scheme \((X'_y)_K\) has \(\geq d\) connected components; details omitted. After replacing \(X\) by \(X'\) we may assume \(X\) is quasi-compact. Then \(f\) is quasi-finite. Let \(x_1, \ldots, x_n\) be the points of \(X\) lying over \(y\). Apply Lemma 02LO to get an étale neighbourhood \((U, u) \to (Y, y)\) and a decomposition \[U \times_Y X = W \amalg \ \coprod\nolimits_{i = 1, \ldots, n} \ \coprod\nolimits_{j = 1, \ldots, m_i} V_{i, j}\] as in locus citatus. Observe that \(n_{X/Y}(y) = \sum_i m_i\) in this situation; some details omitted. Since \(f\) is universally open, we see that \(V_{i, j} \to U\) is open for all \(i, j\). Hence after shrinking \(U\) we may assume \(V_{i, j} \to U\) is surjective for all \(i, j\). This proves that \(n_{U \times_Y X/U} \geq \sum_i m_i = n_{X/Y}(y) \geq d\). Since the construction of \(n_{X/Y}\) is compatible with base change the proof is complete.

Lemma

Let \(f : X \to Y\) be a separated, locally quasi-finite, and universally open morphism of schemes. Let \(n_{X/Y}\) be as in Lemma 0F35. If \(n_{X/Y}\) attains a maximum \(d < \infty\), then the set \[Y_d = \{y \in Y \mid n_{X/Y}(y) = d\}\] is open in \(Y\) and the morphism \(f^{-1}(Y_d) \to Y_d\) is finite.

Proof

The openness of \(Y_d\) is immediate from Lemma 0F36. To prove finiteness over \(Y_d\) we redo the argument of the proof of that lemma. Namely, let \(y \in Y_d\). Then there are at most \(d\) points of \(X\) lying over \(y\). Say \(x_1, \ldots, x_n\) are the points of \(X\) lying over \(y\). Apply Lemma 02LO to get an étale neighbourhood \((U, u) \to (Y, y)\) and a decomposition \[U \times_Y X = W \amalg \ \coprod\nolimits_{i = 1, \ldots, n} \ \coprod\nolimits_{j = 1, \ldots, m_i} V_{i, j}\] as in locus citatus. Observe that \(d = n_{X/Y}(y) = \sum_i m_i\) in this situation; some details omitted. Since \(f\) is universally open, we see that \(V_{i, j} \to U\) is open for all \(i, j\). Hence after shrinking \(U\) we may assume \(V_{i, j} \to U\) is surjective for all \(i, j\) and we may assume \(U\) maps into \(W\). This proves that \(n_{U \times_Y X/U} \geq \sum_i m_i = d\). Since the construction of \(n_{X/Y}\) is compatible with base change we know that \(n_{U \times_Y X/U} = d\). This means that \(W\) has to be empty and we conclude that \(U \times_Y X \to U\) is finite. By Descent, Lemma 02LA this implies that \(X \to Y\) is finite over the image of the open morphism \(U \to Y\). In other words, we see that \(f\) is finite over an open neighbourhood of \(y\) as desired.

Weightings

The material in this section is taken from [SGA4, Exposee XVII, 6.2.4].

Let \(\pi : U \to V\) be a locally quasi-finite morphism of schemes with finite fibres. Given a function \(w : U \to \mathbf{Z}\) we define a function \[\textstyle{\int}_\pi w : V \longrightarrow \mathbf{Z},\quad v \longmapsto \sum\nolimits_{u \in U,\ \pi(u) = v} w(u) [\kappa(u) : \kappa(v)]_s\] Note that the field extensions are finite (Morphisms, Lemma 01TG), \([\kappa' : \kappa]_s\) is the separable degree (Fields, Definition 030L), and the sum is finite as the fibres of \(\pi\) are assumed finite. Another way to compute the value of \(\int_\pi w\) at a point \(v \in V\) is as follows. Choose an algebraically closed field \(k\) and a morphism \(\overline{v} : \Spec(k) \to V\) whose image is \(v\). Then we have \[(\textstyle{\int}_\pi w)(v) = \sum\nolimits_{\overline{u} \in U_{\overline{v}}} w(\overline{u})\] where of course \(w(\overline{u})\) denotes the value of \(w\) at the image \(u\) of the point \(\overline{u}\) under the morphism \(U_{\overline{v}} \to U\). Note that we may view \(\overline{u} \in U_{\overline{v}}\) as morphisms \(\overline{u} : \Spec(k) \to U\) such that \(\pi \circ \overline{u} = \overline{v}\). Namely, since \(U \to V\) is locally quasi-finite with finite fibres, the scheme \(U_{\overline{v}}\) is the spectrum of a finite dimension algebra over \(k\) and all of whose prime ideals are maximal ideals with residue field \(k\). To see that the equality holds, note that the number of morphisms \(\overline{u}\) lying over a given \(u\) is equal to \([\kappa(u) : \kappa(v)]_s\) by Fields, Lemma 09HJ.

Lemma

Given a cartesian square \[\xymatrix{ U \ar[d]_\pi & U' \ar[l]^h \ar[d]^{\pi'} \\ V & V' \ar[l]_g }\] with \(\pi\) locally quasi-finite with finite fibres and a function \(w : U \to \mathbf{Z}\) we have \((\int_\pi w) \circ g = \int_{\pi'} (w \circ h)\).

Proof

This follows immediately from the second description of \(\int_\pi w\) given above. To prove it from the definition, you use that if \(E/F\) is a finite extension of fields and \(F'/F\) is another field extension, then writing \((E \otimes_F F')_{red} = \prod E'_i\) as a product of fields finite over \(F'\), we have \[[E : F]_s = \sum [E'_i : F']_s\] To prove this equality pick an algebraically closed field extension \(\Omega/F'\) and observe that \[\begin{align*} [E : F]_s & = |\Mor_F(E, \Omega)| \\ & = |\Mor_{F'}(E \otimes_F F', \Omega)| \\ & = |\Mor_{F'}((E \otimes_F F')_{red}, \Omega)| \\ & = \sum |\Mor_{F'}(E'_i, \Omega)| \\ & = \sum [E'_i : F']_s \end{align*}\] where we have used Fields, Lemma 09HJ.

Definition

Let \(f : X \to Y\) be a locally quasi-finite morphism. A weighting or a pondération of \(f\) is a map \(w : X \to \mathbf{Z}\) such that for any diagram \[\xymatrix{ X \ar[d]_f & U \ar[l]^h \ar[d]^\pi \\ Y & V \ar[l]_g }\] where \(V \to Y\) is étale, \(U \subset X_V\) is open, and \(U \to V\) finite, the function \(\int_\pi (w \circ h)\) is locally constant.

Of course taking \(w = 0\) we obtain a weighting of any locally quasi-finite morphism \(f\), albeit not a very interesting one. It will turn out that positive weightings, i.e., \(w : X \to \mathbf{Z}_{> 0}\) are the most interesting ones for various purposes.

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism. Let \(w : X \to \mathbf{Z}\) be a weighting. Let \(f' : X' \to Y'\) be the base change of \(f\) by a morphism \(Y' \to Y\). Then the composition \(w' : X' \to \mathbf{Z}\) of \(w\) and the projection \(X' \to X\) is a weighting of \(f'\).

Proof

Consider a diagram \[\xymatrix{ X' \ar[d]_{f'} & U' \ar[l]^{h'} \ar[d]^{\pi'} \\ Y' & V' \ar[l]_{g'} }\] as in Definition 0F3A for the morphism \(f'\). For any \(v' \in V'\) we have to show that \(\int_{\pi'} (w' \circ h')\) is constant in an open neighbourhood of \(v'\). By Lemma 0F39 (and the fact that étale morphisms are open) we may replace \(V'\) by any étale neighbourhood of \(v'\). After replacing \(V'\) by an étale neighbourhood of \(v'\) we may assume that \(U' = U'_1 \amalg \ldots \amalg U'_n\) where each \(U'_i\) has a unique point \(u'_i\) lying over \(v'\) such that \(\kappa(u'_i)/\kappa(v')\) is purely inseparable, see Lemma 02LO. Clearly, it suffices to prove that \(\int_{U'_i \to V'} w'|_{U'_i}\) is constant in a neighbourhood of \(v'\). This reduces us to the case discussed in the next paragraph.

We have \(v' \in V'\) and there is a unique point \(u'\) of \(U'\) lying over \(v'\) with \(\kappa(u')/\kappa(v')\) purely inseparable. Denote \(x \in X\) and \(y \in Y\) the image of \(u'\) and \(v'\). We can find an étale neighbourhood \((V, v) \to (Y, y)\) and an open \(U \subset X_V\) such that \(\pi : U \to V\) is finite and such that there is a unique point \(u \in U\) lying over \(v\) which maps to \(x \in X\) via the projection \(h : U \to X\) such that moreover \(\kappa(u)/\kappa(v)\) is purely inseparable. This is possible by the lemma used above. Consider the morphism \[U'' = U \times_X U' \longrightarrow V \times_Y V' = V''\] Since \(u\) and \(u'\) both map to \(x \in X\) there is a point \(u'' \in U''\) mapping to \((u, u')\). Denote \(v'' \in V''\) the image of \(u''\). After replacing \(V', v'\) by \(V'', v''\) we may assume that the composition \(V' \to Y' \to Y\) factors through a map of étale neighbourhoods \((V', v') \to (V, v)\) such that the induced morphism \(X'_{V'} = X_{V'} \to X_V\) sends \(u'\) to \(u\). Inside the base change \(X'_{V'} = X_{V'}\) we have two open subschemes, namely \(U'\) and the inverse image \(U_{V'}\) of \(U \subset X_V\). By construction both contain a unique point lying over \(v'\), namely \(u'\) for both of them. Thus after shrinking \(V'\) we may assume these open subsets are the same; namely, \(U' \setminus (U' \cap U_{V'})\) and \(U_{V'} \setminus (U' \cap U_{V'})\) have a closed image in \(V'\) and these images do not contain \(v'\). Thus \(U' = U_{V'}\) and we find a cartesian diagram as in Lemma 0F39. Since \(\int_\pi (w \circ h)\) is locally constant by assumption we conclude.

