More on Flatness

Introduction

In this chapter, we discuss some advanced results on flat modules and flat morphisms of schemes and applications. Most of the results on flatness can be found in the paper [GruRay] by Raynaud and Gruson.

Before reading this chapter we advise the reader to take a look at the following results (this list also serves as a pointer to previous results):

1. General discussion on flat modules in Algebra, Section 00H9.

2. The relationship between \(\text{Tor}\)-groups and flatness, see Algebra, Section 00LY.

3. Criteria for flatness, see Algebra, Section 00MD (Noetherian case), Algebra, Section 051E (Artinian case), Algebra, Section 00R3 (non-Noetherian case), and finally More on Morphisms, Section 039A.

4. Generic flatness, see Algebra, Section 051Q and Morphisms, Section 0529.

5. Openness of the flat locus, see Algebra, Section 00R8 and More on Morphisms, Section 0398.

6. Flattening, see More on Algebra, Sections 0521, 05LJ, 05LK, 05LN, and 05LV.

7. Additional results in More on Algebra, Sections 052Y, 0549, 054A, and 0535.

As applications of the material on flatness we discuss the following topics: a non-Noetherian version of Grothendieck’s existence theorem, blowing up and flatness, Nagata’s theorem on compactifications, the h topology, blow up squares and descent, weak normalization, descent of vector bundles in positive characteristic, and the local structure of perfect complexes in the h topology.

Lemmas on étale localization

In this section we list some lemmas on étale localization which will be useful later in this chapter. Please skip this section on a first reading.

Lemma

Let \(i : Z \to X\) be a closed immersion of affine schemes. Let \(Z' \to Z\) be an étale morphism with \(Z'\) affine. Then there exists an étale morphism \(X' \to X\) with \(X'\) affine such that \(Z' \cong Z \times_X X'\) as schemes over \(Z\).

Proof

See Algebra, Lemma 04D1.

Lemma

Let \[\xymatrix{ X \ar[d] & X' \ar[l] \ar[d] \\ S & S' \ar[l] }\] be a commutative diagram of schemes with \(X' \to X\) and \(S' \to S\) étale. Let \(s' \in S'\) be a point. Then \[X' \times_{S'} \Spec(\mathcal{O}_{S', s'}) \longrightarrow X \times_S \Spec(\mathcal{O}_{S', s'})\] is étale.

Proof

This is true because \(X' \to X_{S'}\) is étale as a morphism of schemes étale over \(X\), see Morphisms, Lemma 02GW and the base change of an étale morphism is étale, see Morphisms, Lemma 02GO.

Lemma

Let \(X \to T \to S\) be morphisms of schemes with \(T \to S\) étale. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in X\) be a point. Then \[\mathcal{F}\text{ flat over }S\text{ at }x \Leftrightarrow \mathcal{F}\text{ flat over }T\text{ at }x\] In particular \(\mathcal{F}\) is flat over \(S\) if and only if \(\mathcal{F}\) is flat over \(T\).

Proof

As an étale morphism is a flat morphism (see Morphisms, Lemma 02GS) the implication “\(\Leftarrow\)” follows from Algebra, Lemma 00HC. For the converse assume that \(\mathcal{F}\) is flat at \(x\) over \(S\). Denote \(\tilde x \in X \times_S T\) the point lying over \(x\) in \(X\) and over the image of \(x\) in \(T\) in \(T\). Then \((X \times_S T \to X)^*\mathcal{F}\) is flat at \(\tilde x\) over \(T\) via \(\text{pr}_2 : X \times_S T \to T\), see Morphisms, Lemma 01U8. The diagonal \(\Delta_{T/S} : T \to T \times_S T\) is an open immersion; combine Morphisms, Lemmas 02GE and 02GK. So \(X\) is identified with open subscheme of \(X \times_S T\), the restriction of \(\text{pr}_2\) to this open is the given morphism \(X \to T\), the point \(\tilde x\) corresponds to the point \(x\) in this open, and \((X \times_S T \to X)^*\mathcal{F}\) restricted to this open is \(\mathcal{F}\). Whence we see that \(\mathcal{F}\) is flat at \(x\) over \(T\).

Lemma

Let \(T \to S\) be an étale morphism. Let \(t \in T\) with image \(s \in S\). Let \(M\) be a \(\mathcal{O}_{T, t}\)-module. Then \[M\text{ flat over }\mathcal{O}_{S, s} \Leftrightarrow M\text{ flat over }\mathcal{O}_{T, t}.\]

Proof

We may replace \(S\) by an affine neighbourhood of \(s\) and after that \(T\) by an affine neighbourhood of \(t\). Set \(\mathcal{F} = (\Spec(\mathcal{O}_{T, t}) \to T)_*\widetilde M\). This is a quasi-coherent sheaf (see Schemes, Lemma 01LC or argue directly) on \(T\) whose stalk at \(t\) is \(M\) (details omitted). Apply Lemma 05B9.

Lemma

Let \(S\) be a scheme and \(s \in S\) a point. Denote \(\mathcal{O}_{S, s}^h\) (resp. \(\mathcal{O}_{S, s}^{sh}\)) the henselization (resp. strict henselization), see Algebra, Definition 04GQ. Let \(M^{sh}\) be a \(\mathcal{O}_{S, s}^{sh}\)-module. The following are equivalent

1. \(M^{sh}\) is flat over \(\mathcal{O}_{S, s}\),

2. \(M^{sh}\) is flat over \(\mathcal{O}_{S, s}^h\), and

3. \(M^{sh}\) is flat over \(\mathcal{O}_{S, s}^{sh}\).

If \(M^{sh} = M^h \otimes_{\mathcal{O}_{S, s}^h} \mathcal{O}_{S, s}^{sh}\) this is also equivalent to

1. \(M^h\) is flat over \(\mathcal{O}_{S, s}\), and

2. \(M^h\) is flat over \(\mathcal{O}_{S, s}^h\).

If \(M^h = M \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^h\) this is also equivalent to

1. \(M\) is flat over \(\mathcal{O}_{S, s}\).

Proof

By More on Algebra, Lemma 07QM the local ring maps \(\mathcal{O}_{S, s} \to \mathcal{O}_{S, s}^h \to \mathcal{O}_{S, s}^{sh}\) are faithfully flat. Hence (3) \(\Rightarrow\) (2) \(\Rightarrow\) (1) and (5) \(\Rightarrow\) (4) follow from Algebra, Lemma 00HC. By faithful flatness the equivalences (6) \(\Leftrightarrow\) (5) and (5) \(\Leftrightarrow\) (3) follow from Algebra, Lemma 00HJ. Thus it suffices to show that (1) \(\Rightarrow\) (2) \(\Rightarrow\) (3) and (4) \(\Rightarrow\) (5). To prove these we may assume \(S\) is an affine scheme.

Assume (1). By Lemma 05BA we see that \(M^{sh}\) is flat over \(\mathcal{O}_{T, t}\) for any étale neighbourhood \((T, t) \to (S, s)\). Since \(\mathcal{O}_{S, s}^h\) and \(\mathcal{O}_{S, s}^{sh}\) are directed colimits of local rings of the form \(\mathcal{O}_{T, t}\) (see Algebra, Lemmas 04GV and 04GW) we conclude that \(M^{sh}\) is flat over \(\mathcal{O}_{S, s}^h\) and \(\mathcal{O}_{S, s}^{sh}\) by Algebra, Lemma 05UU. Thus (1) implies (2) and (3). Of course this implies also (2) \(\Rightarrow\) (3) by replacing \(\mathcal{O}_{S, s}\) by \(\mathcal{O}_{S, s}^h\). The same argument applies to prove (4) \(\Rightarrow\) (5).

Lemma

Let \(S\) be a scheme and \(s \in S\) a point. Denote \(\mathcal{O}_{S, s}^h\) (resp. \(\mathcal{O}_{S, s}^{sh}\)) the henselization (resp. strict henselization), see Algebra, Definition 04GQ. Let \(M^{sh}\) be an object of \(D(\mathcal{O}_{S, s}^{sh})\). Let \(a, b \in \mathbf{Z}\). The following are equivalent

1. \(M^{sh}\) has tor amplitude in \([a, b]\) over \(\mathcal{O}_{S, s}\),

2. \(M^{sh}\) has tor amplitude in \([a, b]\) over \(\mathcal{O}_{S, s}^h\), and

3. \(M^{sh}\) has tor amplitude in \([a, b]\) over \(\mathcal{O}_{S, s}^{sh}\).

If \(M^{sh} = M^h \otimes_{\mathcal{O}_{S, s}^h}^\mathbf{L} \mathcal{O}_{S, s}^{sh}\) for \(M^h \in D(\mathcal{O}_{S, s}^h)\) this is also equivalent to

1. \(M^h\) has tor amplitude in \([a, b]\) over \(\mathcal{O}_{S, s}\), and

2. \(M^h\) has tor amplitude in \([a, b]\) over \(\mathcal{O}_{S, s}^h\).

If \(M^h = M \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S, s}^h\) for \(M \in D(\mathcal{O}_{S, s})\) this is also equivalent to

1. \(M\) has tor amplitude in \([a, b]\) over \(\mathcal{O}_{S, s}\).

Proof

By More on Algebra, Lemma 07QM the local ring maps \(\mathcal{O}_{S, s} \to \mathcal{O}_{S, s}^h \to \mathcal{O}_{S, s}^{sh}\) are faithfully flat. Hence (3) \(\Rightarrow\) (2) \(\Rightarrow\) (1) and (5) \(\Rightarrow\) (4) follow from More on Algebra, Lemma 066J. By faithful flatness the equivalences (6) \(\Leftrightarrow\) (5) and (5) \(\Leftrightarrow\) (3) follow from More on Algebra, Lemma 068S. Thus it suffices to show that (1) \(\Rightarrow\) (3), (2) \(\Rightarrow\) (3), and (4) \(\Rightarrow\) (5).

Assume (1). In particular \(M^{sh}\) has vanishing cohomology in degrees \(< a\) and \(> b\). Hence we can represent \(M^{sh}\) by a complex \(P^\bullet\) of free \(\mathcal{O}_{X, x}^{sh}\)-modules with \(P^i = 0\) for \(i > b\) (see for example the very general Derived Categories, Lemma 05T7). Note that \(P^n\) is flat over \(\mathcal{O}_{S, s}\) for all \(n\). Consider \(\Coker(d_P^{a - 1})\). By More on Algebra, Lemma 0653 this is a flat \(\mathcal{O}_{S, s}\)-module. Hence by Lemma 05VL this is a flat \(\mathcal{O}_{S, s}^{sh}\)-module. Thus \(\tau_{\geq a}P^\bullet\) is a complex of flat \(\mathcal{O}_{S, s}^{sh}\)-modules representing \(M^{sh}\) in \(D(\mathcal{O}_{S, s}^{sh}\) and we find that \(M^{sh}\) has tor amplitude in \([a, b]\), see More on Algebra, Lemma 0654. Thus (1) implies (3). Of course this implies also (2) \(\Rightarrow\) (3) by replacing \(\mathcal{O}_{S, s}\) by \(\mathcal{O}_{S, s}^h\). The same argument applies to prove (4) \(\Rightarrow\) (5).

Lemma

Let \(g : T \to S\) be a finite flat morphism of schemes. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_S\)-module. Let \(t \in T\) be a point with image \(s \in S\). Then \[t \in \text{WeakAss}(g^*\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(\mathcal{G})\]

Proof

The implication “\(\Leftarrow\)” follows immediately from Divisors, Lemma 05F0. Assume \(t \in \text{WeakAss}(g^*\mathcal{G})\). Let \(\Spec(A) \subset S\) be an affine open neighbourhood of \(s\). Let \(\mathcal{G}\) be the quasi-coherent sheaf associated to the \(A\)-module \(M\). Let \(\mathfrak p \subset A\) be the prime ideal corresponding to \(s\). As \(g\) is finite flat we have \(g^{-1}(\Spec(A)) = \Spec(B)\) for some finite flat \(A\)-algebra \(B\). Note that \(g^*\mathcal{G}\) is the quasi-coherent \(\mathcal{O}_{\Spec(B)}\)-module associated to the \(B\)-module \(M \otimes_A B\) and \(g_*g^*\mathcal{G}\) is the quasi-coherent \(\mathcal{O}_{\Spec(A)}\)-module associated to the \(A\)-module \(M \otimes_A B\). By Algebra, Lemma 00NZ we have \(B_{\mathfrak p} \cong A_{\mathfrak p}^{\oplus n}\) for some integer \(n \geq 0\). Note that \(n \geq 1\) as we assumed there exists at least one point of \(T\) lying over \(s\). Hence we see by looking at stalks that \[s \in \text{WeakAss}(\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(g_*g^*\mathcal{G})\] Now the assumption that \(t \in \text{WeakAss}(g^*\mathcal{G})\) implies that \(s \in \text{WeakAss}(g_*g^*\mathcal{G})\) by Divisors, Lemma 05EZ and hence by the above \(s \in \text{WeakAss}(\mathcal{G})\).

Lemma

Let \(h : U \to S\) be an étale morphism of schemes. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_S\)-module. Let \(u \in U\) be a point with image \(s \in S\). Then \[u \in \text{WeakAss}(h^*\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(\mathcal{G})\]

Proof

After replacing \(S\) and \(U\) by affine neighbourhoods of \(s\) and \(u\) we may assume that \(g\) is a standard étale morphism of affines, see Morphisms, Lemma 02GT. Thus we may assume \(S = \Spec(A)\) and \(X = \Spec(A[x, 1/g]/(f))\), where \(f\) is monic and \(f'\) is invertible in \(A[x, 1/g]\). Note that \(A[x, 1/g]/(f) = (A[x]/(f))_g\) is also the localization of the finite free \(A\)-algebra \(A[x]/(f)\). Hence we may think of \(U\) as an open subscheme of the scheme \(T = \Spec(A[x]/(f))\) which is finite locally free over \(S\). This reduces us to Lemma 05FN above.

Lemma

Let \(S\) be a scheme and \(s \in S\) a point. Denote \(\mathcal{O}_{S, s}^h\) (resp. \(\mathcal{O}_{S, s}^{sh}\)) the henselization (resp. strict henselization), see Algebra, Definition 04GQ. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_S\)-module. The following are equivalent

1. \(s\) is a weakly associated point of \(\mathcal{F}\),

2. \(\mathfrak m_s\) is a weakly associated prime of \(\mathcal{F}_s\),

3. \(\mathfrak m_s^h\) is a weakly associated prime of \(\mathcal{F}_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^h\), and

4. \(\mathfrak m_s^{sh}\) is a weakly associated prime of \(\mathcal{F}_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^{sh}\).

Proof

The equivalence of (1) and (2) is the definition, see Divisors, Definition 056L. The implications (2) \(\Rightarrow\) (3) \(\Rightarrow\) (4) follows from Divisors, Lemma 05F0 applied to the flat (More on Algebra, Lemma 07QM) morphisms \[\Spec(\mathcal{O}_{S, s}^{sh}) \to \Spec(\mathcal{O}_{S, s}^h) \to \Spec(\mathcal{O}_{S, s})\] and the closed points. To prove (4) \(\Rightarrow\) (2) we may replace \(S\) by an affine neighbourhood. Suppose that \(x \in \mathcal{F}_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^{sh}\) is an element whose annihilator has radical equal to \(\mathfrak m_s^{sh}\). (See Algebra, Lemma 0566.) Since \(\mathcal{O}_{S, s}^{sh}\) is equal to the limit of \(\mathcal{O}_{U, u}\) over étale neighbourhoods \(f : (U, u) \to (S, s)\) by Algebra, Lemma 04GW we may assume that \(x\) is the image of some \(x' \in \mathcal{F}_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{U, u}\). The local ring map \(\mathcal{O}_{U, u} \to \mathcal{O}_{S, s}^{sh}\) is faithfully flat (as it is the strict henselization), hence universally injective (Algebra, Lemma 05CK). It follows that the annihilator of \(x'\) is the inverse image of the annihilator of \(x\). Hence the radical of this annihilator is equal to \(\mathfrak m_u\). Thus \(u\) is a weakly associated point of \(f^*\mathcal{F}\). By Lemma 05FP we see that \(s\) is a weakly associated point of \(\mathcal{F}\).

The local structure of a finite type module

The key technical lemma that makes a lot of the arguments in this chapter work is the geometric Lemma 057S.

Lemma

Let \(f : X \to S\) be a finite type morphism of affine schemes. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in X\) with image \(s = f(x)\) in \(S\). Set \(\mathcal{F}_s = \mathcal{F}|_{X_s}\). Then there exist a closed immersion \(i : Z \to X\) of finite presentation, and a quasi-coherent finite type \(\mathcal{O}_Z\)-module \(\mathcal{G}\) such that \(i_*\mathcal{G} = \mathcal{F}\) and \(Z_s = \text{Supp}(\mathcal{F}_s)\).

Proof

Say the morphism \(f : X \to S\) is given by the ring map \(A \to B\) and that \(\mathcal{F}\) is the quasi-coherent sheaf associated to the \(B\)-module \(M\). By Morphisms, Lemma 01T2 we know that \(A \to B\) is a finite type ring map, and by Properties, Lemma 01PB we know that \(M\) is a finite \(B\)-module. In particular the support of \(\mathcal{F}\) is the closed subscheme of \(\Spec(B)\) cut out by the annihilator \(I = \{x \in B \mid xm = 0\ \forall m \in M\}\) of \(M\), see Algebra, Lemma 00L2. Let \(\mathfrak q \subset B\) be the prime ideal corresponding to \(x\) and let \(\mathfrak p \subset A\) be the prime ideal corresponding to \(s\). Note that \(X_s = \Spec(B \otimes_A \kappa(\mathfrak p))\) and that \(\mathcal{F}_s\) is the quasi-coherent sheaf associated to the \(B \otimes_A \kappa(\mathfrak p)\) module \(M \otimes_A \kappa(\mathfrak p)\). By Morphisms, Lemma 056J the support of \(\mathcal{F}_s\) is equal to \(V(I(B \otimes_A \kappa(\mathfrak p)))\). Since \(B \otimes_A \kappa(\mathfrak p)\) is of finite type over \(\kappa(\mathfrak p)\) there exist finitely many elements \(f_1, \ldots, f_m \in I\) such that \[I(B \otimes_A \kappa(\mathfrak p)) = (f_1, \ldots, f_n)(B \otimes_A \kappa(\mathfrak p)).\] Denote \(i : Z \to X\) the closed subscheme cut out by \((f_1, \ldots, f_m)\), in a formula \(Z = \Spec(B/(f_1, \ldots, f_m))\). Since \(M\) is annihilated by \(I\) we can think of \(M\) as an \(B/(f_1, \ldots, f_m)\)-module. In other words, \(\mathcal{F}\) is the pushforward of a finite type module on \(Z\). As \(Z_s = \text{Supp}(\mathcal{F}_s)\) by construction, this proves the lemma.

Lemma

Let \(f : X \to S\) be morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in X\) with image \(s = f(x)\) in \(S\). Set \(\mathcal{F}_s = \mathcal{F}|_{X_s}\) and \(n = \dim_x(\text{Supp}(\mathcal{F}_s))\). Then we can construct

1. elementary étale neighbourhoods \(g : (X', x') \to (X, x)\), \(e : (S', s') \to (S, s)\),

2. a commutative diagram \[\xymatrix{ X \ar[dd]_f & X' \ar[dd] \ar[l]^g & Z' \ar[l]^i \ar[d]^\pi \\ & & Y' \ar[d]^h \\ S & S' \ar[l]_e & S' \ar@{=}[l] }\]

3. a point \(z' \in Z'\) with \(i(z') = x'\), \(y' = \pi(z')\), \(h(y') = s'\),

4. a finite type quasi-coherent \(\mathcal{O}_{Z'}\)-module \(\mathcal{G}\),

such that the following properties hold

1. \(X'\), \(Z'\), \(Y'\), \(S'\) are affine schemes,

2. \(i\) is a closed immersion of finite presentation,

3. \(i_*(\mathcal{G}) \cong g^*\mathcal{F}\),

4. \(\pi\) is finite and \(\pi^{-1}(\{y'\}) = \{z'\}\),

5. the extension \(\kappa(y')/\kappa(s')\) is purely transcendental,

6. \(h\) is smooth of relative dimension \(n\) with geometrically integral fibres.

Proof

Let \(V \subset S\) be an affine neighbourhood of \(s\). Let \(U \subset f^{-1}(V)\) be an affine neighbourhood of \(x\). Then it suffices to prove the lemma for \(f|_U : U \to V\) and \(\mathcal{F}|_U\). Hence in the rest of the proof we assume that \(X\) and \(S\) are affine.

First, suppose that \(X_s = \text{Supp}(\mathcal{F}_s)\), in particular \(n = \dim_x(X_s)\). Apply More on Morphisms, Lemmas 057K and 057L. This gives us a commutative diagram \[\xymatrix{ X \ar[dd] & X' \ar[l]^g \ar[d]^\pi \\ & Y' \ar[d]^h \\ S & S' \ar[l]_e }\] and point \(x' \in X'\). We set \(Z' = X'\), \(i = \text{id}\), and \(\mathcal{G} = g^*\mathcal{F}\) to obtain a solution in this case.

In general choose a closed immersion \(Z \to X\) and a sheaf \(\mathcal{G}\) on \(Z\) as in Lemma 057Q. Applying the result of the previous paragraph to \(Z \to S\) and \(\mathcal{G}\) we obtain a diagram \[\xymatrix{ X \ar[dd]_f & Z \ar[l] \ar[dd]_{f|_Z} & Z' \ar[l]^g \ar[d]^\pi \\ & & Y' \ar[d]^h \\ S & S \ar@{=}[l] & S' \ar[l]_e }\] and point \(z' \in Z'\) satisfying all the required properties. We will use Lemma 057R to embed \(Z'\) into a scheme étale over \(X\). We cannot apply the lemma directly as we want \(X'\) to be a scheme over \(S'\). Instead we consider the morphisms \[\xymatrix{ Z' \ar[r] & Z \times_S S' \ar[r] & X \times_S S' }\] The first morphism is étale by Morphisms, Lemma 02GW. The second is a closed immersion as a base change of a closed immersion. Finally, as \(X\), \(S\), \(S'\), \(Z\), \(Z'\) are all affine we may apply Lemma 057R to get an étale morphism of affine schemes \(X' \to X \times_S S'\) such that \[Z' = (Z \times_S S') \times_{(X \times_S S')} X' = Z \times_X X'.\] As \(Z \to X\) is a closed immersion of finite presentation, so is \(Z' \to X'\). Let \(x' \in X'\) be the point corresponding to \(z' \in Z'\). Then the completed diagram \[\xymatrix{ X \ar[dd] & X' \ar[dd] \ar[l] & Z' \ar[l]^i \ar[d]^\pi \\ & & Y' \ar[d]^h \\ S & S' \ar[l]_e & S' \ar@{=}[l] }\] is a solution of the original problem.

Lemma

Assumptions and notation as in Lemma 057S. If \(f\) is locally of finite presentation then \(\pi\) is of finite presentation. In this case the following are equivalent

1. \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite presentation in a neighbourhood of \(x\),

2. \(\mathcal{G}\) is an \(\mathcal{O}_{Z'}\)-module of finite presentation in a neighbourhood of \(z'\), and

3. \(\pi_*\mathcal{G}\) is an \(\mathcal{O}_{Y'}\)-module of finite presentation in a neighbourhood of \(y'\).

Still assuming \(f\) locally of finite presentation the following are equivalent to each other

1. \(\mathcal{F}_x\) is an \(\mathcal{O}_{X, x}\)-module of finite presentation,

2. \(\mathcal{G}_{z'}\) is an \(\mathcal{O}_{Z', z'}\)-module of finite presentation, and

3. \((\pi_*\mathcal{G})_{y'}\) is an \(\mathcal{O}_{Y', y'}\)-module of finite presentation.

Proof

Assume \(f\) locally of finite presentation. Then \(Z' \to S\) is locally of finite presentation as a composition of such, see Morphisms, Lemma 01TR. Note that \(Y' \to S\) is also locally of finite presentation as a composition of a smooth and an étale morphism. Hence Morphisms, Lemma 02FV implies \(\pi\) is locally of finite presentation. Since \(\pi\) is finite we conclude that it is also separated and quasi-compact, hence \(\pi\) is actually of finite presentation.

To prove the equivalence of (1), (2), and (3) we also consider: (4) \(g^*\mathcal{F}\) is a \(\mathcal{O}_{X'}\)-module of finite presentation in a neighbourhood of \(x'\). The pullback of a module of finite presentation is of finite presentation, see Modules, Lemma 01BQ. Hence (1) \(\Rightarrow\) (4). The étale morphism \(g\) is open, see Morphisms, Lemma 03WT. Hence for any open neighbourhood \(U' \subset X'\) of \(x'\), the image \(g(U')\) is an open neighbourhood of \(x\) and the map \(\{U' \to g(U')\}\) is an étale covering. Thus (4) \(\Rightarrow\) (1) by Descent, Lemma 05B0. Using Descent, Lemma 05B4 and some easy topological arguments (see More on Morphisms, Lemma 05B8) we see that (4) \(\Leftrightarrow\) (2) \(\Leftrightarrow\) (3).

To prove the equivalence of (a), (b), (c) consider the ring maps \[\mathcal{O}_{X, x} \to \mathcal{O}_{X', x'} \to \mathcal{O}_{Z', z'} \leftarrow \mathcal{O}_{Y', y'}\] The first ring map is faithfully flat. Hence \(\mathcal{F}_x\) is of finite presentation over \(\mathcal{O}_{X, x}\) if and only if \(g^*\mathcal{F}_{x'}\) is of finite presentation over \(\mathcal{O}_{X', x'}\), see Algebra, Lemma 03C4. The second ring map is surjective (hence finite) and finitely presented by assumption, hence \(g^*\mathcal{F}_{x'}\) is of finite presentation over \(\mathcal{O}_{X', x'}\) if and only if \(\mathcal{G}_{z'}\) is of finite presentation over \(\mathcal{O}_{Z', z'}\), see Algebra, Lemma 0564. Because \(\pi\) is finite, of finite presentation, and \(\pi^{-1}(\{y'\}) = \{x'\}\) the ring homomorphism \(\mathcal{O}_{Y', y'} \leftarrow \mathcal{O}_{Z', z'}\) is finite and of finite presentation, see More on Morphisms, Lemma 05B8. Hence \(\mathcal{G}_{z'}\) is of finite presentation over \(\mathcal{O}_{Z', z'}\) if and only if \(\pi_*\mathcal{G}_{y'}\) is of finite presentation over \(\mathcal{O}_{Y', y'}\), see Algebra, Lemma 0564.

Lemma

Assumptions and notation as in Lemma 057S. The following are equivalent

1. \(\mathcal{F}\) is flat over \(S\) in a neighbourhood of \(x\),

2. \(\mathcal{G}\) is flat over \(S'\) in a neighbourhood of \(z'\), and

3. \(\pi_*\mathcal{G}\) is flat over \(S'\) in a neighbourhood of \(y'\).

The following are equivalent also

1. \(\mathcal{F}_x\) is flat over \(\mathcal{O}_{S, s}\),

2. \(\mathcal{G}_{z'}\) is flat over \(\mathcal{O}_{S', s'}\), and

3. \((\pi_*\mathcal{G})_{y'}\) is flat over \(\mathcal{O}_{S', s'}\).

Proof

To prove the equivalence of (1), (2), and (3) we also consider: (4) \(g^*\mathcal{F}\) is flat over \(S\) in a neighbourhood of \(x'\). We will use Lemma 05B9 to equate flatness over \(S\) and \(S'\) without further mention. The étale morphism \(g\) is flat and open, see Morphisms, Lemma 03WT. Hence for any open neighbourhood \(U' \subset X'\) of \(x'\), the image \(g(U')\) is an open neighbourhood of \(x\) and the map \(U' \to g(U')\) is surjective and flat. Thus (4) \(\Leftrightarrow\) (1) by Morphisms, Lemma 02JZ. Note that \[\Gamma(X', g^*\mathcal{F}) = \Gamma(Z', \mathcal{G}) = \Gamma(Y', \pi_*\mathcal{G})\] Hence the flatness of \(g^*\mathcal{F}\), \(\mathcal{G}\) and \(\pi_*\mathcal{G}\) over \(S'\) are all equivalent (this uses that \(X'\), \(Z'\), \(Y'\), and \(S'\) are all affine). Some omitted topological arguments (compare More on Morphisms, Lemma 05B8) regarding affine neighbourhoods now show that (4) \(\Leftrightarrow\) (2) \(\Leftrightarrow\) (3).

To prove the equivalence of (a), (b), (c) consider the commutative diagram of local ring maps \[\xymatrix{ \mathcal{O}_{X', x'} \ar[r]_\iota & \mathcal{O}_{Z', z'} & \mathcal{O}_{Y', y'} \ar[l]^\alpha & \mathcal{O}_{S', s'} \ar[l]^\beta \\ \mathcal{O}_{X, x} \ar[u]^\gamma & & & \mathcal{O}_{S, s} \ar[lll]_\varphi \ar[u]_\epsilon }\] We will use Lemma 05BA to equate flatness over \(\mathcal{O}_{S, s}\) and \(\mathcal{O}_{S', s'}\) without further mention. The map \(\gamma\) is faithfully flat. Hence \(\mathcal{F}_x\) is flat over \(\mathcal{O}_{S, s}\) if and only if \(g^*\mathcal{F}_{x'}\) is flat over \(\mathcal{O}_{S', s'}\), see Algebra, Lemma 0584. As \(\mathcal{O}_{S', s'}\)-modules the modules \(g^*\mathcal{F}_{x'}\), \(\mathcal{G}_{z'}\), and \(\pi_*\mathcal{G}_{y'}\) are all isomorphic, see More on Morphisms, Lemma 05B8. This finishes the proof.

One step dévissage

In this section we explain what is a one step dévissage of a module. A one step dévissage exist étale locally on base and target. We discuss base change, Zariski shrinking and étale localization of a one step dévissage.

Definition

Let \(S\) be a scheme. Let \(X\) be locally of finite type over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. Let \(s \in S\) be a point. A one step dévissage of \(\mathcal{F}/X/S\) over \(s\) is given by morphisms of schemes over \(S\) \[\xymatrix{ X & Z \ar[l]_i \ar[r]^\pi & Y }\] and a quasi-coherent \(\mathcal{O}_Z\)-module \(\mathcal{G}\) of finite type such that

1. \(X\), \(S\), \(Z\) and \(Y\) are affine,

2. \(i\) is a closed immersion of finite presentation,

3. \(\mathcal{F} \cong i_*\mathcal{G}\),

4. \(\pi\) is finite, and

5. the structure morphism \(Y \to S\) is smooth with geometrically irreducible fibres of dimension \(\dim(\text{Supp}(\mathcal{F}_s))\).

In this case we say \((Z, Y, i, \pi, \mathcal{G})\) is a one step dévissage of \(\mathcal{F}/X/S\) over \(s\).

Note that such a one step dévissage can only exist if \(X\) and \(S\) are affine. In the definition above we only require \(X\) to be (locally) of finite type over \(S\) and we continue working in this setting below. In [GruRay] the authors use consistently the setup where \(X \to S\) is locally of finite presentation and \(\mathcal{F}\) quasi-coherent \(\mathcal{O}_X\)-module of finite type. The advantage of this choice is that it “makes sense” to ask for \(\mathcal{F}\) to be of finite presentation as an \(\mathcal{O}_X\)-module, whereas in our setting it “does not make sense”. Please see More on Morphisms, Section 05GX for a discussion; the observations made there show that in our setup we may consider the condition of \(\mathcal{F}\) being “locally of finite presentation relative to \(S\)”, and we could work consistently with this notion. Instead however, we will rely on the results of Lemma 057T and the observations in Remark 05HY to deal with this issue in an ad hoc fashion whenever it comes up.

Definition

Let \(S\) be a scheme. Let \(X\) be locally of finite type over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. Let \(x \in X\) be a point with image \(s\) in \(S\). A one step dévissage of \(\mathcal{F}/X/S\) at \(x\) is a system \((Z, Y, i, \pi, \mathcal{G}, z, y)\), where \((Z, Y, i, \pi, \mathcal{G})\) is a one step dévissage of \(\mathcal{F}/X/S\) over \(s\) and

1. \(\dim_x(\text{Supp}(\mathcal{F}_s)) = \dim(\text{Supp}(\mathcal{F}_s))\),

2. \(z \in Z\) is a point with \(i(z) = x\) and \(\pi(z) = y\),

3. we have \(\pi^{-1}(\{y\}) = \{z\}\),

4. the extension \(\kappa(y)/\kappa(s)\) is purely transcendental.

A one step dévissage of \(\mathcal{F}/X/S\) at \(x\) can only exist if \(X\) and \(S\) are affine. Condition (1) assures us that \(Y \to S\) has relative dimension equal to \(\dim_x(\text{Supp}(\mathcal{F}_s))\) via condition (5) of Definition 05H4.

Lemma

Let \(f : X \to S\) be morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in X\) with image \(s = f(x)\) in \(S\). Then there exists a commutative diagram of pointed schemes \[\xymatrix{ (X, x) \ar[d]_f & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] \\ }\] such that \((S', s') \to (S, s)\) and \((X', x') \to (X, x)\) are elementary étale neighbourhoods, and such that \(g^*\mathcal{F}/X'/S'\) has a one step dévissage at \(x'\).

Proof

This is immediate from Definition 05H5 and Lemma 057S.

Lemma

Let \(S\), \(X\), \(\mathcal{F}\), \(s\) be as in Definition 05H4. Let \((Z, Y, i, \pi, \mathcal{G})\) be a one step dévissage of \(\mathcal{F}/X/S\) over \(s\). Let \((S', s') \to (S, s)\) be any morphism of pointed schemes. Given this data let \(X', Z', Y', i', \pi'\) be the base changes of \(X, Z, Y, i, \pi\) via \(S' \to S\). Let \(\mathcal{F}'\) be the pullback of \(\mathcal{F}\) to \(X'\) and let \(\mathcal{G}'\) be the pullback of \(\mathcal{G}\) to \(Z'\). If \(S'\) is affine, then \((Z', Y', i', \pi', \mathcal{G}')\) is a one step dévissage of \(\mathcal{F}'/X'/S'\) over \(s'\).

Proof

Fibre products of affines are affine, see Schemes, Lemma 01JQ. Base change preserves closed immersions, morphisms of finite presentation, finite morphisms, smooth morphisms, morphisms with geometrically irreducible fibres, and morphisms of relative dimension \(n\), see Morphisms, Lemmas 01QR, 01TS, 01WL, 01VB, 02NK, and More on Morphisms, Lemma 0555. We have \(i'_*\mathcal{G}' \cong \mathcal{F}'\) because pushforward along the finite morphism \(i\) commutes with base change, see Cohomology of Schemes, Lemma 02KG. We have \(\dim(\text{Supp}(\mathcal{F}_s)) = \dim(\text{Supp}(\mathcal{F}'_{s'}))\) by Morphisms, Lemma 02FY because \[\text{Supp}(\mathcal{F}_s) \times_s s' = \text{Supp}(\mathcal{F}'_{s'}).\] This proves the lemma.

