Introduction
In this chapter we continue our study of properties of morphisms of algebraic spaces. A fundamental reference is [Kn].
Conventions
The standing assumption is that all schemes are contained in a big fppf site \(\Sch_{fppf}\). And all rings \(A\) considered have the property that \(\Spec(A)\) is (isomorphic) to an object of this big site.
Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). In this chapter and the following we will write \(X \times_S X\) for the product of \(X\) with itself (in the category of algebraic spaces over \(S\)), instead of \(X \times X\).
Radicial morphisms
It turns out that a radicial morphism is not the same thing as a universally injective morphism, contrary to what happens with morphisms of schemes. In fact it is a bit stronger.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). We say \(f\) is radicial if for any morphism \(\Spec(K) \to Y\) where \(K\) is a field the reduction \((\Spec(K) \times_Y X)_{red}\) is either empty or representable by the spectrum of a purely inseparable field extension of \(K\).
Lemma
A radicial morphism of algebraic spaces is universally injective.
Proof
Let \(S\) be a scheme. Let \(f : X \to Y\) be a radicial morphism of algebraic spaces over \(S\). It is clear from the definition that given a morphism \(\Spec(K) \to Y\) there is at most one lift of this morphism to a morphism into \(X\). Hence we conclude that \(f\) is universally injective by Morphisms of Spaces, Lemma 040X.
Example
It is no longer true that universally injective is equivalent to radicial. For example the morphism \[X = [\Spec(\overline{\mathbf{Q}})/ \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})] \longrightarrow S = \Spec(\mathbf{Q})\] of Spaces, Example 02Z6 is universally injective, but is not radicial in the sense above.
Nonetheless it is often the case that the reverse implication holds.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a universally injective morphism of algebraic spaces over \(S\).
If \(f\) is decent then \(f\) is radicial.
If \(f\) is quasi-separated then \(f\) is radicial.
If \(f\) is locally separated then \(f\) is radicial.
Proof
Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces which is stable under base change and composition and holds for closed immersions. Assume \(f : X \to Y\) has \(\mathcal{P}\) and is universally injective. Then, in the situation of Definition 0481 the morphism \((\Spec(K) \times_Y X)_{red} \to \Spec(K)\) is universally injective and has \(\mathcal{P}\). This reduces the problem of proving \[\mathcal{P} + \text{universally injective} \Rightarrow \text{radicial}\] to the problem of proving that any nonempty reduced algebraic space \(X\) over field whose structure morphism \(X \to \Spec(K)\) is universally injective and \(\mathcal{P}\) is representable by the spectrum of a field. Namely, then \(X \to \Spec(K)\) will be a morphism of schemes and we conclude by the equivalence of radicial and universally injective for morphisms of schemes, see Morphisms, Lemma 01S4.
Let us prove (1). Assume \(f\) is decent and universally injective. By Decent Spaces, Lemmas 03L0, 03L1, and 03M5 (to see that an immersion is decent) we see that the discussion in the first paragraph applies. Let \(X\) be a nonempty decent reduced algebraic space universally injective over a field \(K\). In particular we see that \(|X|\) is a singleton. By Decent Spaces, Lemma 047Z we conclude that \(X \cong \Spec(L)\) for some extension \(K \subset L\) as desired.
A quasi-separated morphism is decent, see Decent Spaces, Lemma 03M5. Hence (1) implies (2).
Let us prove (3). Recall that the separation axioms are stable under base change and composition and that closed immersions are separated, see Morphisms of Spaces, Lemmas 03KL, 03KQ, and 042R. Thus the discussion in the first paragraph of the proof applies. Let \(X\) be a reduced algebraic space universally injective and locally separated over a field \(K\). In particular \(|X|\) is a singleton hence \(X\) is quasi-compact, see Properties of Spaces, Lemma 03E4. We can find a surjective étale morphism \(U \to X\) with \(U\) affine, see Properties of Spaces, Lemma 03H6. Consider the morphism of schemes \[j : U \times_X U \longrightarrow U \times_{\Spec(K)} U\] As \(X \to \Spec(K)\) is universally injective \(j\) is surjective, and as \(X \to \Spec(K)\) is locally separated \(j\) is an immersion. A surjective immersion is a closed immersion, see Schemes, Lemma 01IQ. Hence \(R = U \times_X U\) is affine as a closed subscheme of an affine scheme. In particular \(R\) is quasi-compact. It follows that \(X = U/R\) is quasi-separated, and the result follows from (2).
Remark
Let \(X \to Y\) be a morphism of algebraic spaces. For some applications (of radicial morphisms) it is enough to require that for every \(\Spec(K) \to Y\) where \(K\) is a field
the space \(|\Spec(K) \times_Y X|\) is a singleton,
there exists a monomorphism \(\Spec(L) \to \Spec(K) \times_Y X\), and
\(K \subset L\) is purely inseparable.
If needed later we will may call such a morphism weakly radicial. For example if \(X \to Y\) is a surjective weakly radicial morphism then \(X(k) \to Y(k)\) is surjective for every algebraically closed field \(k\). Note that the base change \(X_{\overline{\mathbf{Q}}} \to \Spec(\overline{\mathbf{Q}})\) of the morphism in Example 0483 is weakly radicial, but not radicial. The analogue of Lemma 0484 is that if \(X \to Y\) has property (\(\beta\)) and is universally injective, then it is weakly radicial (proof omitted).
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume
\(f\) is locally of finite type,
for every étale morphism \(V \to Y\) the map \(|X \times_Y V| \to |V|\) is injective.
Then \(f\) is universally injective.
Proof
The question is étale local on \(Y\) by Morphisms of Spaces, Lemma 03MX. Hence we may assume that \(Y\) is a scheme. Then \(Y\) is in particular decent and by Decent Spaces, Lemma 0ACB we see that \(f\) is locally quasi-finite. Let \(y \in Y\) be a point and let \(X_y\) be the scheme theoretic fibre. Assume \(X_y\) is not empty. By Spaces over Fields, Lemma 06S0 we see that \(X_y\) is a scheme which is locally quasi-finite over \(\kappa(y)\). Since \(|X_y| \subset |X|\) is the fibre of \(|X| \to |Y|\) over \(y\) we see that \(X_y\) has a unique point \(x\). The same is true for \(X_y \times_{\Spec(\kappa(y))} \Spec(k)\) for any finite separable extension \(k/\kappa(y)\) because we can realize \(k\) as the residue field at a point lying over \(y\) in an étale scheme over \(Y\), see More on Morphisms, Lemma 02LF. Thus \(X_y\) is geometrically connected, see Varieties, Lemma 0389. This implies that the finite extension \(\kappa(x)/\kappa(y)\) is purely inseparable.
We conclude (in the case that \(Y\) is a scheme) that for every \(y \in Y\) either the fibre \(X_y\) is empty, or \((X_y)_{red} = \Spec(\kappa(x))\) with \(\kappa(y) \subset \kappa(x)\) purely inseparable. Hence \(f\) is radicial (some details omitted), whence universally injective by Lemma 0482.
Monomorphisms
This section is the continuation of Morphisms of Spaces, Section 042K. We would like to know whether or not every monomorphism of algebraic spaces is representable. If you can prove this is true or have a counterexample, please email stacks.project@gmail.com. For the moment this is known in the following cases
for monomorphisms which are locally of finite type (more generally any separated, locally quasi-finite morphism is representable by Morphisms of Spaces, Lemma 0418 and a monomorphism which is locally of finite type is locally quasi-finite by Morphisms of Spaces, Lemma 0463),
if the target is a disjoint union of spectra of zero dimensional local rings (Decent Spaces, Lemma 06RZ), and
for flat monomorphisms (see below).
Lemma
A flat monomorphism of algebraic spaces is representable by schemes.
Proof
Let \(f : X \to Y\) be a flat monomorphism of algebraic spaces. To prove \(f\) is representable, we have to show \(X \times_Y V\) is a scheme for every scheme \(V\) mapping to \(Y\). Since being a scheme is local (Properties of Spaces, Lemma 03JH), we may assume \(V\) is affine. Thus we may assume \(Y = \Spec(B)\) is an affine scheme. Next, we can assume that \(X\) is quasi-compact by replacing \(X\) by a quasi-compact open. The space \(X\) is separated as \(X \to X \times_{\Spec(B)} X\) is an isomorphism. Applying Limits of Spaces, Lemma 0B80 we reduce to the case where \(B\) is local, \(X \to \Spec(B)\) is a flat monomorphism, and there exists a point \(x \in X\) mapping to the closed point of \(\Spec(B)\). Then \(X \to \Spec(B)\) is surjective as generalizations lift along flat morphisms of separated algebraic spaces, see Decent Spaces, Lemma 0B7W. Hence we see that \(\{X \to \Spec(B)\}\) is an fpqc cover. Then \(X \to \Spec(B)\) is a morphism which becomes an isomorphism after base change by \(X \to \Spec(B)\). Hence it is an isomorphism by fpqc descent, see Descent on Spaces, Lemma 041Y.
The following is (in some sense) a variant of the lemma above.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-compact monomorphism of algebraic spaces such that for every \(T \to Y\) the map \[\mathcal{O}_T \to f_{T,*}\mathcal{O}_{X \times_Y T}\] is injective. Then \(f\) is an isomorphism (and hence representable by schemes).
Proof
The question is étale local on \(Y\), hence we may assume \(Y = \Spec(A)\) is affine. Then \(X\) is quasi-compact and we may choose an affine scheme \(U = \Spec(B)\) and a surjective étale morphism \(U \to X\) (Properties of Spaces, Lemma 03H6). Note that \(U \times_X U = \Spec(B \otimes_A B)\). Hence the category of quasi-coherent \(\mathcal{O}_X\)-modules is equivalent to the category \(DD_{B/A}\) of descent data on modules for \(A \to B\). See Properties of Spaces, Proposition 03M3, Descent, Definition 023G, and Descent, Subsection 08WW. On the other hand, \[A \to B\] is a universally injective ring map. Namely, given an \(A\)-module \(M\) we see that \(A \oplus M \to B \otimes_A (A \oplus M)\) is injective by the assumption of the lemma. Hence \(DD_{B/A}\) is equivalent to the category of \(A\)-modules by Descent, Theorem 08XA. Thus pullback along \(f : X \to \Spec(A)\) determines an equivalence of categories of quasi-coherent modules. In particular \(f^*\) is exact on quasi-coherent modules and we see that \(f\) is flat (small detail omitted). Moreover, it is clear that \(f\) is surjective (for example because \(\Spec(B) \to \Spec(A)\) is surjective). Hence we see that \(\{X \to \Spec(A)\}\) is an fpqc cover. Then \(X \to \Spec(A)\) is a morphism which becomes an isomorphism after base change by \(X \to \Spec(A)\). Hence it is an isomorphism by fpqc descent, see Descent on Spaces, Lemma 041Y.
Lemma
A quasi-compact flat surjective monomorphism of algebraic spaces is an isomorphism.
Proof
Such a morphism satisfies the assumptions of Lemma 0B8B.
Conormal sheaf of an immersion
Let \(S\) be a scheme. Let \(i : Z \to X\) be a closed immersion of algebraic spaces over \(S\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be the corresponding quasi-coherent sheaf of ideals, see Morphisms of Spaces, Lemma 03MB. Consider the short exact sequence \[0 \to \mathcal{I}^2 \to \mathcal{I} \to \mathcal{I}/\mathcal{I}^2 \to 0\] of quasi-coherent sheaves on \(X\). Since the sheaf \(\mathcal{I}/\mathcal{I}^2\) is annihilated by \(\mathcal{I}\) it corresponds to a sheaf on \(Z\) by Morphisms of Spaces, Lemma 04CJ. This quasi-coherent \(\mathcal{O}_Z\)-module is the conormal sheaf of \(Z\) in \(X\) and is often denoted \(\mathcal{I}/\mathcal{I}^2\) by the abuse of notation mentioned in Morphisms of Spaces, Section 04CI.
In case \(i : Z \to X\) is a (locally closed) immersion we define the conormal sheaf of \(i\) as the conormal sheaf of the closed immersion \(i : Z \to X \setminus \partial Z\), see Morphisms of Spaces, Remark 04CE. It is often denoted \(\mathcal{I}/\mathcal{I}^2\) where \(\mathcal{I}\) is the ideal sheaf of the closed immersion \(i : Z \to X \setminus \partial Z\).
Definition
Let \(i : Z \to X\) be an immersion. The conormal sheaf \(\mathcal{C}_{Z/X}\) of \(Z\) in \(X\) or the conormal sheaf of \(i\) is the quasi-coherent \(\mathcal{O}_Z\)-module \(\mathcal{I}/\mathcal{I}^2\) described above.
In [EGA, IV Definition 16.1.2] this sheaf is denoted \(\mathcal{N}_{Z/X}\). We will not follow this convention since we would like to reserve the notation \(\mathcal{N}_{Z/X}\) for the normal sheaf of the immersion. It is defined as \[\mathcal{N}_{Z/X} = \SheafHom_{\mathcal{O}_Z}(\mathcal{C}_{Z/X}, \mathcal{O}_Z) = \SheafHom_{\mathcal{O}_Z}(\mathcal{I}/\mathcal{I}^2, \mathcal{O}_Z)\] provided the conormal sheaf is of finite presentation (otherwise the normal sheaf may not even be quasi-coherent). We will come back to the normal sheaf later (insert future reference here).
Lemma
Let \(S\) be a scheme. Let \(i : Z \to X\) be an immersion. Let \(\varphi : U \to X\) be an étale morphism where \(U\) is a scheme. Set \(Z_U = U \times_X Z\) which is a locally closed subscheme of \(U\). Then \[\mathcal{C}_{Z/X}|_{Z_U} = \mathcal{C}_{Z_U/U}\] canonically and functorially in \(U\).
Proof
Let \(T \subset X\) be a closed subspace such that \(i\) defines a closed immersion into \(X \setminus T\). Let \(\mathcal{I}\) be the quasi-coherent sheaf of ideals on \(X \setminus T\) defining \(Z\). Then the lemma just states that \(\mathcal{I}|_{U \setminus \varphi^{-1}(T)}\) is the sheaf of ideals of the immersion \(Z_U \to U \setminus \varphi^{-1}(T)\). This is clear from the construction of \(\mathcal{I}\) in Morphisms of Spaces, Lemma 03MB.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' }\] be a commutative diagram of algebraic spaces over \(S\). Assume \(i\), \(i'\) immersions. There is a canonical map of \(\mathcal{O}_Z\)-modules \[f^*\mathcal{C}_{Z'/X'} \longrightarrow \mathcal{C}_{Z/X}\]
Proof
First find open subspaces \(U' \subset X'\) and \(U \subset X\) such that \(g(U) \subset U'\) and such that \(i(Z) \subset U\) and \(i(Z') \subset U'\) are closed (proof existence omitted). Replacing \(X\) by \(U\) and \(X'\) by \(U'\) we may assume that \(i\) and \(i'\) are closed immersions. Let \(\mathcal{I}' \subset \mathcal{O}_{X'}\) and \(\mathcal{I} \subset \mathcal{O}_X\) be the quasi-coherent sheaves of ideals associated to \(i'\) and \(i\), see Morphisms of Spaces, Lemma 03MB. Consider the composition \[g^{-1}\mathcal{I}' \to g^{-1}\mathcal{O}_{X'} \xrightarrow{g^\sharp} \mathcal{O}_X \to \mathcal{O}_X/\mathcal{I} = i_*\mathcal{O}_Z\] Since \(g(i(Z)) \subset Z'\) we conclude this composition is zero (see statement on factorizations in Morphisms of Spaces, Lemma 03MB). Thus we obtain a commutative diagram \[\xymatrix{ 0 \ar[r] & \mathcal{I} \ar[r] & \mathcal{O}_X \ar[r] & i_*\mathcal{O}_Z \ar[r] & 0 \\ 0 \ar[r] & g^{-1}\mathcal{I}' \ar[r] \ar[u] & g^{-1}\mathcal{O}_{X'} \ar[r] \ar[u] & g^{-1}i'_*\mathcal{O}_{Z'} \ar[r] \ar[u] & 0 }\] The lower row is exact since \(g^{-1}\) is an exact functor. By exactness we also see that \((g^{-1}\mathcal{I}')^2 = g^{-1}((\mathcal{I}')^2)\). Hence the diagram induces a map \(g^{-1}(\mathcal{I}'/(\mathcal{I}')^2) \to \mathcal{I}/\mathcal{I}^2\). Pulling back (using \(i^{-1}\) for example) to \(Z\) we obtain \(i^{-1}g^{-1}(\mathcal{I}'/(\mathcal{I}')^2) \to \mathcal{C}_{Z/X}\). Since \(i^{-1}g^{-1} = f^{-1}(i')^{-1}\) this gives a map \(f^{-1}\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}\), which induces the desired map.
Lemma
Let \(S\) be a scheme. The conormal sheaf of Definition 04CN, and its functoriality of Lemma 04CP satisfy the following properties:
If \(Z \to X\) is an immersion of schemes over \(S\), then the conormal sheaf agrees with the one from Morphisms, Definition 01R2.
If in Lemma 04CP all the spaces are schemes, then the map \(f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}\) is the same as the one constructed in Morphisms, Lemma 01R4.
Given a commutative diagram \[\xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} \ar[d]_{f'} & X' \ar[d]^{g'} \\ Z'' \ar[r]^{i''} & X'' }\] then the map \((f' \circ f)^*\mathcal{C}_{Z''/X''} \to \mathcal{C}_{Z/X}\) is the same as the composition of \(f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}\) with the pullback by \(f\) of \((f')^*\mathcal{C}_{Z''/X''} \to \mathcal{C}_{Z'/X'}\)
Proof
Omitted. Note that Part (1) is a special case of Lemma 04CO.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' }\] be a fibre product diagram of algebraic spaces over \(S\). Assume \(i\), \(i'\) immersions. Then the canonical map \(f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}\) of Lemma 04CP is surjective. If \(g\) is flat, then it is an isomorphism.
Proof
Choose a commutative diagram \[\xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ U' \ar[r] & X' }\] where \(U\), \(U'\) are schemes and the horizontal arrows are surjective and étale, see Spaces, Lemma 02X1. Then using Lemmas 04CO and 04G2 we see that the question reduces to the case of a morphism of schemes. In the schemes case this is Morphisms, Lemma 0473.
Lemma
Let \(S\) be a scheme. Let \(Z \to Y \to X\) be immersions of algebraic spaces. Then there is a canonical exact sequence \[i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0\] where the maps come from Lemma 04CP and \(i : Z \to Y\) is the first morphism.
Proof
Let \(U\) be a scheme and let \(U \to X\) be a surjective étale morphism. Via Lemmas 04CO and 04G2 the exactness of the sequence translates immediately into the exactness of the corresponding sequence for the immersions of schemes \(Z \times_X U \to Y \times_X U \to U\). Hence the lemma follows from Morphisms, Lemma 062S.
The normal cone of an immersion
Let \(S\) be a scheme. Let \(i : Z \to X\) be a closed immersion of algebraic spaces over \(S\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be the corresponding quasi-coherent sheaf of ideals, see Morphisms of Spaces, Lemma 03MB. Consider the quasi-coherent sheaf of graded \(\mathcal{O}_X\)-algebras \(\bigoplus_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1}\). Since the sheaves \(\mathcal{I}^n/\mathcal{I}^{n + 1}\) are each annihilated by \(\mathcal{I}\) this graded algebra corresponds to a quasi-coherent sheaf of graded \(\mathcal{O}_Z\)-algebras by Morphisms of Spaces, Lemma 04CJ. This quasi-coherent graded \(\mathcal{O}_Z\)-algebra is called the conormal algebra of \(Z\) in \(X\) and is often simply denoted \(\bigoplus_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1}\) by the abuse of notation mentioned in Morphisms of Spaces, Section 04CI.
In case \(i : Z \to X\) is a (locally closed) immersion we define the conormal algebra of \(i\) as the conormal algebra of the closed immersion \(i : Z \to X \setminus \partial Z\), see Morphisms of Spaces, Remark 04CE. It is often denoted \(\bigoplus_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1}\) where \(\mathcal{I}\) is the ideal sheaf of the closed immersion \(i : Z \to X \setminus \partial Z\).
Definition
Let \(i : Z \to X\) be an immersion. The conormal algebra \(\mathcal{C}_{Z/X, *}\) of \(Z\) in \(X\) or the conormal algebra of \(i\) is the quasi-coherent sheaf of graded \(\mathcal{O}_Z\)-algebras \(\bigoplus_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1}\) described above.
Thus \(\mathcal{C}_{Z/X, 1} = \mathcal{C}_{Z/X}\) is the conormal sheaf of the immersion. Also \(\mathcal{C}_{Z/X, 0} = \mathcal{O}_Z\) and \(\mathcal{C}_{Z/X, n}\) is a quasi-coherent \(\mathcal{O}_Z\)-module characterized by the property [09RP]\[\begin{equation} i_*\mathcal{C}_{Z/X, n} = \mathcal{I}^n/\mathcal{I}^{n + 1} \end{equation}\] where \(i : Z \to X \setminus \partial Z\) and \(\mathcal{I}\) is the ideal sheaf of \(i\) as above. Finally, note that there is a canonical surjective map [09RQ]\[\begin{equation} \text{Sym}^*(\mathcal{C}_{Z/X}) \longrightarrow \mathcal{C}_{Z/X, *} \end{equation}\] of quasi-coherent graded \(\mathcal{O}_Z\)-algebras which is an isomorphism in degrees \(0\) and \(1\).
Lemma
Let \(S\) be a scheme. Let \(i : Z \to X\) be an immersion of algebraic spaces over \(S\). Let \(\varphi : U \to X\) be an étale morphism where \(U\) is a scheme. Set \(Z_U = U \times_X Z\) which is a locally closed subscheme of \(U\). Then \[\mathcal{C}_{Z/X, *}|_{Z_U} = \mathcal{C}_{Z_U/U, *}\] canonically and functorially in \(U\).
Proof
Let \(T \subset X\) be a closed subspace such that \(i\) defines a closed immersion into \(X \setminus T\). Let \(\mathcal{I}\) be the quasi-coherent sheaf of ideals on \(X \setminus T\) defining \(Z\). Then the lemma follows from the fact that \(\mathcal{I}|_{U \setminus \varphi^{-1}(T)}\) is the sheaf of ideals of the immersion \(Z_U \to U \setminus \varphi^{-1}(T)\). This is clear from the construction of \(\mathcal{I}\) in Morphisms of Spaces, Lemma 03MB.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' }\] be a commutative diagram of algebraic spaces over \(S\). Assume \(i\), \(i'\) immersions. There is a canonical map of graded \(\mathcal{O}_Z\)-algebras \[f^*\mathcal{C}_{Z'/X', *} \longrightarrow \mathcal{C}_{Z/X, *}\]
Proof
First find open subspaces \(U' \subset X'\) and \(U \subset X\) such that \(g(U) \subset U'\) and such that \(i(Z) \subset U\) and \(i(Z') \subset U'\) are closed (proof existence omitted). Replacing \(X\) by \(U\) and \(X'\) by \(U'\) we may assume that \(i\) and \(i'\) are closed immersions. Let \(\mathcal{I}' \subset \mathcal{O}_{X'}\) and \(\mathcal{I} \subset \mathcal{O}_X\) be the quasi-coherent sheaves of ideals associated to \(i'\) and \(i\), see Morphisms of Spaces, Lemma 03MB. Consider the composition \[g^{-1}\mathcal{I}' \to g^{-1}\mathcal{O}_{X'} \xrightarrow{g^\sharp} \mathcal{O}_X \to \mathcal{O}_X/\mathcal{I} = i_*\mathcal{O}_Z\] Since \(g(i(Z)) \subset Z'\) we conclude this composition is zero (see statement on factorizations in Morphisms of Spaces, Lemma 03MB). Thus we obtain a commutative diagram \[\xymatrix{ 0 \ar[r] & \mathcal{I} \ar[r] & \mathcal{O}_X \ar[r] & i_*\mathcal{O}_Z \ar[r] & 0 \\ 0 \ar[r] & g^{-1}\mathcal{I}' \ar[r] \ar[u] & g^{-1}\mathcal{O}_{X'} \ar[r] \ar[u] & g^{-1}i'_*\mathcal{O}_{Z'} \ar[r] \ar[u] & 0 }\] The lower row is exact since \(g^{-1}\) is an exact functor. By exactness we also see that \((g^{-1}\mathcal{I}')^n = g^{-1}((\mathcal{I}')^n)\) for all \(n \geq 1\). Hence the diagram induces a map \(g^{-1}((\mathcal{I}')^n/(\mathcal{I}')^{n + 1}) \to \mathcal{I}^n/\mathcal{I}^{n + 1}\). Pulling back (using \(i^{-1}\) for example) to \(Z\) we obtain \(i^{-1}g^{-1}((\mathcal{I}')^n/(\mathcal{I}')^{n + 1}) \to \mathcal{C}_{Z/X, n}\). Since \(i^{-1}g^{-1} = f^{-1}(i')^{-1}\) this gives maps \(f^{-1}\mathcal{C}_{Z'/X', n} \to \mathcal{C}_{Z/X, n}\), which induce the desired map.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' }\] be a cartesian square of algebraic spaces over \(S\) with \(i\), \(i'\) immersions. Then the canonical map \(f^*\mathcal{C}_{Z'/X', *} \to \mathcal{C}_{Z/X, *}\) of Lemma 09RS is surjective. If \(g\) is flat, then it is an isomorphism.
Proof
We may check the statement after étale localizing \(X'\). In this case we may assume \(X' \to X\) is a morphism of schemes, hence \(Z\) and \(Z'\) are schemes and the result follows from the case of schemes, see Divisors, Lemma 0635.
We use the same conventions for cones and vector bundles over algebraic spaces as we do for schemes (where we use the conventions of EGA), see Constructions, Sections 062P and 01M1. In particular, a vector bundle is a very general gadget (and not locally isomorphic to an affine space bundle).
Definition
Let \(S\) be a scheme. Let \(i : Z \to X\) be an immersion of algebraic spaces over \(S\). The normal cone \(C_ZX\) of \(Z\) in \(X\) is \[C_ZX = \underline{\Spec}_Z(\mathcal{C}_{Z/X, *})\] see Morphisms of Spaces, Definition 081W. The normal bundle of \(Z\) in \(X\) is the vector bundle \[N_ZX = \underline{\Spec}_Z(\text{Sym}(\mathcal{C}_{Z/X}))\]
Thus \(C_ZX \to Z\) is a cone over \(Z\) and \(N_ZX \to Z\) is a vector bundle over \(Z\). Moreover, the canonical surjection (09RQ) of graded algebras defines a canonical closed immersion [09RV]\[\begin{equation} C_ZX \longrightarrow N_ZX \end{equation}\] of cones over \(Z\).
Sheaf of differentials of a morphism
We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 00RM), the corresponding section in the chapter on morphism of schemes (Morphisms, Section 01UM) as well as Modules on Sites, Section 04BJ. We first show that the notion of sheaf of differentials for a morphism of schemes agrees with the corresponding morphism of small étale (ringed) sites.
To clearly state the following lemma we temporarily go back to denoting \(\mathcal{F}^a\) the sheaf of \(\mathcal{O}_{X_\etale}\)-modules associated to a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) on the scheme \(X\), see Descent, Definition 03DU.
Lemma
Let \(f : X \to Y\) be a morphism of schemes. Let \(f_{small} : X_\etale \to Y_\etale\) be the associated morphism of small étale sites, see Descent, Remark 070R. Then there is a canonical isomorphism \[(\Omega_{X/Y})^a = \Omega_{X_\etale/Y_\etale}\] compatible with universal derivations. Here the first module is the sheaf on \(X_\etale\) associated to the quasi-coherent \(\mathcal{O}_X\)-module \(\Omega_{X/Y}\), see Morphisms, Definition 01UQ, and the second module is the one from Modules on Sites, Definition 04BN.
Proof
Let \(h : U \to X\) be an étale morphism. In this case the natural map \(h^*\Omega_{X/Y} \to \Omega_{U/Y}\) is an isomorphism, see More on Morphisms, Lemma 04BX. This means that there is a natural \(\mathcal{O}_{Y_\etale}\)-derivation \[\text{d}^a : \mathcal{O}_{X_\etale} \longrightarrow (\Omega_{X/Y})^a\] since we have just seen that the value of \((\Omega_{X/Y})^a\) on any object \(U\) of \(X_\etale\) is canonically identified with \(\Gamma(U, \Omega_{U/Y})\). By the universal property of \(\text{d}_{X/Y} : \mathcal{O}_{X_\etale} \to \Omega_{X_\etale/Y_\etale}\) there is a unique \(\mathcal{O}_{X_\etale}\)-linear map \(c : \Omega_{X_\etale/Y_\etale} \to (\Omega_{X/Y})^a\) such that \(\text{d}^a = c \circ \text{d}_{X/Y}\).
Conversely, suppose that \(\mathcal{F}\) is an \(\mathcal{O}_{X_\etale}\)-module and \(D : \mathcal{O}_{X_\etale} \to \mathcal{F}\) is a \(\mathcal{O}_{Y_\etale}\)-derivation. Then we can simply restrict \(D\) to the small Zariski site \(X_{Zar}\) of \(X\). Since sheaves on \(X_{Zar}\) agree with sheaves on \(X\), see Descent, Remark 03FG, we see that \(D|_{X_{Zar}} : \mathcal{O}_X \to \mathcal{F}|_{X_{Zar}}\) is just a “usual” \(Y\)-derivation. Hence we obtain a map \(\psi : \Omega_{X/Y} \longrightarrow \mathcal{F}|_{X_{Zar}}\) such that \(D|_{X_{Zar}} = \psi \circ \text{d}\). In particular, if we apply this with \(\mathcal{F} = \Omega_{X_\etale/Y_\etale}\) we obtain a map \[c' : \Omega_{X/Y} \longrightarrow \Omega_{X_\etale/Y_\etale}|_{X_{Zar}}\] Consider the morphism of ringed sites \(\text{id}_{small, \etale, Zar} : X_\etale \to X_{Zar}\) discussed in Descent, Remark 070R and Lemma 070S. Since the restriction functor \(\mathcal{F} \mapsto \mathcal{F}|_{X_{Zar}}\) is equal to \(\text{id}_{small, \etale, Zar, *}\), since \(\text{id}_{small, \etale, Zar}^*\) is left adjoint to \(\text{id}_{small, \etale, Zar, *}\) and since \((\Omega_{X/Y})^a = \text{id}_{small, \etale, Zar}^*\Omega_{X/Y}\) we see that \(c'\) is adjoint to a map \[c'' : (\Omega_{X/Y})^a \longrightarrow \Omega_{X_\etale/Y_\etale}.\] We claim that \(c''\) and \(c'\) are mutually inverse. This claim finishes the proof of the lemma. To see this it is enough to show that \(c''(\text{d}(f)) = \text{d}_{X/Y}(f)\) and \(c(\text{d}_{X/Y}(f)) = \text{d}(f)\) if \(f\) is a local section of \(\mathcal{O}_X\) over an open of \(X\). We omit the verification.
This clears the way for the following definition. For an alternative, see Remark 04CW.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The sheaf of differentials \(\Omega_{X/Y}\) of \(X\) over \(Y\) is sheaf of differentials (Modules on Sites, Definition 04BQ) for the morphism of ringed topoi \[(f_{small}, f^\sharp) : (X_\etale, \mathcal{O}_X) \to (Y_\etale, \mathcal{O}_Y)\] of Properties of Spaces, Lemma 03G8. The universal \(Y\)-derivation will be denoted \(\text{d}_{X/Y} : \mathcal{O}_X \to \Omega_{X/Y}\).
By Lemma 04CS this does not conflict with the already existing notion in case \(X\) and \(Y\) are representable. From now on, if \(X\) and \(Y\) are representable, we no longer distinguish between the sheaf of differentials defined above and the one defined in Morphisms, Definition 01UQ. We want to relate this to the usual modules of differentials for morphisms of schemes. Here is the key lemma.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Consider any commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_\psi & V \ar[d]^b \\ X \ar[r]^f & Y }\] where the vertical arrows are étale morphisms of algebraic spaces. Then \[\Omega_{X/Y}|_{U_\etale} = \Omega_{U/V}\] In particular, if \(U\), \(V\) are schemes, then this is equal to the usual sheaf of differentials of the morphism of schemes \(U \to V\).
Proof
By Properties of Spaces, Lemma 03LP and Equation (03LQ) we may think of the restriction of a sheaf on \(X_\etale\) to \(U_\etale\) as the pullback by \(a_{small}\). Similarly for \(b\). By Modules on Sites, Lemma 04BO we have \[\Omega_{X/Y}|_{U_\etale} = \Omega_{\mathcal{O}_{U_\etale}/ a_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_\etale}}\] Since \(a_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_\etale} = \psi_{small}^{-1}b_{small}^{-1}\mathcal{O}_{Y_\etale} = \psi_{small}^{-1}\mathcal{O}_{V_\etale}\) we see that the lemma holds.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Then \(\Omega_{X/Y}\) is a quasi-coherent \(\mathcal{O}_X\)-module.
Proof
Choose a diagram as in Lemma 04CU with \(a\) and \(b\) surjective and \(U\) and \(V\) schemes. Then we see that \(\Omega_{X/Y}|_U = \Omega_{U/V}\) which is quasi-coherent (for example by Morphisms, Lemma 08S2). Hence we conclude that \(\Omega_{X/Y}\) is quasi-coherent by Properties of Spaces, Lemma 03M0.
Remark
Now that we know that \(\Omega_{X/Y}\) is quasi-coherent we can attempt to construct it in another manner. For example we can use the result of Properties of Spaces, Section 03M2 to construct the sheaf of differentials by glueing. For example if \(Y\) is a scheme and if \(U \to X\) is a surjective étale morphism from a scheme towards \(X\), then we see that \(\Omega_{U/Y}\) is a quasi-coherent \(\mathcal{O}_U\)-module, and since \(s, t : R \to U\) are étale we get an isomorphism \[\alpha : s^*\Omega_{U/Y} \to \Omega_{R/Y} \to t^*\Omega_{U/Y}\] by using Morphisms, Lemma 02K4. You check that this satisfies the cocycle condition and you’re done. If \(Y\) is not a scheme, then you define \(\Omega_{U/Y}\) as the cokernel of the map \((U \to Y)^*\Omega_{Y/S} \to \Omega_{U/S}\), and proceed as before. This two step process is a little bit ugly. Another possibility is to glue the sheaves \(\Omega_{U/V}\) for any diagram as in Lemma 04CU but this is not very elegant either. Both approaches will work however, and will give a slightly more elementary construction of the sheaf of differentials.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ X' \ar[d] \ar[r]_f & X \ar[d] \\ Y' \ar[r] & Y }\] be a commutative diagram of algebraic spaces. The map \(f^\sharp : \mathcal{O}_X \to f_*\mathcal{O}_{X'}\) composed with the map \(f_*\text{d}_{X'/Y'} : f_*\mathcal{O}_{X'} \to f_*\Omega_{X'/Y'}\) is a \(Y\)-derivation. Hence we obtain a canonical map of \(\mathcal{O}_X\)-modules \(\Omega_{X/Y} \to f_*\Omega_{X'/Y'}\), and by adjointness of \(f_*\) and \(f^*\) a canonical \(\mathcal{O}_{X'}\)-module homomorphism \[c_f : f^*\Omega_{X/Y} \longrightarrow \Omega_{X'/Y'}.\] It is uniquely characterized by the property that \(f^*\text{d}_{X/Y}(t)\) maps to \(\text{d}_{X'/Y'}(f^* t)\) for any local section \(t\) of \(\mathcal{O}_X\).
Proof
This is a special case of Modules on Sites, Lemma 04BR.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ X'' \ar[d] \ar[r]_g & X' \ar[d] \ar[r]_f & X \ar[d] \\ Y'' \ar[r] & Y' \ar[r] & Y }\] be a commutative diagram of algebraic spaces over \(S\). Then we have \[c_{f \circ g} = c_g \circ g^* c_f\] as maps \((f \circ g)^*\Omega_{X/Y} \to \Omega_{X''/Y''}\).
Proof
Omitted. Hint: Use the characterization of \(c_f, c_g, c_{f \circ g}\) in terms of the effect these maps have on local sections.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\), \(g : Y \to B\) be morphisms of algebraic spaces over \(S\). Then there is a canonical exact sequence \[f^*\Omega_{Y/B} \to \Omega_{X/B} \to \Omega_{X/Y} \to 0\] where the maps come from applications of Lemma 04CX.
Proof
Follows from the schemes version, see Morphisms, Lemma 01UX, of this result via étale localization, see Lemma 04CU.
Lemma
Let \(S\) be a scheme. If \(X \to Y\) is an immersion of algebraic spaces over \(S\) then \(\Omega_{X/S}\) is zero.
Proof
Follows from the schemes version, see Morphisms, Lemma 01UY, of this result via étale localization, see Lemma 04CU.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(i : Z \to X\) be an immersion of algebraic spaces over \(B\). There is a canonical exact sequence \[\mathcal{C}_{Z/X} \to i^*\Omega_{X/B} \to \Omega_{Z/B} \to 0\] where the first arrow is induced by \(\text{d}_{X/B}\) and the second arrow comes from Lemma 04CX.
Proof
This is the algebraic spaces version of Morphisms, Lemma 01UZ and will be a consequence of that lemma by étale localization, see Lemmas 04CU and 04CO. However, we should make sure we can define the first arrow globally. Hence we explain the meaning of “induced by \(\text{d}_{X/B}\)” here. Namely, we may assume that \(i\) is a closed immersion after replacing \(X\) by an open subspace. Let \(\mathcal{I} \subset \mathcal{O}_X\) be the quasi-coherent sheaf of ideals corresponding to \(Z \subset X\). Then \(\text{d}_{X/S} : \mathcal{I} \to \Omega_{X/S}\) maps the subsheaf \(\mathcal{I}^2 \subset \mathcal{I}\) to \(\mathcal{I}\Omega_{X/S}\). Hence it induces a map \(\mathcal{I}/\mathcal{I}^2 \to \Omega_{X/S}/\mathcal{I}\Omega_{X/S}\) which is \(\mathcal{O}_X/\mathcal{I}\)-linear. By Morphisms of Spaces, Lemma 04CJ this corresponds to a map \(\mathcal{C}_{Z/X} \to i^*\Omega_{X/S}\) as desired.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(i : Z \to X\) be an immersion of algebraic spaces over \(B\), and assume \(i\) (étale locally) has a left inverse. Then the canonical sequence \[0 \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/B} \to \Omega_{Z/B} \to 0\] of Lemma 05ZA is (étale locally) split exact.
Proof
Clarification: we claim that if \(g : X \to Z\) is a left inverse of \(i\) over \(B\), then \(i^*c_g\) is a right inverse of the map \(i^*\Omega_{X/B} \to \Omega_{Z/B}\). Having said this, the result follows from the corresponding result for morphisms of schemes by étale localization, see Lemmas 04CU and 04CO.
Lemma
Let \(S\) be a scheme. Let \(X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(g : Y' \to Y\) be a morphism of algebraic spaces over \(S\). Let \(X' = X_{Y'}\) be the base change of \(X\). Denote \(g' : X' \to X\) the projection. Then the map \[(g')^*\Omega_{X/Y} \to \Omega_{X'/Y'}\] of Lemma 04CX is an isomorphism.
Proof
Follows from the schemes version, see Morphisms, Lemma 01V0 and étale localization, see Lemma 04CU.
Lemma
Let \(S\) be a scheme. Let \(f : X \to B\) and \(g : Y \to B\) be morphisms of algebraic spaces over \(S\) with the same target. Let \(p : X \times_B Y \to X\) and \(q : X \times_B Y \to Y\) be the projection morphisms. The maps from Lemma 04CX \[p^*\Omega_{X/B} \oplus q^*\Omega_{Y/B} \longrightarrow \Omega_{X \times_B Y/B}\] give an isomorphism.
Proof
Follows from the schemes version, see Morphisms, Lemma 01V1 and étale localization, see Lemma 04CU.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If \(f\) is locally of finite type, then \(\Omega_{X/Y}\) is a finite type \(\mathcal{O}_X\)-module.
Proof
Follows from the schemes version, see Morphisms, Lemma 01V2 and étale localization, see Lemma 04CU.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If \(f\) is locally of finite presentation, then \(\Omega_{X/Y}\) is an \(\mathcal{O}_X\)-module of finite presentation.
Proof
Follows from the schemes version, see Morphisms, Lemma 01V3 and étale localization, see Lemma 04CU.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a smooth morphism of algebraic spaces over \(S\). Then the module of differentials \(\Omega_{X/Y}\) is finite locally free.
Proof
The statement is étale local on \(X\) and \(Y\) by Lemma 04CU. Hence this follows from the case of schemes, see Morphisms, Lemma 02G1.
Topological invariance of the étale site
We show that the site \(X_{spaces, \etale}\) is a “topological invariant”. It then follows that \(X_\etale\), which consists of the representable objects in \(X_{spaces, \etale}\), is a topological invariant too, see Lemma 07VW.
Theorem
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is integral, universally injective and surjective. The functor \[V \longmapsto V_X = X \times_Y V\] defines an equivalence of categories \(Y_{spaces, \etale} \to X_{spaces, \etale}\).
Proof
The morphism \(f\) is representable and a universal homeomorphism, see Morphisms of Spaces, Section 05Z3.
We first prove that the functor is faithful. Suppose that \(V', V\) are objects of \(Y_{spaces, \etale}\) and that \(a, b : V' \to V\) are distinct morphisms over \(Y\). Since \(V', V\) are étale over \(Y\) the equalizer \[E = V' \times_{(a, b), V \times_Y V, \Delta_{V/Y}} V\] of \(a, b\) is étale over \(Y\) also. Hence \(E \to V'\) is an étale monomorphism (i.e., an open immersion) which is an isomorphism if and only if it is surjective. Since \(X \to Y\) is a universal homeomorphism we see that this is the case if and only if \(E_X = V'_X\), i.e., if and only if \(a_X = b_X\).
Next, we prove that the functor is fully faithful. Suppose that \(V', V\) are objects of \(Y_{spaces, \etale}\) and that \(c : V'_X \to V_X\) is a morphism over \(X\). We want to construct a morphism \(a : V' \to V\) over \(Y\) such that \(a_X = c\). Let \(a' : V'' \to V'\) be a surjective étale morphism such that \(V''\) is a separated algebraic space. If we can construct a morphism \(a'' : V'' \to V\) such that \(a''_X = c \circ a'_X\), then the two compositions \[V'' \times_{V'} V'' \xrightarrow{\text{pr}_i} V'' \xrightarrow{a''} V\] will be equal by the faithfulness of the functor proved in the first paragraph. Hence \(a''\) will factor through a unique morphism \(a : V' \to V\) as \(V'\) is (as a sheaf) the quotient of \(V''\) by the equivalence relation \(V'' \times_{V'} V''\). Hence we may assume that \(V'\) is separated. In this case the graph \[\Gamma_c \subset (V' \times_Y V)_X\] is open and closed (details omitted). Since \(X \to Y\) is a universal homeomorphism, there exists an open and closed subspace \(\Gamma \subset V' \times_Y V\) such that \(\Gamma_X = \Gamma_c\). The projection \(\Gamma \to V'\) is an étale morphism whose base change to \(X\) is an isomorphism. Hence \(\Gamma \to V'\) is étale, universally injective, and surjective, so an isomorphism by Morphisms of Spaces, Lemma 05W5. Thus \(\Gamma\) is the graph of a morphism \(a : V' \to V\) as desired.
Finally, we prove that the functor is essentially surjective. Suppose that \(U\) is an object of \(X_{spaces, \etale}\). We have to find an object \(V\) of \(Y_{spaces, \etale}\) such that \(V_X \cong U\). Let \(U' \to U\) be a surjective étale morphism such that \(U' \cong V'_X\) and \(U' \times_U U' \cong V''_X\) for some objects \(V'', V'\) of \(Y_{spaces, \etale}\). Then by fully faithfulness of the functor we obtain morphisms \(s, t : V'' \to V'\) with \(t_X = \text{pr}_0\) and \(s_X = \text{pr}_1\) as morphisms \(U' \times_U U' \to U'\). Using that \((\text{pr}_0, \text{pr}_1) : U' \times_U U' \to U' \times_S U'\) is an étale equivalence relation, and that \(U' \to V'\) and \(U' \times_U U' \to V''\) are universally injective and surjective we deduce that \((t, s) : V'' \to V' \times_S V'\) is an étale equivalence relation. Then the quotient \(V = V'/V''\) (see Spaces, Theorem 02WW) is an algebraic space \(V\) over \(Y\). There is a morphism \(V' \to V\) such that \(V'' = V' \times_V V'\). Thus we obtain a morphism \(V \to Y\) (see Descent on Spaces, Lemma 04P2). On base change to \(X\) we see that we have a morphism \(U' \to V_X\) and a compatible isomorphism \(U' \times_{V_X} U' = U' \times_U U'\), which implies that \(V_X \cong U\) (by the lemma just cited once more).
Pick a scheme \(W\) and a surjective étale morphism \(W \to Y\). Pick a scheme \(U'\) and a surjective étale morphism \(U' \to U \times_X W_X\). Note that \(U'\) and \(U' \times_U U'\) are schemes étale over \(X\) whose structure morphism to \(X\) factors through the scheme \(W_X\). Hence by Étale Cohomology, Theorem 04DZ there exist schemes \(V', V''\) étale over \(W\) whose base change to \(W_X\) is isomorphic to respectively \(U'\) and \(U' \times_U U'\). This finishes the proof.
Lemma
With assumption and notation as in Theorem 05ZH the equivalence of categories \(Y_{spaces, \etale} \to X_{spaces, \etale}\) restricts to equivalences of categories \(Y_\etale \to X_\etale\) and \(Y_{affine, \etale} \to X_{affine, \etale}\).
Proof
This is just the statement that given an object \(V \in Y_{spaces, \etale}\) we have \(V\) is a(n affine) scheme if and only if \(V \times_Y X\) is a(n affine) scheme. Since \(V \times_Y X \to V\) is integral, universally injective, and surjective (as a base change of \(X \to Y\)) this follows from Limits of Spaces, Lemma 07VV and Proposition 07VT.
Remark
A universal homeomorphism of algebraic spaces need not be representable, see Morphisms of Spaces, Example 05Z6. In fact Theorem 05ZH does not hold for universal homeomorphisms. To see this, let \(k\) be an algebraically closed field of characteristic \(0\) and let \[\mathbf{A}^1 \to X \to \mathbf{A}^1\] be as in Morphisms of Spaces, Example 05Z6. Recall that the first morphism is étale and identifies \(t\) with \(-t\) for \(t \in \mathbf{A}^1_k \setminus \{0\}\) and that the second morphism is our universal homeomorphism. Since \(\mathbf{A}^1_k\) has no nontrivial connected finite étale coverings (because \(k\) is algebraically closed of characteristic zero; details omitted), it suffices to construct a nontrivial connected finite étale covering \(Y \to X\). To do this, let \(Y\) be the affine line with zero doubled (Schemes, Example 01JD). Then \(Y = Y_1 \cup Y_2\) with \(Y_i = \mathbf{A}^1_k\) glued along \(\mathbf{A}^1_k \setminus \{0\}\). To define the morphism \(Y \to X\) we use the morphisms \[Y_1 \xrightarrow{1} \mathbf{A}^1_k \to X \quad\text{and}\quad Y_2 \xrightarrow{-1} \mathbf{A}^1_k \to X.\] These glue over \(Y_1 \cap Y_2\) by the construction of \(X\) and hence define a morphism \(Y \to X\). In fact, we claim that \[\xymatrix{ Y \ar[d] & Y_1 \amalg Y_2 \ar[l] \ar[d] \\ X & \mathbf{A}^1_k \ar[l] }\] is a cartesian square. We omit the details; you can use for example Groupoids, Lemma 07S3. Since \(\mathbf{A}^1_k \to X\) is étale and surjective, this proves that \(Y \to X\) is finite étale of degree \(2\) which gives the desired example.
More simply, you can argue as follows. The scheme \(Y\) has a free action of the group \(G = \{+1, -1\}\) where \(-1\) acts by swapping \(Y_1\) and \(Y_2\) and changing the sign of the coordinate. Then \(X = Y/G\) (see Spaces, Definition 02Z3) and hence \(Y \to X\) is finite étale. You can also show directly that there exists a universal homeomorphism \(X \to \mathbf{A}^1_k\) by using \(t \mapsto t^2\) on affine spaces. In fact, this \(X\) is the same as the \(X\) above.
Thickenings
The following terminology may not be completely standard, but it is convenient.
Definition
Thickenings. Let \(S\) be a scheme.
We say an algebraic space \(X'\) is a thickening of an algebraic space \(X\) if \(X\) is a closed subspace of \(X'\) and the associated topological spaces are equal.
We say \(X'\) is a first order thickening of \(X\) if \(X\) is a closed subspace of \(X'\) and the quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_{X'}\) defining \(X\) has square zero.
We say \(X'\) is a finite order thickening of \(X\) if \(X\) is a closed subspace of \(X\) and the quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_{X'}\) defining \(X\) is nilpotent, i.e., there exists an integer \(n \geq 0\) such that \(\mathcal{I}^{n + 1} = 0\).
We say \(X'\) is an \(n\)th order thickening of \(X\) if \(X\) is a closed subspace of \(X\) and \(\mathcal{I}^{n + 1} = 0\) where \(\mathcal{I} \subset \mathcal{O}_{X'}\) is the quasi-coherent sheaf of ideals defining \(X\).
Given two thickenings \(X \subset X'\) and \(Y \subset Y'\) a morphism of thickenings is a morphism \(f' : X' \to Y'\) such that \(f(X) \subset Y\), i.e., such that \(f'|_X\) factors through the closed subspace \(Y\). In this situation we set \(f = f'|_X : X \to Y\) and we say that \((f, f') : (X \subset X') \to (Y \subset Y')\) is a morphism of thickenings.
Let \(B\) be an algebraic space. We similarly define thickenings over \(B\), and morphisms of thickenings over \(B\). This means that the spaces \(X, X', Y, Y'\) above are algebraic spaces endowed with a structure morphism to \(B\), and that the morphisms \(X \to X'\), \(Y \to Y'\) and \(f' : X' \to Y'\) are morphisms over \(B\).
The fundamental equivalence. Note that if \(X \subset X'\) is a thickening, then \(X \to X'\) is integral and universally bijective. This implies that [05ZL]\[\begin{equation} X_{spaces, \etale} = X'_{spaces, \etale} \end{equation}\] via the pullback functor, see Theorem 05ZH. Hence we may think of \(\mathcal{O}_{X'}\) as a sheaf on \(X_{spaces, \etale}\). Thus a canonical equivalence of locally ringed topoi [05ZM]\[\begin{equation} (\Sh(X'_{spaces, \etale}), \mathcal{O}_{X'}) \cong (\Sh(X_{spaces, \etale}), \mathcal{O}_{X'}) \end{equation}\] Below we will frequently combine this with the fully faithfulness result of Properties of Spaces, Theorem 04KL. For example the closed immersion \(i_X : X \to X'\) corresponds to the surjective map \(i_X^\sharp : \mathcal{O}_{X'} \to \mathcal{O}_X\).
Let \(S\) be a scheme, and let \(B\) be an algebraic space over \(S\). Let \((f, f') : (X \subset X') \to (Y \subset Y')\) be a morphism of thickenings over \(B\). Note that the diagram of continuous functors \[\xymatrix{ X_{spaces, \etale} & Y_{spaces, \etale} \ar[l] \\ X'_{spaces, \etale} \ar[u] & Y'_{spaces, \etale} \ar[u] \ar[l] }\] is commutative and the vertical arrows are equivalences. Hence \(f_{spaces, \etale}\), \(f_{small}\), \(f'_{spaces, \etale}\), and \(f'_{small}\) all define the same morphism of topoi. Thus we may think of \[(f')^\sharp : f_{spaces, \etale}^{-1}\mathcal{O}_{Y'} \longrightarrow \mathcal{O}_{X'}\] as a map of sheaves of \(\mathcal{O}_B\)-algebras fitting into the commutative diagram \[\xymatrix{ f_{spaces, \etale}^{-1}\mathcal{O}_Y \ar[r]_-{f^\sharp} \ar[r] & \mathcal{O}_X \\ f_{spaces, \etale}^{-1}\mathcal{O}_{Y'} \ar[r]^-{(f')^\sharp} \ar[u]^{i_Y^\sharp} & \mathcal{O}_{X'} \ar[u]_{i_X^\sharp} }\] Here \(i_X : X \to X'\) and \(i_Y : Y \to Y'\) are the names of the given closed immersions.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(X \subset X'\) and \(Y \subset Y'\) be thickenings of algebraic spaces over \(B\). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(B\). Given any map of \(\mathcal{O}_B\)-algebras \[\alpha : f_{spaces, \etale}^{-1}\mathcal{O}_{Y'} \to \mathcal{O}_{X'}\] such that \[\xymatrix{ f_{spaces, \etale}^{-1}\mathcal{O}_Y \ar[r]_-{f^\sharp} \ar[r] & \mathcal{O}_X \\ f_{spaces, \etale}^{-1}\mathcal{O}_{Y'} \ar[r]^-\alpha \ar[u]^{i_Y^\sharp} & \mathcal{O}_{X'} \ar[u]_{i_X^\sharp} }\] commutes, there exists a unique morphism of \((f, f')\) of thickenings over \(B\) such that \(\alpha = (f')^\sharp\).
Proof
To find \(f'\), by Properties of Spaces, Theorem 04KL, all we have to do is show that the morphism of ringed topoi \[(f_{spaces, \etale}, \alpha) : (\Sh(X_{spaces, \etale}), \mathcal{O}_{X'}) \longrightarrow (\Sh(Y_{spaces, \etale}), \mathcal{O}_{Y'})\] is a morphism of locally ringed topoi. This follows directly from the definition of morphisms of locally ringed topoi (Modules on Sites, Definition 04HA), the fact that \((f, f^\sharp)\) is a morphism of locally ringed topoi (Properties of Spaces, Lemma 04KJ), that \(\alpha\) fits into the given commutative diagram, and the fact that the kernels of \(i_X^\sharp\) and \(i_Y^\sharp\) are locally nilpotent. Finally, the fact that \(f' \circ i_X = i_Y \circ f\) follows from the commutativity of the diagram and another application of Properties of Spaces, Theorem 04KL. We omit the verification that \(f'\) is a morphism over \(B\).
Lemma
Let \(S\) be a scheme. Let \(X \subset X'\) be a thickening of algebraic spaces over \(S\). For any open subspace \(U \subset X\) there exists a unique open subspace \(U' \subset X'\) such that \(U = X \times_{X'} U'\).
Proof
Let \(U' \to X'\) be the object of \(X'_{spaces, \etale}\) corresponding to the object \(U \to X\) of \(X_{spaces, \etale}\) via (05ZL). The morphism \(U' \to X'\) is étale and universally injective, hence an open immersion, see Morphisms of Spaces, Lemma 05W5.
Let \(i_X : X \to X'\) be a thickening of algebraic spaces. Any local section of the kernel \(\mathcal{I} = \Ker(i_X^\sharp) \subset \mathcal{O}_{X'}\) is locally nilpotent. This gives some control over the structure sheaf of \(\mathcal{O}_{X'}\) but technically the class of finite order thickenings \(X \subset X'\) is much easier to handle. Namely, if \(X'\) is an \(n\)th order thickening we have a filtration \[0 \subset \mathcal{I}^n \subset \mathcal{I}^{n - 1} \subset \ldots \subset \mathcal{I} \subset \mathcal{O}_{X'}\] and we see that \(X'\) is filtered by closed subspaces \[X = X_0 \subset X_1 \subset \ldots \subset X_{n - 1} \subset X_{n + 1} = X'\] such that each pair \(X_i \subset X_{i + 1}\) is a first order thickening over \(B\). Using simple induction arguments many results proved for first order thickenings can be rephrased as results on finite order thickenings.