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism. Let \(w : X \to \mathbf{Z}\) be a weighting of \(f\). If \(X' \subset X\) is open, then \(w|_{X'}\) is a weighting of \(f|_{X'} : X' \to Y\).

Proof

Immediate from the definition.

Lemma

Let \(f : X \to Y\) and \(g : Y \to Z\) be locally quasi-finite morphisms. Let \(w_f : X \to \mathbf{Z}\) be a weighting of \(f\) and let \(w_g : Y \to \mathbf{Z}\) be a weighting of \(g\). Then the function \[X \longrightarrow \mathbf{Z},\quad x \longmapsto w_f(x) w_g(f(x))\] is a weighting of \(g \circ f\).

Proof

Let us set \(w_{g \circ f}(x) = w_f(x) w_g(f(x))\) for \(x \in X\). Consider a diagram \[\xymatrix{ X \ar[d]_{g \circ f} & U \ar[l] \ar[d]^\pi \\ Z & W \ar[l] }\] where \(W \to Z\) is étale, \(U \subset X_W\) is open, and \(U \to W\) finite. We have to show that \(\int_\pi w_{g \circ f}|_U\) is locally constant. Choose a point \(w \in W\). By Lemma 0F39 (and the fact that étale morphisms are open) it suffices to show that \(\int_\pi w_{g \circ f}|_U\) is constant after replacing \((W, w)\) by an étale neighbourhood. After replacing \((W, w)\) by an étale neighbourhood we may assume \(U = U_1 \amalg \ldots \amalg U_n\) where each \(U_i\) has a unique point \(u_i\) lying over \(w\) such that \(\kappa(u_i)/\kappa(w)\) is purely inseparable, see Lemma 02LO. Clearly, it suffices to show that \(\int_{U_i \to W} w_{g \circ f}|_{U_i}\) is constant in an étale neighbourhood of \(w\). This reduces us to the case discussed in the next paragraph.

We have \(w \in W\) and there is a unique point \(u \in U\) lying over \(w\) with \(\kappa(u)/\kappa(w)\) purely inseparable. Consider the point \(v = f(u) \in Y\). After replacing \((W, w)\) by an elementary étale neighbourhood we may assume there is an open neighbourhood \(V \subset Y_W\) of \(v\) such that \(V \to W\) is finite, see Lemma 02LK. Then \(f_W^{-1}(V) \cap U\) is an open neighbourhood of \(u\) where \(f_W : X_W \to Y_W\) is the base change of \(f\) to \(W\). Hence after Zariski shrinking \(W\), we may assume \(f_W(U) \subset V\). Thus we obtain morphisms \[U \xrightarrow{a} V \xrightarrow{b} W\] and \(U \to V\) is finite as \(V \to W\) is separated (because finite). Since \(w_f\) and \(w_g\) are weightings of \(f\) and \(g\) we see that \(\int_a w_f|_U\) is locally constant on \(V\) and \(\int_b w_g|_V\) is locally constant on \(W\). Thus after shrinking \(W\) one more time we may assume these functions are constant say with values \(n\) and \(m\). It follows immediately that \(\int_\pi w_{g \circ f}|_U = \int_{b \circ a} w_{g \circ f}|_U\) is constant with value \(nm\) as desired.

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism. Let \(w : X \to \mathbf{Z}\) be a weighting. If \(w(x) > 0\) for all \(x \in X\), then \(f\) is universally open.

Proof

Since the property is preserved by base change, see Lemma 0F3B, it suffices to prove that \(f\) is open. Since we may also replace \(X\) by any open of \(X\), it suffices to prove that \(f(X)\) is open. Let \(y \in f(X)\). Choose \(x \in X\) with \(f(x) = y\). It suffices to prove that \(f(X)\) contains an open neighbourhood of \(y\) and it suffices to do so after replacing \(Y\) by an étale neighbourhood of \(y\). By étale localization of quasi-finite morphisms, see Section 04HF, we may assume there is an open neighbourhood \(U \subset X\) of \(x\) such that \(\pi = f|_U : U \to Y\) is finite. Then \(\int_\pi w|_U\) is locally constant and has positive value at \(y\). Hence \(\pi(U)\) contains an open neighbourhood of \(y\) and the proof is complete.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(f\) is locally quasi-finite, locally of finite presentation, and flat. Then there is a positive weighting \(w : X \to \mathbf{Z}_{> 0}\) of \(f\) given by the rule that sends \(x \in X\) lying over \(y \in Y\) to \[w(x) = \text{length}_{\mathcal{O}_{X, x}} (\mathcal{O}_{X, x}/\mathfrak m_y \mathcal{O}_{X, x}) [\kappa(x) : \kappa(y)]_i\] where \([\kappa' : \kappa]_i\) is the inseparable degree (Fields, Definition 030L).

Proof

Consider a diagram as in Definition 0F3A. Let \(u \in U\) with images \(x, y, v\) in \(X, Y, V\). Then we claim that \[\text{length}_{\mathcal{O}_{X, x}} (\mathcal{O}_{X, x}/\mathfrak m_y \mathcal{O}_{X, x}) = \text{length}_{\mathcal{O}_{U, u}} (\mathcal{O}_{U, u}/\mathfrak m_v \mathcal{O}_{U, u})\] and \[[\kappa(x) : \kappa(y)]_i = [\kappa(u) : \kappa(v)]_i\] The first equality follows as \(\mathcal{O}_{X, x} \to \mathcal{O}_{U, u}\) is a flat local homomorphism such that \(\mathfrak m_y \mathcal{O}_{U, u} = \mathfrak m_v \mathcal{O}_{U, u}\) and \(\mathfrak m_x \mathcal{O}_{U, u} = \mathfrak m_u\) (because \(\mathcal{O}_{Y, y} \to \mathcal{O}_{V, v}\) and \(\mathcal{O}_{X, x} \to \mathcal{O}_{U, u}\) are unramified) and hence the equality by Algebra, Lemma 02M1. The second equality follows because \(\kappa(v)/\kappa(y)\) is a finite separable extension and \(\kappa(u)\) is a factor of \(\kappa(x) \otimes_{\kappa(y)} \kappa(v)\) and hence the inseparable degree is unchanged. Having said this, we see that formation of the function in the lemma commutes with étale base change. This reduces the problem to the discussion of the next paragraph.

Assume that \(f\) is a finite, flat morphism of finite presentation. We have to show that \(\int_f w\) is locally constant on \(Y\). In fact, \(f\) is finite locally free (Morphisms, Lemma 02KB) and we will show that \(\int_f w\) is equal to the degree of \(f\) (which is a locally constant function on \(Y\)). Namely, for \(y \in Y\) we see that \[\begin{align*} (\textstyle{\int}_f w)(y) & = \sum\nolimits_{f(x) = y} \text{length}_{\mathcal{O}_{X, x}} (\mathcal{O}_{X, x}/\mathfrak m_y \mathcal{O}_{X, x}) [\kappa(x) : \kappa(y)]_i [\kappa(x) : \kappa(y)]_s \\ & = \sum\nolimits_{f(x) = y} \text{length}_{\mathcal{O}_{X, x}} (\mathcal{O}_{X, x}/\mathfrak m_y \mathcal{O}_{X, x}) [\kappa(x) : \kappa(y)] \\ & = \text{length}_{\mathcal{O}_{Y, y}}((f_*\mathcal{O}_X)_y/ \mathfrak m_y (f_*\mathcal{O}_X)_y) \end{align*}\] Last equality by Algebra, Lemma 02M0. The final number is the rank of \(f_*\mathcal{O}_X\) at \(y\) as desired.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume

\(f\) is locally quasi-finite, and

\(Y\) is geometrically unibranch and locally Noetherian.

Then there is a weighting \(w : X \to \mathbf{Z}_{\geq 0}\) given by the rule that sends \(x \in X\) lying over \(y \in Y\) to the “generic separable degree” of \(\mathcal{O}_{X, x}^{sh}\) over \(\mathcal{O}_{Y, y}^{sh}\).

Proof

It follows from Algebra, Lemma 05WR that \(\mathcal{O}_{Y, y}^{sh} \to \mathcal{O}_{X, x}^{sh}\) is finite. Since \(Y\) is geometrically unibranch there is a unique minimal prime \(\mathfrak p\) in \(\mathcal{O}_{Y, y}^{sh}\), see More on Algebra, Lemma 06DM. Write \[(\kappa(\mathfrak p) \otimes_{\mathcal{O}_{Y, y}^{sh}} \mathcal{O}_{X, x}^{sh})_{red} = \prod K_i\] as a finite product of fields. We set \(w(x) = \sum [K_i : \kappa(\mathfrak p)]_s\).