Lemma

Let \(S\), \(X\), \(\mathcal{F}\), \(x\), \(s\) be as in Definition 05H5. Let \((Z, Y, i, \pi, \mathcal{G}, z, y)\) be a one step dévissage of \(\mathcal{F}/X/S\) at \(x\). Let \((S', s') \to (S, s)\) be a morphism of pointed schemes which induces an isomorphism \(\kappa(s) = \kappa(s')\). Let \((Z', Y', i', \pi', \mathcal{G}')\) be as constructed in Lemma 05H7 and let \(x' \in X'\) (resp. \(z' \in Z'\), \(y' \in Y'\)) be the unique point mapping to both \(x \in X\) (resp. \(z \in Z\), \(y \in Y\)) and \(s' \in S'\). If \(S'\) is affine, then \((Z', Y', i', \pi', \mathcal{G}', z', y')\) is a one step dévissage of \(\mathcal{F}'/X'/S'\) at \(x'\).

Proof

By Lemma 05H7 \((Z', Y', i', \pi', \mathcal{G}')\) is a one step dévissage of \(\mathcal{F}'/X'/S'\) over \(s'\). Properties (1) – (4) of Definition 05H5 hold for \((Z', Y', i', \pi', \mathcal{G}', z', y')\) as the assumption that \(\kappa(s) = \kappa(s')\) insures that the fibres \(X'_{s'}\), \(Z'_{s'}\), and \(Y'_{s'}\) are isomorphic to \(X_s\), \(Z_s\), and \(Y_s\).

Definition

Let \(S\), \(X\), \(\mathcal{F}\), \(x\), \(s\) be as in Definition 05H5. Let \((Z, Y, i, \pi, \mathcal{G}, z, y)\) be a one step dévissage of \(\mathcal{F}/X/S\) at \(x\). Let us define a standard shrinking of this situation to be given by standard opens \(S' \subset S\), \(X' \subset X\), \(Z' \subset Z\), and \(Y' \subset Y\) such that \(s \in S'\), \(x \in X'\), \(z \in Z'\), and \(y \in Y'\) and such that \[(Z', Y', i|_{Z'}, \pi|_{Z'}, \mathcal{G}|_{Z'}, z, y)\] is a one step dévissage of \(\mathcal{F}|_{X'}/X'/S'\) at \(x\).

Lemma

With assumption and notation as in Definition 05H9 we have:

1. If \(S' \subset S\) is a standard open neighbourhood of \(s\), then setting \(X' = X_{S'}\), \(Z' = Z_{S'}\) and \(Y' = Y_{S'}\) we obtain a standard shrinking.

2. Let \(W \subset Y\) be a standard open neighbourhood of \(y\). Then there exists a standard shrinking with \(Y' = W \times_S S'\).

3. Let \(U \subset X\) be an open neighbourhood of \(x\). Then there exists a standard shrinking with \(X' \subset U\).

Proof

Part (1) is immediate from Lemma 05H8 and the fact that the inverse image of a standard open under a morphism of affine schemes is a standard open, see Algebra, Lemma 00E2.

Let \(W \subset Y\) as in (2). Because \(Y \to S\) is smooth it is open, see Morphisms, Lemma 056G. Hence we can find a standard open neighbourhood \(S'\) of \(s\) contained in the image of \(W\). Then the fibres of \(W_{S'} \to S'\) are nonempty open subschemes of the fibres of \(Y \to S\) over \(S'\) and hence geometrically irreducible too. Setting \(Y' = W_{S'}\) and \(Z' = \pi^{-1}(Y')\) we see that \(Z' \subset Z\) is a standard open neighbourhood of \(z\). Let \(\overline{h} \in \Gamma(Z, \mathcal{O}_Z)\) be a function such that \(Z' = D(\overline{h})\). As \(i : Z \to X\) is a closed immersion, we can find a function \(h \in \Gamma(X, \mathcal{O}_X)\) such that \(i^\sharp(h) = \overline{h}\). Take \(X' = D(h) \subset X\). In this way we obtain a standard shrinking as in (2).

Let \(U \subset X\) be as in (3). We may after shrinking \(U\) assume that \(U\) is a standard open. By More on Morphisms, Lemma 05B8 there exists a standard open \(W \subset Y\) neighbourhood of \(y\) such that \(\pi^{-1}(W) \subset i^{-1}(U)\). Apply (2) to get a standard shrinking \(X', S', Z', Y'\) with \(Y' = W_{S'}\). Since \(Z' \subset \pi^{-1}(W) \subset i^{-1}(U)\) we may replace \(X'\) by \(X' \cap U\) (still a standard open as \(U\) is also standard open) without violating any of the conditions defining a standard shrinking. Hence we win.

Lemma

Let \(S\), \(X\), \(\mathcal{F}\), \(x\), \(s\) be as in Definition 05H5. Let \((Z, Y, i, \pi, \mathcal{G}, z, y)\) be a one step dévissage of \(\mathcal{F}/X/S\) at \(x\). Let \[\xymatrix{ (Y, y) \ar[d] & (Y', y') \ar[l] \ar[d] \\ (S, s) & (S', s') \ar[l] }\] be a commutative diagram of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods. Then there exists a commutative diagram \[\xymatrix{ & & (X'', x'') \ar[lld] \ar[d] & (Z'', z'') \ar[l] \ar[lld] \ar[d] \\ (X, x) \ar[d] & (Z, z) \ar[l] \ar[d] & (S'', s'') \ar[lld] & (Y'', y'') \ar[lld] \ar[l] \\ (S, s) & (Y, y) \ar[l] }\] of pointed schemes with the following properties:

1. \((S'', s'') \to (S', s')\) is an elementary étale neighbourhood and the morphism \(S'' \to S\) is the composition \(S'' \to S' \to S\),

2. \(Y''\) is an open subscheme of \(Y' \times_{S'} S''\),

3. \(Z'' = Z \times_Y Y''\),

4. \((X'', x'') \to (X, x)\) is an elementary étale neighbourhood, and

5. \((Z'', Y'', i'', \pi'', \mathcal{G}'', z'', y'')\) is a one step dévissage at \(x''\) of the sheaf \(\mathcal{F}''\).

Here \(\mathcal{F}''\) (resp. \(\mathcal{G}''\)) is the pullback of \(\mathcal{F}\) (resp. \(\mathcal{G}\)) via the morphism \(X'' \to X\) (resp. \(Z'' \to Z\)) and \(i'' : Z'' \to X''\) and \(\pi'' : Z'' \to Y''\) are as in the diagram.

Proof

Let \((S'', s'') \to (S', s')\) be any elementary étale neighbourhood with \(S''\) affine. Let \(Y'' \subset Y' \times_{S'} S''\) be any affine open neighbourhood containing the point \(y'' = (y', s'')\). Then we obtain an affine \((Z'', z'')\) by (3). Moreover \(Z_{S''} \to X_{S''}\) is a closed immersion and \(Z'' \to Z_{S''}\) is an étale morphism. Hence Lemma 057R applies and we can find an étale morphism \(X'' \to X_{S'}\) of affines such that \(Z'' \cong X'' \times_{X_{S'}} Z_{S'}\). Denote \(i'' : Z'' \to X''\) the corresponding closed immersion. Setting \(x'' = i''(z'')\) we obtain a commutative diagram as in the lemma. Properties (1), (2), (3), and (4) hold by construction. Thus it suffices to show that (5) holds for a suitable choice of \((S'', s'') \to (S', s')\) and \(Y''\).

We first list those properties which hold for any choice of \((S'', s'') \to (S', s')\) and \(Y''\) as in the first paragraph. As we have \(Z'' = X'' \times_X Z\) by construction we see that \(i''_*\mathcal{G}'' = \mathcal{F}''\) (with notation as in the statement of the lemma), see Cohomology of Schemes, Lemma 02KG. Set \(n = \dim(\text{Supp}(\mathcal{F}_s)) = \dim_x(\text{Supp}(\mathcal{F}_s))\). The morphism \(Y'' \to S''\) is smooth of relative dimension \(n\) (because \(Y' \to S'\) is smooth of relative dimension \(n\) as the composition \(Y' \to Y_{S'} \to S'\) of an étale and smooth morphism of relative dimension \(n\) and because base change preserves smooth morphisms of relative dimension \(n\)). We have \(\kappa(y'') = \kappa(y)\) and \(\kappa(s) = \kappa(s'')\) hence \(\kappa(y'')\) is a purely transcendental extension of \(\kappa(s'')\). The morphism of fibres \(X''_{s''} \to X_s\) is an étale morphism of affine schemes over \(\kappa(s) = \kappa(s'')\) mapping the point \(x''\) to the point \(x\) and pulling back \(\mathcal{F}_s\) to \(\mathcal{F}''_{s''}\). Hence \[\dim(\text{Supp}(\mathcal{F}''_{s''})) = \dim(\text{Supp}(\mathcal{F}_s)) = n = \dim_x(\text{Supp}(\mathcal{F}_s)) = \dim_{x''}(\text{Supp}(\mathcal{F}''_{s''}))\] because dimension is invariant under étale localization, see Descent, Lemma 04N4. As \(\pi'' : Z'' \to Y''\) is the base change of \(\pi\) we see that \(\pi''\) is finite and as \(\kappa(y) = \kappa(y'')\) we see that \(\pi^{-1}(\{y''\}) = \{z''\}\).

At this point we have verified all the conditions of Definition 05H4 except we have not verified that \(Y'' \to S''\) has geometrically irreducible fibres. Of course in general this is not going to be true, and it is at this point that we will use that \(\kappa(s) \subset \kappa(y)\) is purely transcendental. Namely, let \(T \subset Y'_{s'}\) be the irreducible component of \(Y'_{s'}\) containing \(y' = (y, s')\). Note that \(T\) is an open subscheme of \(Y'_{s'}\) as this is a smooth scheme over \(\kappa(s')\). By Varieties, Lemma 04KV we see that \(T\) is geometrically connected because \(\kappa(s') = \kappa(s)\) is algebraically closed in \(\kappa(y') = \kappa(y)\). As \(T\) is smooth we see that \(T\) is geometrically irreducible. Hence More on Morphisms, Lemma 057J applies and we can find an elementary étale morphism \((S'', s'') \to (S', s')\) and an affine open \(Y'' \subset Y'_{S''}\) such that all fibres of \(Y'' \to S''\) are geometrically irreducible and such that \(T = Y''_{s''}\). After shrinking (first \(Y''\) and then \(S''\)) we may assume that both \(Y''\) and \(S''\) are affine. This finishes the proof of the lemma.

Lemma

Let \(S\), \(X\), \(\mathcal{F}\), \(s\) be as in Definition 05H4. Let \((Z, Y, i, \pi, \mathcal{G})\) be a one step dévissage of \(\mathcal{F}/X/S\) over \(s\). Let \(\xi \in Y_s\) be the (unique) generic point. Then there exists an integer \(r > 0\) and an \(\mathcal{O}_Y\)-module map \[\alpha : \mathcal{O}_Y^{\oplus r} \longrightarrow \pi_*\mathcal{G}\] such that \[\alpha : \kappa(\xi)^{\oplus r} \longrightarrow (\pi_*\mathcal{G})_\xi \otimes_{\mathcal{O}_{Y, \xi}} \kappa(\xi)\] is an isomorphism. Moreover, in this case we have \[\dim(\text{Supp}(\Coker(\alpha)_s)) < \dim(\text{Supp}(\mathcal{F}_s)).\]

Proof

By assumption the schemes \(S\) and \(Y\) are affine. Write \(S = \Spec(A)\) and \(Y = \Spec(B)\). As \(\pi\) is finite the \(\mathcal{O}_Y\)-module \(\pi_*\mathcal{G}\) is a finite type quasi-coherent \(\mathcal{O}_Y\)-module. Hence \(\pi_*\mathcal{G} = \widetilde{N}\) for some finite \(B\)-module \(N\). Let \(\mathfrak p \subset B\) be the prime ideal corresponding to \(\xi\). To obtain \(\alpha\) set \(r = \dim_{\kappa(\mathfrak p)} N \otimes_B \kappa(\mathfrak p)\) and pick \(x_1, \ldots, x_r \in N\) which form a basis of \(N \otimes_B \kappa(\mathfrak p)\). Take \(\alpha : B^{\oplus r} \to N\) to be the map given by the formula \(\alpha(b_1, \ldots, b_r) = \sum b_ix_i\). It is clear that \(\alpha : \kappa(\mathfrak p)^{\oplus r} \to N \otimes_B \kappa(\mathfrak p)\) is an isomorphism as desired. Finally, suppose \(\alpha\) is any map with this property. Then \(N' = \Coker(\alpha)\) is a finite \(B\)-module such that \(N' \otimes \kappa(\mathfrak p) = 0\). By Nakayama’s lemma (Algebra, Lemma 00DV) we see that \(N'_{\mathfrak p} = 0\). Since the fibre \(Y_s\) is geometrically irreducible of dimension \(n\) with generic point \(\xi\) and since we have just seen that \(\xi\) is not in the support of \(\Coker(\alpha)\) the last assertion of the lemma holds.

Complete dévissage

In this section we explain what is a complete dévissage of a module and prove that such exist. The material in this section is mainly bookkeeping.

Definition

Let \(S\) be a scheme. Let \(X\) be locally of finite type over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. Let \(s \in S\) be a point. A complete dévissage of \(\mathcal{F}/X/S\) over \(s\) is given by a diagram \[\xymatrix{ X & Z_1 \ar[l]^{i_1} \ar[d]^{\pi_1} \\ & Y_1 & Z_2 \ar[l]^{i_2} \ar[d]^{\pi_2} \\ & & Y_2 & Z_3 \ar[l] \ar[d] \\ & & & ... & ... \ar[l] \ar[d] \\ & & & & Y_n }\] of schemes over \(S\), finite type quasi-coherent \(\mathcal{O}_{Z_k}\)-modules \(\mathcal{G}_k\), and \(\mathcal{O}_{Y_k}\)-module maps \[\alpha_k : \mathcal{O}_{Y_k}^{\oplus r_k} \longrightarrow \pi_{k, *}\mathcal{G}_k, \quad k = 1, \ldots, n\] satisfying the following properties:

1. \((Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)\) is a one step dévissage of \(\mathcal{F}/X/S\) over \(s\),

2. the map \(\alpha_k\) induces an isomorphism \[\kappa(\xi_k)^{\oplus r_k} \longrightarrow (\pi_{k, *}\mathcal{G}_k)_{\xi_k} \otimes_{\mathcal{O}_{Y_k, \xi_k}} \kappa(\xi_k)\] where \(\xi_k \in (Y_k)_s\) is the unique generic point,

3. for \(k = 2, \ldots, n\) the system \((Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k)\) is a one step dévissage of \(\Coker(\alpha_{k - 1})/Y_{k - 1}/S\) over \(s\),

4. \(\Coker(\alpha_n) = 0\).

In this case we say that \((Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k)_{k = 1, \ldots, n}\) is a complete dévissage of \(\mathcal{F}/X/S\) over \(s\).

Definition

Let \(S\) be a scheme. Let \(X\) be locally of finite type over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. Let \(x \in X\) be a point with image \(s \in S\). A complete dévissage of \(\mathcal{F}/X/S\) at \(x\) is given by a system \[(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 1, \ldots, n}\] such that \((Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k)\) is a complete dévissage of \(\mathcal{F}/X/S\) over \(s\), and such that

1. \((Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1, z_1, y_1)\) is a one step dévissage of \(\mathcal{F}/X/S\) at \(x\),

2. for \(k = 2, \ldots, n\) the system \((Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, z_k, y_k)\) is a one step dévissage of \(\Coker(\alpha_{k - 1})/Y_{k - 1}/S\) at \(y_{k - 1}\).

Again we remark that a complete dévissage can only exist if \(X\) and \(S\) are affine.

Lemma

Let \(S\), \(X\), \(\mathcal{F}\), \(s\) be as in Definition 05HH. Let \((S', s') \to (S, s)\) be any morphism of pointed schemes. Let \((Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k)_{k = 1, \ldots, n}\) be a complete dévissage of \(\mathcal{F}/X/S\) over \(s\). Given this data let \(X', Z'_k, Y'_k, i'_k, \pi'_k\) be the base changes of \(X, Z_k, Y_k, i_k, \pi_k\) via \(S' \to S\). Let \(\mathcal{F}'\) be the pullback of \(\mathcal{F}\) to \(X'\) and let \(\mathcal{G}'_k\) be the pullback of \(\mathcal{G}_k\) to \(Z'_k\). Let \(\alpha'_k\) be the pullback of \(\alpha_k\) to \(Y'_k\). If \(S'\) is affine, then \((Z'_k, Y'_k, i'_k, \pi'_k, \mathcal{G}'_k, \alpha'_k)_{k = 1, \ldots, n}\) is a complete dévissage of \(\mathcal{F}'/X'/S'\) over \(s'\).

Proof

By Lemma 05H7 we know that the base change of a one step dévissage is a one step dévissage. Hence it suffices to prove that formation of \(\Coker(\alpha_k)\) commutes with base change and that condition (2) of Definition 05HH is preserved by base change. The first is true as \(\pi'_{k, *}\mathcal{G}'_k\) is the pullback of \(\pi_{k, *}\mathcal{G}_k\) (by Cohomology of Schemes, Lemma 02KG) and because \(\otimes\) is right exact. The second because by the same token we have \[(\pi_{k, *}\mathcal{G}_k)_{\xi_k} \otimes_{\mathcal{O}_{Y_k, \xi_k}} \kappa(\xi_k) \otimes_{\kappa(\xi_k)} \kappa(\xi'_k) \cong (\pi'_{k, *}\mathcal{G}'_k)_{\xi'_k} \otimes_{\mathcal{O}_{Y'_k, \xi'_k}} \kappa(\xi'_k)\] with obvious notation.

Lemma

Let \(S\), \(X\), \(\mathcal{F}\), \(x\), \(s\) be as in Definition 05HI. Let \((S', s') \to (S, s)\) be a morphism of pointed schemes which induces an isomorphism \(\kappa(s) = \kappa(s')\). Let \((Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 1, \ldots, n}\) be a complete dévissage of \(\mathcal{F}/X/S\) at \(x\). Let \((Z'_k, Y'_k, i'_k, \pi'_k, \mathcal{G}'_k, \alpha'_k)_{k = 1, \ldots, n}\) be as constructed in Lemma 05HJ and let \(x' \in X'\) (resp. \(z'_k \in Z'\), \(y'_k \in Y'\)) be the unique point mapping to both \(x \in X\) (resp. \(z_k \in Z_k\), \(y_k \in Y_k\)) and \(s' \in S'\). If \(S'\) is affine, then \((Z'_k, Y'_k, i'_k, \pi'_k, \mathcal{G}'_k, \alpha'_k, z'_k, y'_k)_{k = 1, \ldots, n}\) is a complete dévissage of \(\mathcal{F}'/X'/S'\) at \(x'\).

Proof

Combine Lemma 05HJ and Lemma 05H8.

Definition

Let \(S\), \(X\), \(\mathcal{F}\), \(x\), \(s\) be as in Definition 05HI. Consider a complete dévissage \((Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 1, \ldots, n}\) of \(\mathcal{F}/X/S\) at \(x\). Let us define a standard shrinking of this situation to be given by standard opens \(S' \subset S\), \(X' \subset X\), \(Z'_k \subset Z_k\), and \(Y'_k \subset Y_k\) such that \(s_k \in S'\), \(x_k \in X'\), \(z_k \in Z'\), and \(y_k \in Y'\) and such that \[(Z'_k, Y'_k, i'_k, \pi'_k, \mathcal{G}'_k, \alpha'_k, z_k, y_k)_{k = 1, \ldots, n}\] is a one step dévissage of \(\mathcal{F}'/X'/S'\) at \(x\) where \(\mathcal{G}'_k = \mathcal{G}_k|_{Z'_k}\) and \(\mathcal{F}' = \mathcal{F}|_{X'}\).

Lemma

With assumption and notation as in Definition 05HL we have:

1. If \(S' \subset S\) is a standard open neighbourhood of \(s\), then setting \(X' = X_{S'}\), \(Z'_k = Z_{S'}\) and \(Y'_k = Y_{S'}\) we obtain a standard shrinking.

2. Let \(W \subset Y_n\) be a standard open neighbourhood of \(y\). Then there exists a standard shrinking with \(Y'_n = W \times_S S'\).

3. Let \(U \subset X\) be an open neighbourhood of \(x\). Then there exists a standard shrinking with \(X' \subset U\).

Proof

Part (1) is immediate from Lemmas 05HK and 05HA.

Proof of (2). For convenience denote \(X = Y_0\). We apply Lemma 05HA (05HC) to find a standard shrinking \(S', Y'_{n - 1}, Z'_n, Y'_n\) of the one step dévissage of \(\Coker(\alpha_{n - 1})/Y_{n - 1}/S\) at \(y_{n - 1}\) with \(Y'_n = W \times_S S'\). We may repeat this procedure and find a standard shrinking \(S'', Y''_{n - 2}, Z''_{n - 1}, Y''_{n - 1}\) of the one step dévissage of \(\Coker(\alpha_{n - 2})/Y_{n - 2}/S\) at \(y_{n - 2}\) with \(Y''_{n - 1} = Y'_{n - 1} \times_S S''\). We may continue in this manner until we obtain \(S^{(n)}, Y^{(n)}_0, Z^{(n)}_1, Y^{(n)}_1\). At this point it is clear that we obtain our desired standard shrinking by taking \(S^{(n)}\), \(X^{(n)}\), \(Z_k^{(n - k)} \times_S S^{(n)}\), and \(Y_k^{(n - k)} \times_S S^{(n)}\) with the desired property.

Proof of (3). We use induction on the length of the complete dévissage. First we apply Lemma 05HA (05HD) to find a standard shrinking \(S', X', Z'_1, Y'_1\) of the one step dévissage of \(\mathcal{F}/X/S\) at \(x\) with \(X' \subset U\). If \(n = 1\), then we are done. If \(n > 1\), then by induction we can find a standard shrinking \(S''\), \(Y''_1\), \(Z''_k\), and \(Y''_k\) of the complete dévissage \((Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 2, \ldots, n}\) of \(\Coker(\alpha_1)/Y_1/S\) at \(x\) such that \(Y''_1 \subset Y'_1\). Using Lemma 05HA (05HC) we can find \(S''' \subset S'\), \(X''' \subset X'\), \(Z'''_1\) and \(Y'''_1 = Y''_1 \times_S S'''\) which is a standard shrinking. The solution to our problem is to take \[S''', X''', Z'''_1, Y'''_1, Z''_2 \times_S S''', Y''_2 \times_S S''', \ldots, Z''_n \times_S S''', Y''_n \times_S S'''\] This ends the proof of the lemma.

Proposition

Let \(S\) be a scheme. Let \(X\) be locally of finite type over \(S\). Let \(x \in X\) be a point with image \(s \in S\). There exists a commutative diagram \[\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] }\] of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that \(g^*\mathcal{F}/X'/S'\) has a complete dévissage at \(x\).

Proof

We prove this by induction on the integer \(d = \dim_x(\text{Supp}(\mathcal{F}_s))\). By Lemma 05H6 there exists a diagram \[\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] }\] of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that \(g^*\mathcal{F}/X'/S'\) has a one step dévissage at \(x'\). The local nature of the problem implies that we may replace \((X, x) \to (S, s)\) by \((X', x') \to (S', s')\). Thus after doing so we may assume that there exists a one step dévissage \((Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)\) of \(\mathcal{F}/X/S\) at \(x\).

We apply Lemma 05HF to find a map \[\alpha_1 : \mathcal{O}_{Y_1}^{\oplus r_1} \longrightarrow \pi_{1, *}\mathcal{G}_1\] which induces an isomorphism of vector spaces over \(\kappa(\xi_1)\) where \(\xi_1 \in Y_1\) is the unique generic point of the fibre of \(Y_1\) over \(s\). Moreover \(\dim_{y_1}(\text{Supp}(\Coker(\alpha_1)_s)) < d\). It may happen that the stalk of \(\Coker(\alpha_1)_s\) at \(y_1\) is zero. In this case we may shrink \(Y_1\) by Lemma 05HA (05HC) and assume that \(\Coker(\alpha_1) = 0\) so we obtain a complete dévissage of length zero.

Assume now that the stalk of \(\Coker(\alpha_1)_s\) at \(y_1\) is not zero. In this case, by induction, there exists a commutative diagram [05HS]\[\begin{equation} \vcenter{ \xymatrix{ (Y_1, y_1) \ar[d] & (Y'_1, y'_1) \ar[l]^h \ar[d] \\ (S, s) & (S', s') \ar[l] } } \end{equation}\] of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that \(h^*\Coker(\alpha_1)/Y'_1/S'\) has a complete dévissage \[(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 2, \ldots, n}\] at \(y'_1\). (In particular \(i_2 : Z_2 \to Y'_1\) is a closed immersion into \(Y'_2\).) At this point we apply Lemma 05HE to \(S, X, \mathcal{F}, x, s\), the system \((Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)\) and diagram (05HS). We obtain a diagram \[\xymatrix{ & & (X'', x'') \ar[lld] \ar[d] & (Z''_1, z''_1) \ar[l] \ar[lld] \ar[d] \\ (X, x) \ar[d] & (Z_1, z_1) \ar[l] \ar[d] & (S'', s'') \ar[lld] & (Y''_1, y''_1) \ar[lld] \ar[l] \\ (S, s) & (Y_1, y_1) \ar[l] }\] with all the properties as listed in the referenced lemma. In particular \(Y''_1 \subset Y'_1 \times_{S'} S''\). Set \(X_1 = Y'_1 \times_{S'} S''\) and let \(\mathcal{F}_1\) denote the pullback of \(\Coker(\alpha_1)\). By Lemma 05HK the system [05HT]\[\begin{equation} (Z_k \times_{S'} S'', Y_k \times_{S'} S'', i''_k, \pi''_k, \mathcal{G}''_k, \alpha''_k, z''_k, y''_k)_{k = 2, \ldots, n} \end{equation}\] is a complete dévissage of \(\mathcal{F}_1\) to \(X_1\). Again, the nature of the problem allows us to replace \((X, x) \to (S, s)\) by \((X'', x'') \to (S'', s'')\). In this we see that we may assume:

1. There exists a one step dévissage \((Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)\) of \(\mathcal{F}/X/S\) at \(x\),

2. there exists an \(\alpha_1 : \mathcal{O}_{Y_1}^{\oplus r_1} \to \pi_{1, *}\mathcal{G}_1\) such that \(\alpha \otimes \kappa(\xi_1)\) is an isomorphism,

3. \(Y_1 \subset X_1\) is open, \(y_1 = x_1\), and \(\mathcal{F}_1|_{Y_1} \cong \Coker(\alpha_1)\), and

4. there exists a complete dévissage \((Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 2, \ldots, n}\) of \(\mathcal{F}_1/X_1/S\) at \(x_1\).

To finish the proof all we have to do is shrink the one step dévissage and the complete dévissage such that they fit together to a complete dévissage. (We suggest the reader do this on their own using Lemmas 05HA and 05HM instead of reading the proof that follows.) Since \(Y_1 \subset X_1\) is an open neighbourhood of \(x_1\) we may apply Lemma 05HM (05HQ) to find a standard shrinking \(S', X'_1, Z'_2, Y'_2, \ldots, Y'_n\) of the datum (d) so that \(X'_1 \subset Y_1\). Note that \(X'_1\) is also a standard open of the affine scheme \(Y_1\). Next, we shrink the datum (a) as follows: first we shrink the base \(S\) to \(S'\), see Lemma 05HA (05HB) and then we shrink the result to \(S''\), \(X''\), \(Z''_1\), \(Y''_1\) using Lemma 05HA (05HC) such that eventually \(Y''_1 = X'_1 \times_S S''\) and \(S'' \subset S'\). Then we see that \[Z''_1, Y''_1, Z'_2 \times_{S'} S'', Y'_2 \times_{S'} S'', \ldots, Y'_n \times_{S'} S''\] gives the complete dévissage we were looking for.

Some more bookkeeping gives the following consequence.

Lemma

Let \(X \to S\) be a finite type morphism of schemes. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(s \in S\) be a point. There exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and étale morphisms \(h_i : Y_i \to X_{S'}\), \(i = 1, \ldots, n\) such that for each \(i\) there exists a complete dévissage of \(\mathcal{F}_i/Y_i/S'\) over \(s'\), where \(\mathcal{F}_i\) is the pullback of \(\mathcal{F}\) to \(Y_i\) and such that \(X_s = (X_{S'})_{s'} \subset \bigcup h_i(Y_i)\).

Proof

For every point \(x \in X_s\) we can find a diagram \[\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] }\] of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that \(g^*\mathcal{F}/X'/S'\) has a complete dévissage at \(x'\). As \(X \to S\) is of finite type the fibre \(X_s\) is quasi-compact, and since each \(g : X' \to X\) as above is open we can cover \(X_s\) by a finite union of \(g(X'_{s'})\). Thus we can find a finite family of such diagrams \[\vcenter{ \xymatrix{ (X, x) \ar[d] & (X'_i, x'_i) \ar[l]^{g_i} \ar[d] \\ (S, s) & (S'_i, s'_i) \ar[l] } } \quad i = 1, \ldots, n\] such that \(X_s = \bigcup g_i(X'_i)\). Set \(S' = S'_1 \times_S \ldots \times_S S'_n\) and let \(Y_i = X_i \times_{S'_i} S'\) be the base change of \(X'_i\) to \(S'\). By Lemma 05HJ we see that the pullback of \(\mathcal{F}\) to \(Y_i\) has a complete dévissage over \(s\) and we win.

Translation into algebra

It may be useful to spell out algebraically what it means to have a complete dévissage. We introduce the following notion (which is not that useful so we give it an impossibly long name).

Definition

Let \(R \to S\) be a ring map. Let \(\mathfrak q\) be a prime of \(S\) lying over the prime \(\mathfrak p\) of \(R\). A elementary étale localization of the ring map \(R \to S\) at \(\mathfrak q\) is given by a commutative diagram of rings and accompanying primes \[\xymatrix{ S \ar[r] & S' \\ R \ar[u] \ar[r] & R' \ar[u] } \quad\quad \xymatrix{ \mathfrak q \ar@{-}[r] & \mathfrak q' \\ \mathfrak p \ar@{-}[u] \ar@{-}[r] & \mathfrak p' \ar@{-}[u] }\] such that \(R \to R'\) and \(S \to S'\) are étale ring maps and \(\kappa(\mathfrak p) = \kappa(\mathfrak p')\) and \(\kappa(\mathfrak q) = \kappa(\mathfrak q')\).

Definition

Let \(R \to S\) be a finite type ring map. Let \(\mathfrak r\) be a prime of \(R\). Let \(N\) be a finite \(S\)-module. A complete dévissage of \(N/S/R\) over \(\mathfrak r\) is given by \(R\)-algebra maps \[\xymatrix{ & A_1 & & A_2 & & ... & & A_n \\ S \ar[ru] & & B_1 \ar[lu] \ar[ru] & & ... \ar[lu] \ar[ru] & & ... \ar[lu] \ar[ru] & & B_n \ar[lu] }\] finite \(A_i\)-modules \(M_i\) and \(B_i\)-module maps \(\alpha_i : B_i^{\oplus r_i} \to M_i\) such that

1. \(S \to A_1\) is surjective and of finite presentation,

2. \(B_i \to A_{i + 1}\) is surjective and of finite presentation,

3. \(B_i \to A_i\) is finite,

4. \(R \to B_i\) is smooth with geometrically irreducible fibres,

5. \(N \cong M_1\) as \(S\)-modules,

6. \(\Coker(\alpha_i) \cong M_{i + 1}\) as \(B_i\)-modules,

7. \(\alpha_i : \kappa(\mathfrak p_i)^{\oplus r_i} \to M_i \otimes_{B_i} \kappa(\mathfrak p_i)\) is an isomorphism where \(\mathfrak p_i = \mathfrak rB_i\), and

8. \(\Coker(\alpha_n) = 0\).

In this situation we say that \((A_i, B_i, M_i, \alpha_i)_{i = 1, \ldots, n}\) is a complete dévissage of \(N/S/R\) over \(\mathfrak r\).

Remark

Note that the \(R\)-algebras \(B_i\) for all \(i\) and \(A_i\) for \(i \geq 2\) are of finite presentation over \(R\). If \(S\) is of finite presentation over \(R\), then it is also the case that \(A_1\) is of finite presentation over \(R\). In this case all the ring maps in the complete dévissage are of finite presentation. See Algebra, Lemma 00F4. Still assuming \(S\) of finite presentation over \(R\) the following are equivalent

1. \(M\) is of finite presentation over \(S\),

2. \(M_1\) is of finite presentation over \(A_1\),

3. \(M_1\) is of finite presentation over \(B_1\),

4. each \(M_i\) is of finite presentation both as an \(A_i\)-module and as a \(B_i\)-module.

The equivalences (1) \(\Leftrightarrow\) (2) and (2) \(\Leftrightarrow\) (3) follow from Algebra, Lemma 0564. If \(M_1\) is finitely presented, so is \(\Coker(\alpha_1)\) (see Algebra, Lemma 0519) and hence \(M_2\), etc.

Definition

Let \(R \to S\) be a finite type ring map. Let \(\mathfrak q\) be a prime of \(S\) lying over the prime \(\mathfrak r\) of \(R\). Let \(N\) be a finite \(S\)-module. A complete dévissage of \(N/S/R\) at \(\mathfrak q\) is given by a complete dévissage \((A_i, B_i, M_i, \alpha_i)_{i = 1, \ldots, n}\) of \(N/S/R\) over \(\mathfrak r\) and prime ideals \(\mathfrak q_i \subset B_i\) lying over \(\mathfrak r\) such that

1. \(\kappa(\mathfrak r) \subset \kappa(\mathfrak q_i)\) is purely transcendental,

2. there is a unique prime \(\mathfrak q'_i \subset A_i\) lying over \(\mathfrak q_i \subset B_i\),

3. \(\mathfrak q = \mathfrak q'_1 \cap S\) and \(\mathfrak q_i = \mathfrak q'_{i + 1} \cap A_i\),

4. \(R \to B_i\) has relative dimension \(\dim_{\mathfrak q_i}(\text{Supp}(M_i \otimes_R \kappa(\mathfrak r)))\).

Remark

Let \(A \to B\) be a finite type ring map and let \(N\) be a finite \(B\)-module. Let \(\mathfrak q\) be a prime of \(B\) lying over the prime \(\mathfrak r\) of \(A\). Set \(X = \Spec(B)\), \(S = \Spec(A)\) and \(\mathcal{F} = \widetilde{N}\) on \(X\). Let \(x\) be the point corresponding to \(\mathfrak q\) and let \(s \in S\) be the point corresponding to \(\mathfrak p\). Then

1. if there exists a complete dévissage of \(\mathcal{F}/X/S\) over \(s\) then there exists a complete dévissage of \(N/B/A\) over \(\mathfrak p\), and

2. there exists a complete dévissage of \(\mathcal{F}/X/S\) at \(x\) if and only if there exists a complete dévissage of \(N/B/A\) at \(\mathfrak q\).