Lemma
Let \(S\) be a scheme. Let \(X \subset X'\) be a thickening of algebraic spaces over \(S\). Let \(U\) be an affine object of \(X_{spaces, \etale}\). Then \[\Gamma(U, \mathcal{O}_{X'}) \to \Gamma(U, \mathcal{O}_X)\] is surjective where we think of \(\mathcal{O}_{X'}\) as a sheaf on \(X_{spaces, \etale}\) via (05ZM).
Proof
Let \(U' \to X'\) be the étale morphism of algebraic spaces such that \(U = X \times_{X'} U'\), see Theorem 05ZH. By Limits of Spaces, Lemma 07VS we see that \(U'\) is an affine scheme. Hence \(\Gamma(U, \mathcal{O}_{X'}) = \Gamma(U', \mathcal{O}_{U'}) \to \Gamma(U, \mathcal{O}_U)\) is surjective as \(U \to U'\) is a closed immersion of affine schemes. Below we give a direct proof for finite order thickenings which is the case most used in practice.
Proof
We may assume that \(X \subset X'\) is a first order thickening by the principle explained above. Denote \(\mathcal{I}\) the kernel of the surjection \(\mathcal{O}_{X'} \to \mathcal{O}_X\). As \(\mathcal{I}\) is a quasi-coherent \(\mathcal{O}_{X'}\)-module and since \(\mathcal{I}^2 = 0\) by the definition of a first order thickening we may apply Morphisms of Spaces, Lemma 04CJ to see that \(\mathcal{I}\) is a quasi-coherent \(\mathcal{O}_X\)-module. Hence the lemma follows from the long exact cohomology sequence associated to the short exact sequence \[0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\] and the fact that \(H^1_\etale(U, \mathcal{I}) = 0\) as \(\mathcal{I}\) is quasi-coherent, see Descent, Proposition 03DW and Cohomology of Schemes, Lemma 01XB.
Lemma
Let \(S\) be a scheme. Let \(X \subset X'\) be a thickening of algebraic spaces over \(S\). If \(X\) is (representable by) a scheme, then so is \(X'\).
Proof
Note that \(X'_{red} = X_{red}\). Hence if \(X\) is a scheme, then \(X'_{red}\) is a scheme. Thus the result follows from Limits of Spaces, Lemma 07VU. Below we give a direct proof for finite order thickenings which is the case most often used in practice.
Proof
It suffices to prove this when \(X'\) is a first order thickening of \(X\). By Properties of Spaces, Lemma 03JH there is a largest open subspace of \(X'\) which is a scheme. Thus we have to show that every point \(x\) of \(|X'| = |X|\) is contained in an open subspace of \(X'\) which is a scheme. Using Lemma 05ZP we may replace \(X \subset X'\) by \(U \subset U'\) with \(x \in U\) and \(U\) an affine scheme. Hence we may assume that \(X\) is affine. Thus we reduce to the case discussed in the next paragraph.
Assume \(X \subset X'\) is a first order thickening where \(X\) is an affine scheme. Set \(A = \Gamma(X, \mathcal{O}_X)\) and \(A' = \Gamma(X', \mathcal{O}_{X'})\). By Lemma 05ZQ the map \(A' \to A\) is surjective. The kernel \(I\) is an ideal of square zero. By Properties of Spaces, Lemma 05Z1 we obtain a canonical morphism \(f : X' \to \Spec(A')\) which fits into the following commutative diagram \[\xymatrix{ X \ar@{=}[d] \ar[r] & X' \ar[d]^f \\ \Spec(A) \ar[r] & \Spec(A') }\] Because the horizontal arrows are thickenings it is clear that \(f\) is universally injective and surjective. Hence it suffices to show that \(f\) is étale, since then Morphisms of Spaces, Lemma 05W5 will imply that \(f\) is an isomorphism.
To prove that \(f\) is étale choose an affine scheme \(U'\) and an étale morphism \(U' \to X'\). It suffices to show that \(U' \to X' \to \Spec(A')\) is étale, see Properties of Spaces, Definition 03FR. Write \(U' = \Spec(B')\). Set \(U = X \times_{X'} U'\). Since \(U\) is a closed subspace of \(U'\), it is a closed subscheme, hence \(U = \Spec(B)\) with \(B' \to B\) surjective. Denote \(J = \Ker(B' \to B)\) and note that \(J = \Gamma(U, \mathcal{I})\) where \(\mathcal{I} = \Ker(\mathcal{O}_{X'} \to \mathcal{O}_X)\) on \(X_{spaces, \etale}\) as in the proof of Lemma 05ZQ. The morphism \(U' \to X' \to \Spec(A')\) induces a commutative diagram \[\xymatrix{ 0 \ar[r] & J \ar[r] & B' \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & I \ar[r] \ar[u] & A' \ar[r] \ar[u] & A \ar[r] \ar[u] & 0 }\] Now, since \(\mathcal{I}\) is a quasi-coherent \(\mathcal{O}_X\)-module we have \(\mathcal{I} = (\widetilde I)^a\), see Descent, Definition 03DU for notation and Descent, Proposition 03DX for why this is true. Hence we see that \(J = I \otimes_A B\). Finally, note that \(A \to B\) is étale as \(U \to X\) is étale as the base change of the étale morphism \(U' \to X'\). We conclude that \(A' \to B'\) is étale by Algebra, Lemma 05YT.
Lemma
Let \(S\) be a scheme. Let \(X \subset X'\) be a thickening of algebraic spaces over \(S\). The functor \[V' \longmapsto V = X \times_{X'} V'\] defines an equivalence of categories \(X'_\etale \to X_\etale\).
Proof
The functor \(V' \mapsto V\) defines an equivalence of categories \(X'_{spaces, \etale} \to X_{spaces, \etale}\), see Theorem 05ZH. Thus it suffices to show that \(V\) is a scheme if and only if \(V'\) is a scheme. This is the content of Lemma 05ZR.
First order thickening are described as follows.
Lemma
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Consider a short exact sequence \[0 \to \mathcal{I} \to \mathcal{A} \to \mathcal{O}_X \to 0\] of sheaves on \(X_\etale\) where \(\mathcal{A}\) is a sheaf of \(f^{-1}\mathcal{O}_B\)-algebras, \(\mathcal{A} \to \mathcal{O}_X\) is a surjection of sheaves of \(f^{-1}\mathcal{O}_B\)-algebras, and \(\mathcal{I}\) is its kernel. If
\(\mathcal{I}\) is an ideal of square zero in \(\mathcal{A}\), and
\(\mathcal{I}\) is quasi-coherent as an \(\mathcal{O}_X\)-module
then there exists a first order thickening \(X \subset X'\) over \(B\) and an isomorphism \(\mathcal{O}_{X'} \to \mathcal{A}\) of \(f^{-1}\mathcal{O}_B\)-algebras compatible with the surjections to \(\mathcal{O}_X\).
Proof
In this proof we redo some of the arguments used in the proofs of Lemmas 05ZQ and 05ZR. We first handle the case \(B = S = \Spec(\mathbf{Z})\). Let \(U\) be an affine scheme, and let \(U \to X\) be étale. Then \[0 \to \mathcal{I}(U) \to \mathcal{A}(U) \to \mathcal{O}_X(U) \to 0\] is exact as \(H^1(U_\etale, \mathcal{I}) = 0\) as \(\mathcal{I}\) is quasi-coherent, see Descent, Proposition 03DW and Cohomology of Schemes, Lemma 01XB. If \(V \to U\) is a morphism of affine objects of \(X_{spaces, \etale}\) then \[\mathcal{I}(V) = \mathcal{I}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{O}_X(V)\] since \(\mathcal{I}\) is a quasi-coherent \(\mathcal{O}_X\)-module, see Descent, Proposition 03DX. Hence \(\mathcal{A}(U) \to \mathcal{A}(V)\) is an étale ring map, see Algebra, Lemma 05YT. Hence we see that \[U \longmapsto U' = \Spec(\mathcal{A}(U))\] is a functor from \(X_{affine, \etale}\) to the category of affine schemes and étale morphisms. In fact, we claim that this functor can be extended to a functor \(U \mapsto U'\) on all of \(X_\etale\). To see this, if \(U\) is an object of \(X_\etale\), note that \[0 \to \mathcal{I}|_{U_{Zar}} \to \mathcal{A}|_{U_{Zar}} \to \mathcal{O}_X|_{U_{Zar}} \to 0\] and \(\mathcal{I}|_{U_{Zar}}\) is a quasi-coherent sheaf on \(U\), see Descent, Proposition 03LC. Hence by More on Morphisms, Lemma 05YV we obtain a first order thickening \(U \subset U'\) of schemes such that \(\mathcal{O}_{U'}\) is isomorphic to \(\mathcal{A}|_{U_{Zar}}\). It is clear that this construction is compatible with the construction for affines above.
Choose a presentation \(X = U/R\), see Spaces, Definition 0263 so that \(s, t : R \to U\) define an étale equivalence relation. Applying the functor above we obtain an étale equivalence relation \(s', t' : R' \to U'\) in schemes. Consider the algebraic space \(X' = U'/R'\) (see Spaces, Theorem 02WW). The morphism \(X = U/R \to U'/R' = X'\) is a first order thickening. Consider \(\mathcal{O}_{X'}\) viewed as a sheaf on \(X_\etale\). By construction we have an isomorphism \[\gamma : \mathcal{O}_{X'}|_{U_\etale} \longrightarrow \mathcal{A}|_{U_\etale}\] such that \(s^{-1}\gamma\) agrees with \(t^{-1}\gamma\) on \(R_\etale\). Hence by Properties of Spaces, Lemma 05YY this implies that \(\gamma\) comes from a unique isomorphism \(\mathcal{O}_{X'} \to \mathcal{A}\) as desired.
To handle the case of a general base algebraic space \(B\), we first construct \(X'\) as an algebraic space over \(\mathbf{Z}\) as above. Then we use the isomorphism \(\mathcal{O}_{X'} \to \mathcal{A}\) to define \(f^{-1}\mathcal{O}_B \to \mathcal{O}_{X'}\). According to Lemma 05ZN this defines a morphism \(X' \to B\) compatible with the given morphism \(X \to B\) and we are done.
Lemma
Let \(S\) be a scheme. Let \(Y \subset Y'\) be a thickening of algebraic spaces over \(S\). Let \(X' \to Y'\) be a morphism and set \(X = Y \times_{Y'} X'\). Then \((X \subset X') \to (Y \subset Y')\) is a morphism of thickenings. If \(Y \subset Y'\) is a first (resp. finite order) thickening, then \(X \subset X'\) is a first (resp. finite order) thickening.
Proof
Omitted.
Lemma
Let \(S\) be a scheme. If \(X \subset X'\) and \(X' \subset X''\) are thickenings of algebraic spaces over \(S\), then so is \(X \subset X''\).
Proof
Omitted.
Lemma
The property of being a thickening is fpqc local. Similarly for first order thickenings.
Proof
The statement means the following: Let \(S\) be a scheme and let \(X \to X'\) be a morphism of algebraic spaces over \(S\). Let \(\{g_i : X'_i \to X'\}\) be an fpqc covering of algebraic spaces such that the base change \(X_i \to X'_i\) is a thickening for all \(i\). Then \(X \to X'\) is a thickening. Since the morphisms \(g_i\) are jointly surjective we conclude that \(X \to X'\) is surjective. By Descent on Spaces, Lemma 0420 we conclude that \(X \to X'\) is a closed immersion. Thus \(X \to X'\) is a thickening. We omit the proof in the case of first order thickenings.
Morphisms of thickenings
If \((f, f') : (X \subset X') \to (Y \subset Y')\) is a morphism of thickenings of algebraic spaces, then often properties of the morphism \(f\) are inherited by \(f'\). There are several variants.
Lemma
Let \(S\) be a scheme. Let \((f, f') : (X \subset X') \to (Y \subset Y')\) be a morphism of thickenings of algebraic spaces over \(S\). Then
\(f\) is an affine morphism if and only if \(f'\) is an affine morphism,
\(f\) is a surjective morphism if and only if \(f'\) is a surjective morphism,
\(f\) is quasi-compact if and only if \(f'\) quasi-compact,
\(f\) is universally closed if and only if \(f'\) is universally closed,
\(f\) is integral if and only if \(f'\) is integral,
\(f\) is (quasi-)separated if and only if \(f'\) is (quasi-)separated,
\(f\) is universally injective if and only if \(f'\) is universally injective,
\(f\) is universally open if and only if \(f'\) is universally open,
\(f\) is representable if and only if \(f'\) is representable, and
add more here.
Proof
Observe that \(Y \to Y'\) and \(X \to X'\) are integral and universal homeomorphisms. This immediately implies parts (2), (3), (4), (7), and (8). Part (1) follows from Limits of Spaces, Proposition 07VT which tells us that there is a 1-to-1 correspondence between affine schemes étale over \(X\) and \(X'\) and between affine schemes étale over \(Y\) and \(Y'\). Part (5) follows from (1) and (4) by Morphisms of Spaces, Lemma 0415. Finally, note that \[X \times_Y X = X \times_{Y'} X \to X \times_{Y'} X' \to X' \times_{Y'} X'\] is a thickening (the two arrows are thickenings by Lemma 09ZX). Hence applying (3) and (4) to the morphism \((X \subset X') \to (X \times_Y X \to X' \times_{Y'} X')\) we obtain (6). Finally, part (9) follows from the fact that an algebraic space thickening of a scheme is again a scheme, see Lemma 05ZR.
Lemma
Let \(S\) be a scheme. Let \((f, f') : (X \subset X') \to (Y \subset Y')\) be a morphism of thickenings of algebraic spaces over \(S\) such that \(X = Y \times_{Y'} X'\). If \(X \subset X'\) is a finite order thickening, then
\(f\) is a closed immersion if and only if \(f'\) is a closed immersion,
\(f\) is locally of finite type if and only if \(f'\) is locally of finite type,
\(f\) is locally quasi-finite if and only if \(f'\) is locally quasi-finite,
\(f\) is locally of finite type of relative dimension \(d\) if and only if \(f'\) is locally of finite type of relative dimension \(d\),
\(\Omega_{X/Y} = 0\) if and only if \(\Omega_{X'/Y'} = 0\),
\(f\) is unramified if and only if \(f'\) is unramified,
\(f\) is proper if and only if \(f'\) is proper,
\(f\) is a finite morphism if and only if \(f'\) is an finite morphism,
\(f\) is a monomorphism if and only if \(f'\) is a monomorphism,
\(f\) is an immersion if and only if \(f'\) is an immersion, and
add more here.
Proof
Choose a scheme \(V'\) and a surjective étale morphism \(V' \to Y'\). Choose a scheme \(U'\) and a surjective étale morphism \(U' \to X' \times_{Y'} V'\). Set \(V = Y \times_{Y'} V'\) and \(U = X \times_{X'} U'\). Then for étale local properties of morphisms we can reduce to the morphism of thickenings of schemes \((U \subset U') \to (V \subset V')\) and apply More on Morphisms, Lemma 09ZW. This proves (2), (3), (4), (5), and (6).
The properties of morphisms in (1), (7), (8), (9), (10) are stable under base change, hence if \(f'\) has property \(\mathcal{P}\), then so does \(f\). See Spaces, Lemma 02YW, and Morphisms of Spaces, Lemmas 04WP, 03ZS, and 042P.
The interesting direction in (1), (7), (8), (9), (10) is to assume that \(f\) has the property and deduce that \(f'\) has it too. By induction on the order of the thickening we may assume that \(Y \subset Y'\) is a first order thickening, see discussion on finite order thickenings above.
Proof of (1). Choose a scheme \(V'\) and a surjective étale morphism \(V' \to Y'\). Set \(V = Y \times_{Y'} V'\), \(U' = X' \times_{Y'} V'\) and \(U = X \times_Y V\). Then \(U \to V\) is a closed immersion, which implies that \(U\) is a scheme, which in turn implies that \(U'\) is a scheme (Lemma 05ZR). Thus we can apply the lemma in the case of schemes (More on Morphisms, Lemma 09ZW) to \((U \subset U') \to (V \subset V')\) to conclude.
Proof of (7). Follows by combining (2) with results of Lemma 09ZY and the fact that proper equals quasi-compact \(+\) separated \(+\) locally of finite type \(+\) universally closed.
Proof of (8). Follows by combining (2) with results of Lemma 09ZY and using the fact that finite equals integral \(+\) locally of finite type (Morphisms, Lemma 01WJ).
Proof of (9). As \(f\) is a monomorphism we have \(X = X \times_Y X\). We may apply the results proved so far to the morphism of thickenings \((X \subset X') \to (X \times_Y X \subset X' \times_{Y'} X')\). We conclude \(X' \to X' \times_{Y'} X'\) is a closed immersion by (1). In fact, it is a first order thickening as the ideal defining the closed immersion \(X' \to X' \times_{Y'} X'\) is contained in the pullback of the ideal \(\mathcal{I} \subset \mathcal{O}_{Y'}\) cutting out \(Y\) in \(Y'\). Indeed, \(X = X \times_Y X = (X' \times_{Y'} X') \times_{Y'} Y\) is contained in \(X'\). The conormal sheaf of the closed immersion \(\Delta : X' \to X' \times_{Y'} X'\) is equal to \(\Omega_{X'/Y'}\) (this is the analogue of Morphisms, Lemma 08S2 for algebraic spaces and follows either by étale localization or by combining Lemmas 05ZB and 05ZD; some details omitted). Thus it suffices to show that \(\Omega_{X'/Y'} = 0\) which follows from (5) and the corresponding statement for \(X/Y\).
Proof of (10). If \(f : X \to Y\) is an immersion, then it factors as \(X \to V \to Y\) where \(V \to Y\) is an open subspace and \(X \to V\) is a closed immersion, see Morphisms of Spaces, Remark 04CE. Let \(V' \subset Y'\) be the open subspace whose underlying topological space \(|V'|\) is the same as \(|V| \subset |Y| = |Y'|\). Then \(X' \to Y'\) factors through \(V'\) and we conclude that \(X' \to V'\) is a closed immersion by part (1). This finishes the proof.
The following lemma is a variant on the preceding one. Rather than assume that the thickenings involved are finite order (which allows us to transfer the property of being locally of finite type from \(f\) to \(f'\)), we instead take as given that each of \(f\) and \(f'\) is locally of finite type.
Lemma
Let \(S\) be a scheme. Let \((f, f') : (X \subset X') \to (Y \to Y')\) be a morphism of thickenings of algebraic spaces over \(S\). Assume \(f\) and \(f'\) are locally of finite type and \(X = Y \times_{Y'} X'\). Then
\(f\) is locally quasi-finite if and only if \(f'\) is locally quasi-finite,
\(f\) is finite if and only if \(f'\) is finite,
\(f\) is a closed immersion if and only if \(f'\) is a closed immersion,
\(\Omega_{X/Y} = 0\) if and only if \(\Omega_{X'/Y'} = 0\),
\(f\) is unramified if and only if \(f'\) is unramified,
\(f\) is a monomorphism if and only if \(f'\) is a monomorphism,
\(f\) is an immersion if and only if \(f'\) is an immersion,
\(f\) is proper if and only if \(f'\) is proper, and
add more here.
Proof
Choose a scheme \(V'\) and a surjective étale morphism \(V' \to Y'\). Choose a scheme \(U'\) and a surjective étale morphism \(U' \to X' \times_{Y'} V'\). Set \(V = Y \times_{Y'} V'\) and \(U = X \times_{X'} U'\). Then for étale local properties of morphisms we can reduce to the morphism of thickenings of schemes \((U \subset U') \to (V \subset V')\) and apply More on Morphisms, Lemma 0BPG. This proves (1), (4), and (5).
The properties in (2), (3), (6), (7), and (8) are stable under base change, hence if \(f'\) has property \(\mathcal{P}\), then so does \(f\). See Spaces, Lemma 02YW, and Morphisms of Spaces, Lemmas 04WP, 03ZS, and 042P. Hence in each case we need only to prove that if \(f\) has the desired property, so does \(f'\).
Case (2) follows from case (5) of Lemma 09ZY and the fact that the finite morphisms are precisely the integral morphisms that are locally of finite type (Morphisms of Spaces, Lemma 0414).
Case (3). This follows immediately from Limits of Spaces, Lemma 08B2.
Proof of (6). As \(f\) is a monomorphism we have \(X = X \times_Y X\). We may apply the results proved so far to the morphism of thickenings \((X \subset X') \to (X \times_Y X \subset X' \times_{Y'} X')\). We conclude \(\Delta_{X'/Y'} : X' \to X' \times_{Y'} X'\) is a closed immersion by (3). In fact \(\Delta_{X'/Y'}\) induces a bijection \(|X'| \to |X' \times_{Y'} X'|\), hence \(\Delta_{X'/Y'}\) is a thickening. On the other hand \(\Delta_{X'/Y'}\) is locally of finite presentation by Morphisms of Spaces, Lemma 084P. In other words, \(\Delta_{X'/Y'}(X')\) is cut out by a quasi-coherent sheaf of ideals \(\mathcal{J} \subset \mathcal{O}_{X' \times_{Y'} X'}\) of finite type. Since \(\Omega_{X'/Y'} = 0\) by (5) we see that the conormal sheaf of \(X' \to X' \times_{Y'} X'\) is zero. (The conormal sheaf of the closed immersion \(\Delta_{X'/Y'}\) is equal to \(\Omega_{X'/Y'}\); this is the analogue of Morphisms, Lemma 08S2 for algebraic spaces and follows either by étale localization or by combining Lemmas 05ZB and 05ZD; some details omitted.) In other words, \(\mathcal{J}/\mathcal{J}^2 = 0\). This implies \(\Delta_{X'/Y'}\) is an isomorphism, for example by Algebra, Lemma 00EH.
Proof of (7). If \(f : X \to Y\) is an immersion, then it factors as \(X \to V \to Y\) where \(V \to Y\) is an open subspace and \(X \to V\) is a closed immersion, see Morphisms of Spaces, Remark 04CE. Let \(V' \subset Y'\) be the open subspace whose underlying topological space \(|V'|\) is the same as \(|V| \subset |Y| = |Y'|\). Then \(X' \to Y'\) factors through \(V'\) and we conclude that \(X' \to V'\) is a closed immersion by part (3).
Case (8) follows from Lemma 09ZY and the definition of proper morphisms as being the quasi-compact, universally closed, and separated morphisms that are locally of finite type.
Picard groups of thickenings
Some material on Picard groups of thickenings.
Lemma
Let \(S\) be a scheme. Let \(X \subset X'\) be a first order thickening of algebraic spaces over \(S\) with ideal sheaf \(\mathcal{I}\). Then there is a canonical exact sequence \[\xymatrix{ 0 \ar[r] & H^0(X, \mathcal{I}) \ar[r] & H^0(X', \mathcal{O}_{X'}^*) \ar[r] & H^0(X, \mathcal{O}^*_X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^1(X, \mathcal{I}) \ar[r] & \Pic(X') \ar[r] & \Pic(X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^2(X, \mathcal{I}) \ar[r] & \ldots \ar[r] & \ldots }\] of abelian groups.
Proof
Recall that \(X_\etale = X'_\etale\), see Lemma 05ZS and more generally the discussion in Section 05ZJ. The sequence of the lemma is the long exact cohomology sequence associated to the short exact sequence of sheaves of abelian groups \[0 \to \mathcal{I} \to \mathcal{O}_{X'}^* \to \mathcal{O}_X^* \to 0\] on \(X_\etale\) where the first map sends a local section \(f\) of \(\mathcal{I}\) to the invertible section \(1 + f\) of \(\mathcal{O}_{X'}\). We also use the identification of the Picard group of a ringed site with the first cohomology group of the sheaf of invertible functions, see Cohomology on Sites, Lemma 040E.
Infinitesimal neighbourhoods
A natural construction of finite order thickenings is the following. Suppose that \(i : Z \to X\) be an immersion of algebraic spaces. Choose an open subspace \(U \subset X\) such that \(i\) identifies \(Z\) with a closed subspace \(Z \subset U\) (see Morphisms of Spaces, Remark 04CE). Let \(\mathcal{I} \subset \mathcal{O}_U\) be the quasi-coherent sheaf of ideals defining \(Z\) in \(U\), see Morphisms of Spaces, Lemma 03MB. For \(n \geq 1\) we can consider the closed subspace \(Z_n \subset U\) defined by the quasi-coherent sheaf of ideals \(\mathcal{I}^{n + 1}\).
Definition
Let \(i : Z \to X\) be an immersion of algebraic spaces.
The first order infinitesimal neighbourhood of \(Z\) in \(X\) is the first order thickening \(Z \subset Z_1\) over \(X\) described above.
The \(n\)th order infinitesimal neighbourhood of \(Z\) in \(X\) is the \(n\)th order thickening \(Z \subset Z_n\) over \(X\) described above.
This thickening has the following universal property (which will assuage any fears that the construction above depends on the choice of the open \(U\)).
Lemma
Let \(i : Z \to X\) be an immersion of algebraic spaces.
The first order infinitesimal neighbourhood \(Z'\) of \(Z\) in \(X\) has the following universal property: Given any commutative diagram \[\xymatrix{ Z \ar[d]_i & T \ar[l]^a \ar[d] \\ X & T' \ar[l]_b }\] where \(T \subset T'\) is a first order thickening over \(X\), there exists a unique morphism \((a', a) : (T \subset T') \to (Z \subset Z')\) of thickenings over \(X\).
For \(n \geq 1\) the \(n\)th order infinitesimal neighbourhood \(Z_n\) of \(Z\) in \(X\) has the following universal property: Given any commutative diagram \[\xymatrix{ Z \ar[d]_i & T \ar[l]^a \ar[d] \\ X & T' \ar[l]_b }\] where \(T \subset T'\) is an \(n\)th order thickening over \(X\), there exists a unique morphism \((a', a) : (T \subset T') \to (Z \subset Z_n)\) of thickenings over \(X\).
Proof
We will only prove (1). Let \(U \subset X\) be the open subspace used in the construction of \(Z'\), i.e., an open such that \(Z\) is identified with a closed subspace of \(U\) cut out by the quasi-coherent sheaf of ideals \(\mathcal{I}\). Since \(|T| = |T'|\) we see that \(|b|(|T'|) \subset |U|\). Hence we can think of \(b\) as a morphism into \(U\), see Properties of Spaces, Lemma 03IE. Let \(\mathcal{J} \subset \mathcal{O}_{T'}\) be the square zero quasi-coherent sheaf of ideals cutting out \(T\). By the commutativity of the diagram we have \(b|_T = i \circ a\) where \(i : Z \to U\) is the closed immersion. We conclude that \(b^\sharp(b^{-1}\mathcal{I}) \subset \mathcal{J}\) by Morphisms of Spaces, Lemma 03MB. As \(T'\) is a first order thickening of \(T\) we see that \(\mathcal{J}^2 = 0\) hence \(b^\sharp(b^{-1}(\mathcal{I}^2)) = 0\). By Morphisms of Spaces, Lemma 03MB this implies that \(b\) factors through \(Z'\). Letting \(a' : T' \to Z'\) be this factorization we win.
Lemma
Let \(i : Z \to X\) be an immersion of algebraic spaces. Let \(Z \subset Z'\) be the first order infinitesimal neighbourhood of \(Z\) in \(X\). Then the diagram \[\xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Z \ar[r] & X }\] induces a map of conormal sheaves \(\mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Z'}\) by Lemma 04CP. This map is an isomorphism.
Proof
This is clear from the construction of \(Z'\) above.
Formally smooth, étale, unramified transformations
Recall that a ring map \(R \to A\) is called formally smooth, resp. formally étale, resp. formally unramified (see Algebra, Definition 00TI, resp. Definition 00UQ, resp. Definition 00UN) if for every commutative solid diagram \[\xymatrix{ A \ar[r] \ar@{-->}[rd] & B/I \\ R \ar[r] \ar[u] & B \ar[u] }\] where \(I \subset B\) is an ideal of square zero, there exists a, resp. exists a unique, resp. exists at most one dotted arrow which makes the diagram commute. This motivates the following analogue for morphisms of algebraic spaces, and more generally functors.
Definition
Let \(S\) be a scheme. Let \(a : F \to G\) be a transformation of functors \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Consider commutative solid diagrams of the form \[\xymatrix{ F \ar[d]_a & T \ar[d]^i \ar[l] \\ G & T' \ar[l] \ar@{-->}[lu] }\] where \(T\) and \(T'\) are affine schemes and \(i\) is a closed immersion defined by an ideal of square zero.
We say \(a\) is formally smooth if given any solid diagram as above there exists a dotted arrow making the diagram commute1.
We say \(a\) is formally étale if given any solid diagram as above there exists exactly one dotted arrow making the diagram commute.
We say \(a\) is formally unramified if given any solid diagram as above there exists at most one dotted arrow making the diagram commute.
Lemma
Let \(S\) be a scheme. Let \(a : F \to G\) be a transformation of functors \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Then \(a\) is formally étale if and only if \(a\) is both formally smooth and formally unramified.
Proof
Formal from the definition.
Lemma
Composition.
A composition of formally smooth transformations of functors is formally smooth.
A composition of formally étale transformations of functors is formally étale.
A composition of formally unramified transformations of functors is formally unramified.
Proof
This is formal.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\), \(b : H \to G\) be transformations of functors. Consider the fibre product diagram \[\xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G }\]
If \(a\) is formally smooth, then the base change \(a'\) is formally smooth.
If \(a\) is formally étale, then the base change \(a'\) is formally étale.
If \(a\) is formally unramified, then the base change \(a'\) is formally unramified.
Proof
This is formal.
Lemma
Let \(S\) be a scheme. Let \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\) be a representable transformation of functors.
If \(a\) is smooth then \(a\) is formally smooth.
If \(a\) is étale, then \(a\) is formally étale.
If \(a\) is unramified, then \(a\) is formally unramified.
Proof
Consider a solid commutative diagram \[\xymatrix{ F \ar[d]_a & T \ar[d]^i \ar[l] \\ G & T' \ar[l] \ar@{-->}[lu] }\] as in Definition 049S. Then \(F \times_G T'\) is a scheme smooth (resp. étale, resp. unramified) over \(T'\). Hence by More on Morphisms, Lemma 02H6 (resp. Lemma 02HM, resp. Lemma 02HE) we can fill in (resp. uniquely fill in, resp. fill in at most one way) the dotted arrow in the diagram \[\xymatrix{ F \times_G T' \ar[d] & T \ar[d]^i \ar[l] \\ T' & T' \ar[l] \ar@{-->}[lu] }\] an hence we also obtain the corresponding assertion in the first diagram.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\), \(b : G \to H\) be transformations of functors. Assume that \(a\) is representable, surjective, and étale.
If \(b\) is formally smooth, then \(b \circ a\) is formally smooth.
If \(b\) is formally étale, then \(b \circ a\) is formally étale.
If \(b\) is formally unramified, then \(b \circ a\) is formally unramified.
Conversely, consider a solid commutative diagram \[\xymatrix{ G \ar[d]_b & T \ar[d]^i \ar[l] \\ H & T' \ar[l] \ar@{-->}[lu] }\] with \(T'\) an affine scheme over \(S\) and \(i : T \to T'\) a closed immersion defined by an ideal of square zero.
If \(b \circ a\) is formally smooth, then for every \(t \in T\) there exists an étale morphism of affines \(U' \to T'\) and a morphism \(U' \to G\) such that \[\xymatrix{ G \ar[d]_b & T \ar[l] & T \times_{T'} U' \ar[d] \ar[l]\\ H & T' \ar[l] & U' \ar[llu] \ar[l] }\] commutes and \(t\) is in the image of \(U' \to T'\).
If \(b \circ a\) is formally unramified, then there exists at most one dotted arrow in the diagram above, i.e., \(b\) is formally unramified.
If \(b \circ a\) is formally étale, then there exists exactly one dotted arrow in the diagram above, i.e., \(b\) is formally étale.
Proof
Assume \(b\) is formally smooth (resp. formally étale, resp. formally unramified). Since an étale morphism is both smooth and unramified we see that \(a\) is representable and smooth (resp. étale, resp. unramified). Hence parts (1), (2) and (3) follow from a combination of Lemma 04AL and Lemma 049T.
Assume that \(b \circ a\) is formally smooth. Consider a diagram as in the statement of the lemma. Let \(W = F \times_G T\). By assumption \(W\) is a scheme surjective étale over \(T\). By Étale Morphisms, Theorem 039R there exists a scheme \(W'\) étale over \(T'\) such that \(W = T \times_{T'} W'\). Choose an affine open subscheme \(U' \subset W'\) such that \(t\) is in the image of \(U' \to T'\). Because \(b \circ a\) is formally smooth we see that the exist morphisms \(U' \to F\) such that \[\xymatrix{ F \ar[d]_{b \circ a} & W \ar[l] & T \times_{T'} U' \ar[d] \ar[l]\\ H & T' \ar[l] & U' \ar[llu] \ar[l] }\] commutes. Taking the composition \(U' \to F \to G\) gives a map as in part (5) of the lemma.
Assume that \(f, g : T' \to G\) are two dotted arrows fitting into the diagram of the lemma. Let \(W = F \times_G T\). By assumption \(W\) is a scheme surjective étale over \(T\). By Étale Morphisms, Theorem 039R there exists a scheme \(W'\) étale over \(T'\) such that \(W = T \times_{T'} W'\). Since \(a\) is formally étale the compositions \[W' \to T' \xrightarrow{f} G \quad\text{and}\quad W' \to T' \xrightarrow{g} G\] lift to morphisms \(f', g' : W' \to F\) (lift on affine opens and glue by uniqueness). Now if \(b \circ a : F \to H\) is formally unramified, then \(f' = g'\) and hence \(f = g\) as \(W' \to T'\) is an étale covering. This proves part (6) of the lemma.
Assume that \(b \circ a\) is formally étale. Then by part (4) we can étale locally on \(T'\) find a dotted arrow fitting into the diagram and by part (5) this dotted arrow is unique. Hence we may glue the local solutions to get assertion (6). Some details omitted.
Remark
It is tempting to think that in the situation of Lemma 04CY we have “\(b\) formally smooth” \(\Leftrightarrow\) “\(b \circ a\) formally smooth”. However, this is likely not true in general.
Lemma
Let \(S\) be a scheme. Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\), \(b : G \to H\) be transformations of functors. Assume \(b\) is formally unramified.
If \(b \circ a\) is formally unramified then \(a\) is formally unramified.
If \(b \circ a\) is formally étale then \(a\) is formally étale.
If \(b \circ a\) is formally smooth then \(a\) is formally smooth.
Proof
Let \(T \subset T'\) be a closed immersion of affine schemes defined by an ideal of square zero. Let \(g' : T' \to G\) and \(f : T \to F\) be given such that \(g'|_T = a \circ f\). Because \(b\) is formally unramified, there is a one to one correspondence between \[\{f' : T' \to F \mid f = f'|_T\text{ and }a \circ f' = g'\}\] and \[\{f' : T' \to F \mid f = f'|_T\text{ and }b \circ a \circ f' = b \circ g'\}.\] From this the lemma follows formally.
Formally unramified morphisms
In this section we work out what it means that a morphism of algebraic spaces is formally unramified.
Definition
Let \(S\) be a scheme. A morphism \(f : X \to Y\) of algebraic spaces over \(S\) is said to be formally unramified if it is formally unramified as a transformation of functors as in Definition 049S.
We will not restate the results proved in the more general setting of formally unramified transformations of functors in Section 04G3. It turns out we can characterize this property in terms of vanishing of the module of relative differentials, see Lemma 04G9.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
\(f\) is formally unramified,
for every diagram \[\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\) and \(V\) are schemes and the vertical arrows are étale the morphism of schemes \(\psi\) is formally unramified (as in More on Morphisms, Definition 02H8), and
for one such diagram with surjective vertical arrows the morphism \(\psi\) is formally unramified.
Proof
Assume \(f\) is formally unramified. By Lemma 04AL the morphisms \(U \to X\) and \(V \to Y\) are formally unramified. Thus by Lemma 049T the composition \(U \to Y\) is formally unramified. Then it follows from Lemma 04G5 that \(U \to V\) is formally unramified. Thus (1) implies (2). And (2) implies (3) trivially
Assume given a diagram as in (3). By Lemma 04AL the morphism \(V \to Y\) is formally unramified. Thus by Lemma 049T the composition \(U \to Y\) is formally unramified. Then it follows from Lemma 04CY that \(X \to Y\) is formally unramified, i.e., (1) holds.
Lemma
Let \(S\) be a scheme. If \(f : X \to Y\) is a formally unramified morphism of algebraic spaces over \(S\), then given any solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ Y & T' \ar[l] \ar@{-->}[lu] }\] where \(T \subset T'\) is a first order thickening of algebraic spaces over \(S\) there exists at most one dotted arrow making the diagram commute. In other words, in Definition 04G7 the condition that \(T\) be an affine scheme may be dropped.
Proof
This is true because there exists a surjective étale morphism \(U' \to T'\) where \(U'\) is a disjoint union of affine schemes (see Properties of Spaces, Lemma 03FX) and a morphism \(T' \to X\) is determined by its restriction to \(U'\).
Lemma
A composition of formally unramified morphisms is formally unramified.
Proof
This is formal.
Lemma
A base change of a formally unramified morphism is formally unramified.
Proof
This is formal.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
\(f\) is formally unramified, and
\(\Omega_{X/Y} = 0\).
Proof
This is a combination of Lemma 04G8, More on Morphisms, Lemma 02H9, and Lemma 04CU.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
The morphism \(f\) is unramified,
the morphism \(f\) is locally of finite type and \(\Omega_{X/Y} = 0\), and
the morphism \(f\) is locally of finite type and formally unramified.
Proof
Choose a diagram \[\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\) and \(V\) are schemes and the vertical arrows are étale and surjective. Then we see \[\begin{align*} f\text{ unramified} & \Leftrightarrow \psi\text{ unramified} \\ & \Leftrightarrow \psi\text{ locally finite type and }\Omega_{U/V} = 0 \\ & \Leftrightarrow f\text{ locally finite type and }\Omega_{X/Y} = 0 \\ & \Leftrightarrow f\text{ locally finite type and formally unramified} \end{align*}\] Here we have used Morphisms, Lemma 02G5 and Lemma 04G9.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
\(f\) is unramified and a monomorphism,
\(f\) is unramified and universally injective,
\(f\) is locally of finite type and a monomorphism,
\(f\) is universally injective, locally of finite type, and formally unramified.
Moreover, in this case \(f\) is also representable, separated, and locally quasi-finite.
Proof
We have seen in Lemma 04GA that being formally unramified and locally of finite type is the same thing as being unramified. Hence (4) is equivalent to (2). A monomorphism is certainly formally unramified hence (3) implies (4). It is clear that (1) implies (3). Finally, if (2) holds, then \(\Delta : X \to X \times_Y X\) is both an open immersion (Morphisms of Spaces, Lemma 05W1) and surjective (Morphisms of Spaces, Lemma 040X) hence an isomorphism, i.e., \(f\) is a monomorphism. In this way we see that (2) implies (1). Finally, we see that \(f\) is representable, separated, and locally quasi-finite by Morphisms of Spaces, Lemmas 0463 and 0418.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
\(f\) is a closed immersion,
\(f\) is universally closed, unramified, and a monomorphism,
\(f\) is universally closed, unramified, and universally injective,
\(f\) is universally closed, locally of finite type, and a monomorphism,
\(f\) is universally closed, universally injective, locally of finite type, and formally unramified.
Proof
The equivalence of (2) – (5) follows immediately from Lemma 05W6. Moreover, if (2) – (5) are satisfied then \(f\) is representable. Similarly, if (1) is satisfied then \(f\) is representable. Hence the result follows from the case of schemes, see Étale Morphisms, Lemma 04XV.
Universal first order thickenings
Let \(S\) be a scheme. Let \(h : Z \to X\) be a morphism of algebraic spaces over \(S\). A universal first order thickening of \(Z\) over \(X\) is a first order thickening \(Z \subset Z'\) over \(X\) such that given any first order thickening \(T \subset T'\) over \(X\) and a solid commutative diagram [0602]\[\begin{equation} \vcenter{ \xymatrix{ & Z \ar[ld] & & T \ar[rd] \ar[ll]^a \\ Z' \ar[rrd] & & & & T' \ar@{..>}[llll]_{a'} \ar[lld]^b \\ & & X } } \end{equation}\] there exists a unique dotted arrow making the diagram commute. Note that in this situation \((a, a') : (T \subset T') \to (Z \subset Z')\) is a morphism of thickenings over \(X\). Thus if a universal first order thickening exists, then it is unique up to unique isomorphism. In general a universal first order thickening does not exist, but if \(h\) is formally unramified then it does. Before we prove this, let us show that a universal first order thickening in the category of schemes is a universal first order thickening in the category of algebraic spaces.
Lemma
Let \(S\) be a scheme. Let \(h : Z \to X\) be a morphism of algebraic spaces over \(S\). Let \(Z \subset Z'\) be a first order thickening over \(X\). The following are equivalent
Proof
The implications (1) \(\Rightarrow\) (2) \(\Rightarrow\) (3) are formal. Assume (3) a assume given an arbitrary diagram (0602). Choose a presentation \(T' = U'/R'\), see Spaces, Definition 0263. We may assume that \(U' = \coprod U'_i\) is a disjoint union of affines, so \(R' = U' \times_{T'} U' = \coprod_{i, j} U'_i \times_T' U'_j\). For each pair \((i, j)\) choose an affine open covering \(U'_i \times_T' U'_j = \bigcup_k R'_{ijk}\). Denote \(U_i, R_{ijk}\) the fibre products with \(T\) over \(T'\). Then each \(U_i \subset U'_i\) and \(R_{ijk} \subset R'_{ijk}\) is a first order thickening of affine schemes. Denote \(a_i : U_i \to Z\), resp. \(a_{ijk} : R_{ijk} \to Z\) the composition of \(a : T \to Z\) with the morphism \(U_i \to T\), resp. \(R_{ijk} \to T\). By (3) applied to \(a_i : U_i \to Z\) we obtain unique morphisms \(a'_i : U'_i \to Z'\). By (3) applied to \(a_{ijk}\) we see that the two compositions \(R'_{ijk} \to R'_i \to Z'\) and \(R'_{ijk} \to R'_j \to Z'\) are equal. Hence \(a' = \coprod a'_i : U' = \coprod U'_i \to Z'\) descends to the quotient sheaf \(T' = U'/R'\) and we win.
Lemma
Let \(S\) be a scheme. Let \(Z \to Y \to X\) be morphisms of algebraic spaces over \(S\). If \(Z \subset Z'\) is a universal first order thickening of \(Z\) over \(Y\) and \(Y \to X\) is formally étale, then \(Z \subset Z'\) is a universal first order thickening of \(Z\) over \(X\).
Proof
This is formal. Namely, by Lemma 0603 it suffices to consider solid commutative diagrams (0602) with \(T'\) an affine scheme. The composition \(T \to Z \to Y\) lifts uniquely to \(T' \to Y\) as \(Y \to X\) is assumed formally étale. Hence the fact that \(Z \subset Z'\) is a universal first order thickening over \(Y\) produces the desired morphism \(a' : T' \to Z'\).
Lemma
Let \(S\) be a scheme. Let \(Z \to Y \to X\) be morphisms of algebraic spaces over \(S\). Assume \(Z \to Y\) is étale.
If \(Y \subset Y'\) is a universal first order thickening of \(Y\) over \(X\), then the unique étale morphism \(Z' \to Y'\) such that \(Z = Y \times_{Y'} Z'\) (see Theorem 05ZH) is a universal first order thickening of \(Z\) over \(X\).
If \(Z \to Y\) is surjective and \((Z \subset Z') \to (Y \subset Y')\) is an étale morphism of first order thickenings over \(X\) and \(Z'\) is a universal first order thickening of \(Z\) over \(X\), then \(Y'\) is a universal first order thickening of \(Y\) over \(X\).
Proof
Proof of (1). By Lemma 0603 it suffices to consider solid commutative diagrams (0602) with \(T'\) an affine scheme. The composition \(T \to Z \to Y\) lifts uniquely to \(T' \to Y'\) as \(Y'\) is the universal first order thickening. Then the fact that \(Z' \to Y'\) is étale implies (see Lemma 04AL) that \(T' \to Y'\) lifts to the desired morphism \(a' : T' \to Z'\).
Proof of (2). Let \(T \subset T'\) be a first order thickening over \(X\) and let \(a : T \to Y\) be a morphism. Set \(W = T \times_Y Z\) and denote \(c : W \to Z\) the projection Let \(W' \to T'\) be the unique étale morphism such that \(W = T \times_{T'} W'\), see Theorem 05ZH. Note that \(W' \to T'\) is surjective as \(Z \to Y\) is surjective. By assumption we obtain a unique morphism \(c' : W' \to Z'\) over \(X\) restricting to \(c\) on \(W\). By uniqueness the two restrictions of \(c'\) to \(W' \times_{T'} W'\) are equal (as the two restrictions of \(c\) to \(W \times_T W\) are equal). Hence \(c'\) descends to a unique morphism \(a' : T' \to Y'\) and we win.
Lemma
Let \(S\) be a scheme. Let \(h : Z \to X\) be a formally unramified morphism of algebraic spaces over \(S\). There exists a universal first order thickening \(Z \subset Z'\) of \(Z\) over \(X\).
Proof
Choose any commutative diagram \[\xymatrix{ V \ar[d] \ar[r] & U \ar[d] \\ Z \ar[r] & X }\] where \(V\) and \(U\) are schemes and the vertical arrows are étale. Note that \(V \to U\) is a formally unramified morphism of schemes, see Lemma 04G8. Combining Lemma 0603 and More on Morphisms, Lemma 04F3 we see that a universal first order thickening \(V \subset V'\) of \(V\) over \(U\) exists. By Lemma 0604 part (1) \(V'\) is a universal first order thickening of \(V\) over \(X\).
Fix a scheme \(U\) and a surjective étale morphism \(U \to X\). The argument above shows that for any \(V \to Z\) étale with \(V\) a scheme such that \(V \to Z \to X\) factors through \(U\) a universal first order thickening \(V \subset V'\) of \(V\) over \(X\) exists (but does not depend on the chosen factorization of \(V \to X\) through \(U\)). Now we may choose \(V\) such that \(V \to Z\) is surjective étale (see Spaces, Lemma 02X1). Then \(R = V \times_Z V\) a scheme étale over \(Z\) such that \(R \to X\) factors through \(U\) also. Hence we obtain universal first order thickenings \(V \subset V'\) and \(R \subset R'\) over \(X\). As \(V \subset V'\) is a universal first order thickening, the two projections \(s, t : R \to V\) lift to morphisms \(s', t': R' \to V'\). By Lemma 0605 as \(R'\) is the universal first order thickening of \(R\) over \(X\) these morphisms are étale. Then \((t', s') : R' \to V'\) is an étale equivalence relation and we can set \(Z' = V'/R'\). Since \(V' \to Z'\) is surjective étale and \(v'\) is the universal first order thickening of \(V\) over \(X\) we conclude from Lemma 0604 part (2) that \(Z'\) is a universal first order thickening of \(Z\) over \(X\).
Definition
Let \(S\) be a scheme. Let \(h : Z \to X\) be a formally unramified morphism of algebraic spaces over \(S\).
The universal first order thickening of \(Z\) over \(X\) is the thickening \(Z \subset Z'\) constructed in Lemma 0606.
The conormal sheaf of \(Z\) over \(X\) is the conormal sheaf of \(Z\) in its universal first order thickening \(Z'\) over \(X\).
We often denote the conormal sheaf \(\mathcal{C}_{Z/X}\) in this situation.
Thus we see that there is a short exact sequence of sheaves \[0 \to \mathcal{C}_{Z/X} \to \mathcal{O}_{Z'} \to \mathcal{O}_Z \to 0\] on \(Z_\etale\) and \(\mathcal{C}_{Z/X}\) is a quasi-coherent \(\mathcal{O}_Z\)-module. The following lemma proves that there is no conflict between this definition and the definition in case \(Z \to X\) is an immersion.
Lemma
Let \(S\) be a scheme. Let \(i : Z \to X\) be an immersion of algebraic spaces over \(S\). Then
\(i\) is formally unramified,
the universal first order thickening of \(Z\) over \(X\) is the first order infinitesimal neighbourhood of \(Z\) in \(X\) of Definition 05ZV,
the conormal sheaf of \(i\) in the sense of Definition 04CN agrees with the conormal sheaf of \(i\) in the sense of Definition 0607.
Proof
An immersion of algebraic spaces is by definition a representable morphism. Hence by Morphisms, Lemmas 02GB and 02GC an immersion is unramified (via the abstract principle of Spaces, Lemma 02YO). Hence it is formally unramified by Lemma 04GA. The other assertions follow by combining Lemmas 05ZW and 05ZX and the definitions.
Lemma
Let \(S\) be a scheme. Let \(Z \to X\) be a formally unramified morphism of algebraic spaces over \(S\). Then the universal first order thickening \(Z'\) is formally unramified over \(X\).
Proof
Let \(T \subset T'\) be a first order thickening of affine schemes over \(X\). Let \[\xymatrix{ Z' \ar[d] & T \ar[l]^c \ar[d] \\ X & T' \ar[l] \ar[lu]^{a, b} }\] be a commutative diagram. Set \(T_0 = c^{-1}(Z) \subset T\) and \(T'_a = a^{-1}(Z)\) (scheme theoretically). Since \(Z'\) is a first order thickening of \(Z\), we see that \(T'\) is a first order thickening of \(T'_a\). Moreover, since \(c = a|_T\) we see that \(T_0 = T \cap T'_a\) (scheme theoretically). As \(T'\) is a first order thickening of \(T\) it follows that \(T'_a\) is a first order thickening of \(T_0\). Now \(a|_{T'_a}\) and \(b|_{T'_a}\) are morphisms of \(T'_a\) into \(Z'\) over \(X\) which agree on \(T_0\) as morphisms into \(Z\). Hence by the universal property of \(Z'\) we conclude that \(a|_{T'_a} = b|_{T'_a}\). Thus \(a\) and \(b\) are morphism from the first order thickening \(T'\) of \(T'_a\) whose restrictions to \(T'_a\) agree as morphisms into \(Z\). Thus using the universal property of \(Z'\) once more we conclude that \(a = b\). In other words, the defining property of a formally unramified morphism holds for \(Z' \to X\) as desired.
Lemma
Let \(S\) be a scheme Consider a commutative diagram of algebraic spaces over \(S\) \[\xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y }\] with \(h\) and \(h'\) formally unramified. Let \(Z \subset Z'\) be the universal first order thickening of \(Z\) over \(X\). Let \(W \subset W'\) be the universal first order thickening of \(W\) over \(Y\). There exists a canonical morphism \((f, f') : (Z, Z') \to (W, W')\) of thickenings over \(Y\) which fits into the following commutative diagram \[\xymatrix{ & & & Z' \ar[ld] \ar[d]^{f'} \\ Z \ar[rr] \ar[d]_f \ar[rrru] & & X \ar[d] & W' \ar[ld] \\ W \ar[rrru]|!{[rr];[rruu]}\hole \ar[rr] & & Y }\] In particular the morphism \((f, f')\) of thickenings induces a morphism of conormal sheaves \(f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}\).
Proof
The first assertion is clear from the universal property of \(W'\). The induced map on conormal sheaves is the map of Lemma 04CP applied to \((Z \subset Z') \to (W \subset W')\).
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y }\] be a fibre product diagram of algebraic spaces over \(S\) with \(h'\) formally unramified. Then \(h\) is formally unramified and if \(W \subset W'\) is the universal first order thickening of \(W\) over \(Y\), then \(Z = X \times_Y W \subset X \times_Y W'\) is the universal first order thickening of \(Z\) over \(X\). In particular the canonical map \(f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}\) of Lemma 060A is surjective.
Proof
The morphism \(h\) is formally unramified by Lemma 0600. It is clear that \(X \times_Y W'\) is a first order thickening. It is straightforward to check that it has the universal property because \(W'\) has the universal property (by mapping properties of fibre products). See Lemma 04CQ for why this implies that the map of conormal sheaves is surjective.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y }\] be a fibre product diagram of algebraic spaces over \(S\) with \(h'\) formally unramified and \(g\) flat. In this case the corresponding map \(Z' \to W'\) of universal first order thickenings is flat, and \(f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}\) is an isomorphism.
Proof
Flatness is preserved under base change, see Morphisms of Spaces, Lemma 03MO. Hence the first statement follows from the description of \(W'\) in Lemma 060B. It is clear that \(X \times_Y W'\) is a first order thickening. It is straightforward to check that it has the universal property because \(W'\) has the universal property (by mapping properties of fibre products). See Lemma 04CQ for why this implies that the map of conormal sheaves is an isomorphism.
Lemma
Taking the universal first order thickenings commutes with étale localization. More precisely, let \(h : Z \to X\) be a formally unramified morphism of algebraic spaces over a base scheme \(S\). Let \[\xymatrix{ V \ar[d] \ar[r] & U \ar[d] \\ Z \ar[r] & X }\] be a commutative diagram with étale vertical arrows. Let \(Z'\) be the universal first order thickening of \(Z\) over \(X\). Then \(V \to U\) is formally unramified and the universal first order thickening \(V'\) of \(V\) over \(U\) is étale over \(Z'\). In particular, \(\mathcal{C}_{Z/X}|_V = \mathcal{C}_{V/U}\).
Proof
The first statement is Lemma 04G8. The compatibility of universal first order thickenings is a consequence of Lemmas 0604 and 0605.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(h : Z \to X\) be a formally unramified morphism of algebraic spaces over \(B\). Let \(Z \subset Z'\) be the universal first order thickening of \(Z\) over \(X\) with structure morphism \(h' : Z' \to X\). The canonical map \[\text{d}h' : (h')^*\Omega_{X/B} \to \Omega_{Z'/B}\] induces an isomorphism \(h^*\Omega_{X/B} \to \Omega_{Z'/B} \otimes \mathcal{O}_Z\).
Proof
The map \(c_{h'}\) is the map defined in Lemma 04CX. If \(i : Z \to Z'\) is the given closed immersion, then \(i^*c_{h'}\) is a map \(h^*\Omega_{X/S} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z\). Checking that it is an isomorphism reduces to the case of schemes by étale localization, see Lemma 060D and Lemma 04CU. In this case the result is More on Morphisms, Lemma 04FB.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(h : Z \to X\) be a formally unramified morphism of algebraic spaces over \(B\). There is a canonical exact sequence \[\mathcal{C}_{Z/X} \to h^*\Omega_{X/B} \to \Omega_{Z/B} \to 0.\] The first arrow is induced by \(\text{d}_{Z'/B}\) where \(Z'\) is the universal first order neighbourhood of \(Z\) over \(X\).
Proof
We know that there is a canonical exact sequence \[\mathcal{C}_{Z/Z'} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z \to \Omega_{Z/S} \to 0.\] see Lemma 05ZA. Hence the result follows on applying Lemma 060E.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d] \\ & Y }\] be a commutative diagram of algebraic spaces over \(S\) where \(i\) and \(j\) are formally unramified. Then there is a canonical exact sequence \[\mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/Y} \to 0\] where the first arrow comes from Lemma 060A and the second from Lemma 060F.
Proof
Since the maps have been defined, checking the sequence is exact reduces to the case of schemes by étale localization, see Lemma 060D and Lemma 04CU. In this case the result is More on Morphisms, Lemma 067V.
Lemma
Let \(S\) be a scheme. Let \(Z \to Y \to X\) be formally unramified morphisms of algebraic spaces over \(S\).
If \(Z \subset Z'\) is the universal first order thickening of \(Z\) over \(X\) and \(Y \subset Y'\) is the universal first order thickening of \(Y\) over \(X\), then there is a morphism \(Z' \to Y'\) and \(Y \times_{Y'} Z'\) is the universal first order thickening of \(Z\) over \(Y\).