Since this definition is clearly insensitive to étale localization, in order to show that \(w\) is a weighting we reduce to showing that if \(f\) is a finite morphism, then \(\int_f w\) is locally constant. Observe that the value of \(\int_f w\) in a generic point \(\eta\) of \(Y\) is just the number of points of the geometric fibre \(X_{\overline{\eta}}\) of \(X \to Y\) over \(\eta\). Moreover, since \(Y\) is unibranch a point \(y\) of \(Y\) is the specialization of a unique generic point \(\eta\). Hence it suffices to show that \((\int_f w)(y)\) is equal to the number of points of \(X_{\overline{\eta}}\). After passing to an affine neighbourhood of \(y\) we may assume \(X \to Y\) is given by a finite ring map \(A \to B\). Suppose \(\mathcal{O}_{Y, y}^{sh}\) is constructed using a map \(\kappa(y) \to k\) into an algebraically closed field \(k\). Then \[\mathcal{O}_{Y, y}^{sh} \otimes_A B = \prod\nolimits_{f(x) = y} \prod\nolimits_{\varphi \in \Mor_{\kappa(y)}(\kappa(x), k)} \mathcal{O}_{X, x}^{sh}\] by Algebra, Lemma 04GH and the lemma used above. Observe that the minimal prime \(\mathfrak p\) of \(\mathcal{O}_{Y, y}^{sh}\) maps to the prime of \(A\) corresponding to \(\eta\). Hence we see that the desired equality holds because the number of points of a geometric fibre is unchanged by a field extension.

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism of schemes. Let \(w : X \to \mathbf{Z}\) be a weighting of \(f\). Let \(y \leadsto y'\) be a specialization of points of \(Y\). Assume \(w(x) = 0\) for all \(x \in X\) with \(f(x) = y\). Then \(w(x') = 0\) for all \(x' \in X\) with \(f(x') = y'\).

Proof

Let \(x' \in X\) be a point mapping to \(y'\). By Lemma 02LK we can choose a diagram \[\xymatrix{ U \ar[r] \ar[dr]_\pi & X_V \ar[d] \ar[r] & X \ar[d]^f \\ & V \ar[r]^\varphi & Y }\] with \(V \to Y\) étale, with a point \(v' \in V\) mapping to \(y'\), with \(U \subset Y_V\) open, such that \(U \to V\) is finite, and such that there exists an unique point \(u' \in U\) mapping to \(v'\) which moreover maps to \(x'\) in \(X\). Note that since \(V \to Y\) is open, there exists a point \(v \in V\) specializing to \(v'\) and mapping to \(y\). By Definition 0F3A the function \(\int_\pi (w \circ h)\) is locally constant where \(h : U \to X\) is the composition of the top horizontal arrows. Since \(\int_\pi (w \circ h)\) is zero in \(v\) by our assumption on \(w\), we conclude that it is zero at \(v'\). However, the value at \(v'\) is \(w(x')[\kappa(u'):\kappa(v')]_s\) and we conclude.

More on weightings

We prove a few more basic properties of weightings. Although at first it appears that weightings can be very wild, it actually turns out the condition imposed in Definition 0F3A is rather strong.

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism. Let \(w : X \to \mathbf{Z}\) be a weighting of \(f\). Then the level sets of the function \(w\) are locally constructible in \(X\).

Proof

In the proof below we will use Lemmas 0GK8 and 0F3B without further mention. We will also use elementary properties of constructible subsets of schemes and topological spaces, see Topology, Section 04ZC and Properties, Section 054B. Using this the reader sees question is local on \(X\) and \(Y\); details omitted. Hence we may assume \(X\) and \(Y\) are affine. If we can find a surjective morphism \(Y' \to Y\) of finite presentation such that the level sets of \(w\) pull back to locally constructible subsets of \(X' = Y' \times_Y X\), then we conclude by Morphisms, Theorem 054K.

Assume \(X\) and \(Y\) affine. We may choose an immersion \(X \to T\) where \(T \to Y\) is finite, see Lemma 05K0. By Morphisms, Lemma 03HW after replacing \(Y\) by \(Y'\) surjective finite locally free over \(Y\), replacing \(X\) by \(Y' \times_Y X\) and \(T\) by a scheme finite locally free over \(Y'\) containing \(Y' \times_Y T\) as a closed subscheme, we may assume \(T\) is finite locally free over \(Y\), contains closed subschemes \(T_i\) mapping isomorphically to \(Y\) such that \(T = \bigcup_{i = 1, \ldots, n} T_i\) (set theoretically). Since \(T_i \subset T\) is a constructible closed subset (as the image of a finitely presented morphism \(Y \to T\) of schemes), we see that for \(I \subset \{1, \ldots, n\}\) the intersection \(\bigcap_{i \in I} T_i\) is a constructible closed subset of \(T\) and hence maps to a constructible closed subset of \(Y\).

For a disjoint union decomposition \(\{1, \ldots, n\} = I_1 \amalg \ldots \amalg I_r\) with nonempty parts consider the subset \(Y_{I_1, \ldots, I_r} \subset Y\) consisting of points \(y \in Y\) such that \(T_y = \{x_1, \ldots, x_r\}\) consists of exactly \(r\) points with \(x_j \in T_i \Leftrightarrow i \in I_j\). By our remarks above this is a constructible partition of \(Y\). There exists an affine scheme \(Y'\) of finite presentation over \(Y\) such that the image of \(Y' \to Y\) is exactly \(Y_{I_1, \ldots, I_r}\), see Algebra, Lemma 00F8. Hence we may assume that \(Y = Y_{I_1, \ldots, I_r}\) for some disjoint union decomposition \(\{1, \ldots, n\} = I_1 \amalg \ldots \amalg I_r\). In this case \(T = T(1) \amalg \ldots \amalg T(r)\) with \(T(j) = \bigcap_{i \in I_j} T_i\) is a decomposition of \(T\) into disjoint closed (and hence open) subsets. Intersecting with the locally closed subscheme \(X\) we obtain an analogous decomposition \(X = X(1) \amalg \ldots \amalg X(r)\) into open and closed parts. The morphism \(X(j) \to Y\) an immersion. Since \(w\) is a weighting, it follows that \(w|_{X(j)}\) is locally constant¹⁶ and we conclude.

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism of schemes. Let \(w : X \to \mathbf{Z}\) be a weighting of \(f\). Let \(E \subset X\) be a dense subset. Assume \(w(x) = 0\) for all \(x \in E\). Then \(w = 0\).

Proof

We suggest reading Lemma more-morphisms-lemma-weighting-specialization first. Let \(x \in X\) with image \(y \in Y\). We will show that \(w(x) = 0\). By Lemma 02LK we can choose a diagram \[\xymatrix{ U \ar[r] \ar[dr]_\pi & X_V \ar[d] \ar[r] & X \ar[d]^f \\ & V \ar[r]^\varphi & Y }\] with \(V \to Y\) étale, with a point \(v \in V\) mapping to \(y\), with \(U \subset Y_V\) open, such that \(U \to V\) is finite, and such that there exists an unique point \(u \in U\) mapping to \(v\) which moreover maps to \(x\) in \(X\). We may and do assume that \(V\) (and hence \(U\)) is affine. By Definition 0F3A the function \(\int_\pi (w \circ h)\) is locally constant where \(h : U \to X\) is the composition of the top horizontal arrows. The set of points \(U_0\) where \(w \circ h\) is \(0\) is a constructible subset of \(U\) by Lemma 0F3G and dense in \(U\) because it contains \(h^{-1}(E)\) (use that \(h\) is open). By Topology, Lemma topology-lemma-dense-in-constructible we see that \(U_0\) contains a dense open, i.e., the set of points where \(w \circ h\) is nonzero is contained in a nowhere dense closed subset \(Z \subset U\). Then \(\pi(Z)\) is a nowhere dense closed subset of \(V\) by Morphisms, Lemma 03HX. Hence \(\int_\pi (w \circ h)\) is zero on a dense open hence vanishes everywhere. However, the value at \(v\) is \(w(x)[\kappa(u):\kappa(v)]_s\) and we conclude.

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism of finite presentation. Let \(w : X \to \mathbf{Z}\) be a weighting of \(f\). Then the level sets of the function \(\int_f w\) are locally constructible in \(Y\).

Proof

By Lemma 0F39 formation of the function \(\int_f w\) commutes with arbitrary base change and by Lemma 0F3B after base change we still have a weighthing. This means that if we can find \(Y' \to Y\) surjective and of finite presentation, then it suffices to prove the result after base change to \(Y'\), see Morphisms, Theorem 054K.

The question is local on \(Y\) hence we may assume \(Y\) is affine. Then \(X\) is quasi-compact and quasi-separated (as \(f\) is of finite presentation). Suppose that \(X = U \cup V\) are quasi-compact open. Then we have \[\textstyle{\int}_f w = \textstyle{\int}_{f|_U} w|_U + \textstyle{\int}_{f|_V} w|_V - \textstyle{\int}_{f|_{U \cap V}} w|_{U \cap V}\] Thus if we know the result for \(w|_U\), \(w|_V\), \(w|_{U \cap V}\) then we know the result for \(w\). By the induction principle (Cohomology of Schemes, Lemma 08DR) it suffices to prove the lemma when \(X\) is affine.