There is just a small twist in that we omitted the condition on the relative dimension in the formulation of “a complete dévissage of \(N/B/A\) over \(\mathfrak p\)” which is why the implication in (1) only goes in one direction. The notion of a complete dévissage at \(\mathfrak q\) does have this condition built in. In any case we will only use that existence for \(\mathcal{F}/X/S\) implies the existence for \(N/B/A\).

Lemma

Let \(R \to S\) be a finite type ring map. Let \(M\) be a finite \(S\)-module. Let \(\mathfrak q\) be a prime ideal of \(S\). There exists an elementary étale localization \(R' \to S', \mathfrak q', \mathfrak p'\) of the ring map \(R \to S\) at \(\mathfrak q\) such that there exists a complete dévissage of \((M \otimes_S S')/S'/R'\) at \(\mathfrak q'\).

Proof

This is a reformulation of Proposition 05HR via Remark 05I0

Localization and universally injective maps

Lemma

Let \(R \to S\) be a ring map. Let \(N\) be a \(S\)-module. Assume

1. \(R\) is a local ring with maximal ideal \(\mathfrak m\),

2. \(\overline{S} = S/\mathfrak m S\) is Noetherian, and

3. \(\overline{N} = N/\mathfrak m_R N\) is a finite \(\overline{S}\)-module.

Let \(\Sigma \subset S\) be the multiplicative subset of elements which are not a zerodivisor on \(\overline{N}\). Then \(\Sigma^{-1}S\) is a semi-local ring whose spectrum consists of primes \(\mathfrak q \subset S\) contained in an element of \(\text{Ass}_S(\overline{N})\). Moreover, any maximal ideal of \(\Sigma^{-1}S\) corresponds to an associated prime of \(\overline{N}\) over \(\overline{S}\).

Proof

Note that \(\text{Ass}_S(\overline{N}) = \text{Ass}_{\overline{S}}(\overline{N})\), see Algebra, Lemma 05BY. This is a finite set by Algebra, Lemma 00LC. Say \(\{\mathfrak q_1, \ldots, \mathfrak q_r\} = \text{Ass}_S(\overline{N})\). We have \(\Sigma = S \setminus (\bigcup \mathfrak q_i)\) by Algebra, Lemma 00LD. By the description of \(\Spec(\Sigma^{-1}S)\) in Algebra, Lemma 00E3 and by Algebra, Lemma 00DS we see that the primes of \(\Sigma^{-1}S\) correspond to the primes of \(S\) contained in one of the \(\mathfrak q_i\). Hence the maximal ideals of \(\Sigma^{-1}S\) correspond one-to-one with the maximal (w.r.t. inclusion) elements of the set \(\{\mathfrak q_1, \ldots, \mathfrak q_r\}\). This proves the lemma.

Lemma

Assumption and notation as in Lemma 05DE. Assume moreover that

1. \(S\) is local and \(R \to S\) is a local homomorphism,

2. \(S\) is essentially of finite presentation over \(R\),

3. \(N\) is finitely presented over \(S\), and

4. \(N\) is flat over \(R\).

Then each \(s \in \Sigma\) defines a universally injective \(R\)-module map \(s : N \to N\), and the map \(N \to \Sigma^{-1}N\) is \(R\)-universally injective.

Proof

By Algebra, Lemma 046Y the sequence \(0 \to N \to N \to N/sN \to 0\) is exact and \(N/sN\) is flat over \(R\). This implies that \(s : N \to N\) is universally injective, see Algebra, Lemma 00HL. The map \(N \to \Sigma^{-1}N\) is universally injective as the directed colimit of the maps \(s : N \to N\).

Lemma

Let \(R \to S\) be a ring map. Let \(N\) be an \(S\)-module. Let \(S \to S'\) be a ring map. Assume

1. \(R \to S\) is a local homomorphism of local rings

2. \(S\) is essentially of finite presentation over \(R\),

3. \(N\) is of finite presentation over \(S\),

4. \(N\) is flat over \(R\),

5. \(S \to S'\) is flat, and

6. the image of \(\Spec(S') \to \Spec(S)\) contains all primes \(\mathfrak q\) of \(S\) lying over \(\mathfrak m_R\) such that \(\mathfrak q\) is an associated prime of \(N/\mathfrak m_R N\).

Then \(N \to N \otimes_S S'\) is \(R\)-universally injective.

Proof

Set \(N' = N \otimes_R S'\). Consider the commutative diagram \[\xymatrix{ N \ar[d] \ar[r] & N' \ar[d] \\ \Sigma^{-1}N \ar[r] & \Sigma^{-1}N' }\] where \(\Sigma \subset S\) is the set of elements which are not a zerodivisor on \(N/\mathfrak m_R N\). If we can show that the map \(N \to \Sigma^{-1}N'\) is universally injective, then \(N \to N'\) is too (see Algebra, Lemma 05CJ).

By Lemma 05DE the ring \(\Sigma^{-1}S\) is a semi-local ring whose maximal ideals correspond to associated primes of \(N/\mathfrak m_R N\). Hence the image of \(\Spec(\Sigma^{-1}S') \to \Spec(\Sigma^{-1}S)\) contains all these maximal ideals by assumption. By Algebra, Lemma 00HQ the ring map \(\Sigma^{-1}S \to \Sigma^{-1}S'\) is faithfully flat. Hence \(\Sigma^{-1}N \to \Sigma^{-1}N'\), which is the map \[N \otimes_S \Sigma^{-1}S \longrightarrow N \otimes_S \Sigma^{-1}S'\] is universally injective, see Algebra, Lemmas 05CK and 05CH. Finally, we apply Lemma 05DF to see that \(N \to \Sigma^{-1}N\) is universally injective. As the composition of universally injective module maps is universally injective (see Algebra, Lemma 05CI) we conclude that \(N \to \Sigma^{-1}N'\) is universally injective and we win.

Lemma

Let \(R \to S\) be a ring map. Let \(N\) be an \(S\)-module. Let \(S \to S'\) be a ring map. Assume

1. \(R \to S\) is of finite presentation and \(N\) is of finite presentation over \(S\),

2. \(N\) is flat over \(R\),

3. \(S \to S'\) is flat, and

4. the image of \(\Spec(S') \to \Spec(S)\) contains all primes \(\mathfrak q\) such that \(\mathfrak q\) is an associated prime of \(N \otimes_R \kappa(\mathfrak p)\) where \(\mathfrak p\) is the inverse image of \(\mathfrak q\) in \(R\).

Then \(N \to N \otimes_S S'\) is \(R\)-universally injective.

Proof

By Algebra, Lemma 05CL it suffices to show that \(N_{\mathfrak q} \to (N \otimes_R S')_{\mathfrak q}\) is a \(R_{\mathfrak p}\)-universally injective for any prime \(\mathfrak q\) of \(S\) lying over \(\mathfrak p\) in \(R\). Thus we may apply Lemma 05DG to the ring maps \(R_{\mathfrak p} \to S_{\mathfrak q} \to S'_{\mathfrak q}\) and the module \(N_{\mathfrak q}\).

The reader may want to compare the following lemma to Algebra, Lemmas 00ME and 046Y and the results of Section 0ASZ. In each case the conclusion is that the map \(u : M \to N\) is universally injective with flat cokernel.

Lemma

Let \((R, \mathfrak m)\) be a local ring. Let \(u : M \to N\) be an \(R\)-module map. If \(M\) is a projective \(R\)-module, \(N\) is a flat \(R\)-module, and \(\overline{u} : M/\mathfrak mM \to N/\mathfrak mN\) is injective then \(u\) is universally injective.

Proof

By Algebra, Theorem 0593 the module \(M\) is free. If we show the result holds for every finitely generated direct summand of \(M\), then the lemma follows. Hence we may assume that \(M\) is finite free. Write \(N = \colim_i N_i\) as a directed colimit of finite free modules, see Algebra, Theorem 058G. Note that \(u : M \to N\) factors through \(N_i\) for some \(i\) (as \(M\) is finite free). Denote \(u_i : M \to N_i\) the corresponding \(R\)-module map. As \(\overline{u}\) is injective we see that \(\overline{u_i} : M/\mathfrak mM \to N_i/\mathfrak mN_i\) is injective and remains injective on composing with the maps \(N_i/\mathfrak mN_i \to N_{i'}/\mathfrak mN_{i'}\) for all \(i' \geq i\). As \(M\) and \(N_{i'}\) are finite free over the local ring \(R\) this implies that \(M \to N_{i'}\) is a split injection for all \(i' \geq i\). Hence for any \(R\)-module \(Q\) we see that \(M \otimes_R Q \to N_{i'} \otimes_R Q\) is injective for all \(i' \geq i\). As \(- \otimes_R Q\) commutes with colimits we conclude that \(M \otimes_R Q \to N_{i'} \otimes_R Q\) is injective as desired.

Lemma

Assumption and notation as in Lemma 05DE. Assume moreover that \(N\) is projective as an \(R\)-module. Then each \(s \in \Sigma\) defines a universally injective \(R\)-module map \(s : N \to N\), and the map \(N \to \Sigma^{-1}N\) is \(R\)-universally injective.

Proof

Pick \(s \in \Sigma\). By Lemma 05FQ the map \(s : N \to N\) is universally injective. The map \(N \to \Sigma^{-1}N\) is universally injective as the directed colimit of the maps \(s : N \to N\).

Completion and Mittag-Leffler modules

Lemma

Let \(R\) be a ring. Let \(I \subset R\) be an ideal. Let \(A\) be a set. Assume \(R\) is Noetherian and complete with respect to \(I\). The completion \((\bigoplus\nolimits_{\alpha \in A} R)^\wedge\) is flat and Mittag-Leffler.

Proof

By More on Algebra, Lemma 05BC the map \((\bigoplus\nolimits_{\alpha \in A} R)^\wedge \to \prod_{\alpha \in A} R\) is universally injective. Thus, by Algebra, Lemmas 058P and 059N it suffices to show that \(\prod_{\alpha \in A} R\) is flat and Mittag-Leffler. By Algebra, Proposition 05CZ (and Algebra, Lemma 05CY) we see that \(\prod_{\alpha \in A} R\) is flat. Thus we conclude because a product of copies of \(R\) is Mittag-Leffler, see Algebra, Lemma 05D0.

Lemma

Let \(R\) be a ring. Let \(I \subset R\) be an ideal. Let \(M\) be an \(R\)-module. Assume

1. \(R\) is Noetherian and \(I\)-adically complete,

2. \(M\) is flat over \(R\), and

3. \(M/IM\) is a projective \(R/I\)-module.

Then the \(I\)-adic completion \(M^\wedge\) is a flat Mittag-Leffler \(R\)-module.

Proof

Choose a surjection \(F \to M\) where \(F\) is a free \(R\)-module. By Algebra, Lemma 05D3 the module \(M^\wedge\) is a direct summand of the module \(F^\wedge\). Hence it suffices to prove the lemma for \(F\). In this case the lemma follows from Lemma 05DJ.

In Lemmas 05DL and 05DM the assumption that \(S\) be Noetherian holds if \(R \to S\) is of finite type, see Algebra, Lemma 00FN.

Lemma

Let \(R\) be a ring. Let \(I \subset R\) be an ideal. Let \(R \to S\) be a ring map, and \(N\) an \(S\)-module. Assume

1. \(R\) is a Noetherian ring,

2. \(S\) is a Noetherian ring,

3. \(N\) is a finite \(S\)-module, and

4. for any finite \(R\)-module \(Q\), any \(\mathfrak q \in \text{Ass}_S(Q \otimes_R N)\) satisfies \(IS + \mathfrak q \not = S\).

Then the map \(N \to N^\wedge\) of \(N\) into the \(I\)-adic completion of \(N\) is universally injective as a map of \(R\)-modules.

Proof

We have to show that for any finite \(R\)-module \(Q\) the map \(Q \otimes_R N \to Q \otimes_R N^\wedge\) is injective, see Algebra, Theorem 058K. As there is a canonical map \(Q \otimes_R N^\wedge \to (Q \otimes_R N)^\wedge\) it suffices to prove that the canonical map \(Q \otimes_R N \to (Q \otimes_R N)^\wedge\) is injective. Hence we may replace \(N\) by \(Q \otimes_R N\) and it suffices to prove the injectivity for the map \(N \to N^\wedge\).

Let \(K = \Ker(N \to N^\wedge)\). It suffices to show that \(K_{\mathfrak q} = 0\) for \(\mathfrak q \in \text{Ass}(N)\) as \(N\) is a submodule of \(\prod_{\mathfrak q \in \text{Ass}(N)} N_{\mathfrak q}\), see Algebra, Lemma 0311. Pick \(\mathfrak q \in \text{Ass}(N)\). By the last assumption we see that there exists a prime \(\mathfrak q' \supset IS + \mathfrak q\). Since \(K_{\mathfrak q}\) is a localization of \(K_{\mathfrak q'}\) it suffices to prove the vanishing of \(K_{\mathfrak q'}\). Note that \(K = \bigcap I^nN\), hence \(K_{\mathfrak q'} \subset \bigcap I^nN_{\mathfrak q'}\). Hence \(K_{\mathfrak q'} = 0\) by Algebra, Lemma 00IP.

Lemma

Let \(R\) be a ring. Let \(I \subset R\) be an ideal. Let \(R \to S\) be a ring map, and \(N\) an \(S\)-module. Assume

1. \(R\) is a Noetherian ring,

2. \(S\) is a Noetherian ring,

3. \(N\) is a finite \(S\)-module,

4. \(N\) is flat over \(R\), and

5. for any prime \(\mathfrak q \subset S\) which is an associated prime of \(N \otimes_R \kappa(\mathfrak p)\) where \(\mathfrak p = R \cap \mathfrak q\) we have \(IS + \mathfrak q \not = S\).

Then the map \(N \to N^\wedge\) of \(N\) into the \(I\)-adic completion of \(N\) is universally injective as a map of \(R\)-modules.

Proof

This follows from Lemma 05DL because Algebra, Lemma 05C2 and Remark 05E0 guarantee that the set of associated primes of tensor products \(N \otimes_R Q\) are contained in the set of associated primes of the modules \(N \otimes_R \kappa(\mathfrak p)\).

Projective modules

The following lemma can be used to prove projectivity by Noetherian induction on the base, see Lemma 05FS.

Lemma

Let \(R\) be a ring. Let \(I \subset R\) be an ideal. Let \(R \to S\) be a ring map, and \(N\) an \(S\)-module. Assume

1. \(R\) is Noetherian and \(I\)-adically complete,

2. \(R \to S\) is of finite type,

3. \(N\) is a finite \(S\)-module,

4. \(N\) is flat over \(R\),

5. \(N/IN\) is projective as a \(R/I\)-module, and

6. for any prime \(\mathfrak q \subset S\) which is an associated prime of \(N \otimes_R \kappa(\mathfrak p)\) where \(\mathfrak p = R \cap \mathfrak q\) we have \(IS + \mathfrak q \not = S\).

Then \(N\) is projective as an \(R\)-module.

Proof

By Lemma 05DM the map \(N \to N^\wedge\) is universally injective. By Lemma 05DK the module \(N^\wedge\) is Mittag-Leffler. By Algebra, Lemma 059N we conclude that \(N\) is Mittag-Leffler. Hence \(N\) is countably generated, flat and Mittag-Leffler as an \(R\)-module, whence projective by Algebra, Lemma 059X.

Lemma

Let \(R\) be a ring. Let \(R \to S\) be a ring map. Assume

1. \(R\) is Noetherian,

2. \(R \to S\) is of finite type and flat, and

3. every fibre ring \(S \otimes_R \kappa(\mathfrak p)\) is geometrically integral over \(\kappa(\mathfrak p)\).

Then \(S\) is projective as an \(R\)-module.

Proof

Consider the set \[\{I \subset R \mid S/IS\text{ not projective as }R/I\text{-module}\}\] We have to show this set is empty. To get a contradiction assume it is nonempty. Then it contains a maximal element \(I\). Let \(J = \sqrt{I}\) be its radical. If \(I \not = J\), then \(S/JS\) is projective as a \(R/J\)-module, and \(S/IS\) is flat over \(R/I\) and \(J/I\) is a nilpotent ideal in \(R/I\). Applying Algebra, Lemma 05CG we see that \(S/IS\) is a projective \(R/I\)-module, which is a contradiction. Hence we may assume that \(I\) is a radical ideal. In other words we are reduced to proving the lemma in case \(R\) is a reduced ring and \(S/IS\) is a projective \(R/I\)-module for every nonzero ideal \(I\) of \(R\).

Assume \(R\) is a reduced ring and \(S/IS\) is a projective \(R/I\)-module for every nonzero ideal \(I\) of \(R\). By generic flatness, Algebra, Lemma 051R (applied to a localization \(R_g\) which is a domain) or the more general Algebra, Lemma 051Z there exists a nonzero \(f \in R\) such that \(S_f\) is free as an \(R_f\)-module. Denote \(R^\wedge = \lim R/(f^n)\) the \((f)\)-adic completion of \(R\). Note that the ring map \[R \longrightarrow R_f \times R^\wedge\] is a faithfully flat ring map, see Algebra, Lemma 00MB. Hence by faithfully flat descent of projectivity, see Algebra, Theorem 05A9 it suffices to prove that \(S \otimes_R R^\wedge\) is a projective \(R^\wedge\)-module. To see this we will use the criterion of Lemma 05DP. First of all, note that \(S/fS = (S \otimes_R R^\wedge)/f(S \otimes_R R^\wedge)\) is a projective \(R/(f)\)-module and that \(S \otimes_R R^\wedge\) is flat and of finite type over \(R^\wedge\) as a base change of such. Next, suppose that \(\mathfrak p^\wedge\) is a prime ideal of \(R^\wedge\). Let \(\mathfrak p \subset R\) be the corresponding prime of \(R\). As \(R \to S\) has geometrically integral fibre rings, the same is true for the fibre rings of any base change. Hence \(\mathfrak q^\wedge = \mathfrak p^\wedge(S \otimes_R R^\wedge)\), is a prime ideals lying over \(\mathfrak p^\wedge\) and it is the unique associated prime of \(S \otimes_R \kappa(\mathfrak p^\wedge)\). Thus we win if \(f(S \otimes_R R^\wedge) + \mathfrak q^\wedge \not = S \otimes_R R^\wedge\). This is true because \(\mathfrak p^\wedge + fR^\wedge \not = R^\wedge\) as \(f\) lies in the Jacobson radical of the \(f\)-adically complete ring \(R^\wedge\) and because \(R^\wedge \to S \otimes_R R^\wedge\) is surjective on spectra as its fibres are nonempty (irreducible spaces are nonempty).

Lemma

Let \(R\) be a ring. Let \(R \to S\) be a ring map. Assume

1. \(R \to S\) is of finite presentation and flat, and

2. every fibre ring \(S \otimes_R \kappa(\mathfrak p)\) is geometrically integral over \(\kappa(\mathfrak p)\).

Then \(S\) is projective as an \(R\)-module.

Proof

We can find a cocartesian diagram of rings \[\xymatrix{ S_0 \ar[r] & S \\ R_0 \ar[u] \ar[r] & R \ar[u] }\] such that \(R_0\) is of finite type over \(\mathbf{Z}\), the map \(R_0 \to S_0\) is of finite type and flat with geometrically integral fibres, see More on Morphisms, Lemmas 05FE, 05FG, 05FH, and 05FL. By Lemma 05FS we see that \(S_0\) is a projective \(R_0\)-module. Hence \(S = S_0 \otimes_{R_0} R\) is a projective \(R\)-module, see Algebra, Lemma 05A3.

Remark

Lemma 05FT is a key step in the development of results in this chapter. The analogue of this lemma in [GruRay] is [GruRay, I Proposition 3.3.1]: If \(R \to S\) is smooth with geometrically integral fibres, then \(S\) is projective as an \(R\)-module. This is a special case of Lemma 05FT, but as we will later improve on this lemma anyway, we do not gain much from having a stronger result at this point. We briefly sketch the proof of this as it is given in [GruRay].

1. First reduce to the case where \(R\) is Noetherian as above.

2. Since projectivity descends through faithfully flat ring maps, see Algebra, Theorem 05A9 we may work locally in the fppf topology on \(R\), hence we may assume that \(R \to S\) has a section \(\sigma : S \to R\). (Just by the usual trick of base changing to \(S\).) Set \(I = \Ker(S \to R)\).

3. Localizing a bit more on \(R\) we may assume that \(I/I^2\) is a free \(R\)-module and that the completion \(S^\wedge\) of \(S\) with respect to \(I\) is isomorphic to \(R[[t_1, \ldots, t_n]]\), see Morphisms, Lemma 05D9. Here we are using that \(R \to S\) is smooth.

4. To prove that \(S\) is projective as an \(R\)-module, it suffices to prove that \(S\) is flat, countably generated and Mittag-Leffler as an \(R\)-module, see Algebra, Lemma 059X. The first two properties are evident. Thus it suffices to prove that \(S\) is Mittag-Leffler as an \(R\)-module. By Algebra, Lemma 059T the module \(R[[t_1, \ldots, t_n]]\) is Mittag-Leffler over \(R\). Hence Algebra, Lemma 059N shows that it suffices to show that the \(S \to S^\wedge\) is universally injective as a map of \(R\)-modules.

5. Apply Lemma 05DH to see that \(S \to S^\wedge\) is \(R\)-universally injective. Namely, as \(R \to S\) has geometrically integral fibres, any associated point of any fibre ring is just the generic point of the fibre ring which is in the image of \(\Spec(S^\wedge) \to \Spec(S)\).

There is an analogy between the proof as sketched just now, and the development of the arguments leading to the proof of Lemma 05FT. In both a completion plays an essential role, and both times the assumption of having geometrically integral fibres assures one that the map from \(S\) to the completion of \(S\) is \(R\)-universally injective.

Flat finite type modules, Part I

In some cases given a ring map \(R \to S\) of finite presentation and a finite \(S\)-module \(N\) the flatness of \(N\) over \(R\) implies that \(N\) is of finite presentation. In this section we prove this is true “pointwise”. We remark that the first proof of Proposition 05I5 uses the geometric results of Section 057P but not the existence of a complete dévissage.

Lemma

Let \((R, \mathfrak m)\) be a local ring. Let \(R \to S\) be a finitely presented flat ring map with geometrically integral fibres. Write \(\mathfrak p = \mathfrak mS\). Let \(\mathfrak q \subset S\) be a prime ideal lying over \(\mathfrak m\). Let \(N\) be a finite \(S\)-module. There exist \(r \geq 0\) and an \(S\)-module map \[\alpha : S^{\oplus r} \longrightarrow N\] such that \(\alpha : \kappa(\mathfrak p)^{\oplus r} \to N \otimes_S \kappa(\mathfrak p)\) is an isomorphism. For any such \(\alpha\) the following are equivalent:

1. \(N_{\mathfrak q}\) is \(R\)-flat,

2. \(\alpha\) is \(R\)-universally injective and \(\Coker(\alpha)_{\mathfrak q}\) is \(R\)-flat,

3. \(\alpha\) is injective and \(\Coker(\alpha)_{\mathfrak q}\) is \(R\)-flat,

4. \(\alpha_{\mathfrak p}\) is an isomorphism and \(\Coker(\alpha)_{\mathfrak q}\) is \(R\)-flat, and

5. \(\alpha_{\mathfrak q}\) is injective and \(\Coker(\alpha)_{\mathfrak q}\) is \(R\)-flat.

Proof

To obtain \(\alpha\) set \(r = \dim_{\kappa(\mathfrak p)} N \otimes_S \kappa(\mathfrak p)\) and pick \(x_1, \ldots, x_r \in N\) which form a basis of \(N \otimes_S \kappa(\mathfrak p)\). Define \(\alpha(s_1, \ldots, s_r) = \sum s_i x_i\). This proves the existence.

Fix an \(\alpha\). The most interesting implication is (1) \(\Rightarrow\) (2) which we prove first. Assume (1). Because \(S/\mathfrak mS\) is a domain with fraction field \(\kappa(\mathfrak p)\) we see that \((S/\mathfrak mS)^{\oplus r} \to N_{\mathfrak p}/\mathfrak mN_{\mathfrak p} = N \otimes_S \kappa(\mathfrak p)\) is injective. Hence by Lemmas 05FQ and 05FT. the map \(S^{\oplus r} \to N_{\mathfrak p}\) is \(R\)-universally injective. It follows that \(S^{\oplus r} \to N\) is \(R\)-universally injective, see Algebra, Lemma 05CJ. Then also the localization \(\alpha_{\mathfrak q}\) is \(R\)-universally injective, see Algebra, Lemma 05CM. We conclude that \(\Coker(\alpha)_{\mathfrak q}\) is \(R\)-flat by Algebra, Lemma 058P.

The implication (2) \(\Rightarrow\) (3) is immediate. If (3) holds, then \(\alpha_{\mathfrak p}\) is injective as a localization of an injective module map. By Nakayama’s lemma (Algebra, Lemma 00DV) \(\alpha_{\mathfrak p}\) is surjective too. Hence (3) \(\Rightarrow\) (4). If (4) holds, then \(\alpha_{\mathfrak p}\) is an isomorphism, so \(\alpha\) is injective as \(S_{\mathfrak q} \to S_{\mathfrak p}\) is injective. Namely, elements of \(S \setminus \mathfrak p\) are nonzerodivisors on \(S\) by a combination of Lemmas 05FR and 05FT. Hence (4) \(\Rightarrow\) (5). Finally, if (5) holds, then \(N_{\mathfrak q}\) is \(R\)-flat as an extension of flat modules, see Algebra, Lemma 00HM. Hence (5) \(\Rightarrow\) (1) and the proof is finished.

Lemma

Let \((R, \mathfrak m)\) be a local ring. Let \(R \to S\) be a ring map of finite presentation. Let \(N\) be a finite \(S\)-module. Let \(\mathfrak q\) be a prime of \(S\) lying over \(\mathfrak m\). Assume that \(N_{\mathfrak q}\) is flat over \(R\), and assume there exists a complete dévissage of \(N/S/R\) at \(\mathfrak q\). Then \(N\) is a finitely presented \(S\)-module, free as an \(R\)-module, and there exists an isomorphism \[N \cong B_1^{\oplus r_1} \oplus \ldots \oplus B_n^{\oplus r_n}\] as \(R\)-modules where each \(B_i\) is a smooth \(R\)-algebra with geometrically irreducible fibres.

Proof

Let \((A_i, B_i, M_i, \alpha_i, \mathfrak q_i)_{i = 1, \ldots, n}\) be the given complete dévissage. We prove the lemma by induction on \(n\). Note that \(N\) is finitely presented as an \(S\)-module if and only if \(M_1\) is finitely presented as an \(B_1\)-module, see Remark 05HY. Note that \(N_{\mathfrak q} \cong (M_1)_{\mathfrak q_1}\) as \(R\)-modules because (a) \(N_{\mathfrak q} \cong (M_1)_{\mathfrak q'_1}\) where \(\mathfrak q'_1\) is the unique prime in \(A_1\) lying over \(\mathfrak q_1\) and (b) \((A_1)_{\mathfrak q'_1} = (A_1)_{\mathfrak q_1}\) by Algebra, Lemma 00EA, so (c) \((M_1)_{\mathfrak q'_1} \cong (M_1)_{\mathfrak q_1}\). Hence \((M_1)_{\mathfrak q_1}\) is a flat \(R\)-module. Thus we may replace \((S, N)\) by \((B_1, M_1)\) in order to prove the lemma. By Lemma 05I3 the map \(\alpha_1 : B_1^{\oplus r_1} \to M_1\) is \(R\)-universally injective and \(\Coker(\alpha_1)_{\mathfrak q}\) is \(R\)-flat. Note that \((A_i, B_i, M_i, \alpha_i, \mathfrak q_i)_{i = 2, \ldots, n}\) is a complete dévissage of \(\Coker(\alpha_1)/B_1/R\) at \(\mathfrak q_1\). Hence the induction hypothesis implies that \(\Coker(\alpha_1)\) is finitely presented as a \(B_1\)-module, free as an \(R\)-module, and has a decomposition as in the lemma. This implies that \(M_1\) is finitely presented as a \(B_1\)-module, see Algebra, Lemma 0519. It further implies that \(M_1 \cong B_1^{\oplus r_1} \oplus \Coker(\alpha_1)\) as \(R\)-modules, hence a decomposition as in the lemma. Finally, \(B_1\) is projective as an \(R\)-module by Lemma 05FT hence free as an \(R\)-module by Algebra, Theorem 0593. This finishes the proof.

Proposition

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(x \in X\) with image \(s \in S\). Assume that

1. \(f\) is locally of finite presentation,

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

3. \(\mathcal{F}\) is flat at \(x\) over \(S\).

Then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and an open subscheme \[V \subset X \times_S \Spec(\mathcal{O}_{S', s'})\] which contains the unique point of \(X \times_S \Spec(\mathcal{O}_{S', s'})\) mapping to \(x\) such that the pullback of \(\mathcal{F}\) to \(V\) is an \(\mathcal{O}_V\)-module of finite presentation and flat over \(\mathcal{O}_{S', s'}\).

Proof

This proof is longer but does not use the existence of a complete dévissage. The problem is local around \(x\) and \(s\), hence we may assume that \(X\) and \(S\) are affine. During the proof we will finitely many times replace \(S\) by an elementary étale neighbourhood of \((S, s)\). The goal is then to find (after such a replacement) an open \(V \subset X \times_S \Spec(\mathcal{O}_{S, s})\) containing \(x\) such that \(\mathcal{F}|_V\) is flat over \(S\) and finitely presented. Of course we may also replace \(S\) by \(\Spec(\mathcal{O}_{S, s})\) at any point of the proof, i.e., we may assume \(S\) is a local scheme. We will prove the proposition by induction on the integer \(n = \dim_x(\text{Supp}(\mathcal{F}_s))\).

We can choose

1. elementary étale neighbourhoods \(g : (X', x') \to (X, x)\), \(e : (S', s') \to (S, s)\),

2. a commutative diagram \[\xymatrix{ X \ar[dd]_f & X' \ar[dd] \ar[l]^g & Z' \ar[l]^i \ar[d]^\pi \\ & & Y' \ar[d]^h \\ S & S' \ar[l]_e & S' \ar@{=}[l] }\]

3. a point \(z' \in Z'\) with \(i(z') = x'\), \(y' = \pi(z')\), \(h(y') = s'\),

4. a finite type quasi-coherent \(\mathcal{O}_{Z'}\)-module \(\mathcal{G}\),

as in Lemma 057S. We are going to replace \(S\) by \(\Spec(\mathcal{O}_{S', s'})\), see remarks in first paragraph of the proof. Consider the diagram \[\xymatrix{ X_{\mathcal{O}_{S', s'}} \ar[ddr]_f & X'_{\mathcal{O}_{S', s'}} \ar[dd] \ar[l]^g & Z'_{\mathcal{O}_{S', s'}} \ar[l]^i \ar[d]^\pi \\ & & Y'_{\mathcal{O}_{S', s'}} \ar[dl]^h \\ & \Spec(\mathcal{O}_{S', s'}) }\] Here we have base changed the schemes \(X', Z', Y'\) over \(S'\) via \(\Spec(\mathcal{O}_{S', s'}) \to S'\) and the scheme \(X\) over \(S\) via \(\Spec(\mathcal{O}_{S', s'}) \to S\). It is still the case that \(g\) is étale, see Lemma 05H2. After replacing \(X\) by \(X_{\mathcal{O}_{S', s'}}\), \(X'\) by \(X'_{\mathcal{O}_{S', s'}}\), \(Z'\) by \(Z'_{\mathcal{O}_{S', s'}}\), and \(Y'\) by \(Y'_{\mathcal{O}_{S', s'}}\) we may assume we have a diagram as Lemma 057S where in addition \(S = S'\) is a local scheme with closed point \(s\). By Lemmas 057T and 057U the result for \(Y' \to S\), the sheaf \(\pi_*\mathcal{G}\), and the point \(y'\) implies the result for \(X \to S\), \(\mathcal{F}\) and \(x\). Hence we may assume that \(S\) is local and \(X \to S\) is a smooth morphism of affines with geometrically irreducible fibres of dimension \(n\).

The base case of the induction: \(n = 0\). As \(X \to S\) is smooth with geometrically irreducible fibres of dimension \(0\) we see that \(X \to S\) is an open immersion, see Descent, Lemma 02LC. As \(S\) is local and the closed point is in the image of \(X \to S\) we conclude that \(X = S\). Thus we see that \(\mathcal{F}\) corresponds to a finite flat \(\mathcal{O}_{S, s}\) module. In this case the result follows from Algebra, Lemma 00NZ which tells us that \(\mathcal{F}\) is in fact finite free.

The induction step. Assume the result holds whenever the dimension of the support in the closed fibre is \(< n\). Write \(S = \Spec(A)\), \(X = \Spec(B)\) and \(\mathcal{F} = \widetilde{N}\) for some \(B\)-module \(N\). Note that \(A\) is a local ring; denote its maximal ideal \(\mathfrak m\). Then \(\mathfrak p = \mathfrak mB\) is the unique minimal prime lying over \(\mathfrak m\) as \(X \to S\) has geometrically irreducible fibres. Finally, let \(\mathfrak q \subset B\) be the prime corresponding to \(x\). By Lemma 05I3 we can choose a map \[\alpha : B^{\oplus r} \to N\] such that \(\kappa(\mathfrak p)^{\oplus r} \to N \otimes_B \kappa(\mathfrak p)\) is an isomorphism. Moreover, as \(N_{\mathfrak q}\) is \(A\)-flat the lemma also shows that \(\alpha\) is injective and that \(\Coker(\alpha)_{\mathfrak q}\) is \(A\)-flat. Set \(Q = \Coker(\alpha)\). Note that the support of \(Q/\mathfrak mQ\) does not contain \(\mathfrak p\). Hence it is certainly the case that \(\dim_{\mathfrak q}(\text{Supp}(Q/\mathfrak mQ)) < n\). Combining everything we know about \(Q\) we see that the induction hypothesis applies to \(Q\). It follows that there exists an elementary étale morphism \((S', s) \to (S, s)\) such that the conclusion holds for \(Q \otimes_A A'\) over \(B \otimes_A A'\) where \(A' = \mathcal{O}_{S', s'}\). After replacing \(A\) by \(A'\) we have an exact sequence \[0 \to B^{\oplus r} \to N \to Q \to 0\] (here we use that \(\alpha\) is injective as mentioned above) of finite \(B\)-modules and we also get an element \(g \in B\), \(g \not \in \mathfrak q\) such that \(Q_g\) is finitely presented over \(B_g\) and flat over \(A\). Since localization is exact we see that \[0 \to B_g^{\oplus r} \to N_g \to Q_g \to 0\] is still exact. As \(B_g\) and \(Q_g\) are flat over \(A\) we conclude that \(N_g\) is flat over \(A\), see Algebra, Lemma 00HM, and as \(B_g\) and \(Q_g\) are finitely presented over \(B_g\) the same holds for \(N_g\), see Algebra, Lemma 0519.