There is a canonical exact sequence \[i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0\] where the maps come from Lemma 060A and \(i : Z \to Y\) is the first morphism.
Proof
The map \(h : Z' \to Y'\) in (1) comes from Lemma 060A. The assertion that \(Y \times_{Y'} Z'\) is the universal first order thickening of \(Z\) over \(Y\) is clear from the universal properties of \(Z'\) and \(Y'\). By Lemma 06BD we have an exact sequence \[(i')^*\mathcal{C}_{Y \times_{Y'} Z'/Z'} \to \mathcal{C}_{Z/Z'} \to \mathcal{C}_{Z/Y \times_{Y'} Z'} \to 0\] where \(i' : Z \to Y \times_{Y'} Z'\) is the given morphism. By Lemma 04CQ there exists a surjection \(h^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{Y \times_{Y'} Z'/Z'}\). Combined with the equalities \(\mathcal{C}_{Y/Y'} = \mathcal{C}_{Y/X}\), \(\mathcal{C}_{Z/Z'} = \mathcal{C}_{Z/X}\), and \(\mathcal{C}_{Z/Y \times_{Y'} Z'} = \mathcal{C}_{Z/Y}\) this proves the lemma.
Formally étale morphisms
In this section we work out what it means that a morphism of algebraic spaces is formally étale.
Definition
Let \(S\) be a scheme. A morphism \(f : X \to Y\) of algebraic spaces over \(S\) is said to be formally étale if it is formally étale as a transformation of functors as in Definition 049S.
We will not restate the results proved in the more general setting of formally étale transformations of functors in Section 04G3.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
\(f\) is formally étale,
for every diagram \[\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\) and \(V\) are schemes and the vertical arrows are étale the morphism of schemes \(\psi\) is formally étale (as in More on Morphisms, Definition 02HG), and
for one such diagram with surjective vertical arrows the morphism \(\psi\) is formally étale.
Proof
Assume \(f\) is formally étale. By Lemma 04AL the morphisms \(U \to X\) and \(V \to Y\) are formally étale. Thus by Lemma 049T the composition \(U \to Y\) is formally étale. Then it follows from Lemma 04G5 that \(U \to V\) is formally étale. Thus (1) implies (2). And (2) implies (3) trivially
Assume given a diagram as in (3). By Lemma 04AL the morphism \(V \to Y\) is formally étale. Thus by Lemma 049T the composition \(U \to Y\) is formally étale. Then it follows from Lemma 04CY that \(X \to Y\) is formally étale, i.e., (1) holds.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a formally étale morphism of algebraic spaces over \(S\). Then given any solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l]_a \\ Y & T' \ar[l] \ar@{-->}[lu] }\] where \(T \subset T'\) is a first order thickening of algebraic spaces over \(Y\) there exists exactly one dotted arrow making the diagram commute. In other words, in Definition 04GC the condition that \(T\) be affine may be dropped.
Proof
Let \(U' \to T'\) be a surjective étale morphism where \(U' = \coprod U'_i\) is a disjoint union of affine schemes. Let \(U_i = T \times_{T'} U'_i\). Then we get morphisms \(a'_i : U'_i \to X\) such that \(a'_i|_{U_i}\) equals the composition \(U_i \to T \to X\). By uniqueness (see Lemma 05ZY) we see that \(a'_i\) and \(a'_j\) agree on the fibre product \(U'_i \times_{T'} U'_j\). Hence \(\coprod a'_i : U' \to X\) descends to give a unique morphism \(a' : T' \to X\).
Lemma
A composition of formally étale morphisms is formally étale.
Proof
This is formal.
Lemma
A base change of a formally étale morphism is formally étale.
Proof
This is formal.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) The following are equivalent:
\(f\) is formally étale,
\(f\) is formally unramified and the universal first order thickening of \(X\) over \(Y\) is equal to \(X\),
\(f\) is formally unramified and \(\mathcal{C}_{X/Y} = 0\), and
\(\Omega_{X/Y} = 0\) and \(\mathcal{C}_{X/Y} = 0\).
Proof
Actually, the last assertion only make sense because \(\Omega_{X/Y} = 0\) implies that \(\mathcal{C}_{X/Y}\) is defined via Lemma 04G9 and Definition 0607. This also makes it clear that (3) and (4) are equivalent.
Either of the assumptions (1), (2), and (3) imply that \(f\) is formally unramified. Hence we may assume \(f\) is formally unramified. The equivalence of (1), (2), and (3) follow from the universal property of the universal first order thickening \(X'\) of \(X\) over \(S\) and the fact that \(X = X' \Leftrightarrow \mathcal{C}_{X/Y} = 0\) since after all by definition \(\mathcal{C}_{X/Y} = \mathcal{C}_{X/X'}\) is the ideal sheaf of \(X\) in \(X'\).
Lemma
An unramified flat morphism is formally étale.
Proof
Follows from the case of schemes, see More on Morphisms, Lemma 04FF and étale localization, see Lemmas 04G8 and 04GD and Morphisms of Spaces, Lemma 03MM.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
The morphism \(f\) is étale, and
the morphism \(f\) is locally of finite presentation and formally étale.
Proof
Follows from the case of schemes, see More on Morphisms, Lemma 02HM and étale localization, see Lemma 04GD and Morphisms of Spaces, Lemmas 0410 and 03XT.
Infinitesimal deformations of maps
In this section we explain how a derivation can be used to infinitesimally move a map. Throughout this section we use that a sheaf on a thickening \(X'\) of \(X\) can be seen as a sheaf on \(X\), see Equations (05ZL) and (05ZM).
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(X \subset X'\) and \(Y \subset Y'\) be two first order thickenings of algebraic spaces over \(B\). Let \((a, a'), (b, b') : (X \subset X') \to (Y \subset Y')\) be two morphisms of thickenings over \(B\). Assume that
\(a = b\), and
the two maps \(a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) (Lemma 04CP) are equal.
Then the map \((a')^\sharp - (b')^\sharp\) factors as \[\mathcal{O}_{Y'} \to \mathcal{O}_Y \xrightarrow{D} a_*\mathcal{C}_{X/X'} \to a_*\mathcal{O}_{X'}\] where \(D\) is an \(\mathcal{O}_B\)-derivation.
Proof
Instead of working on \(Y\) we work on \(X\). The advantage is that the pullback functor \(a^{-1}\) is exact. Using (1) and (2) we obtain a commutative diagram with exact rows \[\xymatrix{ 0 \ar[r] & \mathcal{C}_{X/X'} \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_X \ar[r] & 0 \\ 0 \ar[r] & a^{-1}\mathcal{C}_{Y/Y'} \ar[r] \ar[u] & a^{-1}\mathcal{O}_{Y'} \ar[r] \ar@<1ex>[u]^{(a')^\sharp} \ar@<-1ex>[u]_{(b')^\sharp} & a^{-1}\mathcal{O}_Y \ar[r] \ar[u] & 0 }\] Now it is a general fact that in such a situation the difference of the \(\mathcal{O}_B\)-algebra maps \((a')^\sharp\) and \((b')^\sharp\) is an \(\mathcal{O}_B\)-derivation from \(a^{-1}\mathcal{O}_Y\) to \(\mathcal{C}_{X/X'}\). By adjointness of the functors \(a^{-1}\) and \(a_*\) this is the same thing as an \(\mathcal{O}_B\)-derivation from \(\mathcal{O}_Y\) into \(a_*\mathcal{C}_{X/X'}\). Some details omitted.
Note that in the situation of the lemma above we may write \(D\) as [0619]\[\begin{equation} D = \text{d}_{Y/B} \circ \theta \end{equation}\] where \(\theta\) is an \(\mathcal{O}_Y\)-linear map \(\theta : \Omega_{Y/B} \to a_*\mathcal{C}_{X/X'}\). Of course, then by adjunction again we may view \(\theta\) as an \(\mathcal{O}_X\)-linear map \(\theta : a^*\Omega_{Y/B} \to \mathcal{C}_{X/X'}\).
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \((a, a') : (X \subset X') \to (Y \subset Y')\) be a morphism of first order thickenings over \(B\). Let \[\theta : a^*\Omega_{Y/B} \to \mathcal{C}_{X/X'}\] be an \(\mathcal{O}_X\)-linear map. Then there exists a unique morphism of pairs \((b, b') : (X \subset X') \to (Y \subset Y')\) such that (1) and (2) of Lemma 0618 hold and the derivation \(D\) and \(\theta\) are related by Equation (0619).
Proof
Consider the map \[\alpha = (a')^\sharp + D : a^{-1}\mathcal{O}_{Y'} \to \mathcal{O}_{X'}\] where \(D\) is as in Equation (0619). As \(D\) is an \(\mathcal{O}_B\)-derivation it follows that \(\alpha\) is a map of sheaves of \(\mathcal{O}_B\)-algebras. By construction we have \(i_X^\sharp \circ \alpha = a^\sharp \circ i_Y^\sharp\) where \(i_X : X \to X'\) and \(i_Y : Y \to Y'\) are the given closed immersions. By Lemma 05ZN we obtain a unique morphism \((a, b') : (X \subset X') \to (Y \subset Y')\) of thickenings over \(B\) such that \(\alpha = (b')^\sharp\). Setting \(b = a\) we win.
Remark
Assumptions and notation as in Lemma 04D0. The action of a local section \(\theta\) on \(a'\) is sometimes indicated by \(\theta \cdot a'\). Note that this means nothing else than the fact that \((a')^\sharp\) and \((\theta \cdot a')^\sharp\) differ by a derivation \(D\) which is related to \(\theta\) by Equation (0619).
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(X \subset X'\) and \(Y \subset Y'\) be first order thickenings over \(B\). Assume given a morphism \(a : X \to Y\) and a map \(A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) of \(\mathcal{O}_X\)-modules. For an object \(U'\) of \((X')_{spaces, \etale}\) with \(U = X \times_{X'} U'\) consider morphisms \(a' : U' \to Y'\) such that
\(a'\) is a morphism over \(B\),
\(a'|_U = a|_U\), and
the induced map \(a^*\mathcal{C}_{Y/Y'}|_U \to \mathcal{C}_{X/X'}|_U\) is the restriction of \(A\) to \(U\).
Then the rule [061B]\[\begin{equation} U' \mapsto \{a' : U' \to Y'\text{ such that (1), (2), (3) hold.}\} \end{equation}\] defines a sheaf of sets on \((X')_{spaces, \etale}\).
Proof
Denote \(\mathcal{F}\) the rule of the lemma. The restriction mapping \(\mathcal{F}(U') \to \mathcal{F}(V')\) for \(V' \subset U' \subset X'\) of \(\mathcal{F}\) is really the restriction map \(a' \mapsto a'|_{V'}\). With this definition in place it is clear that \(\mathcal{F}\) is a sheaf since morphisms of algebraic spaces satisfy étale descent, see Descent on Spaces, Lemma 04P2.
Lemma
Same notation and assumptions as in Lemma 061A. We identify sheaves on \(X\) and \(X'\) via (05ZL). There is an action of the sheaf \[\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/B}, \mathcal{C}_{X/X'})\] on the sheaf (061B). Moreover, the action is simply transitive for any object \(U'\) of \((X')_{spaces, \etale}\) over which the sheaf (061B) has a section.
Proof
Remark
A special case of Lemmas 0618, 04D0, 061A, and 061C is where \(Y = Y'\). In this case the map \(A\) is always zero. The sheaf of Lemma 061A is just given by the rule \[U' \mapsto \{a' : U' \to Y\text{ over }B\text{ with } a'|_U = a|_U\}\] and we act on this by the sheaf \(\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/B}, \mathcal{C}_{X/X'})\).
Remark
Another special case of Lemmas 0618, 04D0, 061A, and 061C is where \(B\) itself is a thickening \(Z \subset Z' = B\) and \(Y = Z \times_{Z'} Y'\). Picture \[\xymatrix{ (X \subset X') \ar@{..>}[rr]_{(a, ?)} \ar[rd]_{(g, g')} & & (Y \subset Y') \ar[ld]^{(h, h')} \\ & (Z \subset Z') }\] In this case the map \(A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) is determined by \(a\): the map \(h^*\mathcal{C}_{Z/Z'} \to \mathcal{C}_{Y/Y'}\) is surjective (because we assumed \(Y = Z \times_{Z'} Y'\)), hence the pullback \(g^*\mathcal{C}_{Z/Z'} = a^*h^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'}\) is surjective, and the composition \(g^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) has to be the canonical map induced by \(g'\). Thus the sheaf of Lemma 061A is just given by the rule \[U' \mapsto \{a' : U' \to Y'\text{ over }Z'\text{ with } a'|_U = a|_U\}\] and we act on this by the sheaf \(\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/Z}, \mathcal{C}_{X/X'})\).
Lemma
Let \(S\) be a scheme. Consider a commutative diagram of first order thickenings \[\vcenter{ \xymatrix{ (T_2 \subset T_2') \ar[d]_{(h, h')} \ar[rr]_{(a_2, a_2')} & & (X_2 \subset X_2') \ar[d]^{(f, f')} \\ (T_1 \subset T_1') \ar[rr]^{(a_1, a_1')} & & (X_1 \subset X_1') } } \quad \begin{matrix} \text{and a commutative} \\ \text{diagram} \end{matrix} \quad \vcenter{ \xymatrix{ X_2' \ar[r] \ar[d] & B_2 \ar[d] \\ X_1' \ar[r] & B_1 } }\] of algebraic spaces over \(S\) with \(X_2 \to X_1\) and \(B_2 \to B_1\) étale. For any \(\mathcal{O}_{T_1}\)-linear map \(\theta_1 : a_1^*\Omega_{X_1/B_1} \to \mathcal{C}_{T_1/T'_1}\) let \(\theta_2\) be the composition \[\xymatrix{ a_2^*\Omega_{X_2/B_2} \ar@{=}[r] & h^*a_1^*\Omega_{X_1/B_1} \ar[r]^-{h^*\theta_1} & h^*\mathcal{C}_{T_1/T'_1} \ar[r] & \mathcal{C}_{T_2/T'_2} }\] (equality sign is explained in the proof). Then the diagram \[\xymatrix{ T_2' \ar[rr]_{\theta_2 \cdot a_2'} \ar[d] & & X'_2 \ar[d] \\ T_1' \ar[rr]^{\theta_1 \cdot a_1'} & & X'_1 }\] commutes where the actions \(\theta_2 \cdot a_2'\) and \(\theta_1 \cdot a_1'\) are as in Remark 0CK6.
Proof
The equality sign comes from the identification \(f^*\Omega_{X_1/S_1} = \Omega_{X_2/S_2}\) we get as the construction of the sheaf of differentials is compatible with étale localization (both on source and target), see Lemma 04CU. Namely, using this we have \(a_2^*\Omega_{X_2/S_2} = a_2^*f^*\Omega_{X_1/S_1} = h^*a_1^*\Omega_{X_1/S_1}\) because \(f \circ a_2 = a_1 \circ h\). Having said this, the commutativity of the diagram may be checked on étale locally. Thus we may assume \(T'_i\), \(X'_i\), \(B_2\), and \(B_1\) are schemes and in this case the lemma follows from More on Morphisms, Lemma 04BY. Alternative proof: using Lemma 05ZN it suffices to show a certain diagram of sheaves of rings on \(X_1'\) is commutative; then argue exactly as in the proof of the aforementioned More on Morphisms, Lemma 04BY to see that this is indeed the case.
Infinitesimal deformations of algebraic spaces
The following simple lemma is often a convenient tool to check whether an infinitesimal deformation of a map is flat.
Lemma
Let \(S\) be a scheme. Let \((f, f') : (X \subset X') \to (Y \subset Y')\) be a morphism of first order thickenings of algebraic spaces over \(S\). Assume that \(f\) is flat. Then the following are equivalent
\(f'\) is flat and \(X = Y \times_{Y'} X'\), and
the canonical map \(f^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) is an isomorphism.
Proof
Choose a scheme \(V'\) and a surjective étale morphism \(V' \to Y'\). Choose a scheme \(U'\) and a surjective étale morphism \(U' \to X' \times_{Y'} V'\). Set \(U = X \times_{X'} U'\) and \(V = Y \times_{Y'} V'\). According to our definition of a flat morphism of algebraic spaces we see that the induced map \(g : U \to V\) is a flat morphism of schemes and that \(f'\) is flat if and only if the corresponding morphism \(g' : U' \to V'\) is flat. Also, \(X = Y \times_{Y'} X'\) if and only if \(U = V \times_{V'} V'\). Finally, the map \(f^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}\) is an isomorphism if and only if \(g^*\mathcal{C}_{V/V'} \to \mathcal{C}_{U/U'}\) is an isomorphism. Hence the lemma follows from its analogue for morphisms of schemes, see More on Morphisms, Lemma 063Y.
The following lemma is the “nilpotent” version of the “critère de platitude par fibres”, see Section 05WX.
Lemma
Let \(S\) be a scheme. Consider a commutative diagram \[\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (B \subset B') }\] of thickenings of algebraic spaces over \(S\). Assume
\(X'\) is flat over \(B'\),
\(f\) is flat,
\(B \subset B'\) is a finite order thickening, and
\(X = B \times_{B'} X'\) and \(Y = B \times_{B'} Y'\).
Then \(f'\) is flat and \(Y'\) is flat over \(B'\) at all points in the image of \(f'\).
Proof
Choose a scheme \(U'\) and a surjective étale morphism \(U' \to B'\). Choose a scheme \(V'\) and a surjective étale morphism \(V' \to U' \times_{B'} Y'\). Choose a scheme \(W'\) and a surjective étale morphism \(W' \to V' \times_{Y'} X'\). Let \(U, V, W\) be the base change of \(U', V', W'\) by \(B \to B'\). Then flatness of \(f'\) is equivalent to flatness of \(W' \to V'\) and we are given that \(W \to V\) is flat. Hence we may apply the lemma in the case of schemes to the diagram \[\xymatrix{ (W \subset W') \ar[rr] \ar[rd] & & (V \subset V') \ar[ld] \\ & (U \subset U') }\] of thickenings of schemes. See More on Morphisms, Lemma 06AF. The statement about flatness of \(Y'/B'\) at points in the image of \(f'\) follows in the same manner.
Many properties of morphisms of schemes are preserved under flat deformations.
Lemma
Let \(S\) be a scheme. Consider a commutative diagram \[\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (B \subset B') }\] of thickenings of algebraic spaces over \(S\). Assume \(B \subset B'\) is a finite order thickening, \(X'\) flat over \(B'\), \(X = B \times_{B'} X'\), and \(Y = B \times_{B'} Y'\). Then
\(f\) is representable if and only if \(f'\) is representable, [0CG7]
\(f\) is flat if and only if \(f'\) is flat, [0CG8]
\(f\) is an isomorphism if and only if \(f'\) is an isomorphism, [0CG9]
\(f\) is an open immersion if and only if \(f'\) is an open immersion, [0CGA]
\(f\) is quasi-compact if and only if \(f'\) is quasi-compact, [0CGB]
\(f\) is universally closed if and only if \(f'\) is universally closed, [0CGC]
\(f\) is (quasi-)separated if and only if \(f'\) is (quasi-)separated, [0CGD]
\(f\) is a monomorphism if and only if \(f'\) is a monomorphism, [0CGE]
\(f\) is surjective if and only if \(f'\) is surjective, [0CGF]
\(f\) is universally injective if and only if \(f'\) is universally injective, [0CGG]
\(f\) is affine if and only if \(f'\) is affine, [0CGH]
\(f\) is locally of finite type if and only if \(f'\) is locally of finite type,
\(f\) is locally quasi-finite if and only if \(f'\) is locally quasi-finite, [0CGJ]
\(f\) is locally of finite presentation if and only if \(f'\) is locally of finite presentation,
\(f\) is locally of finite type of relative dimension \(d\) if and only if \(f'\) is locally of finite type of relative dimension \(d\),
\(f\) is universally open if and only if \(f'\) is universally open, [0CGM]
\(f\) is syntomic if and only if \(f'\) is syntomic, [0CGN]
\(f\) is smooth if and only if \(f'\) is smooth, [0CGP]
\(f\) is unramified if and only if \(f'\) is unramified, [0CGQ]
\(f\) is étale if and only if \(f'\) is étale, [0CGR]
\(f\) is proper if and only if \(f'\) is proper, [0CGS]
\(f\) is integral if and only if \(f'\) is integral, [0CGT]
\(f\) is finite if and only if \(f'\) is finite, [0CGU]
\(f\) is finite locally free (of rank \(d\)) if and only if \(f'\) is finite locally free (of rank \(d\)), and
add more here.
Proof
Case (0CG7) follows from Lemma 09ZY.
Choose a scheme \(U'\) and a surjective étale morphism \(U' \to B'\). Choose a scheme \(V'\) and a surjective étale morphism \(V' \to U' \times_{B'} Y'\). Choose a scheme \(W'\) and a surjective étale morphism \(W' \to V' \times_{Y'} X'\). Let \(U, V, W\) be the base change of \(U', V', W'\) by \(B \to B'\). Consider the diagram \[\xymatrix{ (W \subset W') \ar[rr] \ar[rd] & & (V \subset V') \ar[ld] \\ & (U \subset U') }\] of thickenings of schemes. For any of the properties which are étale local on the source-and-target the result follows immediately from the corresponding result for morphisms of thickenings of schemes applied to the diagram above. Thus cases (0CG8), (0CGI), (0CGJ), (0CGK), (0CGL), (0CGN), (0CGP), (0CGQ), (0CGR) follow from the corresponding cases of More on Morphisms, Lemma 06AG.
Since \(X \to X'\) and \(Y \to Y'\) are universal homeomorphisms we see that any question about the topology of the maps \(X \to Y\) and \(X' \to Y'\) has the same answer. Thus we see that cases (0CGB), (0CGC), (0CGF), (0CGG), and (0CGM) hold.
In each of the remaining cases we only prove the implication \(f\text{ has }P \Rightarrow f'\text{ has }P\) since the other implication follows from the fact that \(P\) is stable under base change, see Spaces, Lemma 02YW and Morphisms of Spaces, Lemmas 03KL, 042P, 03WI, 04WP, 03ZS, and 03ZY.
The case (0CGA). Assume \(f\) is an open immersion. Then \(f'\) is étale by (0CGR) and universally injective by (0CGG) hence \(f'\) is an open immersion, see Morphisms of Spaces, Lemma 05W5. You can avoid using this lemma at the cost of first using (0CG7) to reduce to the case of schemes.
The case (0CG9). Follows from cases (0CGA) and (0CGF).
The case (0CGD). See Lemma 09ZY.
The case (0CGE). Assume \(f\) is a monomorphism. Consider the diagonal morphism \(\Delta_{X'/Y'} : X' \to X' \times_{Y'} X'\). The base change of \(\Delta_{X'/Y'}\) by \(B \to B'\) is \(\Delta_{X/Y}\) which is an isomorphism by assumption. By (0CG9) we conclude that \(\Delta_{X'/Y'}\) is an isomorphism and hence \(f'\) is a monomorphism.
The case (0CGH). See Lemma 09ZY.
The case (0CGS). See Lemma 09ZZ.
The case (0CGT). See Lemma 09ZY.
The case (0CGU). See Lemma 09ZZ.
The case (0CGV). Assume \(f\) finite locally free. By (0CGU) we see that \(f'\) is finite. By (0CG8) we see that \(f'\) is flat. By (0CGK) \(f'\) is locally of finite presentation. Hence \(f'\) is finite locally free by Morphisms of Spaces, Lemma 0416.
The following lemma is the “locally nilpotent” version of the “critère de platitude par fibres”, see Section 05WX.
Lemma
Let \(S\) be a scheme. Consider a commutative diagram \[\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (B \subset B') }\] of thickenings of algebraic spaces over \(S\). Assume
\(Y' \to B'\) is locally of finite type,
\(X' \to B'\) is flat and locally of finite presentation,
\(f\) is flat, and
\(X = B \times_{B'} X'\) and \(Y = B \times_{B'} Y'\).
Then \(f'\) is flat and for all \(y' \in |Y'|\) in the image of \(|f'|\) the morphism \(Y' \to B'\) is flat at \(y'\).
Proof
Choose a scheme \(U'\) and a surjective étale morphism \(U' \to B'\). Choose a scheme \(V'\) and a surjective étale morphism \(V' \to U' \times_{B'} Y'\). Choose a scheme \(W'\) and a surjective étale morphism \(W' \to V' \times_{Y'} X'\). Let \(U, V, W\) be the base change of \(U', V', W'\) by \(B \to B'\). Then flatness of \(f'\) is equivalent to flatness of \(W' \to V'\) and we are given that \(W \to V\) is flat. Hence we may apply the lemma in the case of schemes to the diagram \[\xymatrix{ (W \subset W') \ar[rr] \ar[rd] & & (V \subset V') \ar[ld] \\ & (U \subset U') }\] of thickenings of schemes. See More on Morphisms, Lemma 0CF3. The statement about flatness of \(Y'/B'\) at points in the image of \(f'\) follows in the same manner.
Many properties of morphisms of schemes are preserved under flat deformations as in the lemma above.
Lemma
Let \(S\) be a scheme. Consider a commutative diagram \[\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (B \subset B') }\] of thickenings of algebraic spaces over \(S\). Assume \(Y' \to B'\) locally of finite type, \(X' \to B'\) flat and locally of finite presentation, \(X = B \times_{B'} X'\), and \(Y = B \times_{B'} Y'\). Then
\(f\) is representable if and only if \(f'\) is representable, [0CGY]
\(f\) is flat if and only if \(f'\) is flat, [0CGZ]
\(f\) is an isomorphism if and only if \(f'\) is an isomorphism, [0CH0]
\(f\) is an open immersion if and only if \(f'\) is an open immersion, [0CH1]
\(f\) is quasi-compact if and only if \(f'\) is quasi-compact, [0CH2]
\(f\) is universally closed if and only if \(f'\) is universally closed, [0CH3]
\(f\) is (quasi-)separated if and only if \(f'\) is (quasi-)separated, [0CH4]
\(f\) is a monomorphism if and only if \(f'\) is a monomorphism, [0CH5]
\(f\) is surjective if and only if \(f'\) is surjective, [0CH6]
\(f\) is universally injective if and only if \(f'\) is universally injective, [0CH7]
\(f\) is affine if and only if \(f'\) is affine, [0CH8]
\(f\) is locally quasi-finite if and only if \(f'\) is locally quasi-finite, [0CH9]
\(f\) is locally of finite type of relative dimension \(d\) if and only if \(f'\) is locally of finite type of relative dimension \(d\),
\(f\) is universally open if and only if \(f'\) is universally open, [0CHB]
\(f\) is syntomic if and only if \(f'\) is syntomic, [0CHC]
\(f\) is smooth if and only if \(f'\) is smooth, [0CHD]
\(f\) is unramified if and only if \(f'\) is unramified, [0CHE]
\(f\) is étale if and only if \(f'\) is étale, [0CHF]
\(f\) is proper if and only if \(f'\) is proper, [0CHG]
\(f\) is finite if and only if \(f'\) is finite, [0CHH]
\(f\) is finite locally free (of rank \(d\)) if and only if \(f'\) is finite locally free (of rank \(d\)), and
add more here.
Proof
Case (0CGY) follows from Lemma 09ZY.
Choose a scheme \(U'\) and a surjective étale morphism \(U' \to B'\). Choose a scheme \(V'\) and a surjective étale morphism \(V' \to U' \times_{B'} Y'\). Choose a scheme \(W'\) and a surjective étale morphism \(W' \to V' \times_{Y'} X'\). Let \(U, V, W\) be the base change of \(U', V', W'\) by \(B \to B'\). Consider the diagram \[\xymatrix{ (W \subset W') \ar[rr] \ar[rd] & & (V \subset V') \ar[ld] \\ & (U \subset U') }\] of thickenings of schemes. For any of the properties which are étale local on the source-and-target the result follows immediately from the corresponding result for morphisms of thickenings of schemes applied to the diagram above. Thus cases (0CGZ), (0CH9), (0CHA), (0CHC), (0CHD), (0CHE), (0CHF) follow from the corresponding cases of More on Morphisms, Lemma 0CF4.
Since \(X \to X'\) and \(Y \to Y'\) are universal homeomorphisms we see that any question about the topology of the maps \(X \to Y\) and \(X' \to Y'\) has the same answer. Thus we see that cases (0CH2), (0CH3), (0CH6), (0CH7), and (0CHB) hold.
In each of the remaining cases we only prove the implication \(f\text{ has }P \Rightarrow f'\text{ has }P\) since the other implication follows from the fact that \(P\) is stable under base change, see Spaces, Lemma 02YW and Morphisms of Spaces, Lemmas 03KL, 042P, 03WI, 04WP, 03ZS, and 03ZY.
The case (0CH1). Assume \(f\) is an open immersion. Then \(f'\) is étale by (0CHF) and universally injective by (0CH7) hence \(f'\) is an open immersion, see Morphisms of Spaces, Lemma 05W5. You can avoid using this lemma at the cost of first using (0CGY) to reduce to the case of schemes.
The case (0CH0). Follows from cases (0CH1) and (0CH6).
The case (0CH4). See Lemma 09ZY.
The case (0CH5). Assume \(f\) is a monomorphism. Consider the diagonal morphism \(\Delta_{X'/Y'} : X' \to X' \times_{Y'} X'\). The base change of \(\Delta_{X'/Y'}\) by \(B \to B'\) is \(\Delta_{X/Y}\) which is an isomorphism by assumption. By (0CH0) we conclude that \(\Delta_{X'/Y'}\) is an isomorphism and hence \(f'\) is a monomorphism.
The case (0CH8). See Lemma 09ZY.
The case (0CHG). See Lemma 0BPJ.
The case (0CHH). See Lemma 0BPJ.
The case (0CHI). Assume \(f\) finite locally free. By (0CHH) we see that \(f'\) is finite. By (0CGZ) we see that \(f'\) is flat. Also \(f'\) is locally finite presentation by Morphisms of Spaces, Lemma 05WT. Hence \(f'\) is finite locally free by Morphisms of Spaces, Lemma 0416.
Formally smooth morphisms
In this section we introduce the notion of a formally smooth morphism \(X \to Y\) of algebraic spaces. Such a morphism is characterized by the property that \(T\)-valued points of \(X\) lift to infinitesimal thickenings of \(T\) provided \(T\) is affine. The main result is that a morphism which is formally smooth and locally of finite presentation is smooth, see Lemma 04AM. It turns out that this criterion is often easier to use than the Jacobian criterion.
Definition
Let \(S\) be a scheme. A morphism \(f : X \to Y\) of algebraic spaces over \(S\) is said to be formally smooth if it is formally smooth as a transformation of functors as in Definition 049S.
In the cases of formally unramified and formally étale morphisms the condition that \(T'\) be affine could be dropped, see Lemmas 05ZY and 0611. This is no longer true in the case of formally smooth morphisms. In fact, a slightly more natural condition would be that we should be able to fill in the dotted arrow étale locally on \(T'\). In fact, analyzing the proof of Lemma 04AM shows that this would be equivalent to the definition as it currently stands. It is also true that requiring the existence of the dotted arrow fppf locally on \(T'\) would be sufficient, but that is slightly more difficult to prove.
We will not restate the results proved in the more general setting of formally smooth transformations of functors in Section 04G3.
Lemma
A composition of formally smooth morphisms is formally smooth.
Proof
Omitted.
Lemma
A base change of a formally smooth morphism is formally smooth.
Proof
Omitted, but see Algebra, Lemma 00TJ for the algebraic version.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Then \(f\) is formally étale if and only if \(f\) is formally smooth and formally unramified.
Proof
Omitted.
Here is a helper lemma which will be superseded by Lemma 061K.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y }\] be a commutative diagram of morphisms of algebraic spaces over \(S\). If the vertical arrows are étale and \(f\) is formally smooth, then \(\psi\) is formally smooth.
Proof
By Lemma 04AL the morphisms \(U \to X\) and \(V \to Y\) are formally étale. By Lemma 049T the composition \(U \to Y\) is formally smooth. By Lemma 04G5 we see \(\psi : U \to V\) is formally smooth.
The following lemma is the main result of this section. It implies, combined with Limits of Spaces, Proposition 04AK, that we can recognize whether a morphism of algebraic spaces \(f : X \to Y\) is smooth in terms of “simple” properties of the transformation of functors \(X \to Y\).
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
The morphism \(f\) is smooth.
The morphism \(f\) is locally of finite presentation, and formally smooth.
Proof
Assume \(f : X \to S\) is locally of finite presentation and formally smooth. Consider a commutative diagram \[\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\) and \(V\) are schemes and the vertical arrows are étale and surjective. By Lemma 061H we see \(\psi : U \to V\) is formally smooth. By Morphisms of Spaces, Lemma 0410 the morphism \(\psi\) is locally of finite presentation. Hence by the case of schemes the morphism \(\psi\) is smooth, see More on Morphisms, Lemma 02H6. Hence \(f\) is smooth, see Morphisms of Spaces, Lemma 03ZF.
Conversely, assume that \(f : X \to Y\) is smooth. Consider a solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l]^a \\ Y & T' \ar[l] \ar@{-->}[lu] }\] as in Definition 060G. We will show the dotted arrow exists thereby proving that \(f\) is formally smooth. Let \(\mathcal{F}\) be the sheaf of sets on \((T')_{spaces, \etale}\) of Lemma 061A as in the special case discussed in Remark 061D. Let \[\mathcal{H} = \SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/Y}, \mathcal{C}_{T/T'})\] be the sheaf of \(\mathcal{O}_T\)-modules on \(T_{spaces, \etale}\) with action \(\mathcal{H} \times \mathcal{F} \to \mathcal{F}\) as in Lemma 061C. The action \(\mathcal{H} \times \mathcal{F} \to \mathcal{F}\) turns \(\mathcal{F}\) into a pseudo \(\mathcal{H}\)-torsor, see Cohomology on Sites, Definition 03AH. Our goal is to show that \(\mathcal{F}\) is a trivial \(\mathcal{H}\)-torsor. There are two steps: (I) To show that \(\mathcal{F}\) is a torsor we have to show that \(\mathcal{F}\) has étale locally a section. (II) To show that \(\mathcal{F}\) is the trivial torsor it suffices to show that \(H^1(T_\etale, \mathcal{H}) = 0\), see Cohomology on Sites, Lemma 03AJ.
First we prove (I). To see this choose a commutative diagram \[\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\) and \(V\) are schemes and the vertical arrows are étale and surjective. As \(f\) is assumed smooth we see that \(\psi\) is smooth and hence formally smooth by Lemma 04AL. By the same lemma the morphism \(V \to Y\) is formally étale. Thus by Lemma 049T the composition \(U \to Y\) is formally smooth. Then (I) follows from Lemma 04CY part (4).
Finally we prove (II). By Lemma 05ZF we see that \(\Omega_{X/S}\) is of finite presentation. Hence \(a^*\Omega_{X/S}\) is of finite presentation (see Properties of Spaces, Section 05VR). Hence the sheaf \(\mathcal{H} = \SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/Y}, \mathcal{C}_{T/T'})\) is quasi-coherent by Properties of Spaces, Lemma 03M1. Thus by Descent, Proposition 03DW and Cohomology of Schemes, Lemma 01XB we have \[H^1(T_{spaces, \etale}, \mathcal{H}) = H^1(T_\etale, \mathcal{H}) = H^1(T, \mathcal{H}) = 0\] as desired.
Smooth morphisms satisfy strong local lifting property, see Lemma 0CHJ. If in the lemma we assume \(T'\) is affine, then we do not know if it is necessary to take an étale covering. More precisely, if we have a commutative diagram \[\xymatrix{ X \ar[d] & T \ar[l] \ar[d] \\ Y & T' \ar[l] \ar@{..>}[lu] }\] of algebraic spaces where \(X \to Y\) is smooth and \(T \to T'\) is a thickening of affine schemes, the does a dotted arrow making the diagram commute always exist? If you know the answer, or if you have a reference, please email stacks.project@gmail.com.
Lemma
Let \(S\) be a scheme. Consider a commutative diagram \[\xymatrix{ X \ar[d] & T \ar[l] \ar[d] \\ Y & T' \ar[l] }\] of algebraic spaces over \(S\) where \(X \to Y\) is smooth and \(T \to T'\) is a thickening. Then there exists an étale covering \(\{T'_i \to T'\}\) such that we can find the dotted arrow in \[\xymatrix{ X \ar[d] & T \ar[l] \ar[d] & T \times_{T'} T'_i \ar[l] \ar[d] \\ Y & T' \ar[l] & T'_i \ar[l] \ar@{..>}[llu] }\] making the diagram commute (for all \(i\)).
Proof
Choose an étale covering \(\{Y_i \to Y\}\) with each \(Y_i\) affine. After replacing \(T'\) by the induced étale covering we may assume \(Y\) is affine.
Assume \(Y\) is affine. Choose an étale covering \(\{X_i \to X\}\). This gives rise to an étale covering of \(T\). This étale covering of \(T\) comes from an étale covering of \(T'\) (by Theorem 05ZH, see discussion in Section 05ZJ). Hence we may assume \(X\) is affine.
Assume \(X\) and \(Y\) are affine. We can do one more étale covering of \(T'\) and assume \(T'\) is affine. In this case the lemma follows from Algebra, Lemma 07K4.
We do a bit more work to show that being formally smooth is étale local on the source. To begin we show that a formally smooth morphism has a nice sheaf of differentials. The notion of a locally projective quasi-coherent module is defined in Properties of Spaces, Section 060P.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a formally smooth morphism of algebraic spaces over \(S\). Then \(\Omega_{X/Y}\) is locally projective on \(X\).
Proof
Choose a diagram \[\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\) and \(V\) are affine(!) schemes and the vertical arrows are étale. By Lemma 061H we see \(\psi : U \to V\) is formally smooth. Hence \(\Gamma(V, \mathcal{O}_V) \to \Gamma(U, \mathcal{O}_U)\) is a formally smooth ring map, see More on Morphisms, Lemma 02H4. Hence by Algebra, Lemma 031I the \(\Gamma(U, \mathcal{O}_U)\)-module \(\Omega_{\Gamma(U, \mathcal{O}_U)/\Gamma(V, \mathcal{O}_V)}\) is projective. Hence \(\Omega_{U/V}\) is locally projective, see Properties, Section 05JN. Since \(\Omega_{X/Y}|_U = \Omega_{U/V}\) we see that \(\Omega_{X/Y}\) is locally projective too. (Because we can find an étale covering of \(X\) by the affine \(U\)’s fitting into diagrams as above – details omitted.)
Lemma
Let \(T\) be an affine scheme. Let \(\mathcal{F}\), \(\mathcal{G}\) be quasi-coherent \(\mathcal{O}_T\)-modules on \(T_\etale\). Consider the internal hom sheaf \(\mathcal{H} = \SheafHom_{\mathcal{O}_T}(\mathcal{F}, \mathcal{G})\) on \(T_\etale\). If \(\mathcal{F}\) is locally projective, then \(H^1(T_\etale, \mathcal{H}) = 0\).
Proof
By the definition of a locally projective sheaf on an algebraic space (see Properties of Spaces, Definition 060R) we see that \(\mathcal{F}_{Zar} = \mathcal{F}|_{T_{Zar}}\) is a locally projective sheaf on the scheme \(T\). Thus \(\mathcal{F}_{Zar}\) is a direct summand of a free \(\mathcal{O}_{T_{Zar}}\)-module. Whereupon we conclude (as \(\mathcal{F} = (\mathcal{F}_{Zar})^a\), see Descent, Proposition 03DX) that \(\mathcal{F}\) is a direct summand of a free \(\mathcal{O}_T\)-module on \(T_\etale\). Hence we may assume that \(\mathcal{F} = \bigoplus_{i \in I} \mathcal{O}_T\) is a free module. In this case \(\mathcal{H} = \prod_{i \in I} \mathcal{G}\) is a product of quasi-coherent modules. By Cohomology on Sites, Lemma 060L we conclude that \(H^1 = 0\) because the cohomology of a quasi-coherent sheaf on an affine scheme is zero, see Descent, Proposition 03DW and Cohomology of Schemes, Lemma 01XB.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
\(f\) is formally smooth,
for every diagram \[\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\) and \(V\) are schemes and the vertical arrows are étale the morphism of schemes \(\psi\) is formally smooth (as in More on Morphisms, Definition 02H8), and
for one such diagram with surjective vertical arrows the morphism \(\psi\) is formally smooth.
Proof
We have seen that (1) implies (2) and (3) in Lemma 061H. Assume (3). The proof that \(f\) is formally smooth is entirely similar to the proof of (1) \(\Rightarrow\) (2) of Lemma 04AM.
Consider a solid commutative diagram \[\xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l]^a \\ Y & T' \ar[l] \ar@{-->}[lu] }\] as in Definition 060G. We will show the dotted arrow exists thereby proving that \(f\) is formally smooth. Let \(\mathcal{F}\) be the sheaf of sets on \((T')_{spaces, \etale}\) of Lemma 061A as in the special case discussed in Remark 061D. Let \[\mathcal{H} = \SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/Y}, \mathcal{C}_{T/T'})\] be the sheaf of \(\mathcal{O}_T\)-modules on \(T_{spaces, \etale}\) with action \(\mathcal{H} \times \mathcal{F} \to \mathcal{F}\) as in Lemma 061C. The action \(\mathcal{H} \times \mathcal{F} \to \mathcal{F}\) turns \(\mathcal{F}\) into a pseudo \(\mathcal{H}\)-torsor, see Cohomology on Sites, Definition 03AH. Our goal is to show that \(\mathcal{F}\) is a trivial \(\mathcal{H}\)-torsor. There are two steps: (I) To show that \(\mathcal{F}\) is a torsor we have to show that \(\mathcal{F}\) has étale locally a section. (II) To show that \(\mathcal{F}\) is the trivial torsor it suffices to show that \(H^1(T_\etale, \mathcal{H}) = 0\), see Cohomology on Sites, Lemma 03AJ.
First we prove (I). To see this consider a diagram (which exists because we are assuming (3)) \[\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\) and \(V\) are schemes, the vertical arrows are étale and surjective, and \(\psi\) is formally smooth. By Lemma 04AL the morphism \(V \to Y\) is formally étale. Thus by Lemma 049T the composition \(U \to Y\) is formally smooth. Then (I) follows from Lemma 04CY part (4).
Finally we prove (II). By Lemma 061I we see that \(\Omega_{U/V}\) locally projective. Hence \(\Omega_{X/Y}\) is locally projective, see Descent on Spaces, Lemma 060Y. Hence \(a^*\Omega_{X/Y}\) is locally projective, see Properties of Spaces, Lemma 060S. Hence \[H^1(T_\etale, \mathcal{H}) = H^1(T_\etale, \SheafHom_{\mathcal{O}_T}(a^*\Omega_{X/Y}, \mathcal{C}_{T/T'}) = 0\] by Lemma 061J as desired.
Lemma
The property \(\mathcal{P}(f) =\)“\(f\) is formally smooth” is fpqc local on the base.
Proof
Let \(f : X \to Y\) be a morphism of algebraic spaces over a scheme \(S\). Choose an index set \(I\) and diagrams \[\xymatrix{ U_i \ar[d] \ar[r]_{\psi_i} & V_i \ar[d] \\ X \ar[r]^f & Y }\] with étale vertical arrows and \(U_i\), \(V_i\) affine schemes. Moreover, assume that \(\coprod U_i \to X\) and \(\coprod V_i \to Y\) are surjective, see Properties of Spaces, Lemma 03FX. By Lemma 061K we see that \(f\) is formally smooth if and only if each of the morphisms \(\psi_i\) are formally smooth. Hence we reduce to the case of a morphism of affine schemes. In this case the result follows from Algebra, Lemma 06CM. Some details omitted.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\), \(g : Y \to Z\) be morphisms of algebraic spaces over \(S\). Assume \(f\) is formally smooth. Then \[0 \to f^*\Omega_{Y/Z} \to \Omega_{X/Z} \to \Omega_{X/Y} \to 0\] Lemma 05Z8 is short exact.
Proof
Follows from the case of schemes, see More on Morphisms, Lemma 06B6, by étale localization, see Lemmas 061K and 04CU.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(h : Z \to X\) be a formally unramified morphism of algebraic spaces over \(B\). Assume that \(Z\) is formally smooth over \(B\). Then the canonical exact sequence \[0 \to \mathcal{C}_{Z/X} \to h^*\Omega_{X/B} \to \Omega_{Z/B} \to 0\] of Lemma 060F is short exact.
Proof
Let \(Z \to Z'\) be the universal first order thickening of \(Z\) over \(X\). From the proof of Lemma 060F we see that our sequence is identified with the sequence \[\mathcal{C}_{Z/Z'} \to \Omega_{Z'/B} \otimes \mathcal{O}_Z \to \Omega_{Z/B} \to 0.\] Since \(Z \to S\) is formally smooth we can étale locally on \(Z'\) find a left inverse \(Z' \to Z\) over \(B\) to the inclusion map \(Z \to Z'\). Thus the sequence is étale locally split, see Lemma 05ZB.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d]^f \\ & Y }\] be a commutative diagram of algebraic spaces over \(S\) where \(i\) and \(j\) are formally unramified and \(f\) is formally smooth. Then the canonical exact sequence \[0 \to \mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/Y} \to 0\] of Lemma 06BE is exact and locally split.
Proof
Denote \(Z \to Z'\) the universal first order thickening of \(Z\) over \(X\). Denote \(Z \to Z''\) the universal first order thickening of \(Z\) over \(Y\). By Lemma 060F here is a canonical morphism \(Z' \to Z''\) so that we have a commutative diagram \[\xymatrix{ Z \ar[r]_{i'} \ar[rd]_{j'} & Z' \ar[r]_a \ar[d]^k & X \ar[d]^f \\ & Z'' \ar[r]^b & Y }\] The sequence above is identified with the sequence \[\mathcal{C}_{Z/Z''} \to \mathcal{C}_{Z/Z'} \to (i')^*\Omega_{Z'/Z''} \to 0\] via our definitions concerning conormal sheaves of formally unramified morphisms. Let \(U'' \to Z''\) be an étale morphism with \(U''\) affine. Denote \(U \to Z\) and \(U' \to Z'\) the corresponding affine schemes étale over \(Z\) and \(Z'\). As \(f\) is formally smooth there exists a morphism \(h : U'' \to X\) which agrees with \(i\) on \(U\) and such that \(f \circ h\) equals \(b|_{U''}\). Since \(Z'\) is the universal first order thickening we obtain a unique morphism \(g : U'' \to Z'\) such that \(g = a \circ h\). The universal property of \(Z''\) implies that \(k \circ g\) is the inclusion map \(U'' \to Z''\). Hence \(g\) is a left inverse to \(k\). Picture \[\xymatrix{ U \ar[d] \ar[r] & Z' \ar[d]^k \\ U'' \ar[r] \ar[ru]^g & Z'' }\] Thus \(g\) induces a map \(\mathcal{C}_{Z/Z'}|_U \to \mathcal{C}_{Z/Z''}|_U\) which is a left inverse to the map \(\mathcal{C}_{Z/Z''} \to \mathcal{C}_{Z/Z'}\) over \(U\).
Smoothness over a Noetherian base
This section is the analogue of More on Morphisms, Section 02HW.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(x \in |X|\). Assume that \(Y\) is locally Noetherian and \(f\) locally of finite type. The following are equivalent:
\(f\) is smooth at \(x\),
for every solid commutative diagram \[\xymatrix{ X \ar[d]_f & \Spec(B) \ar[d]^i \ar[l]^-\alpha \\ Y & \Spec(B') \ar[l]_-{\beta} \ar@{-->}[lu] }\] where \(B' \to B\) is a surjection of local rings with \(\Ker(B' \to B)\) of square zero, and \(\alpha\) mapping the closed point of \(\Spec(B)\) to \(x\) there exists a dotted arrow making the diagram commute, and
same as in (2) but with \(B' \to B\) ranging over small extensions (see Algebra, Definition 02HS).
Proof
Condition (1) means there is an open subspace \(X' \subset X\) such that \(X' \to Y\) is smooth. Hence (1) implies conditions (2) and (3) by Lemma 04AM. Condition (2) implies condition (3) trivially. Assume (3). Choose a commutative diagram \[\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \\ Y & V \ar[l] }\] with \(U\) and \(V\) affine, horizontal arrows étale and such that there is a point \(u \in U\) mapping to \(x\). Next, consider a diagram \[\xymatrix{ X \ar[d] & U \ar[l] \ar[d] & \Spec(B) \ar[d]^i \ar[l]^-\alpha \\ Y & V \ar[l] & \Spec(B') \ar[l]_-{\beta} }\] as in (3) but for \(u \in U \to V\). Let \(\gamma : \Spec(B') \to X\) be the arrow we get from our assumption that (3) holds for \(X\). Because \(U \to X\) is étale and hence formally étale (Lemma 0616) the morphism \(\gamma\) has a unique lift to \(U\) compatible with \(\alpha\). Then because \(V \to Y\) is étale hence formally étale this lift is compatible with \(\beta\). Hence (3) holds for \(u \in U \to V\) and we conclude that \(U \to V\) is smooth at \(u\) by More on Morphisms, Lemma 02HX. This proves that \(X \to Y\) is smooth at \(x\), thereby finishing the proof.
Sometimes it is useful to know that one only needs to check the lifting criterion for small extensions “centered” at points of finite type (see Morphisms of Spaces, Section 06EE).
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(Y\) is locally Noetherian and \(f\) locally of finite type. The following are equivalent:
\(f\) is smooth,
for every solid commutative diagram \[\xymatrix{ X \ar[d]_f & \Spec(B) \ar[d]^i \ar[l]^-\alpha \\ Y & \Spec(B') \ar[l]_-{\beta} \ar@{-->}[lu] }\] where \(B' \to B\) is a small extension of Artinian local rings and \(\beta\) of finite type (!) there exists a dotted arrow making the diagram commute.
Proof
If \(f\) is smooth, then the infinitesimal lifting criterion (Lemma 04AM) says \(f\) is formally smooth and (2) holds.
Assume \(f\) is not smooth. The set of points \(x \in X\) where \(f\) is not smooth forms a closed subset \(T\) of \(|X|\). By Morphisms of Spaces, Lemma 06EK, there exists a point \(x \in T \subset X\) with \(x \in X_{\text{ft-pts}}\). Choose a commutative diagram \[\xymatrix{ X \ar[d] & U \ar[l] \ar[d] & u \ar@{|->}[d] \\ Y & V \ar[l] & v }\] with \(U\) and \(V\) affine, horizontal arrows étale and such that there is a point \(u \in U\) mapping to \(x\). Then \(u\) is a finite type point of \(U\). Since \(U \to V\) is not smooth at the point \(u\), by More on Morphisms, Lemma 02HX there is a diagram \[\xymatrix{ X \ar[d] & U \ar[l] \ar[d] & \Spec(B) \ar[d]^i \ar[l]^-\alpha \\ Y & V \ar[l] & \Spec(B') \ar[l]_-{\beta} \ar@{-->}[lu] }\] with \(B' \to B\) a small extension of (Artinian) local rings such that the residue field of \(B\) is equal to \(\kappa(v)\) and such that the dotted arrow does not exist. Since \(U \to V\) is of finite type, we see that \(v\) is a finite type point of \(V\). By Morphisms, Lemma 02HV the morphism \(\beta\) is of finite type, hence the composition \(\Spec(B) \to Y\) is of finite type also. Arguing exactly as in the proof of Lemma 0APN (using that \(U \to X\) and \(V \to Y\) are étale hence formally étale) we see that there cannot be an arrow \(\Spec(B) \to X\) fitting into the outer rectangle of the last displayed diagram. In other words, (2) doesn’t hold and the proof is complete.
Here is a useful application.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is locally of finite type and \(Y\) locally Noetherian. Let \(Z \subset Y\) be a closed subspace with \(n\)th infinitesimal neighbourhood \(Z_n \subset Y\). Set \(X_n = Z_n \times_Y X\).
If \(X_n \to Z_n\) is smooth for all \(n\), then \(f\) is smooth at every point of \(f^{-1}(Z)\).
If \(X_n \to Z_n\) is étale for all \(n\), then \(f\) is étale at every point of \(f^{-1}(Z)\).
Proof
Assume \(X_n \to Z_n\) is smooth for all \(n\). Let \(x \in X\) be a point lying over a point of \(Z\). Given a small extension \(B' \to B\) and morphisms \(\alpha\), \(\beta\) as in Lemma 0APN part (3) the maximal ideal of \(B'\) is nilpotent (as \(B'\) is Artinian) and hence the morphism \(\beta\) factors through \(Z_n\) and \(\alpha\) factors through \(X_n\) for a suitable \(n\). Thus the lifting property for \(X_n \to Z_n\) kicks in to get the desired dotted arrow in the diagram. This proves (1). Part (2) follows from (1) and the fact that a morphism is étale if and only if it is smooth of relative dimension \(0\).
The naive cotangent complex
This section is the continuation of Modules on Sites, Section 08TT which in turn continues the discussion in Algebra, Section 00S0.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The naive cotangent complex of \(f\) is the complex defined in Modules on Sites, Definition 08U0 for the morphism of ringed topoi \(f_{small}\) between the small étale sites of \(X\) and \(Y\), see Properties of Spaces, Lemma 03G8. Notation: \(\NL_f\) or \(\NL_{X/Y}\).
The next lemmas show this definition is compatible with the definition for ring maps and for schemes and that \(\NL_{X/Y}\) is an object of \(D_\QCoh(\mathcal{O}_X)\).
Lemma
Let \(S\) be a scheme. Consider a commutative diagram \[\xymatrix{ U \ar[d]_p \ar[r]_g & V \ar[d]^q \\ X \ar[r]^f & Y }\] of algebraic spaces over \(S\) with \(p\) and \(q\) étale. Then there is a canonical identification \(\NL_{X/Y}|_{U_\etale} = \NL_{U/V}\) in \(D(\mathcal{O}_U)\).
Proof
Formation of the naive cotangent complex commutes with pullback (Modules on Sites, Lemma 08TZ) and we have \(p_{small}^{-1}\mathcal{O}_X = \mathcal{O}_U\) and \(g_{small}^{-1}\mathcal{O}_{V_\etale} = p_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_\etale}\) because \(q_{small}^{-1}\mathcal{O}_{Y_\etale} = \mathcal{O}_{V_\etale}\) by Properties of Spaces, Lemma 03LV. Tracing through the definitions we conclude that \(\NL_{X/Y}|_{U_\etale} = \NL_{U/V}\).
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(X\) and \(Y\) representable by schemes \(X_0\) and \(Y_0\). Then there is a canonical identification \(\NL_{X/Y} = \epsilon^*\NL_{X_0/Y_0}\) in \(D(\mathcal{O}_X)\) where \(\epsilon\) is as in Derived Categories of Spaces, Section 071P and \(\NL_{X_0/Y_0}\) is as in More on Morphisms, Definition 0D0H.