Assume \(X\) and \(Y\) are affine. We may choose an open immersion \(X \to T\) where \(T \to Y\) is finite, see Lemma 05K0. Because we may still base change with a suitable \(Y' \to Y\) we can use Morphisms, Lemma 03HW to reduce to the case where all residue field extensions induced by the morphism \(T \to Y\) (and a foriori induced by \(X \to Y\)) are trivial. In this situation \(\int_f w\) is just taking the sums of the values of \(w\) in fibres. The level sets of \(w\) are locally constructible in \(X\) (Lemma 0F3G). The function \(w\) only takes a finite number of values by Properties, Lemma 0F2M. Hence we conclude by Morphisms, Theorem 054K and some elementary arguments on sums of integers.

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism. Let \(w : X \to \mathbf{Z}_{> 0}\) be a positive weighting of \(f\). Then \(w\) is upper semi-continuous.

Proof

Let \(x \in X\) with image \(y \in Y\). Choose an étale neighbourhood \((V, v) \to (Y, y)\) and an open \(U \subset X_V\) such that \(\pi : U \to V\) is finite and there is a unique point \(u \in U\) mapping to \(v\) with \(\kappa(u)/\kappa(v)\) purely inseparable. See Lemma 02LM. Then \((\int_\pi w|_U)(v) = w(u)\). It follows from Definition 0F3A that after replacing \(V\) by a neighbourhood of \(v\) we we have \(w|_U(u') \leq w|_U(u) = w(x)\) for all \(u' \in U\). Namely, \(w|_U(u')\) occurs as a summand in the expression for \((\int_\pi w|_U)(\pi(u'))\). This proves the lemma because the étale morphism \(U \to X\) is open.

Lemma

Let \(f : X \to Y\) be a separated, locally quasi-finite morphism with finite fibres. Let \(w : X \to \mathbf{Z}_{> 0}\) be a positive weighting of \(f\). Then \(\int_f w\) is lower semi-continuous.

Proof

Let \(y \in Y\). Let \(x_1, \ldots, x_r \in X\) be the points lying over \(y\). Apply Lemma 02LO to get an étale neighbourhood \((U, u) \to (Y, y)\) and a decomposition \[U \times_Y X = W \amalg \ \coprod\nolimits_{i = 1, \ldots, n} \ \coprod\nolimits_{j = 1, \ldots, m_i} V_{i, j}\] as in locus citatus. Observe that \((\int_f w)(y) = \sum w(v_{i, j})\) where \(w(v_{i, j}) = w(x_i)\). Since \(\int_{V_{i, j} \to U} w|_{V_{i, j}}\) is locally constant by definition, we may after shrinking \(U\) assume these functions are constant with value \(w(v_{i, j})\). We conclude that \[\textstyle{\int}_{U \times_Y X \to U} w|_{U \times_Y X} = \textstyle{\int}_{W \to U} w|_W + \sum \textstyle{\int}_{V_{i, j} \to U} w|_{V_{i, j}} = \textstyle{\int}_{W \to U} w|_W + (\int_f w)(y)\] This is \(\geq (\int_f w)(y)\) and we conclude because \(U \to Y\) is open and formation of the integral commutes with base change (Lemma 0F39).

Lemma

Let \(f : X \to Y\) be a locally quasi-finite morphism with \(X\) quasi-compact. Let \(w : X \to \mathbf{Z}\) be a weighting of \(f\). Then \(\int_f w\) attains its maximum.

Proof

It follows from Lemma 0F3G and Properties, Lemma 0F2M that \(w\) only takes a finite number of values on \(X\). It follows from Morphisms, Lemma 03JA that \(X \to Y\) has bounded geometric fibres. This shows that \(\int_f w\) is bounded.

Lemma

Let \(f : X \to Y\) be a separated, locally quasi-finite morphism. Let \(w : X \to \mathbf{Z}_{> 0}\) be a positive weighting of \(f\). Assume \(\int_w f\) attains its maximum \(d\) and let \(Y_d \subset Y\) be the open set of points \(y\) with \((\int_f w)(y) = d\). Then the morphism \(f^{-1}(Y_d) \to Y_d\) is finite.

Proof

Observe that \(Y_d\) is open by Lemma 0F3J. Let \(y \in Y_d\). Say \(x_1, \ldots, x_n\) are the points of \(X\) lying over \(y\). Apply Lemma 02LO to get an étale neighbourhood \((U, u) \to (Y, y)\) and a decomposition \[U \times_Y X = W \amalg \ \coprod\nolimits_{i = 1, \ldots, n} \ \coprod\nolimits_{j = 1, \ldots, m_i} V_{i, j}\] as in locus citatus. Observe that \(d = \sum w(v_{i, j})\) where \(w(v_{i, j}) = w(x_i)\). Since \(\int_{V_{i, j} \to U} w|_{V_{i, j}}\) is locally constant by definition, we may after shrinking \(U\) assume these functions are constant with value \(w(v_{i, j})\). We conclude that \[\textstyle{\int}_{U \times_Y X \to U} w|_{U \times_Y X} = \textstyle{\int}_{W \to U} w|_W + \sum \textstyle{\int}_{V_{i, j} \to U} w|_{V_{i, j}} = \textstyle{\int}_{W \to U} w|_W + (\int_f w)(y)\] This is \(\geq (\int_f w)(y) = d\) and we conclude that \(W\) must be the emptyset. Thus \(U \times_Y X \to U\) is finite. By Descent, Lemma 02LA this implies that \(X \to Y\) is finite over the image of the open morphism \(U \to Y\). In other words, we see that \(f\) is finite over an open neighbourhood of \(y\) as desired.

Lemma

Let \(A \to B\) be a ring map which is finite and of finite presentation. There exists a finitely presented ring map \(A \to A_{univ}\) and an idempotent \(e_{univ} \in B \otimes_A A_{univ}\) such that for any ring map \(A \to A'\) and idempotent \(e \in B \otimes_A A'\) there is a ring map \(A_{univ} \to A'\) mapping \(e_{univ}\) to \(e\).

Proof

Choose \(b_1, \ldots, b_n \in B\) generating \(B\) as an \(A\)-module. For each \(i\) choose a monic \(P_i \in A[x]\) such that \(P_i(b_i) = 0\) in \(B\), see Algebra, Lemma 00GK. Thus \(B\) is a quotient of the finite free \(A\)-algebra \(B' = A[x_1, \ldots, x_n]/(P_1(x_1), \ldots, P_n(x_n))\). Let \(J \subset B'\) be the kernel of the surjection \(B' \to B\). Then \(J =(f_1, \ldots, f_m)\) is finitely generated as \(B\) is a finitely generated \(A\)-algebra, see Algebra, Lemma 00F4. Choose an \(A\)-basis \(b'_1, \ldots, b'_N\) of \(B'\). Consider the algebra \[A_{univ} = A[z_1, \ldots, z_N, y_1, \ldots, y_m]/I\] where \(I\) is the ideal generated by the coefficients in \(A[z_1, \ldots, z_n, y_1, \ldots, y_m]\) of the basis elements \(b'_1, \ldots, b'_N\) of the expression \[(\sum z_j b'_j)^2 - \sum z_j b'_j + \sum y_k f_k\] in \(B'[z_1, \ldots, z_N, y_1, \ldots, y_m]\). By construction the element \(\sum z_j b'_j\) maps to an idempotent \(e_{univ}\) in the algebra \(B \otimes_A A_{univ}\). Moreover, if \(e \in B \otimes_A A'\) is an idempotent, then we can lift \(e\) to an element of the form \(\sum b'_j \otimes a'_j\) in \(B' \otimes_A A'\) and we can find \(a''_k \in A'\) such that \[(\sum b'_j \otimes a'_j)^2 - \sum b'_j \otimes a'_j + \sum f_k \otimes a''_k\] is zero in \(B' \otimes_A A'\). Hence we get an \(A\)-algebra map \(A_{univ} \to A\) sending \(z_j\) to \(a'_j\) and \(y_k\) to \(a''_k\) mapping \(e_{univ}\) to \(e\). This finishes the proof.

Lemma

Let \(X \to Y\) be a morphism of affine schemes which is quasi-finite and of finite presentation. There exists a morphism \(Y_{univ} \to Y\) of finite presentation and an open subscheme \(U_{univ} \subset Y_{univ} \times_Y X\) such that \(U_{univ} \to Y_{univ}\) is finite with the following property: given any morphism \(Y' \to Y\) of affine schemes and an open subscheme \(U' \subset Y' \times_Y X\) such that \(U' \to Y'\) is finite, there exists a morphism \(Y' \to Y_{univ}\) such that the inverse image of \(U_{univ}\) is \(U'\).