Proof

We apply Proposition 05HR to find a commutative diagram \[\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] }\] of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that \(g^*\mathcal{F}/X'/S'\) has a complete dévissage at \(x\). (In particular \(S'\) and \(X'\) are affine.) By Morphisms, Lemma 02JZ we see that \(g^*\mathcal{F}\) is flat at \(x'\) over \(S\) and by Lemma 05B9 we see that it is flat at \(x'\) over \(S'\). Via Remark 05I0 we deduce that \[\Gamma(X', g^*\mathcal{F})/ \Gamma(X', \mathcal{O}_{X'})/ \Gamma(S', \mathcal{O}_{S'})\] has a complete dévissage at the prime of \(\Gamma(X', \mathcal{O}_{X'})\) corresponding to \(x'\). We may base change this complete dévissage to the local ring \(\mathcal{O}_{S', s'}\) of \(\Gamma(S', \mathcal{O}_{S'})\) at the prime corresponding to \(s'\). Thus Lemma 05I4 implies that \[\Gamma(X', \mathcal{F}') \otimes_{\Gamma(S', \mathcal{O}_{S'})} \mathcal{O}_{S', s'}\] is flat over \(\mathcal{O}_{S', s'}\) and of finite presentation over \(\Gamma(X', \mathcal{O}_{X'}) \otimes_{\Gamma(S', \mathcal{O}_{S'})} \mathcal{O}_{S', s'}\). In other words, the restriction of \(\mathcal{F}\) to \(X' \times_{S'} \Spec(\mathcal{O}_{S', s'})\) is of finite presentation and flat over \(\mathcal{O}_{S', s'}\). Since the morphism \(X' \times_{S'} \Spec(\mathcal{O}_{S', s'}) \to X \times_S \Spec(\mathcal{O}_{S', s'})\) is étale (Lemma 05H2) its image \(V \subset X \times_S \Spec(\mathcal{O}_{S', s'})\) is an open subscheme, and by étale descent the restriction of \(\mathcal{F}\) to \(V\) is of finite presentation and flat over \(\mathcal{O}_{S', s'}\). (Results used: Morphisms, Lemma 03WT, Descent, Lemma 05B0, and Morphisms, Lemma 02JZ.)

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 \(s \in S\). Then the set \[\{x \in X_s \mid \mathcal{F} \text{ flat over }S\text{ at }x\}\] is open in the fibre \(X_s\).

Proof

Suppose \(x \in U\). Choose an elementary étale neighbourhood \((S', s') \to (S, s)\) and open \(V \subset X \times_S \Spec(\mathcal{O}_{S', s'})\) as in Proposition 05I5. Note that \(X_{s'} = X_s\) as \(\kappa(s) = \kappa(s')\). If \(x' \in V \cap X_{s'}\), then the pullback of \(\mathcal{F}\) to \(X \times_S S'\) is flat over \(S'\) at \(x'\). Hence \(\mathcal{F}\) is flat at \(x'\) over \(S\), see Morphisms, Lemma 02JZ. In other words \(X_s \cap V \subset U\) is an open neighbourhood of \(x\) in \(U\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(x \in X\) with image \(s \in S\). Assume that

1. \(f\) is locally of finite type,

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

3. \(\mathcal{F}\) is flat at \(x\) over \(S\).

Then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and an open subscheme \[V \subset X \times_S \Spec(\mathcal{O}_{S', s'})\] which contains the unique point of \(X \times_S \Spec(\mathcal{O}_{S', s'})\) mapping to \(x\) such that the pullback of \(\mathcal{F}\) to \(V\) is flat over \(\mathcal{O}_{S', s'}\).

Proof

(The only difference between this and Proposition 05I5 is that we do not assume \(f\) is of finite presentation.) The question is local on \(X\) and \(S\), hence we may assume \(X\) and \(S\) are affine. Write \(X = \Spec(B)\), \(S = \Spec(A)\) and write \(B = A[x_1, \ldots, x_n]/I\). In other words we obtain a closed immersion \(i : X \to \mathbf{A}^n_S\). Denote \(t = i(x) \in \mathbf{A}^n_S\). We may apply Proposition 05I5 to \(\mathbf{A}^n_S \to S\), the sheaf \(i_*\mathcal{F}\) and the point \(t\). We obtain an elementary étale neighbourhood \((S', s') \to (S, s)\) and an open subscheme \[W \subset \mathbf{A}^n_{\mathcal{O}_{S', s'}}\] such that the pullback of \(i_*\mathcal{F}\) to \(W\) is flat over \(\mathcal{O}_{S', s'}\). This means that \(V := W \cap \big(X \times_S \Spec(\mathcal{O}_{S', s'})\big)\) is the desired open subscheme.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(s \in S\). Assume that

1. \(f\) is of finite presentation,

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

3. \(\mathcal{F}\) is flat over \(S\) at every point of the fibre \(X_s\).

Then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and an open subscheme \[V \subset X \times_S \Spec(\mathcal{O}_{S', s'})\] which contains the fibre \(X_s = X \times_S s'\) such that the pullback of \(\mathcal{F}\) to \(V\) is an \(\mathcal{O}_V\)-module of finite presentation and flat over \(\mathcal{O}_{S', s'}\).

Proof

For every point \(x \in X_s\) we can use Proposition 05I5 to find an elementary étale neighbourhood \((S_x, s_x) \to (S, s)\) and an open \(V_x \subset X \times_S \Spec(\mathcal{O}_{S_x, s_x})\) such that \(x \in X_s = X \times_S s_x\) is contained in \(V_x\) and such that the pullback of \(\mathcal{F}\) to \(V_x\) is an \(\mathcal{O}_{V_x}\)-module of finite presentation and flat over \(\mathcal{O}_{S_x, s_x}\). In particular we may view the fibre \((V_x)_{s_x}\) as an open neighbourhood of \(x\) in \(X_s\). Because \(X_s\) is quasi-compact we can find a finite number of points \(x_1, \ldots, x_n \in X_s\) such that \(X_s\) is the union of the \((V_{x_i})_{s_{x_i}}\). Choose an elementary étale neighbourhood \((S' , s') \to (S, s)\) which dominates each of the neighbourhoods \((S_{x_i}, s_{x_i})\), see More on Morphisms, Lemma 057B. Set \(V = \bigcup V_i\) where \(V_i\) is the inverse images of the open \(V_{x_i}\) via the morphism \[X \times_S \Spec(\mathcal{O}_{S', s'}) \longrightarrow X \times_S \Spec(\mathcal{O}_{S_{x_i}, s_{x_i}})\] By construction \(V\) contains \(X_s\) and by construction the pullback of \(\mathcal{F}\) to \(V\) is an \(\mathcal{O}_V\)-module of finite presentation and flat over \(\mathcal{O}_{S', s'}\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(s \in S\). Assume that

1. \(f\) is of finite type,

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

3. \(\mathcal{F}\) is flat over \(S\) at every point of the fibre \(X_s\).

Then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and an open subscheme \[V \subset X \times_S \Spec(\mathcal{O}_{S', s'})\] which contains the fibre \(X_s = X \times_S s'\) such that the pullback of \(\mathcal{F}\) to \(V\) is flat over \(\mathcal{O}_{S', s'}\).

Proof

(The only difference between this and Lemma 05KU is that we do not assume \(f\) is of finite presentation.) For every point \(x \in X_s\) we can use Lemma 05KT to find an elementary étale neighbourhood \((S_x, s_x) \to (S, s)\) and an open \(V_x \subset X \times_S \Spec(\mathcal{O}_{S_x, s_x})\) such that \(x \in X_s = X \times_S s_x\) is contained in \(V_x\) and such that the pullback of \(\mathcal{F}\) to \(V_x\) is flat over \(\mathcal{O}_{S_x, s_x}\). In particular we may view the fibre \((V_x)_{s_x}\) as an open neighbourhood of \(x\) in \(X_s\). Because \(X_s\) is quasi-compact we can find a finite number of points \(x_1, \ldots, x_n \in X_s\) such that \(X_s\) is the union of the \((V_{x_i})_{s_{x_i}}\). Choose an elementary étale neighbourhood \((S' , s') \to (S, s)\) which dominates each of the neighbourhoods \((S_{x_i}, s_{x_i})\), see More on Morphisms, Lemma 057B. Set \(V = \bigcup V_i\) where \(V_i\) is the inverse images of the open \(V_{x_i}\) via the morphism \[X \times_S \Spec(\mathcal{O}_{S', s'}) \longrightarrow X \times_S \Spec(\mathcal{O}_{S_{x_i}, s_{x_i}})\] By construction \(V\) contains \(X_s\) and by construction the pullback of \(\mathcal{F}\) to \(V\) is flat over \(\mathcal{O}_{S', s'}\).

Lemma

Let \(S\) be a scheme. Let \(X\) be locally of finite type over \(S\). Let \(x \in X\) with image \(s \in S\). If \(X\) is flat at \(x\) over \(S\), then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and an open subscheme \[V \subset X \times_S \Spec(\mathcal{O}_{S', s'})\] which contains the unique point of \(X \times_S \Spec(\mathcal{O}_{S', s'})\) mapping to \(x\) such that \(V \to \Spec(\mathcal{O}_{S', s'})\) is flat and of finite presentation.

Proof

The question is local on \(X\) and \(S\), hence we may assume \(X\) and \(S\) are affine. Write \(X = \Spec(B)\), \(S = \Spec(A)\) and write \(B = A[x_1, \ldots, x_n]/I\). In other words we obtain a closed immersion \(i : X \to \mathbf{A}^n_S\). Denote \(t = i(x) \in \mathbf{A}^n_S\). We may apply Proposition 05I5 to \(\mathbf{A}^n_S \to S\), the sheaf \(\mathcal{F} = i_*\mathcal{O}_X\) and the point \(t\). We obtain an elementary étale neighbourhood \((S', s') \to (S, s)\) and an open subscheme \[W \subset \mathbf{A}^n_{\mathcal{O}_{S', s'}}\] such that the pullback of \(i_*\mathcal{O}_X\) is flat and of finite presentation. This means that \(V := W \cap \big(X \times_S \Spec(\mathcal{O}_{S', s'})\big)\) is the desired open subscheme.

Lemma

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 of finite type. If \(x \in X\) and \(\mathcal{F}\) is flat at \(x\) over \(S\), then \(\mathcal{F}_x\) is an \(\mathcal{O}_{X, x}\)-module of finite presentation.

Proof

Let \(s = f(x)\). By Proposition 05I5 there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) such that the pullback of \(\mathcal{F}\) to \(X \times_S \Spec(\mathcal{O}_{S', s'})\) is of finite presentation in a neighbourhood of the point \(x' \in X_{s'} = X_s\) corresponding to \(x\). The ring map \[\mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{X \times_S \Spec(\mathcal{O}_{S', s'}), x'} = \mathcal{O}_{X \times_S S', x'}\] is flat and local as a localization of an étale ring map. Hence \(\mathcal{F}_x\) is of finite presentation over \(\mathcal{O}_{X, x}\) by descent, see Algebra, Lemma 03C4 (and also that a flat local ring map is faithfully flat, see Algebra, Lemma 00HR).

Lemma

Let \(f : X \to S\) be a morphism which is locally of finite type. Let \(x \in X\) with image \(s \in S\). If \(f\) is flat at \(x\) over \(S\), then \(\mathcal{O}_{X, x}\) is essentially of finite presentation over \(\mathcal{O}_{S, s}\).

Proof

We may assume \(X\) and \(S\) affine. Write \(X = \Spec(B)\), \(S = \Spec(A)\) and write \(B = A[x_1, \ldots, x_n]/I\). In other words we obtain a closed immersion \(i : X \to \mathbf{A}^n_S\). Denote \(t = i(x) \in \mathbf{A}^n_S\). We may apply Lemma 05I7 to \(\mathbf{A}^n_S \to S\), the sheaf \(\mathcal{F} = i_*\mathcal{O}_X\) and the point \(t\). We conclude that \(\mathcal{O}_{X, x}\) is of finite presentation over \(\mathcal{O}_{\mathbf{A}^n_S, t}\) which implies what we want.

Extending properties from an open

In this section we collect a number of results of the form: If \(f : X \to S\) is a flat morphism of schemes and \(f\) satisfies some property over a dense open of \(S\), then \(f\) satisfies the same property over all of \(S\).

Lemma

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

1. \(f\) is locally of finite presentation,

2. \(\mathcal{F}\) is of finite type and flat over \(S\),

3. \(U \subset S\) is retrocompact and scheme theoretically dense,

4. \(\mathcal{F}|_{f^{-1}U}\) is of finite presentation.

Then \(\mathcal{F}\) is of finite presentation.

Proof

The problem is local on \(X\) and \(S\), hence we may assume \(X\) and \(S\) affine. Write \(S = \Spec(A)\) and \(X = \Spec(B)\). Let \(N\) be a finite \(B\)-module such that \(\mathcal{F}\) is the quasi-coherent sheaf associated to \(N\). We have \(U = D(f_1) \cup \ldots \cup D(f_n)\) for some \(f_i \in A\), see Algebra, Lemma 00F6. As \(U\) is schematically dense the map \(A \to A_{f_1} \times \ldots \times A_{f_n}\) is injective. Pick a prime \(\mathfrak q \subset B\) lying over \(\mathfrak p \subset A\) corresponding to \(x \in X\) mapping to \(s \in S\). By Lemma 05I7 the module \(N_\mathfrak q\) is of finite presentation over \(B_\mathfrak q\). Choose a surjection \(\varphi : B^{\oplus m} \to N\) of \(B\)-modules. Choose \(k_1, \ldots, k_t \in \Ker(\varphi)\) and set \(N' = B^{\oplus m}/\sum Bk_j\). There is a canonical surjection \(N' \to N\) and \(N\) is the filtered colimit of the \(B\)-modules \(N'\) constructed in this manner. Thus we see that we can choose \(k_1, \ldots, k_t\) such that (a) \(N'_{f_i} \cong N_{f_i}\), \(i = 1, \ldots, n\) and (b) \(N'_\mathfrak q \cong N_\mathfrak q\). This in particular implies that \(N'_\mathfrak q\) is flat over \(A\). By openness of flatness, see Algebra, Theorem 00RC we conclude that there exists a \(g \in B\), \(g \not \in \mathfrak q\) such that \(N'_g\) is flat over \(A\). Consider the commutative diagram \[\xymatrix{ N'_g \ar[r] \ar[d] & N_g \ar[d] \\ \prod N'_{gf_i} \ar[r] & \prod N_{gf_i} }\] The bottom arrow is an isomorphism by choice of \(k_1, \ldots, k_t\). The left vertical arrow is an injective map as \(A \to \prod A_{f_i}\) is injective and \(N'_g\) is flat over \(A\). Hence the top horizontal arrow is injective, hence an isomorphism. This proves that \(N_g\) is of finite presentation over \(B_g\). We conclude by applying Algebra, Lemma 00EO.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(U \subset S\) be open. Assume

1. \(f\) is locally of finite type and flat,

2. \(U \subset S\) is retrocompact and scheme theoretically dense,

3. \(f|_{f^{-1}U} : f^{-1}U \to U\) is locally of finite presentation.

Then \(f\) is of locally of finite presentation.

Proof

The question is local on \(X\) and \(S\), hence we may assume \(X\) and \(S\) affine. Choose a closed immersion \(i : X \to \mathbf{A}^n_S\) and apply Lemma 081N to \(i_*\mathcal{O}_X\). Some details omitted.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat and locally of finite type. Let \(U \subset S\) be a dense open such that \(X_U \to U\) has relative dimension \(\leq e\), see Morphisms, Definition 02NJ. If also either

1. \(f\) is locally of finite presentation, or

2. \(U \subset S\) is retrocompact,

then \(f\) has relative dimension \(\leq e\).

Proof

Proof in case (1). Let \(W \subset X\) be the open subscheme constructed and studied in More on Morphisms, Lemmas 045U and 054T. Note that every generic point of every fibre is contained in \(W\), hence it suffices to prove the result for \(W\). Since \(W = \bigcup_{d \geq 0} U_d\), it suffices to prove that \(U_d = \emptyset\) for \(d > e\). Since \(f\) is flat and locally of finite presentation it is open hence \(f(U_d)\) is open (Morphisms, Lemma 01UA). Thus if \(U_d\) is not empty, then \(f(U_d) \cap U \not = \emptyset\) as desired.

Proof in case (2). We may replace \(S\) by its reduction. Then \(U\) is scheme theoretically dense. Hence \(f\) is locally of finite presentation by Lemma 081P. In this way we reduce to case (1).

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat and proper. Let \(U \subset S\) be a dense open such that \(X_U \to U\) is finite. If also either \(f\) is locally of finite presentation or \(U \subset S\) is retrocompact, then \(f\) is finite.

Proof

By Lemma 081L the fibres of \(f\) have dimension zero. Hence \(f\) is quasi-finite (Morphisms, Lemma 0397) whence has finite fibres (Morphisms, Lemma 02NH). Hence \(f\) is finite by More on Morphisms, Lemma 02LS.

Lemma

Let \(f : X \to S\) be a morphism of schemes and \(U \subset S\) an open. If

1. \(f\) is separated, locally of finite type, and flat,

2. \(f^{-1}(U) \to U\) is an isomorphism, and

3. \(U \subset S\) is retrocompact and scheme theoretically dense,

then \(f\) is an open immersion.

Proof

By Lemma 081P the morphism \(f\) is locally of finite presentation. The image \(f(X) \subset S\) is open (Morphisms, Lemma 01UA) hence we may replace \(S\) by \(f(X)\). Thus we have to prove that \(f\) is an isomorphism. We may assume \(S\) is affine. We can reduce to the case that \(X\) is quasi-compact because it suffices to show that any quasi-compact open \(X' \subset X\) whose image is \(S\) maps isomorphically to \(S\). Thus we may assume \(f\) is quasi-compact. All the fibers of \(f\) have dimension \(0\), see Lemma 081L. Hence \(f\) is quasi-finite, see Morphisms, Lemma 0397. Let \(s \in S\). Choose an elementary étale neighbourhood \(g : (T, t) \to (S, s)\) such that \(X \times_S T = V \amalg W\) with \(V \to T\) finite and \(W_t = \emptyset\), see More on Morphisms, Lemma 02LP. Denote \(\pi : V \amalg W \to T\) the given morphism. Since \(\pi\) is flat and locally of finite presentation, we see that \(\pi(V)\) is open in \(T\) (Morphisms, Lemma 01UA). After shrinking \(T\) we may assume that \(T = \pi(V)\). Since \(f\) is an isomorphism over \(U\) we see that \(\pi\) is an isomorphism over \(g^{-1}U\). Since \(\pi(V) = T\) this implies that \(\pi^{-1}g^{-1}U\) is contained in \(V\). By Morphisms, Lemma 081H we see that \(\pi^{-1}g^{-1}U \subset V \amalg W\) is scheme theoretically dense. Hence we deduce that \(W = \emptyset\). Thus \(X \times_S T = V\) is finite over \(T\). This implies that \(f\) is finite (after replacing \(S\) by an open neighbourhood of \(s\)), for example by Descent, Lemma 02LA. Then \(f\) is finite locally free (Morphisms, Lemma 02KB) and after shrinking \(S\) to a smaller open neighbourhood of \(s\) we see that \(f\) is finite locally free of some degree \(d\) (Morphisms, Lemma 04MH). But \(d = 1\) as is clear from the fact that the degree is \(1\) over the dense open \(U\). Hence \(f\) is an isomorphism.

Flat finitely presented modules

In some cases given a ring map \(R \to S\) of finite presentation and a finitely presented \(S\)-module \(N\) the flatness of \(N\) over \(R\) implies that \(N\) is projective as an \(R\)-module, at least after replacing \(S\) by an étale extension. In this section we collect a some results of this nature.

Lemma

Let \(R\) be a ring. Let \(R \to S\) be a finitely presented flat ring map with geometrically integral fibres. Let \(\mathfrak q \subset S\) be a prime ideal lying over the prime \(\mathfrak r \subset R\). Set \(\mathfrak p = \mathfrak r S\). Let \(N\) be a finitely presented \(S\)-module. There exists \(r \geq 0\) and an \(S\)-module map \[\alpha : S^{\oplus r} \longrightarrow N\] such that \(\alpha : \kappa(\mathfrak p)^{\oplus r} \to N \otimes_S \kappa(\mathfrak p)\) is an isomorphism. For any such \(\alpha\) the following are equivalent:

1. \(N_{\mathfrak q}\) is \(R\)-flat,

2. there exists an \(f \in R\), \(f \not \in \mathfrak r\) such that \(\alpha_f : S_f^{\oplus r} \to N_f\) is \(R_f\)-universally injective and a \(g \in S\), \(g \not \in \mathfrak q\) such that \(\Coker(\alpha)_g\) is \(R\)-flat,

3. \(\alpha_{\mathfrak r}\) is \(R_{\mathfrak r}\)-universally injective and \(\Coker(\alpha)_{\mathfrak q}\) is \(R\)-flat

4. \(\alpha_{\mathfrak r}\) is injective and \(\Coker(\alpha)_{\mathfrak q}\) is \(R\)-flat,

5. \(\alpha_{\mathfrak p}\) is an isomorphism and \(\Coker(\alpha)_{\mathfrak q}\) is \(R\)-flat, and

6. \(\alpha_{\mathfrak q}\) is injective and \(\Coker(\alpha)_{\mathfrak q}\) is \(R\)-flat.

Proof

To obtain \(\alpha\) set \(r = \dim_{\kappa(\mathfrak p)} N \otimes_S \kappa(\mathfrak p)\) and pick \(x_1, \ldots, x_r \in N\) which form a basis of \(N \otimes_S \kappa(\mathfrak p)\). Define \(\alpha(s_1, \ldots, s_r) = \sum s_i x_i\). This proves the existence.

Fix a choice of \(\alpha\). We may apply Lemma 05I3 to the map \(\alpha_{\mathfrak r} : S_{\mathfrak r}^{\oplus r} \to N_{\mathfrak r}\). Hence we see that (1), (3), (4), (5), and (6) are all equivalent. Since it is also clear that (2) implies (3) we see that all we have to do is show that (1) implies (2).

Assume (1). By openness of flatness, see Algebra, Theorem 00RC, the set \[U_1 = \{\mathfrak q' \subset S \mid N_{\mathfrak q'}\text{ is flat over }R\}\] is open in \(\Spec(S)\). It contains \(\mathfrak q\) by assumption and hence \(\mathfrak p\). Because \(S^{\oplus r}\) and \(N\) are finitely presented \(S\)-modules the set \[U_2 = \{\mathfrak q' \subset S \mid \alpha_{\mathfrak q'}\text{ is an isomorphism}\}\] is open in \(\Spec(S)\), see Algebra, Lemma 05GF. It contains \(\mathfrak p\) by (5). As \(R \to S\) is finitely presented and flat the map \(\Phi : \Spec(S) \to \Spec(R)\) is open, see Algebra, Proposition 00I1. For any prime \(\mathfrak r' \in \Phi(U_1 \cap U_2)\) we see that there exists a prime \(\mathfrak q'\) lying over \(\mathfrak r'\) such that \(N_{\mathfrak q'}\) is flat and such that \(\alpha_{\mathfrak q'}\) is an isomorphism, which implies that \(\alpha \otimes \kappa(\mathfrak p')\) is an isomorphism where \(\mathfrak p' = \mathfrak r' S\). Thus \(\alpha_{\mathfrak r'}\) is \(R_{\mathfrak r'}\)-universally injective by the implication (1) \(\Rightarrow\) (3). Hence if we pick \(f \in R\), \(f \not \in \mathfrak r\) such that \(D(f) \subset \Phi(U_1 \cap U_2)\) then we conclude that \(\alpha_f\) is \(R_f\)-universally injective, see Algebra, Lemma 05CL. The same reasoning also shows that for any \(\mathfrak q' \in U_1 \cap \Phi^{-1}(\Phi(U_1 \cap U_2))\) the module \(\Coker(\alpha)_{\mathfrak q'}\) is \(R\)-flat. Note that \(\mathfrak q \in U_1 \cap \Phi^{-1}(\Phi(U_1 \cap U_2))\). Hence we can find a \(g \in S\), \(g \not \in \mathfrak q\) such that \(D(g) \subset U_1 \cap \Phi^{-1}(\Phi(U_1 \cap U_2))\) and we win.

Lemma

Let \(R \to S\) be a ring map of finite presentation. Let \(N\) be a finitely presented \(S\)-module flat over \(R\). Let \(\mathfrak r \subset R\) be a prime ideal. Assume there exists a complete dévissage of \(N/S/R\) over \(\mathfrak r\). Then there exists an \(f \in R\), \(f \not \in \mathfrak r\) such that \[N_f \cong B_1^{\oplus r_1} \oplus \ldots \oplus B_n^{\oplus r_n}\] as \(R\)-modules where each \(B_i\) is a smooth \(R_f\)-algebra with geometrically irreducible fibres. Moreover, \(N_f\) is projective as an \(R_f\)-module.

Proof

Let \((A_i, B_i, M_i, \alpha_i)_{i = 1, \ldots, n}\) be the given complete dévissage. We prove the lemma by induction on \(n\). Note that the assertions of the lemma are entirely about the structure of \(N\) as an \(R\)-module. Hence we may replace \(N\) by \(M_1\), and we may think of \(M_1\) as a \(B_1\)-module. See Remark 05HY in order to see why \(M_1\) is of finite presentation as a \(B_1\)-module. By Lemma 05IA we may, after replacing \(R\) by \(R_f\) for some \(f \in R\), \(f \not \in \mathfrak r\), assume the map \(\alpha_1 : B_1^{\oplus r_1} \to M_1\) is \(R\)-universally injective. Since \(M_1\) and \(B_1^{\oplus r_1}\) are \(R\)-flat and finitely presented as \(B_1\)-modules we see that \(\Coker(\alpha_1)\) is \(R\)-flat (Algebra, Lemma 058P) and finitely presented as a \(B_1\)-module. Note that \((A_i, B_i, M_i, \alpha_i)_{i = 2, \ldots, n}\) is a complete dévissage of \(\Coker(\alpha_1)\). Hence the induction hypothesis implies that, after replacing \(R\) by \(R_f\) for some \(f \in R\), \(f \not \in \mathfrak r\), we may assume that \(\Coker(\alpha_1)\) has a decomposition as in the lemma and is projective. In particular \(M_1 = B_1^{\oplus r_1} \oplus \Coker(\alpha_1)\). This proves the statement regarding the decomposition. The statement on projectivity follows as \(B_1\) is projective as an \(R\)-module by Lemma 05FT.

Remark

There is a variant of Lemma 05IB where we weaken the flatness condition by assuming only that \(N\) is flat at some given prime \(\mathfrak q\) lying over \(\mathfrak r\) but where we strengthen the dévissage condition by assuming the existence of a complete dévissage at \(\mathfrak q\). Compare with Lemma 05I4.

The following is the main result of this section.

Proposition

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(x \in X\) with image \(s \in S\). Assume that

1. \(f\) is locally of finite presentation,

2. \(\mathcal{F}\) is of finite presentation, and

3. \(\mathcal{F}\) is flat at \(x\) over \(S\).

Then there exists a commutative diagram of pointed schemes \[\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] }\] whose horizontal arrows are elementary étale neighbourhoods such that \(X'\), \(S'\) are affine and such that \(\Gamma(X', g^*\mathcal{F})\) is a projective \(\Gamma(S', \mathcal{O}_{S'})\)-module.

Proof

By openness of flatness, see More on Morphisms, Theorem 0399 we may replace \(X\) by an open neighbourhood of \(x\) and assume that \(\mathcal{F}\) is flat over \(S\). Next, we apply Proposition 05HR to find a diagram as in the statement of the proposition such that \(g^*\mathcal{F}/X'/S'\) has a complete dévissage over \(s'\). (In particular \(S'\) and \(X'\) are affine.) By Morphisms, Lemma 02JZ we see that \(g^*\mathcal{F}\) is flat over \(S\) and by Lemma 05B9 we see that it is flat over \(S'\). Via Remark 05I0 we deduce that \[\Gamma(X', g^*\mathcal{F})/ \Gamma(X', \mathcal{O}_{X'})/ \Gamma(S', \mathcal{O}_{S'})\] has a complete dévissage over the prime of \(\Gamma(S', \mathcal{O}_{S'})\) corresponding to \(s'\). Thus Lemma 05IB implies that the result of the proposition holds after replacing \(S'\) by a standard open neighbourhood of \(s'\).

In the rest of this section we prove a number of variants on this result. The first is a “global” version.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(s \in S\). Assume that

1. \(f\) is of finite presentation,

2. \(\mathcal{F}\) is of finite presentation, and

3. \(\mathcal{F}\) is flat over \(S\) at every point of the fibre \(X_s\).

Then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and a commutative diagram of schemes \[\xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & S' \ar[l] }\] such that \(g\) is étale, \(X_s \subset g(X')\), the schemes \(X'\), \(S'\) are affine, and such that \(\Gamma(X', g^*\mathcal{F})\) is a projective \(\Gamma(S', \mathcal{O}_{S'})\)-module.

Proof

For every point \(x \in X_s\) we can use Proposition 05ID to find a commutative diagram \[\xymatrix{ (X, x) \ar[d] & (Y_x, y_x) \ar[l]^{g_x} \ar[d] \\ (S, s) & (S_x, s_x) \ar[l] }\] whose horizontal arrows are elementary étale neighbourhoods such that \(Y_x\), \(S_x\) are affine and such that \(\Gamma(Y_x, g_x^*\mathcal{F})\) is a projective \(\Gamma(S_x, \mathcal{O}_{S_x})\)-module. In particular \(g_x(Y_x) \cap X_s\) is an open neighbourhood of \(x\) in \(X_s\). Because \(X_s\) is quasi-compact we can find a finite number of points \(x_1, \ldots, x_n \in X_s\) such that \(X_s\) is the union of the \(g_{x_i}(Y_{x_i}) \cap X_s\). Choose an elementary étale neighbourhood \((S' , s') \to (S, s)\) which dominates each of the neighbourhoods \((S_{x_i}, s_{x_i})\), see More on Morphisms, Lemma 057B. We may also assume that \(S'\) is affine. Set \(X' = \coprod Y_{x_i} \times_{S_{x_i}} S'\) and endow it with the obvious morphism \(g : X' \to X\). By construction \(g(X')\) contains \(X_s\) and \[\Gamma(X', g^*\mathcal{F}) = \bigoplus \Gamma(Y_{x_i}, g_{x_i}^*\mathcal{F}) \otimes_{\Gamma(S_{x_i}, \mathcal{O}_{S_{x_i}})} \Gamma(S', \mathcal{O}_{S'}).\] This is a projective \(\Gamma(S', \mathcal{O}_{S'})\)-module, see Algebra, Lemma 05A3.

The following two lemmas are reformulations of the results above in case \(\mathcal{F} = \mathcal{O}_X\).

Lemma

Let \(f : X \to S\) be locally of finite presentation. Let \(x \in X\) with image \(s \in S\). If \(f\) is flat at \(x\) over \(S\), then there exists a commutative diagram of pointed schemes \[\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] }\] whose horizontal arrows are elementary étale neighbourhoods such that \(X'\), \(S'\) are affine and such that \(\Gamma(X', \mathcal{O}_{X'})\) is a projective \(\Gamma(S', \mathcal{O}_{S'})\)-module.

Proof

This is a special case of Proposition 05ID.

Lemma

Let \(f : X \to S\) be of finite presentation. Let \(s \in S\). If \(X\) is flat over \(S\) at all points of \(X_s\), then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and a commutative diagram of schemes \[\xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & S' \ar[l] }\] with \(g\) étale, \(X_s \subset g(X')\), such that \(X'\), \(S'\) are affine, and such that \(\Gamma(X', \mathcal{O}_{X'})\) is a projective \(\Gamma(S', \mathcal{O}_{S'})\)-module.

Proof

This is a special case of Lemma 05KW.

The following lemmas explain consequences of Proposition 05ID in case we only assume the morphism and the sheaf are of finite type (and not necessarily of finite presentation).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(x \in X\) with image \(s \in S\). Assume that

1. \(f\) is locally of finite presentation,

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

3. \(\mathcal{F}\) is flat at \(x\) over \(S\).

Then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and a commutative diagram of pointed schemes \[\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (\Spec(\mathcal{O}_{S', s'}), s') \ar[l] }\] such that \(X' \to X \times_S \Spec(\mathcal{O}_{S', s'})\) is étale, \(\kappa(x) = \kappa(x')\), the scheme \(X'\) is affine of finite presentation over \(\mathcal{O}_{S', s'}\), the sheaf \(g^*\mathcal{F}\) is of finite presentation over \(\mathcal{O}_{X'}\), and such that \(\Gamma(X', g^*\mathcal{F})\) is a free \(\mathcal{O}_{S', s'}\)-module.

Proof

To prove the lemma we may replace \((S, s)\) by any elementary étale neighbourhood, and we may also replace \(S\) by \(\Spec(\mathcal{O}_{S, s})\). Hence by Proposition 05I5 we may assume that \(\mathcal{F}\) is finitely presented and flat over \(S\) in a neighbourhood of \(x\). In this case the result follows from Proposition 05ID because Algebra, Theorem 0593 assures us that projective \(=\) free over a local ring.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(x \in X\) with image \(s \in S\). Assume that

1. \(f\) is locally of finite type,

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

3. \(\mathcal{F}\) is flat at \(x\) over \(S\).

Then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and a commutative diagram of pointed schemes \[\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (\Spec(\mathcal{O}_{S', s'}), s') \ar[l] }\] such that \(X' \to X \times_S \Spec(\mathcal{O}_{S', s'})\) is étale, \(\kappa(x) = \kappa(x')\), the scheme \(X'\) is affine, and such that \(\Gamma(X', g^*\mathcal{F})\) is a free \(\mathcal{O}_{S', s'}\)-module.

Proof

(The only difference with Lemma 05KY is that we do not assume \(f\) is of finite presentation.) The problem is local on \(X\) and \(S\). Hence we may assume \(X\) and \(S\) are affine, say \(X = \Spec(B)\) and \(S = \Spec(A)\). Since \(B\) is a finite type \(A\)-algebra we can find a surjection \(A[x_1, \ldots, x_n] \to B\). In other words, we can choose a closed immersion \(i : X \to \mathbf{A}^n_S\). Set \(t = i(x)\) and \(\mathcal{G} = i_*\mathcal{F}\). Note that \(\mathcal{G}_t \cong \mathcal{F}_x\) are \(\mathcal{O}_{S, s}\)-modules. Hence \(\mathcal{G}\) is flat over \(S\) at \(t\). We apply Lemma 05KY to the morphism \(\mathbf{A}^n_S \to S\), the point \(t\), and the sheaf \(\mathcal{G}\). Thus we can find an elementary étale neighbourhood \((S', s') \to (S, s)\) and a commutative diagram of pointed schemes \[\xymatrix{ (\mathbf{A}^n_S, t) \ar[d] & (Y, y) \ar[l]^h \ar[d] \\ (S, s) & (\Spec(\mathcal{O}_{S', s'}), s') \ar[l] }\] such that \(Y \to \mathbf{A}^n_{\mathcal{O}_{S', s'}}\) is étale, \(\kappa(t) = \kappa(y)\), the scheme \(Y\) is affine, and such that \(\Gamma(Y, h^*\mathcal{G})\) is a projective \(\mathcal{O}_{S', s'}\)-module. Then a solution to the original problem is given by the closed subscheme \(X' = Y \times_{\mathbf{A}^n_S} X\) of \(Y\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(s \in S\). Assume that

1. \(f\) is of finite presentation,

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

3. \(\mathcal{F}\) is flat over \(S\) at all points of \(X_s\).