Proof
Let \(f_0 : X_0 \to Y_0\) be the morphism of schemes corresponding to \(f\). There is a canonical map \(\epsilon^{-1}f_0^{-1}\mathcal{O}_{Y_0} \to f_{small}^{-1}\mathcal{O}_Y\) compatible with \(\epsilon^\sharp : \epsilon^{-1}\mathcal{O}_{X_0} \to \mathcal{O}_X\) because there is a commutative diagram \[\xymatrix{ X_{0, Zar} \ar[d]_{f_0} & X_\etale \ar[l]^\epsilon \ar[d]^f \\ Y_{0, Zar} & Y_\etale \ar[l]_\epsilon }\] see Derived Categories of Spaces, Remark 08GH. Thus we obtain a canonical map \[\epsilon^{-1}\NL_{X_0/Y_0} = \epsilon^{-1}\NL_{\mathcal{O}_{X_0}/f_0^{-1}\mathcal{O}_{Y_0}} = \NL_{\epsilon^{-1}\mathcal{O}_{X_0}/\epsilon^{-1}f_0^{-1}\mathcal{O}_{Y_0}} \to \NL_{\mathcal{O}_X/f^{-1}_{small}\mathcal{O}_Y} = \NL_{X/Y}\] by functoriality of the naive cotangent complex. To see that the induced map \(\epsilon^*\NL_{X_0/Y_0} \to \NL_{X/Y}\) is an isomorphism in \(D(\mathcal{O}_X)\) we may check on stalks at geometric points (Properties of Spaces, Theorem 04K5). Let \(\overline{x} : \Spec(k) \to X_0\) be a geometric point lying over \(x \in X_0\), with \(\overline{y} = f \circ \overline{x}\) lying over \(y \in Y_0\). Then \[\NL_{X/Y, \overline{x}} = \NL_{\mathcal{O}_{X, \overline{x}}/\mathcal{O}_{Y, \overline{y}}}\] This is true because taking stalks at \(\overline{x}\) is the same as taking inverse image via \(\overline{x} : \Spec(k) \to X\) and we may apply Modules on Sites, Lemma 08TZ. On the other hand we have \[(\epsilon^*\NL_{X_0/Y_0})_{\overline{x}} = \NL_{X_0/Y_0, x} \otimes_{\mathcal{O}_{X_0, x}} \mathcal{O}_{X, \overline{x}} = \NL_{\mathcal{O}_{X_0, x}/\mathcal{O}_{Y_0, y}} \otimes_{\mathcal{O}_{X_0, x}} \mathcal{O}_{X, \overline{x}}\] Some details omitted (hint: use that the stalk of a pullback is the stalk at the image point, see Sites, Lemma 05V1, as well as the corresponding result for modules, see Modules on Sites, Lemma 05V5). Observe that \(\mathcal{O}_{X, \overline{x}}\) is the strict henselization of \(\mathcal{O}_{X_0, x}\) and similarly for \(\mathcal{O}_{Y, \overline{y}}\) (Properties of Spaces, Lemma 04KF). Thus the result follows from More on Algebra, Lemma 0D08.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The cohomology sheaves of the complex \(\NL_{X/Y}\) are quasi-coherent, zero outside degrees \(-1\), \(0\) and equal to \(\Omega_{X/Y}\) in degree \(0\).
Proof
By construction of the naive cotangent complex in Modules on Sites, Section 08TT we have that \(\NL_{X/Y}\) is a complex sitting in degrees \(-1\), \(0\) and that its cohomology in degree \(0\) is \(\Omega_{X/Y}\) (by our construction of \(\Omega_{X/Y}\) in Section 04CR). The sheaf of differentials is quasi-coherent (by Lemma 04CV). To finish the proof it suffices to show that \(H^{-1}(\NL_{X/Y})\) is quasi-coherent. This follows by checking étale locally (allowed by Lemma 0D0W and Properties of Spaces, Lemma 03M0) reducing to the case of schemes (Lemma 0D0X) and finally using the result in the case of schemes (More on Morphisms, Lemma 0D0J).
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If \(f\) is locally of finite presentation, then \(\NL_{X/Y}\) is étale locally on \(X\) quasi-isomorphic to a complex \[\ldots \to 0 \to \mathcal{F}^{-1} \to \mathcal{F}^0 \to 0 \to \ldots\] of quasi-coherent \(\mathcal{O}_X\)-modules with \(\mathcal{F}^0\) of finite presentation and \(\mathcal{F}^{-1}\) of finite type.
Proof
Formation of the naive cotangent complex commutes with étale localization by Lemma 0D0W. This reduces us to the case of schemes by Lemma 0D0X. The result in the case of schemes is More on Morphisms, Lemma 0D0K.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent
\(f\) is formally smooth,
\(H^{-1}(\NL_{X/Y}) = 0\) and \(H^0(\NL_{X/Y}) = \Omega_{X/Y}\) is locally projective.
Proof
This follows from Lemma 061K, Lemma 0D0W, Lemma 0D0X and the case of schemes which is More on Morphisms, Lemma 0D0L.
Lemma
Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent
\(f\) is formally étale,
\(H^{-1}(\NL_{X/Y}) = H^0(\NL_{X/Y}) = 0\).
Proof
Assume (1). A formally étale morphism is a formally smooth morphism. Thus \(H^{-1}(\NL_{X/Y}) = 0\) by Lemma 0D10. On the other hand, a formally étale morphism if formally unramified hence we have \(\Omega_{X/Y} = 0\) by Lemma 04G9. Conversely, if (2) holds, then \(f\) is formally smooth by Lemma 0D10 and formally unramified by Lemma 04G9 and hence formally étale by Lemmas 061G.
Lemma
Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent
\(f\) is smooth, and
\(f\) is locally of finite presentation, \(H^{-1}(\NL_{X/Y}) = 0\), and \(H^0(\NL_{X/Y}) = \Omega_{X/Y}\) is finite locally free.
Proof
This follows from Lemma 061K, Lemma 0D0W, Lemma 0D0X and the case of schemes which is More on Morphisms, Lemma 0D0N.
Openness of the flat locus
This section is analogue of More on Morphisms, Section 0398. Note that we have defined the notion of flatness for quasi-coherent modules on algebraic spaces in Morphisms of Spaces, Section 05VT.
Theorem
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Assume \(f\) is locally of finite presentation and that \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module which is locally of finite presentation. Then \[\{x \in |X| : \mathcal{F}\text{ is flat over }Y\text{ at }x\}\] is open in \(|X|\).
Proof
Choose a commutative diagram \[\xymatrix{ U \ar[d]_p \ar[r]_\alpha & V \ar[d]^q \\ X \ar[r]^a & Y }\] with \(U\), \(V\) schemes and \(p\), \(q\) surjective and étale as in Spaces, Lemma 02X1. By More on Morphisms, Theorem 0399 the set \(U' = \{u \in |U| : p^*\mathcal{F}\text{ is flat over }V\text{ at }u\}\) is open in \(U\). By Morphisms of Spaces, Definition 05VV the image of \(U'\) in \(|X|\) is the set of the theorem. Hence we are done because the map \(|U| \to |X|\) is open, see Properties of Spaces, Lemma 03BX.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] be a cartesian diagram of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume \(g\) is flat, \(f\) is locally of finite presentation, and \(\mathcal{F}\) is locally of finite presentation. Then \[\{x' \in |X'| : (g')^*\mathcal{F}\text{ is flat over }Y'\text{ at }x'\}\] is the inverse image of the open subset of Theorem 05WV under the continuous map \(|g'| : |X'| \to |X|\).
Proof
This follows from Morphisms of Spaces, Lemma 05VW.
Critère de platitude par fibres
Let \(S\) be a scheme. Consider a commutative diagram of algebraic spaces over \(S\) \[\xymatrix{ X \ar[rr]_f \ar[dr]_g & & Y \ar[dl]^h \\ & Z }\] and a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\). Given a point \(x \in |X|\) we consider the question as to whether \(\mathcal{F}\) is flat over \(Y\) at \(x\). If \(\mathcal{F}\) is flat over \(Z\) at \(x\), then the theorem below states this question is intimately related to the question of whether the restriction of \(\mathcal{F}\) to the fibre of \(X \to Z\) over \(g(x)\) is flat over the fibre of \(Y \to Z\) over \(g(x)\). To make sense out of this we offer the following preliminary lemma.
Lemma
In the situation above the following are equivalent
Pick a geometric point \(\overline{x}\) of \(X\) lying over \(x\). Set \(\overline{y} = f \circ \overline{x}\) and \(\overline{z} = g \circ \overline{x}\). Then the module \(\mathcal{F}_{\overline{x}}/ \mathfrak m_{\overline{z}}\mathcal{F}_{\overline{x}}\) is flat over \(\mathcal{O}_{Y, \overline{y}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{Y, \overline{y}}\).
Pick a morphism \(x : \Spec(K) \to X\) in the equivalence class of \(x\). Set \(z = g \circ x\), \(X_z = \Spec(K) \times_{z, Z} X\), \(Y_z = \Spec(K) \times_{z, Z} Y\), and \(\mathcal{F}_z\) the pullback of \(\mathcal{F}\) to \(X_z\). Then \(\mathcal{F}_z\) is flat at \(x\) over \(Y_z\) (as defined in Morphisms of Spaces, Definition 05VV).
Pick a commutative diagram \[\xymatrix{ & & & U \ar[llld]_a \ar[rr] \ar[dr] & & V \ar[llld]_>>>>>>>b \ar[dl] \\ X \ar[rr]_f \ar[dr]_g & & Y \ar[dl]^h & & W \ar[llld]_c \\ & Z }\] where \(U, V, W\) are schemes, and \(a, b, c\) are étale, and a point \(u \in U\) mapping to \(x\). Let \(w \in W\) be the image of \(u\). Let \(\mathcal{F}_w\) be the pullback of \(\mathcal{F}\) to the fibre \(U_w\) of \(U \to W\) at \(w\). Then \(\mathcal{F}_w\) is flat over \(V_w\) at \(u\).
Proof
Note that in (2) the morphism \(x : \Spec(K) \to X\) defines a \(K\)-rational point of \(X_z\), hence the statement makes sense. Moreover, the condition in (2) is independent of the choice of \(\Spec(K) \to X\) in the equivalence class of \(x\) (details omitted; this will also follow from the arguments below because the other conditions do not depend on this choice). Also note that we can always choose a diagram as in (3) by: first choosing a scheme \(W\) and a surjective étale morphism \(W \to Z\), then choosing a scheme \(V\) and a surjective étale morphism \(V \to W \times_Z Y\), and finally choosing a scheme \(U\) and a surjective étale morphism \(U \to V \times_Y X\). Having made these choices we set \(U \to W\) equal to the composition \(U \to V \to W\) and we can pick a point \(u \in U\) mapping to \(x\) because the morphism \(U \to X\) is surjective.
Suppose given both a diagram as in (3) and a geometric point \(\overline{x} : \Spec(k) \to X\) as in (1). By Properties of Spaces, Lemma 05VN we can choose a geometric point \(\overline{u} : \Spec(k) \to U\) lying over \(u\) such that \(\overline{x} = a \circ \overline{u}\). Denote \(\overline{v} : \Spec(k) \to V\) and \(\overline{w} : \Spec(k) \to W\) the induced geometric points of \(V\) and \(W\). In this setting we know that \(\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}\) and similarly for \(Y\) and \(Z\), see Properties of Spaces, Lemma 04KF. In the same vein we have \[\mathcal{F}_{\overline{x}} = (a^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{U, u}^{sh}\] see Properties of Spaces, Lemma 05VP. Note that the stalk of \(\mathcal{F}_w\) at \(u\) is given by \[(\mathcal{F}_w)_u = (a^*\mathcal{F})_u/\mathfrak m_w(a^*\mathcal{F})_u\] and the local ring of \(V_w\) at \(v\) is given by \[\mathcal{O}_{V_w, v} = \mathcal{O}_{V, v}/\mathfrak m_w\mathcal{O}_{V, v}.\] Since \(\mathfrak m_{\overline{z}} = \mathfrak m_w \mathcal{O}_{Z, \overline{z}} = \mathfrak m_w \mathcal{O}_{W, w}^{sh}\) we see that \[\begin{align*} \mathcal{F}_{\overline{x}}/ \mathfrak m_{\overline{z}}\mathcal{F}_{\overline{x}} & = (a^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{X, \overline{x}} \\ & = (\mathcal{F}_w)_u \otimes_{\mathcal{O}_{U_w, u}} \mathcal{O}_{U, u}^{sh}/\mathfrak m_w\mathcal{O}_{U, u}^{sh} \\ & = (\mathcal{F}_w)_u \otimes_{\mathcal{O}_{U_w, u}} \mathcal{O}_{U_w, \overline{u}}^{sh} \\ & = (\mathcal{F}_w)_{\overline{u}} \end{align*}\] the penultimate equality by Algebra, Lemma 05WS and the last equality by Properties of Spaces, Lemma 05VP. The same arguments applied to the structure sheaves of \(V\) and \(Y\) show that \[\mathcal{O}_{V_w, \overline{v}}^{sh} = \mathcal{O}_{V, v}^{sh}/\mathfrak m_w \mathcal{O}_{V, v}^{sh} = \mathcal{O}_{Y, \overline{y}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{Y, \overline{y}}.\] OK, and now we can use Morphisms of Spaces, Lemma 05VU to see that (1) is equivalent to (3).
Finally we prove the equivalence of (2) and (3). To do this we pick a field extension \(\tilde K\) of \(K\) and a morphism \(\tilde x : \Spec(\tilde K) \to U\) which lies over \(u\) (this is possible because \(u \times_{X, x} \Spec(K)\) is a nonempty scheme). Set \(\tilde z : \Spec(\tilde K) \to U \to W\) be the composition. We obtain a commutative diagram \[\xymatrix{ & & & U_w \times_w \tilde z \ar[llld]_a \ar[rr] \ar[dr] & & V_w \times_w \tilde z \ar[llld]_>>>>>>>b \ar[dl] \\ X_z \ar[rr]_f \ar[dr]_g & & Y_z \ar[dl]^h & & \tilde z \ar[llld]_c \\ & z }\] where \(z = \Spec(K)\) and \(w = \Spec(\kappa(w))\). Now it is clear that \(\mathcal{F}_w\) and \(\mathcal{F}_z\) pull back to the same module on \(U_w \times_w \tilde z\). This leads to a commutative diagram \[\xymatrix{ X_z \ar[d] & U_w \times_w \tilde z \ar[l] \ar[d] \ar[r] & U_w \ar[d] \\ Y_z & V_w \times_w \tilde z \ar[l] \ar[r] & V_w }\] both of whose squares are cartesian and whose bottom horizontal arrows are flat: the lower left horizontal arrow is the composition of the morphism \(Y \times_Z \tilde z \to Y \times_Z z = Y_z\) (base change of a flat morphism), the étale morphism \(V \times_Z \tilde z \to Y \times_Z \tilde z\), and the étale morphism \(V \times_W \tilde z \to V \times_Z \tilde z\). Thus it follows from Morphisms of Spaces, Lemma 05VW that \[\mathcal{F}_z\text{ flat at }x\text{ over }Y_z \Leftrightarrow \mathcal{F}|_{U_w \times_w \tilde z} \text{ flat at }\tilde x\text{ over }V_w \times_w \tilde z \Leftrightarrow \mathcal{F}_w\text{ flat at }u\text{ over }V_w\] and we win.
Definition
Let \(S\) be a scheme. Let \(X \to Y \to Z\) be morphisms of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in |X|\) be a point and denote \(z \in |Z|\) its image.
We say the restriction of \(\mathcal{F}\) to its fibre over \(z\) is flat at \(x\) over the fibre of \(Y\) over \(z\) if the equivalent conditions of Lemma 05WY are satisfied.
We say the fibre of \(X\) over \(z\) is flat at \(x\) over the fibre of \(Y\) over \(z\) if the equivalent conditions of Lemma 05WY hold with \(\mathcal{F} = \mathcal{O}_X\).
We say the fibre of \(X\) over \(z\) is flat over the fibre of \(Y\) over \(z\) if for all \(x \in |X|\) lying over \(z\) the fibre of \(X\) over \(z\) is flat at \(x\) over the fibre of \(Y\) over \(z\)
With this definition in hand we can state a version of the criterion as follows. The Noetherian version can be found in Section 08VN.
Theorem
Let \(S\) be a scheme. Let \(f : X \to Y\) and \(Y \to Z\) be morphisms of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume
\(X\) is locally of finite presentation over \(Z\),
\(\mathcal{F}\) an \(\mathcal{O}_X\)-module of finite presentation, and
\(Y\) is locally of finite type over \(Z\).
Let \(x \in |X|\) and let \(y \in |Y|\) and \(z \in |Z|\) be the images of \(x\). If \(\mathcal{F}_{\overline{x}} \not = 0\), then the following are equivalent:
\(\mathcal{F}\) is flat over \(Z\) at \(x\) and the restriction of \(\mathcal{F}\) to its fibre over \(z\) is flat at \(x\) over the fibre of \(Y\) over \(z\), and
\(Y\) is flat over \(Z\) at \(y\) and \(\mathcal{F}\) is flat over \(Y\) at \(x\).
Moreover, the set of points \(x\) where (1) and (2) hold is open in \(\text{Supp}(\mathcal{F})\).
Proof
Choose a diagram as in Lemma 05WY part (3). It follows from the definitions that this reduces to the corresponding theorem for the morphisms of schemes \(U \to V \to W\), the quasi-coherent sheaf \(a^*\mathcal{F}\), and the point \(u \in U\). Thus the theorem follows from the corresponding result for schemes which is More on Morphisms, Theorem 039C.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) and \(Y \to Z\) be a morphism of algebraic spaces over \(S\). Assume
\(X\) is locally of finite presentation over \(Z\),
\(X\) is flat over \(Z\),
for every \(z \in |Z|\) the fibre of \(X\) over \(z\) is flat over the fibre of \(Y\) over \(z\), and
\(Y\) is locally of finite type over \(Z\).
Then \(f\) is flat. If \(f\) is also surjective, then \(Y\) is flat over \(Z\).
Proof
This is a special case of Theorem 05X0.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) and \(Y \to Z\) be morphisms of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume
\(X\) is locally of finite presentation over \(Z\),
\(\mathcal{F}\) an \(\mathcal{O}_X\)-module of finite presentation,
\(\mathcal{F}\) is flat over \(Z\), and
\(Y\) is locally of finite type over \(Z\).
Then the set \[A = \{x \in |X| : \mathcal{F} \text{ flat at }x \text{ over }Y\}.\] is open in \(|X|\) and its formation commutes with arbitrary base change: If \(Z' \to Z\) is a morphism of algebraic spaces, and \(A'\) is the set of points of \(X' = X \times_Z Z'\) where \(\mathcal{F}' = \mathcal{F} \times_Z Z'\) is flat over \(Y' = Y \times_Z Z'\), then \(A'\) is the inverse image of \(A\) under the continuous map \(|X'| \to |X|\).
Proof
One way to prove this is to translate the proof as given in More on Morphisms, Lemma 039E into the category of algebraic spaces. Instead we will prove this by reducing to the case of schemes. Namely, choose a diagram as in Lemma 05WY part (3) such that \(a\), \(b\), and \(c\) are surjective. It follows from the definitions that this reduces to the corresponding theorem for the morphisms of schemes \(U \to V \to W\), the quasi-coherent sheaf \(a^*\mathcal{F}\), and the point \(u \in U\). The only minor point to make is that given a morphism of algebraic spaces \(Z' \to Z\) we choose a scheme \(W'\) and a surjective étale morphism \(W' \to W \times_Z Z'\). Then we set \(U' = W' \times_W U\) and \(V' = W' \times_W V\). We write \(a', b', c'\) for the morphisms from \(U', V', W'\) to \(X', Y', Z'\). In this case \(A\), resp. \(A'\) are images of the open subsets of \(U\), resp. \(U'\) associated to \(a^*\mathcal{F}\), resp. \((a')^*\mathcal{F}'\). This indeed does reduce the lemma to More on Morphisms, Lemma 039E.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) and \(Y \to Z\) be a morphism of algebraic spaces over \(S\). Assume
\(X\) is locally of finite presentation over \(Z\),
\(X\) is flat over \(Z\), and
\(Y\) is locally of finite type over \(Z\).
Then the set \[\{x \in |X| : X\text{ flat at }x \text{ over }Y\}.\] is open in \(|X|\) and its formation commutes with arbitrary base change \(Z' \to Z\).
Proof
This is a special case of Lemma 05X2.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite presentation. Let \(\mathcal{F}\) be a finitely presented \(\mathcal{O}_X\)-module. Let \(x \in |X|\) with image \(y \in |Y|\). If \(\mathcal{F}\) is flat at \(x\) over \(Y\), then the following are equivalent
\((\mathcal{F}_{\overline{y}})_{\overline{x}}\) is a flat \(\mathcal{O}_{X_{\overline{y}}, \overline{x}}\)-module,
\((\mathcal{F}_{\overline{y}})_{\overline{x}}\) is a free \(\mathcal{O}_{X_{\overline{y}}, \overline{x}}\)-module,
\(\mathcal{F}_{\overline{y}}\) is finite free in an étale neighbourhood of \(\overline{x}\) in \(X_{\overline{y}}\), and
\(\mathcal{F}\) is finite free in an étale neighbourhood of \(x\) in \(X\).
Here \(\overline{x}\) is a geometric point of \(X\) lying over \(x\) and \(\overline{y} = f \circ \overline{x}\).
Proof
Pick a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\) and \(V\) are schemes and the vertical arrows are étale such that there is a point \(u \in U\) mapping to \(x\). Let \(v \in V\) be the image of \(u\). Applying Lemma 05WY to \(\text{id} : X \to X\) over \(Y\) we see that (1) translates into the condition “\(\mathcal{F}|_{U_v}\) is flat over \(U_v\) at \(u\)”. In other words, (1) is equivalent to \((\mathcal{F}|_{U_v})_u\) being a flat \(\mathcal{O}_{U_v, u}\)-module. By the case of schemes (More on Morphisms, Lemma 080Q), we find that this implies that \(\mathcal{F}|_U\) is finite free in an open neighbourhood of \(u\). In this way we see that (1) implies (4). The implications (4) \(\Rightarrow\) (3) and (2) \(\Rightarrow\) (1) are immediate. For the implication (3) \(\Rightarrow\) (2) use the description of local rings and stalks in Properties of Spaces, Lemmas 04KF and 05VP.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite presentation. Let \(\mathcal{F}\) be a finitely presented \(\mathcal{O}_X\)-module flat over \(Y\). Then the set \[\{x \in |X| : \mathcal{F}\text{ free in an \'etale neighbourhood of }x\}\] is open in \(|X|\) and its formation commutes with arbitrary base change \(Y' \to Y\).
Proof
Openness holds trivially. Let \(Y' \to Y\) be a morphism of algebraic spaces, set \(X' = Y' \times_Y X\), and let \(x' \in |X'|\) be a point lying over \(x \in |X|\). By Lemma 0CZS we see that \(x\) is in our set if and only if \((\mathcal{F}_{\overline{y}})_{\overline{x}}\) is a flat \(\mathcal{O}_{X_{\overline{y}}, \overline{x}}\)-module. Similarly, \(x'\) is in the analogue of our set for the pullback \(\mathcal{F}'\) of \(\mathcal{F}\) to \(X'\) if and only if \((\mathcal{F}'_{\overline{y}'})_{\overline{x}'}\) is a flat \(\mathcal{O}_{X'_{\overline{y}'}, \overline{x}'}\)-module (with obvious notation). These two assertions are equivalent by Lemma 05WY applied to the morphism \(\text{id} : X \to X\) over \(Y\). Thus the statement on base change holds.
Flatness over a Noetherian base
Here is the “Critère de platitude par fibres” in the Noetherian case.
Theorem
Let \(S\) be a scheme. Let \(f : X \to Y\) and \(Y \to Z\) be morphisms of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume
\(X\), \(Y\), \(Z\) locally Noetherian, and
\(\mathcal{F}\) a coherent \(\mathcal{O}_X\)-module.
Let \(x \in |X|\) and let \(y \in |Y|\) and \(z \in |Z|\) be the images of \(x\). If \(\mathcal{F}_{\overline{x}} \not = 0\), then the following are equivalent:
\(\mathcal{F}\) is flat over \(Z\) at \(x\) and the restriction of \(\mathcal{F}\) to its fibre over \(z\) is flat at \(x\) over the fibre of \(Y\) over \(z\), and
\(Y\) is flat over \(Z\) at \(y\) and \(\mathcal{F}\) is flat over \(Y\) at \(x\).
Proof
Choose a diagram as in Lemma 05WY part (3). It follows from the definitions that this reduces to the corresponding theorem for the morphisms of schemes \(U \to V \to W\), the quasi-coherent sheaf \(a^*\mathcal{F}\), and the point \(u \in U\). Thus the theorem follows from the corresponding result for schemes which is More on Morphisms, Theorem 039B.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) and \(Y \to Z\) be a morphism of algebraic spaces over \(S\). Assume
\(X\), \(Y\), \(Z\) locally Noetherian,
\(X\) is flat over \(Z\),
for every \(z \in |Z|\) the fibre of \(X\) over \(z\) is flat over the fibre of \(Y\) over \(z\).
Then \(f\) is flat. If \(f\) is also surjective, then \(Y\) is flat over \(Z\).
Proof
This is a special case of Theorem 0APR.
Just like for checking smoothness, if the base is Noetherian it suffices to check flatness over Artinian rings. Here is a sample statement.
Lemma
Let \(A\) be a Noetherian ring. Let \(I \subset A\) be an ideal. Let \(X\) be an algebraic space locally of finite presentation over \(S = \Spec(A)\). For \(n \geq 1\) set \(S_n = \Spec(A/I^n)\) and \(X_n = S_n \times_S X\). Let \(\mathcal{F}\) be coherent \(\mathcal{O}_X\)-module. If for every \(n \geq 1\) the pullback \(\mathcal{F}_n\) of \(\mathcal{F}\) to \(X\) is flat over \(S_n\), then the (open) locus where \(\mathcal{F}\) is flat over \(X\) contains the inverse image of \(V(I)\) under \(X \to S\).
Proof
The locus where \(\mathcal{F}\) is flat over \(S\) is open in \(|X|\) by Theorem 05WV. The statement is insensitive to replacing \(X\) by the members of an étale covering, hence we may assume \(X\) is an affine scheme. In this case the result follows immediately from Algebra, Lemma 0523. Some details omitted.
Normalization revisited
Normalization commutes with smooth base change.
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a smooth morphism of algebraic spaces over \(S\). Let \(\mathcal{A}\) be a quasi-coherent sheaf of \(\mathcal{O}_X\)-algebras. The integral closure of \(\mathcal{O}_Y\) in \(f^*\mathcal{A}\) is equal to \(f^*\mathcal{A}'\) where \(\mathcal{A}' \subset \mathcal{A}\) is the integral closure of \(\mathcal{O}_X\) in \(\mathcal{A}\).
Proof
By our construction of the integral closure, see Morphisms of Spaces, Definition 0821, this reduces immediately to the case where \(X\) and \(Y\) are affine. In this case the result is Algebra, Lemma 03GG.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ Y_2 \ar[r] \ar[d] & Y_1 \ar[d]^f \\ X_2 \ar[r]^\varphi & X_1 }\] be a fibre square of algebraic spaces over \(S\). Assume \(f\) is quasi-compact and quasi-separated and \(\varphi\) is smooth. Let \(Y_i \to X_i' \to X_i\) be the normalization of \(X_i\) in \(Y_i\). Then \(X_2' \cong X_2 \times_{X_1} X_1'\).
Proof
The base change of the factorization \(Y_1 \to X_1' \to X_1\) to \(X_2\) is a factorization \(Y_2 \to X_2 \times_{X_1} X_1' \to X_1\) and \(X_2 \times_{X_1} X_1' \to X_1\) is integral (Morphisms of Spaces, Lemma 03ZS). Hence we get a morphism \(h : X_2' \to X_2 \times_{X_1} X_1'\) by the universal property of Morphisms of Spaces, Lemma 0823. Observe that \(X_2'\) is the relative spectrum of the integral closure of \(\mathcal{O}_{X_2}\) in \(f_{2, *}\mathcal{O}_{Y_2}\). If \(\mathcal{A}' \subset f_{1, *}\mathcal{O}_{Y_1}\) denotes the integral closure of \(\mathcal{O}_{X_2}\), then \(X_2 \times_{X_1} X_1'\) is the relative spectrum of \(\varphi^*\mathcal{A}'\) as the construction of the relative spectrum commutes with arbitrary base change. By Cohomology of Spaces, Lemma 073K we know that \(f_{2, *}\mathcal{O}_{Y_2} = \varphi^*f_{1, *}\mathcal{O}_{Y_1}\). Hence the result follows from Lemma 082E.
Cohen-Macaulay morphisms
This is the analogue of More on Morphisms, Section 045Q.
Lemma
The property of morphisms of germs of schemes \[\begin{align*} & \mathcal{P}((X, x) \to (S, s)) = \\ & \text{the local ring } \mathcal{O}_{X_s, x} \text{ of the fibre is Noetherian and Cohen-Macaulay} \end{align*}\] is étale local on the source-and-target (Descent, Definition 04NB).
Proof
Given a diagram as in Descent, Definition 04NB we obtain an étale morphism of fibres \(U'_{v'} \to U_v\) mapping \(u'\) to \(u\), see Descent, Lemma 04NI. Thus the strict henselizations of the local rings \(\mathcal{O}_{U'_{v'}, u'}\) and \(\mathcal{O}_{U_v, u}\) are the same. We conclude by More on Algebra, Lemma 06LM.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume the fibres of \(f\) are locally Noetherian (Divisors on Spaces, Definition 0CUX).
Let \(x \in |X|\), and \(y = f(x)\). We say that \(f\) is Cohen-Macaulay at \(x\) if \(f\) is flat at \(x\) and the equivalent conditions of Morphisms of Spaces, Lemma 04NC hold for the property \(\mathcal{P}\) described in Lemma 0E0U.
We say \(f\) is a Cohen-Macaulay morphism if \(f\) is Cohen-Macaulay at every point of \(X\).
Here is a translation.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume the fibres of \(f\) are locally Noetherian. The following are equivalent
\(f\) is Cohen-Macaulay,
\(f\) is flat and for some surjective étale morphism \(V \to Y\) where \(V\) is a scheme, the fibres of \(X_V \to V\) are Cohen-Macaulay algebraic spaces, and
\(f\) is flat and for any étale morphism \(V \to Y\) where \(V\) is a scheme, the fibres of \(X_V \to V\) are Cohen-Macaulay algebraic spaces.
Given \(x \in |X|\) with image \(y \in |Y|\) the following are equivalent
\(f\) is Cohen-Macaulay at \(x\), and
\(\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}\) is flat and \(\mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}}\) is Cohen-Macaulay.
Proof
Given an étale morphism \(V \to Y\) where \(V\) is a scheme choose a scheme \(U\) and a surjective étale morphism \(U \to X \times_Y V\). Consider the commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] Let \(u \in U\) with images \(x \in |X|\), \(y \in |Y|\), and \(v \in V\). Then \(f\) is Cohen-Macaulay at \(x\) if and only if \(U \to V\) is Cohen-Macaulay at \(u\) (by definition). Moreover the morphism \(U_v \to X_v = (X_V)_v\) is surjective étale. Hence the scheme \(U_v\) is Cohen-Macaulay if and only if the algebraic space \(X_v\) is Cohen-Macaulay. Thus the equivalence of (1), (2), and (3) follows from the corresponding equivalence for morphisms of schemes, see More on Morphisms, Lemma 045S by a formal argument.
Proof of equivalence of (a) and (b). The corresponding equivalence for flatness is Morphisms of Spaces, Lemma 04NG. Thus we may assume \(f\) is flat at \(x\) when proving the equivalence. Consider a diagram and \(x, y, u, v\) as above. Then \(\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}\) is equal to the map \(\mathcal{O}_{V, v}^{sh} \to \mathcal{O}_{U, u}^{sh}\) on strict henselizations of local rings, see Properties of Spaces, Lemma 04KF. Thus we have \[\mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}} = (\mathcal{O}_{U, u}/\mathfrak m_v \mathcal{O}_{U, u})^{sh}\] by Algebra, Lemma 05WS. Thus we have to show that the Noetherian local ring \(\mathcal{O}_{U, u}/\mathfrak m_v \mathcal{O}_{U, u}\) is Cohen-Macaulay if and only if its strict henselization is. This is More on Algebra, Lemma 06LM.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of algebraic spaces over \(S\). Assume that the fibres of \(f\), \(g\), and \(g \circ f\) are locally Noetherian. Let \(x \in |X|\) with images \(y \in |Y|\) and \(z \in |Z|\).
If \(f\) is Cohen-Macaulay at \(x\) and \(g\) is Cohen-Macaulay at \(f(x)\), then \(g \circ f\) is Cohen-Macaulay at \(x\).
If \(f\) and \(g\) are Cohen-Macaulay, then \(g \circ f\) is Cohen-Macaulay.
If \(g \circ f\) is Cohen-Macaulay at \(x\) and \(f\) is flat at \(x\), then \(f\) is Cohen-Macaulay at \(x\) and \(g\) is Cohen-Macaulay at \(f(x)\).
If \(f \circ g\) is Cohen-Macaulay and \(f\) is flat, then \(f\) is Cohen-Macaulay and \(g\) is Cohen-Macaulay at every point in the image of \(f\).
Proof
Working étale locally this follows from the corresponding result for schemes, see More on Morphisms, Lemma 0C0W. Alternatively, we can use the equivalence of (a) and (b) in Lemma 0E0W. Thus we consider the local homomorphism of Noetherian local rings \[\mathcal{O}_{Y, \overline{y}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{Y, \overline{y}} \longrightarrow \mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{X, \overline{x}}\] whose fibre is \[\mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}}\] and we use Algebra, Lemma 045J.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a flat morphism of locally Noetherian algebraic spaces over \(S\). If \(X\) is Cohen-Macaulay, then \(f\) is Cohen-Macaulay and \(\mathcal{O}_{Y, f(\overline{x})}\) is Cohen-Macaulay for all \(x \in |X|\).
Proof
After translating into algebra using Lemma 0E0W (compare with the proof of Lemma 0E0X) this follows from Algebra, Lemma 045J.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume the fibres of \(f\) are locally Noetherian. Let \(Y' \to Y\) be locally of finite type. Let \(f' : X' \to Y'\) be the base change of \(f\). Let \(x' \in |X'|\) be a point with image \(x \in |X|\).
If \(f\) is Cohen-Macaulay at \(x\), then \(f' : X' \to Y'\) is Cohen-Macaulay at \(x'\).
If \(f\) is flat at \(x\) and \(f'\) is Cohen-Macaulay at \(x'\), then \(f\) is Cohen-Macaulay at \(x\).
If \(Y' \to Y\) is flat at \(f'(x')\) and \(f'\) is Cohen-Macaulay at \(x'\), then \(f\) is Cohen-Macaulay at \(x\).
Proof
Denote \(y \in |Y|\) and \(y' \in |Y'|\) the image of \(x'\). Choose a surjective étale morphism \(V \to Y\) where \(V\) is a scheme. Choose a surjective étale morphism \(U \to X \times_Y V\) where \(U\) is a scheme. Choose a surjectiev étale morphism \(V' \to Y' \times_Y V\) where \(V'\) is a scheme. Then \(U' = U \times_V V'\) is a scheme which comes equipped with a surjective étale morphism \(U' \to X'\). Choose \(u' \in U'\) mapping to \(x'\). Denote \(u \in U\) the image of \(u'\). Then the lemma follows from the lemma for \(U \to V\) and its base change \(U' \to V'\) and the points \(u'\) and \(u\) (this follows from the definitions). Thus the lemma follows from the case of schemes, see More on Morphisms, Lemma 045T.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and locally of finite presentation. Let \[W = \{x \in |X| : f\text{ is Cohen-Macaulay at }x\}\] Then \(W\) is open in \(|X|\) and the formation of \(W\) commutes with arbitrary base change of \(f\): For any morphism \(g : Y' \to Y\), consider the base change \(f' : X' \to Y'\) of \(f\) and the projection \(g' : X' \to X\). Then the corresponding set \(W'\) for the morphism \(f'\) is equal to \(W' = (g')^{-1}(W)\).
Proof
Choose a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] with étale vertical arrows and \(U\) and \(V\) schemes. Let \(u \in U\) with image \(x \in |X|\). Then \(f\) is Cohen-Macaulay at \(x\) if and only if \(U \to V\) is Cohen-Macaulay at \(u\) (by definition). Thus we reduce to the case of the morphism \(U \to V\). See More on Morphisms, Lemma 045U.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume that \(f\) is locally of finite presentation and Cohen-Macaulay. Then there exist open and closed subschemes \(X_d \subset X\) such that \(X = \coprod_{d \geq 0} X_d\) and \(f|_{X_d} : X_d \to Y\) has relative dimension \(d\).
Proof
Choose a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] with étale vertical arrows and \(U\) and \(V\) schemes. Then \(U \to V\) is locally of finite presentation and Cohen-Macaulay (immediate from our definitions). Thus we have a decomposition \(U = \coprod_{d \geq 0} U_d\) into open and closed subschemes with \(f|_{U_d} : U_d \to V\) of relative dimension \(d\), see Morphisms, Lemma 02NM. Let \(u \in U\) with image \(x \in |X|\). Then \(f\) has relative dimension \(d\) at \(x\) if and only if \(U \to V\) has relative dimension \(d\) at \(u\) (this follows from our definitions). In this way we see that \(U_d\) is the inverse image of a subset \(X_d \subset |X|\) which is necessarily open and closed. Denoting \(X_d\) the corresponding open and closed algebraic subspace of \(X\) we see that the lemma is true.
Gorenstein morphisms
This is the analogue of Duality for Schemes, Section 0C02.
Lemma
The property of morphisms of germs of schemes \[\begin{align*} & \mathcal{P}((X, x) \to (S, s)) = \\ & \text{the local ring } \mathcal{O}_{X_s, x} \text{ of the fibre is Noetherian and Gorenstein} \end{align*}\] is étale local on the source-and-target (Descent, Definition 04NB).
Proof
Given a diagram as in Descent, Definition 04NB we obtain an étale morphism of fibres \(U'_{v'} \to U_v\) mapping \(u'\) to \(u\), see Descent, Lemma 04NI. Thus \(\mathcal{O}_{U_v, u} \to \mathcal{O}_{U'_{v'}, u'}\) is the localization of an étale ring map. Hence the first is Noetherian if and only if the second is Noetherian, see More on Algebra, Lemma 0AGZ. Then, since \(\mathcal{O}_{U'_{v'}, u'}/\mathfrak m_u \mathcal{O}_{U'_{v'}, u'} = \kappa(u')\) (Algebra, Lemma 00U4) is a Gorenstein ring, we see that \(\mathcal{O}_{U_v, u}\) is Gorenstein if and only if \(\mathcal{O}_{U'_{v'}, u'}\) is Gorenstein by Dualizing Complexes, Lemma 0BJL.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume the fibres of \(f\) are locally Noetherian (Divisors on Spaces, Definition 0CUX).
Let \(x \in |X|\), and \(y = f(x)\). We say that \(f\) is Gorenstein at \(x\) if \(f\) is flat at \(x\) and the equivalent conditions of Morphisms of Spaces, Lemma 04NC hold for the property \(\mathcal{P}\) described in Lemma 0E13.
We say \(f\) is a Gorenstein morphism if \(f\) is Gorenstein at every point of \(X\).
Here is a translation.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume the fibres of \(f\) are locally Noetherian. The following are equivalent
\(f\) is Gorenstein,
\(f\) is flat and for some surjective étale morphism \(V \to Y\) where \(V\) is a scheme, the fibres of \(X_V \to V\) are Gorenstein algebraic spaces, and
\(f\) is flat and for any étale morphism \(V \to Y\) where \(V\) is a scheme, the fibres of \(X_V \to V\) are Gorenstein algebraic spaces.
Given \(x \in |X|\) with image \(y \in |Y|\) the following are equivalent
\(f\) is Gorenstein at \(x\), and
\(\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}\) is flat and \(\mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}}\) is Gorenstein.
Proof
Given an étale morphism \(V \to Y\) where \(V\) is a scheme choose a scheme \(U\) and a surjective étale morphism \(U \to X \times_Y V\). Consider the commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] Let \(u \in U\) with images \(x \in |X|\), \(y \in |Y|\), and \(v \in V\). Then \(f\) is Gorenstein at \(x\) if and only if \(U \to V\) is Gorenstein at \(u\) (by definition). Moreover the morphism \(U_v \to X_v = (X_V)_v\) is surjective étale. Hence the scheme \(U_v\) is Gorenstein if and only if the algebraic space \(X_v\) is Gorenstein. Thus the equivalence of (1), (2), and (3) follows from the corresponding equivalence for morphisms of schemes, see Duality for Schemes, Lemma 0BFQ by a formal argument.
Proof of equivalence of (a) and (b). The corresponding equivalence for flatness is Morphisms of Spaces, Lemma 04NG. Thus we may assume \(f\) is flat at \(x\) when proving the equivalence. Consider a diagram and \(x, y, u, v\) as above. Then \(\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}\) is equal to the map \(\mathcal{O}_{V, v}^{sh} \to \mathcal{O}_{U, u}^{sh}\) on strict henselizations of local rings, see Properties of Spaces, Lemma 04KF. Thus we have \[\mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}} = (\mathcal{O}_{U, u}/\mathfrak m_v \mathcal{O}_{U, u})^{sh}\] by Algebra, Lemma 05WS. Thus we have to show that the Noetherian local ring \(\mathcal{O}_{U, u}/\mathfrak m_v \mathcal{O}_{U, u}\) is Gorenstein if and only if its strict henselization is. This follows immediately from Dualizing Complexes, Lemma 0DWD and the definition of a Gorenstein local ring as a Noetherian local ring which is a dualizing complex over itself.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of algebraic spaces over \(S\). Assume that the fibres of \(f\), \(g\), and \(g \circ f\) are locally Noetherian. Let \(x \in |X|\) with images \(y \in |Y|\) and \(z \in |Z|\).
If \(f\) is Gorenstein at \(x\) and \(g\) is Gorenstein at \(f(x)\), then \(g \circ f\) is Gorenstein at \(x\).
If \(f\) and \(g\) are Gorenstein, then \(g \circ f\) is Gorenstein.
If \(g \circ f\) is Gorenstein at \(x\) and \(f\) is flat at \(x\), then \(f\) is Gorenstein at \(x\) and \(g\) is Gorenstein at \(f(x)\).
If \(f \circ g\) is Gorenstein and \(f\) is flat, then \(f\) is Gorenstein and \(g\) is Gorenstein at every point in the image of \(f\).
Proof
Working étale locally this follows from the corresponding result for schemes, see Duality for Schemes, Lemma 0C11. Alternatively, we can use the equivalence of (a) and (b) in Lemma 0E15. Thus we consider the local homomorphism of Noetherian local rings \[\mathcal{O}_{Y, \overline{y}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{Y, \overline{y}} \longrightarrow \mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{X, \overline{x}}\] whose fibre is \[\mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}}\] and we use Dualizing Complexes, Lemma 0BJL.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a flat morphism of locally Noetherian algebraic spaces over \(S\). If \(X\) is Gorenstein, then \(f\) is Gorenstein and \(\mathcal{O}_{Y, f(\overline{x})}\) is Gorenstein for all \(x \in |X|\).
Proof
After translating into algebra using Lemma 0E15 (compare with the proof of Lemma 0E16) this follows from Dualizing Complexes, Lemma 0BJL.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume the fibres of \(f\) are locally Noetherian. Let \(Y' \to Y\) be locally of finite type. Let \(f' : X' \to Y'\) be the base change of \(f\). Let \(x' \in |X'|\) be a point with image \(x \in |X|\).
If \(f\) is Gorenstein at \(x\), then \(f' : X' \to Y'\) is Gorenstein at \(x'\).
If \(f\) is flat at \(x\) and \(f'\) is Gorenstein at \(x'\), then \(f\) is Gorenstein at \(x\).
If \(Y' \to Y\) is flat at \(f'(x')\) and \(f'\) is Gorenstein at \(x'\), then \(f\) is Gorenstein at \(x\).
Proof
Denote \(y \in |Y|\) and \(y' \in |Y'|\) the image of \(x'\). Choose a surjective étale morphism \(V \to Y\) where \(V\) is a scheme. Choose a surjective étale morphism \(U \to X \times_Y V\) where \(U\) is a scheme. Choose a surjectiev étale morphism \(V' \to Y' \times_Y V\) where \(V'\) is a scheme. Then \(U' = U \times_V V'\) is a scheme which comes equipped with a surjective étale morphism \(U' \to X'\). Choose \(u' \in U'\) mapping to \(x'\). Denote \(u \in U\) the image of \(u'\). Then the lemma follows from the lemma for \(U \to V\) and its base change \(U' \to V'\) and the points \(u'\) and \(u\) (this follows from the definitions). Thus the lemma follows from the case of schemes, see Duality for Schemes, Lemma 0C07.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and locally of finite presentation. Let \[W = \{x \in |X| : f\text{ is Gorenstein at }x\}\] Then \(W\) is open in \(|X|\) and the formation of \(W\) commutes with arbitrary base change of \(f\): For any morphism \(g : Y' \to Y\), consider the base change \(f' : X' \to Y'\) of \(f\) and the projection \(g' : X' \to X\). Then the corresponding set \(W'\) for the morphism \(f'\) is equal to \(W' = (g')^{-1}(W)\).
Proof
Choose a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] Let \(u \in U\) with image \(x \in |X|\). Then \(f\) is Gorenstein at \(x\) if and only if \(U \to V\) is Gorenstein at \(u\) (by definition). Thus we reduce to the case of the morphism \(U \to V\) of schemes. Openness is proven in Duality for Schemes, Lemma 0C09 and compatibility with base change in Duality for Schemes, Lemma 0E0Q.
Slicing Cohen-Macaulay morphisms
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(f_1, \ldots, f_r \in \Gamma(X, \mathcal{O}_X)\). In this case we denote \(V(f_1, \ldots, f_r)\) the closed subspace of \(X\) cut out by \(f_1, \ldots, f_r\). More precisely, we can define \(V(f_1, \ldots, f_r)\) as the closed subspace of \(X\) corresponding to the quasi-coherent sheaf of ideals generated by \(f_1, \ldots, f_r\), see Morphisms of Spaces, Lemma 03MB. Alternatively, we can choose a presentation \(X = U/R\) and consider the closed subscheme \(Z \subset U\) cut out by \(f_1|U, \ldots, f_r|_U\). It is clear that \(Z\) is an \(R\)-invariant (see Groupoids, Definition 03BC) closed subscheme and we may set \(V(f_1, \ldots, f_r) = Z/R_Z\).
Lemma
Let \(S\) be a scheme. Consider a cartesian diagram \[\xymatrix{ X \ar[d] & F \ar[l]^p \ar[d] \\ Y & \Spec(k) \ar[l] }\] where \(X \to Y\) is a morphism of algebraic spaces over \(S\) which is flat and locally of finite presentation, and where \(k\) is a field over \(S\). Let \(f_1, \ldots, f_r \in \Gamma(X, \mathcal{O}_X)\) and \(z \in |F|\) such that \(f_1, \ldots, f_r\) map to a regular sequence in the local ring \(\mathcal{O}_{F, \overline{z}}\). Then, after replacing \(X\) by an open subspace containing \(p(z)\), the morphism \[V(f_1, \ldots, f_r) \longrightarrow Y\] is flat and locally of finite presentation.
Proof
Set \(Z = V(f_1, \ldots, f_r)\). It is clear that \(Z \to X\) is locally of finite presentation, hence the composition \(Z \to Y\) is locally of finite presentation, see Morphisms of Spaces, Lemma 03XQ. Hence it suffices to show that \(Z \to Y\) is flat in a neighbourhood of \(p(z)\). Let \(k'/k\) be an extension field. Then \(F' = F \times_{\Spec(k)} \Spec(k')\) is surjective and flat over \(F\), hence we can find a point \(z' \in |F'|\) mapping to \(z\) and the local ring map \(\mathcal{O}_{F, \overline{z}} \to \mathcal{O}_{F', \overline{z}'}\) is flat, see Morphisms of Spaces, Lemma 04NG. Hence the image of \(f_1, \ldots, f_r\) in \(\mathcal{O}_{F', \overline{z}'}\) is a regular sequence too, see Algebra, Lemma 00LM. Thus, during the proof we may replace \(k\) by an extension field. In particular, we may assume that \(z \in |F|\) comes from a section \(z : \Spec(k) \to F\) of the structure morphism \(F \to \Spec(k)\).
Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to X \times_Y V\). After possibly enlarging \(k\) once more we may assume that \(\Spec(k) \to F \to X\) factors through \(U\) (as \(U \to X\) is surjective). Let \(u : \Spec(k) \to U\) be such a factorization and denote \(v \in V\) the image of \(u\). Note that the morphisms \[U_v \times_{\Spec(\kappa(v))} \Spec(k) = U \times_V \Spec(k) \to U \times_Y \Spec(k) \to F\] are étale (the first as the base change of \(V \to V \times_Y V\) and the second as the base change of \(U \to X\)). Moreover, by construction the point \(u : \Spec(k) \to U\) gives a point of the left most space which maps to \(z\) on the right. Hence the elements \(f_1, \ldots, f_r\) map to a regular sequence in the local ring on the right of the following map \[\mathcal{O}_{U_v, u} \longrightarrow \mathcal{O}_{U_v \times_{\Spec(\kappa(v)} \Spec(k), \overline{u}} = \mathcal{O}_{U \times_V \Spec(k), \overline{u}}.\] But since the displayed arrow is flat (combine More on Flatness, Lemma 05VL and Morphisms of Spaces, Lemma 04NG) we see from Algebra, Lemma 00LM that \(f_1, \ldots, f_r\) maps to a regular sequence in \(\mathcal{O}_{U_v, u}\). By More on Morphisms, Lemma 06LI we conclude that the morphism of schemes \[V(f_1, \ldots, f_r) \times_X U = V(f_1|_U, \ldots, f_r|_U) \to V\] is flat in an open neighbourhood \(U'\) of \(u\). Let \(X' \subset X\) be the open subspace corresponding to the image of \(|U'| \to |X|\) (see Properties of Spaces, Lemmas 03BX and 03BZ). We conclude that \(V(f_1, \ldots, f_r) \cap X' \to Y\) is flat (see Morphisms of Spaces, Definition 03ML) as we have the commutative diagram \[\xymatrix{ V(f_1, \ldots, f_r) \times_X U' \ar[d]_a \ar[r] & V \ar[d]^b \\ V(f_1, \ldots, f_r) \cap X' \ar[r] & Y }\] with \(a, b\) étale and \(a\) surjective.
Reduced fibres
This section is the analogue of More on Morphisms, Section 0574.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(y \in |Y|\). The following are equivalent
for some morphism \(\Spec(k) \to Y\) in the equivalence class of \(y\) the algebraic space \(X_k\) is geometrically reduced over \(k\),
for every morphism \(\Spec(k) \to Y\) in the equivalence class of \(y\) the algebraic space \(X_k\) is geometrically reduced over \(k\),
for every morphism \(\Spec(k) \to Y\) in the equivalence class of \(y\) the algebraic space \(X_k\) is reduced.
Proof
This follows immediately from Spaces over Fields, Lemma 0E04 and the definition of the equivalence relation defining \(|X|\) given in Properties of Spaces, Section 03BT.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(y \in |Y|\). We say the fibre of \(f : X \to Y\) at \(y\) is geometrically reduced if the equivalent conditions of Lemma 0E07 hold.
Here are the obligatory lemmas.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) and \(g : Y' \to Y\) be morphisms of algebraic spaces over \(S\). Denote \(f' : X' \to Y'\) the base change of \(f\) by \(g\). Then \[\begin{align*} \{y' \in |Y'| : \text{the fibre of }f' : X' \to Y'\text{ at }y' \text{ is geometrically reduced}\} \\ = g^{-1}(\{y \in |Y| : \text{the fibre of }f : X \to Y\text{ at }y \text{ is geometrically reduced}\}). \end{align*}\]
Proof
For \(y' \in |Y'|\) choose a morphism \(\Spec(k) \to Y'\) in the equivalence class of \(y'\). Then \(g(y')\) is represented by the composition \(\Spec(k) \to Y' \to Y\). Hence \(X' \times_{Y'} \Spec(k) = X \times_Y \Spec(k)\) and the result follows from the definition.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is quasi-compact and locally of finite presentation. Then the set \[E = \{y \in |Y| : \text{the fibre of }f : X \to Y\text{ at }y \text{ is geometrically reduced}\}\] is étale locally constructible.
Proof
Choose an affine scheme \(V\) and an étale morphism \(V \to Y\). The meaning of the statement is that the inverse image of \(E\) in \(|V|\) is constructible. By Lemma 0E09 we may replace \(Y\) by \(V\), i.e., we may assume that \(Y\) is an affine scheme. Then \(X\) is quasi-compact. Choose an affine scheme \(U\) and a surjective étale morphism \(U \to X\). For a morphism \(\Spec(k) \to Y\) the morphism between fibres \(U_k \to X_k\) is surjective étale. Hence \(U_k\) is geometrically reduced over \(k\) if and only if \(X_k\) is geometrically reduced over \(k\), see Spaces over Fields, Lemma 0E05. Thus the set \(E\) for \(X \to Y\) is the same as the set \(E\) for \(U \to Y\). In this way we see that the lemma follows from the case of schemes, see More on Morphisms, Lemma 0579.
Lemma
Let \(X\) be an algebraic space over a discrete valuation ring \(R\) whose structure morphism \(X \to \Spec(R)\) is proper and flat. If the special fibre is reduced, then both \(X\) and the generic fibre \(X_\eta\) are reduced.
Proof
Choose an étale morphism \(U \to X\) where \(U\) is an affine scheme. Then \(U\) is of finite type over \(R\). Let \(u \in U\) be in the special fibre. The local ring \(A = \mathcal{O}_{U, u}\) is essentially of finite type over \(R\), hence Noetherian. Let \(\pi \in R\) be a uniformizer. Since \(X\) is flat over \(R\), we see that \(\pi \in \mathfrak m_A\) is a nonzerodivisor on \(A\) and since the special fibre of \(X\) is reduced, we have that \(A/\pi A\) is reduced. If \(a \in A\), \(a \not = 0\) then there exists an \(n \geq 0\) and an element \(a' \in A\) such that \(a = \pi^n a'\) and \(a' \not \in \pi A\). This follows from Krull intersection theorem (Algebra, Lemma 00IP). If \(a\) is nilpotent, so is \(a'\), because \(\pi\) is a nonzerodivisor. But \(a'\) maps to a nonzero element of the reduced ring \(A/\pi A\) so this is impossible. Hence \(A\) is reduced. It follows that there exists an open neighbourhood of \(u\) in \(U\) which is reduced (small detail omitted; use that \(U\) is Noetherian). Thus we can find an étale morphism \(U \to X\) with \(U\) a reduced scheme, such that every point of the special fibre of \(X\) is in the image. Since \(X\) is proper over \(R\) it follows that \(U \to X\) is surjective. Hence \(X\) is reduced. Since the generic fibre of \(U \to \Spec(R)\) is reduced as well (on affine pieces it is computed by taking localizations), we conclude the same thing is true for the generic fibre.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If \(f\) is flat, proper, and of finite presentation, then the set \[E = \{y \in |Y| : \text{the fibre of }f : X \to Y\text{ at }y \text{ is geometrically reduced}\}\] is open in \(|Y|\).
Proof
By Lemma 0E09 formation of \(E\) commutes with base change. To check a subset of \(|Y|\) is open, we may replace \(Y\) by the members of an étale covering. Thus we may assume \(Y\) is affine. Then \(Y\) is a cofiltered limit of affine schemes of finite type over \(\mathbf{Z}\). Hence we can assume \(X \to Y\) is the base change of \(X_0 \to Y_0\) where \(Y_0\) is the spectrum of a finite type \(\mathbf{Z}\)-algebra and \(X_0 \to Y_0\) is flat and proper. See Limits of Spaces, Lemma 07SK, 08K0, and 08K1. Since the formation of \(E\) commutes with base change (see above), we may assume the base is Noetherian.
Assume \(Y\) is Noetherian. The set is constructible by Lemma 0E0A. Hence it suffices to show the set is stable under generalization (Topology, Lemma 0542). By Properties, Lemma 054F we reduce to the case where \(Y = \Spec(R)\), \(R\) is a discrete valuation ring, and the closed fibre \(X_y\) is geometrically reduced. To show: the generic fibre \(X_\eta\) is geometrically reduced.