Proof

Recall that a finite type morphism is quasi-finite if and only if it has relative dimension \(0\), see Morphisms, Lemma 0397. By Lemma 05FJ applied with \(d = 0\) we reduce to the case where \(X\) and \(Y\) are Noetherian. We may choose an open immersion \(X \to X'\) such that \(X' \to Y\) is finite, see Algebra, Lemma 00QB. Note that if we have \(Y' \to Y\) and \(U'\) as in (2), then \[U' \to Y' \times_Y X \to Y' \times_Y X'\] is open immersion between schemes finite over \(Y'\) and hence is closed as well. We conclude that \(U'\) corresponds to an idempotent in \[\Gamma(Y', \mathcal{O}_{Y'}) \otimes_{\Gamma(Y, \mathcal{O}_Y)} \Gamma(X', \mathcal{O}_{X'})\] whose corresponding open and closed subset is contained in the open \(Y' \times_Y X\). Let \(Y'_{univ} \to Y\) and idempotent \[e'_{univ} \in \Gamma(Y_{univ}, \mathcal{O}_{Y_{univ}}) \otimes_{\Gamma(Y, \mathcal{O}_Y)} \Gamma(X', \mathcal{O}_{X'})\] be the pair constructed in Lemma 0F3M for the ring map \(\Gamma(Y, \mathcal{O}_Y) \to \Gamma(X', \mathcal{O}_{X'})\) (here we use that \(Y\) is Noetherian to see that \(X'\) is of finite presentation over \(Y\)). Let \(U'_{univ} \subset Y'_{univ} \times_Y X'\) be the corresponding open and closed subscheme. Then we see that \[U'_{univ} \setminus Y'_{univ} \times_Y X\] is a closed subset of \(U'_{univ}\) and hence has closed image \(T \subset Y'_{univ}\). If we set \(Y_{univ} = Y'_{univ} \setminus T\) and \(U_{univ}\) the restriction of \(U'_{univ}\) to \(Y_{univ} \times_Y X\), then we see that the lemma is true.

Lemma

Let \(Y = \lim Y_i\) be a directed limit of affine schemes. Let \(0 \in I\) and let \(f_0 : X_0 \to Y_0\) be a morphism of affine schemes which is quasi-finite and of finite presentation. Let \(f : X \to Y\) and \(f_i : X_i \to Y_i\) for \(i \geq 0\) be the base changes of \(f_0\). If \(w : X \to \mathbf{Z}\) is a weighting of \(f\), then for sufficiently large \(i\) there exists a weighting \(w_i : X_i \to \mathbf{Z}\) of \(f_i\) whose pullback to \(X\) is \(w\).

Proof

By Lemma 0F3G the level sets of \(w\) are constructible subsets \(E_k\) of \(X\). This implies the function \(w\) only takes a finite number of values by Properties, Lemma 0F2M. Thus there exists an \(i\) such that \(E_k\) descends to a construcible subset \(E_{i, k}\) in \(X_i\) for all \(k\); moreover, we may assume \(X_i = \coprod E_{i, k}\). This follows as the topological space of \(X\) is the limit in the category of topological spaces of the spectral spaces \(X_i\) along a directed system with spectral transition maps. See Limits, Section 081A and Topology, Section 0A2U. We define \(w_i : X_i \to \mathbf{Z}\) such that its level sets are the constructible sets \(E_{i, k}\).

Choose \(Y_{i, univ} \to Y_i\) and \(U_{i, univ} \subset Y_{i, univ} \times_{Y_i} X_i\) as in Lemma 0F3N. By the universal property of the construction, in order to show that \(w_i\) is a weighting, it would suffice to show that \[\tau_i = \textstyle{\int}_{U_{i, univ} \to Y_{i, univ}} w_i|_{U_{i, univ}}\] is locally constant on \(Y_{i, univ}\). By Lemma 0F3H this function has constructible level sets but it may not (yet) be locally constant. Set \(Y_{univ} = Y_{i, univ} \times_{Y_i} Y\) and let \(U_{univ} \subset Y_{univ} \times_Y X\) be the inverse image of \(U_{i, univ}\). Then, since the pullback of \(w\) to \(Y_{univ} \times_Y X\) is a weighting for \(Y_{univ} \times_Y X \to Y_{univ}\) (Lemma 0F3B) we do have that \[\tau = \textstyle{\int}_{U_{univ} \to Y_{univ}} w_i|_{U_{univ}}\] is locally constant on \(Y_{univ}\). Thus the level sets of \(\tau\) are open and closed. Finally, we have \(Y_{univ} = \lim_{i' \geq i} Y_{i', univ}\) and the level sets of \(\tau\) are the inverse limits of the level sets of \(\tau_{i'}\) (similarly defined). Hence the references above imply that for sufficiently large \(i'\) the level sets of \(\tau_{i'}\) are open as well. For such an index \(i'\) we conclude that \(w_{i'}\) is a weighting of \(f_{i'}\) as desired.

Weightings and affine stratification numbers

In this section we give a bound for the affine stratification number of a scheme which has a certain kind of cover by an affine scheme.

Lemma

Let \(f : X \to Y\) be a morphism of affine schemes which is quasi-finite and of finite presentation. Let \(w : X \to \mathbf{Z}_{> 0}\) be a positive weighting of \(f\). Let \(d < \infty\) be the maximum value of \(\int_f w\). The open \[Y_d = \{y \in Y \mid (\textstyle{\int}_f w)(y) = d \}\] of \(Y\) is affine.

Proof

Observe that \(\int_f w\) attains its maximum by Lemma 0F3K. The set \(Y_d\) is open by Lemma 0F3J. Thus the statement of the lemma makes sense.

Reduction to the Noetherian case; please skip this paragraph. Recall that a finite type morphism is quasi-finite if and only if it has relative dimension \(0\), see Morphisms, Lemma 0397. By Lemma 05FJ applied with \(d = 0\) we can find a quasi-finite morphism \(f_0 : X_0 \to Y_0\) of affine Noetherian schemes and a morphism \(Y \to Y_0\) such that \(f\) is the base change of \(f_0\). Then we can write \(Y = \lim Y_i\) as a directed limit of affine schemes of finite type over \(Y_0\), see Algebra, Lemma 00QN. By Lemma 0F3P we can find an \(i\) such that our weighting \(w\) descends to a weighting \(w_i\) of the base change \(f_i : X_i \to Y_i\) of \(f_0\). Now if the lemma holds for \(f_i, w_i\), then it implies the lemma for \(f\) as formation of \(\int_f w\) commutes with base change, see Lemma 0F39.

Assume \(X\) and \(Y\) Noetherian. Let \(X' \to Y'\) be the base change of \(f\) by a morphism \(g : Y' \to Y\). The formation of \(\int_f w\) and hence the open \(Y_d\) commute with base change. If \(g\) is finite and surjective, then \(Y'_d \to Y_d\) is finite and surjective. In this case proving that \(Y_d\) is affine is equivalent to showing that \(Y'_d\) is affine, see Cohomology of Schemes, Lemma 01YQ.

We may choose an immersion \(X \to T\) with \(T\) finite over \(Y\), see Lemma 05K0. We are going to apply Morphisms, Lemma 03HW to the finite morphism \(T \to Y\). This lemma tells us that there is a finite surjective morphism \(Y' \to Y\) such that \(Y' \times_Y T\) is a closed subscheme of a scheme \(T'\) finite over \(Y'\) which has a special form. By the discussion in the first paragraph, we may replace \(Y\) by \(Y'\), \(T\) by \(T'\), and \(X\) by \(Y' \times_Y X\). Thus we may assume there is an immersion \(X \to T\) (not necessarily open or closed) and closed subschemes \(T_i \subset T\), \(i = 1, \ldots, n\) where

\(T \to Y\) is finite (and locally free),

\(T_i \to Y\) is an isomorphism, and

\(T = \bigcup_{i = 1, \ldots, n} T_i\) set theoretically.

Let \(Y' = \coprod Y_k\) be the disjoint union of the irreducible components of \(Y\) (viewed as integral closed subschemes of \(Y\)). Then we may base change once more by \(Y' \to Y\); here we are using that \(Y\) is Noetherian. Thus we may in addition assume \(Y\) is integral and Noetherian.

We also may and do assume that \(T_i \not = T_j\) if \(i \not = j\) by removing repeats. Since \(Y\) and hence all \(T_i\) are integral, this means that if \(T_i\) and \(T_j\) intersect, then they intersect in a closed subset which maps to a proper closed subset of \(Y\).

Observe that \(V_i = X \cap T_i\) is a locally closed subset which is in addition a closed subscheme of \(X\) hence affine. Let \(\eta \in Y\) and \(\eta_i \in T_i\) be the generic points. If \(\eta \not \in Y_d\), then \(Y_d = \emptyset\) and we’re done. Assume \(\eta \in Y_d\). Denote \(I \in \{1, \ldots, n\}\) the subset of indices \(i\) such that \(\eta_i \in V_i\). For \(i \in I\) the locally closed subset \(V_i \subset T_i\) contains the generic point of the irreducible space \(T_i\) and hence is open. On the other hand, since \(f\) is open (Lemma 0F3C), for any \(x \in X\) we can find an \(i \in I\) and a specialization \(\eta_i \leadsto x\). It follows that \(x \in T_i\) and hence \(x \in V_i\). In other words, we see that \(X = \bigcup_{i \in I} V_i\) set theoretically. We claim that \(Y_d = \bigcap_{i \in I} \Im(V_i \to Y)\); this will finish the proof as the intersection of affine opens \(\Im(V_i \to Y)\) of \(Y\) is affine.