Then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and a commutative diagram of schemes \[\xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & \Spec(\mathcal{O}_{S', s'}) \ar[l] }\] such that \(X' \to X \times_S \Spec(\mathcal{O}_{S', s'})\) is étale, \(X_s = g((X')_{s'})\), the scheme \(X'\) is affine of finite presentation over \(\mathcal{O}_{S', s'}\), the sheaf \(g^*\mathcal{F}\) is of finite presentation over \(\mathcal{O}_{X'}\), and such that \(\Gamma(X', g^*\mathcal{F})\) is a free \(\mathcal{O}_{S', s'}\)-module.

Proof

For every point \(x \in X_s\) we can use Lemma 05KY to find an elementary étale neighbourhood \((S_x , s_x) \to (S, s)\) and a commutative diagram \[\xymatrix{ (X, x) \ar[d] & (Y_x, y_x) \ar[l]^{g_x} \ar[d] \\ (S, s) & (\Spec(\mathcal{O}_{S_x, s_x}), s_x) \ar[l] }\] such that \(Y_x \to X \times_S \Spec(\mathcal{O}_{S_x, s_x})\) is étale, \(\kappa(x) = \kappa(y_x)\), the scheme \(Y_x\) is affine of finite presentation over \(\mathcal{O}_{S_x, s_x}\), the sheaf \(g_x^*\mathcal{F}\) is of finite presentation over \(\mathcal{O}_{Y_x}\), and such that \(\Gamma(Y_x, g_x^*\mathcal{F})\) is a free \(\mathcal{O}_{S_x, s_x}\)-module. In particular \(g_x((Y_x)_{s_x})\) is an open neighbourhood of \(x\) in \(X_s\). Because \(X_s\) is quasi-compact we can find a finite number of points \(x_1, \ldots, x_n \in X_s\) such that \(X_s\) is the union of the \(g_{x_i}((Y_{x_i})_{s_{x_i}})\). Choose an elementary étale neighbourhood \((S' , s') \to (S, s)\) which dominates each of the neighbourhoods \((S_{x_i}, s_{x_i})\), see More on Morphisms, Lemma 057B. Set \[X' = \coprod Y_{x_i} \times_{\Spec(\mathcal{O}_{S_{x_i}, s_{x_i}})} \Spec(\mathcal{O}_{S', s'})\] and endow it with the obvious morphism \(g : X' \to X\). By construction \(X_s = g(X'_{s'})\) and \[\Gamma(X', g^*\mathcal{F}) = \bigoplus \Gamma(Y_{x_i}, g_{x_i}^*\mathcal{F}) \otimes_{\mathcal{O}_{S_{x_i}, s_{x_i}}} \mathcal{O}_{S', s'}.\] This is a free \(\mathcal{O}_{S', s'}\)-module as a direct sum of base changes of free modules. Some minor details omitted.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(s \in S\). Assume that

1. \(f\) is of finite type,

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

3. \(\mathcal{F}\) is flat over \(S\) at all points of \(X_s\).

Then there exists an elementary étale neighbourhood \((S', s') \to (S, s)\) and a commutative diagram of schemes \[\xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & \Spec(\mathcal{O}_{S', s'}) \ar[l] }\] such that \(X' \to X \times_S \Spec(\mathcal{O}_{S', s'})\) is étale, \(X_s = g((X')_{s'})\), the scheme \(X'\) is affine, and such that \(\Gamma(X', g^*\mathcal{F})\) is a free \(\mathcal{O}_{S', s'}\)-module.

Proof

(The only difference with Lemma 05L0 is that we do not assume \(f\) is of finite presentation.) For every point \(x \in X_s\) we can use Lemma 05KZ to find an elementary étale neighbourhood \((S_x , s_x) \to (S, s)\) and a commutative diagram \[\xymatrix{ (X, x) \ar[d] & (Y_x, y_x) \ar[l]^{g_x} \ar[d] \\ (S, s) & (\Spec(\mathcal{O}_{S_x, s_x}), s_x) \ar[l] }\] such that \(Y_x \to X \times_S \Spec(\mathcal{O}_{S_x, s_x})\) is étale, \(\kappa(x) = \kappa(y_x)\), the scheme \(Y_x\) is affine, and such that \(\Gamma(Y_x, g_x^*\mathcal{F})\) is a free \(\mathcal{O}_{S_x, s_x}\)-module. In particular \(g_x((Y_x)_{s_x})\) is an open neighbourhood of \(x\) in \(X_s\). Because \(X_s\) is quasi-compact we can find a finite number of points \(x_1, \ldots, x_n \in X_s\) such that \(X_s\) is the union of the \(g_{x_i}((Y_{x_i})_{s_{x_i}})\). Choose an elementary étale neighbourhood \((S' , s') \to (S, s)\) which dominates each of the neighbourhoods \((S_{x_i}, s_{x_i})\), see More on Morphisms, Lemma 057B. Set \[X' = \coprod Y_{x_i} \times_{\Spec(\mathcal{O}_{S_{x_i}, s_{x_i}})} \Spec(\mathcal{O}_{S', s'})\] and endow it with the obvious morphism \(g : X' \to X\). By construction \(X_s = g(X'_{s'})\) and \[\Gamma(X', g^*\mathcal{F}) = \bigoplus \Gamma(Y_{x_i}, g_{x_i}^*\mathcal{F}) \otimes_{\mathcal{O}_{S_{x_i}, s_{x_i}}} \mathcal{O}_{S', s'}.\] This is a free \(\mathcal{O}_{S', s'}\)-module as a direct sum of base changes of free modules.

Flat finite type modules, Part II

We will need the following lemma.

Lemma

Let \(R \to S\) be a ring map of finite presentation. Let \(N\) be a finitely presented \(S\)-module. Let \(\mathfrak q \subset S\) be a prime ideal lying over \(\mathfrak p \subset R\). Set \(\overline{S} = S \otimes_R \kappa(\mathfrak p)\), \(\overline{\mathfrak q} = \mathfrak q \overline{S}\), and \(\overline{N} = N \otimes_R \kappa(\mathfrak p)\). Then we can find a \(g \in S\) with \(g \not \in \mathfrak q\) such that \(\overline{g} \in \mathfrak r\) for all \(\mathfrak r \in \text{Ass}_{\overline{S}}(\overline{N})\) such that \(\mathfrak r \not \subset \overline{\mathfrak q}\).

Proof

Namely, if \(\text{Ass}_{\overline{S}}(\overline{N}) = \{\mathfrak r_1, \ldots, \mathfrak r_n\}\) (finiteness by Algebra, Lemma 00LC), then after renumbering we may assume that \[\mathfrak r_1 \subset \overline{\mathfrak q}, \ldots, \mathfrak r_r \subset \overline{\mathfrak q}, \quad \mathfrak r_{r + 1} \not \subset \overline{\mathfrak q}, \ldots, \mathfrak r_n \not \subset \overline{\mathfrak q}\] Since \(\overline{\mathfrak q}\) is a prime ideal we see that the product \(\mathfrak r_{r + 1} \ldots \mathfrak r_n\) is not contained in \(\overline{\mathfrak q}\) and hence we can pick an element \(a\) of \(\overline{S}\) contained in \(\mathfrak r_{r + 1}, \ldots, \mathfrak r_n\) but not in \(\overline{\mathfrak q}\). If there exists \(g \in S\) mapping to \(a\), then \(g\) works. In general we can find a nonzero element \(\lambda \in \kappa(\mathfrak p)\) such that \(\lambda a\) is the image of a \(g \in S\).

The following lemma has a slightly stronger variant Lemma 05II below.

Lemma

Let \(R \to S\) be a ring map of finite presentation. Let \(N\) be a finitely presented \(S\)-module which is flat as an \(R\)-module. Let \(M\) be an \(R\)-module. Let \(\mathfrak q\) be a prime of \(S\) lying over \(\mathfrak p \subset R\). Then \[\mathfrak q \in \text{WeakAss}_S(M \otimes_R N) \Leftrightarrow \Big( \mathfrak p \in \text{WeakAss}_R(M) \text{ and } \overline{\mathfrak q} \in \text{Ass}_{\overline{S}}(\overline{N}) \Big)\] Here \(\overline{S} = S \otimes_R \kappa(\mathfrak p)\), \(\overline{\mathfrak q} = \mathfrak q \overline{S}\), and \(\overline{N} = N \otimes_R \kappa(\mathfrak p)\).

Proof

Pick \(g \in S\) as in Lemma 0CU6. Apply Proposition 05ID to the morphism of schemes \(\Spec(S_g) \to \Spec(R)\), the quasi-coherent module associated to \(N_g\), and the points corresponding to the primes \(\mathfrak qS_g\) and \(\mathfrak p\). Translating into algebra we obtain a commutative diagram of rings \[\xymatrix{ S \ar[r] & S_g \ar[r] & S' \\ & R \ar[lu] \ar[u] \ar[r] & R' \ar[u] } \quad\quad \xymatrix{ \mathfrak q \ar@{-}[r] \ar@{-}[rd] & \mathfrak qS_g \ar@{-}[d] \ar@{-}[r] & \mathfrak q' \ar@{-}[d] \\ & \mathfrak p \ar@{-}[r] & \mathfrak p' }\] endowed with primes as shown, the horizontal arrows are étale, and \(N \otimes_S S'\) is projective as an \(R'\)-module. Set \(N' = N \otimes_S S'\), \(M' = M \otimes_R R'\), \(\overline{S}' = S' \otimes_{R'} \kappa(\mathfrak q')\), \(\overline{\mathfrak q}' = \mathfrak q' \overline{S}'\), and \[\overline{N}' = N' \otimes_{R'} \kappa(\mathfrak p') = \overline{N} \otimes_{\overline{S}} \overline{S}'\] By Lemma 05FP we have \[\begin{align*} \text{WeakAss}_{S'}(M' \otimes_{R'} N') & = (\Spec(S') \to \Spec(S))^{-1}\text{WeakAss}_S(M \otimes_R N) \\ \text{WeakAss}_{R'}(M') & = (\Spec(R') \to \Spec(R))^{-1}\text{WeakAss}_R(M) \\ \text{Ass}_{\overline{S}'}(\overline{N}') & = (\Spec(\overline{S}') \to \Spec(\overline{S}))^{-1} \text{Ass}_{\overline{S}}(\overline{N}) \end{align*}\] Use Algebra, Lemma 058A for \(\overline{N}\) and \(\overline{N}'\). In particular we have \[\begin{align*} \mathfrak q \in \text{WeakAss}_S(M \otimes_R N) & \Leftrightarrow \mathfrak q' \in \text{WeakAss}_{S'}(M' \otimes_{R'} N') \\ \mathfrak p \in \text{WeakAss}_R(M) & \Leftrightarrow \mathfrak p' \in \text{WeakAss}_{R'}(M') \\ \overline{\mathfrak q} \in \text{Ass}_{\overline{S}}(\overline{N}) & \Leftrightarrow \overline{\mathfrak q}' \in \text{WeakAss}_{\overline{S}'}(\overline{N}') \end{align*}\] Our careful choice of \(g\) and the formula for \(\text{Ass}_{\overline{S}'}(\overline{N}')\) above shows that [0CU7]\[\begin{equation} \text{if }\mathfrak r' \in \text{Ass}_{\overline{S}'}(\overline{N}') \text{ lies over }\mathfrak r \subset \overline{S}\text{ then } \mathfrak r \subset \overline{\mathfrak q} \end{equation}\] This will be a key observation later in the proof. We will use the characterization of weakly associated primes given in Algebra, Lemma 0566 without further mention.

Suppose that \(\overline{\mathfrak q} \not \in \text{Ass}_{\overline{S}}(\overline{N})\). Then \(\overline{\mathfrak q}' \not \in \text{Ass}_{\overline{S}'}(\overline{N}')\). By Algebra, Lemmas 00LD, 00LC, and 00DS there exists an element \(\overline{a}' \in \overline{\mathfrak q}'\) which is not a zerodivisor on \(\overline{N}'\). After replacing \(\overline{a}'\) by \(\lambda \overline{a}'\) for some nonzero \(\lambda \in \kappa(\mathfrak p)\) we can find \(a' \in \mathfrak q'\) mapping to \(\overline{a}'\). By Lemma 05FR the map \(a' : N'_{\mathfrak p'} \to N'_{\mathfrak p'}\) is \(R'_{\mathfrak p'}\)-universally injective. In particular we see that \(a' : M' \otimes_{R'} N' \to M' \otimes_{R'} N'\) is injective after localizing at \(\mathfrak p'\) and hence after localizing at \(\mathfrak q'\). Clearly this implies that \(\mathfrak q' \not \in \text{WeakAss}_{S'}(M' \otimes_{R'} N')\). We conclude that \(\mathfrak q \in \text{WeakAss}_S(M \otimes_R N)\) implies \(\overline{\mathfrak q} \in \text{Ass}_{\overline{S}}(\overline{N})\).

Assume \(\mathfrak q \in \text{WeakAss}_S(M \otimes_R N)\). We want to show \(\mathfrak p \in \text{WeakAss}_S(M)\). Let \(z \in M \otimes_R N\) be an element such that \(\mathfrak q\) is minimal over \(J = \text{Ann}_S(z)\). Let \(f_i \in \mathfrak p\), \(i \in I\) be a set of generators of the ideal \(\mathfrak p\). Since \(\mathfrak q\) lies over \(\mathfrak p\), for every \(i\) we can choose an \(n_i \geq 1\) and \(g_i \in S\), \(g_i \not \in \mathfrak q\) with \(g_i f_i^{n_i} \in J\), i.e., \(g_i f_i^{n_i} z = 0\). Let \(z' \in (M' \otimes_{R'} N')_{\mathfrak p'}\) be the image of \(z\). Observe that \(z'\) is nonzero because \(z\) has nonzero image in \((M \otimes_R N)_\mathfrak q\) and because \(S_\mathfrak q \to S'_{\mathfrak q'}\) is faithfully flat. We claim that \(f_i^{n_i} z' = 0\).

Proof of the claim: Let \(g'_i \in S'\) be the image of \(g_i\). By the key observation (0CU7) we find that the image \(\overline{g}'_i \in \overline{S}'\) is not contained in \(\mathfrak r'\) for any \(\mathfrak r' \in \text{Ass}_{\overline{S}'}(\overline{N})\). Hence by Lemma 05FR we see that \(g'_i : N'_{\mathfrak p'} \to N'_{\mathfrak p'}\) is \(R'_{\mathfrak p'}\)-universally injective. In particular we see that \(g'_i : M' \otimes_{R'} N' \to M' \otimes_{R'} N'\) is injective after localizating at \(\mathfrak p'\). The claim follows because \(g_i f_i^{n_i} z' = 0\).

Our claim shows that the annihilator of \(z'\) in \(R'_{\mathfrak p'}\) contains the elements \(f_i^{n_i}\). As \(R \to R'\) is étale we have \(\mathfrak p'R'_{\mathfrak p'} = \mathfrak pR'_{\mathfrak p'}\) by Algebra, Lemma 00U4. Hence the annihilator of \(z'\) in \(R'_{\mathfrak p'}\) has radical equal to \(\mathfrak p' R_{\mathfrak p'}\) (here we use \(z'\) is not zero). On the other hand \[z' \in (M' \otimes_{R'} N')_{\mathfrak p'} = M'_{\mathfrak p'} \otimes_{R'_{\mathfrak p'}} N'_{\mathfrak p'}\] The module \(N'_{\mathfrak p'}\) is projective over the local ring \(R'_{\mathfrak p'}\) and hence free (Algebra, Theorem 0593). Thus we can find a finite free direct summand \(F' \subset N'_{\mathfrak p'}\) such that \(z' \in M'_{\mathfrak p'} \otimes_{R'_{\mathfrak p'}} F'\). If \(F'\) has rank \(n\), then we deduce that \(\mathfrak p' R'_{\mathfrak p'} \in \text{WeakAss}_{R'_{\mathfrak p'}}({M'_{\mathfrak p'}}^{\oplus n})\). This implies \(\mathfrak p'R'_{\mathfrak p'} \in \text{WeakAss}(M'_{\mathfrak p'})\) for example by Algebra, Lemma 0548. Then \(\mathfrak p' \in \text{WeakAss}_{R'}(M')\) which in turn gives \(\mathfrak p \in \text{WeakAss}_R(M)\). This finishes the proof of the implication “\(\Rightarrow\)” of the equivalence of the lemma.

Assume that \(\mathfrak p \in \text{WeakAss}_R(M)\) and \(\overline{\mathfrak q} \in \text{Ass}_{\overline{S}}(\overline{N})\). We want to show that \(\mathfrak q\) is weakly associated to \(M \otimes_R N\). Note that \(\overline{\mathfrak q}'\) is a maximal element of \(\text{Ass}_{\overline{S}'}(\overline{N}')\). This is a consequence of (0CU7) and the fact that there are no inclusions among the primes of \(\overline{S}'\) lying over \(\overline{\mathfrak q}\) (as fibres of étale morphisms are discrete Morphisms, Lemma 02GL). Thus, after replacing \(R, S, \mathfrak p, \mathfrak q, M, N\) by \(R', S', \mathfrak p', \mathfrak q', M', N'\) we may assume, in addition to the assumptions of the lemma, that

1. \(\mathfrak p \in \text{WeakAss}_R(M)\),

2. \(\overline{\mathfrak q} \in \text{Ass}_{\overline{S}}(\overline{N})\),

3. \(N\) is projective as an \(R\)-module, and

4. \(\overline{\mathfrak q}\) is maximal in \(\text{Ass}_{\overline{S}}(\overline{N})\).

There is one more reduction, namely, we may replace \(R, S, M, N\) by their localizations at \(\mathfrak p\). This leads to one more condition, namely,

1. \(R\) is a local ring with maximal ideal \(\mathfrak p\).

We will finish by showing that (1) – (5) imply \(\mathfrak q \in \text{WeakAss}(M \otimes_R N)\).

Since \(R\) is local and \(\mathfrak p \in \text{WeakAss}_R(M)\) we can pick a \(y \in M\) whose annihilator \(I\) has radical equal to \(\mathfrak p\). Write \(\overline{\mathfrak q} = (\overline{g}_1, \ldots, \overline{g}_n)\) for some \(\overline{g}_i \in \overline{S}\). Choose \(g_i \in S\) mapping to \(\overline{g}_i\). Then \(\mathfrak q = \mathfrak pS + g_1S + \ldots + g_nS\). Consider the map \[\Psi : N/IN \longrightarrow (N/IN)^{\oplus n}, \quad z \longmapsto (g_1z, \ldots, g_nz).\] This is a homomorphism of projective \(R/I\)-modules. The local ring \(R/I\) is auto-associated (More on Algebra, Definition 05GM) as \(\mathfrak p/I\) is locally nilpotent. The map \(\Psi \otimes \kappa(\mathfrak p)\) is not injective, because \(\overline{\mathfrak q} \in \text{Ass}_{\overline{S}}(\overline{N})\). Hence More on Algebra, Lemma 05GQ implies \(\Psi\) is not injective. Pick \(z \in N/IN\) nonzero in the kernel of \(\Psi\). The annihilator \(J = \text{Ann}_S(z)\) contains \(IS\) and \(g_i\) by construction. Thus \(\sqrt{J} \subset S\) contains \(\mathfrak q\). Let \(\mathfrak s \subset S\) be a prime minimal over \(J\). Then \(\mathfrak q \subset \mathfrak s\), \(\mathfrak s\) lies over \(\mathfrak p\), and \(\mathfrak s \in \text{WeakAss}_S(N/IN)\). The last fact by definition of weakly associated primes. Apply the “\(\Rightarrow\)” part of the lemma (which we’ve already proven) to the ring map \(R \to S\) and the modules \(R/I\) and \(N\) to conclude that \(\overline{\mathfrak s} \in \text{Ass}_{\overline{S}}(\overline{N})\). Since \(\overline{\mathfrak q} \subset \overline{\mathfrak s}\) the maximality of \(\overline{\mathfrak q}\), see condition (4) above, implies that \(\overline{\mathfrak q} = \overline{\mathfrak s}\). This shows that \(\mathfrak q = \mathfrak s\) and we conlude what we want.

Lemma

Let \(S\) be a scheme. Let \(f : X \to S\) be locally of finite type. Let \(x \in X\) with image \(s \in S\). Let \(\mathcal{F}\) be a finite type quasi-coherent sheaf on \(X\). Let \(\mathcal{G}\) be a quasi-coherent sheaf on \(S\). If \(\mathcal{F}\) is flat at \(x\) over \(S\), then \[x \in \text{WeakAss}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}_S(\mathcal{G}) \text{ and } x \in \text{Ass}_{X_s}(\mathcal{F}_s).\]

Proof

In this paragraph we reduce to \(f\) being of finite presentation. The question is local on \(X\) and \(S\), hence we may assume \(X\) and \(S\) are affine. Write \(X = \Spec(B)\), \(S = \Spec(A)\) and write \(B = A[x_1, \ldots, x_n]/I\). In other words we obtain a closed immersion \(i : X \to \mathbf{A}^n_S\) over \(S\). Denote \(t = i(x) \in \mathbf{A}^n_S\). Note that \(i_*\mathcal{F}\) is a finite type quasi-coherent sheaf on \(\mathbf{A}^n_S\) which is flat at \(t\) over \(S\) and note that \[i_*(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = i_*\mathcal{F} \otimes_{\mathcal{O}_{\mathbf{A}^n_S}} p^*\mathcal{G}\] where \(p : \mathbf{A}^n_S \to S\) is the projection. Note that \(t\) is a weakly associated point of \(i_*(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})\) if and only if \(x\) is a weakly associated point of \(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}\), see Divisors, Lemma 05EZ. Similarly \(x \in \text{Ass}_{X_s}(\mathcal{F}_s)\) if and only if \(t \in \text{Ass}_{\mathbf{A}^n_s}((i_*\mathcal{F})_s)\) (see Algebra, Lemma 05BY). Hence it suffices to prove the lemma in case \(X = \mathbf{A}^n_S\). Thus we may assume that \(X \to S\) is of finite presentation.

In this paragraph we reduce to \(\mathcal{F}\) being of finite presentation and flat over \(S\). Choose an elementary étale neighbourhood \(e : (S', s') \to (S, s)\) and an open \(V \subset X \times_S \Spec(\mathcal{O}_{S', s'})\) as in Proposition 05I5. Let \(x' \in X' = X \times_S S'\) be the unique point mapping to \(x\) and \(s'\). Then it suffices to prove the statement for \(X' \to S'\), \(x'\), \(s'\), \((X' \to X)^*\mathcal{F}\), and \(e^*\mathcal{G}\), see Lemma 05FP. Let \(v \in V\) the unique point mapping to \(x'\) and let \(s' \in \Spec(\mathcal{O}_{S', s'})\) be the closed point. Then \(\mathcal{O}_{V, v} = \mathcal{O}_{X', x'}\) and \(\mathcal{O}_{\Spec(\mathcal{O}_{S', s'}), s'} = \mathcal{O}_{S', s'}\) and similarly for the stalks of pullbacks of \(\mathcal{F}\) and \(\mathcal{G}\). Also \(V_{s'} \subset X'_{s'}\) is an open subscheme. Since the condition of being a weakly associated point depend only on the stalk of the sheaf, we may replace \(X' \to S'\), \(x'\), \(s'\), \((X' \to X)^*\mathcal{F}\), and \(e^*\mathcal{G}\) by \(V \to \Spec(\mathcal{O}_{S', s'})\), \(v\), \(s'\), \((V \to X)^*\mathcal{F}\), and \((\Spec(\mathcal{O}_{S', s'}) \to S)^*\mathcal{G}\). Thus we may assume that \(f\) is of finite presentation and \(\mathcal{F}\) of finite presentation and flat over \(S\).

Assume \(f\) is of finite presentation and \(\mathcal{F}\) of finite presentation and flat over \(S\). After shrinking \(X\) and \(S\) to affine neighbourhoods of \(x\) and \(s\), this case is handled by Lemma 05IG.

Lemma

Let \(R \to S\) be a ring map which is essentially of finite type. Let \(N\) be a localization of a finite \(S\)-module flat over \(R\). Let \(M\) be an \(R\)-module. Then \[\text{WeakAss}_S(M \otimes_R N) = \bigcup\nolimits_{\mathfrak p \in \text{WeakAss}_R(M)} \text{Ass}_{S \otimes_R \kappa(\mathfrak p)}(N \otimes_R \kappa(\mathfrak p))\]

Proof

This lemma is a translation of Lemma 05IH into algebra. Details of translation omitted.

Lemma

Let \(f : X \to S\) be a morphism which is locally of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent sheaf on \(X\) which is flat over \(S\). Let \(\mathcal{G}\) be a quasi-coherent sheaf on \(S\). Then we have \[\text{WeakAss}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = \bigcup\nolimits_{s \in \text{WeakAss}_S(\mathcal{G})} \text{Ass}_{X_s}(\mathcal{F}_s)\]

Proof

Immediate consequence of Lemma 05IH.

Theorem

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

1. \(X \to S\) is locally of finite presentation,

2. \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite type, and

3. the set of weakly associated points of \(S\) is locally finite in \(S\).

Then \(U = \{x \in X \mid \mathcal{F}\text{ flat at }x\text{ over }S\}\) is open in \(X\) and \(\mathcal{F}|_U\) is an \(\mathcal{O}_U\)-module of finite presentation and flat over \(S\).

Proof

Let \(x \in X\) be such that \(\mathcal{F}\) is flat at \(x\) over \(S\). We have to find an open neighbourhood of \(x\) such that \(\mathcal{F}\) restricts to a \(S\)-flat finitely presented module on this neighbourhood. The problem is local on \(X\) and \(S\), hence we may assume that \(X\) and \(S\) are affine. As \(\mathcal{F}_x\) is a finitely presented \(\mathcal{O}_{X, x}\)-module by Lemma 05I7 we conclude from Algebra, Lemma 05GJ there exists a finitely presented \(\mathcal{O}_X\)-module \(\mathcal{F}'\) and a map \(\varphi : \mathcal{F}' \to \mathcal{F}\) which induces an isomorphism \(\varphi_x : \mathcal{F}'_x \to \mathcal{F}_x\). In particular we see that \(\mathcal{F}'\) is flat over \(S\) at \(x\), hence by openness of flatness More on Morphisms, Theorem 0399 we see that after shrinking \(X\) we may assume that \(\mathcal{F}'\) is flat over \(S\). As \(\mathcal{F}\) is of finite type after shrinking \(X\) we may assume that \(\varphi\) is surjective, see Modules, Lemma 01B8 or alternatively use Nakayama’s lemma (Algebra, Lemma 00DV). By Lemma 05IJ we have \[\text{WeakAss}_X(\mathcal{F}') \subset \bigcup\nolimits_{s \in \text{WeakAss}(S)} \text{Ass}_{X_s}(\mathcal{F}'_s)\] As \(\text{WeakAss}(S)\) is finite by assumption and since \(\text{Ass}_{X_s}(\mathcal{F}'_s)\) is finite by Divisors, Lemma 05AF we conclude that \(\text{WeakAss}_X(\mathcal{F}')\) is finite. Using Algebra, Lemma 00DS we may, after shrinking \(X\) once more, assume that \(\text{WeakAss}_X(\mathcal{F}')\) is contained in the generalization of \(x\). Now consider \(\mathcal{K} = \Ker(\varphi)\). We have \(\text{WeakAss}_X(\mathcal{K}) \subset \text{WeakAss}_X(\mathcal{F}')\) (by Divisors, Lemma 05AN) but on the other hand, \(\varphi_x\) is an isomorphism, also \(\varphi_{x'}\) is an isomorphism for all \(x' \leadsto x\). We conclude that \(\text{WeakAss}_X(\mathcal{K}) = \emptyset\) whence \(\mathcal{K} = 0\) by Divisors, Lemma 05AP.

Lemma

Let \(R \to S\) be a ring map of finite presentation. Let \(M\) be a finite \(S\)-module. Assume \(\text{WeakAss}_R(R)\) is finite. Then \[U = \{\mathfrak q \subset S \mid M_{\mathfrak q}\text{ flat over }R\}\] is open in \(\Spec(S)\) and for every \(g \in S\) such that \(D(g) \subset U\) the localization \(M_g\) is a finitely presented \(S_g\)-module flat over \(R\).

Proof

Follows immediately from Theorem 05IK.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Assume the set of weakly associated points of \(S\) is locally finite in \(S\). Then the set of points \(x \in X\) where \(f\) is flat is an open subscheme \(U \subset X\) and \(U \to S\) is flat and locally of finite presentation.

Proof

The problem is local on \(X\) and \(S\), hence we may assume that \(X\) and \(S\) are affine. Then \(X \to S\) corresponds to a finite type ring map \(A \to B\). Choose a surjection \(A[x_1, \ldots, x_n] \to B\) and consider \(B\) as an \(A[x_1, \ldots, x_n]\)-module. An application of Lemma 05IL finishes the proof.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type and flat. If \(S\) is integral, then \(f\) is locally of finite presentation.

Proof

Special case of Lemma 05IM.

Proposition

Let \(R\) be a domain. Let \(R \to S\) be a ring map of finite type. Let \(M\) be a finite \(S\)-module.

1. If \(S\) is flat over \(R\), then \(S\) is a finitely presented \(R\)-algebra.

2. If \(M\) is flat as an \(R\)-module, then \(M\) is finitely presented as an \(S\)-module.

Proof

Part (1) is a special case of Lemma 05IN. For Part (2) choose a surjection \(R[x_1, \ldots, x_n] \to S\). By Lemma 05IL we find that \(M\) is finitely presented as an \(R[x_1, \ldots, x_n]\)-module. We conclude by Algebra, Lemma 0561.

Lemma

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

1. \(X \to S\) is locally of finite type,

2. \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite type, and

3. the set of weakly associated points of \(S\) is locally finite in \(S\).

Then \(U = \{x \in X \mid \mathcal{F}\text{ flat at }x\text{ over }S\}\) is open in \(X\) and \(\mathcal{F}|_U\) is flat over \(S\) and locally finitely presented relative to \(S\) (see More on Morphisms, Definition 05H1).

Proof

The question is local on \(X\) and \(S\). Thus we may assume \(X\) and \(S\) are affine. Then we may choose a closed immersion \(i : X \to \mathbf{A}^n_S\). We apply Theorem 05IK to \(X' = \mathbf{A}^n_S \to S\) and the quasi-coherent module \(\mathcal{F}' = i_*\mathcal{F}\) of finite type and we find that \[U' = \{x' \in X' \mid \mathcal{F}'\text{ flat at }x'\text{ over }S\}\] is open in \(X'\) and that \(\mathcal{F}'|_{U'}\) is of finite presentation. Since \(\mathcal{F}'\) restricts to zero on \(X' \setminus i(X)\) and since \(\mathcal{F}'_{i(x)} \cong \mathcal{F}_x\) for all \(x \in X\) we see that \[U' = i(U) \amalg (X' \setminus i(X))\] Hence \(U = i^{-1}(U')\) is open. Moreover, it is clear that \(\mathcal{F}'|_{U'} = (i|_U)_*(\mathcal{F}|_U)\). Hence we conclude that \(\mathcal{F}|_U\) is finitely presented relative to \(S\) by More on Morphisms, Lemmas 09T8 and 09T9.

Lemma

Let \(R \to S\) be a ring map of finite type. Let \(M\) be a finite \(S\)-module. Assume \(\text{WeakAss}_R(R)\) is finite. Then \[U = \{\mathfrak q \subset S \mid M_{\mathfrak q}\text{ flat over }R\}\] is open in \(\Spec(S)\) and for every \(g \in S\) such that \(D(g) \subset U\) the localization \(M_g\) is flat over \(R\) and an \(S_g\)-module finitely presented relative to \(R\) (see More on Algebra, Definition 05GZ).

Proof

This is Lemma 05IQ translated into algebra.

Examples of relatively pure modules

In the short section we discuss some examples of results that will serve as motivation for the notion of a relatively pure module and the concept of an impurity which we will introduce later. Each of the examples is stated as a lemma. Note the similarity with the condition on associated primes to the conditions appearing in Lemmas 05DH, 05DL, 05DM, and 05DP. See also Algebra, Lemma 05GB for a discussion.

Lemma

Let \(R\) be a local ring with maximal ideal \(\mathfrak m\). Let \(R \to S\) be a ring map. Let \(N\) be an \(S\)-module. Assume

1. \(N\) is projective as an \(R\)-module, and

2. \(S/\mathfrak mS\) is Noetherian and \(N/\mathfrak mN\) is a finite \(S/\mathfrak mS\)-module.

Then for any prime \(\mathfrak q \subset S\) which is an associated prime of \(N \otimes_R \kappa(\mathfrak p)\) where \(\mathfrak p = R \cap \mathfrak q\) we have \(\mathfrak q + \mathfrak m S \not = S\).

Proof

Note that the hypotheses of Lemmas 05DE and 05FR are satisfied. We will use the conclusions of these lemmas without further mention. Let \(\Sigma \subset S\) be the multiplicative set of elements which are not zerodivisors on \(N/\mathfrak mN\). The map \(N \to \Sigma^{-1}N\) is \(R\)-universally injective. Hence we see that any \(\mathfrak q \subset S\) which is an associated prime of \(N \otimes_R \kappa(\mathfrak p)\) is also an associated prime of \(\Sigma^{-1}N \otimes_R \kappa(\mathfrak p)\). Clearly this implies that \(\mathfrak q\) corresponds to a prime of \(\Sigma^{-1}S\). Thus \(\mathfrak q \subset \mathfrak q'\) where \(\mathfrak q'\) corresponds to an associated prime of \(N/\mathfrak mN\) and we win.

The following lemma gives another (slightly silly) example of this phenomenon.

Lemma

Let \(R\) be a ring. Let \(I \subset R\) be an ideal. Let \(R \to S\) be a ring map. Let \(N\) be an \(S\)-module. If \(N\) is \(I\)-adically complete, then for any \(R\)-module \(M\) and for any prime \(\mathfrak q \subset S\) which is an associated prime of \(N \otimes_R M\) we have \(\mathfrak q + I S \not = S\).

Proof

Let \(S^\wedge\) denote the \(I\)-adic completion of \(S\). Note that \(N\) is an \(S^\wedge\)-module, hence also \(N \otimes_R M\) is an \(S^\wedge\)-module. Let \(z \in N \otimes_R M\) be an element such that \(\mathfrak q = \text{Ann}_S(z)\). Since \(z \not = 0\) we see that \(\text{Ann}_{S^\wedge}(z) \not = S^\wedge\). Hence \(\mathfrak q S^\wedge \not = S^\wedge\). Hence there exists a maximal ideal \(\mathfrak m \subset S^\wedge\) with \(\mathfrak q S^\wedge \subset \mathfrak m\). Since \(IS^\wedge \subset \mathfrak m\) by Algebra, Lemma 05GI we win.