If not then there exists a finite extension \(L\) of the fraction field of \(R\) such that \(X_L\) is not reduced, see Spaces over Fields, Lemmas 0E02 (characteristic zero) and 0E03 (positive characteristic). There exists a discrete valuation ring \(R' \subset L\) with fraction field \(L\) dominating \(R\), see Algebra, Lemma 09IG. After replacing \(R\) by \(R'\) we reduce to Lemma 0E0B.
Connected components of fibres
This section is the analogue of More on Morphisms, Section 055C.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \[n_{X/Y} : |Y| \to \{0, 1, 2, 3, \ldots, \infty\}\] be the function which associates to \(y \in Y\) the number of connected components of \(X_k\) where \(\Spec(k) \to Y\) is in the equivalence class of \(y\) with \(k\) algebraically closed. This is well defined and if \(g : Y' \to Y\) is a morphism then \[n_{X'/Y'} = n_{X/Y} \circ g\] where \(X' \to Y'\) is the base change of \(f\).
Proof
Suppose that \(y' \in Y'\) has image \(y \in Y\). Let \(\Spec(k') \to Y'\) be in the equivalence class of \(y'\) with \(k'\) algebraically closed. Then we can choose a commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[rd] & \Spec(k') \ar[r] & Y' \ar[d] \\ & \Spec(k) \ar[r] & Y }\] where \(K\) is an algebraically closed field. The result follows as the morphisms of schemes \[\xymatrix{ X'_{k'} & (X'_{k'})_K = (X_k)_K \ar[l] \ar[r] & X_k }\] induce bijections between connected components, see Spaces over Fields, Lemma 0A12. To use this to prove the function is well defined take \(Y' = Y\).
Dimension of fibres
This section is the analogue of More on Morphisms, Section 05F6.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a finite type morphism of algebraic spaces over \(S\). Let \(y \in |Y|\). The following quantities are the same
\(d = -\infty\) if \(y\) is not in the image of \(|f|\) and otherwise the minimal integer \(d\) such that \(f\) has relative dimension \(\leq d\) at every \(x \in |X|\) mapping to \(y\),
the dimension of the algebraic space \(X_k = \Spec(k) \times_Y X\) for any morphism \(\Spec(k) \to Y\) in the equivalence class defining \(y\).
Proof
To parse this one has to consult Morphisms of Spaces, Definition 04NM, Properties of Spaces, Definition 04N6, Properties of Spaces, Definition 04N5. We will show that the numbers in (1) and (2) are equal for a fixed morphism \(\Spec(k) \to Y\). Choose an étale morphism \(V \to Y\) where \(V\) is an affine scheme and a point \(v \in V\) mapping to \(y\). Since \(V \times_Y \Spec(k) \to \Spec(k)\) is surjective étale (by Properties of Spaces, Lemma 03H4) we can find a finite separable extension \(k'/k\) (by Morphisms, Lemma 02GL) and a commutative diagram \[\xymatrix{ \Spec(k') \ar[r] \ar[d] & V \ar[d] \\ \Spec(k) \ar[r] & Y }\] We may replace \(X \to Y\) by \(V \times_Y X \to V\) and \(X_k\) by \(X_{k'} = \Spec(k') \times_V (V \times_Y X)\) because this does not change the dimensions in question by Properties of Spaces, Lemma 0A4H and Morphisms of Spaces, Lemma 04NS. Thus we may assume that \(Y\) is an affine scheme. In this case we may assume that \(k = \kappa(y)\) because the dimension of \(X_{\kappa(y)}\) and \(X_k\) are the same by the aforementioned Morphisms of Spaces, Lemma 04NS and the fact that for an algebraic space \(Z\) over a field \(K\) the relative dimension of \(Z\) at a point \(z \in |Z|\) is the same as \(\dim_z(Z)\) by definition. Assume \(Y\) is affine and \(k = \kappa(y)\). Then \(X\) is quasi-compact we can choose an affine scheme \(U\) and an surjective étale morphism \(U \to X\). Then \(\dim(X_k) = \dim(U_k) = \max \dim_u(U_k)\) is equal to the number given in (1) by definition.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a finite type morphism of algebraic spaces over \(S\). Let \[n_{X/Y} : |Y| \to \{-\infty, 0, 1, 2, 3, \ldots\}\] be the function which associates to \(y \in |Y|\) the integer discussed in Lemma 0D4M. If \(g : Y' \to Y\) is a morphism then \[n_{X'/Y'} = n_{X/Y} \circ |g|\] where \(X' \to Y'\) is the base change of \(f\).
Proof
This follows immediately from Lemma 0D4M.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a flat morphism of finite presentation of algebraic spaces over \(S\). Let \(n_{X/Y}\) be the function on \(Y\) giving the dimension of fibres of \(f\) introduced in Lemma 0D4N. Then \(n_{X/Y}\) is lower semi-continuous.
Proof
Let \(V \to Y\) be a surjective étale morphism where \(V\) is a scheme. If we can show that the composition \(n_{X/Y} \circ |g|\) is lower semi-continuous, then the lemma follows as \(|g|\) is open. Hence we may assume \(Y\) is a scheme. Working locally we may assume \(V\) is an affine scheme. Then we can choose an affine scheme \(U\) and a surjective étale morphism \(U \to X\). Then \(n_{X/Y} = n_{U/Y}\). Hence we may assume \(X\) and \(Y\) are both schemes. In this case the lemma follows from More on Morphisms, Lemma 0D4H.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper morphism of algebraic spaces over \(S\). Let \(n_{X/Y}\) be the function on \(Y\) giving the dimension of fibres of \(f\) introduced in Lemma 0D4N. Then \(n_{X/Y}\) is upper semi-continuous.
Proof
Let \(Z_d = \{x \in |X| : \text{the fibre of }f\text{ at }x\text{ has dimension }> d\}\). Then \(Z_d\) is a closed subset of \(|X|\) by Morphisms of Spaces, Lemma 04NT. Since \(f\) is proper \(f(Z_d)\) is closed in \(|Y|\). Since \(y \in f(Z_d) \Leftrightarrow n_{X/Y}(y) > d\) we see that the lemma is true.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper, flat, finitely presented morphism of algebraic spaces over \(S\). Let \(n_{X/Y}\) be the function on \(Y\) giving the dimension of fibres of \(f\) introduced in Lemma 0D4N. Then \(n_{X/Y}\) is locally constant.
Proof
Catenary algebraic spaces
This section continues the discussion started in Decent Spaces, Section 0ED3. The following lemma will be used in the proof of the next one.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be an integral morphism of algebraic spaces over \(S\). Let \(y \in |Y|\) be a point which can be represented by a closed immersion \(y : \Spec(k) \to Y\). Then there exists a factorization \(X \to X' \to Y\) of \(f\) such that
\(X' \to Y\) is integral,
\(X \to X'\) is an isomorphism over \(X' \setminus X'_y\),
\(X'_y\) has a unique point \(x'\) with \(\kappa(x') = k\).
Moreover, if \(f\) is finite and \(Y\) is locally Noetherian, then \(X' \to Y\) is finite.
Proof
By Morphisms of Spaces, Lemma 03M9 the sheaves \(f_*\mathcal{O}_X\), \((X_y \to Y)_*\mathcal{O}_{X_y}\), and \(y_*\mathcal{O}_{\Spec(k)}\) are quasi-coherent sheaves of \(\mathcal{O}_Y\)-algebras. Consider the maps \[f_*\mathcal{O}_Y \longrightarrow (X_y \to Y)_*\mathcal{O}_{X_y} \longleftarrow y_*\mathcal{O}_{\Spec(k)}\] The fibre product is a quasi-coherent sheaf of \(\mathcal{O}_Y\)-algebras \(\mathcal{A}'\) and we can define \(X' \to Y\) as the relative spectrum of \(\mathcal{A}'\) over \(Y\), see Morphisms, Lemma 01SA. This construction commutes with arbitrary change of base. In particular, it is clear that over the open subspace \(|Y| \setminus \{y\}\) the morphism \(X \to X'\) is an isomorphism and over \(|Y| \setminus \{y\}\) the morphism \(X' \to Y\) is integral. It remains to prove the statements in a small neighbourhood of \(y\). Choose an affine scheme \(V = \Spec(R)\) and an étale morphism \(\varphi : V \to Y\) such that \(y\) is in the image of \(\varphi\). Then \(V_y\) is a closed subscheme of \(V\) étale over \(k\), whence consists of finitely many points each with residue field separable over \(k\) (see Decent Spaces, Remark 03II). After shrinking \(V\) we may assume there is a unique closed point \(v = \Spec(l) \to V\) mapping to \(y\) with \(l/k\) finite separable. We may write \(V \times_Y X = \Spec(C)\) with \(R \to C\) an integral ring map. The stated compatibility with base change gives us that \(U \times_X Y' = \Spec(C')\) where \[C' = C \times_{C \otimes_R l} l\] Since \(R \to l\) is surjective, also \(C \to C \otimes_R l\) is surjective and we see that this is a fibre product of the kind studied in More on Algebra, Situation 08KH (with \(A, A', B, B'\) corresponding to \(C \otimes_R l, C, l, C'\)). Observe that \(C'\) is an \(R\)-subalgebra of \(C\) and hence is integral over \(R\); this proves (1). Finally, More on Algebra, Lemma 0B7J shows that \(V \times_X Y' = \Spec(C')\) has a unique point \(y''\) lying over \(v\) with residue \(l\) (this corresponds with the obvious surjective map \(C' \to l\)). Thus \(X_y \times_{\Spec(k)} \Spec(l)\) has a unique point with residue field \(l\). Since \(l/k\) is finite separable, this implies \(X'_y\) has a unique point with residue field \(k\), i.e., (3) holds.
To prove the final statement, observe that if \(Y\) is locally Noetherian, then \(R\) is a Noetherian ring and if \(f\) is finite, then \(R \to C\) is finite. Then \(C'\) is a finite type \(R\)-algebra by More on Algebra, Lemma 00IT. This proves that \(X' \to Y\) is finite.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(\delta : |B| \to \mathbf{Z}\) be a function. Assume \(B\) is decent, locally Noetherian, and universally catenary and \(\delta\) is a dimension function. If \(X\) is a decent algebraic space over \(B\) whose structure morphism \(f : X \to B\) is locally of finite type we define \(\delta_X : |X| \to \mathbf{Z}\) by the rule \[\delta_X(x) = \delta(f(x)) + \text{transcendence degreeof }x/f(x)\] (Morphisms of Spaces, Definition 04NM). Then \(\delta_X\) is a dimension function.
Proof
The problem is local on \(B\). Thus we may assume \(B\) is quasi-compact. By Decent Spaces, Lemma 0BB6 we see \(B\) is quasi-separated. By Limits of Spaces, Proposition 09YC we can choose a finite surjective morphism \(\pi : Y \to X\) where \(Y\) is a scheme. Claim: \(\delta_Y\) is a dimension function.
The claim implies the lemma. With \(X \to B\) as in the lemma set \(Z = Y \times_B X\) with projections \(p : Z \to Y\) and \(q : Z \to X\). Then we have \[\delta_Z(z) = \delta_Y(p(z)) + \text{transcendence degreeof }z/p(z)\] and \(\delta_Z(z) = \delta_X(q(z))\). This follows from Morphisms of Spaces, Lemma 04NR and the fact that these transcendence degrees are zero for finite morphisms. By Decent Spaces, Lemma 0ED5 and the claim we find that \(\delta_Z\) is a dimension function. Then we find that \(\delta_X\) is a dimension function by Decent Spaces, Lemma 0ED9.
Proof of the claim. Consider a specialization \(y \leadsto y'\), \(y \not = y'\) of points of the Noetherian scheme \(Y\). Then \(\delta_Y(y) > \delta_Y(y')\) because there are no specializations between points in fibres of \(Y\) (see Decent Spaces, Lemma 0ACK). Using this for a chain of specializations we find \[\delta_Y(y) - \delta_Y(y') \geq \text{codim}(\overline{\{y'\}}, \overline{\{y\}})\] Our task is to show equality. By Properties, Lemma 02IL we can choose a specialization \(y' \leadsto y_0\). It suffices to show \(\delta_Y(y) - \delta_Y(y_0) = \text{codim}(\overline{\{y_0\}}, \overline{\{y\}})\) because this will imply the equality for both \(y \leadsto y'\) and \(y' \leadsto y_0\).
Choose a maximal chain \(y = y_c \leadsto y_{c - 1} \leadsto \ldots \leadsto y_0\) of specializations in \(Y\). Set \(b = \pi(y)\) and \(b_0 = \pi(y_0)\). Choose a maximal chain \(b = b_e \leadsto b_{e - 1} \leadsto \ldots \leadsto b_0\) of specializations in \(|B|\). We have to show \(e = c\). Since \(\pi\) is closed (Morphisms of Spaces, Lemma 04NZ) we can find a sequence of specializations \(y = y'_e \leadsto y'_{e - 1} \leadsto \ldots \leadsto y'_0\) mapping to \(b = b_e \leadsto b_{e - 1} \leadsto \ldots \leadsto b_0\). Observe that \(y'_e \leadsto y'_{e - 1} \leadsto \ldots \leadsto y'_0\) is a maximal chain as well. If \(y_0 = y'_0\), then because \(Y\) is catenary, we conclude that \(e = c\) as desired. In the next paragraph we reduce to this case by sleight of hand and we conclude in the same manner.
Since \(\pi\) is closed we see that \(b_0\) is a closed point of \(|B|\). By Decent Spaces, Lemma 0AHB we can represent \(b_0\) by a closed immersion \(b_0 : \Spec(k) \to B\). By Lemma 0EDM we can find a factorization \[Y \to Y' \to X\] with \(\pi' : Y' \to X\) finite and \(Y \to Y'\) a morphism which map \(y_0\) and \(y'_0\) to the same point and is an isomorphism away from the inverse image of \(b_0\). (Of course \(Y'\) won’t be a scheme but this doesn’t matter for the argument that follows.) Clearly the maximal chains of specializations \(y_c \leadsto y_{c - 1} \leadsto \ldots \leadsto y_0\) and \(y'_e \leadsto y'_{e - 1} \leadsto \ldots \leadsto y'_0\) map to maximal chains of specializations in \(Y'\) having the same start and end. Since \(B\) is universally catenary, we see that \(|Y'|\) is catenary and we conclude as before.
Étale localization of morphisms
The section is the analogue of More on Morphisms, Section 04HF.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(y \in |Y|\). Let \(x_1, \ldots, x_n \in |X|\) mapping to \(y\). Assume that
\(f\) is locally of finite type,
\(f\) is separated,
\(f\) is quasi-finite at \(x_1, \ldots, x_n\), and
\(f\) is quasi-compact or \(Y\) is decent.
Then there exists an étale morphism \((U, u) \to (Y, y)\) of pointed algebraic spaces and a decomposition \[U \times_Y X = W \amalg V\] into open and closed subspaces such that the morphism \(V \to U\) is finite, every point of the fibre of \(|V| \to |U|\) over \(u\) maps to an \(x_i\), and the fibre of \(|W| \to |U|\) over \(u\) contains no point mapping to an \(x_i\).
Proof
Let \((U, u) \to (Y, y)\) be an étale morphism of algebraic spaces and consider the set of \(w \in |U \times_Y X|\) mapping to \(u \in |U|\) and one of the \(x_i \in |X|\). By Decent Spaces, Lemma 0AC6 (if \(f\) is of finite type) or Decent Spaces, Lemma 0AC7 (if \(Y\) is decent) this set is finite. It follows that we may replace \(f\) by the base change \(U \times_Y X \to U\) and \(x_1, \ldots, x_n\) by the set of these \(w\). In particular we may and do assume that \(Y\) is an affine scheme, whence \(X\) is a separated algebraic space.
Choose an affine scheme \(Z\) and an étale morphism \(Z \to X\) such that \(x_1, \ldots, x_n\) are in the image of \(|Z| \to |X|\). The fibres of \(|Z| \to |X|\) are finite, see Properties of Spaces, Lemma 03IJ (or the more general discussion in Decent Spaces, Section 03I7). Let \(\{z_1, \ldots, z_m\} \subset |Z|\) be the preimage of \(\{x_1, \ldots, x_n\}\). By More on Morphisms, Lemma 02LN there exists an étale morphism \((U, u) \to (Y, y)\) such that \(U \times_Y Z = Z_1 \amalg Z_2\) with \(Z_1 \to U\) finite and \((Z_1)_y = \{z_1, \ldots, z_m\}\). We may assume that \(U\) is affine and hence \(Z_1\) is affine too.
Since \(f\) is separated, the image \(V\) of \(Z_1 \to X\) is both open and closed (Morphisms of Spaces, Lemma 04NX). Set \(W = X \setminus V\) to get a decomposition as in the lemma. To finish the proof we have to show that \(V \to U\) is finite. As \(Z_1 \to V\) is surjective and étale, \(V\) is the quotient of \(Z_1\) by the étale equivalence relation \(R = Z_1 \times_V Z_1\), see Spaces, Lemma 0262. Since \(f\) is separated, \(V \to U\) is separated and \(R\) is closed in \(Z_1 \times_U Z_1\). Since \(Z_1 \to U\) is finite, the projections \(s, t : R \to Z_1\) are finite. Thus \(V\) is an affine scheme by Groupoids, Proposition 03BM. By Morphisms, Lemma 03GN we conclude that \(V \to U\) is proper and by Morphisms, Lemma 01WN we conclude that \(V \to U\) is finite, thereby finishing the proof.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(x \in |X|\) with image \(y \in |Y|\). Assume that
\(f\) is locally of finite type,
\(f\) is separated, and
\(f\) is quasi-finite at \(x\).
Then there exists an étale morphism \((U, u) \to (Y, y)\) of pointed algebraic spaces and a decomposition \[U \times_Y X = W \amalg V\] into open and closed subspaces such that the morphism \(V \to U\) is finite and there exists a point \(v \in |V|\) which maps to \(x\) in \(|X|\) and \(u\) in \(|U|\).
Proof
Pick a scheme \(U\), a point \(u \in U\), and an étale morphism \(U \to Y\) mapping \(u\) to \(y\). There exists a point \(x' \in |U \times_Y X|\) mapping to \(x\) in \(|X|\) and \(u\) in \(|U|\) (Properties of Spaces, Lemma 03H4). To finish, apply Lemma 082H to the morphism \(U \times_Y X \to U\) and the point \(x'\). It applies because \(U\) is a scheme and hence \(u\) comes from the monomorphism \(\Spec(\kappa(u)) \to U\).
Zariski’s Main Theorem
In this section we apply the results of the previous section to prove Zariski’s main theorem for morphisms of algebraic spaces. This section is the analogue of More on Morphisms, Section 02LQ.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is of finite type and separated. Let \(Y'\) be the normalization of \(Y\) in \(X\). Picture: \[\xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & Y' \ar[ld]^\nu \\ & Y & }\] Then there exists an open subspace \(U' \subset Y'\) such that
\((f')^{-1}(U') \to U'\) is an isomorphism, and
\((f')^{-1}(U') \subset X\) is the set of points at which \(f\) is quasi-finite.
Proof
By Morphisms of Spaces, Lemma 04NW there is an open subspace \(U \subset X\) corresponding to the points of \(|X|\) where \(f\) is quasi-finite. We have to prove
the image of \(|U| \to |Y'|\) is \(|U'|\) for some open subspace \(U'\) of \(Y'\),
\(U = f^{-1}(U')\), and
\(U \to U'\) is an isomorphism.
Since formation of \(U\) commutes with arbitrary base change (Morphisms of Spaces, Lemma 04NW), since formation of the normalization \(Y'\) commutes with smooth base change (Lemma 082F), since étale morphisms are open, and since “being an isomorphism” is fpqc local on the base (Descent on Spaces, Lemma 041Y), it suffices to prove (a), (b), (c) étale locally on \(Y\) (some details omitted). Thus we may assume \(Y\) is an affine scheme. This implies that \(Y'\) is an (affine) scheme as well.
Let \(x \in |U|\). Claim: there exists an open neighbourhood \(f'(x) \in V \subset Y'\) such that \((f')^{-1}V \to V\) is an isomorphism. We first prove the claim implies the lemma. Namely, then \((f')^{-1}V \cong V\) is a scheme (as an open of \(Y'\)), locally of finite type over \(Y\) (as an open subspace of \(X\)), and for \(v \in V\) the residue field extension \(\kappa(v)/\kappa(\nu(v))\) is algebraic (as \(V \subset Y'\) and \(Y'\) is integral over \(Y\)). Hence the fibres of \(V \to Y\) are discrete (Morphisms, Lemma 01TE) and \((f')^{-1}V \to Y\) is locally quasi-finite (Morphisms, Lemma 06RT). This implies \((f')^{-1}V \subset U\) and \(V \subset U'\). Since \(x\) was arbitrary we see that (a), (b), and (c) are true.
Let \(y = f(x) \in |Y|\). Let \((T, t) \to (Y, y)\) be an étale morphism of pointed schemes. Denote by a subscript \({}_T\) the base change to \(T\). Let \(z \in X_T\) be a point in the fibre \(X_t\) lying over \(x\). Note that \(U_T \subset X_T\) is the set of points where \(f_T\) is quasi-finite, see Morphisms of Spaces, Lemma 04NW. Note that \[X_T \xrightarrow{f'_T} Y'_T \xrightarrow{\nu_T} T\] is the normalization of \(T\) in \(X_T\), see Lemma 082F. Suppose that the claim holds for \(z \in U_T \subset X_T \to Y'_T \to T\), i.e., suppose that we can find an open neighbourhood \(f'_T(z) \in V' \subset Y'_T\) such that \((f'_T)^{-1}V' \to V'\) is an isomorphism. The morphism \(Y'_T \to Y'\) is étale hence the image \(V \subset Y'\) of \(V'\) is open. Observe that \(f'(x) \in V\) as \(f'_T(z) \in V'\). Observe that \[\xymatrix{ (f'_T)^{-1}V' \ar[r] \ar[d] & (f')^{-1}(V) \ar[d] \\ V' \ar[r] & V }\] is a fibre square (as \(Y'_T \times_{Y'} X = X_T\)). Since the left vertical arrow is an isomorphism and \(\{V' \to V\}\) is a étale covering, we conclude that the right vertical arrow is an isomorphism by Descent on Spaces, Lemma 041Y. In other words, the claim holds for \(x \in U \subset X \to Y' \to Y\).
By the result of the previous paragraph to prove the claim for \(x \in |U|\), we may replace \(Y\) by an étale neighbourhood \(T\) of \(y = f(x)\) and \(x\) by any point lying over \(x\) in \(T \times_Y X\). Thus we may assume there is a decomposition \[X = V \amalg W\] into open and closed subspaces where \(V \to Y\) is finite and \(x \in V\), see Lemma 082H. Since \(X\) is a disjoint union of \(V\) and \(W\) over \(Y\) and since \(V \to Y\) is finite we see that the normalization of \(Y\) in \(X\) is the morphism \[X = V \amalg W \longrightarrow V \amalg W' \longrightarrow S\] where \(W'\) is the normalization of \(Y\) in \(W\), see Morphisms of Spaces, Lemmas 0824, 0414, and 0825. The claim follows and we win.
The following lemma is a duplicate of Morphisms of Spaces, Lemma 0ABS. The reason for having two copies of the same lemma is that the proofs are somewhat different. The proof given below rests on Zariski’s Main Theorem for nonrepresentable morphisms of algebraic spaces as presented above, whereas the proof of Morphisms of Spaces, Lemma 0ABS rests on Morphisms of Spaces, Proposition 03XX to reduce to the case of morphisms of schemes.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is quasi-finite and separated. Let \(Y'\) be the normalization of \(Y\) in \(X\). Picture: \[\xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & Y' \ar[ld]^\nu \\ & Y & }\] Then \(f'\) is a quasi-compact open immersion and \(\nu\) is integral. In particular \(f\) is quasi-affine.
Proof
This follows from Lemma 082I. Namely, by that lemma there exists an open subspace \(U' \subset Y'\) such that \((f')^{-1}(U') = X\) (!) and \(X \to U'\) is an isomorphism! In other words, \(f'\) is an open immersion. Note that \(f'\) is quasi-compact as \(f\) is quasi-compact and \(\nu : Y' \to Y\) is separated (Morphisms of Spaces, Lemma 03KS). Hence for every affine scheme \(Z\) and morphism \(Z \to Y\) the fibre product \(Z \times_Y X\) is a quasi-compact open subscheme of the affine scheme \(Z \times_Y Y'\). Hence \(f\) is quasi-affine by definition.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is quasi-finite and separated and assume that \(Y\) is quasi-compact and quasi-separated. Then there exists a factorization \[\xymatrix{ X \ar[rd]_f \ar[rr]_j & & T \ar[ld]^\pi \\ & Y & }\] where \(j\) is a quasi-compact open immersion and \(\pi\) is finite.
Proof
Let \(X \to Y' \to Y\) be as in the conclusion of Lemma 082J. By Limits of Spaces, Lemma 082C we can write \(\nu_*\mathcal{O}_{Y'} = \colim_{i \in I} \mathcal{A}_i\) as a directed colimit of finite quasi-coherent \(\mathcal{O}_X\)-algebras \(\mathcal{A}_i \subset \nu_*\mathcal{O}_{Y'}\). Then \(\pi_i : T_i = \underline{\Spec}_Y(\mathcal{A}_i) \to Y\) is a finite morphism for each \(i\). Note that the transition morphisms \(T_{i'} \to T_i\) are affine and that \(Y' = \lim T_i\).
By Limits of Spaces, Lemma 0827 there exists an \(i\) and a quasi-compact open \(U_i \subset T_i\) whose inverse image in \(Y'\) equals \(f'(X)\). For \(i' \geq i\) let \(U_{i'}\) be the inverse image of \(U_i\) in \(T_{i'}\). Then \(X \cong f'(X) = \lim_{i' \geq i} U_{i'}\), see Limits of Spaces, Lemma 07SF. By Limits of Spaces, Lemma 0828 we see that \(X \to U_{i'}\) is a closed immersion for some \(i' \geq i\). (In fact \(X \cong U_{i'}\) for sufficiently large \(i'\) but we don’t need this.) Hence \(X \to T_{i'}\) is an immersion. By Morphisms of Spaces, Lemma 081U we can factor this as \(X \to T \to T_{i'}\) where the first arrow is an open immersion and the second a closed immersion. Thus we win.
Lemma
With notation and hypotheses as in Lemma 082K. Assume moreover that \(f\) is locally of finite presentation. Then we can choose the factorization such that \(T\) is finite and of finite presentation over \(Y\).
Proof
By Limits of Spaces, Lemma 07VR we can write \(T = \lim T_i\) where all \(T_i\) are finite and of finite presentation over \(Y\) and the transition morphisms \(T_{i'} \to T_i\) are closed immersions. By Limits of Spaces, Lemma 0827 there exists an \(i\) and an open subscheme \(U_i \subset T_i\) whose inverse image in \(T\) is \(X\). By Limits of Spaces, Lemma 0828 we see that \(X \cong U_i\) for large enough \(i\). Replacing \(T\) by \(T_i\) finishes the proof.
Applications of Zariski’s Main Theorem, I
A first application is the characterization of finite morphisms as proper morphisms with finite fibres.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
\(f\) is finite,
\(f\) is proper and locally quasi-finite,
\(f\) is proper and \(|X_k|\) is a discrete space for every morphism \(\Spec(k) \to Y\) where \(k\) is a field,
\(f\) is universally closed, separated, locally of finite type and \(|X_k|\) is a discrete space for every morphism \(\Spec(k) \to Y\) where \(k\) is a field.
Proof
We have (1) \(\Rightarrow\) (2) by Morphisms of Spaces, Lemmas 04NZ, 04NY. We have (2) \(\Rightarrow\) (3) by Morphisms of Spaces, Lemma 06RW. By definition (3) implies (4).
Assume (4). Since \(f\) is universally closed it is quasi-compact (Morphisms of Spaces, Lemma 04XW). Pick a point \(y\) of \(|Y|\). We represent \(y\) by a morphism \(\Spec(k) \to Y\). Note that \(|X_k|\) is finite discrete as a quasi-compact discrete space. The map \(|X_k| \to |X|\) surjects onto the fibre of \(|X| \to |Y|\) over \(y\) (Properties of Spaces, Lemma 03H4). By Morphisms of Spaces, Lemma 06LS we see that \(X \to Y\) is quasi-finite at all the points of the fibre of \(|X| \to |Y|\) over \(y\). Choose an elementary étale neighbourhood \((U, u) \to (Y, y)\) and decomposition \(X_U = V \amalg W\) as in Lemma 082H adapted to all the points of \(|X|\) lying over \(y\). Note that \(W_u = \emptyset\) because we used all the points in the fibre of \(|X| \to |Y|\) over \(y\). Since \(f\) is universally closed we see that the image of \(|W|\) in \(|U|\) is a closed set not containing \(u\). After shrinking \(U\) we may assume that \(W = \emptyset\). In other words we see that \(X_U = V\) is finite over \(U\). Since \(y \in |Y|\) was arbitrary this means there exists a family \(\{U_i \to Y\}\) of étale morphisms whose images cover \(Y\) such that the base changes \(X_{U_i} \to U_i\) are finite. We conclude that \(f\) is finite by Morphisms of Spaces, Lemma 03ZQ.
As a consequence we have the following useful result.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(y \in |Y|\). Assume
\(f\) is proper, and
\(f\) is quasi-finite at all \(x \in |X|\) lying over \(y\) (Decent Spaces, Lemma 0ACK).
Then there exists an open neighbourhood \(V \subset Y\) of \(y\) such that \(f|_{f^{-1}(V)} : f^{-1}(V) \to V\) is finite.
Proof
By Morphisms of Spaces, Lemma 04NW the set of points at which \(f\) is quasi-finite is an open \(U \subset X\). Let \(Z = X \setminus U\). Then \(y \not \in f(Z)\). Since \(f\) is proper the set \(f(Z) \subset Y\) is closed. Choose any open neighbourhood \(V \subset Y\) of \(y\) with \(Z \cap V = \emptyset\). Then \(f^{-1}(V) \to V\) is locally quasi-finite and proper. Hence \(f^{-1}(V) \to V\) is finite by Lemma 0A4X.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ X \ar[rr]_h \ar[rd]_f & & Y \ar[ld]^g \\ & B }\] be a commutative diagram of morphism of algebraic spaces over \(S\). Let \(b \in B\) and let \(\Spec(k) \to B\) be a morphism in the equivalence class of \(b\). Assume
\(X \to B\) is a proper morphism,
\(Y \to B\) is separated and locally of finite type,
one of the following is true
the image of \(|X_k| \to |Y_k|\) is finite,
the image of \(|f|^{-1}(\{b\})\) in \(|Y|\) is finite and \(B\) is decent.
Then there is an open subspace \(B' \subset B\) containing \(b\) such that \(X_{B'} \to Y_{B'}\) factors through a closed subspace \(Z \subset Y_{B'}\) finite over \(B'\).
Proof
Let \(Z \subset Y\) be the scheme theoretic image of \(h\), see Morphisms of Spaces, Section 082W. By Morphisms of Spaces, Lemma 0AGD the morphism \(X \to Z\) is surjective and \(Z \to B\) is proper. Thus \[\{x \in |X|\text{ lying over }b\} \to \{z \in |Z|\text{ lying over }b\}\] and \(|X_k| \to |Z_k|\) are surjective. We see that either (3)(a) or (3)(b) imply that \(Z \to B\) is quasi-finite all points of \(|Z|\) lying over \(b\) by Decent Spaces, Lemma 0ACK. Hence \(Z \to B\) is finite in an open neighbourhood of \(b\) by Lemma 0A4Y.
Stein factorization
Stein factorization is the statement that a proper morphism \(f : X \to S\) with \(f_*\mathcal{O}_X = \mathcal{O}_S\) has connected fibres.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a universally closed and quasi-separated morphism of algebraic spaces over \(S\). There exists a factorization \[\xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & Y' \ar[dl]^\pi \\ & Y & }\] with the following properties:
the morphism \(f'\) is universally closed, quasi-compact, quasi-separated, and surjective,
the morphism \(\pi : Y' \to Y\) is integral,
we have \(f'_*\mathcal{O}_X = \mathcal{O}_{Y'}\),
we have \(Y' = \underline{\Spec}_Y(f_*\mathcal{O}_X)\), and
\(Y'\) is the normalization of \(Y\) in \(X\) as defined in Morphisms of Spaces, Definition 0822.
Formation of the factorization \(f = \pi \circ f'\) commutes with flat base change.
Proof
By Morphisms of Spaces, Lemma 04XW the morphism \(f\) is quasi-compact. We just define \(Y'\) as the normalization of \(Y\) in \(X\), so (5) and (2) hold automatically. By Morphisms of Spaces, Lemma 0A0Q we see that (4) holds. The morphism \(f'\) is universally closed by Morphisms of Spaces, Lemma 04NX. It is quasi-compact by Morphisms of Spaces, Lemma 03KS and quasi-separated by Morphisms of Spaces, Lemma 03KR.
To show the remaining statements we may assume the base \(Y\) is affine (as taking normalization commutes with étale localization). Say \(Y = \Spec(R)\). Then \(Y' = \Spec(A)\) with \(A = \Gamma(X, \mathcal{O}_X)\) an integral \(R\)-algebra. Thus it is clear that \(f'_*\mathcal{O}_X\) is \(\mathcal{O}_{Y'}\) (because \(f'_*\mathcal{O}_X\) is quasi-coherent, by Morphisms of Spaces, Lemma 03M9, and hence equal to \(\widetilde{A}\)). This proves (3).
Let us show that \(f'\) is surjective. As \(f'\) is universally closed (see above) the image of \(f'\) is a closed subset \(V(I) \subset Y' = \Spec(A)\). Pick \(h \in I\). Then \(h|_X = f^\sharp(h)\) is a global section of the structure sheaf of \(X\) which vanishes at every point. As \(X\) is quasi-compact this means that \(h|_X\) is a nilpotent section, i.e., \(h^n|X = 0\) for some \(n > 0\). But \(A = \Gamma(X, \mathcal{O}_X)\), hence \(h^n = 0\). In other words \(I\) is contained in the Jacobson radical of \(A\) and we conclude that \(V(I) = Y'\) as desired.
Lemma
In Lemma 0A19 assume in addition that \(f\) is locally of finite type and \(Y\) affine. Then for \(y \in Y\) the fibre \(\pi^{-1}(\{y\}) = \{y_1, \ldots, y_n\}\) is finite and the field extensions \(\kappa(y_i)/\kappa(y)\) are finite.
Proof
Recall that there are no specializations among the points of \(\pi^{-1}(\{y\})\), see Algebra, Lemma 00GT. As \(f'\) is surjective, we find that \(|X_y| \to \pi^{-1}(\{y\})\) is surjective. Observe that \(X_y\) is a quasi-separated algebraic space of finite type over a field (quasi-compactness was shown in the proof of the referenced lemma). Thus \(|X_y|\) is a Noetherian topological space (Morphisms of Spaces, Lemma 04ZL). A topological argument (omitted) now shows that \(\pi^{-1}(\{y\})\) is finite. For each \(i\) we can pick a finite type point \(x_i \in |X_y|\) mapping to \(y_i\) (Morphisms of Spaces, Lemma 06EK). We conclude that \(\kappa(y_i)/\kappa(y)\) is finite: \(x_i\) can be represented by a morphism \(\Spec(k_i) \to X_y\) of finite type (by our definition of finite type points) and hence \(\Spec(k_i) \to y = \Spec(\kappa(y))\) is of finite type (as a composition of finite type morphisms), hence \(k_i/\kappa(y)\) is finite (Morphisms, Lemma 01TA).
Let \(f : X \to Y\) be a morphism of algebraic spaces and let \(\overline{y} : \Spec(k) \to Y\) be a geometric point. Then the fibre of \(f\) over \(\overline{y}\) is the algebraic space \(X_{\overline{y}} = X \times_{Y, \overline{y}} \Spec(k)\) over \(k\). If \(Y\) is a scheme and \(y \in Y\) is a point, then we denote \(X_y = X \times_Y \Spec(\kappa(y))\) the fibre as usual.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\overline{y}\) be a geometric point of \(Y\). Then \(X_{\overline{y}}\) is connected, if and only if for every étale neighbourhood \((V, \overline{v}) \to (Y, \overline{y})\) where \(V\) is a scheme the base change \(X_V \to V\) has connected fibre \(X_v\).
Proof
Since the category of étale neighbourhoods of \(\overline{y}\) is cofiltered and contains a cofinal collection of schemes (Properties of Spaces, Lemma 04JW) we may replace \(Y\) by one of these neighbourhoods and assume that \(Y\) is a scheme. Let \(y \in Y\) be the point corresponding to \(\overline{y}\). Then \(X_y\) is geometrically connected over \(\kappa(y)\) if and only if \(X_{\overline{y}}\) is connected and if and only if \((X_y)_{k'}\) is connected for every finite separable extension \(k'\) of \(\kappa(y)\). See Spaces over Fields, Section 0A0Y and especially Lemma 0A17. By More on Morphisms, Lemma 02LF there exists an affine étale neighbourhood \((V, v) \to (Y, y)\) such that \(\kappa(s) \subset \kappa(u)\) is identified with \(\kappa(s) \subset k'\) any given finite separable extension. The lemma follows.
Theorem
Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper morphism of algebraic spaces over \(S\) with \(Y\) locally Noetherian. There exists a factorization \[\xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & Y' \ar[dl]^\pi \\ & Y & }\] with the following properties:
the morphism \(f'\) is proper with connected geometric fibres,
the morphism \(\pi : Y' \to Y\) is finite,
we have \(f'_*\mathcal{O}_X = \mathcal{O}_{Y'}\),
we have \(Y' = \underline{\Spec}_Y(f_*\mathcal{O}_X)\), and
\(Y'\) is the normalization of \(Y\) in \(X\), see Morphisms, Definition 035H.
Proof
Let \(f = \pi \circ f'\) be the factorization of Lemma 0A19. Note that besides the conclusions of Lemma 0A19 we also have that \(f'\) is separated (Morphisms of Spaces, Lemma 03KR) and finite type (Morphisms of Spaces, Lemma 0462). Hence \(f'\) is proper. By Cohomology of Spaces, Lemma 08AR we see that \(f_*\mathcal{O}_X\) is a coherent \(\mathcal{O}_Y\)-module. Hence we see that \(\pi\) is finite, i.e., (2) holds.
This proves all but the most interesting assertion, namely that the geometric fibres of \(f'\) are connected. It is clear from the discussion above that we may replace \(Y\) by \(Y'\). Then \(Y\) is locally Noetherian, \(f : X \to Y\) is proper, and \(f_*\mathcal{O}_X = \mathcal{O}_Y\). Let \(\overline{y}\) be a geometric point of \(Y\). At this point we apply the theorem on formal functions, more precisely Cohomology of Spaces, Lemma 08B1. It tells us that \[\mathcal{O}^\wedge_{Y, \overline{y}} = \lim_n H^0(X_n, \mathcal{O}_{X_n})\] where \(X_n = \Spec(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^n) \times_Y X\). Note that \(X_1 = X_{\overline{y}} \to X_n\) is a (finite order) thickening and hence the underlying topological space of \(X_n\) is equal to that of \(X_{\overline{y}}\). Thus, if \(X_{\overline{y}} = T_1 \amalg T_2\) is a disjoint union of nonempty open and closed subspaces, then similarly \(X_n = T_{1, n} \amalg T_{2, n}\) for all \(n\). And this in turn means \(H^0(X_n, \mathcal{O}_{X_n})\) contains a nontrivial idempotent \(e_{1, n}\), namely the function which is identically \(1\) on \(T_{1, n}\) and identically \(0\) on \(T_{2, n}\). It is clear that \(e_{1, n + 1}\) restricts to \(e_{1, n}\) on \(X_n\). Hence \(e_1 = \lim e_{1, n}\) is a nontrivial idempotent of the limit. This contradicts the fact that \(\mathcal{O}^\wedge_{Y, \overline{y}}\) is a local ring. Thus the assumption was wrong, i.e., \(X_{\overline{y}}\) is connected as desired.
Theorem
Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper morphism of algebraic spaces over \(S\). There exists a factorization \[\xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & Y' \ar[dl]^\pi \\ & Y & }\] with the following properties:
the morphism \(f'\) is proper with connected geometric fibres,
the morphism \(\pi : Y' \to Y\) is integral,
we have \(f'_*\mathcal{O}_X = \mathcal{O}_{Y'}\),
we have \(Y' = \underline{\Spec}_Y(f_*\mathcal{O}_X)\), and
\(Y'\) is the normalization of \(Y\) in \(X\) (Morphisms of Spaces, Definition 0822).
Proof
We may apply Lemma 0A19 to get the morphism \(f' : X \to Y'\). Note that besides the conclusions of Lemma 0A19 we also have that \(f'\) is separated (Morphisms of Spaces, Lemma 03KR) and finite type (Morphisms of Spaces, Lemma 0462). Hence \(f'\) is proper. At this point we have proved all of the statements except for the statement that \(f'\) has connected geometric fibres.
It is clear from the discussion that we may replace \(Y\) by \(Y'\). Then \(f : X \to Y\) is proper and \(f_*\mathcal{O}_X = \mathcal{O}_Y\). Note that these conditions are preserved under flat base change (Morphisms of Spaces, Lemma 04WP and Cohomology of Spaces, Lemma 073K). Let \(\overline{y}\) be a geometric point of \(Y\). By Lemma 0A1A and the remark just made we reduce to the case where \(Y\) is a scheme, \(y \in Y\) is a point, \(f : X \to Y\) is a proper algebraic space over \(Y\) with \(f_*\mathcal{O}_X = \mathcal{O}_Y\), and we have to show the fibre \(X_y\) is connected. Replacing \(Y\) by an affine neighbourhood of \(y\) we may assume that \(Y = \Spec(R)\) is affine. Then \(f_*\mathcal{O}_X = \mathcal{O}_Y\) signifies that the ring map \(R \to \Gamma(X, \mathcal{O}_X)\) is bijective.
By Limits of Spaces, Lemma 0A0X we can write \((X \to Y) = \lim (X_i \to Y_i)\) with \(X_i \to Y_i\) proper and of finite presentation and \(Y_i\) Noetherian. For \(i\) large enough \(Y_i\) is affine (Limits of Spaces, Lemma 07SQ). Say \(Y_i = \Spec(R_i)\). Let \(R'_i = \Gamma(X_i, \mathcal{O}_{X_i})\). Observe that we have ring maps \(R_i \to R_i' \to R\). Namely, we have the first because \(X_i\) is an algebraic space over \(R_i\) and the second because we have \(X \to X_i\) and \(R = \Gamma(X, \mathcal{O}_X)\). Note that \(R = \colim R'_i\) by Limits of Spaces, Lemma 07SI. Then \[\xymatrix{ X \ar[d] \ar[r] & X_i \ar[d] \\ Y \ar[r] & Y'_i \ar[r] & Y_i }\] is commutative with \(Y'_i = \Spec(R'_i)\). Let \(y'_i \in Y'_i\) be the image of \(y\). We have \(X_y = \lim X_{i, y'_i}\) because \(X = \lim X_i\), \(Y = \lim Y'_i\), and \(\kappa(y) = \colim \kappa(y'_i)\). Now let \(X_y = U \amalg V\) with \(U\) and \(V\) open and closed. Then \(U, V\) are the inverse images of opens \(U_i, V_i\) in \(X_{i, y'_i}\) (Limits of Spaces, Lemma 0827). By Theorem 0A1B the fibres of \(X_i \to Y'_i\) are connected, hence either \(U\) or \(V\) is empty. This finishes the proof.
Here is an application.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume
\(f\) is proper,
\(Y\) is integral (Spaces over Fields, Definition 0AD4) with generic point \(\xi\),
\(Y\) is normal,
\(X\) is reduced,
every generic point of an irreducible component of \(|X|\) maps to \(\xi\),
we have \(H^0(X_\xi, \mathcal{O}) = \kappa(\xi)\).
Then \(f_*\mathcal{O}_X = \mathcal{O}_Y\) and \(f\) has geometrically connected fibres.
Proof
Apply Theorem 0A1C to get a factorization \(X \to Y' \to Y\). It is enough to show that \(Y' = Y\). It suffices to show that \(Y' \times_Y V \to V\) is an isomorphism, where \(V \to Y\) is an étale morphism and \(V\) an affine integral scheme, see Spaces over Fields, Lemma 0AYH. The formation of \(Y'\) commutes with étale base change, see Morphisms of Spaces, Lemma 0ABP. The generic points of \(X \times_Y V\) lie over the generic points of \(X\) (Decent Spaces, Lemma 0ABV) hence map to the generic point of \(V\) by assumption (5). Moreover, condition (6) is preserved under the base change by \(V \to Y\), for example by flat base change (Cohomology of Spaces, Lemma 073K). Thus it suffices to prove the lemma in case \(Y\) is a normal integral affine scheme.
Assume \(Y\) is a normal integral affine scheme. We will show \(Y' \to Y\) is an isomorphism by an application of Morphisms, Lemma 0AB1. Namely, \(Y'\) is reduced because \(X\) is reduced (Morphisms of Spaces, Lemma 0AYF). The morphism \(Y' \to Y\) is integral by the theorem cited above. Since \(Y\) is decent and \(X \to Y\) is separated, we see that \(X\) is decent too; to see this use Decent Spaces, Lemmas 03M5 and 0ABY. By assumption (5), Morphisms of Spaces, Lemma 0AYG, and Decent Spaces, Lemma 0ABV we see that every generic point of an irreducible component of \(|Y'|\) maps to \(\xi\). On the other hand, since \(Y'\) is the relative spectrum of \(f_*\mathcal{O}_X\) we see that the scheme theoretic fibre \(Y'_\xi\) is the spectrum of \(H^0(X_\xi, \mathcal{O})\) which is equal to \(\kappa(\xi)\) by assumption. Hence \(Y'\) is an integral scheme with function field equal to the function field of \(Y\). This finishes the proof.
Here is another application.
Lemma
Let \(S\) be a scheme. Let \(X \to Y\) be a morphism of algebraic spaces over \(S\). If \(f\) is proper, flat, and of finite presentation, then the function \(n_{X/Y} : |Y| \to \mathbf{Z}\) counting the number of geometric connected components of fibres of \(f\) (Lemma 0E1B) is lower semi-continuous.
Proof
The question is étale local on \(Y\), hence we may and do assume \(Y\) is an affine scheme. Let \(y \in Y\). Set \(n = n_{X/S}(y)\). Note that \(n < \infty\) as the geometric fibre of \(X \to Y\) at \(y\) is a proper algebraic space over a field, hence Noetherian, hence has a finite number of connected components. We have to find an open neighbourhood \(V\) of \(y\) such that \(n_{X/S}|_V \geq n\). Let \(X \to Y' \to Y\) be the Stein factorization as in Theorem 0A1C. By Lemma 0E1C there are finitely many points \(y'_1, \ldots, y'_m \in Y'\) lying over \(y\) and the extensions \(\kappa(y'_i)/\kappa(y)\) are finite. More on Morphisms, Lemma 0BSR tells us that after replacing \(Y\) by an étale neighbourhood of \(y\) we may assume \(Y' = V_1 \amalg \ldots \amalg V_m\) as a scheme with \(y'_i \in V_i\) and \(\kappa(y'_i)/\kappa(y)\) purely inseparable. Then the algebraic spaces \(X_{y_i'}\) are geometrically connected over \(\kappa(y)\), hence \(m = n\). The algebraic spaces \(X_i = (f')^{-1}(V_i)\), \(i = 1, \ldots, n\) are flat and of finite presentation over \(Y\). Hence the image of \(X_i \to Y\) is open (Morphisms of Spaces, Lemma 042S). Thus in a neighbourhood of \(y\) we see that \(n_{X/Y}\) is at least \(n\).
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume
\(f\) is proper, flat, and of finite presentation, and
the geometric fibres of \(f\) are reduced.
Then the function \(n_{X/S} : |Y| \to \mathbf{Z}\) counting the numbers of geometric connected components of fibres of \(f\) (Lemma 0E1B) is locally constant.
Proof
By Lemma 0E1D the function \(n_{X/Y}\) is lower semincontinuous. Thus it suffices to show it is upper semi-continuous. To do this we may work étale locally on \(Y\), hence we may assume \(Y\) is an affine scheme. For \(y \in Y\) consider the \(\kappa(y)\)-algebra \[A = H^0(X_y, \mathcal{O}_{X_y})\] By Spaces over Fields, Lemma 0DMY and the fact that \(X_y\) is geometrically reduced \(A\) is finite product of finite separable extensions of \(\kappa(y)\). Hence \(A \otimes_{\kappa(y)} \kappa(\overline{y})\) is a product of \(\beta_0(y) = \dim_{\kappa(y)} A\) copies of \(\kappa(\overline{y})\). Thus \(X_{\overline{y}}\) has \(\beta_0(y)\) connected components. In other words, we have \(n_{X/S} = \beta_0\) as functions on \(Y\). Thus \(n_{X/Y}\) is upper semi-continuous by Derived Categories of Spaces, Lemma 0E0R. This finishes the proof.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper morphism of algebraic spaces over \(S\). Let \(X \to Y' \to Y\) be the Stein factorization of \(f\) (Theorem 0A1C). If \(f\) is of finite presentation, flat, with geometrically reduced fibres (Definition 0E08), then \(Y' \to Y\) is finite étale.
Proof
Formation of the Stein factorization commutes with flat base change, see Lemma 0A19. Thus we may work étale locally on \(Y\) and we may assume \(Y\) is an affine scheme. Then \(Y'\) is an affine scheme and \(Y' \to Y\) is integral.
Let \(y \in Y\). Set \(n\) be the number of connected components of the geometric fibre \(X_{\overline{y}}\). Note that \(n < \infty\) as the geometric fibre of \(X \to Y\) at \(y\) is a proper algebraic space over a field, hence Noetherian, hence has a finite number of connected components. By Lemma 0E1C there are finitely many points \(y'_1, \ldots, y'_m \in Y'\) lying over \(y\) and for each \(i\) we can pick a finite type point \(x_i \in |X_y|\) mapping to \(y'_i\) the extension \(\kappa(y'_i)/\kappa(y)\) is finite. Thus More on Morphisms, Lemma 0BSR tells us that after replacing \(Y\) by an étale neighbourhood of \(y\) we may assume \(Y' = V_1 \amalg \ldots \amalg V_m\) as a scheme with \(y'_i \in V_i\) and \(\kappa(y'_i)/\kappa(y)\) purely inseparable. In this case the algebraic spaces \(X_{y_i'}\) are geometrically connected over \(\kappa(y)\), hence \(m = n\). The algebraic spaces \(X_i = (f')^{-1}(V_i)\), \(i = 1, \ldots, n\) are proper, flat, of finite presentation, with geometrically reduced fibres over \(Y\). It suffices to prove the lemma for each of the morphisms \(X_i \to Y\). This reduces us to the case where \(X_{\overline{y}}\) is connected.
Assume that \(X_{\overline{y}}\) is connected. By Lemma 0E1E we see that \(X \to Y\) has geometrically connected fibres in a neighbourhood of \(y\). Thus we may assume the fibres of \(X \to Y\) are geometrically connected. Then \(f_*\mathcal{O}_X = \mathcal{O}_Y\) by Derived Categories of Spaces, Lemma 0E0S which finishes the proof.
The proof of the following lemma uses Stein factorization for schemes which is why it ended up in this section.
Lemma
Let \((A, I)\) be a henselian pair. Let \(X\) be an algebraic space separated and of finite type over \(A\). Set \(X_0 = X \times_{\Spec(A)} \Spec(A/I)\). Let \(Y \subset X_0\) be an open and closed subspace such that \(Y \to \Spec(A/I)\) is proper. Then there exists an open and closed subspace \(W \subset X\) which is proper over \(A\) with \(W \times_{\Spec(A)} \Spec(A/I) = Y\).
Proof
We will denote \(T \mapsto T_0\) the base change by \(\Spec(A/I) \to \Spec(A)\). By a weak version of Chow’s lemma (in the form of Cohomology of Spaces, Lemma 089J) there exists a surjective proper morphism \(\varphi : X' \to X\) such that \(X'\) admits an immersion into \(\mathbf{P}^n_A\). Set \(Y' = \varphi^{-1}(Y)\). This is an open and closed subscheme of \(X'_0\). The lemma holds for \((X', Y')\) by More on Morphisms, Lemma 0CT9. Let \(W' \subset X'\) be the open and closed subscheme proper over \(A\) such that \(Y' = W'_0\). By Morphisms of Spaces, Lemma 04NX \(Q_1 = \varphi(|W'|) \subset |X|\) and \(Q_2 = \varphi(|X' \setminus W'|) \subset |X|\) are closed subsets and by Morphisms of Spaces, Lemma 08AJ any closed subspace structure on \(Q_1\) is proper over \(A\). The image of \(Q_1 \cap Q_2\) in \(\Spec(A)\) is closed. Since \((A, I)\) is henselian, if \(Q_1 \cap Q_2\) is nonempty, then we find that \(Q_1 \cap Q_2\) has a point lying over \(\Spec(A/I)\). This is impossible as \(W'_0 = Y' = \varphi^{-1}(Y)\). We conclude that \(Q_1\) is open and closed in \(|X|\). Let \(W \subset X\) be the corresponding open and closed subspace. Then \(W\) is proper over \(A\) with \(W_0 = Y\).
Extending properties from an open
In this section we collect a number of results of the form: If \(f : X \to Y\) is a flat morphism of algebraic spaces and \(f\) satisfies some property over a dense open of \(Y\), then \(f\) satisfies the same property over all of \(Y\).
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(V \subset Y\) be an open subspace. Assume
\(f\) is locally of finite presentation,
\(\mathcal{F}\) is of finite type and flat over \(Y\),
\(V \to Y\) is quasi-compact and scheme theoretically dense,
\(\mathcal{F}|_{f^{-1}V}\) is of finite presentation.
Then \(\mathcal{F}\) is of finite presentation.
Proof
It suffices to prove the pullback of \(\mathcal{F}\) to a scheme surjective and étale over \(X\) is of finite presentation. Hence we may assume \(X\) is a scheme. Similarly, we can replace \(Y\) by a scheme surjective and étale and over \(Y\) (the inverse image of \(V\) in this scheme is scheme theoretically dense, see Morphisms of Spaces, Section 0831). Thus we reduce to the case of schemes which is More on Flatness, Lemma 081N.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(V \subset Y\) be an open subspace. Assume
\(f\) is locally of finite type and flat,
\(V \to Y\) is quasi-compact and scheme theoretically dense,
\(f|_{f^{-1}V} : f^{-1}V \to V\) is locally of finite presentation.
Then \(f\) is of locally of finite presentation.
Proof
The proof is identical to the proof of Lemma 0876 except one uses More on Flatness, Lemma 081P.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and locally of finite type. Let \(V \subset Y\) be an open subspace such that \(|V| \subset |Y|\) is dense and such that \(X_V \to V\) has relative dimension \(\leq d\). If also either
\(f\) is locally of finite presentation, or
\(V \to Y\) is quasi-compact,
then \(f : X \to Y\) has relative dimension \(\leq d\).
Proof
We may replace \(Y\) by its reduction, hence we may assume \(Y\) is reduced. Then \(V\) is scheme theoretically dense in \(Y\), see Morphisms of Spaces, Lemma 088G. By definition the property of having relative dimension \(\leq d\) can be checked on an étale covering, see Morphisms of Spaces, Sections 04NH. Thus it suffices to prove \(f\) has relative dimension \(\leq d\) after replacing \(X\) by a scheme surjective and étale over \(X\). Similarly, we can replace \(Y\) by a scheme surjective and étale and over \(Y\). The inverse image of \(V\) in this scheme is scheme theoretically dense, see Morphisms of Spaces, Section 0831. Since a scheme theoretically dense open of a scheme is in particular dense, we reduce to the case of schemes which is More on Flatness, Lemma 081L.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and proper. Let \(V \to Y\) be an open subspace with \(|V| \subset |Y|\) dense such that \(X_V \to V\) is finite. If also either \(f\) is locally of finite presentation or \(V \to Y\) is quasi-compact, then \(f\) is finite.