For \(y \in Y\) let \(f^{-1}(\{y\}) = \{x_1, \ldots, x_r\}\) in \(X\). For each \(i \in I\) there is at most one \(j(i) \in \{1, \ldots, x_r\}\) such that \(\eta_i \leadsto x_{j(i)}\). In fact, \(j(i)\) exists and is equal to \(j\) if and only if \(x_j \in V_i\). If \(i \in I\) is such that \(j = j(i)\) exists, then \(V_i \to Y\) is an isomorphism in a neighbourhood of \(x_j \mapsto y\). Hence \(\bigcup_{i \in I,\ j(i) = j} V_i \to Y\) is finite after replacing source and target by neighbourhoods of \(x_j \mapsto y\). Thus the definition of a weighting tells us that \(w(x_j) = \sum_{i \in I,\ j(i) = j} w(\eta_i)\). Thus we see that \[(\textstyle{\int}_f w)(\eta) = \sum\nolimits_{i \in I} w(\eta_i) \geq \sum\nolimits_{j(i)\text{ exists}} w(\eta_i) = \sum\nolimits_j w(x_j) = (\textstyle{\int}_f w)(y)\] Thus equality holds if and only if \(y\) is contained in \(\bigcap_{i \in I} \Im(V_i \to Y)\) which is what we wanted to show.

Proposition

Let \(f : X \to Y\) be a surjective quasi-finite morphism of schemes. Let \(w : X \to \mathbf{Z}_{> 0}\) be a positive weighting of \(f\). Assume \(X\) affine and \(Y\) separated¹⁷ and nonempty. Then the affine stratification number of \(Y\) is at most the number of distinct values of \(\int_f w\) minus \(1\).

Proof

Note that since \(Y\) is separated, the morphism \(X \to Y\) is affine (Morphisms, Lemma 01SG). The function \(\int_f w\) attains its maximum \(d\) by Lemma 0F3K. We will use induction on \(d\). Consider the open subscheme \(Y_d = \{y \in Y \mid (\int_f w)(y) = d\}\) of \(Y\) and recall that \(f^{-1}(Y_d) \to Y_d\) is finite, see Lemma 0F3L. By Lemma 0F3R for every affine open \(W \subset Y\) we have that \(Y_d \cap W\) is affine (this uses that \(W \times_Y X\) is affine, being affine over \(X\)). Hence \(Y_d \to Y\) is an affine morphism of schemes. We conclude that \(f^{-1}(Y_d) = Y_d \times_Y X\) is an affine scheme being affine over \(X\). Then \(f^{-1}(Y_d) \to Y_d\) is surjective and hence \(Y_d\) is affine by Limits, Lemma 01ZT. If \(Y_d = Y\), then the affine stratification number of \(Y\) is \(0\) and we conclude the lemma is true. If not, then let \(Y' = Y \setminus Y_d\) viewed as a reduced closed subscheme of \(Y\). Set \(X' = Y' \times_Y X\). Since \(X'\) is closed in \(X\) it is affine. Since \(Y'\) is closed in \(Y\) it is separated. The morphism \(f' : X' \to Y'\) is surjective and \(w\) induces a weighting \(w'\) of \(f'\) (see Lemma 0F3B) and \(\int_{f'} w'\) is the restriction of \(\int_f w\) to \(Y'\). By induction \(Y'\) has an affine stratification of length \(\leq\) the number of distinct values of \(\int_{f'} w'\) minus \(1\) and the proof is complete.

Completely decomposed morphisms

Nishnevich studied the notion of a completely decomposed family of étale morphisms, in order to define what is now called the Nishnevich topology, see for example [Nishnevich].

Definition

A morphism \(f : X \to Y\) of schemes is said to be completely decomposed¹⁸ if for all points \(y \in Y\) there is a point \(x \in X\) with \(f(x) = y\) such that the field extension \(\kappa(x)/\kappa(y)\) is trivial. A family of morphisms \(\{f_i : X_i \to Y\}_{i \in I}\) of schemes with fixed target is said to be completely decomposed if \(\coprod f_i : \coprod Y_i \to X\) is completely decomposed.

We start with some basic lemmas.

Lemma

The composition of two completely decomposed morphisms of schemes is completely decomposed. If \(\{f_i : X_i \to Y\}_{i \in I}\) is completely decomposed and for each \(i\) we have a family \(\{X_{ij} \to X_i\}_{j \in J_i}\) which is completely decomposed, then the family \(\{X_{ij} \to Y\}_{i \in I, j \in J_i}\) is completely decomposed.

Proof

Omitted.

Lemma

The base change of a completely decomposed morphism of schemes is completely decomposed. If \(\{f_i : X_i \to Y\}_{i \in I}\) is completely decomposed and \(Y' \to Y\) is a morphism of schemes, then \(\{X_i \times_Y Y' \to Y'\}_{i \in I}\) is completely decomposed.

Proof

Let \(f : X \to Y\) and \(g : Y' \to Y\) be morphisms of schemes. Let \(y' \in Y'\) be a point with image \(y = g(y')\) in \(Y\). If \(x \in X\) is a point such that \(f(x) = y\) and \(\kappa(x) = \kappa(y)\), then there exists a unique point \(x' \in X' = X \times_Y Y'\) which maps to \(y'\) in \(Y'\) and to \(x\) in \(X\) and moreover \(\kappa(x') = \kappa(y')\), see Schemes, Lemma 01JT. From this fact the lemma follows easily; we omit the details.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(f\) is completely decomposed, \(f\) is locally of finite presentation, and \(Y\) is quasi-compact and quasi-separated. Then there exist \(n \geq 0\) and morphisms \(Z_i \to Y\), \(i = 1, \ldots, n\) with the following properties

\(\coprod Z_i \to Y\) is surjective,

\(Z_i \to Y\) is an immersion for all \(i\),

\(Z_i \to Y\) is of finite presentation for all \(i\), and

the base change \(X \times_Y Z_i \to Z_i\) has a section for all \(i\).

Proof

Let \(y \in Y\). By assumption there is a morphism \(\sigma : \Spec(\kappa(y)) \to X\) over \(Y\). We can write \(\Spec(\kappa(y))\) as a directed limit of affine schemes \(Z\) over \(Y\) such that \(Z \to Y\) is an immersion of finite presentation. Namely, choose an affine open \(y \in \Spec(A) \subset Y\) and say \(y\) corresponds to the prime ideal \(\mathfrak p\) of \(A\). Then \(\kappa(\mathfrak p)\) is the filtered colimit of the rings \((A/I)_f\) where \(I \subset \mathfrak p\) is a finitely generated ideal and \(f \in A\), \(f \not \in \mathfrak p\). The morphisms \(Z = \Spec((A/I)_f) \to Y\) are immersions of finite presentation; quasi-compactness of \(Z \to Y\) follows as \(Y\) is quasi-separated, see Schemes, Lemma 03GI. By Limits, Proposition 01ZC for some such \(Z\) there is a morphism \(\sigma' : Z \to X\) over \(Y\) agreeing with \(\sigma\) on the spectrum of \(\kappa(\mathfrak p)\). Since \(\sigma'\) is a morphism over \(Y\), we obtain a section of the projection \(X \times_Y Z \to Z\)

We conclude that \(Y\) is the union of the images of immersions \(Z \to Y\) of finite presentation such that \(X \times_Y Z \to Z\) has a section. Since the image of \(Z \to Y\) is constructible (Morphisms, Lemma 054J) and since \(Y\) is compact in the constructible topology (Properties, Lemma 094L and Topology, Lemma 0901), we see that a finite number of these suffice.

Lemma

Let \(S = \lim_{\lambda \in \Lambda} S_\lambda\) be a limit of a directed system of schemes with affine transition morphisms. Let \(0 \in \Lambda\) and let \(f_0 : X_0 \to Y_0\) be a morphism of schemes over \(S_0\). For \(\lambda \geq 0\) let \(f_\lambda : X_\lambda \to Y_\lambda\) be the base change of \(f_0\) to \(S_\lambda\) and let \(f : X \to Y\) be the base change of \(f_0\) to \(S\). If

\(f\) is completely decomposed,

\(Y_0\) is quasi-compact and quasi-separated, and

\(f_0\) is locally of finite presentation,

then there exists an \(\lambda \geq 0\) such that \(f_\lambda\) is completely decomposed.

Proof

Since \(Y_0\) is quasi-compact and quasi-separated, the scheme \(Y\), which is affine over \(Y_0\), is quasi-compact and quasi-separated. Choose \(n \geq 0\) and \(Z_i \to Y\), \(i = 1, \ldots, n\) as in Lemma 0GTL. Denote \(\sigma_i : Z_i \to X\) morphisms over \(Y\) which exist by our choice of \(Z_i\). After increasing \(0 \in \Lambda\) we may assume there exist morphisms \(Z_{i, 0} \to Y_0\) of finite presentation whose base changes to \(S\) are the morphisms \(Z_i \to Y\), see Limits, Lemma 01ZM. By Limits, Lemma 0GTB we may assume, after possibly increasing \(0\), that \(Z_{i, 0} \to Y_0\) is an immersion. Since \(\coprod Z_i \to Y\) is surjective, we may assume, after possibly increasing \(0\), that \(\coprod Z_{i, 0} \to Y_0\) is surjective, see Limits, Lemma 07RR. Observe that \(Z_i = \lim_{\lambda \geq 0} Z_{i, \lambda}\) where \(Z_{i, \lambda} = Y_\lambda \times_{Y_0} Z_{i, 0}\). Let us view the compositions \[Z_i \xrightarrow{\sigma_i} X \to X_0\] as morphisms over \(Y_0\). Since \(f_0\) is locally of finite presentation by Limits, Proposition 01ZC we can find a \(\lambda \geq 0\) such that there exist morphisms \(\sigma'_{i, \lambda} : Z_{i, \lambda} \to X_0\) over \(Y_0\) whose precomposition with \(Z_i \to Z_{i, \lambda}\) are the displayed arrows. Of course, then \(\sigma'_{i, \lambda}\) determines a morphism \(\sigma_{i, \lambda} : Z_{i, \lambda} \to X_\lambda = X_0 \times_{Y_0} Y_\lambda\) over \(Y_\lambda\). Since \(\coprod Z_{i, \lambda} \to Y_\lambda\) is surjective we conclude that \(X_\lambda \to Y_\lambda\) is completely decomposed.