Note that the following lemma gives an alternative proof of Lemma 05FV as a projective module over a local ring is free, see Algebra, Theorem 0593.

Lemma

Let \(R\) be a local ring with maximal ideal \(\mathfrak m\). Let \(R \to S\) be a ring map. Let \(N\) be an \(S\)-module. Assume \(N\) is isomorphic as an \(R\)-module to a direct sum of finite \(R\)-modules. Then for any \(R\)-module \(M\) and for any prime \(\mathfrak q \subset S\) which is an associated prime of \(N \otimes_R M\) we have \(\mathfrak q + \mathfrak m S \not = S\).

Proof

Write \(N = \bigoplus_{i \in I} M_i\) with each \(M_i\) a finite \(R\)-module. Let \(M\) be an \(R\)-module and let \(\mathfrak q \subset S\) be an associated prime of \(N \otimes_R M\) such that \(\mathfrak q + \mathfrak m S = S\). Let \(z \in N \otimes_R M\) be an element with \(\mathfrak q = \text{Ann}_S(z)\). After modifying the direct sum decomposition a little bit we may assume that \(z \in M_1 \otimes_R M\) for some element \(1 \in I\). Write \(1 = f + \sum x_j g_j\) for some \(f \in \mathfrak q\), \(x_j \in \mathfrak m\), and \(g_j \in S\). For any \(g \in S\) denote \(g'\) the \(R\)-linear map \[M_1 \to N \xrightarrow{g} N \to M_1\] where the first arrow is the inclusion map, the second arrow is multiplication by \(g\) and the third arrow is the projection map. Because each \(x_j \in R\) we obtain the equality \[f' + \sum x_j g'_j = \text{id}_{M_1} \in \text{End}_R(M_1)\] By Nakayama’s lemma (Algebra, Lemma 00DV) we see that \(f'\) is surjective, hence by Algebra, Lemma 05G8 we see that \(f'\) is an isomorphism. In particular the map \[M_1 \otimes_R M \to N \otimes_R M \xrightarrow{f} N \otimes_R M \to M_1 \otimes_R M\] is an isomorphism. This contradicts the assumption that \(fz = 0\).

Lemma

Let \(R\) be a henselian local ring with maximal ideal \(\mathfrak m\). Let \(R \to S\) be a ring map. Let \(N\) be an \(S\)-module. Assume \(N\) is countably generated and Mittag-Leffler as an \(R\)-module. Then for any \(R\)-module \(M\) and for any prime \(\mathfrak q \subset S\) which is an associated prime of \(N \otimes_R M\) we have \(\mathfrak q + \mathfrak m S \not = S\).

Proof

This lemma reduces to Lemma 05IU by Algebra, Lemma 05D6.

Suppose \(f : X \to S\) is a morphism of schemes and \(\mathcal{F}\) is a quasi-coherent module on \(X\). Let \(\xi \in \text{Ass}_{X/S}(\mathcal{F})\) and let \(Z = \overline{\{\xi\}}\). Picture \[\xymatrix{ \xi \ar@{|->}[d] & Z \ar[r] \ar[rd] & X \ar[d]^f \\ f(\xi) & & S }\] Note that \(f(Z) \subset \overline{\{f(\xi)\}}\) and that \(f(Z)\) is closed if and only if equality holds, i.e., \(f(Z) = \overline{\{f(\xi)\}}\). It follows from Lemma 05FV that if \(S\), \(X\) are affine, the fibres \(X_s\) are Noetherian, \(\mathcal{F}\) is of finite type, and \(\Gamma(X, \mathcal{F})\) is a projective \(\Gamma(S, \mathcal{O}_S)\)-module, then \(f(Z) = \overline{\{f(\xi)\}}\) is a closed subset. Slightly different analogous statements holds for the cases described in Lemmas 05IT, 05IU, and 05IV.

Impurities

We want to formalize the phenomenon of which we gave examples in Section 05IS in terms of specializations of points of \(\text{Ass}_{X/S}(\mathcal{F})\). We also want to work locally around a point \(s \in S\). In order to do so we make the following definitions.

Situation

Here \(S\), \(X\) are schemes and \(f : X \to S\) is a finite type morphism. Also, \(\mathcal{F}\) is a finite type quasi-coherent \(\mathcal{O}_X\)-module. Finally \(s\) is a point of \(S\).

In this situation consider a morphism \(g : T \to S\), a point \(t \in T\) with \(g(t) = s\), a specialization \(t' \leadsto t\), and a point \(\xi \in X_T\) in the base change of \(X\) lying over \(t'\). Picture [05IX]\[\begin{equation} \vcenter{ \xymatrix{ \xi \ar@{|->}[d] & \\ t' \ar@{~>}[r] & t \ar@{|->}[r] & s } } \quad\quad \vcenter{ \xymatrix{ X_T \ar[d] \ar[r] & X \ar[d] \\ T \ar[r]^g & S } } \end{equation}\] Moreover, denote \(\mathcal{F}_T\) the pullback of \(\mathcal{F}\) to \(X_T\).

Definition

In Situation 05FW we say a diagram (05IX) defines an impurity of \(\mathcal{F}\) above \(s\) if \(\xi \in \text{Ass}_{X_T/T}(\mathcal{F}_T)\) and \(\overline{\{\xi\}} \cap X_t = \emptyset\). We will indicate this by saying “let \((g : T \to S, t' \leadsto t, \xi)\) be an impurity of \(\mathcal{F}\) above \(s\)”.

Lemma

In Situation 05FW. If there exists an impurity of \(\mathcal{F}\) above \(s\), then there exists an impurity \((g : T \to S, t' \leadsto t, \xi)\) of \(\mathcal{F}\) above \(s\) such that \(g\) is locally of finite presentation and \(t\) a closed point of the fibre of \(g\) above \(s\).

Proof

Let \((g : T \to S, t' \leadsto t, \xi)\) be any impurity of \(\mathcal{F}\) above \(s\). We apply Limits, Lemma 05BD to \(t \in T\) and \(Z = \overline{\{\xi\}}\) to obtain an open neighbourhood \(V \subset T\) of \(t\), a commutative diagram \[\xymatrix{ V \ar[d] \ar[r]_a & T' \ar[d]^b \\ T \ar[r]^g & S, }\] and a closed subscheme \(Z' \subset X_{T'}\) such that

1. the morphism \(b : T' \to S\) is locally of finite presentation,

2. we have \(Z' \cap X_{a(t)} = \emptyset\), and

3. \(Z \cap X_V\) maps into \(Z'\) via the morphism \(X_V \to X_{T'}\).

As \(t'\) specializes to \(t\) we may replace \(T\) by the open neighbourhood \(V\) of \(t\). Thus we have a commutative diagram \[\xymatrix{ X_T \ar[d] \ar[r] & X_{T'} \ar[d] \ar[r] & X \ar[d] \\ T \ar[r]^a & T' \ar[r]^b & S }\] where \(b \circ a = g\). Let \(\xi' \in X_{T'}\) denote the image of \(\xi\). By Divisors, Lemma 05DC we see that \(\xi' \in \text{Ass}_{X_{T'}/T'}(\mathcal{F}_{T'})\). Moreover, by construction the closure of \(\overline{\{\xi'\}}\) is contained in the closed subset \(Z'\) which avoids the fibre \(X_{a(t)}\). In this way we see that \((T' \to S, a(t') \leadsto a(t), \xi')\) is an impurity of \(\mathcal{F}\) above \(s\).

Thus we may assume that \(g : T \to S\) is locally of finite presentation. Let \(Z = \overline{\{\xi\}}\). By assumption \(Z_t = \emptyset\). By More on Morphisms, Lemma 054W this means that \(Z_{t''} = \emptyset\) for \(t''\) in an open subset of \(\overline{\{t\}}\). Since the fibre of \(T \to S\) over \(s\) is a Jacobson scheme, see Morphisms, Lemma 02J6 we find that there exist a closed point \(t'' \in \overline{\{t\}}\) such that \(Z_{t''} = \emptyset\). Then \((g : T \to S, t' \leadsto t'', \xi)\) is the desired impurity.

Lemma

In Situation 05FW. Let \((g : T \to S, t' \leadsto t, \xi)\) be an impurity of \(\mathcal{F}\) above \(s\). Assume \(T = \lim_{i \in I} T_i\) is a directed limit of affine schemes over \(S\). Then for some \(i\) the triple \((T_i \to S, t'_i \leadsto t_i, \xi_i)\) is an impurity of \(\mathcal{F}\) above \(s\).

Proof

The notation in the statement means this: Let \(p_i : T \to T_i\) be the projection morphisms, let \(t_i = p_i(t)\) and \(t'_i = p_i(t')\). Finally \(\xi_i \in X_{T_i}\) is the image of \(\xi\). By Divisors, Lemma 05DC it is true that \(\xi_i\) is a point of the relative assassin of \(\mathcal{F}_{T_i}\) over \(T_i\). Thus the only point is to show that \(\overline{\{\xi_i\}} \cap X_{t_i} = \emptyset\) for some \(i\).

First proof. Let \(Z_i = \overline{\{\xi_i\}} \subset X_{T_i}\) and \(Z = \overline{\{\xi\}} \subset X_T\) endowed with the reduced induced scheme structure. Then \(Z = \lim Z_i\) by Limits, Lemma 0CUG. Choose a field \(k\) and a morphism \(\Spec(k) \to T\) whose image is \(t\). Then \[\emptyset = Z \times_T \Spec(k) = (\lim Z_i) \times_{(\lim T_i)} \Spec(k) = \lim Z_i \times_{T_i} \Spec(k)\] because limits commute with fibred products (limits commute with limits). Each \(Z_i \times_{T_i} \Spec(k)\) is quasi-compact because \(X_{T_i} \to T_i\) is of finite type and hence \(Z_i \to T_i\) is of finite type. Hence \(Z_i \times_{T_i} \Spec(k)\) is empty for some \(i\) by Limits, Lemma 01Z2. Since the image of the composition \(\Spec(k) \to T \to T_i\) is \(t_i\) we obtain what we want.

Second proof. Set \(Z = \overline{\{\xi\}}\). Apply Limits, Lemma 05BD to this situation to obtain an open neighbourhood \(V \subset T\) of \(t\), a commutative diagram \[\xymatrix{ V \ar[d] \ar[r]_a & T' \ar[d]^b \\ T \ar[r]^g & S, }\] and a closed subscheme \(Z' \subset X_{T'}\) such that

1. the morphism \(b : T' \to S\) is locally of finite presentation,

2. we have \(Z' \cap X_{a(t)} = \emptyset\), and

3. \(Z \cap X_V\) maps into \(Z'\) via the morphism \(X_V \to X_{T'}\).

We may assume \(V\) is an affine open of \(T\), hence by Limits, Lemmas 01Z4 and 01Z6 we can find an \(i\) and an affine open \(V_i \subset T_i\) with \(V = f_i^{-1}(V_i)\). By Limits, Proposition 01ZC after possibly increasing \(i\) a bit we can find a morphism \(a_i : V_i \to T'\) such that \(a = a_i \circ f_i|_V\). The induced morphism \(X_{V_i} \to X_{T'}\) maps \(\xi_i\) into \(Z'\). As \(Z' \cap X_{a(t)} = \emptyset\) we conclude that \((T_i \to S, t'_i \leadsto t_i, \xi_i)\) is an impurity of \(\mathcal{F}\) above \(s\).

Lemma

In Situation 05FW. If there exists an impurity \((g : T \to S, t' \leadsto t, \xi)\) of \(\mathcal{F}\) above \(s\) with \(g\) quasi-finite at \(t\), then there exists an impurity \((g : T \to S, t' \leadsto t, \xi)\) such that \((T, t) \to (S, s)\) is an elementary étale neighbourhood.

Proof

Let \((g : T \to S, t' \leadsto t, \xi)\) be an impurity of \(\mathcal{F}\) above \(s\) such that \(g\) is quasi-finite at \(t\). After shrinking \(T\) we may assume that \(g\) is locally of finite type. Apply More on Morphisms, Lemma 02LK to \(T \to S\) and \(t \mapsto s\). This gives us a diagram \[\xymatrix{ T \ar[d] & T \times_S U \ar[l] \ar[d] & V \ar[l] \ar[ld] \\ S & U \ar[l] }\] where \((U, u) \to (S, s)\) is an elementary étale neighbourhood and \(V \subset T \times_S U\) is an open neighbourhood of \(v = (t, u)\) such that \(V \to U\) is finite and such that \(v\) is the unique point of \(V\) lying over \(u\). Since the morphism \(V \to T\) is étale hence flat we see that there exists a specialization \(v' \leadsto v\) such that \(v' \mapsto t'\). Note that \(\kappa(t') \subset \kappa(v')\) is finite separable. Pick any point \(\zeta \in X_{v'}\) mapping to \(\xi \in X_{t'}\). By Divisors, Lemma 05DC we see that \(\zeta \in \text{Ass}_{X_V/V}(\mathcal{F}_V)\). Moreover, the closure \(\overline{\{\zeta\}}\) does not meet the fibre \(X_v\) as by assumption the closure \(\overline{\{\xi\}}\) does not meet \(X_t\). In other words \((V \to S, v' \leadsto v, \zeta)\) is an impurity of \(\mathcal{F}\) above \(S\).

Next, let \(u' \in U'\) be the image of \(v'\) and let \(\theta \in X_U\) be the image of \(\zeta\). Then \(\theta \mapsto u'\) and \(u' \leadsto u\). By Divisors, Lemma 05DC we see that \(\theta \in \text{Ass}_{X_U/U}(\mathcal{F})\). Moreover, as \(\pi : X_V \to X_U\) is finite we see that \(\pi\big(\overline{\{\zeta\}}\big) = \overline{\{\pi(\zeta)\}}\). Since \(v\) is the unique point of \(V\) lying over \(u\) we see that \(X_u \cap \overline{\{\pi(\zeta)\}} = \emptyset\) because \(X_v \cap \overline{\{\zeta\}} = \emptyset\). In this way we conclude that \((U \to S, u' \leadsto u, \theta)\) is an impurity of \(\mathcal{F}\) above \(s\) and we win.

Lemma

In Situation 05FW. Assume that \(S\) is locally Noetherian. If there exists an impurity of \(\mathcal{F}\) above \(s\), then there exists an impurity \((g : T \to S, t' \leadsto t, \xi)\) of \(\mathcal{F}\) above \(s\) such that \(g\) is quasi-finite at \(t\).

Proof

We may replace \(S\) by an affine neighbourhood of \(s\). By Lemma 05FX we may assume that we have an impurity \((g : T \to S, t' \leadsto t, \xi)\) of such that \(g\) is locally of finite type and \(t\) a closed point of the fibre of \(g\) above \(s\). We may replace \(T\) by the reduced induced scheme structure on \(\overline{\{t'\}}\). Let \(Z = \overline{\{\xi\}} \subset X_T\). By assumption \(Z_t = \emptyset\) and the image of \(Z \to T\) contains \(t'\). By More on Morphisms, Lemma 05F1 there exists a nonempty open \(V \subset Z\) such that for any \(w \in f(V)\) any generic point \(\xi'\) of \(V_w\) is in \(\text{Ass}_{X_T/T}(\mathcal{F}_T)\). By More on Morphisms, Lemma 05F5 there exists a nonempty open \(W \subset T\) with \(W \subset f(V)\). By More on Morphisms, Lemma 05GU there exists a closed subscheme \(T' \subset T\) such that \(t \in T'\), \(T' \to S\) is quasi-finite at \(t\), and there exists a point \(z \in T' \cap W\), \(z \leadsto t\) which does not map to \(s\). Choose any generic point \(\xi'\) of the nonempty scheme \(V_z\). Then \((T' \to S, z \leadsto t, \xi')\) is the desired impurity.

In the following we will use the henselization \(S^h = \Spec(\mathcal{O}_{S, s}^h)\) of \(S\) at \(s\), see Étale Cohomology, Definition 03PS. Since \(S^h \to S\) maps to closed point of \(S^h\) to \(s\) and induces an isomorphism of residue fields, we will indicate \(s \in S^h\) this closed point also. Thus \((S^h, s) \to (S, s)\) is a morphism of pointed schemes.

Lemma

In Situation 05FW. If there exists an impurity \((S^h \to S, s' \leadsto s, \xi)\) of \(\mathcal{F}\) above \(s\) then there exists an impurity \((T \to S, t' \leadsto t, \xi)\) of \(\mathcal{F}\) above \(s\) where \((T, t) \to (S, s)\) is an elementary étale neighbourhood.

Proof

We may replace \(S\) by an affine neighbourhood of \(s\). Say \(S = \Spec(A)\) and \(s\) corresponds to the prime \(\mathfrak p \subset A\). Then \(\mathcal{O}_{S, s}^h = \colim_{(T, t)} \Gamma(T, \mathcal{O}_T)\) where the limit is over the opposite of the cofiltered category of affine elementary étale neighbourhoods \((T, t)\) of \((S, s)\), see More on Morphisms, Lemma 05KS and its proof. Hence \(S^h = \lim_i T_i\) and we win by Lemma 05IZ.

Lemma

In Situation 05FW the following are equivalent

1. there exists an impurity \((S^h \to S, s' \leadsto s, \xi)\) of \(\mathcal{F}\) above \(s\) where \(S^h\) is the henselization of \(S\) at \(s\),

2. there exists an impurity \((T \to S, t' \leadsto t, \xi)\) of \(\mathcal{F}\) above \(s\) such that \((T, t) \to (S, s)\) is an elementary étale neighbourhood, and

3. there exists an impurity \((T \to S, t' \leadsto t, \xi)\) of \(\mathcal{F}\) above \(s\) such that \(T \to S\) is quasi-finite at \(t\).

Proof

As an étale morphism is locally quasi-finite it is clear that (2) implies (3). We have seen that (3) implies (2) in Lemma 05J0. We have seen that (1) implies (2) in Lemma 05J2. Finally, if \((T \to S, t' \leadsto t, \xi)\) is an impurity of \(\mathcal{F}\) above \(s\) such that \((T, t) \to (S, s)\) is an elementary étale neighbourhood, then we can choose a factorization \(S^h \to T \to S\) of the structure morphism \(S^h \to S\). Choose any point \(s' \in S^h\) mapping to \(t'\) and choose any \(\xi' \in X_{s'}\) mapping to \(\xi \in X_{t'}\). Then \((S^h \to S, s' \leadsto s, \xi')\) is an impurity of \(\mathcal{F}\) above \(s\). We omit the details.

Relatively pure modules

The notion of a module pure relative to a base was introduced in [GruRay].

Definition

Let \(f : X \to S\) be a morphism of schemes which is of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module.

1. Let \(s \in S\). We say \(\mathcal{F}\) is pure along \(X_s\) if there is no impurity \((g : T \to S, t' \leadsto t, \xi)\) of \(\mathcal{F}\) above \(s\) with \((T, t) \to (S, s)\) an elementary étale neighbourhood.

2. We say \(\mathcal{F}\) is universally pure along \(X_s\) if there does not exist any impurity of \(\mathcal{F}\) above \(s\).

3. We say that \(X\) is pure along \(X_s\) if \(\mathcal{O}_X\) is pure along \(X_s\).

4. We say \(\mathcal{F}\) is universally \(S\)-pure, or universally pure relative to \(S\) if \(\mathcal{F}\) is universally pure along \(X_s\) for every \(s \in S\).

5. We say \(\mathcal{F}\) is \(S\)-pure, or pure relative to \(S\) if \(\mathcal{F}\) is pure along \(X_s\) for every \(s \in S\).

6. We say that \(X\) is \(S\)-pure or pure relative to \(S\) if \(\mathcal{O}_X\) is pure relative to \(S\).

We intentionally restrict ourselves here to morphisms which are of finite type and not just morphisms which are locally of finite type, see Remark 05J5 for a discussion. In the situation of the definition Lemma 05J3 tells us that the following are equivalent

1. \(\mathcal{F}\) is pure along \(X_s\),

2. there is no impurity \((g : T \to S, t' \leadsto t, \xi)\) with \(g\) quasi-finite at \(t\),

3. there does not exist any impurity of the form \((S^h \to S, s' \leadsto s, \xi)\), where \(S^h\) is the henselization of \(S\) at \(s\).

If we denote \(X^h = X \times_S S^h\) and \(\mathcal{F}^h\) the pullback of \(\mathcal{F}\) to \(X^h\), then we can formulate the last condition in the following more positive way:

1. All points of \(\text{Ass}_{X^h/S^h}(\mathcal{F}^h)\) specialize to points of \(X_s\).

In particular, it is clear that \(\mathcal{F}\) is pure along \(X_s\) if and only if the pullback of \(\mathcal{F}\) to \(X \times_S \Spec(\mathcal{O}_{S, s})\) is pure along \(X_s\).

Remark

Let \(f : X \to S\) be a morphism which is locally of finite type and \(\mathcal{F}\) a quasi-coherent finite type \(\mathcal{O}_X\)-module. In this case it is still true that (1) and (2) above are equivalent because the proof of Lemma 05J0 does not use that \(f\) is quasi-compact. It is also clear that (3) and (4) are equivalent. However, we don’t know if (1) and (3) are equivalent. In this case it may sometimes be more convenient to define purity using the equivalent conditions (3) and (4) as is done in [GruRay]. On the other hand, for many applications it seems that the correct notion is really that of being universally pure.

A natural question to ask is if the property of being pure relative to the base is preserved by base change, i.e., if being pure is the same thing as being universally pure. It turns out that this is true over Noetherian base schemes (see Lemma 05J8), or if the sheaf is flat (see Lemmas 05L5 and 05L6). It is not true in general, even if the morphism and the sheaf are of finite presentation, see Examples, Section 05JJ for a counter example. First we match our usage of “universally” to the usual notion.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(s \in S\). The following are equivalent

1. \(\mathcal{F}\) is universally pure along \(X_s\), and

2. for every morphism of pointed schemes \((S', s') \to (S, s)\) the pullback \(\mathcal{F}_{S'}\) is pure along \(X_{s'}\).

In particular, \(\mathcal{F}\) is universally pure relative to \(S\) if and only if every base change \(\mathcal{F}_{S'}\) of \(\mathcal{F}\) is pure relative to \(S'\).

Proof

This is formal.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(s \in S\). Let \((S', s') \to (S, s)\) be a morphism of pointed schemes. If \(S' \to S\) is quasi-finite at \(s'\) and \(\mathcal{F}\) is pure along \(X_s\), then \(\mathcal{F}_{S'}\) is pure along \(X_{s'}\).

Proof

It \((T \to S', t' \leadsto t, \xi)\) is an impurity of \(\mathcal{F}_{S'}\) above \(s'\) with \(T \to S'\) quasi-finite at \(t\), then \((T \to S, t' \to t, \xi)\) is an impurity of \(\mathcal{F}\) above \(s\) with \(T \to S\) quasi-finite at \(t\), see Morphisms, Lemma 01TL. Hence the lemma follows immediately from the characterization (2) of purity given following Definition 05J4.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(s \in S\). If \(\mathcal{O}_{S, s}\) is Noetherian then \(\mathcal{F}\) is pure along \(X_s\) if and only if \(\mathcal{F}\) is universally pure along \(X_s\).

Proof

First we may replace \(S\) by \(\Spec(\mathcal{O}_{S, s})\), i.e., we may assume that \(S\) is Noetherian. Next, use Lemma 05J1 and characterization (2) of purity given in discussion following Definition 05J4 to conclude.

Purity satisfies flat descent.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(s \in S\). Let \((S', s') \to (S, s)\) be a morphism of pointed schemes. Assume \(S' \to S\) is flat at \(s'\).

1. If \(\mathcal{F}_{S'}\) is pure along \(X_{s'}\), then \(\mathcal{F}\) is pure along \(X_s\).

2. If \(\mathcal{F}_{S'}\) is universally pure along \(X_{s'}\), then \(\mathcal{F}\) is universally pure along \(X_s\).

Proof

Let \((T \to S, t' \leadsto t, \xi)\) be an impurity of \(\mathcal{F}\) above \(s\). Set \(T_1 = T \times_S S'\), and let \(t_1\) be the unique point of \(T_1\) mapping to \(t\) and \(s'\). Since \(T_1 \to T\) is flat at \(t_1\), see Morphisms, Lemma 01U9, there exists a specialization \(t'_1 \leadsto t_1\) lying over \(t' \leadsto t\), see Algebra, Section 00HU. Choose a point \(\xi_1 \in X_{t'_1}\) which corresponds to a generic point of \(\Spec(\kappa(t'_1) \otimes_{\kappa(t')} \kappa(\xi))\), see Schemes, Lemma 01JT. By Divisors, Lemma 05DC we see that \(\xi_1 \in \text{Ass}_{X_{T_1}/T_1}(\mathcal{F}_{T_1})\). As the Zariski closure of \(\{\xi_1\}\) in \(X_{T_1}\) maps into the Zariski closure of \(\{\xi\}\) in \(X_T\) we conclude that this closure is disjoint from \(X_{t_1}\). Hence \((T_1 \to S', t'_1 \leadsto t_1, \xi_1)\) is an impurity of \(\mathcal{F}_{S'}\) above \(s'\). In other words we have proved the contrapositive to part (2) of the lemma. Finally, if \((T, t) \to (S, s)\) is an elementary étale neighbourhood, then \((T_1, t_1) \to (S', s')\) is an elementary étale neighbourhood too, and in this way we see that (1) holds.

Lemma

Let \(i : Z \to X\) be a closed immersion of schemes of finite type over a scheme \(S\). Let \(s \in S\). Let \(\mathcal{F}\) be a finite type, quasi-coherent sheaf on \(Z\). Then \(\mathcal{F}\) is (universally) pure along \(Z_s\) if and only if \(i_*\mathcal{F}\) is (universally) pure along \(X_s\).

Proof

This follows from Divisors, Lemma 0CUD.

Examples of relatively pure sheaves

Here are some example cases where it is possible to see what purity means.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module.

1. If the support of \(\mathcal{F}\) is proper over \(S\), then \(\mathcal{F}\) is universally pure relative to \(S\).

2. If \(f\) is proper, then \(\mathcal{F}\) is universally pure relative to \(S\).

3. If \(f\) is proper, then \(X\) is universally pure relative to \(S\).

Proof

First we reduce (1) to (2). Namely, let \(Z \subset X\) be the scheme theoretic support of \(\mathcal{F}\). Let \(i : Z \to X\) be the corresponding closed immersion and write \(\mathcal{F} = i_*\mathcal{G}\) for some finite type quasi-coherent \(\mathcal{O}_Z\)-module \(\mathcal{G}\), see Morphisms, Section 056H. In case (1) \(Z \to S\) is proper by assumption. Thus by Lemma 05K1 case (1) reduces to case (2).

Assume \(f\) is proper. Let \((g : T \to S, t' \leadsto t, \xi)\) be an impurity of \(\mathcal{F}\) above \(s \in S\). Since \(f\) is proper, it is universally closed. Hence \(f_T : X_T \to T\) is closed. Since \(f_T(\xi) = t'\) this implies that \(t \in f(\overline{\{\xi\}})\) which is a contradiction.

Lemma

Let \(f : X \to S\) be a separated, finite type morphism of schemes. Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. Assume that \(\text{Supp}(\mathcal{F}_s)\) is finite for every \(s \in S\). Then the following are equivalent

1. \(\mathcal{F}\) is pure relative to \(S\),

2. the scheme theoretic support of \(\mathcal{F}\) is finite over \(S\), and

3. \(\mathcal{F}\) is universally pure relative to \(S\).

In particular, given a quasi-finite separated morphism \(X \to S\) we see that \(X\) is pure relative to \(S\) if and only if \(X \to S\) is finite.

Proof

Let \(Z \subset X\) be the scheme theoretic support of \(\mathcal{F}\), see Morphisms, Definition 05JV. Then \(Z \to S\) is a separated, finite type morphism of schemes with finite fibres. Hence it is separated and quasi-finite, see Morphisms, Lemma 02NH. By Lemma 05K1 it suffices to prove the lemma for \(Z \to S\) and the sheaf \(\mathcal{F}\) viewed as a finite type quasi-coherent module on \(Z\). Hence we may assume that \(X \to S\) is separated and quasi-finite and that \(\text{Supp}(\mathcal{F}) = X\).

It follows from Lemma 05K3 and Morphisms, Lemma 01WN that (2) implies (3). Trivially (3) implies (1). Assume (1) holds. We will prove that (2) holds. It is clear that we may assume \(S\) is affine. By More on Morphisms, Lemma 05K0 we can find a diagram \[\xymatrix{ X \ar[rd]_f \ar[rr]_j & & T \ar[ld]^\pi \\ & S & }\] with \(\pi\) finite and \(j\) a quasi-compact open immersion. If we show that \(j\) is closed, then \(j\) is a closed immersion and we conclude that \(f = \pi \circ j\) is finite. To show that \(j\) is closed it suffices to show that specializations lift along \(j\), see Schemes, Lemma 01K9. Let \(x \in X\), set \(t' = j(x)\) and let \(t' \leadsto t\) be a specialization. We have to show \(t \in j(X)\). Set \(s' = f(x)\) and \(s = \pi(t)\) so \(s' \leadsto s\). By More on Morphisms, Lemma 02LN we can find an elementary étale neighbourhood \((U, u) \to (S, s)\) and a decomposition \[T_U = T \times_S U = V \amalg W\] into open and closed subschemes, such that \(V \to U\) is finite and there exists a unique point \(v\) of \(V\) mapping to \(u\), and such that \(v\) maps to \(t\) in \(T\). As \(V \to T\) is étale, we can lift generalizations, see Morphisms, Lemmas 03HV and 02GS. Hence there exists a specialization \(v' \leadsto v\) such that \(v'\) maps to \(t' \in T\). In particular we see that \(v' \in X_U \subset T_U\). Denote \(u' \in U\) the image of \(t'\). Note that \(v' \in \text{Ass}_{X_U/U}(\mathcal{F})\) because \(X_{u'}\) is a finite discrete set and \(X_{u'} = \text{Supp}(\mathcal{F}_{u'})\). As \(\mathcal{F}\) is pure relative to \(S\) we see that \(v'\) must specialize to a point in \(X_u\). Since \(v\) is the only point of \(V\) lying over \(u\) (and since no point of \(W\) can be a specialization of \(v'\)) we see that \(v \in X_u\). Hence \(t \in X\).

Lemma

Let \(f : X \to S\) be a finite type, flat morphism of schemes with geometrically integral fibres. Then \(X\) is universally pure over \(S\).

Proof

Let \(\xi \in X\) with \(s' = f(\xi)\) and \(s' \leadsto s\) a specialization of \(S\). If \(\xi\) is an associated point of \(X_{s'}\), then \(\xi\) is the unique generic point because \(X_{s'}\) is an integral scheme. Let \(\xi_0\) be the unique generic point of \(X_s\). As \(X \to S\) is flat we can lift \(s' \leadsto s\) to a specialization \(\xi' \leadsto \xi_0\) in \(X\), see Morphisms, Lemma 03HV. The \(\xi \leadsto \xi'\) because \(\xi\) is the generic point of \(X_{s'}\) hence \(\xi \leadsto \xi_0\). This means that \((\text{id}_S, s' \to s, \xi)\) is not an impurity of \(\mathcal{O}_X\) above \(s\). Since the assumption that \(f\) is finite type, flat with geometrically integral fibres is preserved under base change, we see that there doesn’t exist an impurity after any base change. In this way we see that \(X\) is universally \(S\)-pure.

Lemma

Let \(f : X \to S\) be a finite type, affine morphism of schemes. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module such that \(f_*\mathcal{F}\) is locally projective on \(S\), see Properties, Definition 05JP. Then \(\mathcal{F}\) is universally pure over \(S\).

Proof

After reducing to the case where \(S\) is the spectrum of a henselian local ring this follows from Lemma 05FV.

A criterion for purity

We first prove that given a flat family of finite type quasi-coherent sheaves the points in the relative assassin specialize to points in the relative assassins of nearby fibres (if they specialize at all).

Lemma

Let \(f : X \to S\) be a morphism of schemes of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. Let \(s \in S\). Assume that \(\mathcal{F}\) is flat over \(S\) at all points of \(X_s\). Let \(x' \in \text{Ass}_{X/S}(\mathcal{F})\) with \(f(x') = s'\) such that \(s' \leadsto s\) is a specialization in \(S\). If \(x'\) specializes to a point of \(X_s\), then \(x' \leadsto x\) with \(x \in \text{Ass}_{X_s}(\mathcal{F}_s)\).

Proof

Say \(x' \leadsto t\) with \(t \in X_s\). Then we can find specializations \(x' \leadsto x \leadsto t\) with \(x\) corresponding to a generic point of an irreducible component of \(\overline{\{x'\}} \cap f^{-1}(\{s\})\). By assumption \(\mathcal{F}\) is flat over \(S\) at \(x\). By More on Morphisms, Lemma 0GSI we see that \(x \in \text{Ass}_{X/S}(\mathcal{F})\) as desired.

Lemma

Let \(f : X \to S\) be a morphism of schemes of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. Let \(s \in S\). Let \((S', s') \to (S, s)\) be an elementary étale neighbourhood and let \[\xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & S' \ar[l] }\] be a commutative diagram of morphisms of schemes. Assume

1. \(\mathcal{F}\) is flat over \(S\) at all points of \(X_s\),

2. \(X' \to S'\) is of finite type,

3. \(g^*\mathcal{F}\) is pure along \(X'_{s'}\),

4. \(g : X' \to X\) is étale, and

5. \(g(X')\) contains \(\text{Ass}_{X_s}(\mathcal{F}_s)\).

In this situation \(\mathcal{F}\) is pure along \(X_s\) if and only if the image of \(X' \to X \times_S S'\) contains the points of \(\text{Ass}_{X \times_S S'/S'}(\mathcal{F} \times_S S')\) lying over points in \(S'\) which specialize to \(s'\).

Proof

Since the morphism \(S' \to S\) is étale, we see that if \(\mathcal{F}\) is pure along \(X_s\), then \(\mathcal{F} \times_S S'\) is pure along \(X_s\), see Lemma 05J7. Since purity satisfies flat descent, see Lemma 05J9, we see that if \(\mathcal{F} \times_S S'\) is pure along \(X_{s'}\), then \(\mathcal{F}\) is pure along \(X_s\). Hence we may replace \(S\) by \(S'\) and assume that \(S = S'\) so that \(g : X' \to X\) is an étale morphism between schemes of finite type over \(S\). Moreover, we may replace \(S\) by \(\Spec(\mathcal{O}_{S, s})\) and assume that \(S\) is local.