Proof
By Lemma 0878 the fibres of \(f\) have dimension zero. By Morphisms of Spaces, Lemma 04NV this implies that \(f\) is locally quasi-finite. By Morphisms of Spaces, Lemma 0418 this implies that \(f\) is representable. We can check whether \(f\) is finite étale locally on \(Y\), hence we may assume \(Y\) is a scheme. Since \(f\) is representable, we reduce to the case of schemes which is More on Flatness, Lemma 0B48.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(V \subset Y\) be an open subspace. If
\(f\) is separated, locally of finite type, and flat,
\(f^{-1}(V) \to V\) is an isomorphism, and
\(V \to Y\) is quasi-compact and scheme theoretically dense,
then \(f\) is an open immersion.
Proof
Applying Lemma 0877 we see that \(f\) is locally of finite presentation. Applying Lemma 0878 we see that \(f\) has relative dimension \(\leq 0\). By Morphisms of Spaces, Lemma 04NV this implies that \(f\) is locally quasi-finite. By Morphisms of Spaces, Lemma 0418 this implies that \(f\) is representable. By Descent on Spaces, Lemma 041X we can check whether \(f\) is an open immersion étale locally on \(Y\). Hence we may assume that \(Y\) is a scheme. Since \(f\) is representable, we reduce to the case of schemes which is More on Flatness, Lemma 081M.
Blowing up and flatness
Instead of redoing the work in More on Flatness, Section 080X we prove an analogue of More on Flatness, Lemma 0813 which tells us that the problem of finding a suitable blowup is often étale local on the base.
Lemma
Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(\varphi : W \to X\) be a quasi-compact separated étale morphism. Let \(U \subset X\) be a quasi-compact open subspace. Let \(\mathcal{I} \subset \mathcal{O}_W\) be a finite type quasi-coherent sheaf of ideals such that \(V(\mathcal{I}) \cap \varphi^{-1}(U) = \emptyset\). Then there exists a finite type quasi-coherent sheaf of ideals \(\mathcal{J} \subset \mathcal{O}_X\) such that
\(V(\mathcal{J}) \cap U = \emptyset\), and
\(\varphi^{-1}(\mathcal{J})\mathcal{O}_W = \mathcal{I} \mathcal{I}'\) for some finite type quasi-coherent ideal \(\mathcal{I}' \subset \mathcal{O}_W\).
Proof
Choose a factorization \(W \to Y \to X\) where \(j : W \to Y\) is a quasi-compact open immersion and \(\pi : Y \to X\) is a finite morphism of finite presentation (Lemma 0874). Let \(V = j(W) \cup \pi^{-1}(U) \subset Y\). Note that \(\mathcal{I}\) on \(W \cong j(W)\) and \(\mathcal{O}_{\pi^{-1}(U)}\) glue to a finite type quasi-coherent sheaf of ideals \(\mathcal{I}_1 \subset \mathcal{O}_V\). By Limits of Spaces, Lemma 0853 there exists a finite type quasi-coherent sheaf of ideals \(\mathcal{I}_2 \subset \mathcal{O}_Y\) such that \(\mathcal{I}_2|_V = \mathcal{I}_1\). In other words, \(\mathcal{I}_2 \subset \mathcal{O}_Y\) is a finite type quasi-coherent sheaf of ideals such that \(V(\mathcal{I}_2)\) is disjoint from \(\pi^{-1}(U)\) and \(j^{-1}\mathcal{I}_2 = \mathcal{I}\). Denote \(i : Z \to Y\) the corresponding closed immersion which is of finite presentation (Morphisms of Spaces, Lemma 084Q). In particular the composition \(\tau = \pi \circ i : Z \to X\) is finite and of finite presentation (Morphisms of Spaces, Lemmas 03XQ and 03ZR).
Let \(\mathcal{F} = \tau_*\mathcal{O}_Z\) which we think of as a quasi-coherent \(\mathcal{O}_X\)-module. By Descent on Spaces, Lemma 0610 we see that \(\mathcal{F}\) is a finitely presented \(\mathcal{O}_X\)-module. Let \(\mathcal{J} = \text{Fit}_0(\mathcal{F})\). (Insert reference to fitting modules on ringed topoi here.) This is a finite type quasi-coherent sheaf of ideals on \(X\) (as \(\mathcal{F}\) is of finite presentation, see More on Algebra, Lemma 07ZA). Part (1) of the lemma holds because \(|\tau|(|Z|) \cap |U| = \emptyset\) by our choice of \(\mathcal{I}_2\) and because the \(0\)th Fitting ideal of the trivial module equals the structure sheaf. To prove (2) note that \(\varphi^{-1}(\mathcal{J})\mathcal{O}_W = \text{Fit}_0(\varphi^*\mathcal{F})\) because taking Fitting ideals commutes with base change. On the other hand, as \(\varphi : W \to X\) is separated and étale we see that \((1, j) : W \to W \times_X Y\) is an open and closed immersion. Hence \(W \times_Y Z = V(\mathcal{I}) \amalg Z'\) for some finite and finitely presented morphism of algebraic spaces \(\tau' : Z' \to W\). Thus we see that \[\begin{align*} \text{Fit}_0(\varphi^*\mathcal{F}) & = \text{Fit}_0((W \times_Y Z \to W)_*\mathcal{O}_{W \times_Y Z}) \\ & = \text{Fit}_0(\mathcal{O}_W/\mathcal{I}) \cdot \text{Fit}_0(\tau'_*\mathcal{O}_{Z'}) \\ & = \mathcal{I} \cdot \text{Fit}_0(\tau'_*\mathcal{O}_{Z'}) \end{align*}\] the second equality by More on Algebra, Lemma 07ZA translated in sheaves on ringed topoi. Setting \(\mathcal{I}' = \text{Fit}_0(\tau'_*\mathcal{O}_{Z'})\) finishes the proof of the lemma.
Theorem
Let \(S\) be a scheme. Let \(B\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(X\) be an algebraic space over \(B\). Let \(\mathcal{F}\) be a quasi-coherent module on \(X\). Let \(U \subset B\) be a quasi-compact open subspace. Assume
\(X\) is quasi-compact,
\(X\) is locally of finite presentation over \(B\),
\(\mathcal{F}\) is a module of finite type,
\(\mathcal{F}_U\) is of finite presentation, and
\(\mathcal{F}_U\) is flat over \(U\).
Then there exists a \(U\)-admissible blowup \(B' \to B\) such that the strict transform \(\mathcal{F}'\) of \(\mathcal{F}\) is an \(\mathcal{O}_{X \times_B B'}\)-module of finite presentation and flat over \(B'\).
Proof
Choose an affine scheme \(V\) and a surjective étale morphism \(V \to X\). Because strict transform commutes with étale localization (Divisors on Spaces, Lemma 0863) it suffices to prove the result with \(X\) replaced by \(V\). Hence we may assume that \(X \to B\) is representable (in addition to the hypotheses of the lemma).
Assume that \(X \to B\) is representable. Choose an affine scheme \(W\) and a surjective étale morphism \(\varphi : W \to B\). Note that \(X \times_B W\) is a scheme. By the case of schemes (More on Flatness, Theorem 0815) we can find a finite type quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_W\) such that (a) \(|V(\mathcal{I})| \cap |\varphi^{-1}(U)| = \emptyset\) and (b) the strict transform of \(\mathcal{F}|_{X \times_B W}\) with respect to the blowing up \(W' \to W\) in \(\mathcal{I}\) becomes flat over \(W'\) and is a module of finite presentation. Choose a finite type sheaf of ideals \(\mathcal{J} \subset \mathcal{O}_B\) as in Lemma 087B. Let \(B' \to B\) be the blowing up of \(\mathcal{J}\). We claim that this blowup works. Namely, it is clear that \(B' \to B\) is \(U\)-admissible by our choice of ideal \(\mathcal{J}\). Moreover, the base change \(B' \times_B W \to W\) is the blowup of \(W\) in \(\varphi^{-1}\mathcal{J} = \mathcal{I}\mathcal{I}'\) (compatibility of blowup with flat base change, see Divisors on Spaces, Lemma 085S). Hence there is a factorization \[W \times_B B' \to W' \to W\] where the first morphism is a blowup as well, see Divisors on Spaces, Lemma 085Y). The restriction of \(\mathcal{F}'\) (which lives on \(B' \times_B X\)) to \(W \times_B B' \times_B X\) is the strict transform of \(\mathcal{F}|_{X \times_B W}\) (Divisors on Spaces, Lemma 0863) and hence is the twice repeated strict transform of \(\mathcal{F}|_{X \times_B W}\) by the two blowups displayed above (Divisors on Spaces, Lemma 0868). After the first blowup our sheaf is already flat over the base and of finite presentation (by construction). Whence this holds after the second strict transform as well (since this is a pullback by Divisors on Spaces, Lemma 0865). Thus we see that the restriction of \(\mathcal{F}'\) to an étale cover of \(B' \times_B X\) has the desired properties and the theorem is proved.
Applications
In this section we apply the result on flattening by blowing up.
Lemma
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(U \subset B\) be an open subspace. Assume
\(B\) is quasi-compact and quasi-separated,
\(U\) is quasi-compact,
\(f : X \to B\) is of finite type and quasi-separated, and
\(f^{-1}(U) \to U\) is flat and locally of finite presentation.
Then there exists a \(U\)-admissible blowup \(B' \to B\) such that the strict transform \(X'\) of \(X\) is flat and of finite presentation over \(B'\).
Proof
Let \(B' \to B\) be a \(U\)-admissible blowup. Note that the strict transform of \(X\) is quasi-compact and quasi-separated over \(B'\) as \(X\) is quasi-compact and quasi-separated over \(B\). Hence we only need to worry about finding a \(U\)-admissible blowup such that the strict transform becomes flat and locally of finite presentation. We cannot directly apply Theorem 087C because \(X\) is not locally of finite presentation over \(B\).
Choose an affine scheme \(V\) and a surjective étale morphism \(V \to X\). (This is possible as \(X\) is quasi-compact as a finite type space over the quasi-compact space \(B\).) Then it suffices to show the result for the morphism \(V \to B\) (as strict transform commutes with étale localization, see Divisors on Spaces, Lemma 0863). Hence we may assume that \(X \to B\) is separated as well as finite type. In this case we can find a closed immersion \(i : X \to Y\) with \(Y \to B\) separated and of finite presentation, see Limits of Spaces, Proposition 0873.
Apply Theorem 087C to \(\mathcal{F} = i_*\mathcal{O}_X\) on \(Y/B\). We find a \(U\)-admissible blowup \(B' \to B\) such that strict transform of \(\mathcal{F}\) is flat over \(B'\) and of finite presentation. Let \(X'\) be the strict transform of \(X\) under the blowup \(B' \to B\). Let \(i' : X' \to Y \times_B B'\) be the induced morphism. Since taking strict transform commutes with pushforward along affine morphisms (Divisors on Spaces, Lemma 0866), we see that \(i'_*\mathcal{O}_{X'}\) is flat over \(B'\) and of finite presentation as a \(\mathcal{O}_{Y \times_B B'}\)-module. Thus \(X' \to B'\) is flat and locally of finite presentation. This implies the lemma by our earlier remarks.
Lemma
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(U \subset B\) be an open subspace. Assume
\(B\) is quasi-compact and quasi-separated,
\(U\) is quasi-compact,
\(f : X \to B\) is proper, and
\(f^{-1}(U) \to U\) is finite locally free.
Then there exists a \(U\)-admissible blowup \(B' \to B\) such that the strict transform \(X'\) of \(X\) is finite locally free over \(B'\).
Proof
By Lemma 087E we may assume that \(X \to B\) is flat and of finite presentation. After replacing \(B\) by a \(U\)-admissible blowup if necessary, we may assume that \(U \subset B\) is scheme theoretically dense. Then \(f\) is finite by Lemma 0B4J. Hence \(f\) is finite locally free by Morphisms of Spaces, Lemma 0416.
Lemma
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(U \subset B\) be an open subspace. Assume
\(B\) is quasi-compact and quasi-separated,
\(U\) is quasi-compact,
\(f : X \to B\) is proper, and
\(f^{-1}(U) \to U\) is an isomorphism.
Then there exists a \(U\)-admissible blowup \(B' \to B\) such that the strict transform \(X'\) of \(X\) maps isomorphically to \(B'\).
Proof
By Lemma 087E we may assume that \(X \to B\) is flat and of finite presentation. After replacing \(B\) by a \(U\)-admissible blowup if necessary, we may assume that \(U \subset B\) is scheme theoretically dense. Then \(f\) is finite by Lemma 0B4J and an open immersion by Lemma 0879. Hence \(f\) is an open immersion whose image is closed and contains the dense open \(U\), whence \(f\) is an isomorphism.
Lemma
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(U \subset B\) be an open subspace. Assume
\(B\) quasi-compact and quasi-separated,
\(U\) is quasi-compact,
\(f\) is of finite type
\(f^{-1}(U) \to U\) is an isomorphism.
Then there exists a \(U\)-admissible blowup \(B' \to B\) such that \(U\) is scheme theoretically dense in \(B'\) and such that the strict transform \(X'\) of \(X\) maps isomorphically to an open subspace of \(B'\).
Proof
This lemma is a generalization of Lemma 0GUW. As the composition of \(U\)-admissible blowups is \(U\)-admissible (Divisors on Spaces, Lemma 086C) we can proceed in stages. Pick a finite type quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_B\) with \(|B| \setminus |U| = |V(\mathcal{I})|\). Replace \(B\) by the blowup of \(B\) in \(\mathcal{I}\) and \(X\) by the strict transform of \(X\). After this replacement \(B \setminus U\) is the support of an effective Cartier divisor \(D\) (Divisors on Spaces, Lemma 085T). In particular \(U\) is scheme theoretically dense in \(B\) (Divisors on Spaces, Lemma 083S). Next, we do another \(U\)-admissible blowup to get to the situation where \(X \to B\) is flat and of finite presentation, see Lemma 087E. Note that \(U\) is still scheme theoretically dense in \(B\). Hence \(X \to B\) is an open immersion by Lemma 0879.
The following lemma says that a modification can be dominated by a blowup.
Lemma
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(U \subset B\) be an open subspace. Assume
\(B\) is quasi-compact and quasi-separated,
\(U\) is quasi-compact,
\(f : X \to B\) is proper,
\(f^{-1}(U) \to U\) us an isomorphism.
Then there exists a \(U\)-admissible blowup \(B' \to B\) which dominates \(X\), i.e., such that there exists a factorization \(B' \to X \to B\) of the blowup morphism.
Proof
By Lemma 0GUW we may find a \(U\)-admissible blowup \(B' \to B\) such that the strict transform \(X'\) maps isomorphically to \(B'\). Then we can use \(B' = X' \to X\) as the factorization.
Lemma
Let \(S\) be a scheme. Let \(X\), \(Y\) be algebraic spaces over \(S\). Let \(U \subset W \subset Y\) be open subspaces. Let \(f : X \to W\) and let \(s : U \to X\) be morphisms such that \(f \circ s = \text{id}_U\). Assume
\(f\) is proper,
\(Y\) is quasi-compact and quasi-separated, and
\(U\) and \(W\) are quasi-compact.
Then there exists a \(U\)-admissible blowup \(b : Y' \to Y\) and a morphism \(s' : b^{-1}(W) \to X\) extending \(s\) with \(f \circ s' = b|_{b^{-1}(W)}\).
Proof
We may and do replace \(X\) by the scheme theoretic image of \(s\). Then \(X \to W\) is an isomorphism over \(U\), see Morphisms of Spaces, Lemma 0CP3. By Lemma 087G there exists a \(U\)-admissible blowup \(W' \to W\) and an extension \(W' \to X\) of \(s\). We finish the proof by applying Divisors on Spaces, Lemma 086D to extend \(W' \to W\) to a \(U\)-admissible blowup of \(Y\).
Chow’s lemma
In this section we prove Chow’s lemma (Lemma 088U). We encourage the reader to take a look at Cohomology of Spaces, Section 089I for a weak version of Chow’s lemma that is easy to prove and sufficient for many applications.
Since we have yet to define projective morphisms of algebraic spaces, the statements of lemmas (see for example Lemma 088R) will involve representable proper morphisms, rather than projective ones.
Lemma
Let \(S\) be a scheme. Let \(Y\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(U \to X_1\) and \(U \to X_2\) be open immersions of algebraic spaces over \(Y\) and assume \(U\), \(X_1\), \(X_2\) of finite type and separated over \(Y\). Then there exists a commutative diagram \[\xymatrix{ X_1' \ar[d] \ar[r] & X & X_2' \ar[l] \ar[d] \\ X_1 & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & X_2 }\] of algebraic spaces over \(Y\) where \(X_i' \to X_i\) is a \(U\)-admissible blowup, \(X_i' \to X\) is an open immersion, and \(X\) is separated and finite type over \(Y\).
Proof
Throughout the proof all the algebraic spaces will be separated of finite type over \(Y\). This in particular implies these algebraic spaces are quasi-compact and quasi-separated and that the morphisms between them will be quasi-compact and separated. See Morphisms of Spaces, Sections 03HJ and 03HC. We will use that if \(U \to W\) is an immersion of such spaces over \(Y\), then the scheme theoretic image \(Z\) of \(U\) in \(W\) is a closed subspace of \(W\) and \(U \to Z\) is an open immersion, \(U \subset Z\) is scheme theoretically dense, and \(|U| \subset |Z|\) is dense. See Morphisms of Spaces, Lemma 088G.
Let \(X_{12} \subset X_1 \times_Y X_2\) be the scheme theoretic image of \(U \to X_1 \times_Y X_2\). The projections \(p_i : X_{12} \to X_i\) induce isomorphisms \(p_i^{-1}(U) \to U\) by Morphisms of Spaces, Lemma 0CP3. Choose a \(U\)-admissible blowup \(X_i^i \to X_i\) such that the strict transform \(X_{12}^i\) of \(X_{12}\) is isomorphic to an open subspace of \(X_i^i\), see Lemma 087F. Let \(\mathcal{I}_i \subset \mathcal{O}_{X_i}\) be the corresponding finite type quasi-coherent sheaf of ideals. Recall that \(X_{12}^i \to X_{12}\) is the blowup in \(p_i^{-1}\mathcal{I}_i \mathcal{O}_{X_{12}}\), see Divisors on Spaces, Lemma 0864. Let \(X_{12}'\) be the blowup of \(X_{12}\) in \(p_1^{-1}\mathcal{I}_1 p_2^{-1}\mathcal{I}_2 \mathcal{O}_{X_{12}}\), see Divisors on Spaces, Lemma 085Y for what this entails. We obtain a commutative diagram \[\xymatrix{ X_{12}' \ar[d] \ar[r] & X_{12}^2 \ar[d] \\ X_{12}^1 \ar[r] & X_{12} }\] where all the morphisms are \(U\)-admissible blowing ups. Since \(X_{12}^i \subset X_i^i\) is an open we may choose a \(U\)-admissible blowup \(X_i' \to X_i^i\) restricting to \(X_{12}' \to X_{12}^i\), see Divisors on Spaces, Lemma 086D. Then \(X_{12}' \subset X_i'\) is an open subspace and the diagram \[\xymatrix{ X_{12}' \ar[d] \ar[r] & X_i' \ar[d] \\ X_{12}^i \ar[r] & X_i^i }\] is commutative with vertical arrows blowing ups and horizontal arrows open immersions. Note that \(X'_{12} \to X_1' \times_Y X_2'\) is an immersion and proper (use that \(X'_{12} \to X_{12}\) is proper and \(X_{12} \to X_1 \times_Y X_2\) is closed and \(X_1' \times_Y X_2' \to X_1 \times_Y X_2\) is separated and apply Morphisms of Spaces, Lemma 04NX). Thus \(X'_{12} \to X_1' \times_Y X_2'\) is a closed immersion. If we define \(X\) by glueing \(X_1'\) and \(X_2'\) along the common open subspace \(X_{12}'\), then \(X \to Y\) is of finite type and separated2. As compositions of \(U\)-admissible blowups are \(U\)-admissible blowups (Divisors on Spaces, Lemma 086C) the lemma is proved.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(U \subset X\) be an open subspace. Assume
\(U\) is quasi-compact,
\(Y\) is quasi-compact and quasi-separated,
there exists an immersion \(U \to \mathbf{P}^n_Y\) over \(Y\),
\(f\) is of finite type and separated.
Then there exists a commutative diagram \[\xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & Y & \mathbf{P}^n_Y \ar[l] }\] where the arrows with source \(U\) are open immersions, \(X' \to X\) is a \(U\)-admissible blowup, \(X' \to Z'\) is an open immersion, \(Z' \to Y\) is a proper and representable morphism of algebraic spaces. More precisely, \(Z' \to Z\) is a \(U\)-admissible blowup and \(Z \to \mathbf{P}^n_Y\) is a closed immersion.
Proof
Let \(Z \subset \mathbf{P}^n_Y\) be the scheme theoretic image of the immersion \(U \to \mathbf{P}^n_Y\). Since \(U \to \mathbf{P}^n_Y\) is quasi-compact we see that \(U \subset Z\) is a (scheme theoretically) dense open subspace (Morphisms of Spaces, Lemma 088G). Apply Lemma 088Q to find a diagram \[\xymatrix{ X' \ar[d] \ar[r] & \overline{X}' & Z' \ar[l] \ar[d] \\ X & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & Z }\] with properties as listed in the statement of that lemma. As \(X' \to X\) and \(Z' \to Z\) are \(U\)-admissible blowups we find that \(U\) is a scheme theoretically dense open of both \(X'\) and \(Z'\) (see Divisors on Spaces, Lemmas 085T and 083S). Since \(Z' \to Z \to Y\) is proper we see that \(Z' \subset \overline{X}'\) is a closed subspace (see Morphisms of Spaces, Lemma 04NX). It follows that \(X' \subset Z'\) (scheme theoretically), hence \(X'\) is an open subspace of \(Z'\) (small detail omitted) and the lemma is proved.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) separated, of finite type, and \(Y\) Noetherian. Then there exists a dense open subspace \(U \subset X\) and a commutative diagram \[\xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & Y & \mathbf{P}^n_Y \ar[l] }\] where the arrows with source \(U\) are open immersions, \(X' \to X\) is a \(U\)-admissible blowup, \(X' \to Z'\) is an open immersion, \(Z' \to Y\) is a proper and representable morphism of algebraic spaces. More precisely, \(Z' \to Z\) is a \(U\)-admissible blowup and \(Z \to \mathbf{P}^n_Y\) is a closed immersion.
Proof
By Limits of Spaces, Lemma 088N there exists a dense open subspace \(U \subset X\) and an immersion \(U \to \mathbf{A}^n_Y\) over \(Y\). Composing with the open immersion \(\mathbf{A}^n_Y \to \mathbf{P}^n_Y\) we obtain a situation as in Lemma 088R and the result follows.
Remark
In Lemmas 088R and 088S the morphism \(g : Z' \to Y\) is a composition of projective morphisms. Presumably (by the analogue for algebraic spaces of Morphisms, Lemma 0C4K) there exists a \(g\)-ample invertible sheaf on \(Z'\). If we ever need this, then we will state and prove this here.
The following result is [Kn, IV Theorem 3.1]. Note that the immersion \(X' \to \mathbf{P}^n_Y\) is quasi-compact, hence can be factored as \(X' \to Z' \to \mathbf{P}^n_Y\) where the first morphism is an open immersion and the second morphism a closed immersion (Morphisms of Spaces, Lemma 088G).
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) separated of finite type, and \(Y\) separated and Noetherian. Then there exists a commutative diagram \[\xymatrix{ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & \mathbf{P}^n_Y \ar[ld] \\ & Y }\] where \(X' \to X\) is a \(U\)-admissible blowup for some dense open \(U \subset X\) and the morphism \(X' \to \mathbf{P}^n_Y\) is an immersion.
Proof
In this first paragraph of the proof we reduce the lemma to the case where \(Y\) is of finite type over \(\Spec(\mathbf{Z})\). We may and do replace the base scheme \(S\) by \(\Spec(\mathbf{Z})\). We can write \(Y = \lim Y_i\) as a directed limit of separated algebraic spaces of finite type over \(\Spec(\mathbf{Z})\), see Limits of Spaces, Proposition 07SU and Lemma 084T. For all \(i\) sufficiently large we can find a separated finite type morphism \(X_i \to Y_i\) such that \(X = Y \times_{Y_i} X_i\), see Limits of Spaces, Lemmas 07SK and 0851. Let \(\eta_1, \ldots, \eta_n\) be the generic points of the irreducible components of \(|X|\) (\(X\) is Noetherian as a finite type separated algebraic space over the Noetherian algebraic space \(Y\) and therefore \(|X|\) is a Noetherian topological space). By Limits of Spaces, Lemma 086V we find that the images of \(\eta_1, \ldots, \eta_n\) in \(|X_i|\) are distinct for \(i\) large enough. We may replace \(X_i\) by the scheme theoretic image of the (quasi-compact, in fact affine) morphism \(X \to X_i\). After this replacement we see that the images of \(\eta_1, \ldots, \eta_n\) in \(|X_i|\) are the generic points of the irreducible components of \(|X_i|\), see Morphisms of Spaces, Lemma 082Z. Having said this, suppose we can find a diagram \[\xymatrix{ X_i \ar[rd] & X_i' \ar[l] \ar[d] \ar[r] & \mathbf{P}^n_{Y_i} \ar[ld] \\ & Y }\] where \(X_i' \to X_i\) is a \(U_i\)-admissible blowup for some dense open \(U_i \subset X_i\) and the morphism \(X_i' \to \mathbf{P}^n_{Y_i}\) is an immersion. Then the strict transform \(X' \to X\) of \(X\) relative to \(X_i' \to X_i\) is a \(U\)-admissible blowing up where \(U \subset X\) is the inverse image of \(U_i\) in \(X\). Because of our carefully chosen index \(i\) it follows that \(\eta_1, \ldots, \eta_n \in |U|\) and \(U \subset X\) is dense. Moreover, \(X' \to \mathbf{P}^n_Y\) is an immersion as \(X'\) is closed in \(X_i' \times_{X_i} X = X_i' \times_{Y_i} Y\) which comes with an immersion into \(\mathbf{P}^n_Y\). Thus we have reduced to the situation of the following paragraph.
Assume that \(Y\) is separated of finite type over \(\Spec(\mathbf{Z})\). Then \(X \to \Spec(\mathbf{Z})\) is separated of finite type as well. We apply Lemma 088S to \(X \to \Spec(\mathbf{Z})\) to find a dense open subspace \(U \subset X\) and a commutative diagram \[\xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & \Spec(\mathbf{Z}) & \mathbf{P}^n_\mathbf{Z} \ar[l] }\] with all the properties listed in the lemma. Note that \(Z\) has an ample invertible sheaf, namely \(\mathcal{O}_{\mathbf{P}^n}(1)|_Z\). Hence \(Z' \to Z\) is a H-projective morphism by Morphisms, Lemma 087S. It follows that \(Z' \to \Spec(\mathbf{Z})\) is H-projective by Morphisms, Lemma 01WE. Thus there exists a closed immersion \(Z' \to \mathbf{P}^m_{\Spec(\mathbf{Z})}\) for some \(m \geq 0\). It follows that the diagonal morphism \[X' \to Y \times \mathbf{P}^m_\mathbf{Z} = \mathbf{P}^m_Y\] is an immersion (because the composition with the projection to \(\mathbf{P}^m_\mathbf{Z}\) is an immersion) and we win.
Variants of Chow’s Lemma
In this section we prove a number of variants of Chow’s lemma dealing with morphisms between non-Noetherian algebraic spaces. The Noetherian versions are Lemma 088S and Lemma 088U.
Lemma
Let \(S\) be a scheme. Let \(Y\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(f : X \to Y\) be a separated morphism of finite type. Then there exists a commutative diagram \[\xymatrix{ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & \overline{X}' \ar[ld] \\ & Y }\] where \(X' \to X\) is proper surjective, \(X' \to \overline{X}'\) is an open immersion, and \(\overline{X}' \to Y\) is proper and representable morphism of algebraic spaces.
Proof
By Limits of Spaces, Proposition 0873 we can find a closed immersion \(X \to X_1\) where \(X_1\) is separated and of finite presentation over \(Y\). Clearly, if we prove the assertion for \(X_1 \to Y\), then the result follows for \(X\). Hence we may assume that \(X\) is of finite presentation over \(Y\).
We may and do replace the base scheme \(S\) by \(\Spec(\mathbf{Z})\). Write \(Y = \lim_i Y_i\) as a directed limit of quasi-separated algebraic spaces of finite type over \(\Spec(\mathbf{Z})\), see Limits of Spaces, Proposition 07SU. By Limits of Spaces, Lemma 07SK we can find an index \(i \in I\) and a scheme \(X_i \to Y_i\) of finite presentation so that \(X = Y \times_{Y_i} X_i\). By Limits of Spaces, Lemma 0851 we may assume that \(X_i \to Y_i\) is separated. Clearly, if we prove the assertion for \(X_i\) over \(Y_i\), then the assertion holds for \(X\). The case \(X_i \to Y_i\) is treated by Lemma 088S.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) separated of finite type, and \(Y\) separated and quasi-compact. Then there exists a commutative diagram \[\xymatrix{ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & \mathbf{P}^n_Y \ar[ld] \\ & Y }\] where \(X' \to X\) is proper surjective morphism and the morphism \(X' \to \mathbf{P}^n_Y\) is an immersion.
Proof
By Limits of Spaces, Proposition 0873 we can find a closed immersion \(X \to X_1\) where \(X_1\) is separated and of finite presentation over \(Y\). Clearly, if we prove the assertion for \(X_1 \to Y\), then the result follows for \(X\). Hence we may assume that \(X\) is of finite presentation over \(Y\).
We may and do replace the base scheme \(S\) by \(\Spec(\mathbf{Z})\). Write \(Y = \lim_i Y_i\) as a directed limit of quasi-separated algebraic spaces of finite type over \(\Spec(\mathbf{Z})\), see Limits of Spaces, Proposition 07SU. By Limits of Spaces, Lemma 084T we may assume that \(Y_i\) is separated for all \(i\). By Limits of Spaces, Lemma 07SK we can find an index \(i \in I\) and a scheme \(X_i \to Y_i\) of finite presentation so that \(X = Y \times_{Y_i} X_i\). By Limits of Spaces, Lemma 0851 we may assume that \(X_i \to Y_i\) is separated. Clearly, if we prove the assertion for \(X_i\) over \(Y_i\), then the assertion holds for \(X\). The case \(X_i \to Y_i\) is treated by Lemma 088U.
Grothendieck’s existence theorem
In this section we discuss Grothendieck’s existence theorem for algebraic spaces. Instead of developing a theory of “formal algebraic spaces” we temporarily develop a bit of language that replaces the notion of a “coherent module on a Noetherian adic formal space”.
Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Below we will consider inverse systems \((\mathcal{F}_n)\) of coherent \(\mathcal{O}_X\)-modules such that
\(\mathcal{F}_n\) is annihilated by \(\mathcal{I}^n\), and
the transition maps induce isomorphisms \(\mathcal{F}_{n + 1}/\mathcal{I}^n\mathcal{F}_{n + 1} \to \mathcal{F}_n\).
A morphism \(\alpha : (\mathcal{F}_n) \to (\mathcal{G}_n)\) of such inverse systems is simply a compatible system of morphisms \(\alpha_n : \mathcal{F}_n \to \mathcal{G}_n\). Let us denote the category of these inverse systems with \(\textit{Coh}(X, \mathcal{I})\). We will develop some theory regarding these systems that will parallel to the corresponding results in the case of schemes, see Cohomology of Schemes, Sections 087V, 0886, 0CYW, and 0898.
Functoriality. Let \(f : X \to Y\) be a morphism of Noetherian algebraic spaces over a scheme \(S\), and let \(\mathcal{J} \subset \mathcal{O}_Y\) be a quasi-coherent sheaf of ideals. Set \(\mathcal{I} = f^{-1}\mathcal{J}\mathcal{O}_X\). In this situation there is a functor \[f^* : \textit{Coh}(Y, \mathcal{J}) \longrightarrow \textit{Coh}(X, \mathcal{I})\] which sends \((\mathcal{G}_n)\) to \((f^*\mathcal{G}_n)\). Compare with Cohomology of Schemes, Lemma 0887. If \(f\) is étale, then we may think of this as simply the restriction of the system to \(X\), see Properties of Spaces, Equation 03LW.
Étale descent. Let \(S\) be a scheme. Let \(U_0 \to X\) be a surjective étale morphism of Noetherian algebraic spaces. Set \(U_1 = U_0 \times_X U_0\) and \(U_2 = U_0 \times_X U_0 \times_X U_0\). Let \(\mathcal{I} \subset \mathcal{O}_{X}\) be a quasi-coherent sheaf of ideals. Set \(\mathcal{I}_i = \mathcal{I}|_{U_i}\). In this situation we obtain a diagram of categories \[\xymatrix{ \textit{Coh}(X, \mathcal{I}) \ar[r] & \textit{Coh}(U_0, \mathcal{I}_0) \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \textit{Coh}(U_1, \mathcal{I}_1) \ar@<1ex>[r] \ar[r] \ar@<-1ex>[r] & \textit{Coh}(U_2, \mathcal{I}_2) }\] an the first arrow presents \(\textit{Coh}(X, \mathcal{I})\) as the homotopy limit of the right part of the diagram. More precisely, given a descent datum, i.e., a pair \(((\mathcal{G}_n), \varphi)\) where \((\mathcal{G}_n)\) is an object of \(\textit{Coh}(U_0, \mathcal{I}_0)\) and \(\varphi : \text{pr}_0^*(\mathcal{G}_n) \to \text{pr}_1^*(\mathcal{G}_n)\) is an isomorphism in \(\textit{Coh}(U_1, \mathcal{I}_1)\) satisfying the cocycle condition in \(\textit{Coh}(U_2, \mathcal{I}_2)\), then there exists a unique object \((\mathcal{F}_n)\) of \(\textit{Coh}(X, \mathcal{I})\) whose associated canonical descent datum is isomorphic to \(((\mathcal{G}_n), \varphi)\). Compare with Descent on Spaces, Definition 04W5. The proof of this statement follows immediately by applying Descent on Spaces, Proposition 04W8 to the descent data \((\mathcal{G}_n, \varphi_n)\) for varying \(n\).
Lemma
Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\) and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals.
The category \(\textit{Coh}(X, \mathcal{I})\) is abelian.
Exactness in \(\textit{Coh}(X, \mathcal{I})\) can be checked étale locally.
For any flat morphism \(f : X' \to X\) of Noetherian algebraic spaces the functor \(f^* : \textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(X', f^{-1}\mathcal{I}\mathcal{O}_{X'})\) is exact.
Proof
Proof of (1). Choose an affine scheme \(U_0\) and a surjective étale morphism \(U_0 \to X\). Set \(U_1 = U_0 \times_X U_0\) and \(U_2 = U_0 \times_X U_0 \times_X U_0\) as in our discussion of étale descent above. The categories \(\textit{Coh}(U_i, \mathcal{I}_i)\) are abelian (Cohomology of Schemes, Lemma 087X) and the pullback functors are exact functors \(\textit{Coh}(U_0, \mathcal{I}_0) \to \textit{Coh}(U_1, \mathcal{I}_1)\) and \(\textit{Coh}(U_1, \mathcal{I}_1) \to \textit{Coh}(U_2, \mathcal{I}_2)\) (Cohomology of Schemes, Lemma 0887). The lemma then follows formally from the description of \(\textit{Coh}(X, \mathcal{I})\) as a category of descent data. Some details omitted; compare with the proof of Groupoids, Lemma 077S.
Part (2) follows immediately from the discussion in the previous paragraph. In the situation of (3) choose a commutative diagram \[\xymatrix{ U' \ar[d] \ar[r] & U \ar[d] \\ X' \ar[r] & X }\] where \(U'\) and \(U\) are affine schemes and the vertical morphisms are surjective étale. Then \(U' \to U\) is a flat morphism of Noetherian schemes (Morphisms of Spaces, Lemma 03MM) whence the pullback functor \(\textit{Coh}(U, \mathcal{I}\mathcal{O}_U) \to \textit{Coh}(U', \mathcal{I}\mathcal{O}_{U'})\) is exact by Cohomology of Schemes, Lemma 0887. Since we can check exactness in \(\textit{Coh}(X, \mathcal{O}_X)\) on \(U\) and similarly for \(X', U'\) the assertion follows.
Lemma
Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\) and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. A map \((\mathcal{F}_n) \to (\mathcal{G}_n)\) is surjective in \(\textit{Coh}(X, \mathcal{I})\) if and only if \(\mathcal{F}_1 \to \mathcal{G}_1\) is surjective.
Proof
We can check on an affine étale cover of \(X\) by Lemma 089P. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 087Y.
Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\) and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. There is a functor [08B4]\[\begin{equation} \textit{Coh}(\mathcal{O}_X) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \end{equation}\] which associates to the coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) the object \(\mathcal{F}^\wedge = (\mathcal{F}/\mathcal{I}^n\mathcal{F})\) of \(\textit{Coh}(X, \mathcal{I})\).
Lemma
The functor (08B4) is exact.
Proof
It suffices to check this étale locally on \(X\), see Lemma 089P. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 0881.
Lemma
Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\) and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. Set \(\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\). Then \[\lim H^0(X, \mathcal{H}/\mathcal{I}^n\mathcal{H}) = \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{F}^\wedge, \mathcal{G}^\wedge).\]
Proof
Since \(\mathcal{H}\) is a sheaf on \(X_\etale\) and since we have étale descent for objects of \(\textit{Coh}(X, \mathcal{I})\) it suffices to prove this étale locally. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 0882.
We introduce the setting that we will focus on throughout the rest of this section.
Situation
Here \(A\) is a Noetherian ring complete with respect to an ideal \(I\). Also \(f : X \to \Spec(A)\) is a finite type separated morphism of algebraic spaces and \(\mathcal{I} = I\mathcal{O}_X\).
In this situation we denote \[\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_X)\] be the full subcategory of \(\textit{Coh}(\mathcal{O}_X)\) consisting of those coherent \(\mathcal{O}_X\)-modules whose support is proper over \(\Spec(A)\), or equivalently whose scheme theoretic support is proper over \(\Spec(A)\), see Derived Categories of Spaces, Lemma 0CZI. Similarly, we let \[\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})\] be the full subcategory of \(\textit{Coh}(X, \mathcal{I})\) consisting of those objects \((\mathcal{F}_n)\) such that the support of \(\mathcal{F}_1\) is proper over \(\Spec(A)\). Since the support of a quotient module is contained in the support of the module, it follows that (08B4) induces a functor [08B8]\[\begin{equation} \textit{Coh}_{\text{support proper over }A}(\mathcal{O}_X) \longrightarrow \textit{Coh}_{\text{support proper over }A}(X, \mathcal{I}) \end{equation}\] Our first result is that this functor is fully faithful.
Lemma
In Situation 08B7. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. Assume that the intersection of the supports of \(\mathcal{F}\) and \(\mathcal{G}\) is proper over \(\Spec(A)\). Then the map \[\Mor_{\textit{Coh}(\mathcal{O}_X)}(\mathcal{F}, \mathcal{G}) \longrightarrow \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{F}^\wedge, \mathcal{G}^\wedge)\] coming from (08B4) is a bijection. In particular, (08B8) is fully faithful.
Proof
Let \(\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})\). This is a coherent \(\mathcal{O}_X\)-module because its restriction of schemes étale over \(X\) is coherent by Modules, Lemma 01CQ. By Lemma 08B6 the map \[\lim_n H^0(X, \mathcal{H}/\mathcal{I}^n\mathcal{H}) \to \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge, \mathcal{F}^\wedge)\] is bijective. Let \(i : Z \to X\) be the scheme theoretic support of \(\mathcal{H}\). It is clear that \(Z\) is a closed subspace such that \(|Z|\) is contained in the intersection of the supports of \(\mathcal{F}\) and \(\mathcal{G}\). Hence \(Z \to \Spec(A)\) is proper by assumption (see Derived Categories of Spaces, Section 0CZB). Write \(\mathcal{H} = i_*\mathcal{H}'\) for some coherent \(\mathcal{O}_Z\)-module \(\mathcal{H}'\). We have \(i_*(\mathcal{H}'/I^n\mathcal{H}') = \mathcal{H}/I^n\mathcal{H}\). Hence we obtain \[\begin{align*} \lim_n H^0(X, \mathcal{H}/\mathcal{I}^n\mathcal{H}) & = \lim_n H^0(Z, \mathcal{H}'/\mathcal{I}^n\mathcal{H}') \\ & = H^0(Z, \mathcal{H}') \\ & = H^0(X, \mathcal{H}) \\ & = \Mor_{\textit{Coh}(\mathcal{O}_X)}(\mathcal{F}, \mathcal{G}) \end{align*}\] the second equality by the theorem on formal functions (Cohomology of Spaces, Lemma 08B0). This proves the lemma.
Remark
Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\) and let \(\mathcal{I}, \mathcal{K} \subset \mathcal{O}_X\) be quasi-coherent sheaves of ideals. Let \(\alpha : (\mathcal{F}_n) \to (\mathcal{G}_n)\) be a morphism of \(\textit{Coh}(X, \mathcal{I})\). Given an affine scheme \(U = \Spec(A)\) and a surjective étale morphism \(U \to X\) denote \(I, K \subset A\) the ideals corresponding to the restrictions \(\mathcal{I}|_U, \mathcal{K}|_U\). Denote \(\alpha_U : M \to N\) of finite \(A^\wedge\)-modules which corresponds to \(\alpha|_U\) via Cohomology of Schemes, Lemma 087W. We claim the following are equivalent
there exists an integer \(t \geq 1\) such that \(\Ker(\alpha_n)\) and \(\Coker(\alpha_n)\) are annihilated by \(\mathcal{K}^t\) for all \(n \geq 1\),
for any (or some) affine open \(\Spec(A) = U \subset X\) as above the modules \(\Ker(\alpha_U)\) and \(\Coker(\alpha_U)\) are annihilated by \(K^t\) for some integer \(t \geq 1\).
If these equivalent conditions hold we will say that \(\alpha\) is a map whose kernel and cokernel are annihilated by a power of \(\mathcal{K}\). To see the equivalence we refer to Cohomology of Schemes, Remark 0888.
Lemma
Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\) and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(\mathcal{G}\) be a coherent \(\mathcal{O}_X\)-module, \((\mathcal{F}_n)\) an object of \(\textit{Coh}(X, \mathcal{I})\), and \(\alpha : (\mathcal{F}_n) \to \mathcal{G}^\wedge\) a map whose kernel and cokernel are annihilated by a power of \(\mathcal{I}\). Then there exists a unique (up to unique isomorphism) triple \((\mathcal{F}, a, \beta)\) where
\(\mathcal{F}\) is a coherent \(\mathcal{O}_X\)-module,
\(a : \mathcal{F} \to \mathcal{G}\) is an \(\mathcal{O}_X\)-module map whose kernel and cokernel are annihilated by a power of \(\mathcal{I}\),
\(\beta : (\mathcal{F}_n) \to \mathcal{F}^\wedge\) is an isomorphism, and
\(\alpha = a^\wedge \circ \beta\).
Proof
The uniqueness and étale descent for objects of \(\textit{Coh}(X, \mathcal{I})\) and \(\textit{Coh}(\mathcal{O}_X)\) implies it suffices to construct \((\mathcal{F}, a, \beta)\) étale locally on \(X\). Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 0889.
Lemma
In Situation 08B7. Let \(\mathcal{K} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(X_e \subset X\) be the closed subspace cut out by \(\mathcal{K}^e\). Let \(\mathcal{I}_e = \mathcal{I}\mathcal{O}_{X_e}\). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})\). Assume
the functor \(\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_{X_e}) \to \textit{Coh}_{\text{support proper over } A}(X_e, \mathcal{I}_e)\) is an equivalence for all \(e \geq 1\), and
there exists an object \(\mathcal{H}\) of \(\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_X)\) and a map \(\alpha : (\mathcal{F}_n) \to \mathcal{H}^\wedge\) whose kernel and cokernel are annihilated by a power of \(\mathcal{K}\).
Then \((\mathcal{F}_n)\) is in the essential image of (08B8).
Proof
During this proof we will use without further mention that for a closed immersion \(i : Z \to X\) the functor \(i_*\) gives an equivalence between the category of coherent modules on \(Z\) and coherent modules on \(X\) annihilated by the ideal sheaf of \(Z\), see Cohomology of Spaces, Lemma 08AM. In particular we think of \[\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_{X_e}) \subset \textit{Coh}_{\text{support proper over } A}(\mathcal{O}_X)\] as the full subcategory of consisting of modules annihilated by \(\mathcal{K}^e\) and \[\textit{Coh}_{\text{support proper over } A}(X_e, \mathcal{I}_e) \subset \textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})\] as the full subcategory of objects annihilated by \(\mathcal{K}^e\). Moreover (1) tells us these two categories are equivalent under the completion functor (08B8).
Applying this equivalence we get a coherent \(\mathcal{O}_X\)-module \(\mathcal{G}_e\) annihilated by \(\mathcal{K}^e\) corresponding to the system \((\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n)\) of \(\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})\). The maps \(\mathcal{F}_n/\mathcal{K}^{e + 1}\mathcal{F}_n \to \mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n\) correspond to canonical maps \(\mathcal{G}_{e + 1} \to \mathcal{G}_e\) which induce isomorphisms \(\mathcal{G}_{e + 1}/\mathcal{K}^e\mathcal{G}_{e + 1} \to \mathcal{G}_e\). We obtain an object \((\mathcal{G}_e)\) of the category \(\textit{Coh}_{\text{support proper over } A}(X, \mathcal{K})\). The map \(\alpha\) induces a system of maps \[\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n \longrightarrow \mathcal{H}/(\mathcal{I}^n + \mathcal{K}^e)\mathcal{H}\] whence maps \(\mathcal{G}_e \to \mathcal{H}/\mathcal{K}^e\mathcal{H}\) (by the equivalence of categories again). Let \(t \geq 1\) be an integer, which exists by assumption (2), such that \(\mathcal{K}^t\) annihilates the kernel and cokernel of all the maps \(\mathcal{F}_n \to \mathcal{H}/\mathcal{I}^n\mathcal{H}\). Then \(\mathcal{K}^{2t}\) annihilates the kernel and cokernel of the maps \(\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n \to \mathcal{H}/(\mathcal{I}^n + \mathcal{K}^e)\mathcal{H}\) (details omitted; see Cohomology of Schemes, Remark 0888). Whereupon we conclude that \(\mathcal{K}^{4t}\) annihilates the kernel and the cokernel of the maps \[\mathcal{G}_e \longrightarrow \mathcal{H}/\mathcal{K}^e\mathcal{H},\] (details omitted; see Cohomology of Schemes, Remark 0888). We apply Lemma 08BB to obtain a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\), a map \(a : \mathcal{F} \to \mathcal{H}\) and an isomorphism \(\beta : (\mathcal{G}_e) \to (\mathcal{F}/\mathcal{K}^e\mathcal{F})\) in \(\textit{Coh}(X, \mathcal{K})\). Working backwards, for a given \(n\) the triple \((\mathcal{F}/\mathcal{I}^n\mathcal{F}, a \bmod \mathcal{I}^n, \beta \bmod \mathcal{I}^n)\) is a triple as in the lemma for the morphism \(\alpha_n \bmod \mathcal{K}^e : (\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n) \to (\mathcal{H}/(\mathcal{I}^n + \mathcal{K}^e)\mathcal{H})\) of \(\textit{Coh}(X, \mathcal{K})\). Thus the uniqueness in Lemma 08BB gives a canonical isomorphism \(\mathcal{F}/\mathcal{I}^n\mathcal{F} \to \mathcal{F}_n\) compatible with all the morphisms in sight.
To finish the proof of the lemma we still have to show that the support of \(\mathcal{F}\) is proper over \(A\). By construction the kernel of \(a : \mathcal{F} \to \mathcal{H}\) is annihilated by a power of \(\mathcal{K}\). Hence the support of this kernel is contained in the support of \(\mathcal{G}_1\). Since \(\mathcal{G}_1\) is an object of \(\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_{X_1})\) we see this is proper over \(A\). Combined with the fact that the support of \(\mathcal{H}\) is proper over \(A\) we conclude that the support of \(\mathcal{F}\) is proper over \(A\) by Derived Categories of Spaces, Lemma 0CZH.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a representable proper morphism of Noetherian algebraic spaces over \(S\). Let \(\mathcal{J}, \mathcal{K} \subset \mathcal{O}_Y\) be quasi-coherent sheaves of ideals. Assume \(f\) is an isomorphism over \(V = Y \setminus V(\mathcal{K})\). Set \(\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_X\). Let \((\mathcal{G}_n)\) be an object of \(\textit{Coh}(Y, \mathcal{J})\), let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module, and let \(\beta : (f^*\mathcal{G}_n) \to \mathcal{F}^\wedge\) be an isomorphism in \(\textit{Coh}(X, \mathcal{I})\). Then there exists a map \[\alpha : (\mathcal{G}_n) \longrightarrow (f_*\mathcal{F})^\wedge\] in \(\textit{Coh}(Y, \mathcal{J})\) whose kernel and cokernel are annihilated by a power of \(\mathcal{K}\).
Proof
Since \(f\) is a proper morphism we see that \(f_*\mathcal{F}\) is a coherent \(\mathcal{O}_Y\)-module (Cohomology of Spaces, Lemma 08AR). Thus the statement of the lemma makes sense. Consider the compositions \[\gamma_n : \mathcal{G}_n \to f_*f^*\mathcal{G}_n \to f_*(\mathcal{F}/\mathcal{I}^n\mathcal{F}).\] Here the first map is the adjunction map and the second is \(f_*\beta_n\). We claim that there exists a unique \(\alpha\) as in the lemma such that the compositions \[\mathcal{G}_n \xrightarrow{\alpha_n} f_*\mathcal{F}/\mathcal{J}^nf_*\mathcal{F} \to f_*(\mathcal{F}/\mathcal{I}^n\mathcal{F})\] equal \(\gamma_n\) for all \(n\). Because of the uniqueness and étale descent for \(\textit{Coh}(Y, \mathcal{J})\) it suffices to prove this étale locally on \(Y\). Thus we may assume \(Y\) is the spectrum of a Noetherian ring. As \(f\) is representable we see that \(X\) is a scheme as well. Thus we reduce to the case of schemes, see proof of Cohomology of Schemes, Lemma 088B.
Theorem
Proof
We will use the equivalence of categories of Cohomology of Spaces, Lemma 08AM without further mention in the proof of the theorem. By Lemma 08B9 the functor is fully faithful. Thus we need to prove the functor is essentially surjective.
Consider the collection \(\Xi\) of quasi-coherent sheaves of ideals \(\mathcal{K} \subset \mathcal{O}_X\) such that the statement holds for every object \((\mathcal{F}_n)\) of \(\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})\) annihilated by \(\mathcal{K}\). We want to show \((0)\) is in \(\Xi\). If not, then since \(X\) is Noetherian there exists a maximal quasi-coherent sheaf of ideals \(\mathcal{K}\) not in \(\Xi\), see Cohomology of Spaces, Lemma 07UJ. After replacing \(X\) by the closed subscheme of \(X\) corresponding to \(\mathcal{K}\) we may assume that every nonzero \(\mathcal{K}\) is in \(\Xi\). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})\). We will show that this object is in the essential image, thereby completing the proof of the theorem.
Apply Chow’s lemma (Lemma 088U) to find a proper surjective morphism \(f : Y \to X\) which is an isomorphism over a dense open \(U \subset X\) such that \(Y\) is H-quasi-projective over \(A\). Note that \(Y\) is a scheme and \(f\) representable. Choose an open immersion \(j : Y \to Y'\) with \(Y'\) projective over \(A\), see Morphisms, Lemma 01WA. Let \(T_n\) be the scheme theoretic support of \(\mathcal{F}_n\). Note that \(|T_n| = |T_1|\), hence \(T_n\) is proper over \(A\) for all \(n\) (Morphisms of Spaces, Lemma 08AJ). Then \(f^*\mathcal{F}_n\) is supported on the closed subscheme \(f^{-1}T_n\) which is proper over \(A\) (by Morphisms of Spaces, Lemma 04XY and properness of \(f\)). In particular, the composition \(f^{-1}T_n \to Y \to Y'\) is closed (Morphisms, Lemma 01W6). Let \(T'_n \subset Y'\) be the corresponding closed subscheme; it is contained in the open subscheme \(Y\) and equal to \(f^{-1}T_n\) as a closed subscheme of \(Y\). Let \(\mathcal{F}_n'\) be the coherent \(\mathcal{O}_{Y'}\)-module corresponding to \(f^*\mathcal{F}_n\) viewed as a coherent module on \(Y'\) via the closed immersion \(f^{-1}T_n = T'_n \subset Y'\). Then \((\mathcal{F}_n')\) is an object of \(\textit{Coh}(Y', I\mathcal{O}_{Y'})\). By the projective case of Grothendieck’s existence theorem (Cohomology of Schemes, Lemma 0885) there exists a coherent \(\mathcal{O}_{Y'}\)-module \(\mathcal{F}'\) and an isomorphism \((\mathcal{F}')^\wedge \cong (\mathcal{F}'_n)\) in \(\textit{Coh}(Y', I\mathcal{O}_{Y'})\). Let \(Z' \subset Y'\) be the scheme theoretic support of \(\mathcal{F}'\). Since \(\mathcal{F}'/I\mathcal{F}' = \mathcal{F}'_1\) we see that \(Z' \cap V(I\mathcal{O}_{Y'}) = T'_1\) set-theoretically. The structure morphism \(p' : Y' \to \Spec(A)\) is proper, hence \(p'(Z' \cap (Y' \setminus Y))\) is closed in \(\Spec(A)\). If nonempty, then it would contain a point of \(V(I)\) as \(I\) is contained in the Jacobson radical of \(A\) (Algebra, Lemma 05GI). But we’ve seen above that \(Z' \cap (p')^{-1}V(I) = T'_1 \subset Y\) hence we conclude that \(Z' \subset Y\). Thus \(\mathcal{F}'|_Y\) is supported on a closed subscheme of \(Y\) proper over \(A\).
Let \(\mathcal{K}\) be the quasi-coherent sheaf of ideals cutting out the reduced complement \(X \setminus U\). By Cohomology of Spaces, Lemma 08AR the \(\mathcal{O}_X\)-module \(\mathcal{H} = f_*\mathcal{F}'\) is coherent and by Lemma 08BD there exists a morphism \(\alpha : (\mathcal{F}_n) \to \mathcal{H}^\wedge\) in the category \(\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})\) whose kernel and cokernel are annihilated by a power of \(\mathcal{K}\). Let \(Z_0 \subset X\) be the scheme theoretic support of \(\mathcal{H}\). It is clear that \(|Z_0| \subset f(|Z'|)\). Hence \(Z_0 \to \Spec(A)\) is proper (Morphisms of Spaces, Lemma 08AJ). Thus \(\mathcal{H}\) is an object of \(\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_X)\). Since each of the sheaves of ideals \(\mathcal{K}^e\) is an element of \(\Xi\) we see that the assumptions of Lemma 08BC are satisfied and we conclude.
Remark
Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Write \(S = \Spec(A)\) and \(S_n = \Spec(A/I^n)\). Let \(X \to S\) be a morphism of algebraic spaces that is separated and of finite type. For \(n \geq 1\) we set \(X_n = X \times_S S_n\). Picture: \[\xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots & X \ar[d] \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots & S }\] In this situation we consider systems \((\mathcal{F}_n, \varphi_n)\) where
\(\mathcal{F}_n\) is a coherent \(\mathcal{O}_{X_n}\)-module,
\(\varphi_n : i_n^*\mathcal{F}_{n + 1} \to \mathcal{F}_n\) is an isomorphism, and
\(\text{Supp}(\mathcal{F}_1)\) is proper over \(S_1\).