Families of ample invertible modules

We continue the discussion from Morphisms, Section 0FXQ.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume

\(Y\) has an ample family of invertible modules,

there exists an \(f\)-ample invertible module on \(X\).

Then \(X\) has an ample family of invertible modules.

Proof

Let \(\mathcal{L}\) be an \(f\)-ample invertible module on \(X\). This in particular implies that \(f\) is quasi-compact, see Morphisms, Definition 01VH. Since \(Y\) is quasi-compact by Morphisms, Definition 0FXR we see that \(X\) is quasi-compact (and hence \(X\) itself satisfies the first condition of Morphisms, Definition 0FXR). Let \(x \in X\) with image \(y \in Y\). By assumption (2) we can find an invertible \(\mathcal{O}_Y\)-module \(\mathcal{N}\) and a section \(t \in \Gamma(Y, \mathcal{N})\) such that the locus \(Y_t\) where \(t\) does not vanish is affine. Then \(\mathcal{L}\) is ample over \(f^{-1}(Y_t) = X_{f^*t}\) and hence we can find a section \(s \in \Gamma(X_{f^*t}, \mathcal{L})\) such that \((X_{f^*t})_s\) is affine and contains \(x\). By Properties, Lemma 01PW for some \(n \geq 0\) the product \((f^*t)^n s\) extends to a section \(s' \in \Gamma(X, f^*\mathcal{N}^{\otimes n} \otimes \mathcal{L})\). Then finally the section \(s'' = f^* ts'\) of \(f^*\mathcal{N}^{\otimes n + 1} \otimes \mathcal{L}\) vanishes at every point of \(X \setminus X_{f^*t}\) hence we see that \(X_{s''} = (X_{f^*t})_s\) is affine as desired.

Lemma

Let \(f : X \to Y\) be an affine or quasi-affine morphism of schemes. If \(Y\) has an ample family of invertible modules, so does \(X\).

Proof

By Morphisms, Lemma 0891 this is a special case of Lemma 0GTP.

Blowing up and ample families of invertible modules

We prove a result from [Gross-thesis].

Lemma

Let \(X\) be a scheme. Suppose given effective Cartier divisors \(D_1, \ldots, D_m\) on \(X\) and invertible modules \(\mathcal{L}_1, \ldots, \mathcal{L}_m\) such that \(\bigcap D_i = \emptyset\) and \(\mathcal{L}_i|_{X \setminus D_i}\) is ample. Then \(X\) has an ample family of invertible modules.

Proof

Let \(x \in X\). Choose an index \(i \in \{1, \ldots, m\}\) such that \(x \not \in D_i\). Set \(U_i = X \setminus D_i\). Since \(\mathcal{L}_i|_{U_i}\) we can find an \(n \geq 1\) and a section \(s \in \Gamma(U_i, \mathcal{L}_i^{\otimes n})\) such that the locus \((U_i)_s\) where \(s\) doesn’t vanish is affine (Properties, Definition 01PS). Since \(U_i\) is the locus where the canonical section \(1 \in \mathcal{O}_X(D_i)\) doesn’t vanish, we see from Properties, Lemma 01PW there exists an \(N \geq 0\) such that \(s\) extends to a section \[s' \in \Gamma(X, \mathcal{L}_i^{\otimes n} \otimes_{\mathcal{O}_X} \mathcal{O}_X(N D_i))\] After replacing \(N\) by \(N + 1\) we see that \(s'\) vanishes at every point of \(D_i\) and hence that \(X_{s'} = (U_i)_s\) is affine. This proves that \(X\) has an ample family of invertible modules, see Morphisms, Definition 0FXR.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme with finitely many irreducible components. There exists a quasi-compact dense open \(U \subset X\) and a \(U\)-admissible blowing up \(X' \to X\) such that the scheme \(X'\) has an ample family of invertible modules.

Proof

Let \(\eta_1, \ldots, \eta_n \in X\) be the generic points of the irreducible components of \(X\). By Properties, Lemma 01ZX and the fact that \(X\) is quasi-compact we can find a finite affine open covering \(X = U_1 \cup \ldots \cup U_m\) such that each \(U_i\) contains \(\eta_1, \ldots, \eta_n\). In particular the quasi-compact open subset \(U = U_1 \cap \ldots \cap U_m\) is dense in \(X\). Let \(\mathcal{I}_i \subset \mathcal{O}_X\) be a finite type quasi-coherent ideal sheaf such that \(U_i = X \setminus Z_i\) where \(Z_i = V(\mathcal{I}_i)\), see Properties, Lemma 01PH. Let \[f : X' \longrightarrow X\] be the blowing up of \(X\) in the ideal sheaf \(\mathcal{I} = \mathcal{I}_1 \cdots \mathcal{I}_m\). Note that \(f\) is a \(U\)-admissible blowing up as \(V(\mathcal{I})\) is (set theoretically) the union of the \(Z_i\) which are disjoint from \(U\). Also, \(f\) is a projective morphism and \(\mathcal{O}_{X'}(1)\) is \(f\)-relatively ample, see Divisors, Lemma 02NS. By Divisors, Lemma 080A for each \(i\) the morphism \(f'\) factors as \(X' \to X'_i \to X\) where \(X'_i \to X\) is the blowing up in \(\mathcal{I}_i\) and \(X' \to X'_i\) is another blowing up (namely in the pullback of the products of the ideals \(\mathcal{I}_j\) omitting \(\mathcal{I}_i\)). It follows from this that \(D_i = f^{-1}(Z_i) \subset X'\) is an effective Cartier divisor, see Divisors, Lemmas 0809 and 02OS. We have \(X' \setminus D_i = f^{-1}(U_i)\). As \(\mathcal{O}_{X'}(1)\) is \(f\)-ample, the restriction of \(\mathcal{O}_{X'}(1)\) to \(X' \setminus D_i\) is ample. It follows from Lemma 0GTS that \(X'\) has an ample family of invertible modules.

Proposition

Let \(X\) be a quasi-compact and quasi-separated scheme. There exists a morphism \(f : Y \to X\) which is of finite presentation, proper, and completely decomposed (Definition 0GTI) such that the scheme \(Y\) has an ample family of invertible modules.

Proof

By Limits, Proposition 01ZA there exists an affine morphism \(X \to X_0\) where \(X_0\) is a scheme of finite type over \(\mathbf{Z}\). Below we produce a morphism \(Y_0 \to X_0\) with all the desired properties. Then setting \(Y = X \times_{X_0} Y_0\) and \(f\) equal to the projection \(f : Y \to X\) we conclude. To see this observe that \(f\) is of finite presentation (Morphisms, Lemma 01TS), \(f\) is proper (Morphisms, Lemma 01W4), \(f\) is completely decomposed (Lemma 0GTK). Finally, since \(Y \to Y_0\) is affine (as the base change of \(X \to X_0\)) we see that \(Y\) has an ample family of invertible modules by Lemma 0GTQ. This reduces us to the case discussed in the next paragraph.

Assume \(X\) is of finite type over \(\mathbf{Z}\). In particular \(\dim(X) < \infty\). We will argue by induction on \(\dim(X)\). If \(\dim(X) = 0\), then \(X\) is affine and has the resolution property. In general, there exists a dense open \(U \subset X\) and a \(U\)-admissible blowing up \(X' \to X\) such that \(X'\) has an ample family of invertible modules, see Lemma 0GTT. Since \(f : X' \to X\) is an isomorphism over \(U\) we see that every point of \(U\) lifts to a point of \(X'\) with the same residue field. Let \(Z = X \setminus U\) with the reduced induced scheme structure. Then \(\dim(Z) < \dim(X)\) as \(U\) is dense in \(X\) (see above). By induction we find a proper, completely decomposed morphism \(W \to Z\) such that \(W\) has an ample family of invertible modules. Then it follows that \(Y = X' \amalg W \to X\) is the desired morphism.

The extensive criterion for closed immersions

In this section, we give a criterion for a morphism of schemes to be a closed immersion.

Lemma

A morphism \(f : X \to Y\) of affine schemes is a closed immersion if and only if for every injective ring map \(A \to B\) and commutative square \[\xymatrix{ \Spec(B) \ar[d] \ar[r] & X \ar[d]^f \\ \Spec(A) \ar[r] \ar@{..>}[ur] & Y }\] there exists a lift \(\Spec(A) \to X\) making the two triangles commute.

Proof

Let the morphism \(f\) be given by the ring map \(\phi : R \to S\). Then \(f\) is a closed immersion if and only if \(\phi\) is surjective.