First, assume that \(\mathcal{F}\) is pure along \(X_s\). In this case every point of \(\text{Ass}_{X/S}(\mathcal{F})\) specializes to a point of \(X_s\) by purity. Hence by Lemma 05L3 we see that every point of \(\text{Ass}_{X/S}(\mathcal{F})\) specializes to a point of \(\text{Ass}_{X_s}(\mathcal{F}_s)\). Thus every point of \(\text{Ass}_{X/S}(\mathcal{F})\) is in the image of \(g\) (as the image is open and contains \(\text{Ass}_{X_s}(\mathcal{F}_s)\)).

Conversely, assume that \(g(X')\) contains \(\text{Ass}_{X/S}(\mathcal{F})\). Let \(S^h = \Spec(\mathcal{O}_{S, s}^h)\) be the henselization of \(S\) at \(s\). Denote \(g^h : (X')^h \to X^h\) the base change of \(g\) by \(S^h \to S\), and denote \(\mathcal{F}^h\) the pullback of \(\mathcal{F}\) to \(X^h\). By Divisors, Lemma 05DC and Remark 05KL the relative assassin \(\text{Ass}_{X^h/S^h}(\mathcal{F}^h)\) is the inverse image of \(\text{Ass}_{X/S}(\mathcal{F})\) via the projection \(X^h \to X\). As we have assumed that \(g(X')\) contains \(\text{Ass}_{X/S}(\mathcal{F})\) we conclude that the base change \(g^h((X')^h) = g(X') \times_S S^h\) contains \(\text{Ass}_{X^h/S^h}(\mathcal{F}^h)\). In this way we reduce to the case where \(S\) is the spectrum of a henselian local ring. Let \(x \in \text{Ass}_{X/S}(\mathcal{F})\). To finish the proof of the lemma we have to show that \(x\) specializes to a point of \(X_s\), see criterion (4) for purity in discussion following Definition 05J4. By assumption there exists a \(x' \in X'\) such that \(g(x') = x\). As \(g : X' \to X\) is étale, we see that \(x' \in \text{Ass}_{X'/S}(g^*\mathcal{F})\), see Lemma 05FP (applied to the morphism of fibres \(X'_w \to X_w\) where \(w \in S\) is the image of \(x'\)). Since \(g^*\mathcal{F}\) is pure along \(X'_s\) we see that \(x' \leadsto y\) for some \(y \in X'_s\). Hence \(x = g(x') \leadsto g(y)\) and \(g(y) \in X_s\) as desired.

Lemma

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

1. \(f\) is of finite type,

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

3. \(\mathcal{F}\) is flat over \(S\) at all points of \(X_s\), and

4. \(\mathcal{F}\) is pure along \(X_s\).

Then \(\mathcal{F}\) is universally pure along \(X_s\).

Proof

We first make a preliminary remark. Suppose that \((S', s') \to (S, s)\) is an elementary étale neighbourhood. Denote \(\mathcal{F}'\) the pullback of \(\mathcal{F}\) to \(X' = X \times_S S'\). By the discussion following Definition 05J4 we see that \(\mathcal{F}'\) is pure along \(X'_{s'}\). Moreover, \(\mathcal{F}'\) is flat over \(S'\) along \(X'_{s'}\). Then it suffices to prove that \(\mathcal{F}'\) is universally pure along \(X'_{s'}\). Namely, given any morphism \((T, t) \to (S, s)\) of pointed schemes the fibre product \((T', t') = (T \times_S S', (t, s'))\) is flat over \((T, t)\) and hence if \(\mathcal{F}_{T'}\) is pure along \(X_{t'}\) then \(\mathcal{F}_T\) is pure along \(X_t\) by Lemma 05J9. Thus during the proof we may always replace \((s, S)\) by an elementary étale neighbourhood. We may also replace \(S\) by \(\Spec(\mathcal{O}_{S, s})\) due to the local nature of the problem.

Choose an elementary étale neighbourhood \((S', s') \to (S, s)\) and a commutative diagram \[\xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & \Spec(\mathcal{O}_{S', s'}) \ar[l] }\] such that \(X' \to X \times_S \Spec(\mathcal{O}_{S', s'})\) is étale, \(X_s = g((X')_{s'})\), the scheme \(X'\) is affine, and such that \(\Gamma(X', g^*\mathcal{F})\) is a free \(\mathcal{O}_{S', s'}\)-module, see Lemma 05L1. Note that \(X' \to \Spec(\mathcal{O}_{S', s'})\) is of finite type (as a quasi-compact morphism which is the composition of an étale morphism and the base change of a finite type morphism). By our preliminary remarks in the first paragraph of the proof we may replace \(S\) by \(\Spec(\mathcal{O}_{S', s'})\). Hence we may assume there exists a commutative diagram \[\xymatrix{ X \ar[dr] & & X' \ar[ll]^g \ar[ld] \\ & S & }\] of schemes of finite type over \(S\), where \(g\) is étale, \(X_s \subset g(X')\), with \(S\) local with closed point \(s\), with \(X'\) affine, and with \(\Gamma(X', g^*\mathcal{F})\) a free \(\Gamma(S, \mathcal{O}_S)\)-module. Note that in this case \(g^*\mathcal{F}\) is universally pure over \(S\), see Lemma 05K6.

In this situation we apply Lemma 05L4 to deduce that \(\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')\) from our assumption that \(\mathcal{F}\) is pure along \(X_s\) and flat over \(S\) along \(X_s\). By Divisors, Lemma 05DC and Remark 05KL we see that for any morphism of pointed schemes \((T, t) \to (S, s)\) we have \[\text{Ass}_{X_T/T}(\mathcal{F}_T) \subset (X_T \to X)^{-1}(\text{Ass}_{X/S}(\mathcal{F})) \subset g(X') \times_S T = g_T(X'_T).\] Hence by Lemma 05L4 applied to the base change of our displayed diagram to \((T, t)\) we conclude that \(\mathcal{F}_T\) is pure along \(X_t\) as desired.

Lemma

Let \(f : X \to S\) be a finite type morphism of schemes. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Assume \(\mathcal{F}\) is flat over \(S\). In this case \(\mathcal{F}\) is pure relative to \(S\) if and only if \(\mathcal{F}\) is universally pure relative to \(S\).

Proof

Immediate consequence of Lemma 05L5 and the definitions.

Lemma

Let \(I\) be a directed set. Let \((S_i, g_{ii'})\) be an inverse system of affine schemes over \(I\). Set \(S = \lim_i S_i\) and \(s \in S\). Denote \(g_i : S \to S_i\) the projections and set \(s_i = g_i(s)\). Suppose that \(f : X \to S\) is a morphism of finite presentation, \(\mathcal{F}\) a quasi-coherent \(\mathcal{O}_X\)-module of finite presentation which is pure along \(X_s\) and flat over \(S\) at all points of \(X_s\). Then there exists an \(i \in I\), a morphism of finite presentation \(X_i \to S_i\), a quasi-coherent \(\mathcal{O}_{X_i}\)-module \(\mathcal{F}_i\) of finite presentation which is pure along \((X_i)_{s_i}\) and flat over \(S_i\) at all points of \((X_i)_{s_i}\) such that \(X \cong X_i \times_{S_i} S\) and such that the pullback of \(\mathcal{F}_i\) to \(X\) is isomorphic to \(\mathcal{F}\).

Proof

Let \(U \subset X\) be the set of points where \(\mathcal{F}\) is flat over \(S\). By More on Morphisms, Theorem 0399 this is an open subscheme of \(X\). By assumption \(X_s \subset U\). As \(X_s\) is quasi-compact, we can find a quasi-compact open \(U' \subset U\) with \(X_s \subset U'\). By Limits, Lemma 01ZM we can find an \(i \in I\) and a morphism of finite presentation \(f_i : X_i \to S_i\) whose base change to \(S\) is isomorphic to \(f_i\). Fix such a choice and set \(X_{i'} = X_i \times_{S_i} S_{i'}\). Then \(X = \lim_{i'} X_{i'}\) with affine transition morphisms. By Limits, Lemma 01ZR we can, after possible increasing \(i\) assume there exists a quasi-coherent \(\mathcal{O}_{X_i}\)-module \(\mathcal{F}_i\) of finite presentation whose base change to \(S\) is isomorphic to \(\mathcal{F}\). By Limits, Lemma 01Z4 after possibly increasing \(i\) we may assume there exists an open \(U'_i \subset X_i\) whose inverse image in \(X\) is \(U'\). Note that in particular \((X_i)_{s_i} \subset U'_i\). By Limits, Lemma 05LY (after increasing \(i\) once more) we may assume that \(\mathcal{F}_i\) is flat on \(U'_i\). In particular we see that \(\mathcal{F}_i\) is flat along \((X_i)_{s_i}\).

Next, we use Lemma 05KW to choose an elementary étale neighbourhood \((S_i', s_i') \to (S_i, s_i)\) and a commutative diagram of schemes \[\xymatrix{ X_i \ar[d] & X_i' \ar[l]^{g_i} \ar[d] \\ S_i & S_i' \ar[l] }\] such that \(g_i\) is étale, \((X_i)_{s_i} \subset g_i(X_i')\), the schemes \(X_i'\), \(S_i'\) are affine, and such that \(\Gamma(X_i', g_i^*\mathcal{F}_i)\) is a projective \(\Gamma(S_i', \mathcal{O}_{S_i'})\)-module. Note that \(g_i^*\mathcal{F}_i\) is universally pure over \(S'_i\), see Lemma 05K6. We may base change the diagram above to a diagram with morphisms \((S'_{i'}, s'_{i'}) \to (S_{i'}, s_{i'})\) and \(g_{i'} : X'_{i'} \to X_{i'}\) over \(S_{i'}\) for any \(i' \geq i\) and we may base change the diagram to a diagram with morphisms \((S', s') \to (S, s)\) and \(g : X' \to X\) over \(S\).

At this point we can use our criterion for purity. Set \(W'_i \subset X_i \times_{S_i} S'_i\) equal to the image of the étale morphism \(X'_i \to X_i \times_{S_i} S'_i\). For every \(i' \geq i\) we have similarly the image \(W'_{i'} \subset X_{i'} \times_{S_{i'}} S'_{i'}\) and we have the image \(W' \subset X \times_S S'\). Taking images commutes with base change, hence \(W'_{i'} = W'_i \times_{S'_i} S'_{i'}\) and \(W' = W_i \times_{S'_i} S'\). Because \(\mathcal{F}\) is pure along \(X_s\) the Lemma 05L4 implies that [05MB]\[\begin{equation} f^{-1}(\Spec(\mathcal{O}_{S', s'})) \cap \text{Ass}_{X \times_S S'/S'}(\mathcal{F} \times_S S') \subset W' \end{equation}\] By More on Morphisms, Lemma 05KR we see that \[E = \{t \in S' \mid \text{Ass}_{X_t}(\mathcal{F}_t) \subset W' \} \quad\text{and}\quad E_{i'} = \{t \in S'_{i'} \mid \text{Ass}_{X_t}(\mathcal{F}_{i', t}) \subset W'_{i'} \}\] are locally constructible subsets of \(S'\) and \(S'_{i'}\). By More on Morphisms, Lemma 05KQ we see that \(E_{i'}\) is the inverse image of \(E_i\) under the morphism \(S'_{i'} \to S'_i\) and that \(E\) is the inverse image of \(E_i\) under the morphism \(S' \to S'_i\). Thus Equation (05MB) is equivalent to the assertion that \(\Spec(\mathcal{O}_{S', s'})\) maps into \(E_i\). As \(\mathcal{O}_{S', s'} = \colim_{i' \geq i} \mathcal{O}_{S'_{i'}, s'_{i'}}\) we see that \(\Spec(\mathcal{O}_{S'_{i'}, s'_{i'}})\) maps into \(E_i\) for some \(i' \geq i\), see Limits, Lemma 05F4. Then, applying Lemma 05L4 to the situation over \(S_{i'}\), we conclude that \(\mathcal{F}_{i'}\) is pure along \((X_{i'})_{s_{i'}}\).

Lemma

Let \(f : X \to S\) be a morphism of finite presentation. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite presentation flat over \(S\). Then the set \[U = \{s \in S \mid \mathcal{F}\text{ is pure along }X_s\}\] is open in \(S\).

Proof

Let \(s \in U\). Using Lemma 05KW we can find an elementary étale neighbourhood \((S', s') \to (S, s)\) and a commutative diagram \[\xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & S' \ar[l] }\] such that \(g\) is étale, \(X_s \subset g(X')\), the schemes \(X'\), \(S'\) are affine, and such that \(\Gamma(X', g^*\mathcal{F})\) is a projective \(\Gamma(S', \mathcal{O}_{S'})\)-module. Note that \(g^*\mathcal{F}\) is universally pure over \(S'\), see Lemma 05K6. Set \(W' \subset X \times_S S'\) equal to the image of the étale morphism \(X' \to X \times_S S'\). Note that \(W\) is open and quasi-compact over \(S'\). Set \[E = \{t \in S' \mid \text{Ass}_{X_t}(\mathcal{F}_t) \subset W' \}.\] By More on Morphisms, Lemma 05KR \(E\) is a constructible subset of \(S'\). By Lemma 05L4 we see that \(\Spec(\mathcal{O}_{S', s'}) \subset E\). By Morphisms, Lemma 05LW we see that \(E\) contains an open neighbourhood \(V'\) of \(s'\). Applying Lemma 05L4 once more we see that for any point \(s_1\) in the image of \(V'\) in \(S\) the sheaf \(\mathcal{F}\) is pure along \(X_{s_1}\). Since \(S' \to S\) is étale the image of \(V'\) in \(S\) is open and we win.

How purity is used

Here are some examples of how purity can be used. The first lemma actually uses a slightly weaker form of purity.

Lemma

Let \(f : X \to S\) be a morphism of finite type. Let \(\mathcal{F}\) be a quasi-coherent sheaf of finite type on \(X\). Assume \(S\) is local with closed point \(s\). Assume \(\mathcal{F}\) is pure along \(X_s\) and that \(\mathcal{F}\) is flat over \(S\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) of quasi-coherent \(\mathcal{O}_X\)-modules. Then the following are equivalent

1. the map on stalks \(\varphi_x\) is injective for all \(x \in \text{Ass}_{X_s}(\mathcal{F}_s)\), and

2. \(\varphi\) is injective.

Proof

Let \(\mathcal{K} = \Ker(\varphi)\). Our goal is to prove that \(\mathcal{K} = 0\). In order to do this it suffices to prove that \(\text{WeakAss}_X(\mathcal{K}) = \emptyset\), see Divisors, Lemma 05AP. We have \(\text{WeakAss}_X(\mathcal{K}) \subset \text{WeakAss}_X(\mathcal{F})\), see Divisors, Lemma 05AN. As \(\mathcal{F}\) is flat we see from Lemma 05IJ that \(\text{WeakAss}_X(\mathcal{F}) \subset \text{Ass}_{X/S}(\mathcal{F})\). By purity any point \(x'\) of \(\text{Ass}_{X/S}(\mathcal{F})\) is a generalization of a point of \(X_s\), and hence is the specialization of a point \(x \in \text{Ass}_{X_s}(\mathcal{F}_s)\), by Lemma 05L3. Hence the injectivity of \(\varphi_x\) implies the injectivity of \(\varphi_{x'}\), whence \(\mathcal{K}_{x'} = 0\).

Proposition

Let \(f : X \to S\) be an affine, finitely presented morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite presentation, flat over \(S\). Then the following are equivalent

1. \(f_*\mathcal{F}\) is locally projective on \(S\), and

2. \(\mathcal{F}\) is pure relative to \(S\).

In particular, given a ring map \(A \to B\) of finite presentation and a finitely presented \(B\)-module \(N\) flat over \(A\) we have: \(N\) is projective as an \(A\)-module if and only if \(\widetilde{N}\) on \(\Spec(B)\) is pure relative to \(\Spec(A)\).

Proof

The implication (1) \(\Rightarrow\) (2) is Lemma 05K6. Assume \(\mathcal{F}\) is pure relative to \(S\). Note that by Lemma 05L5 this implies \(\mathcal{F}\) remains pure after any base change. By Descent, Lemma 05JZ it suffices to prove \(f_*\mathcal{F}\) is fpqc locally projective on \(S\). Pick \(s \in S\). We will prove that the restriction of \(f_*\mathcal{F}\) to an étale neighbourhood of \(s\) is locally projective. Namely, by Lemma 05KW, after replacing \(S\) by an affine elementary étale neighbourhood of \(s\), we may assume there exists a diagram \[\xymatrix{ X \ar[dr] & & X' \ar[ll]^g \ar[ld] \\ & S & }\] of schemes affine and of finite presentation over \(S\), where \(g\) is étale, \(X_s \subset g(X')\), and with \(\Gamma(X', g^*\mathcal{F})\) a projective \(\Gamma(S, \mathcal{O}_S)\)-module. Note that in this case \(g^*\mathcal{F}\) is universally pure over \(S\), see Lemma 05K6. Hence by Lemma 05L4 we see that the open \(g(X')\) contains the points of \(\text{Ass}_{X/S}(\mathcal{F})\) lying over \(\Spec(\mathcal{O}_{S, s})\). Set \[E = \{t \in S \mid \text{Ass}_{X_t}(\mathcal{F}_t) \subset g(X') \}.\] By More on Morphisms, Lemma 05KR \(E\) is a constructible subset of \(S\). We have seen that \(\Spec(\mathcal{O}_{S, s}) \subset E\). By Morphisms, Lemma 05LW we see that \(E\) contains an open neighbourhood of \(s\). Hence after replacing \(S\) by an affine neighbourhood of \(s\) we may assume that \(\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')\). By Lemma 05DH this means that \[\Gamma(X, \mathcal{F}) \longrightarrow \Gamma(X', g^*\mathcal{F})\] is \(\Gamma(S, \mathcal{O}_S)\)-universally injective. By Algebra, Lemma 059N we conclude that \(\Gamma(X, \mathcal{F})\) is Mittag-Leffler as an \(\Gamma(S, \mathcal{O}_S)\)-module. Since \(\Gamma(X, \mathcal{F})\) is countably generated and flat as a \(\Gamma(S, \mathcal{O}_S)\)-module, we conclude it is projective by Algebra, Lemma 059X.

We can use the proposition to improve some of our earlier results. The following lemma is an improvement of Proposition 05ID.

Lemma

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 of finite presentation. Let \(x \in X\) with \(s = f(x) \in S\). If \(\mathcal{F}\) is flat at \(x\) over \(S\) there exists an affine elementary étale neighbourhood \((S', s') \to (S, s)\) and an affine open \(U' \subset X \times_S S'\) which contains \(x' = (x, s')\) such that \(\Gamma(U', \mathcal{F}|_{U'})\) is a projective \(\Gamma(S', \mathcal{O}_{S'})\)-module.

Proof

During the proof we may replace \(X\) by an open neighbourhood of \(x\) and we may replace \(S\) by an elementary étale neighbourhood of \(s\). Hence, by openness of flatness (see More on Morphisms, Theorem 0399) we may assume that \(\mathcal{F}\) is flat over \(S\). We may assume \(S\) and \(X\) are affine. After shrinking \(X\) some more we may assume that any point of \(\text{Ass}_{X_s}(\mathcal{F}_s)\) is a generalization of \(x\). This property is preserved on replacing \((S, s)\) by an elementary étale neighbourhood. Hence we may apply Lemma 05KW to arrive at the situation where there exists a diagram \[\xymatrix{ X \ar[dr] & & X' \ar[ll]^g \ar[ld] \\ & S & }\] of schemes affine and of finite presentation over \(S\), where \(g\) is étale, \(X_s \subset g(X')\), and with \(\Gamma(X', g^*\mathcal{F})\) a projective \(\Gamma(S, \mathcal{O}_S)\)-module. Note that in this case \(g^*\mathcal{F}\) is universally pure over \(S\), see Lemma 05K6.

Let \(U \subset g(X')\) be an affine open neighbourhood of \(x\). We claim that \(\mathcal{F}|_U\) is pure along \(U_s\). If we prove this, then the lemma follows because \(\mathcal{F}|_U\) will be pure relative to \(S\) after shrinking \(S\), see Lemma 05MC, whereupon the projectivity follows from Proposition 05MD. To prove the claim we have to show, after replacing \((S, s)\) by an arbitrary elementary étale neighbourhood, that any point \(\xi\) of \(\text{Ass}_{U/S}(\mathcal{F}|_U)\) lying over some \(s' \in S\), \(s' \leadsto s\) specializes to a point of \(U_s\). Since \(U \subset g(X')\) we can find a \(\xi' \in X'\) with \(g(\xi') = \xi\). Because \(g^*\mathcal{F}\) is pure over \(S\), using Lemma 05L3, we see there exists a specialization \(\xi' \leadsto x'\) with \(x' \in \text{Ass}_{X'_s}(g^*\mathcal{F}_s)\). Then \(g(x') \in \text{Ass}_{X_s}(\mathcal{F}_s)\) (see for example Lemma 05FP applied to the étale morphism \(X'_s \to X_s\) of Noetherian schemes) and hence \(g(x') \leadsto x\) by our choice of \(X\) above! Since \(x \in U\) we conclude that \(g(x') \in U\). Thus \(\xi = g(\xi') \leadsto g(x') \in U_s\) as desired.

The following lemma is an improvement of Lemma 05KZ.

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 which is of finite type. Let \(x \in X\) with \(s = f(x) \in S\). If \(\mathcal{F}\) is flat at \(x\) over \(S\) there exists an affine elementary étale neighbourhood \((S', s') \to (S, s)\) and an affine open \(U' \subset X \times_S \Spec(\mathcal{O}_{S', s'})\) which contains \(x' = (x, s')\) such that \(\Gamma(U', \mathcal{F}|_{U'})\) is a free \(\mathcal{O}_{S', s'}\)-module.

Proof

The question is Zariski local on \(X\) and \(S\). Hence we may assume that \(X\) and \(S\) are affine. Then we can find a closed immersion \(i : X \to \mathbf{A}^n_S\) over \(S\). It is clear that it suffices to prove the lemma for the sheaf \(i_*\mathcal{F}\) on \(\mathbf{A}^n_S\) and the point \(i(x)\). In this way we reduce to the case where \(X \to S\) is of finite presentation. After replacing \(S\) by \(\Spec(\mathcal{O}_{S', s'})\) and \(X\) by an open of \(X \times_S \Spec(\mathcal{O}_{S', s'})\) we may assume that \(\mathcal{F}\) is of finite presentation, see Proposition 05I5. In this case we may appeal to Lemma 05ME and Algebra, Theorem 0593 to conclude.

Lemma

Let \(A \to B\) be a local ring map of local rings which is essentially of finite type. Let \(N\) be a finite \(B\)-module which is flat as an \(A\)-module. If \(A\) is henselian, then \(N\) is a filtered colimit \[N = \colim_i F_i\] of free \(A\)-modules \(F_i\) such that all transition maps \(u_i : F_i \to F_{i'}\) of the system induce injective maps \(\overline{u}_i : F_i/\mathfrak m_AF_i \to F_{i'}/\mathfrak m_AF_{i'}\). Also, \(N\) is a Mittag-Leffler \(A\)-module.

Proof

We can find a morphism of finite type \(X \to S = \Spec(A)\) and a point \(x \in X\) lying over the closed point \(s\) of \(S\) and a finite type quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) such that \(\mathcal{F}_x \cong N\) as an \(A\)-module. After shrinking \(X\) we may assume that each point of \(\text{Ass}_{X_s}(\mathcal{F}_s)\) specializes to \(x\). By Lemma 05MF we see that there exists a fundamental system of affine open neighbourhoods \(U_i \subset X\) of \(x\) such that \(\Gamma(U_i, \mathcal{F})\) is a free \(A\)-module \(F_i\). Note that if \(U_{i'} \subset U_i\), then \[F_i/\mathfrak m_AF_i = \Gamma(U_{i, s}, \mathcal{F}_s) \longrightarrow \Gamma(U_{i', s}, \mathcal{F}_s) = F_{i'}/\mathfrak m_AF_{i'}\] is injective because a section of the kernel would be supported at a closed subset of \(X_s\) not meeting \(x\) which is a contradiction to our choice of \(X\) above. Since the maps \(F_i \to F_{i'}\) are \(A\)-universally injective (Lemma 05FQ) it follows that \(N\) is Mittag-Leffler by Algebra, Lemma 0AS7.

The following lemma should be skipped if reading through for the first time.

Lemma

Let \(A \to B\) be a local ring map of local rings which is essentially of finite type. Let \(N\) be a finite \(B\)-module which is flat as an \(A\)-module. If \(A\) is a valuation ring, then any element of \(N\) has a content ideal \(I \subset A\) (More on Algebra, Definition 0ASA). Also, \(I\) is a principal ideal.

Proof

The final statement follows from the fact that \(I\) is a finitely generated ideal by More on Algebra, Lemma 0ASB and Algebra, Lemma 090Q.

Proof of existence of \(I\). Let \(A \subset A^h\) be the henselization. Let \(B'\) be the localization of \(B \otimes_A A^h\) at the maximal ideal \(\mathfrak m_B \otimes A^h + B \otimes \mathfrak m_{A^h}\). Then \(B \to B'\) is flat, hence faithfully flat. Let \(N' = N \otimes_B B'\). Let \(x \in N\) and let \(x' \in N'\) be the image. We claim that for an ideal \(I \subset A\) we have \(x \in IN \Leftrightarrow x' \in IN'\). Namely, \(N/IN \to N'/IN'\) is the tensor product of \(B \to B'\) with \(N/IN\) and \(B \to B'\) is universally injective by Algebra, Lemma 05CK. By More on Algebra, Lemma 0ASK and Algebra, Lemma 00IH the map \(A \to A^h\) defines an inclusion preserving bijection \(I \mapsto IA^h\) on sets of ideals. We conclude that \(x\) has a content ideal in \(A\) if and only if \(x'\) has a content ideal in \(A^h\). The assertion for \(x' \in N'\) follows from Lemma 05U7 and Algebra, Lemma 0AS6.

An application is the following.

Lemma

Let \(X \to \Spec(R)\) be a proper flat morphism where \(R\) is a valuation ring. If the special fibre is reduced, then \(X\) and every fibre of \(X \to \Spec(R)\) is reduced.

Proof

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 \(X\) is 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 \(a \in A\) be nonzero and let \(I = (\pi) \subset R\) be its content ideal, see Lemma 0ASX. Then \(a = \pi a'\) and \(a'\) maps to a nonzero element of \(A/\mathfrak mA\) where \(\mathfrak m \subset R\) is the maximal ideal. If \(a\) is nilpotent, so is \(a'\), because \(\pi\) is a nonzerodivisor by flatness of \(A\) over \(R\). But \(a'\) maps to a nonzero element of the reduced ring \(A/\mathfrak m A = \mathcal{O}_{X_s, x'}\). This is a contradiction unless \(A\) is reduced, which is what we wanted to show.

Of course, if \(X\) is reduced, so is the generic fibre of \(X\) over \(R\). If \(\mathfrak p \subset R\) is a prime ideal, then \(R/\mathfrak p\) is a valuation ring by Algebra, Lemma 088Y. Hence redoing the argument with the base change of \(X\) to \(R/\mathfrak p\) proves the fibre over \(\mathfrak p\) is reduced.

Flattening functors

Let \(S\) be a scheme. Recall that a functor \(F : (\Sch/S)^{opp} \to \textit{Sets}\) is called limit preserving if for every directed inverse system \(\{T_i\}_{i \in I}\) of affine schemes with limit \(T\) we have \(F(T) = \colim_i F(T_i)\).

Situation

Let \(f : X \to S\) be a morphism of schemes. Let \(u : \mathcal{F} \to \mathcal{G}\) be a homomorphism of quasi-coherent \(\mathcal{O}_X\)-modules. For any scheme \(T\) over \(S\) we will denote \(u_T : \mathcal{F}_T \to \mathcal{G}_T\) the base change of \(u\) to \(T\), in other words, \(u_T\) is the pullback of \(u\) via the projection morphism \(X_T = X \times_S T \to X\). In this situation we can consider the functor [05MI]\[\begin{equation} F_{iso} : (\Sch/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{*\} & \text{if} & u_T \text{ is an isomorphism}, \\ \emptyset & \text{else.} & \end{matrix} \right. \end{equation}\] There are variants \(F_{inj}\), \(F_{surj}\), \(F_{zero}\) where we ask that \(u_T\) is injective, surjective, or zero.

Lemma

In Situation 05MH.

1. Each of the functors \(F_{iso}\), \(F_{inj}\), \(F_{surj}\), \(F_{zero}\) satisfies the sheaf property for the fpqc topology.

2. If \(f\) is quasi-compact and \(\mathcal{G}\) is of finite type, then \(F_{surj}\) is limit preserving.

3. If \(f\) is quasi-compact and \(\mathcal{F}\) of finite type, then \(F_{zero}\) is limit preserving.

4. If \(f\) is quasi-compact, \(\mathcal{F}\) is of finite type, and \(\mathcal{G}\) is of finite presentation, then \(F_{iso}\) is limit preserving.

Proof

Let \(\{T_i \to T\}_{i \in I}\) be an fpqc covering of schemes over \(S\). Set \(X_i = X_{T_i} = X \times_S T_i\) and \(u_i = u_{T_i}\). Note that \(\{X_i \to X_T\}_{i \in I}\) is an fpqc covering of \(X_T\), see Topologies, Lemma 022D. In particular, for every \(x \in X_T\) there exists an \(i \in I\) and an \(x_i \in X_i\) mapping to \(x\). Since \(\mathcal{O}_{X_T, x} \to \mathcal{O}_{X_i, x_i}\) is flat, hence faithfully flat (see Algebra, Lemma 00HR) we conclude that \((u_i)_{x_i}\) is injective, surjective, bijective, or zero if and only if \((u_T)_x\) is injective, surjective, bijective, or zero. Whence part (1) of the lemma.

Proof of (2). Assume \(f\) quasi-compact and \(\mathcal{G}\) of finite type. Let \(T = \lim_{i \in I} T_i\) be a directed limit of affine \(S\)-schemes and assume that \(u_T\) is surjective. Set \(X_i = X_{T_i} = X \times_S T_i\) and \(u_i = u_{T_i} : \mathcal{F}_i = \mathcal{F}_{T_i} \to \mathcal{G}_i = \mathcal{G}_{T_i}\). To prove part (2) we have to show that \(u_i\) is surjective for some \(i\). Pick \(i_0 \in I\) and replace \(I\) by \(\{i \mid i \geq i_0\}\). Since \(f\) is quasi-compact the scheme \(X_{i_0}\) is quasi-compact. Hence we may choose affine opens \(W_1, \ldots, W_m \subset X\) and an affine open covering \(X_{i_0} = U_{1, i_0} \cup \ldots \cup U_{m, i_0}\) such that \(U_{j, i_0}\) maps into \(W_j\) under the projection morphism \(X_{i_0} \to X\). For any \(i \in I\) let \(U_{j, i}\) be the inverse image of \(U_{j, i_0}\). Setting \(U_j = \lim_i U_{j, i}\) we see that \(X_T = U_1 \cup \ldots \cup U_m\) is an affine open covering of \(X_T\). Now it suffices to show, for a given \(j \in \{1, \ldots, m\}\) that \(u_i|_{U_{j, i}}\) is surjective for some \(i = i(j) \in I\). Using Properties, Lemma 01PB this translates into the following algebra problem: Let \(A\) be a ring and let \(u : M \to N\) be an \(A\)-module map. Suppose that \(R = \colim_{i \in I} R_i\) is a directed colimit of \(A\)-algebras. If \(N\) is a finite \(A\)-module and if \(u \otimes 1 : M \otimes_A R \to N \otimes_A R\) is surjective, then for some \(i\) the map \(u \otimes 1 : M \otimes_A R_i \to N \otimes_A R_i\) is surjective. This is Algebra, Lemma 05LI part (2).

Proof of (3). Exactly the same arguments as given in the proof of (2) reduces this to the following algebra problem: Let \(A\) be a ring and let \(u : M \to N\) be an \(A\)-module map. Suppose that \(R = \colim_{i \in I} R_i\) is a directed colimit of \(A\)-algebras. If \(M\) is a finite \(A\)-module and if \(u \otimes 1 : M \otimes_A R \to N \otimes_A R\) is zero, then for some \(i\) the map \(u \otimes 1 : M \otimes_A R_i \to N \otimes_A R_i\) is zero. This is Algebra, Lemma 05LI part (1).

Proof of (4). Assume \(f\) quasi-compact and \(\mathcal{F}, \mathcal{G}\) of finite presentation. Arguing in exactly the same manner as in the previous paragraph (using in addition also Properties, Lemma 01PC) part (3) translates into the following algebra statement: Let \(A\) be a ring and let \(u : M \to N\) be an \(A\)-module map. Suppose that \(R = \colim_{i \in I} R_i\) is a directed colimit of \(A\)-algebras. Assume \(M\) is a finite \(A\)-module, \(N\) is a finitely presented \(A\)-module, and \(u \otimes 1 : M \otimes_A R \to N \otimes_A R\) is an isomorphism. Then for some \(i\) the map \(u \otimes 1 : M \otimes_A R_i \to N \otimes_A R_i\) is an isomorphism. This is Algebra, Lemma 05LI part (3).

Situation

Let \((A, \mathfrak m_A)\) be a local ring. Denote \(\mathcal{C}\) the category whose objects are \(A\)-algebras \(A'\) which are local rings such that the algebra structure \(A \to A'\) is a local homomorphism of local rings. A morphism between objects \(A', A''\) of \(\mathcal{C}\) is a local homomorphism \(A' \to A''\) of \(A\)-algebras. Let \(A \to B\) be a local ring map of local rings and let \(M\) be a \(B\)-module. If \(A'\) is an object of \(\mathcal{C}\) we set \(B' = B \otimes_A A'\) and we set \(M' = M \otimes_A A'\) as a \(B'\)-module. Given \(A' \in \Ob(\mathcal{C})\), consider the condition [05ML]\[\begin{equation} \forall \mathfrak q \in V(\mathfrak m_{A'}B' + \mathfrak m_B B') \subset \Spec(B') : M'_{\mathfrak q}\text{ is flat over }A'. \end{equation}\] Note the similarity with More on Algebra, Equation (05LQ). In particular, if \(A' \to A''\) is a morphism of \(\mathcal{C}\) and (05ML) holds for \(A'\), then it holds for \(A''\), see More on Algebra, Lemma 05LR. Hence we obtain a functor [05MM]\[\begin{equation} F_{lf} : \mathcal{C} \longrightarrow \textit{Sets}, \quad A' \longrightarrow \left\{ \begin{matrix} \{*\} & \text{if }(\href{flat.html#flat-equation-flat-at-primes}{05ML})\text{ holds}, \\ \emptyset & \text{else.} & \end{matrix} \right. \end{equation}\]

Lemma

In Situation 05MK.

1. If \(A' \to A''\) is a flat morphism in \(\mathcal{C}\) then \(F_{lf}(A') = F_{lf}(A'')\).

2. If \(A \to B\) is essentially of finite presentation and \(M\) is a \(B\)-module of finite presentation, then \(F_{lf}\) is limit preserving: If \(\{A_i\}_{i \in I}\) is a directed system of objects of \(\mathcal{C}\), then \(F_{lf}(\colim_i A_i) = \colim_i F_{lf}(A_i)\).