Theorem 08BE says that the completion functor \[\begin{matrix} \text{coherent }\mathcal{O}_X\text{-modules }\mathcal{F} \\ \text{with support proper over }A \end{matrix} \quad \longrightarrow \quad \begin{matrix} \text{systems }(\mathcal{F}_n) \\ \text{as above} \end{matrix}\] is an equivalence of categories. In the special case that \(X\) is proper over \(A\) we can omit the conditions on the supports.
Grothendieck’s algebraization theorem
This section is the analogue of Cohomology of Schemes, Section 0898. However, this section is missing the result on algebraization of deformations of proper algebraic spaces endowed with ample invertible sheaves, as a proper algebraic space which comes with an ample invertible sheaf is already a scheme. We do have an algebraization result on proper algebraic spaces of relative dimension \(1\). Our first result is a translation of Grothendieck’s existence theorem in terms of closed subschemes and finite morphisms.
Lemma
Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Write \(S = \Spec(A)\) and \(S_n = \Spec(A/I^n)\). Let \(X \to S\) be a morphism of algebraic spaces that is separated and of finite type. For \(n \geq 1\) we set \(X_n = X \times_S S_n\). Suppose given a commutative diagram \[\xymatrix{ Z_1 \ar[r] \ar[d] & Z_2 \ar[r] \ar[d] & Z_3 \ar[r] \ar[d] & \ldots \\ X_1 \ar[r]^{i_1} & X_2 \ar[r]^{i_2} & X_3 \ar[r] & \ldots }\] of algebraic spaces with cartesian squares. Assume that
\(Z_1 \to X_1\) is a closed immersion, and
\(Z_1 \to S_1\) is proper.
Then there exists a closed immersion of algebraic spaces \(Z \to X\) such that \(Z_n = Z \times_S S_n\) for all \(n \geq 1\). Moreover, \(Z\) is proper over \(S\).
Proof
Let’s write \(j_n : Z_n \to X_n\) for the vertical morphisms. As the squares in the statement are cartesian we see that the base change of \(j_n\) to \(X_1\) is \(j_1\). Thus Limits of Spaces, Lemma 08B2 shows that \(j_n\) is a closed immersion. Set \(\mathcal{F}_n = j_{n, *}\mathcal{O}_{Z_n}\), so that \(j_n^\sharp\) is a surjection \(\mathcal{O}_{X_n} \to \mathcal{F}_n\). Again using that the squares are cartesian we see that the pullback of \(\mathcal{F}_{n + 1}\) to \(X_n\) is \(\mathcal{F}_n\). Hence Grothendieck’s existence theorem, as reformulated in Remark 08BF, tells us there exists a map \(\mathcal{O}_X \to \mathcal{F}\) of coherent \(\mathcal{O}_X\)-modules whose restriction to \(X_n\) recovers \(\mathcal{O}_{X_n} \to \mathcal{F}_n\). Moreover, the support of \(\mathcal{F}\) is proper over \(S\). As the completion functor is exact (Lemma 08B5) we see that \(\mathcal{O}_X \to \mathcal{F}\) is surjective. Thus \(\mathcal{F} = \mathcal{O}_X/\mathcal{J}\) for some quasi-coherent sheaf of ideals \(\mathcal{J}\). Setting \(Z = V(\mathcal{J})\) finishes the proof.
Lemma
Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Write \(S = \Spec(A)\) and \(S_n = \Spec(A/I^n)\). Let \(X \to S\) be a morphism of algebraic spaces that is separated and of finite type. For \(n \geq 1\) we set \(X_n = X \times_S S_n\). Suppose given a commutative diagram \[\xymatrix{ Y_1 \ar[r] \ar[d] & Y_2 \ar[r] \ar[d] & Y_3 \ar[r] \ar[d] & \ldots \\ X_1 \ar[r]^{i_1} & X_2 \ar[r]^{i_2} & X_3 \ar[r] & \ldots }\] of algebraic spaces with cartesian squares. Assume that
\(Y_1 \to X_1\) is a finite morphism, and
\(Y_1 \to S_1\) is proper.
Then there exists a finite morphism of algebraic spaces \(Y \to X\) such that \(Y_n = Y \times_S S_n\) for all \(n \geq 1\). Moreover, \(Y\) is proper over \(S\).
Proof
Let’s write \(f_n : Y_n \to X_n\) for the vertical morphisms. As the squares in the statement are cartesian we see that the base change of \(f_n\) to \(X_1\) is \(f_1\). Thus Lemma 09ZZ shows that \(f_n\) is a finite morphism. Set \(\mathcal{F}_n = f_{n, *}\mathcal{O}_{Y_n}\). Using that the squares are cartesian we see that the pullback of \(\mathcal{F}_{n + 1}\) to \(X_n\) is \(\mathcal{F}_n\). Hence Grothendieck’s existence theorem, as reformulated in Remark 08BF, tells us there exists a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) whose restriction to \(X_n\) recovers \(\mathcal{F}_n\). Moreover, the support of \(\mathcal{F}\) is proper over \(S\). As the completion functor is fully faithful (Theorem 08BE) we see that the multiplication maps \(\mathcal{F}_n \otimes_{\mathcal{O}_{X_n}} \mathcal{F}_n \to \mathcal{F}_n\) fit together to give an algebra structure on \(\mathcal{F}\). Setting \(Y = \underline{\Spec}_X(\mathcal{F})\) finishes the proof.
Lemma
Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Write \(S = \Spec(A)\) and \(S_n = \Spec(A/I^n)\). Let \(X\), \(Y\) be algebraic spaces over \(S\). For \(n \geq 1\) we set \(X_n = X \times_S S_n\) and \(Y_n = Y \times_S S_n\). Suppose given a compatible system of commutative diagrams \[\xymatrix{ & & X_{n + 1} \ar[rd] \ar[rr]_{g_{n + 1}} & & Y_{n + 1} \ar[ld] \\ X_n \ar[rru] \ar[rd] \ar[rr]_{g_n} & & Y_n \ar[rru] \ar[ld] & S_{n + 1} \\ & S_n \ar[rru] }\] Assume that
\(X \to S\) is proper, and
\(Y \to S\) is separated of finite type.
Then there exists a unique morphism of algebraic spaces \(g : X \to Y\) over \(S\) such that \(g_n\) is the base change of \(g\) to \(S_n\).
Proof
The morphisms \((1, g_n) : X_n \to X_n \times_S Y_n\) are closed immersions because \(Y_n \to S_n\) is separated (Morphisms of Spaces, Lemma 03KP). Thus by Lemma 08BG there exists a closed subspace \(Z \subset X \times_S Y\) proper over \(S\) whose base change to \(S_n\) recovers \(X_n \subset X_n \times_S Y_n\). The first projection \(p : Z \to X\) is a proper morphism (as \(Z\) is proper over \(S\), see Morphisms of Spaces, Lemma 04NX) whose base change to \(S_n\) is an isomorphism for all \(n\). In particular, \(p : Z \to X\) is quasi-finite on an open subspace of \(Z\) containing every point of \(Z_0\) for example by Morphisms of Spaces, Lemma 04NW. As \(Z\) is proper over \(S\) this open neighbourhood is all of \(Z\). We conclude that \(p : Z \to X\) is finite by Zariski’s main theorem (for example apply Lemma 082K and use properness of \(Z\) over \(X\) to see that the immersion is a closed immersion). Applying the equivalence of Theorem 08BE we see that \(p_*\mathcal{O}_Z = \mathcal{O}_X\) as this is true modulo \(I^n\) for all \(n\). Hence \(p\) is an isomorphism and we obtain the morphism \(g\) as the composition \(X \cong Z \to Y\). We omit the proof of uniqueness.
Remark
We can ask if in Grothendieck’s algebraization theorem (in the form of Lemma 0A4Z), we can get by with weaker separation axioms on the target. Let us be more precise. Let \(A\), \(I\), \(S\), \(S_n\), \(X\), \(Y\), \(X_n\), \(Y_n\), and \(g_n\) be as in the statement of Lemma 0A4Z and assume that
\(X \to S\) is proper, and
\(Y \to S\) is locally of finite type.
Does there exist a morphism of algebraic spaces \(g : X \to Y\) over \(S\) such that \(g_n\) is the base change of \(g\) to \(S_n\)? We don’t know the answer in general; if you do please email stacks.project@gmail.com. If \(Y \to S\) is separated, then the result holds by the lemma (there is an immediate reduction to the case where \(X\) is finite type over \(S\), by choosing a quasi-compact open containing the image of \(g_1\)). If we only assume \(Y \to S\) is quasi-separated, then the result is true as well. First, as before we may assume \(Y\) is quasi-compact as well as quasi-separated. Then we can use either [Bhatt-Algebraize] or from [Hall-Rydh-coherent] to algebraize \((g_n)\). Namely, to apply the first reference, we use \[D_{perf}(X) \to \lim D_{perf}(X_n) \xrightarrow{\lim Lg_n^*} \lim D_{perf}(Y_n) = D_{perf}(Y)\] where the last step uses a Grothendieck existence result for the derived category of the proper algebraic space \(Y\) over \(R\) (compare with Flatness on Spaces, Remark 0DIR). The paper cited shows that this arrow determines a morphism \(Y \to X\) as desired. To apply the second reference we use the same argument with coherent modules: \[\textit{Coh}(\mathcal{O}_X) \to \lim \textit{Coh}(\mathcal{O}_{X_n}) \xrightarrow{\lim g_n^*} \lim \textit{Coh}(\mathcal{O}_{Y_n}) = \textit{Coh}(\mathcal{O}_Y)\] where the final equality is a consequence of Grothendieck’s existence theorem (Theorem 08BE). The second reference tells us that this functor corresponds to a morphism \(Y \to X\) over \(R\). If we ever need this generalization we will precisely state and carefully prove the result here.
Lemma
Let \((A, \mathfrak m, \kappa)\) be a complete local Noetherian ring. Set \(S = \Spec(A)\) and \(S_n = \Spec(A/\mathfrak m^n)\). Consider a commutative diagram \[\xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots }\] of algebraic spaces with cartesian squares. If \(\dim(X_1) \leq 1\), then there exists a projective morphism of schemes \(X \to S\) and isomorphisms \(X_n \cong X \times_S S_n\) compatible with \(i_n\).
Proof
By Spaces over Fields, Lemma 0ADD the algebraic space \(X_1\) is a scheme. Hence \(X_1\) is a proper scheme of dimension \(\leq 1\) over \(\kappa\). By Varieties, Lemma 0A26 we see that \(X_1\) is H-projective over \(\kappa\). Let \(\mathcal{L}_1\) be an ample invertible sheaf on \(X_1\).
We are going to show that \(\mathcal{L}_1\) lifts to a compatible system \(\{\mathcal{L}_n\}\) of invertible sheaves on \(\{X_n\}\). Observe that \(X_n\) is a scheme too by Lemma 05ZR. Recall that \(X_1 \to X_n\) induces homeomorphisms of underlying topological spaces. In the rest of the proof we do not distinguish between sheaves on \(X_n\) and sheaves on \(X_1\). Suppose, given a lift \(\mathcal{L}_n\) to \(X_n\). We consider the exact sequence \[1 \to (1 + \mathfrak m^n\mathcal{O}_{X_{n + 1}})^* \to \mathcal{O}_{X_{n + 1}}^* \to \mathcal{O}_{X_n}^* \to 1\] of sheaves on \(X_{n + 1}\). The class of \(\mathcal{L}_n\) in \(H^1(X_n, \mathcal{O}_{X_n}^*)\) (see Cohomology, Lemma 09NU) can be lifted to an element of \(H^1(X_{n + 1}, \mathcal{O}_{X_{n + 1}}^*)\) if and only if the obstruction in \(H^2(X_{n + 1}, (1 + \mathfrak m^n\mathcal{O}_{X_{n + 1}})^*)\) is zero. As \(X_1\) is a Noetherian scheme of dimension \(\leq 1\) this cohomology group vanishes (Cohomology, Proposition 02UZ).
By Grothendieck’s algebraization theorem (Cohomology of Schemes, Theorem 089A) we find a projective morphism of schemes \(X \to S = \Spec(A)\) and a compatible system of isomorphisms \(X_n = S_n \times_S X\).
Lemma
Let \((A, \mathfrak m, \kappa)\) be a complete Noetherian local ring. Let \(X\) be an algebraic space over \(\Spec(A)\). If \(X \to \Spec(A)\) is proper and \(\dim(X_\kappa) \leq 1\), then \(X\) is a scheme projective over \(A\).
Proof
Set \(X_n = X \times_{\Spec(A)} \Spec(A/\mathfrak m^n)\). By Lemma 0E7R there exists a projective morphism \(Y \to \Spec(A)\) and compatible isomorphisms \(Y \times_{\Spec(A)} \Spec(A/\mathfrak m^n) \cong X \times_{\Spec(A)} \Spec(A/\mathfrak m^n)\). By Lemma 0A4Z we see that \(X \cong Y\) and the proof is complete.
Regular immersions
This section is the analogue of Divisors, Section 0638 for morphisms of algebraic spaces. The reader is encouraged to read up on regular immersions of schemes in that section first.
In Divisors, Section 0638 we defined four types of regular immersions for morphisms of schemes. Of these only three are (as far as we know) local on the target for the étale topology; as usual plain old regular immersions aren’t. This is why for morphisms of algebraic spaces we cannot actually define regular immersions. (These kinds of annoyances prompted Grothendieck and his school to replace original notion of a regular immersion by a Koszul-regular immersions, see [SGA6, Exposee VII, Definition 1.4].) But we can define Koszul-regular, \(H_1\)-regular, and quasi-regular immersions. Another remark is that since Koszul-regular immersions are not preserved by arbitrary base change, we cannot use the strategy of Morphisms of Spaces, Section 03HA to define them. Similarly, as Koszul-regular immersions are not étale local on the source, we cannot use Morphisms of Spaces, Lemma 03MJ to define them either. We replace this lemma instead by the following.
Lemma
Let \(\mathcal{P}\) be a property of morphisms of schemes which is étale local on the target. Let \(S\) be a scheme. Let \(f : X \to Y\) be a representable morphism of algebraic spaces over \(S\). Consider commutative diagrams \[\xymatrix{ X \times_Y V \ar[d] \ar[r] & V \ar[d] \\ X \ar[r]^f & Y }\] where \(V\) is a scheme and \(V \to Y\) is étale. The following are equivalent
for any diagram as above the projection \(X \times_Y V \to V\) has property \(\mathcal{P}\), and
for some diagram as above with \(V \to Y\) surjective the projection \(X \times_Y V \to V\) has property \(\mathcal{P}\).
If \(X\) and \(Y\) are representable, then this is also equivalent to \(f\) (as a morphism of schemes) having property \(\mathcal{P}\).
Proof
Let us prove the equivalence of (1) and (2). The implication (1) \(\Rightarrow\) (2) is immediate. Assume \[\xymatrix{ X \times_Y V \ar[d] \ar[r] & V \ar[d] \\ X \ar[r]^f & Y } \quad\quad \xymatrix{ X \times_Y V' \ar[d] \ar[r] & V' \ar[d] \\ X \ar[r]^f & Y }\] are two diagrams as in the lemma. Assume \(V \to Y\) is surjective and \(X \times_Y V \to V\) has property \(\mathcal{P}\). To show that (2) implies (1) we have to prove that \(X \times_Y V' \to V'\) has \(\mathcal{P}\). To do this consider the diagram \[\xymatrix{ X \times_Y V \ar[d] & (X \times_Y V) \times_X (X \times_Y V') \ar[l] \ar[d] \ar[r] & X \times_Y V' \ar[d] \\ V & V \times_Y V' \ar[l] \ar[r] & V' }\] By our assumption that \(\mathcal{P}\) is étale local on the source, we see that \(\mathcal{P}\) is preserved under étale base change, see Descent, Lemma 04QU. Hence if the left vertical arrow has \(\mathcal{P}\) the so does the middle vertical arrow. Since \(U \times_X U' \to U'\) is surjective and étale (hence defines an étale covering of \(U'\)) this implies (as \(\mathcal{P}\) is assumed local for the étale topology on the target) that the left vertical arrow has \(\mathcal{P}\).
If \(X\) and \(Y\) are representable, then we can take \(\text{id}_Y : Y \to Y\) as our étale covering to see the final statement of the lemma is true.
Note that “being a Koszul-regular (resp. \(H_1\)-regular, resp. quasi-regular) immersion” is a property of morphisms of schemes which is fpqc local on the target, see Descent, Lemma 0694. Hence the following definition now makes sense.
Definition
Let \(S\) be a scheme. Let \(i : X \to Y\) be a morphism of algebraic spaces over \(S\).
We say \(i\) is a Koszul-regular immersion if \(i\) is representable and the equivalent conditions of Lemma 06BM hold with \(\mathcal{P}(f) =\)“\(f\) is a Koszul-regular immersion”.
We say \(i\) is an \(H_1\)-regular immersion if \(i\) is representable and the equivalent conditions of Lemma 06BM hold with \(\mathcal{P}(f) =\)“\(f\) is an \(H_1\)-regular immersion”.
We say \(i\) is a quasi-regular immersion if \(i\) is representable and the equivalent conditions of Lemma 06BM hold with \(\mathcal{P}(f) =\)“\(f\) is a quasi-regular immersion”.
Lemma
Let \(S\) be a scheme. Let \(i : Z \to X\) be an immersion of algebraic spaces over \(S\). We have the following implications: \(i\) is Koszul-regular \(\Rightarrow\) \(i\) is \(H_1\)-regular \(\Rightarrow\) \(i\) is quasi-regular.
Proof
Via the definition this lemma immediately reduces to Divisors, Lemma 063K.
Lemma
Let \(S\) be a scheme. Let \(i : Z \to X\) be an immersion of algebraic spaces over \(S\). Assume \(X\) is locally Noetherian. Then \(i\) is Koszul-regular \(\Leftrightarrow\) \(i\) is \(H_1\)-regular \(\Leftrightarrow\) \(i\) is quasi-regular.
Proof
Via Definition 06BN (and the definition of a locally Noetherian algebraic space in Properties of Spaces, Section 03E5) this immediately translates to the case of schemes which is Divisors, Lemma 063L.
Lemma
Let \(S\) be a scheme. Let \(i : Z \to X\) be a Koszul-regular, \(H_1\)-regular, or quasi-regular immersion of algebraic spaces over \(S\). Let \(X' \to X\) be a flat morphism of algebraic spaces over \(S\). Then the base change \(i' : Z \times_X X' \to X'\) is a Koszul-regular, \(H_1\)-regular, or quasi-regular immersion.
Proof
Via Definition 06BN (and the definition of a flat morphism of algebraic spaces in Morphisms of Spaces, Section 03MK) this lemma reduces to the case of schemes, see Divisors, Lemma 067P.
Lemma
Let \(S\) be a scheme. Let \(i : Z \to X\) be an immersion of algebraic spaces over \(S\). Then \(i\) is a quasi-regular immersion if and only if the following conditions are satisfied
\(i\) is locally of finite presentation,
the conormal sheaf \(\mathcal{C}_{Z/X}\) is finite locally free, and
the map (09RQ) is an isomorphism.
Proof
Follows from the case of schemes (Divisors, Lemma 063M) via étale localization (use Definition 06BN and Lemma 09RR).
Lemma
Let \(S\) be a scheme. Let \(Z \to Y \to X\) be immersions of algebraic spaces over \(S\). Assume that \(Z \to Y\) is \(H_1\)-regular. Then the canonical sequence of Lemma 06BD \[0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0\] is exact and (étale) locally split.
Proof
Since \(\mathcal{C}_{Z/Y}\) is finite locally free (see Lemma 09RY and Lemma 06BP) it suffices to prove that the sequence is exact. It suffices to show that the first map is injective as the sequence is already right exact in general. After étale localization on \(X\) this reduces to the case of schemes, see Divisors, Lemma 063N.
A composition of quasi-regular immersions may not be quasi-regular, see Algebra, Remark 065M. The other types of regular immersions are preserved under composition.
Lemma
Let \(S\) be a scheme. Let \(i : Z \to Y\) and \(j : Y \to X\) be immersions of algebraic spaces over \(S\).
If \(i\) and \(j\) are Koszul-regular immersions, so is \(j \circ i\).
If \(i\) and \(j\) are \(H_1\)-regular immersions, so is \(j \circ i\).
If \(i\) is an \(H_1\)-regular immersion and \(j\) is a quasi-regular immersion, then \(j \circ i\) is a quasi-regular immersion.
Proof
Immediate from the case of schemes, see Divisors, Lemma 067Q.
Lemma
Let \(S\) be a scheme. Let \(i : Z \to Y\) and \(j : Y \to X\) be immersions of algebraic spaces over \(S\). Assume \(j\) is locally of finite presentation and that the sequence \[0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0\] of Lemma 06BD is exact and locally split.
If \(j \circ i\) is a quasi-regular immersion, so is \(i\).
If \(j \circ i\) is a \(H_1\)-regular immersion, so is \(i\).
If both \(j\) and \(j \circ i\) are Koszul-regular immersions, so is \(i\).
Proof
Immediate from the case of schemes, see Divisors, Lemma 068Z.
Lemma
Let \(S\) be a scheme. Let \(i : Z \to Y\) and \(j : Y \to X\) be immersions of algebraic spaces over \(S\). Assume \(X\) is locally Noetherian. The following are equivalent
\(i\) and \(j\) are Koszul regular immersions,
\(i\) and \(j \circ i\) are Koszul regular immersions,
\(j \circ i\) is a Koszul regular immersion and the conormal sequence \[0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0\] is exact and locally split.
Proof
Immediate from the case of schemes, see Divisors, Lemma 0690.
Relative pseudo-coherence
This section is the analogue of More on Morphisms, Section 09UH. However, in the treatment of this material for algebraic spaces we have decided to work exclusively with objects in the derived category whose cohomology sheaves are quasi-coherent. There are two reasons for this: (1) it greatly simplifies the exposition and (2) we currently have no use for the more general notion.
Remark
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of representable algebraic spaces over \(S\) which is locally of finite type. Let \(f_0 : X_0 \to Y_0\) be a morphism of schemes representing \(f\) (awkward but temporary notation). Then \(f_0\) is locally of finite type. If \(E\) is an object of \(D_\QCoh(\mathcal{O}_X)\), then \(E\) is the pullback of a unique object \(E_0\) in \(D_\QCoh(\mathcal{O}_{X_0})\), see Derived Categories of Spaces, Lemma 071Q. In this situation the phrase “\(E\) is \(m\)-pseudo-coherent relative to \(Y\)” will be taken to mean “\(E_0\) is \(m\)-pseudo-coherent relative to \(Y_0\)” as defined in More on Morphisms, Section 09UH.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite type. Let \(m \in \mathbf{Z}\). Let \(E \in D_\QCoh(\mathcal{O}_X)\). With notation as explained in Remark 0CSW the following are equivalent:
for every commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\), \(V\) are schemes and the vertical arrows are étale, the complex \(E|_U\) is \(m\)-pseudo-coherent relative to \(V\),
for some commutative diagram as in (1) with \(U \to X\) surjective, the complex \(E|_U\) is \(m\)-pseudo-coherent relative to \(V\),
for every commutative diagram as in (1) with \(U\) and \(V\) affine the complex \(R\Gamma(U, E)\) of \(\mathcal{O}_X(U)\)-modules is \(m\)-pseudo-coherent relative to \(\mathcal{O}_Y(V)\).
Proof
Part (1) implies (3) by More on Morphisms, Lemma 09VF.
Assume (3). Pick any commutative diagram as in (1) with \(U \to X\) surjective. Choose an affine open covering \(V = \bigcup V_j\) and affine open coverings \((U \to V)^{-1}(V_j) = \bigcup U_{ij}\). By (3) and More on Morphisms, Lemma 09VF we see that \(E|_U\) is \(m\)-pseudo-coherent relative to \(V\). Thus (3) implies (2).
Assume (2). Choose a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\), \(V\) are schemes, the vertical arrows are étale, the morphism \(U \to X\) is surjective, and \(E|_U\) is \(m\)-pseudo-coherent relative to \(V\). Next, suppose given a second commutative diagram \[\xymatrix{ U' \ar[d] \ar[r] & V' \ar[d] \\ X \ar[r] & Y }\] with étale vertical arrows and \(U', V'\) schemes. We want to show that \(E|_{U'}\) is \(m\)-pseudo-coherent relative to \(V'\). The morphism \(U'' = U \times_X U' \to U'\) is surjective étale and \(U'' \to V'\) factors through \(V'' = V' \times_Y V\) which is étale over \(V'\). Hence it suffices to show that \(E|_{U''}\) is \(m\)-pseudo-coherent relative to \(V''\), see More on Morphisms, Lemmas 0CSN and 0CSP. Using the second lemma once more it suffices to show that \(E|_{U''}\) is \(m\)-pseudo-coherent relative to \(V\). This is true by More on Morphisms, Lemma 09US and the fact that an étale morphism of schemes is pseudo-coherent by More on Morphisms, Lemma 0695.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite type. Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Fix \(m \in \mathbf{Z}\).
We say \(E\) is \(m\)-pseudo-coherent relative to \(Y\) if the equivalent conditions of Lemma 0CSX are satisfied.
We say \(E\) is pseudo-coherent relative to \(Y\) if \(E\) is \(m\)-pseudo-coherent relative to \(Y\) for all \(m \in \mathbf{Z}\).
We say \(\mathcal{F}\) is \(m\)-pseudo-coherent relative to \(Y\) if \(\mathcal{F}\) viewed as an object of \(D_\QCoh(\mathcal{O}_X)\) is \(m\)-pseudo-coherent relative to \(Y\).
We say \(\mathcal{F}\) is pseudo-coherent relative to \(Y\) if \(\mathcal{F}\) viewed as an object of \(D_\QCoh(\mathcal{O}_X)\) is pseudo-coherent relative to \(Y\).
Most of the properties of pseudo-coherent complexes relative to a base will follow immediately from the corresponding properties in the case of schemes. We will add the relevant lemmas here as needed.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(E\) in \(D_\QCoh(\mathcal{O}_X)\). If \(f\) is flat and locally of finite presentation, then the following are equivalent
\(E\) is pseudo-coherent relative to \(Y\), and
\(E\) is pseudo-coherent on \(X\).
Proof
By étale localization and the definitions we may assume \(X\) and \(Y\) are schemes. For the case of schemes this follows from More on Morphisms, Lemma 09UU.
Pseudo-coherent morphisms
This section is the analogue of More on Morphisms, Section 067X for morphisms of schemes. The reader is encouraged to read up on pseudo-coherent morphisms of schemes in that section first.
The property “pseudo-coherent” of morphisms of schemes is étale local on the source-and-target. To see this use More on Morphisms, Lemmas 0696 and 0699 and Descent, Lemma 04R2. By Morphisms of Spaces, Lemma 03MJ we may define the notion of a pseudo-coherent morphism of algebraic spaces as follows and it agrees with the already existing notion defined in More on Morphisms, Section 067X when the algebraic spaces in question are representable.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\).
We say \(f\) is pseudo-coherent if the equivalent conditions of Morphisms of Spaces, Lemma 03MJ hold with \(\mathcal{P} =\)“pseudo-coherent”.
Let \(x \in |X|\). We say \(f\) is pseudo-coherent at \(x\) if there exists an open neighbourhood \(X' \subset X\) of \(x\) such that \(f|_{X'} : X' \to Y\) is pseudo-coherent.
Beware that a base change of a pseudo-coherent morphism is not pseudo-coherent in general.
Lemma
A flat base change of a pseudo-coherent morphism is pseudo-coherent.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 0680.
Lemma
A composition of pseudo-coherent morphisms is pseudo-coherent.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 0681.
Lemma
A pseudo-coherent morphism is locally of finite presentation.
Proof
Immediate from the definitions.
Lemma
A flat morphism which is locally of finite presentation is pseudo-coherent.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 0695.
Lemma
Let \(f : X \to Y\) be a morphism of algebraic spaces pseudo-coherent over a base algebraic space \(B\). Then \(f\) is pseudo-coherent.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 0683.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If \(Y\) is locally Noetherian, then \(f\) is pseudo-coherent if and only if \(f\) is locally of finite type.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 0684.
Perfect morphisms
This section is the analogue of More on Morphisms, Section 0685 for morphisms of schemes. The reader is encouraged to read up on perfect morphisms of schemes in that section first.
The property “perfect” of morphisms of schemes is étale local on the source-and-target. To see this use More on Morphisms, Lemmas 069B and 069D and Descent, Lemma 04R2. By Morphisms of Spaces, Lemma 03MJ we may define the notion of a perfect morphism of algebraic spaces as follows and it agrees with the already existing notion defined in More on Morphisms, Section 0685 when the algebraic spaces in question are representable.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\).
We say \(f\) is perfect if the equivalent conditions of Morphisms of Spaces, Lemma 03MJ hold with \(\mathcal{P} =\)“perfect”.
Let \(x \in |X|\). We say \(f\) is perfect at \(x\) if there exists an open neighbourhood \(X' \subset X\) of \(x\) such that \(f|_{X'} : X' \to Y\) is perfect.
Note that a perfect morphism is pseudo-coherent, hence locally of finite presentation. Beware that a base change of a perfect morphism is not perfect in general.
Lemma
A flat base change of a perfect morphism is perfect.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 0688.
Lemma
A composition of perfect morphisms is perfect.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 0689.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent
\(f\) is flat and perfect, and
\(f\) is flat and locally of finite presentation.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 068A.
Lemma
Let \(S\) be a scheme. Let \(Y\) be a Noetherian algebraic space over \(S\). Let \(f : X \to Y\) be a perfect proper morphism of algebraic spaces. Let \(E \in D(\mathcal{O}_X)\) be perfect. Then \(Rf_*E\) is a perfect object of \(D(\mathcal{O}_Y)\).
Proof
We claim that Derived Categories of Spaces, Lemma 08IS applies. Conditions (1) and (2) are immediate. Condition (3) is local on \(X\). Thus we may assume \(X\) and \(Y\) affine and \(E\) represented by a strictly perfect complex of \(\mathcal{O}_X\)-modules. Thus it suffices to show that \(\mathcal{O}_X\) has finite tor dimension as a sheaf of \(f^{-1}\mathcal{O}_Y\)-modules on the étale site. By Derived Categories of Spaces, Lemma 0DK7 it suffices to check this on the Zariski site. This is equivalent to being perfect for finite type morphisms of schemes by More on Morphisms, Lemma 069C.
Local complete intersection morphisms
This section is the analogue of More on Morphisms, Section 068E for morphisms of schemes. The reader is encouraged to read up on local complete intersection morphisms of schemes in that section first.
The property “being a local complete intersection morphism” of morphisms of schemes is étale local on the source-and-target. To see this use More on Morphisms, Lemmas 069N and 069P and Descent, Lemma 04R2. By Morphisms of Spaces, Lemma 03MJ we may define the notion of a local complete intersection morphism of algebraic spaces as follows and it agrees with the already existing notion defined in More on Morphisms, Section 068E when the algebraic spaces in question are representable.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\).
We say \(f\) is a Koszul morphism, or that \(f\) is a local complete intersection morphism if the equivalent conditions of Morphisms of Spaces, Lemma 03MJ hold with \(\mathcal{P}(f) =\)“\(f\) is a local complete intersection morphism”.
Let \(x \in |X|\). We say \(f\) is Koszul at \(x\) if there exists an open neighbourhood \(X' \subset X\) of \(x\) such that \(f|_{X'} : X' \to Y\) is a local complete intersection morphism.
In some sense the defining property of a local complete intersection morphism is the result of the following lemma.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a local complete intersection morphism of algebraic spaces over \(S\). Let \(P\) be an algebraic space smooth over \(Y\). Let \(U \to X\) be an étale morphism of algebraic spaces and let \(i : U \to P\) an immersion of algebraic spaces over \(Y\). Picture: \[\xymatrix{ X \ar[rd] & U \ar[l] \ar[d] \ar[r]_i & P \ar[ld] \\ & Y }\] Then \(i\) is a Koszul-regular immersion of algebraic spaces.
Proof
Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(W\) and a surjective étale morphism \(W \to P \times_Y V\). Set \(U' = U \times_P W\), which is a scheme étale over \(U\). We have to show that \(U' \to W\) is a Koszul-regular immersion of schemes, see Definition 06BN. By Definition 06C4 above the morphism of schemes \(U' \to V\) is a local complete intersection morphism. Hence the result follows from More on Morphisms, Lemma 069G.
It seems like a good idea to collect here some properties in common with all Koszul morphisms.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a local complete intersection morphism of algebraic spaces over \(S\). Then
\(f\) is locally of finite presentation,
\(f\) is pseudo-coherent, and
\(f\) is perfect.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 069H.
Beware that a base change of a Koszul morphism is not Koszul in general.
Lemma
A flat base change of a local complete intersection morphism is a local complete intersection morphism.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 069I.
Lemma
A composition of local complete intersection morphisms is a local complete intersection morphism.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 069J.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent
\(f\) is flat and a local complete intersection morphism, and
\(f\) is syntomic.
Proof
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 069K.
Lemma
Let \(S\) be a scheme. A Koszul-regular immersion of algebraic spaces over \(S\) is a local complete intersection morphism.
Proof
Let \(i : X \to Y\) be a Koszul-regular immersion of algebraic spaces over \(S\). By definition there exists a surjective étale morphism \(V \to Y\) where \(V\) is a scheme such that \(X \times_Y V\) is a scheme and the base change \(X \times_Y V \to V\) is a Koszul-regular immersion of schemes. By More on Morphisms, Lemma 069L we see that \(X \times_Y V \to V\) is a local complete intersection morphism. From Definition 06C4 we conclude that \(i\) is a local complete intersection morphism of algebraic spaces.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & Z }\] be a commutative diagram of morphisms of algebraic spaces over \(S\). Assume \(Y \to Z\) is smooth and \(X \to Z\) is a local complete intersection morphism. Then \(f : X \to Y\) is a local complete intersection morphism.
Proof
Choose a scheme \(W\) and a surjective étale morphism \(W \to Z\). Choose a scheme \(V\) and a surjective étale morphism \(V \to W \times_Z Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to V \times_Y X\). Then \(U \to W\) is a local complete intersection morphism of schemes and \(V \to W\) is a smooth morphism of schemes. By the result for schemes (More on Morphisms, Lemma 069M) we conclude that \(U \to V\) is a local complete intersection morphism. By definition this means that \(f\) is a local complete intersection morphism.
Lemma
The property \(\mathcal{P}(f) =\)“\(f\) is a local complete intersection morphism” is fpqc local on the base.
Proof
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\{Y_i \to Y\}\) be an fpqc covering (Topologies on Spaces, Definition 03MQ). Let \(f_i : X_i \to Y_i\) be the base change of \(f\) by \(Y_i \to Y\). If \(f\) is a local complete intersection morphism, then each \(f_i\) is a local complete intersection morphism by Lemma 06C7.
Conversely, assume each \(f_i\) is a local complete intersection morphism. We may replace the covering by a refinement (again because flat base change preserves the property of being a local complete intersection morphism). Hence we may assume \(Y_i\) is a scheme for each \(i\), see Topologies on Spaces, Lemma 0419. Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to V \times_Y X\). We have to show that \(U \to V\) is a local complete intersection morphism of schemes. By Topologies on Spaces, Lemma 03MS we have that \(\{Y_i \times_Y V \to V\}\) is an fpqc covering of schemes. By the case of schemes (More on Morphisms, Lemma 069N) it suffices to prove the base change \[U \times_Y Y_i = U \times_V (V \times_Y Y_i) \longrightarrow V\] of \(U \to V\) by \(V \times_Y Y_i \to V\) is a local complete intersection morphism. We can write this as the composition \[U \times_Y Y_i \longrightarrow (V \times_Y X) \times_Y Y_i = V \times_Y X_i \longrightarrow V \times_Y Y_i\] The first arrow is an étale morphism of schemes (as a base change of \(U \to V \times_Y X\)) and the second arrow is a local complete intersection morphism of schemes as a flat base change of \(f_i\). The result follows as being a local complete intersection morphism is syntomic local on the source and since étale morphisms are syntomic (More on Morphisms, Lemma 069P and Morphisms, Lemma 02GQ).
Lemma
The property \(\mathcal{P}(f) =\)“\(f\) is a local complete intersection morphism” is syntomic local on the source.
Proof
This follows from Descent on Spaces, Lemma 06ER and More on Morphisms, Lemma 069P.
Lemma
Let \(S\) be a scheme. Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z }\] of algebraic spaces over \(S\). Assume that both \(p\) and \(q\) are flat and locally of finite presentation. Then there exists an open subspace \(U(f) \subset X\) such that \(|U(f)| \subset |X|\) is the set of points where \(f\) is Koszul. Moreover, for any morphism of algebraic spaces \(Z' \to Z\), if \(f' : X' \to Y'\) is the base change of \(f\) by \(Z' \to Z\), then \(U(f')\) is the inverse image of \(U(f)\) under the projection \(X' \to X\).
Proof
This lemma is the analogue of More on Morphisms, Lemma 06B8 and in fact we will deduce the lemma from it. By Definition 06C4 the set \(\{x \in |X| : f \text{ is Koszul at }x\}\) is open in \(|X|\) hence by Properties of Spaces, Lemma 03BZ it corresponds to an open subspace \(U(f)\) of \(X\). Hence we only need to prove the final statement.
Choose a scheme \(W\) and a surjective étale morphism \(W \to Z\). Choose a scheme \(V\) and a surjective étale morphism \(V \to W \times_Z Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to V \times_Y X\). Finally, choose a scheme \(W'\) and a surjective étale morphism \(W' \to W \times_Z Z'\). Set \(V' = W' \times_W V\) and \(U' = W' \times_W U\), so that we obtain surjective étale morphisms \(V' \to Y'\) and \(U' \to X'\). We will use without further mention an étale morphism of algebraic spaces induces an open map of associated topological spaces (see Properties of Spaces, Lemma 03IR). Note that by definition \(U(f)\) is the image in \(|X|\) of the set \(T\) of points in \(U\) where the morphism of schemes \(U \to V\) is Koszul. Similarly, \(U(f')\) is the image in \(|X'|\) of the set \(T'\) of points in \(U'\) where the morphism of schemes \(U' \to V'\) is Koszul. Now, by construction the diagram \[\xymatrix{ U' \ar[r] \ar[d] & U \ar[d] \\ V' \ar[r] & V }\] is cartesian (in the category of schemes). Hence the aforementioned More on Morphisms, Lemma 06B8 applies to show that \(T'\) is the inverse image of \(T\). Since \(|U'| \to |X'|\) is surjective this implies the lemma.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a local complete intersection morphism of algebraic spaces over \(S\). Then \(f\) is unramified if and only if \(f\) is formally unramified and in this case the conormal sheaf \(\mathcal{C}_{X/Y}\) is finite locally free on \(X\).
Proof
This follows from the corresponding result for morphisms of schemes, see More on Morphisms, Lemma 06B9, by étale localization, see Lemma 060D. (Note that in the situation of this lemma the morphism \(V \to U\) is unramified and a local complete intersection morphism by definition.)
Lemma
Let \(S\) be a scheme. Let \(Z \to Y \to X\) be formally unramified morphisms of algebraic spaces over \(S\). Assume that \(Z \to Y\) is a local complete intersection morphism. The exact sequence \[0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0\] of Lemma 06BD is short exact.
Proof
Choose a scheme \(U\) and a surjective étale morphism \(U \to X\). Choose a scheme \(V\) and a surjective étale morphism \(V \to U \times_X Y\). Choose a scheme \(W\) and a surjective étale morphism \(W \to V \times_Y Z\). By Lemma 060D the morphisms \(W \to V\) and \(V \to U\) are formally unramified. Moreover the sequence \(i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0\) restricts to the corresponding sequence \(i^*\mathcal{C}_{V/U} \to \mathcal{C}_{W/U} \to \mathcal{C}_{W/V} \to 0\) for \(W \to V \to U\). Hence the result follows from the result for schemes (More on Morphisms, Lemma 06BA) as by definition the morphism \(W \to V\) is a local complete intersection morphism.
When is a morphism an isomorphism?
More generally we can ask: “When does a morphism have property \(\mathcal{P}\)?” A more precise question is the following. Suppose given a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z }\] of algebraic spaces. Does there exist a monomorphism of algebraic spaces \(W \to Z\) with the following two properties:
the base change \(f_W : X_W \to Y_W\) has property \(\mathcal{P}\), and
any morphism \(Z' \to Z\) of algebraic spaces factors through \(W\) if and only if the base change \(f_{Z'} : X_{Z'} \to Y_{Z'}\) has property \(\mathcal{P}\).
In many cases, if \(W \to Z\) exists, then it is an immersion, open immersion, or closed immersion.
The answer to this question may depend on auxiliary properties of the morphisms \(f\), \(p\), and \(q\). An example is \(\mathcal{P}(f) =\)“\(f\) is flat” which we have discussed for morphisms of schemes in the case \(Y = S\) in great detail in the chapter “More on Flatness”, starting with More on Flatness, Section 05MG.
Lemma
Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z }\] of algebraic spaces. Assume that \(p\) is locally of finite type and closed. Then there exists an open subspace \(W \subset Z\) such that a morphism \(Z' \to Z\) factors through \(W\) if and only if the base change \(f_{Z'} : X_{Z'} \to Y_{Z'}\) is unramified.
Proof
By Morphisms of Spaces, Lemma 05W2 there exists an open subspace \(U(f) \subset X\) which is the set of points where \(f\) is unramified. Moreover, formation of \(U(f)\) commutes with arbitrary base change. Let \(W \subset Z\) be the open subspace (see Properties of Spaces, Lemma 03BZ) with underlying set of points \[|W| = |Z| \setminus |p|\left(|X| \setminus |U(f)|\right)\] i.e., \(z \in |Z|\) is a point of \(W\) if and only if \(f\) is unramified at every point of \(X\) above \(z\). Note that this is open because we assumed that \(p\) is closed. Since the formation of \(U(f)\) commutes with arbitrary base change we immediately see (using Properties of Spaces, Lemma 03IE) that \(W\) has the desired universal property.
Lemma
Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z }\] of algebraic spaces. Assume that
\(p\) is locally of finite type,
\(p\) is closed, and
\(p_2 : X \times_Y X \to Z\) is closed.
Then there exists an open subspace \(W \subset Z\) such that a morphism \(Z' \to Z\) factors through \(W\) if and only if the base change \(f_{Z'} : X_{Z'} \to Y_{Z'}\) is unramified and universally injective.
Proof
After replacing \(Z\) by the open subspace found in Lemma 05X8 we may assume that \(f\) is already unramified; note that this does not destroy assumption (2) or (3). By Morphisms of Spaces, Lemma 05W1 we see that \(\Delta_{X/Y} : X \to X \times_Y X\) is an open immersion. This remains true after any base change. Hence by Morphisms of Spaces, Lemma 040X we see that \(f_{Z'}\) is universally injective if and only if the base change of the diagonal \(X_{Z'} \to (X \times_Y X)_{Z'}\) is an isomorphism. Let \(W \subset Z\) be the open subspace (see Properties of Spaces, Lemma 03BZ) with underlying set of points \[|W| = |Z| \setminus |p_2|\left(|X \times_Y X| \setminus \Im(|\Delta_{X/Y}|)\right)\] i.e., \(z \in |Z|\) is a point of \(W\) if and only if the fibre of \(|X \times_Y X| \to |Z|\) over \(z\) is in the image of \(|X| \to |X \times_Y X|\). Then it is clear from the discussion above that the restriction \(p^{-1}(W) \to q^{-1}(W)\) of \(f\) is unramified and universally injective.
Conversely, suppose that \(f_{Z'}\) is unramified and universally injective. In order to show that \(Z' \to Z\) factors through \(W\) it suffices to show that \(|Z'| \to |Z|\) has image contained in \(|W|\), see Properties of Spaces, Lemma 03IE. Hence it suffices to prove the result when \(Z'\) is the spectrum of a field. Denote \(z \in |Z|\) the image of \(|Z'| \to |Z|\). The discussion above shows that \[|X_{Z'}| \longrightarrow |(X \times_Y X)_{Z'}|\] is surjective. By Properties of Spaces, Lemma 03H4 in the commutative diagram \[\xymatrix{ |X_{Z'}| \ar[d] \ar[r] & |(X \times_Y X)_{Z'}| \ar[d] \\ |p|^{-1}(\{z\}) \ar[r] & |p_2|^{-1}(\{z\}) }\] the vertical arrows are surjective. It follows that \(z \in |W|\) as desired.
Lemma
Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z }\] of algebraic spaces. Assume that
\(p\) is locally of finite type,
\(p\) is universally closed, and
\(q : Y \to Z\) is separated.
Then there exists an open subspace \(W \subset Z\) such that a morphism \(Z' \to Z\) factors through \(W\) if and only if the base change \(f_{Z'} : X_{Z'} \to Y_{Z'}\) is a closed immersion.
Proof
We will use the characterization of closed immersions as universally closed, unramified, and universally injective morphisms, see Lemma 05W8. First, note that since \(p\) is universally closed and \(q\) is separated, we see that \(f\) is universally closed, see Morphisms of Spaces, Lemma 04NX. It follows that any base change of \(f\) is universally closed, see Morphisms of Spaces, Lemma 03IS. Thus to finish the proof of the lemma it suffices to prove that the assumptions of Lemma 05X9 are satisfied. The projection \(\text{pr}_0 : X \times_Y X \to X\) is universally closed as a base change of \(f\), see Morphisms of Spaces, Lemma 03IS. Hence \(X \times_Y X \to Z\) is universally closed as a composition of universally closed morphisms (see Morphisms of Spaces, Lemma 03IU). This finishes the proof of the lemma.
Lemma
Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z }\] of algebraic spaces. Assume that
\(p\) is locally of finite presentation,
\(p\) is flat,
\(p\) is closed, and
\(q\) is locally of finite type.
Then there exists an open subspace \(W \subset Z\) such that a morphism \(Z' \to Z\) factors through \(W\) if and only if the base change \(f_{Z'} : X_{Z'} \to Y_{Z'}\) is flat.
Proof
By Lemma 05X3 the set \[A = \{x \in |X| : X\text{ flat at }x \text{ over }Y\}.\] is open in \(|X|\) and its formation commutes with arbitrary base change. Let \(W \subset Z\) be the open subspace (see Properties of Spaces, Lemma 03BZ) with underlying set of points \[|W| = |Z| \setminus |p|\left(|X| \setminus A\right)\] i.e., \(z \in |Z|\) is a point of \(W\) if and only if the whole fibre of \(|X| \to |Z|\) over \(z\) is contained in \(A\). This is open because \(p\) is closed. Since the formation of \(A\) commutes with arbitrary base change it follows that \(W\) works.
Lemma
Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z }\] of algebraic spaces. Assume that
\(p\) is locally of finite presentation,
\(p\) is flat,
\(p\) is closed,
\(q\) is locally of finite type, and
\(q\) is closed.
Then there exists an open subspace \(W \subset Z\) such that a morphism \(Z' \to Z\) factors through \(W\) if and only if the base change \(f_{Z'} : X_{Z'} \to Y_{Z'}\) is surjective and flat.
Proof
By Lemma 05XB we may assume that \(f\) is flat. Note that \(f\) is locally of finite presentation by Morphisms of Spaces, Lemma 05WT. Hence \(f\) is open, see Morphisms of Spaces, Lemma 042S. Let \(W \subset Z\) be the open subspace (see Properties of Spaces, Lemma 03BZ) with underlying set of points \[|W| = |Z| \setminus |q|\left(|Y| \setminus |f|(|X|)\right).\] in other words for \(z \in |Z|\) we have \(z \in |W|\) if and only if the whole fibre of \(|Y| \to |Z|\) over \(z\) is in the image of \(|X| \to |Y|\). Since \(q\) is closed this set is open in \(|Z|\). The morphism \(X_W \to Y_W\) is surjective by construction. Finally, suppose that \(X_{Z'} \to Y_{Z'}\) is surjective. In order to show that \(Z' \to Z\) factors through \(W\) it suffices to show that \(|Z'| \to |Z|\) has image contained in \(|W|\), see Properties of Spaces, Lemma 03IE. Hence it suffices to prove the result when \(Z'\) is the spectrum of a field. Denote \(z \in |Z|\) the image of \(|Z'| \to |Z|\). By Properties of Spaces, Lemma 03H4 in the commutative diagram \[\xymatrix{ |X_{Z'}| \ar[d] \ar[r] & |Y_{Z'}| \ar[d] \\ |p|^{-1}(\{z\}) \ar[r] & |q|^{-1}(\{z\}) }\] the vertical arrows are surjective. It follows that \(z \in |W|\) as desired.
Lemma
Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z }\] of algebraic spaces. Assume that
\(p\) is locally of finite presentation,
\(p\) is flat,
\(p\) is universally closed,
\(q\) is locally of finite type,
\(q\) is closed, and
\(q\) is separated.
Then there exists an open subspace \(W \subset Z\) such that a morphism \(Z' \to Z\) factors through \(W\) if and only if the base change \(f_{Z'} : X_{Z'} \to Y_{Z'}\) is an isomorphism.
Proof
By Lemma 05XC there exists an open subspace \(W_1 \subset Z\) such that \(f_{Z'}\) is surjective and flat if and only if \(Z' \to Z\) factors through \(W_1\). By Lemma 05XA there exists an open subspace \(W_2 \subset Z\) such that \(f_{Z'}\) is a closed immersion if and only if \(Z' \to Z\) factors through \(W_2\). We claim that \(W = W_1 \cap W_2\) works. Certainly, if \(f_{Z'}\) is an isomorphism, then \(Z' \to Z\) factors through \(W\). Hence it suffices to show that \(f_W\) is an isomorphism. By construction \(f_W\) is a surjective flat closed immersion. In particular \(f_W\) is representable. Since a surjective flat closed immersion of schemes is an isomorphism (see Morphisms, Lemma 04PW) we win. (Note that actually \(f_W\) is locally of finite presentation, whence open, so you can avoid the use of this lemma if you like.)
Lemma
Consider a commutative diagram \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z }\] of algebraic spaces. Assume that
\(p\) is flat and locally of finite presentation,
\(p\) is closed, and
\(q\) is flat and locally of finite presentation,
Then there exists an open subspace \(W \subset Z\) such that a morphism \(Z' \to Z\) factors through \(W\) if and only if the base change \(f_{Z'} : X_{Z'} \to Y_{Z'}\) is a local complete intersection morphism.
Proof
By Lemma 06CA there exists an open subspace \(U(f) \subset X\) which is the set of points where \(f\) is Koszul. Moreover, formation of \(U(f)\) commutes with arbitrary base change. Let \(W \subset Z\) be the open subspace (see Properties of Spaces, Lemma 03BZ) with underlying set of points \[|W| = |Z| \setminus |p|\left(|X| \setminus |U(f)|\right)\] i.e., \(z \in |Z|\) is a point of \(W\) if and only if \(f\) is Koszul at every point of \(X\) above \(z\). Note that this is open because we assumed that \(p\) is closed. Since the formation of \(U(f)\) commutes with arbitrary base change we immediately see (using Properties of Spaces, Lemma 03IE) that \(W\) has the desired universal property.
Exact sequences of differentials and conormal sheaves
In this section we collect some results on exact sequences of conormal sheaves and sheaves of differentials. In some sense these are all realizations of the triangle of cotangent complexes associated to composable morphisms of algebraic spaces.
In the sequences below each of the maps are as constructed in either Lemma 04CX or Lemma 060A. Let \(S\) be a scheme. Let \(g : Z \to Y\) and \(f : Y \to X\) be morphisms of algebraic spaces over \(S\).
There is a canonical exact sequence \[g^*\Omega_{Y/X} \to \Omega_{Z/X} \to \Omega_{Z/Y} \to 0,\] see Lemma 05Z8. If \(g : Z \to Y\) is formally smooth, then this sequence is a short exact sequence, see Lemma 06BI.
If \(g\) is formally unramified, then there is a canonical exact sequence \[\mathcal{C}_{Z/Y} \to g^*\Omega_{Y/X} \to \Omega_{Z/X} \to 0,\] see Lemma 060F. If \(f \circ g : Z \to X\) is formally smooth, then this sequence is a short exact sequence, see Lemma 06BJ.
if \(g\) and \(f \circ g\) are formally unramified, then there is a canonical exact sequence \[\mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to g^*\Omega_{Y/X} \to 0,\] see Lemma 06BE. If \(f : Y \to X\) is formally smooth, then this sequence is a short exact sequence, see Lemma 06BK.
if \(g\) and \(f\) are formally unramified, then there is a canonical exact sequence \[g^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0.\] see Lemma 06BF. If \(g : Z \to Y\) is a local complete intersection morphism, then this sequence is a short exact sequence, see Lemma 06CC.
Characterizing pseudo-coherent complexes, II
In this section we discuss a characterization of pseudo-coherent complexes in terms of cohomology. Earlier material on pseudo-coherent complexes on algebraic spaces may be found in Derived Categories of Spaces, Section 08HC and in Derived Categories of Spaces, Section 0DKA. The analogue of this section for schemes is More on Morphisms, Section 0CSI. A basic tool will be to reduce to the case of projective space using a derived version of Chow’s lemma, see Lemma 0CTR.
Lemma
Let \(S\) be a scheme. Consider a commutative diagram of algebraic spaces \[\xymatrix{ Z' \ar[d] \ar[r] & Y' \ar[d] \\ X' \ar[r] & B' }\] over \(S\). Let \(B \to B'\) be a morphism. Denote by \(X\) and \(Y\) the base changes of \(X'\) and \(Y'\) to \(B\). Assume \(Y' \to B'\) and \(Z' \to X'\) are flat. Then \(X \times_B Y\) and \(Z'\) are Tor independent over \(X' \times_{B'} Y'\).
Proof
By Derived Categories of Spaces, Lemma 08IQ we may check tor independence étale locally on \(X \times_B Y\) and \(Z'\). This3 reduces the lemma to the case of schemes which is More on Morphisms, Lemma 0CTA.
Lemma
Let \(A\) be a ring. Let \(X\) be a separated algebraic space of finite presentation over \(A\). Let \(x \in |X|\). Then there exist an \(n \geq 0\), a closed subspace \(Z \subset X \times_A \mathbf{P}^n_A\), a point \(z \in |Z|\), an open \(V \subset \mathbf{P}^n_A\), and an object \(E\) in \(D(\mathcal{O}_{X \times_A \mathbf{P}^n_A})\) such that
\(Z \to X \times_A \mathbf{P}^n_A\) is of finite presentation,
\(c : Z \to \mathbf{P}^n_A\) is a closed immersion over \(V\), set \(W = c^{-1}(V)\),
the restriction of \(b : Z \to X\) to \(W\) is étale, \(z \in W\), and \(b(z) = x\),
\(E|_{X \times_A V} \cong (b, c)_*\mathcal{O}_Z|_{X \times_A V}\),
\(E\) is pseudo-coherent and supported on \(Z\).