First, we assume that \(\phi\) is surjective. Let \(\psi : A \to B\) be an injective ring map, and suppose we are given a commutative diagram \[\xymatrix{ R \ar[r]^\alpha \ar[d]^\phi & A \ar[d]^\psi \\ S \ar[r]^\beta \ar@{..>}[ur] & B }\] Then we define a lift \(S \to A\) by \(s \mapsto \alpha(r)\), where \(r \in R\) is such that \(\phi(r) = s\). This is well-defined because \(\psi\) is injective and the square commutes. Since taking the ring spectrum defines an anti-equivalence between commutative rings and affine schemes, the desired lifting property for \(f\) holds.

Next, we assume that \(\phi\) has lifts against all injective ring maps \(\psi: A \to B\). Note that \(\phi(R)\) is a subring of \(S\), so we obtain a commutative square \[\xymatrix{ R \ar[r] \ar[d]^\phi & \phi(R) \ar[d] \\ S \ar@{=}[r] \ar@{..>}[ur] & S }\] in which a lift \(S \to \phi(R)\) exists. Hence, the inclusion \(\phi(R) \to S\) must be an isomorphism, which shows that \(\phi\) is surjective, and we win.

Lemma

Let \(X\) be a scheme. If the canonical morphism \(X \to \Spec(\Gamma(X, \mathcal{O}_X))\) of Schemes, Lemma 01I1 has a retraction, then \(X\) is an affine scheme.

Proof

Write \(S = \Spec(\Gamma(X, \mathcal{O}_X))\) and \(f : X \to S\) the morphism given in the lemma. Let \(s : S \to X\) be a retraction; so \(\text{id}_X = sf\). Then \(f s f = \text{id}_S f\). Since \(f\) induces an isomorphism \(\Gamma(S, \mathcal{O}_S) \to \Gamma(X, \mathcal{O}_X)\) this means that \(fs\) and \(\text{id}_S\) induce the same map on \(\Gamma(S, \mathcal{O}_S)\). Whence \(f s = \text{id}_S\) as \(S\) is affine. Hence \(f\) is an isomorphism and \(X\) is an affine scheme, as was to be shown.

Lemma

Let \(X\) be a scheme. Let \(f : X \to S = \Spec(\Gamma(X, \mathcal{O}_X))\) be the canonical morphism of Schemes, Lemma 01I1. The largest quasi-coherent \(\mathcal{O}_S\)-module contained in the kernel of \(f^\sharp : \mathcal{O}_S \to f_*\mathcal{O}_X\) is zero. If \(X\) is quasi-compact, then \(f^\sharp\) is injective. In particular, if \(X\) is quasi-compact, then \(f\) is a dominant morphism.

Proof

Let \(M \subset \Gamma(S, \mathcal{O}_S)\) be the submodule corresponding to the largest quasi-coherent \(\mathcal{O}_S\)-module contained in the kernel of \(f^\sharp\). Then any element \(a \in M\) is mapped to zero by \(f^\sharp\). However, \(f^\sharp(a)\) is the element of \[\Gamma(S, f_*\mathcal{O}_X) = \Gamma(X, \mathcal{O}_X) = \Gamma(S, \mathcal{O}_S)\] corresponding to \(a\) itself! Thus \(a = 0\). Hence \(M = 0\) which proves the first assertion. Note that this is equivalent to the morphism \(f : X \to S\) being scheme-theoretically surjective.

If \(X\) is quasi-compact, then \(\Ker(f^\sharp)\) is quasi-coherent by Morphisms, Lemma 01R8. Hence \(\Ker(f^\sharp) = 0\) and \(f^\sharp\) is injective. In this case, \(f\) is a dominant morphism by part (4) of Morphisms, Lemma 01R8.

Lemma

Let \(f: X \to Y\) be a quasi-compact morphism of schemes. Then \(f\) is a closed immersion if and only if for every injective ring map \(A \to B\) and commutative square \[\xymatrix{ \Spec(B) \ar[d] \ar[r] & X \ar[d]^f \\ \Spec(A) \ar[r] \ar@{..>}[ur] & Y }\] there exists a lift \(\Spec A \to X\) making the diagram commute.

Proof

Assume that \(f\) is a closed immersion. Let \(A \to B\) be an injective ring map and consider a commutative square \[\xymatrix{ \Spec(B) \ar[d] \ar[r] & X \ar[d]^f \\ \Spec(A) \ar[r] \ar@{..>}[ur] & Y }\] Then \(\Spec(A) \times_Y X \to \Spec(A)\) is a closed immersion and hence we get an ideal \(I \subset A\) and a commutative diagram \[\xymatrix{ \Spec(B) \ar[d] \ar[r] & \Spec(A/I) \ar[r] \ar[d] & X \ar[d]^f \\ \Spec(A) \ar[r] \ar@{..>}[ur] & \Spec(A) \ar[r] & Y }\] We obtain a lift by Lemma 0H2P.

Assume that \(f\) has the lifting property stated in the lemma. To prove that \(f\) is a closed immersion is local on \(Y\), hence we may and do assume \(Y\) is affine. In particular, \(Y\) is quasi-compact and therefore \(X\) is quasi-compact. Hence there exists a finite affine open covering \(X = U_1 \cup \ldots \cup U_n\). The source of the morphism \[\pi : U = \coprod U_i \longrightarrow X\] is affine and the induced ring map \(\Gamma(X, \mathcal{O}_X) \to \Gamma(U, \mathcal{O}_U)\) is injective. By assumption, there exists a lift in the diagram \[\xymatrix{ U \ar[r]^\pi \ar[d] & X \ar[d]^f \\ \Spec(\Gamma(X, \mathcal{O}_X)) \ar[r]^-{f'} \ar@{..>}[ur]^h & Y }\] where \(f'\) is the morphism of affine schemes corresponding to the ring map \(\Gamma(Y, \mathcal{O}_Y) \to \Gamma(X, \mathcal{O}_X)\). It follows from the fact that \(\pi\) is an epimorphism that the morphism \(h\) is a retraction of the canonical morphism \(X \to \Spec(\Gamma(X, \mathcal{O}_X))\); details omitted. Hence \(X\) is affine by Lemma 0H2Q. By Lemma 0H2P we conclude that \(f\) is a closed immersion.

With the same properties as those enjoyed by \(X' \to S'\) and \(\mathcal{L}'\), i.e., \(X'_0 \to \Spec(R'_0)\) is flat and proper and \(\mathcal{L}'_0\) is ample.↩︎
Actually, one can reduce to this case by doing more limit arguments.↩︎
The other types are coprof \(\leq k\), Cohen-Macaulay, \((S_k)\), regular, \((R_k)\), and reduced. See [EGA, IV Definition 6.8.1.]. Gorenstein morphisms will be defined in Duality for Schemes, Section 0AWV.↩︎
If \(S\) is quasi-separated, then \(g\) will be quasi-finite.↩︎
We will deal with the finite field case in the last paragraph of the proof.↩︎
In fact, it would suffice if \(\kappa(x)\) is geometrically irreducible over \(\kappa(s)\). If we ever need this we will add a detailed proof.↩︎
In the presence of (1) this means that \(f\) is quasi-finite at \(x\), see Morphisms, Lemma 01TH.↩︎
The proof actually gives an open \(X' \subset S' \times_S X\).↩︎
It similarly follows that \(\pi_*\mathcal{O}_L(i) = \bigoplus_{m \geq -i} \mathcal{O}_P(m)\).↩︎
Often \(L\) is a line bundle over \(P\), see below.↩︎
Parts (1) and (2) are clear. To see (3), note that if \(a \in A_d\), then \(\sigma(a) = \sigma(f) \psi(a/f)\). For (4) note that \(b/g^m\) is in the kernel of \(\tau\) if and only if \(b \in A_{\geq md}\) maps to zero in \(A_{md}\). Thus it suffices to show if \(m' > md\) and \(a \in A_{m'}\), then some power of \(a^{(md)}/g^m\) is in the ideal generated by \(\sigma(f)\). Take \(e\) such that \(em' - emd \geq d\). Then \[(a^{(md)}/g^m)^e = (a^e)^{(emd)}/g^{em} = (fa^e)^{(emd + d)}/g^{em + 1} = \sigma(f) \cdot (a^e)^{(emd + d)}/g^{em + 1}\] as desired (apologies for the terrible notation). To see (5) argue as before and note that \(a^{(md)}/g^m = \sigma(f) \cdot a^{(md + 1)}/g^{m + 1}\) if \(d = 1\).↩︎
This means \(f\) is pseudo-coherent, see Definition 067Z.↩︎
This means \(Y \to S\) is pseudo-coherent, see Definition 067Z.↩︎
It suffices for \(g\) to be a \(H_1\)-regular immersion. Observe that an immersion which is a local complete intersection morphism is Koszul regular.↩︎
To be precise \(X, Y, Z, Y \amalg_Z X, X', Y', Z', Y' \amalg_{Z'} X'\) correspond to \(A', B, A, B', C', D, C, D'\).↩︎
In fact, if \(f : X \to Y\) is an immersion and \(w\) is a weighting of \(f\), then \(f\) restricts to an open map on the locus where \(w\) is nonzero.↩︎
It suffices if the diagonal of \(Y\) is affine.↩︎
This may be nonstandard terminology.↩︎