Proof

Part (1) is a special case of More on Algebra, Lemma 05LS. Part (2) is a special case of More on Algebra, Lemma 05LT.

Lemma

In Situation 05MK. Let \(B \to C\) is a local map of local \(A\)-algebras and \(N\) a \(C\)-module. Denote \(F'_{lf} : \mathcal{C} \to \textit{Sets}\) the functor associated to the pair \((C, N)\). If \(M \cong N\) as \(B\)-modules and \(B \to C\) is finite, then \(F_{lf} = F'_{lf}\).

Proof

Let \(A'\) be an object of \(\mathcal{C}\). Set \(C' = C \otimes_A A'\) and \(N' = N \otimes_A A'\) similarly to the definitions of \(B'\), \(M'\) in Situation 05MK. Note that \(M' \cong N'\) as \(B'\)-modules. The assumption that \(B \to C\) is finite has two consequences: (a) \(\mathfrak m_C = \sqrt{\mathfrak m_B C}\) and (b) \(B' \to C'\) is finite. Consequence (a) implies that \[V(\mathfrak m_{A'}C' + \mathfrak m_C C') = \left( \Spec(C') \to \Spec(B') \right)^{-1}V(\mathfrak m_{A'}B' + \mathfrak m_B B').\] Suppose \(\mathfrak q \subset V(\mathfrak m_{A'}B' + \mathfrak m_B B')\). Then \(M'_{\mathfrak q}\) is flat over \(A'\) if and only if the \(C'_{\mathfrak q}\)-module \(N'_{\mathfrak q}\) is flat over \(A'\) (because these are isomorphic as \(A'\)-modules) if and only if for every maximal ideal \(\mathfrak r\) of \(C'_{\mathfrak q}\) the module \(N'_{\mathfrak r}\) is flat over \(A'\) (see Algebra, Lemma 00HT). As \(B'_{\mathfrak q} \to C'_{\mathfrak q}\) is finite by (b), the maximal ideals of \(C'_{\mathfrak q}\) correspond exactly to the primes of \(C'\) lying over \(\mathfrak q\) (see Algebra, Lemma 00GU) and these primes are all contained in \(V(\mathfrak m_{A'}C' + \mathfrak m_C C')\) by the displayed equation above. Thus the result of the lemma holds.

Lemma

In Situation 05MK suppose that \(B \to C\) is a flat local homomorphism of local rings. Set \(N = M \otimes_B C\). Denote \(F'_{lf} : \mathcal{C} \to \textit{Sets}\) the functor associated to the pair \((C, N)\). Then \(F_{lf} = F'_{lf}\).

Proof

Let \(A'\) be an object of \(\mathcal{C}\). Set \(C' = C \otimes_A A'\) and \(N' = N \otimes_A A' = M' \otimes_{B'} C'\) similarly to the definitions of \(B'\), \(M'\) in Situation 05MK. Note that \[V(\mathfrak m_{A'}B' + \mathfrak m_B B') = \Spec( \kappa(\mathfrak m_B) \otimes_A \kappa(\mathfrak m_{A'}) )\] and similarly for \(V(\mathfrak m_{A'}C' + \mathfrak m_C C')\). The ring map \[\kappa(\mathfrak m_B) \otimes_A \kappa(\mathfrak m_{A'}) \longrightarrow \kappa(\mathfrak m_C) \otimes_A \kappa(\mathfrak m_{A'})\] is faithfully flat, hence \(V(\mathfrak m_{A'}C' + \mathfrak m_C C') \to V(\mathfrak m_{A'}B' + \mathfrak m_B B')\) is surjective. Finally, if \(\mathfrak r \in V(\mathfrak m_{A'}C' + \mathfrak m_C C')\) maps to \(\mathfrak q \in V(\mathfrak m_{A'}B' + \mathfrak m_B B')\), then \(M'_{\mathfrak q}\) is flat over \(A'\) if and only if \(N'_{\mathfrak r}\) is flat over \(A'\) because \(B' \to C'\) is flat, see Algebra, Lemma 0584. The lemma follows formally from these remarks.

Situation

Let \(f : X \to S\) be a smooth morphism with geometrically irreducible fibres. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. For any scheme \(T\) over \(S\) we will denote \(\mathcal{F}_T\) the base change of \(\mathcal{F}\) to \(T\), in other words, \(\mathcal{F}_T\) is the pullback of \(\mathcal{F}\) via the projection morphism \(X_T = X \times_S T \to X\). Note that \(X_T \to T\) is smooth with geometrically irreducible fibres, see Morphisms, Lemma 01VB and More on Morphisms, Lemma 0555. Let \(p \geq 0\) be an integer. Given a point \(t \in T\) consider the condition [05MQ]\[\begin{equation} \mathcal{F}_T \text{ is free of rank }p\text{ in a neighbourhood of }\xi_t \end{equation}\] where \(\xi_t\) is the generic point of the fibre \(X_t\). This condition for all \(t \in T\) is stable under base change, and hence we obtain a functor [05MR]\[\begin{equation} H_p : (\Sch/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ satisfies (\href{flat.html#flat-equation-free-at-generic-point-fibre}{05MQ}) }\forall t\in T, \\ \emptyset & \text{else.} \end{matrix} \right. \end{equation}\]

Lemma

In Situation 05MP.

1. The functor \(H_p\) satisfies the sheaf property for the fpqc topology.

2. If \(\mathcal{F}\) is of finite presentation, then functor \(H_p\) is limit preserving.

Proof

Let \(\{T_i \to T\}_{i \in I}\) be an fpqc¹ covering of schemes over \(S\). Set \(X_i = X_{T_i} = X \times_S T_i\) and denote \(\mathcal{F}_i\) the pullback of \(\mathcal{F}\) to \(X_i\). Assume that \(\mathcal{F}_i\) satisfies (05MQ) for all \(i\). Pick \(t \in T\) and let \(\xi_t \in X_T\) denote the generic point of \(X_t\). We have to show that \(\mathcal{F}\) is free in a neighbourhood of \(\xi_t\). For some \(i \in I\) we can find a \(t_i \in T_i\) mapping to \(t\). Let \(\xi_i \in X_i\) denote the generic point of \(X_{t_i}\), so that \(\xi_i\) maps to \(\xi_t\). The fact that \(\mathcal{F}_i\) is free of rank \(p\) in a neighbourhood of \(\xi_i\) implies that \((\mathcal{F}_i)_{x_i} \cong \mathcal{O}_{X_i, x_i}^{\oplus p}\) which implies that \(\mathcal{F}_{T, \xi_t} \cong \mathcal{O}_{X_T, \xi_t}^{\oplus p}\) as \(\mathcal{O}_{X_T, \xi_t} \to \mathcal{O}_{X_i, x_i}\) is flat, see for example Algebra, Lemma 00O1. Thus there exists an affine neighbourhood \(U\) of \(\xi_t\) in \(X_T\) and a surjection \(\mathcal{O}_U^{\oplus p} \to \mathcal{F}_U = \mathcal{F}_T|_U\), see Modules, Lemma 01B8. After shrinking \(T\) we may assume that \(U \to T\) is surjective. Hence \(U \to T\) is a smooth morphism of affines with geometrically irreducible fibres. Moreover, for every \(t' \in T\) we see that the induced map \[\alpha : \mathcal{O}_{U, \xi_{t'}}^{\oplus p} \longrightarrow \mathcal{F}_{U, \xi_{t'}}\] is an isomorphism (since by the same argument as before the module on the right is free of rank \(p\)). It follows from Lemma 05I3 that \[\Gamma(U, \mathcal{O}_U^{\oplus p}) \otimes_{\Gamma(T, \mathcal{O}_T)} \mathcal{O}_{T, t'} \longrightarrow \Gamma(U, \mathcal{F}_U) \otimes_{\Gamma(T, \mathcal{O}_T)} \mathcal{O}_{T, t'}\] is injective for every \(t' \in T\). Hence we see the surjection \(\alpha\) is an isomorphism. This finishes the proof of (1).

Assume that \(\mathcal{F}\) is of finite presentation. Let \(T = \lim_{i \in I} T_i\) be a directed limit of affine \(S\)-schemes and assume that \(\mathcal{F}_T\) satisfies (05MQ). Set \(X_i = X_{T_i} = X \times_S T_i\) and denote \(\mathcal{F}_i\) the pullback of \(\mathcal{F}\) to \(X_i\). Let \(U \subset X_T\) denote the open subscheme of points where \(\mathcal{F}_T\) is flat over \(T\), see More on Morphisms, Theorem 0399. By assumption every generic point of every fibre is a point of \(U\), i.e., \(U \to T\) is a smooth surjective morphism with geometrically irreducible fibres. We may shrink \(U\) a bit and assume that \(U\) is quasi-compact. Using Limits, Lemma 01Z4 we can find an \(i \in I\) and a quasi-compact open \(U_i \subset X_i\) whose inverse image in \(X_T\) is \(U\). After increasing \(i\) we may assume that \(\mathcal{F}_i|_{U_i}\) is flat over \(T_i\), see Limits, Lemma 05LY. In particular, \(\mathcal{F}_i|_{U_i}\) is finite locally free hence defines a locally constant rank function \(\rho : U_i \to \{0, 1, 2, \ldots \}\). Let \((U_i)_p \subset U_i\) denote the open and closed subset where \(\rho\) has value \(p\). Let \(V_i \subset T_i\) be the image of \((U_i)_p\); note that \(V_i\) is open and quasi-compact. By assumption the image of \(T \to T_i\) is contained in \(V_i\). Hence there exists an \(i' \geq i\) such that \(T_{i'} \to T_i\) factors through \(V_i\) by Limits, Lemma 01Z4. Then \(\mathcal{F}_{i'}\) satisfies (05MQ) as desired. Some details omitted.

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 \(n \geq 0\). The following are equivalent

1. for \(s \in S\) the closed subset \(Z \subset X_s\) of points where \(\mathcal{F}\) is not flat over \(S\) (see Lemma 05M9) satisfies \(\dim(Z) < n\), and

2. for \(x \in X\) such that \(\mathcal{F}\) is not flat at \(x\) over \(S\) we have \(\text{trdeg}_{\kappa(f(x))}(\kappa(x)) < n\).

If this is true, then it remains true after any base change.

Proof

Let \(x \in X\) be a point over \(s \in S\). Then the dimension of the closure of \(\{x\}\) in \(X_s\) is \(\text{trdeg}_{\kappa(s)}(\kappa(x))\) by Varieties, Lemma 0A21. Conversely, if \(Z \subset X_s\) is a closed subset of dimension \(d\), then there exists a point \(x \in Z\) with \(\text{trdeg}_{\kappa(s)}(\kappa(x)) = d\) (same reference). Therefore the equivalence of (1) and (2) holds (even fibre by fibre). The statement on base change follows from Morphisms, Lemmas 01U8 and 02FY.

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 of finite type. Let \(n \geq 0\). We say \(\mathcal{F}\) is flat over \(S\) in dimensions \(\geq n\) if the equivalent conditions of Lemma 0CWF are satisfied.

Situation

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. For any scheme \(T\) over \(S\) we will denote \(\mathcal{F}_T\) the base change of \(\mathcal{F}\) to \(T\), in other words, \(\mathcal{F}_T\) is the pullback of \(\mathcal{F}\) via the projection morphism \(X_T = X \times_S T \to X\). Note that \(X_T \to T\) is of finite type and that \(\mathcal{F}_T\) is an \(\mathcal{O}_{X_T}\)-module of finite type (Morphisms, Lemma 01T4 and Modules, Lemma 01B6). Let \(n \geq 0\). By Definition 0CWG and Lemma 0CWF we obtain a functor [05MU]\[\begin{equation} F_n : (\Sch/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ is flat over }T\text{ in }\dim \geq n, \\ \emptyset & \text{else.} \end{matrix} \right. \end{equation}\]

Lemma

In Situation 05MT.

1. The functor \(F_n\) satisfies the sheaf property for the fpqc topology.

2. If \(f\) is quasi-compact and locally of finite presentation and \(\mathcal{F}\) is of finite presentation, then the functor \(F_n\) is limit preserving.

Proof

Let \(\{T_i \to T\}_{i \in I}\) be an fpqc covering of schemes over \(S\). Set \(X_i = X_{T_i} = X \times_S T_i\) and denote \(\mathcal{F}_i\) the pullback of \(\mathcal{F}\) to \(X_i\). Assume that \(\mathcal{F}_i\) is flat over \(T_i\) in dimensions \(\geq n\) for all \(i\). Let \(t \in T\). Choose an index \(i\) and a point \(t_i \in T_i\) mapping to \(t\). Consider the cartesian diagram \[\xymatrix{ X_{\Spec(\mathcal{O}_{T, t})} \ar[d] & X_{\Spec(\mathcal{O}_{T_i, t_i})} \ar[d] \ar[l] \\ \Spec(\mathcal{O}_{T, t}) & \Spec(\mathcal{O}_{T_i, t_i}) \ar[l] }\] As the lower horizontal morphism is flat we see from More on Morphisms, Lemma 047C that the set \(Z_i \subset X_{t_i}\) where \(\mathcal{F}_i\) is not flat over \(T_i\) and the set \(Z \subset X_t\) where \(\mathcal{F}_T\) is not flat over \(T\) are related by the rule \(Z_i = Z_{\kappa(t_i)}\). Hence we see that \(\mathcal{F}_T\) is flat over \(T\) in dimensions \(\geq n\) by Morphisms, Lemma 02FY.

Assume that \(f\) is quasi-compact and locally of finite presentation and that \(\mathcal{F}\) is of finite presentation. In this paragraph we first reduce the proof of (2) to the case where \(f\) is of finite presentation. Let \(T = \lim_{i \in I} T_i\) be a directed limit of affine \(S\)-schemes and assume that \(\mathcal{F}_T\) is flat in dimensions \(\geq n\). Set \(X_i = X_{T_i} = X \times_S T_i\) and denote \(\mathcal{F}_i\) the pullback of \(\mathcal{F}\) to \(X_i\). We have to show that \(\mathcal{F}_i\) is flat in dimensions \(\geq n\) for some \(i\). Pick \(i_0 \in I\) and replace \(I\) by \(\{i \mid i \geq i_0\}\). Since \(T_{i_0}\) is affine (hence quasi-compact) there exist finitely many affine opens \(W_j \subset S\), \(j = 1, \ldots, m\) and an affine open overing \(T_{i_0} = \bigcup_{j = 1, \ldots, m} V_{j, i_0}\) such that \(T_{i_0} \to S\) maps \(V_{j, i_0}\) into \(W_j\). For \(i \geq i_0\) denote \(V_{j, i}\) the inverse image of \(V_{j, i_0}\) in \(T_i\). If we can show, for each \(j\), that there exists an \(i\) such that \(\mathcal{F}_{V_{j, i_0}}\) is flat in dimensions \(\geq n\), then we win. In this way we reduce to the case that \(S\) is affine. In this case \(X\) is quasi-compact and we can choose a finite affine open covering \(X = W_1 \cup \ldots \cup W_m\). In this case the result for \((X \to S, \mathcal{F})\) is equivalent to the result for \((\coprod W_j, \coprod \mathcal{F}|_{W_j})\). Hence we may assume that \(f\) is of finite presentation.

Assume \(f\) is of finite presentation and \(\mathcal{F}\) is of finite presentation. Let \(U \subset X_T\) denote the open subscheme of points where \(\mathcal{F}_T\) is flat over \(T\), see More on Morphisms, Theorem 0399. By assumption the dimension of every fibre of \(Z = X_T \setminus U\) over \(T\) has dimension \(< n\). By Limits, Lemma 05M6 we can find a closed subscheme \(Z \subset Z' \subset X_T\) such that \(\dim(Z'_t) < n\) for all \(t \in T\) and such that \(Z' \to X_T\) is of finite presentation. By Limits, Lemmas 01ZM and 01ZP there exists an \(i \in I\) and a closed subscheme \(Z'_i \subset X_i\) of finite presentation whose base change to \(T\) is \(Z'\). By Limits, Lemma 05M5 we may assume all fibres of \(Z'_i \to T_i\) have dimension \(< n\). By Limits, Lemma 05LY we may assume that \(\mathcal{F}_i|_{X_i \setminus T'_i}\) is flat over \(T_i\). This implies that \(\mathcal{F}_i\) is flat in dimensions \(\geq n\); here we use that \(Z' \to X_T\) is of finite presentation, and hence the complement \(X_T \setminus Z'\) is quasi-compact! Thus part (2) is proved and the proof of the lemma is complete.

Situation

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. For any scheme \(T\) over \(S\) we will denote \(\mathcal{F}_T\) the base change of \(\mathcal{F}\) to \(T\), in other words, \(\mathcal{F}_T\) is the pullback of \(\mathcal{F}\) via the projection morphism \(X_T = X \times_S T \to X\). Since the base change of a flat module is flat we obtain a functor [05MX]\[\begin{equation} F_{flat} : (\Sch/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{*\} & \text{if } \mathcal{F}_T \text{ is flat over }T, \\ \emptyset & \text{else.} \end{matrix} \right. \end{equation}\]

Lemma

In Situation 05MW.

1. The functor \(F_{flat}\) satisfies the sheaf property for the fpqc topology.

2. If \(f\) is quasi-compact and locally of finite presentation and \(\mathcal{F}\) is of finite presentation, then the functor \(F_{flat}\) is limit preserving.

Proof

Part (1) follows from the following statement: If \(T' \to T\) is a surjective flat morphism of schemes over \(S\), then \(\mathcal{F}_{T'}\) is flat over \(T'\) if and only if \(\mathcal{F}_T\) is flat over \(T\), see More on Morphisms, Lemma 047C. Part (2) follows from Limits, Lemma 05LY after reducing to the case where \(X\) and \(S\) are affine (compare with the proof of Lemma 05MV).

Flattening stratifications

Just the definitions. The reader looking for a “generic flatness stratification”, should consult More on Morphisms, Section 0H3Y.

Definition

Let \(X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. We say that the universal flattening of \(\mathcal{F}\) exists if the functor \(F_{flat}\) defined in Situation 05MW is representable by a scheme \(S'\) over \(S\). We say that the universal flattening of \(X\) exists if the universal flattening of \(\mathcal{O}_X\) exists.

Note that if the universal flattening \(S'\)² of \(\mathcal{F}\) exists, then the morphism \(S' \to S\) is a surjective monomorphism of schemes such that \(\mathcal{F}_{S'}\) is flat over \(S'\) and such that a morphism \(T \to S\) factors through \(S'\) if and only if \(\mathcal{F}_T\) is flat over \(T\).

Example

Let \(X = S = \Spec(k[x, y])\) where \(k\) is a field. Let \(\mathcal{F} = \widetilde{M}\) where \(M = k[x, x^{-1}, y]/(y)\). For a \(k[x, y]\)-algebra \(A\) set \(F_{flat}(A) = F_{flat}(\Spec(A))\). Then \(F_{flat}(k[x, y]/(x, y)^n) = \{*\}\) for all \(n\), while \(F_{flat}(k[[x, y]]) = \emptyset\). This means that \(F_{flat}\) isn’t representable (even by an algebraic space, see Formal Spaces, Lemma 0AQH). Thus the universal flattening does not exist in this case.

We define (compare with Topology, Remark 09Y2) a (locally finite, scheme theoretic) stratification of a scheme \(S\) to be given by closed subschemes \(Z_i \subset S\) indexed by a partially ordered set \(I\) such that \(S = \bigcup Z_i\) (set theoretically), such that every point of \(S\) has a neighbourhood meeting only a finite number of \(Z_i\), and such that \[Z_i \cap Z_j = \bigcup\nolimits_{k \leq i, j} Z_k.\] Setting \(S_i = Z_i \setminus \bigcup_{j < i} Z_j\) the actual stratification is the decomposition \(S = \coprod S_i\) into locally closed subschemes. We often only indicate the strata \(S_i\) and leave the construction of the closed subschemes \(Z_i\) to the reader. Given a stratification we obtain a monomorphism \[S' = \coprod\nolimits_{i \in I} S_i \longrightarrow S.\] We will call this the monomorphism associated to the stratification. With this terminology we can define what it means to have a flattening stratification.

Definition

Let \(X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. We say that \(\mathcal{F}\) has a flattening stratification if the functor \(F_{flat}\) defined in Situation 05MW is representable by a monomorphism \(S' \to S\) associated to a stratification of \(S\) by locally closed subschemes. We say that \(X\) has a flattening stratification if \(\mathcal{O}_X\) has a flattening stratification.

When a flattening stratification exists, it is often important to understand the index set labeling the strata and its partial ordering. This often has to do with ranks of modules. For example if \(X = S\) and \(\mathcal{F}\) is a finitely presented \(\mathcal{O}_S\)-module, then the flattening stratification exists and is given by the Fitting ideals of \(\mathcal{F}\), see Divisors, Lemma 05P9.

Flattening stratification over an Artinian ring

A flatting stratification exists when the base scheme is the spectrum of an Artinian ring.

Lemma

Let \(S\) be the spectrum of an Artinian ring. For any scheme \(X\) over \(S\), and any quasi-coherent \(\mathcal{O}_X\)-module there exists a universal flattening. In fact the universal flattening is given by a closed immersion \(S' \to S\), and hence is a flattening stratification for \(\mathcal{F}\) as well.

Proof

Choose an affine open covering \(X = \bigcup U_i\). Then \(F_{flat}\) is the product of the functors associated to each of the pairs \((U_i, \mathcal{F}|_{U_i})\). Hence it suffices to prove the result for each \((U_i, \mathcal{F}|_{U_i})\). In the affine case the lemma follows immediately from More on Algebra, Lemma 0525.

Flattening a map

Theorem 05PF is the key to further flattening statements.

Lemma

Let \(S\) be a scheme. Let \(g : X' \to X\) be a flat morphism of schemes over \(S\) with \(X\) locally of finite type over \(S\). Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module which is flat over \(S\). If \(\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')\) then the canonical map \[\mathcal{F} \longrightarrow g_*g^*\mathcal{F}\] is injective, and remains injective after any base change.

Proof

The final assertion means that \(\mathcal{F}_T \to (g_T)_*g_T^*\mathcal{F}_T\) is injective for any morphism \(T \to S\). The assumption \(\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')\) is preserved by base change, see Divisors, Lemma 05DC and Remark 05KL. The same holds for the assumption of flatness and finite type. Hence it suffices to prove the injectivity of the displayed arrow. Let \(\mathcal{K} = \Ker(\mathcal{F} \to g_*g^*\mathcal{F})\). Our goal is to prove that \(\mathcal{K} = 0\). In order to do this it suffices to prove that \(\text{WeakAss}_X(\mathcal{K}) = \emptyset\), see Divisors, Lemma 05AP. We have \(\text{WeakAss}_X(\mathcal{K}) \subset \text{WeakAss}_X(\mathcal{F})\), see Divisors, Lemma 05AN. As \(\mathcal{F}\) is flat we see from Lemma 05IJ that \(\text{WeakAss}_X(\mathcal{F}) \subset \text{Ass}_{X/S}(\mathcal{F})\). By assumption any point \(x\) of \(\text{Ass}_{X/S}(\mathcal{F})\) is the image of some \(x' \in X'\). Since \(g\) is flat the local ring map \(\mathcal{O}_{X, x} \to \mathcal{O}_{X', x'}\) is faithfully flat, hence the map \[\mathcal{F}_x \longrightarrow g^*\mathcal{F}_{x'} = \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{X', x'}\] is injective (see Algebra, Lemma 05CK). This implies that \(\mathcal{K}_x = 0\) as desired.

Lemma

Let \(A\) be a ring. Let \(u : M \to N\) be a surjective map of \(A\)-modules. If \(M\) is projective as an \(A\)-module, then there exists an ideal \(I \subset A\) such that for any ring map \(\varphi : A \to B\) the following are equivalent

1. \(u \otimes 1 : M \otimes_A B \to N \otimes_A B\) is an isomorphism, and

2. \(\varphi(I) = 0\).

Proof

As \(M\) is projective we can find a projective \(A\)-module \(C\) such that \(F = M \oplus C\) is a free \(A\)-module. By replacing \(u\) by \(u \oplus 1 : F = M \oplus C \to N \oplus C\) we see that we may assume \(M\) is free. In this case let \(I\) be the ideal of \(A\) generated by coefficients of all the elements of \(\Ker(u)\) with respect to some (fixed) basis of \(M\). The reason this works is that, since \(u\) is surjective and \(\otimes_A B\) is right exact, \(\Ker(u \otimes 1)\) is the image of \(\Ker(u) \otimes_A B\) in \(M \otimes_A B\).

Theorem

In Situation 05MH assume

1. \(f\) is of finite presentation,

2. \(\mathcal{F}\) is of finite presentation, flat over \(S\), and pure relative to \(S\), and

3. \(u\) is surjective.

Then \(F_{iso}\) is representable by a closed immersion \(Z \to S\). Moreover \(Z \to S\) is of finite presentation if \(\mathcal{G}\) is of finite presentation.

Proof

We will use without further mention that \(\mathcal{F}\) is universally pure over \(S\), see Lemma 05L5. By Lemma 05MJ and Descent, Lemmas 03I0 and 02W5 the question is local for the étale topology on \(S\). Hence it suffices to prove, given \(s \in S\), that there exists an étale neighbourhood of \((S, s)\) so that the theorem holds.

Using Lemma 05KW and after replacing \(S\) by an elementary étale neighbourhood of \(s\) we may assume there exists a commutative diagram \[\xymatrix{ X \ar[dr] & & X' \ar[ll]^g \ar[ld] \\ & S & }\] of schemes of finite presentation over \(S\), where \(g\) is étale, \(X_s \subset g(X')\), the schemes \(X'\) and \(S\) are affine, \(\Gamma(X', g^*\mathcal{F})\) a projective \(\Gamma(S, \mathcal{O}_S)\)-module. Note that \(g^*\mathcal{F}\) is universally pure over \(S\), see Lemma 05K6. Hence by Lemma 05L4 we see that the open \(g(X')\) contains the points of \(\text{Ass}_{X/S}(\mathcal{F})\) lying over \(\Spec(\mathcal{O}_{S, s})\). Set \[E = \{t \in S \mid \text{Ass}_{X_t}(\mathcal{F}_t) \subset g(X') \}.\] By More on Morphisms, Lemma 05KR \(E\) is a constructible subset of \(S\). We have seen that \(\Spec(\mathcal{O}_{S, s}) \subset E\). By Morphisms, Lemma 05LW we see that \(E\) contains an open neighbourhood of \(s\). Hence after replacing \(S\) by a smaller affine neighbourhood of \(s\) we may assume that \(\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')\).

Since we have assumed that \(u\) is surjective we have \(F_{iso} = F_{inj}\). From Lemma 05PD it follows that \(u : \mathcal{F} \to \mathcal{G}\) is injective if and only if \(g^*u : g^*\mathcal{F} \to g^*\mathcal{G}\) is injective, and the same remains true after any base change. Hence we have reduced to the case where, in addition to the assumptions in the theorem, \(X \to S\) is a morphism of affine schemes and \(\Gamma(X, \mathcal{F})\) is a projective \(\Gamma(S, \mathcal{O}_S)\)-module. This case follows immediately from Lemma 05PE.

To see that \(Z\) is of finite presentation if \(\mathcal{G}\) is of finite presentation, combine Lemma 05MJ part (4) with Limits, Remark 05LX.

Lemma

Let \(f:X\to S\) be a morphism of schemes which is of finite presentation, flat, and pure. Let \(Y\) be a closed subscheme of \(X\). Let \(F=f_*Y\) be the Weil restriction functor of \(Y\) along \(f\), defined by \[F : (\Sch/S)^{opp} \to \textit{Sets}, \quad T \mapsto \left\{ \begin{matrix} \{*\} & \text{if} & Y_T\to X_T \text{ is an isomorphism, }\\ \emptyset & \text{else.} & \end{matrix} \right.\] Then \(F\) is representable by a closed immersion \(Z\to S\). Moreover \(Z\to S\) is of finite presentation if \(Y\to S\) is.

Proof

Let \(\mathcal{I}\) be the ideal sheaf defining \(Y\) in \(X\) and let \(u:\mathcal{O}_X\to\mathcal{O}_X/\mathcal{I}\) be the surjection. Then for an \(S\)-scheme \(T\), the closed immersion \(Y_T\to X_T\) is an isomorphism if and only if \(u_T\) is an isomorphism. Hence the result follows from Theorem 05PF.

Flattening in the local case

In this section we start applying the earlier material to obtain a shadow of the flattening stratification.

Theorem

In Situation 05MK assume \(A\) is henselian, \(B\) is essentially of finite type over \(A\), and \(M\) is a finite \(B\)-module. Then there exists an ideal \(I \subset A\) such that \(A/I\) corepresents the functor \(F_{lf}\) on the category \(\mathcal{C}\). In other words given a local homomorphism of local rings \(\varphi : A \to A'\) with \(B' = B \otimes_A A'\) and \(M' = M \otimes_A A'\) the following are equivalent:

1. \(\forall \mathfrak q \in V(\mathfrak m_{A'}B' + \mathfrak m_B B') \subset \Spec(B') : M'_{\mathfrak q}\text{ is flat over }A'\), and

2. \(\varphi(I) = 0\).

If \(B\) is essentially of finite presentation over \(A\) and \(M\) of finite presentation over \(B\), then \(I\) is a finitely generated ideal.

Proof

Choose a finite type ring map \(A \to C\) and a finite \(C\)-module \(N\) and a prime \(\mathfrak q\) of \(C\) such that \(B = C_{\mathfrak q}\) and \(M = N_{\mathfrak q}\). In the following, when we say “the theorem holds for \((N/C/A, \mathfrak q)\) we mean that it holds for \((A \to B, M)\) where \(B = C_{\mathfrak q}\) and \(M = N_{\mathfrak q}\). By Lemma 05P5 the functor \(F_{lf}\) is unchanged if we replace \(B\) by a local ring flat over \(B\). Hence, since \(A\) is henselian, we may apply Lemma 05I1 and assume that there exists a complete dévissage of \(N/C/A\) at \(\mathfrak q\).

Let \((A_i, B_i, M_i, \alpha_i, \mathfrak q_i)_{i = 1, \ldots, n}\) be such a complete dévissage of \(N/C/A\) at \(\mathfrak q\). Let \(\mathfrak q'_i \subset A_i\) be the unique prime lying over \(\mathfrak q_i \subset B_i\) as in Definition 05HZ. Since \(C \to A_1\) is surjective and \(N \cong M_1\) as \(C\)-modules, we see by Lemma 05P4 it suffices to prove the theorem holds for \((M_1/A_1/A, \mathfrak q'_1)\). Since \(B_1 \to A_1\) is finite and \(\mathfrak q_1\) is the only prime of \(B_1\) over \(\mathfrak q'_1\) we see that \((A_1)_{\mathfrak q'_1} \to (B_1)_{\mathfrak q_1}\) is finite (see Algebra, Lemma 00EA or More on Morphisms, Lemma 05B8). Hence by Lemma 05P4 it suffices to prove the theorem holds for \((M_1/B_1/A, \mathfrak q_1)\).

At this point we may assume, by induction on the length \(n\) of the dévissage, that the theorem holds for \((M_2/B_2/A, \mathfrak q_2)\). (If \(n = 1\), then \(M_2 = 0\) which is flat over \(A\).) Reversing the last couple of steps of the previous paragraph, using that \(M_2 \cong \Coker(\alpha_2)\) as \(B_1\)-modules, we see that the theorem holds for \((\Coker(\alpha_1)/B_1/A, \mathfrak q_1)\).

Let \(A'\) be an object of \(\mathcal{C}\). At this point we use Lemma 05I3 to see that if \((M_1 \otimes_A A')_{\mathfrak q'}\) is flat over \(A'\) for a prime \(\mathfrak q'\) of \(B_1 \otimes_A A'\) lying over \(\mathfrak m_{A'}\), then \((\Coker(\alpha_1) \otimes_A A')_{\mathfrak q'}\) is flat over \(A'\). Hence we conclude that \(F_{lf}\) is a subfunctor of the functor \(F'_{lf}\) associated to the module \(\Coker(\alpha_1)_{\mathfrak q_1}\) over \((B_1)_{\mathfrak q_1}\). By the previous paragraph we know \(F'_{lf}\) is corepresented by \(A/J\) for some ideal \(J \subset A\). Hence we may replace \(A\) by \(A/J\) and assume that \(\Coker(\alpha_1)_{\mathfrak q_1}\) is flat over \(A\).

Since \(\Coker(\alpha_1)\) is a \(B_1\)-module for which there exist a complete dévissage of \(N_1/B_1/A\) at \(\mathfrak q_1\) and since \(\Coker(\alpha_1)_{\mathfrak q_1}\) is flat over \(A\) by Lemma 05I4 we see that \(\Coker(\alpha_1)\) is free as an \(A\)-module, in particular flat as an \(A\)-module. Hence Lemma 05I3 implies \(F_{lf}(A')\) is nonempty if and only if \(\alpha \otimes 1_{A'}\) is injective. Let \(N_1 = \Im(\alpha_1) \subset M_1\) so that we have exact sequences \[0 \to N_1 \to M_1 \to \Coker(\alpha_1) \to 0 \quad\text{and}\quad B_1^{\oplus r_1} \to N_1 \to 0\] The flatness of \(\Coker(\alpha_1)\) implies the first sequence is universally exact (see Algebra, Lemma 058M). Hence \(\alpha \otimes 1_{A'}\) is injective if and only if \(B_1^{\oplus r_1} \otimes_A A' \to N_1 \otimes_A A'\) is an isomorphism. Finally, Theorem 05PF applies to show this functor is corepresentable by \(A/I\) for some ideal \(I\) and we conclude \(F_{lf}\) is corepresentable by \(A/I\) also.

To prove the final statement, suppose that \(A \to B\) is essentially of finite presentation and \(M\) of finite presentation over \(B\). Let \(I \subset A\) be the ideal such that \(F_{lf}\) is corepresented by \(A/I\). Write \(I = \bigcup I_\lambda\) where \(I_\lambda\) ranges over the finitely generated ideals contained in \(I\). Then, since \(F_{lf}(A/I) = \{*\}\) we see that \(F_{lf}(A/I_\lambda) = \{*\}\) for some \(\lambda\), see Lemma 05MN part (2). Clearly this implies that \(I = I_\lambda\).

Remark

Here is a scheme theoretic reformulation of Theorem 05PG. Let \((X, x) \to (S, s)\) be a morphism of pointed schemes which is locally of finite type. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Assume \(S\) henselian local with closed point \(s\). There exists a closed subscheme \(Z \subset S\) with the following property: for any morphism of pointed schemes \((T, t) \to (S, s)\) the following are equivalent

1. \(\mathcal{F}_T\) is flat over \(T\) at all points of the fibre \(X_t\) which map to \(x \in X_s\), and

2. \(\Spec(\mathcal{O}_{T, t}) \to S\) factors through \(Z\).

Moreover, if \(X \to S\) is of finite presentation at \(x\) and \(\mathcal{F}_x\) of finite presentation over \(\mathcal{O}_{X, x}\), then \(Z \to S\) is of finite presentation.