Proof
We can find a finite type \(\mathbf{Z}\)-subalgebra \(A' \subset A\) and an algebraic space \(X'\) separated and of finite presentation over \(A'\) whose base change to \(A\) is \(X\). See Limits of Spaces, Lemmas 07SK and 0851. Let \(x' \in |X'|\) be the image of \(x\). If we can prove the lemma for \((X'/A', x')\), then the lemma follows for \((X/A, x)\). Namely, if \(n', Z', z', V', E'\) provide the solution for \((X'/A', x')\), then we can let \(n = n'\), let \(Z \subset X \times \mathbf{P}^n\) be the inverse image of \(Z'\), let \(z \in Z\) be the unique point mapping to \(x\), let \(V \subset \mathbf{P}^n_A\) be the inverse image of \(V'\), and let \(E\) be the derived pullback of \(E'\). Observe that \(E\) is pseudo-coherent by Cohomology on Sites, Lemma 08H4. It only remains to check (5). To see this set \(W = c^{-1}(V)\) and \(W' = (c')^{-1}(V')\) and consider the cartesian square \[\xymatrix{ W \ar[d]_{(b, c)} \ar[r] & W' \ar[d]^{(b', c')} \\ X \times_A V \ar[r] & X' \times_{A'} V' }\] By Lemma 0CTQ \(X \times_A V\) and \(W'\) are tor-independent over \(X' \times_{A'} V'\). Thus the derived pullback of \((b', c')_*\mathcal{O}_{W'}\) to \(X \times_A V\) is \((b, c)_*\mathcal{O}_W\) by Derived Categories of Spaces, Lemma 08IR. This also uses that \(R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'}\) because \((b', c')\) is a closed immersion and similarly for \((b, c)_*\mathcal{O}_Z\). Since \(E'|_{U' \times_{A'} V'} = (b', c')_*\mathcal{O}_{W'}\) we obtain \(E|_{U \times_A V} = (b, c)_*\mathcal{O}_W\) and (5) holds. This reduces us to the situation described in the next paragraph.
Assume \(A\) is of finite type over \(\mathbf{Z}\). Choose an étale morphism \(U \to X\) where \(U\) is an affine scheme and a point \(u \in U\) mapping to \(x\). Then \(U\) is of finite type over \(A\). Choose a closed immersion \(U \to \mathbf{A}^n_A\) and denote \(j : U \to \mathbf{P}^n_A\) the immersion we get by composing with the open immersion \(\mathbf{A}^n_A \to \mathbf{P}^n_A\). Let \(Z\) be the scheme theoretic closure of \[(\text{id}_U, j) : U \longrightarrow X \times_A \mathbf{P}^n_A\] Let \(z \in Z\) be the image of \(u\). Let \(Y \subset \mathbf{P}^n_A\) be the scheme theoretic closure of \(j\). Then it is clear that \(Z \subset X \times_A Y\) is the scheme theoretic closure of \((\text{id}_U, j) : U \to X \times_A Y\). As \(X\) is separated, the morphism \(X \times_A Y \to Y\) is separated as well. Hence we see that \(Z \to Y\) is an isomorphism over the open subscheme \(j(U) \subset Y\) by Morphisms of Spaces, Lemma 0CP3. Choose \(V \subset \mathbf{P}^n_A\) open with \(V \cap Y = j(U)\). Then we see that (2) holds, that \(W = (\text{id}_U, j)(U)\), and hence that (3) holds. Part (1) holds because \(A\) is Noetherian.
Because \(A\) is Noetherian we see that \(X\) and \(X \times_A \mathbf{P}^n_A\) are Noetherian algebraic spaces. Hence we can take \(E = (b, c)_*\mathcal{O}_Z\) in this case: (4) is clear and for (5) see Derived Categories of Spaces, Lemma 08IK. This finishes the proof.
Lemma
Let \(X/A\), \(x \in |X|\), and \(n, Z, z, V, E\) be as in Lemma 0CTR. For any \(K \in D_\QCoh(\mathcal{O}_X)\) we have \[Rq_*(Lp^*K \otimes^\mathbf{L} E)|_V = R(W \to V)_*K|_W\] where \(p : X \times_A \mathbf{P}^n_A \to X\) and \(q : X \times_A \mathbf{P}^n_A \to \mathbf{P}^n_A\) are the projections and where the morphism \(W \to V\) is the finitely presented closed immersion \(c|_W : W \to V\).
Proof
Since \(W = c^{-1}(V)\) and since \(c\) is a closed immersion over \(V\), we see that \(c|_W\) is a closed immersion. It is of finite presentation because \(W\) and \(V\) are of finite presentation over \(A\), see Morphisms of Spaces, Lemma 05WT. First we have \[Rq_*(Lp^*K \otimes^\mathbf{L} E)|_V = Rq'_*\left((Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V}\right)\] where \(q' : X \times_A V \to V\) is the projection because formation of total direct image commutes with localization. Denote \(i = (b, c)|_W : W \to X \times_A V\) the given closed immersion. Then \[Rq'_*\left((Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V}\right) = Rq'_*(Lp^*K|_{X \times_A V} \otimes^\mathbf{L} i_*\mathcal{O}_W)\] by property (5). Since \(i\) is a closed immersion we have \(i_*\mathcal{O}_W = Ri_*\mathcal{O}_W\). Using Derived Categories of Spaces, Lemma 08IN we can rewrite this as \[Rq'_* Ri_* Li^* Lp^*K|_{X \times_A V} = R(q' \circ i)_* Lb^*K|_W = R(W \to V)_* K|_W\] which is what we want. (Note that restricting to \(W\) and derived pulling back via \(W \to X\) is the same thing as \(W\) is étale over \(X\).)
Lemma
Let \(A\) be a ring. Let \(X\) be an algebraic space separated and of finite presentation over \(A\). Let \(K \in D_\QCoh(\mathcal{O}_X)\). If \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is pseudo-coherent in \(D(A)\) for every pseudo-coherent \(E\) in \(D(\mathcal{O}_X)\), then \(K\) is pseudo-coherent relative to \(A\) (Definition 0CSY).
Proof
Assume \(K \in D_\QCoh(\mathcal{O}_X)\) and \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is pseudo-coherent in \(D(A)\) for every pseudo-coherent \(E\) in \(D(\mathcal{O}_X)\). Let \(x \in |X|\). We will show that \(K\) is pseudo-coherent relative to \(A\) in an étale neighbourhood of \(x\). This will prove the lemma by our definition of relative pseudo-coherence.
Choose \(n, Z, z, V, E\) as in Lemma 0CTR. Denote \(p : X \times \mathbf{P}^n \to X\) and \(q : X \times \mathbf{P}^n \to \mathbf{P}^n_A\) the projections. Then for any \(i \in \mathbf{Z}\) we have \[\begin{align*} & R\Gamma(\mathbf{P}^n_A, Rq_*(Lp^*K \otimes^\mathbf{L} E) \otimes^\mathbf{L} \mathcal{O}_{\mathbf{P}^n_A}(i)) \\ & = R\Gamma(X \times \mathbf{P}^n, Lp^*K \otimes^\mathbf{L} E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i)) \\ & = R\Gamma(X, K \otimes^\mathbf{L} Rp_*(E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i))) \end{align*}\] by Derived Categories of Spaces, Lemma 08IN. By Derived Categories of Spaces, Lemma 0CTN the complex \(Rp_*(E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i))\) is pseudo-coherent on \(X\). Hence the assumption tells us the expression in the displayed formula is a pseudo-coherent object of \(D(A)\). By Derived Categories of Schemes, Lemma 0CSG we conclude that \(Rq_*(Lp^*K \otimes^\mathbf{L} E)\) is pseudo-coherent on \(\mathbf{P}^n_A\). By Lemma 0CTS we have \[Rq_*(Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V} = R(W \to V)_*K|_W\] Since \(W \to V\) is a closed immersion into an open subscheme of \(\mathbf{P}^n_A\) this means \(K|_W\) is pseudo-coherent relative to \(A\) for example by More on Morphisms, Lemma 09UU.
Lemma
Let \(A\) be a ring. Let \(X\) be an algebraic space separated and of finite presentation over \(A\). Let \(K \in D_\QCoh(\mathcal{O}_X)\). If \(R \Gamma (X, E \otimes ^{\mathbf{L}} K)\) is pseudo-coherent in \(D(A)\) for every perfect \(E \in D(\mathcal{O}_X)\), then \(K\) is pseudo-coherent relative to \(A\).
Proof
In view of Lemma 0CTT, it suffices to show \(R \Gamma (X, E \otimes ^{\mathbf{L}} K)\) is pseudo-coherent in \(D(A)\) for every pseudo-coherent \(E \in D(\mathcal{O}_X)\). By Derived Categories of Spaces, Proposition 0GFH it follows that \(K \in D^-_\QCoh (\mathcal{O}_X)\). Now the result follows by Derived Categories of Spaces, Lemma 0CWH.
Relatively perfect objects
In this section we introduce a notion from [lieblich-complexes]. This notion has been discussed for morphisms of schemes in Derived Categories of Schemes, Section 0DHZ.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and locally of finite presentation. An object \(E\) of \(D(\mathcal{O}_X)\) is perfect relative to \(Y\) or \(Y\)-perfect if \(E\) is pseudo-coherent (Cohomology on Sites, Definition 08FT) and \(E\) locally has finite tor dimension as an object of \(D(f^{-1}\mathcal{O}_Y)\) (Cohomology on Sites, Definition 08FZ).
Please see Derived Categories of Schemes, Remark 0DI9 for a discussion; here we just mention that \(E\) being pseudo-coherent is the same thing as \(E\) being pseudo-coherent relative to \(Y\) by Lemma 0DII. Moreover, pseudo-coherence of \(E\) implies \(E \in D_\QCoh(\mathcal{O}_X)\), see Derived Categories of Spaces, Lemma 08JL.
Example
Let \(k\) be a field. Let \(X\) be an algebraic space of finite presentation over \(k\) (in particular \(X\) is quasi-compact). Then an object \(E\) of \(D(\mathcal{O}_X)\) is \(k\)-perfect if and only if it is bounded and pseudo-coherent (by definition), i.e., if and only if it is in \(D^b_{\textit{Coh}}(X)\) (by Derived Categories of Spaces, Lemma 08IK). Thus being relatively perfect does not mean “perfect on the fibres”.
The corresponding algebra concept is studied in More on Algebra, Section 0DHR. We can link the notion for algebraic spaces with the algebraic notion as follows.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and locally of finite presentation. Let \(E \in D_\QCoh(\mathcal{O}_X)\). The following are equivalent:
\(E\) is \(Y\)-perfect,
for every commutative diagram \[\xymatrix{ U \ar[d] \ar[r]_g & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\), \(V\) are schemes and the vertical arrows are étale, the complex \(E|_U\) is \(V\)-perfect in the sense of Derived Categories of Schemes, Definition 0DI0,
for some commutative diagram as in (2) with \(U \to X\) surjective, the complex \(E|_U\) is \(V\)-perfect in the sense of Derived Categories of Schemes, Definition 0DI0,
for every commutative diagram as in (2) with \(U\) and \(V\) affine the complex \(R\Gamma(U, E)\) is \(\mathcal{O}_Y(V)\)-perfect.
Proof
To make sense of parts (2), (3), (4) of the lemma, observe that the object \(E|_U\) of \(D_\QCoh(\mathcal{O}_U)\) corresponds to an object \(E_0\) of \(D_\QCoh(\mathcal{O}_{U_0})\) where \(U_0\) denotes the scheme underlying \(U\), see Derived Categories of Spaces, Lemma 071Q. Moreover, in this case \(E_0\) is pseudo-coherent if and only if \(E|_U\) is pseudo-coherent, see Derived Categories of Spaces, Lemma 08HE. Also, \(E|_U\) locally has finite tor dimension over \(f^{-1}\mathcal{O}_Y|_U = g^{-1}\mathcal{O}_V\) if and only if \(E_0\) locally has finite tor dimension over \(g_0^{-1}\mathcal{O}_{V_0}\) by Derived Categories of Spaces, Lemma 0DK7. Here \(g_0 : U_0 \to V_0\) is the morphism of schemes representing \(g : U \to V\) (notation as in Derived Categories of Spaces, Remark 08GH). Finally, observe that “being pseudo-coherent” is étale local and of course “having locally finite tor dimension” is étale local. Thus we see that it suffices to check \(Y\)-perfectness étale locally and by the above discussion we see that (1) implies (2) and (3) implies (1). Since part (4) is equivalent to (2) and (3) by Derived Categories of Schemes, Lemma 0DI2 the proof is complete.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and locally of finite presentation. The full subcategory of \(D(\mathcal{O}_X)\) consisting of \(Y\)-perfect objects is a saturated4 triangulated subcategory.
Proof
This follows from Cohomology on Sites, Lemmas 08FV, 08FW, 08G2, and 08G3.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and locally of finite presentation. A perfect object of \(D(\mathcal{O}_X)\) is \(Y\)-perfect. If \(K, M \in D(\mathcal{O}_X)\), then \(K \otimes_{\mathcal{O}_X}^\mathbf{L} M\) is \(Y\)-perfect if \(K\) is perfect and \(M\) is \(Y\)-perfect.
Proof
Reduce to the case of schemes using Lemma 0DKQ and then apply Derived Categories of Schemes, Lemma 0DI4.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and locally of finite presentation. Let \(g : Y' \to Y\) be a morphism of algebraic spaces over \(S\). Set \(X' = Y' \times_Y X\) and denote \(g' : X' \to X\) the projection. If \(K \in D(\mathcal{O}_X)\) is \(Y\)-perfect, then \(L(g')^*K\) is \(Y'\)-perfect.
Proof
Reduce to the case of schemes using Lemma 0DKQ and then apply Derived Categories of Schemes, Lemma 0DI5.
Situation
Let \(S\) be a scheme. Let \(Y = \lim_{i \in I} Y_i\) be a limit of a directed system of algebraic spaces over \(S\) with affine transition morphisms \(g_{i'i} : Y_{i'} \to Y_i\). We assume that \(Y_i\) is quasi-compact and quasi-separated for all \(i \in I\). We denote \(g_i : Y \to Y_i\) the projection. We fix an element \(0 \in I\) and a flat morphism of finite presentation \(X_0 \to Y_0\). We set \(X_i = Y_i \times_{Y_0} X_0\) and \(X = Y \times_{Y_0} X_0\) and we denote the transition morphisms \(f_{i'i} : X_{i'} \to X_i\) and \(f_i : X \to X_i\) the projections.
Lemma
In Situation 0DKU. Let \(K_0\) and \(L_0\) be objects of \(D(\mathcal{O}_{X_0})\). Set \(K_i = Lf_{i0}^*K_0\) and \(L_i = Lf_{i0}^*L_0\) for \(i \geq 0\) and set \(K = Lf_0^*K_0\) and \(L = Lf_0^*L_0\). Then the map \[\colim_{i \geq 0} \Hom_{D(\mathcal{O}_{X_i})}(K_i, L_i) \longrightarrow \Hom_{D(\mathcal{O}_X)}(K, L)\] is an isomorphism if \(K_0\) is pseudo-coherent and \(L_0 \in D_\QCoh(\mathcal{O}_{X_0})\) has (locally) finite tor dimension as an object of \(D((X_0 \to Y_0)^{-1}\mathcal{O}_{Y_0})\)
Proof
For every quasi-compact and quasi-separated object \(U_0\) of \((X_0)_{spaces, \etale}\) consider the condition \(P\) that \[\colim_{i \geq 0} \Hom_{D(\mathcal{O}_{U_i})}(K_i|_{U_i}, L_i|_{U_i}) \longrightarrow \Hom_{D(\mathcal{O}_U)}(K|_U, L|_U)\] is an isomorphism where \(U = X \times_{X_0} U_0\) and \(U_i = X_i \times_{X_0} U_0\). We will prove \(P\) holds for each \(U_0\).
Suppose that \((U_0 \subset W_0, V_0 \to W_0)\) is an elementary distinguished square in \((X_0)_{spaces, \etale}\) and \(P\) holds for \(U_0, V_0, U_0 \times_{W_0} V_0\). Then \(P\) holds for \(W_0\) by Mayer-Vietoris for hom in the derived category, see Derived Categories of Spaces, Lemma 08H9.
We first consider \(U_0 = W_0 \times_{Y_0} X_0\) with \(W_0\) a quasi-compact and quasi-separated object of \((Y_0)_{spaces, \etale}\). By the induction principle of Derived Categories of Spaces, Lemma 08GP applied to these \(W_0\) and the previous paragraph, we find that it is enough to prove \(P\) for \(U_0 = W_0 \times_{Y_0} X_0\) with \(W_0\) affine. In other words, we have reduced to the case where \(Y_0\) is affine. Next, we apply the induction principle again, this time to all quasi-compact and quasi-separated opens of \(X_0\), to reduce to the case where \(X_0\) is affine as well.
If \(X_0\) and \(Y_0\) are affine, then we are back in the case of schemes which is proved in Derived Categories of Schemes, Lemma 0DI7. The reader may use Derived Categories of Spaces, Lemmas 08JL, 071Q, 08HE, and 0DK7 to accomplish the translation of the statement into a statement involving only schemes and derived categories of modules on schemes.
Lemma
In Situation 0DKU the category of \(Y\)-perfect objects of \(D(\mathcal{O}_X)\) is the colimit of the categories of \(Y_i\)-perfect objects of \(D(\mathcal{O}_{X_i})\).
Proof
For every quasi-compact and quasi-separated object \(U_0\) of \((X_0)_{spaces, \etale}\) consider the condition \(P\) that the functor \[\colim_{i \geq 0} D_{Y_i\text{-perfect}}(\mathcal{O}_{U_i}) \longrightarrow D_{Y\text{-perfect}}(\mathcal{O}_U)\] is an equivalence where \(U = X \times_{X_0} U_0\) and \(U_i = X_i \times_{X_0} U_0\). We observe that we already know this functor is fully faithful by Lemma 0DKV. Thus it suffices to prove essential surjectivity.
Suppose that \((U_0 \subset W_0, V_0 \to W_0)\) is an elementary distinguished square in \((X_0)_{spaces, \etale}\) and \(P\) holds for \(U_0, V_0, U_0 \times_{W_0} V_0\). We claim that \(P\) holds for \(W_0\). We will use the notation \(U_i = X_i \times_{X_0} U_0\), \(U = X \times_{X_0} U_0\), and similarly for \(V_0\) and \(W_0\). We will abusively use the symbol \(f_i\) for all the morphisms \(U \to U_i\), \(V \to V_i\), \(U \times_W V \to U_i \times_{W_i} V_i\), and \(W \to W_i\). Suppose \(E\) is an \(Y\)-perfect object of \(D(\mathcal{O}_W)\). Goal: show \(E\) is in the essential image of the functor. By assumption, we can find \(i \geq 0\), an \(Y_i\)-perfect object \(E_{U, i}\) on \(U_i\), an \(Y_i\)-perfect object \(E_{V, i}\) on \(V_i\), and isomorphisms \(Lf_i^*E_{U, i} \to E|_U\) and \(Lf_i^*E_{V, i} \to E|_V\). Let \[a : E_{U, i} \to (Rf_{i, *}E)|_{U_i} \quad\text{and}\quad b : E_{V, i} \to (Rf_{i, *}E)|_{V_i}\] the maps adjoint to the isomorphisms \(Lf_i^*E_{U, i} \to E|_U\) and \(Lf_i^*E_{V, i} \to E|_V\). By fully faithfulness, after increasing \(i\), we can find an isomorphism \(c : E_{U, i}|_{U_i \times_{W_i} V_i} \to E_{V, i}|_{U_i \times_{W_i} V_i}\) which pulls back to the identifications \[Lf_i^*E_{U, i}|_{U \times_W V} \to E|_{U \times_W V} \to Lf_i^*E_{V, i}|_{U \times_W V}.\] Apply Derived Categories of Spaces, Lemma 08HB to get an object \(E_i\) on \(W_i\) and a map \(d : E_i \to Rf_{i, *}E\) which restricts to the maps \(a\) and \(b\) over \(U_i\) and \(V_i\). Then it is clear that \(E_i\) is \(Y_i\)-perfect (because being relatively perfect is an étale local property) and that \(d\) is adjoint to an isomorphism \(Lf_i^*E_i \to E\).
By exactly the same argument as used in the proof of Lemma 0DKV using the induction principle (Derived Categories of Spaces, Lemma 08GP) we reduce to the case where both \(X_0\) and \(Y_0\) are affine: first work with quasi-compact and quasi-separated objects in \((Y_0)_{spaces, \etale}\) to reduce to \(Y_0\) affine, then work with quasi-compact and quasi-separated object in \((X_0)_{spaces, \etale}\) to reduce to \(X_0\) affine. In the affine case the result follows from the case of schemes which is Derived Categories of Schemes, Lemma 0DI8. The translation into the case for schemes is done by Lemma 0DKQ.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat, proper, and of finite presentation. Let \(E \in D(\mathcal{O}_X)\) be \(Y\)-perfect. Then \(Rf_*E\) is a perfect object of \(D(\mathcal{O}_Y)\) and its formation commutes with arbitrary base change.
Proof
The statement on base change is Derived Categories of Spaces, Lemma 0A1K (with \(\mathcal{G}^\bullet\) equal to \(\mathcal{O}_X\) in degree \(0\)). Thus it suffices to show that \(Rf_*E\) is a perfect object. We will reduce to the case where \(Y\) is Noetherian affine by a limit argument.
The question is étale local on \(Y\), hence we may assume \(Y\) is affine. Say \(Y = \Spec(R)\). We write \(R = \colim R_i\) as a filtered colimit of Noetherian rings \(R_i\). By Limits of Spaces, Lemma 07SK there exists an \(i\) and an algebraic space \(X_i\) of finite presentation over \(R_i\) whose base change to \(R\) is \(X\). By Limits of Spaces, Lemmas 08K1 and 08K0 we may assume \(X_i\) is proper and flat over \(R_i\). By Lemma 0DKW we may assume there exists a \(R_i\)-perfect object \(E_i\) of \(D(\mathcal{O}_{X_i})\) whose pullback to \(X\) is \(E\). Applying Derived Categories of Spaces, Lemma 08IS to \(X_i \to \Spec(R_i)\) and \(E_i\) and using the base change property already shown we obtain the result.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(E, K \in D(\mathcal{O}_X)\). Assume
\(Y\) is quasi-compact and quasi-separated,
\(f\) is proper, flat, and of finite presentation,
\(E\) is \(Y\)-perfect,
\(K\) is pseudo-coherent.
Then there exists a pseudo-coherent \(L \in D(\mathcal{O}_Y)\) such that \[Rf_*R\SheafHom(K, E) = R\SheafHom(L, \mathcal{O}_Y)\] and the same is true after arbitrary base change: given \[\vcenter{ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } } \quad\quad \begin{matrix} \text{cartesian, then we have } \\ Rf'_*R\SheafHom(L(g')^*K, L(g')^*E) \\ = R\SheafHom(Lg^*L, \mathcal{O}_{Y'}) \end{matrix}\]
Proof
Since \(Y\) is quasi-compact and quasi-separated, the same is true for \(X\). By Derived Categories of Spaces, Lemma 0DKB we can write \(K = \text{hocolim} K_n\) with \(K_n\) perfect and \(K_n \to K\) inducing an isomorphism on truncations \(\tau_{\geq -n}\). Let \(K_n^\vee\) be the dual perfect complex (Cohomology on Sites, Lemma 08JJ). We obtain an inverse system \(\ldots \to K_3^\vee \to K_2^\vee \to K_1^\vee\) of perfect objects. By Lemma 0DKS we see that \(K_n^\vee \otimes_{\mathcal{O}_X} E\) is \(Y\)-perfect. Thus we may apply Lemma 0DKX to \(K_n^\vee \otimes_{\mathcal{O}_X} E\) and we obtain an inverse system \[\ldots \to M_3 \to M_2 \to M_1\] of perfect complexes on \(Y\) with \[M_n = Rf_*(K_n^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E) = Rf_*R\SheafHom(K_n, E)\] Moreover, the formation of these complexes commutes with any base change, namely \(Lg^*M_n = Rf'_*((L(g')^*K_n)^\vee \otimes_{\mathcal{O}_{X'}}^\mathbf{L} L(g')^*E) = Rf'_*R\SheafHom(L(g')^*K_n, L(g')^*E)\).
As \(K_n \to K\) induces an isomorphism on \(\tau_{\geq -n}\), we see that \(K_n \to K_{n + 1}\) induces an isomorphism on \(\tau_{\geq -n}\). It follows that \(K_{n + 1}^\vee \to K_n^\vee\) induces an isomorphism on \(\tau_{\leq n}\) as \(K_n^\vee = R\SheafHom(K_n, \mathcal{O}_X)\). Suppose that \(E\) has tor amplitude in \([a, b]\) as a complex of \(f^{-1}\mathcal{O}_Y\)-modules. Then the same is true after any base change, see Derived Categories of Spaces, Lemma 0DKD. We find that \(K_{n + 1}^\vee \otimes_{\mathcal{O}_X} E \to K_n^\vee \otimes_{\mathcal{O}_X} E\) induces an isomorphism on \(\tau_{\leq n + a}\) and the same is true after any base change. Applying the right derived functor \(Rf_*\) we conclude the maps \(M_{n + 1} \to M_n\) induce isomorphisms on \(\tau_{\leq n + a}\) and the same is true after any base change. Choose a distinguished triangle \[M_{n + 1} \to M_n \to C_n \to M_{n + 1}[1]\] Pick \(y \in |Y|\). Choose an elementary étale neighbourhood \((U, u) \to (Y, y)\); this is possible by Decent Spaces, Lemma 0BBP. Take \(Y'\) equal to the spectrum of the residue field at \(u\). Pull back to see that \(C_n|_U \otimes_{\mathcal{O}_U}^\mathbf{L} \kappa(u)\) has nonzero cohomology only in degrees \(\geq n + a\). By More on Algebra, Lemma 0BCD we see that the perfect complex \(C_n|_U\) has tor amplitude in \([n + a, m_n]\) for some integer \(m_n\) and after possibly shrinking \(U\). Thus \(C_n\) has tor amplitude in \([n + a, m_n]\) for some integer \(m_n\) (because \(Y\) is quasi-compact). In particular, the dual perfect complex \(C_n^\vee\) has tor amplitude in \([-m_n, -n - a]\).
Let \(L_n = M_n^\vee\) be the dual perfect complex. The conclusion from the discussion in the previous paragraph is that \(L_n \to L_{n + 1}\) induces isomorphisms on \(\tau_{\geq -n - a}\). Thus \(L = \text{hocolim} L_n\) is pseudo-coherent, see Derived Categories of Spaces, Lemma 0DKB. Since we have \[R\SheafHom(K, E) = R\SheafHom(\text{hocolim} K_n, E) = R\lim R\SheafHom(K_n, E) = R\lim K_n^\vee \otimes_{\mathcal{O}_X} E\] (Cohomology on Sites, Lemma 0A0A) and since \(R\lim\) commutes with \(Rf_*\) we find that \[Rf_*R\SheafHom(K, E) = R\lim M_n = R\lim R\SheafHom(L_n, \mathcal{O}_Y) = R\SheafHom(L, \mathcal{O}_Y)\] This proves the formula over \(Y\). Since the construction of \(M_n\) is compatible with base chance, the formula continues to hold after any base change.
Remark
The reader may have noticed the similarity between Lemma 0DKY and Derived Categories of Spaces, Lemma 08JR. Indeed, the pseudo-coherent complex \(L\) of Lemma 0DKY may be characterized as the unique pseudo-coherent complex on \(Y\) such that there are functorial isomorphisms \[\Ext^i_{\mathcal{O}_Y}(L, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(K, E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F})\] compatible with boundary maps for \(\mathcal{F}\) ranging over \(\QCoh(\mathcal{O}_Y)\). If we ever need this we will formulate a precise result here and give a detailed proof.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\) such that the structure morphism \(f : X \to S\) is flat and locally of finite presentation. Let \(E\) be a pseudo-coherent object of \(D(\mathcal{O}_X)\). The following are equivalent
\(E\) is \(S\)-perfect, and
\(E\) is locally bounded below and for every point \(s \in S\) the object \(L(X_s \to X)^*E\) of \(D(\mathcal{O}_{X_s})\) is locally bounded below.
Proof
Since everything is local we immediately reduce to the case that \(X\) and \(S\) are affine, see Lemma 0DKQ. This case is handled by Derived Categories of Schemes, Lemma 0GEH.
Lemma
Let \(A\) be a ring. Let \(X\) be an algebraic space separated, of finite presentation, and flat over \(A\). Let \(K \in D_\QCoh(\mathcal{O}_X)\). If \(R \Gamma (X, E \otimes^\mathbf{L} K)\) is perfect in \(D(A)\) for every perfect \(E \in D(\mathcal{O}_X)\), then \(K\) is \(\Spec(A)\)-perfect.
Proof
By Lemma 0GFJ, \(K\) is pseudo-coherent relative to \(A\). By Lemma 0DII, \(K\) is pseudo-coherent in \(D(\mathcal{O}_X)\). By Derived Categories of Spaces, Proposition 0GFI we see that \(K\) is in \(D^-(\mathcal{O}_X)\). Let \(\mathfrak{p}\) be a prime ideal of \(A\) and denote \(i : Y \to X\) the inclusion of the scheme theoretic fibre over \(\mathfrak{p}\), i.e., \(Y\) is a scheme over \(\kappa(\mathfrak p)\). By Lemma 0GFK, we will be done if we can show \(Li^*(K)\) is bounded below. Let \(G \in D_{perf} (\mathcal{O}_X)\) be a perfect complex which generates \(D_\QCoh (\mathcal{O}_X)\), see Derived Categories of Spaces, Theorem 09IY. We have \[\begin{align*} R\Hom _{\mathcal{O}_Y}(Li^*(G), Li^*(K)) & = R\Gamma(Y, Li^*(G ^\vee \otimes ^\mathbf{L} K)) \\ & = R\Gamma(X, G^\vee \otimes ^{\mathbf{L}} K) \otimes^\mathbf{L}_A \kappa(\mathfrak{p}) \end{align*}\] The first equality uses that \(Li^*\) preserves perfect objects and duals and Cohomology on Sites, Lemma 08JJ; we omit some details. The second equality follows from Derived Categories of Spaces, Lemma 08IR as \(X\) is flat over \(A\). It follows from our hypothesis that this is a perfect object of \(D(\kappa(\mathfrak{p}))\). The object \(Li^*(G) \in D_{perf}(\mathcal{O}_Y)\) generates \(D_\QCoh(\mathcal{O}_Y)\) by Derived Categories of Spaces, Remark 0E4R. Hence Derived Categories of Spaces, Proposition 0GFI now implies that \(Li^*(K)\) is bounded below and we win.
Theorem of the cube
This section is the analogue of More on Morphisms, Section 0BEZ. The following lemma tells us that the diagonal of the Picard functor is representable by locally closed immersions under the assumptions made in the lemma.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a flat, proper morphism of finite presentation of algebraic spaces over \(S\). Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module. For a morphism \(g : Y' \to Y\) consider the base change diagram \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] Assume \(\mathcal{O}_{Y'} \to f'_*\mathcal{O}_{X'}\) is an isomorphism for all \(g : Y' \to Y\). Then there exists an immersion \(j : Z \to Y\) of finite presentation such that a morphism \(g : Y' \to Y\) factors through \(Z\) if and only if there exists a finite locally free \(\mathcal{O}_{Y'}\)-module \(\mathcal{N}\) with \((f')^*\mathcal{N} \cong (g')^*\mathcal{L}\).
Proof
Let \(y : \Spec(k) \to Y\) be a field valued point. Then the fibre \(X_y\) of \(f\) at \(y\) is connected by our assumption that \(H^0(X_y, \mathcal{O}_{X_y}) = k\). Thus the rank of \(\mathcal{E}\) is constant on the fibres. Since \(f\) is open (Morphisms of Spaces, Lemma 042S) and closed we conclude that there is a decomposition \(Y = \coprod Y_r\) of \(Y\) into open and closed subspaces such that \(\mathcal{E}\) has constant rank \(r\) on the inverse image of \(Y_r\). Thus we may assume \(\mathcal{E}\) has constant rank \(r\). We will denote \(\mathcal{E}^\vee = \SheafHom(\mathcal{E}, \mathcal{O}_X)\) the dual rank \(r\) module.
By cohomology and base change (more precisely by Derived Categories of Spaces, Lemma 0CTM) we see that \(E = Rf_*\mathcal{E}\) is a perfect object of the derived category of \(Y\) and that its formation commutes with arbitrary change of base. Similarly for \(E' = Rf_*\mathcal{E}^\vee\). Since there is never any cohomology in degrees \(< 0\), we see that \(E\) and \(E'\) have (locally) tor-amplitude in \([0, b]\) for some \(b\). Observe that for any \(g : Y' \to Y\) we have \(f'_*((g')^*\mathcal{E}) = H^0(Lg^*E)\) and \(f'_*((g')^*\mathcal{E}^\vee) = H^0(Lg^*E')\). Let \(j : Z \to Y\) and \(j' : Z' \to Y\) be the locally closed immersions constructed in Derived Categories of Spaces, Lemma 0D21 for \(E\) and \(E'\) with \(a = 0\) and \(r = r\); these are characterized by the property that \(H^0(Lj^*E)\) and \(H^0((j')^*E')\) are locally free modules of rank \(r\) compatible with pullback.
Let \(g : Y' \to Y\) be a morphism. If there exists an \(\mathcal{N}\) as in the lemma, then, using the projection formula Cohomology on Sites, Lemma 0944, we see that the modules \[f'_*((g')^*\mathcal{E}) \cong f'_*((f')^*\mathcal{N}) \cong \mathcal{N} \otimes_{\mathcal{O}_{Y'}} f'_*\mathcal{O}_{X'} \cong \mathcal{N}\quad\text{and similarly }\quad f'_*((g')^*\mathcal{E}^\vee) \cong \mathcal{N}^\vee\] are locally free of rank \(r\) and remain locally free of rank \(r\) after any further base change \(Y'' \to Y'\). Hence in this case \(g : Y' \to Y\) factors through \(j\) and through \(j'\). Thus we may replace \(Y\) by \(Z \times_Y Z'\) and assume that \(f_*\mathcal{E}\) and \(f_*\mathcal{E}^\vee\) are locally free \(\mathcal{O}_Y\)-modules of rank \(r\) whose formation commutes with arbitrary change of base.
In this situation if \(g : Y' \to Y\) is a morphism and there exists an \(\mathcal{N}\) as in the lemma, then the map (cup product in degree \(0\)) \[f'_*((g')^*\mathcal{E}) \otimes_{\mathcal{O}_{Y'}} f'_*((g')^*\mathcal{E}^\vee) \longrightarrow \mathcal{O}_{Y'}\] is a perfect pairing. Conversely, if this cup product map is a perfect pairing, then we see that locally on \(Y'\) we have a basis of sections \(\sigma_1, \ldots, \sigma_r\) in \(f'_*((g')^*\mathcal{L})\) and \(\tau_1, \ldots, \tau_r\) in \(f'_*((g')^*\mathcal{E}^\vee)\) whose products satisfy \(\sigma_i \tau_j = \delta_{ij}\). Thinking of \(\sigma_i\) as a section of \((g')^*\mathcal{L}\) on \(X'\) and \(\tau_j\) as a section of \((g')^*\mathcal{L}^\vee\) on \(X'\), we conclude that \[\sigma_1, \ldots, \sigma_r : \mathcal{O}_{X'}^{\oplus r} \longrightarrow (g')^*\mathcal{E}\] is an isomorphism with inverse given by \[\tau_1, \ldots, \tau_r : (g')^*\mathcal{E} \longrightarrow \mathcal{O}_{X'}^{\oplus r}\] In other words, we see that \((f')^*f'_*(g')^*\mathcal{E} \cong (g')^*\mathcal{E}\). But the condition that the cup product is nondegenerate picks out a retrocompact open subscheme (namely, the locus where a suitable determinant is nonzero) and the proof is complete.
Descent of finiteness properties of complexes
This section is the analogue of More on Morphisms, Section 0CSM and Derived Categories of Schemes, Section 09UC.
Lemma
Let \(S\) be a scheme. Let \(\{f_i : X_i \to X\}\) be an fpqc covering of algebraic spaces over \(S\). Let \(E \in D_\QCoh(\mathcal{O}_X)\). Let \(m \in \mathbf{Z}\). Then \(E\) is \(m\)-pseudo-coherent if and only if each \(Lf_i^*E\) is \(m\)-pseudo-coherent.
Proof
Pullback always preserves \(m\)-pseudo-coherence, see Cohomology on Sites, Lemma 08H4. Thus it suffices to assume \(Lf_i^*E\) is \(m\)-pseudo-coherent and to prove that \(E\) is \(m\)-pseudo-coherent. Then first we may assume \(X_i\) is a scheme for all \(i\), see Topologies on Spaces, Lemma 0419. Next, choose a surjective étale morphism \(U \to X\) where \(U\) is a scheme. Then \(U_i = U \times_X X_i\) is a scheme and we obtain an fpqc covering \(\{U_i \to U\}\) of schemes, see Topologies on Spaces, Lemma 03MS. We know the result is true for \(\{U_i \to U\}_{i \in I}\) by the case for schemes, see Derived Categories of Schemes, Lemma 09UE. On the other hand, the restriction \(E|_U\) comes from an object of \(D_\QCoh(\mathcal{O}_U)\) (defined using the Zariski topology and the “usual” structure sheaf of \(U\)), see Derived Categories of Spaces, Lemma 071Q. The lemma follows as the two notions of pseudo-coherent (étale and Zariski) agree by Derived Categories of Spaces, Lemma 08HE.
Lemma
Let \(S\) be a scheme. Let \(\{g_i : Y_i \to Y\}\) be an fpqc covering of algebraic spaces over \(S\). Let \(f : X \to Y\) be a morphism of algebraic spaces and set \(X_i = Y_i \times_Y X\) with projections \(f_i : X_i \to Y_i\) and \(g'_i : X_i \to X\). Let \(E \in D_\QCoh(\mathcal{O}_X)\). Let \(a, b \in \mathbf{Z}\). Then the following are equivalent
\(E\) has tor amplitude in \([a, b]\) as an object of \(D(f^{-1}\mathcal{O}_Y)\), and
\(L(g'_i)^*E\) has tor amplitude in \([a, b]\) as a object of \(D(f_i^{-1}\mathcal{O}_{Y_i})\) for all \(i\).
Also true if “tor amplitude in \([a, b]\)” is replaced by “locally finite tor dimension”.
Proof
Pullback preserves “tor amplitude in \([a, b]\)” by Derived Categories of Spaces, Lemma 0DKD Observe that \(Y_i\) and \(X\) are tor independent over \(Y\) as \(Y_i \to Y\) is flat. Let us assume (2) and prove (1). We can compute tor dimension at stalks, see Cohomology on Sites, Lemma 0DJJ and Properties of Spaces, Theorem 04K5. Let \(\overline{x}\) be a geometric point of \(X\). Choose an \(i\) and a geometric point \(\overline{x}_i\) in \(X_i\) with image \(\overline{x}\) in \(X\). Then \[(L(g_i')^*E)_{\overline{x}_i} = E_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}}^\mathbf{L} \mathcal{O}_{X_, \overline{x}_i}\] Let \(\overline{y}_i\) in \(Y_i\) and \(\overline{y}\) in \(Y\) be the image of \(\overline{x}_i\) and \(\overline{x}\). Since \(X\) and \(Y_i\) are tor independent over \(Y\), we can apply More on Algebra, Lemma 0661 to see that the right hand side of the displayed formula is equal to \(E_{\overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}}^\mathbf{L} \mathcal{O}_{Y_i, \overline{y}_i}\) in \(D(\mathcal{O}_{Y_i, \overline{y}_i})\). Since we have assume the tor amplitude of this is in \([a, b]\), we conclude that the tor amplitude of \(E_{\overline{x}}\) in \(D(\mathcal{O}_{Y, \overline{y}})\) is in \([a, b]\) by More on Algebra, Lemma 068S. Thus (1) follows.
Using some elementary topology the case “locally finite tor dimension” follows too.
The following lemmas do not really belong in this section.
Lemma
Let \(S\) be a scheme. Let \(i : X \to X'\) be a finite order thickening of algebraic spaces. Let \(K' \in D(\mathcal{O}_{X'})\) be an object such that \(K = Li^*K'\) is pseudo-coherent. Then \(K'\) is pseudo-coherent.
Proof
We first prove \(K'\) has quasi-coherent cohomology sheaves; we urge the reader to skip this part. To do this, we may reduce to the case of a first order thickening, see Section 05ZJ. Let \(\mathcal{I} \subset \mathcal{O}_{X'}\) be the quasi-coherent sheaf of ideals cutting out \(X\). Tensoring the short exact sequence \[0 \to \mathcal{I} \to \mathcal{O}_{X'} \to i_*\mathcal{O}_X \to 0\] with \(K'\) we obtain a distinguished triangle \[K' \otimes_{\mathcal{O}_{X'}}^\mathbf{L} \mathcal{I} \to K' \to K' \otimes_{\mathcal{O}_{X'}}^\mathbf{L} i_*\mathcal{O}_X \to (K' \otimes_{\mathcal{O}_{X'}}^\mathbf{L} \mathcal{I})[1]\] Since \(i_* = Ri_*\) and since we may view \(\mathcal{I}\) as a quasi-coherent \(\mathcal{O}_X\)-module (as we have a first order thickening) we may rewrite this as \[i_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I}) \to K' \to i_*K \to i_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I})[1]\] Please use Cohomology of Spaces, Lemma 0DK5 to identify the terms. Since \(K\) is in \(D_\QCoh(\mathcal{O}_X)\) we conclude that \(K'\) is in \(D_\QCoh(\mathcal{O}_{X'})\); this uses Derived Categories of Spaces, Lemmas 08JL, 08F5, and 08FA.
Assume \(K'\) is in \(D_\QCoh(\mathcal{O}_{X'})\). The question is étale local on \(X'\) hence we may assume \(X'\) is affine. In this case the result follows from the case of schemes (More on Morphisms, Lemma 0DJX). The translation into the language of schemes uses Derived Categories of Spaces, Lemmas 071Q and 08HE and Remark 08GH.
Lemma
Let \(S\) be a scheme. Consider a cartesian diagram \[\xymatrix{ X \ar[r]_i \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r]^j & Y' }\] of algebraic spaces over \(S\). Assume \(X' \to Y'\) is flat and locally of finite presentation and \(Y \to Y'\) is a finite order thickening. Let \(E' \in D(\mathcal{O}_{X'})\). If \(E = Li^*(E')\) is \(Y\)-perfect, then \(E'\) is \(Y'\)-perfect.
Proof
Recall that being \(Y\)-perfect for \(E\) means \(E\) is pseudo-coherent and locally has finite tor dimension as a complex of \(f^{-1}\mathcal{O}_Y\)-modules (Definition 0DKN). By Lemma 0DL3 we find that \(E'\) is pseudo-coherent. In particular, \(E'\) is in \(D_\QCoh(\mathcal{O}_{X'})\), see Derived Categories of Spaces, Lemma 08JL. By Lemma 0DKQ this reduces us to the case of schemes. The case of schemes is More on Morphisms, Lemma 0DJY.
Lemma
Let \((R, I)\) be a pair consisting of a ring and an ideal \(I\) contained in the Jacobson radical. Set \(S = \Spec(R)\) and \(S_0 = \Spec(R/I)\). Let \(X\) be an algebraic space over \(R\) whose structure morphism \(f : X \to S\) is proper, flat, and of finite presentation. Denote \(X_0 = S_0 \times_S X\). Let \(E \in D(\mathcal{O}_X)\) be pseudo-coherent. If the derived restriction \(E_0\) of \(E\) to \(X_0\) is \(S_0\)-perfect, then \(E\) is \(S\)-perfect.
Proof
Choose a surjective étale morphism \(U \to X\) with \(U\) affine. Choose a closed immersion \(U \to \mathbf{A}^d_S\). Set \(U_0 = S_0 \times_S U\). The complex \(E_0|_{U_0}\) has tor amplitude in \([a, b]\) for some \(a, b \in \mathbf{Z}\). Let \(\overline{x}\) be a geometric point of \(X\). We will show that the tor amplitude of \(E_{\overline{x}}\) over \(R\) is in \([a - d, b]\). This will finish the proof as the tor amplitude can be read off from the stalks by Cohomology on Sites, Lemma 0DJJ and Properties of Spaces, Theorem 04K5.
Let \(x \in |X|\) be the point determined by \(\overline{x}\). Recall that \(|X| \to |S|\) is closed (by definition of proper morphisms). Since \(I\) is contained in the Jacobson radical, any nonempty closed subset of \(S\) contains a point of the closed subscheme \(S_0\). Hence we can find a specialization \(x \leadsto x_0\) in \(|X|\) with \(x_0 \in |X_0|\). Choose \(u_0 \in U_0\) mapping to \(x_0\). By Decent Spaces, Lemma 0B7W (or by Decent Spaces, Lemma 03K2 which applies directly to étale morphisms) we find a specialization \(u \leadsto u_0\) in \(U\) such that \(u\) maps to \(x\). We may lift \(\overline{x}\) to a geometric point \(\overline{u}\) of \(U\) lying over \(u\). Then we have \(E_{\overline{x}} = (E|_U)_{\overline{u}}\).
Write \(U = \Spec(A)\). Then \(A\) is a flat, finitely presented \(R\)-algebra which is a quotient of a polynomial \(R\)-algebra in \(d\)-variables. The restriction \(E|_U\) corresponds (by Derived Categories of Spaces, Lemmas 08JL, 071Q, and 08HE and Derived Categories of Schemes, Lemma 06Z0 and 08E7) to a pseudo-coherent object \(K\) of \(D(A)\). Observe that \(E_0\) corresponds to \(K \otimes_A^\mathbf{L} A/IA\). Let \(\mathfrak q \subset \mathfrak q_0 \subset A\) be the prime ideals corresponding to \(u \leadsto u_0\). Then \[E_{\overline{x}} = (E|_U)_{\overline{u}} = E_u \otimes_{\mathcal{O}_{U, u}}^\mathbf{L} \mathcal{O}_{U, \overline{u}} = K_{\mathfrak q} \otimes_{A_\mathfrak q}^\mathbf{L} A_{\mathfrak q}^{sh}\] (some details omitted). Since \(A_\mathfrak q \to A_\mathfrak q^{sh}\) is flat, the tor amplitude of this as an \(R\)-module is the same as the tor amplitude of \(K_\mathfrak q\) as an \(R\)-module (More on Algebra, Lemma 0DJF). Also, \(K_{\mathfrak q}\) is a localization of \(K_{\mathfrak q_0}\). Hence it suffices to show that \(K_{\mathfrak q_0}\) has tor amplitude in \([a - d, b]\) as a complex of \(R\)-modules.
Let \(I \subset \mathfrak p_0 \subset R\) be the prime ideal corresponding to \(f(x_0)\). Then we have \[\begin{align*} K \otimes_R^\mathbf{L} \kappa(\mathfrak p_0) & = (K \otimes_R^\mathbf{L} R/I) \otimes_{R/I}^\mathbf{L} \kappa(\mathfrak p_0) \\ & = (K \otimes_A^\mathbf{L} A/IA) \otimes_{R/I}^\mathbf{L} \kappa(\mathfrak p_0) \end{align*}\] the second equality because \(R \to A\) is flat. By our choice of \(a, b\) this complex has cohomology only in degrees in the interval \([a, b]\). Thus we may finally apply More on Algebra, Lemma 0DJH to \(R \to A\), \(\mathfrak q_0\), \(\mathfrak p_0\) and \(K\) to conclude.
Families of nodal curves
This section is the continuation of Algebraic Curves, Section 0C58. Please also see that section for our choice of terminology.
The property “at-worst-nodal of relative dimension \(1\)” of morphisms of schemes is étale local on the source-and-target, see Descent, Lemma 04R2 and Algebraic Curves, Lemmas 0C5D, 0C5E, and 0CD8. It is also stable under base change and fpqc local on the target, see Algebraic Curves, Lemmas 0C5B and 0C5E. Hence, by Morphisms of Spaces, Lemma 03MJ we may define the notion of an at-worst-nodal morphism of relative dimension \(1\) for algebraic spaces as follows and it agrees with the already existing notion defined in Morphisms of Spaces, Section 03HA when the morphism is representable.
Definition
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). We say \(f\) is at-worst-nodal of relative dimension \(1\) if the equivalent conditions of Morphisms of Spaces, Lemma 03MJ hold with \(\mathcal{P} =\)“at-worst-nodal of relative dimension \(1\)”.
Lemma
The property of being at-worst-nodal of relative dimension \(1\) is preserved under base change.
Proof
See Morphisms of Spaces, Remark 0AMM and Algebraic Curves, Lemma 0C5B.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:
\(f\) is at-worst-nodal of relative dimension \(1\),
for every scheme \(Z\) and any morphism \(Z \to Y\) the morphism \(Z \times_Y X \to Z\) is at-worst-nodal of relative dimension \(1\),
for every affine scheme \(Z\) and any morphism \(Z \to Y\) the morphism \(Z \times_Y X \to Z\) is at-worst-nodal of relative dimension \(1\),
there exists a scheme \(V\) and a surjective étale morphism \(V \to Y\) such that \(V \times_Y X \to V\) is at-worst-nodal of relative dimension \(1\),
there exists a scheme \(U\) and a surjective étale morphism \(\varphi : U \to X\) such that the composition \(f \circ \varphi\) is at-worst-nodal of relative dimension \(1\),
for every commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\), \(V\) are schemes and the vertical arrows are étale the top horizontal arrow is at-worst-nodal of relative dimension \(1\),
there exists a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\), \(V\) are schemes, the vertical arrows are étale, and \(U \to X\) is surjective such that the top horizontal arrow is at-worst-nodal of relative dimension \(1\), and
there exist Zariski coverings \(Y = \bigcup_{i \in I} Y_i\), and \(f^{-1}(Y_i) = \bigcup X_{ij}\) such that each morphism \(X_{ij} \to Y_i\) is at-worst-nodal of relative dimension \(1\).
Proof
Omitted.
The following lemma tells us that we can check whether a morphism is at-worst-nodal of relative dimension \(1\) on the fibres.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is flat and locally of finite presentation. Then there is a maximal open subspace \(X' \subset X\) such that \(f|_{X'} : X' \to Y\) is at-worst-nodal of relative dimension \(1\). Moreover, formation of \(X'\) commutes with arbitrary base change.
Proof
Choose a commutative diagram \[\xymatrix{ U \ar[d] \ar[r]_h & V \ar[d] \\ X \ar[r]^f & Y }\] where \(U\), \(V\) are schemes, the vertical arrows are étale, and \(U \to X\) is surjective. By the lemma for the case of schemes (Algebraic Curves, Lemma 0DSC) we find a maximal open subscheme \(U' \subset U\) such that \(h|_{U'} : U' \to V\) is at-worst-nodal of relative dimension \(1\) and such that formation of \(U'\) commutes with base change. Let \(X' \subset X\) be the open subspace whose points correspond to the open subset \(\Im(|U'| \to |X|)\). By Lemma 0DSG we see that \(X' \to Y\) is at-worst-nodal of relative dimension \(1\) and that \(X'\) is the largest open subspace with this property (this also implies that \(U'\) is the inverse image of \(X'\) in \(U\), but we do not need this). Since the same is true after base change the proof is complete.
The resolution property
We continue the discussion in Derived Categories of Spaces, Section 0GUR.
Situation
Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(V \to X\) be a surjective étale morphism where \(V\) is an affine scheme (such a thing exists by Properties of Spaces, Lemma 03H6). Choose a commutative diagram \[\xymatrix{ V \ar[rd]_\varphi \ar[rr]_j & & Y \ar[ld]^\pi \\ & X }\] where \(j\) is an open immersion and \(\pi\) is a finite morphism of algebraic spaces (such a diagram exists by Lemma 082K). Let \(\mathcal{I} \subset \mathcal{O}_Y\) be a finite type quasi-coherent sheaf of ideals on \(Y\) with \(V(\mathcal{I}) = Y \setminus j(V)\) (such a sheaf of ideals exists by Limits of Spaces, Lemma 0855).
Lemma
In Situation 0GUY, assume \(X\) is Noetherian. Then for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) there exist \(r \geq 0\), integers \(n_1, \ldots, n_r \geq 0\), and a surjection \[\bigoplus\nolimits_{i = 1, \ldots, r} \pi_*(\mathcal{I}^{n_i}) \longrightarrow \mathcal{F}\] of \(\mathcal{O}_X\)-modules.
Proof
Denote \(\omega_{Y/X}\) the coherent \(\mathcal{O}_Y\)-module such that there is an isomorphism \[\pi_*\omega_{Y/X} \cong \SheafHom_{\mathcal{O}_X}(\pi_*\mathcal{O}_Y, \mathcal{O}_X)\] of \(\pi_*\mathcal{O}_Y\)-modules, see Morphisms of Spaces, Lemma 08AI and Descent on Spaces, Lemma 060Z. The canonical map \(\mathcal{O}_X \to \pi_*\mathcal{O}_Y\) produces a canonical map \[\text{Tr}_\pi : \pi_*\omega_{Y/X} \longrightarrow \mathcal{O}_X\] Since \(V\) is Noetherian affine we may choose sections \[s_1, \ldots, s_r \in \Gamma(V, \pi^*\mathcal{F} \otimes_{\mathcal{O}_Y} \omega_{Y/X})\] generating the coherent module \(\pi^*\mathcal{F} \otimes_{\mathcal{O}_X} \omega_{Y/X}\) over \(V\). By Cohomology of Spaces, Lemma 07UM we can choose integers \(n_i \geq 0\) such that \(s_i\) extends to a map \(s_i' : \mathcal{I}^{n_i} \to \pi^*\mathcal{F} \otimes_{\mathcal{O}_Y} \omega_{Y/X}\). Pushing to \(X\) we obtain maps \[\sigma_i : \pi_*\mathcal{I}^{n_i} \xrightarrow{\pi_*s'_i} \pi_*(\pi^*\mathcal{F} \otimes_{\mathcal{O}_Y} \omega_{Y/X}) = \mathcal{F} \otimes_{\mathcal{O}_X} \pi_*\omega_{Y/X} \xrightarrow{\text{Tr}_\pi} \mathcal{F}\] where the equality sign is Cohomology of Spaces, Lemma 0DK4. To finish the proof we will show that the sum of these maps is surjective.
Let \(x \in |X|\) be a point of \(X\). Let \(v \in |V|\) be a point mapping to \(x\). We may choose an étale neighbourhood \((U, u) \to (X, x)\) such that \[U \times_X Y = W \coprod W'\] (disjoint union of algebraic spaces) such that \(W \to U\) is an isomorphism and such that the unique point \(w \in W\) lying over \(u\) maps to \(v\) in \(V \subset Y\). To see this is true use Lemma 0ADU and Étale Morphisms, Lemma 04HL. After shrinking \(U\) further if necessary we may assume \(W\) maps into \(V \subset Y\) by the projection. Since the formation of \(\omega_{Y/X}\) commutes with étale localization we see that \[\pi_*\omega_{Y/X}|_U = (\pi|_W)_*\omega_{W/U} \oplus (\pi|_{W'})_*\omega_{W'/U}\] We have \((\pi|_W)_*\omega_{W/U} = \mathcal{O}_U\) and this isomorphism is given by the trace map \(\text{Tr}_\pi|_U\) restricted to the first summand in the decomposition above. Since \(W\) maps into \(V\) we see that \(\mathcal{I}^{n_i}|_W = \mathcal{O}_W\). Hence \[\pi_*(\mathcal{I}^{n_i})|_U = \mathcal{O}_U \oplus (W' \to U)_*(\mathcal{I}^{n_i}|_{W'})\] Chasing diagrams the reader sees (details omitted) that \(\sigma_i|_U\) on the summand \(\mathcal{O}_U\) is the map \(\mathcal{O}_U \to \mathcal{F}\) corresponding to the section \[s_i|_W \in \Gamma(W,\pi^*\mathcal{F} \otimes_{\mathcal{O}_Y} \omega_{Y/X}) = \Gamma(W, \mathcal{F}|_W \otimes_{\mathcal{O}_W} \omega_{W/U}) = \Gamma(U, \mathcal{F})\] Since the sections \(s_i\) generate the module \(\pi^*\mathcal{F} \otimes_{\mathcal{O}_Y} \omega_{Y/X}\) over \(V\) and since \(W\) maps into \(V\) we conclude that the restriction of \(\bigoplus \sigma_i\) to \(U\) is surjective. Since \(x\) was an arbitrary point the proof is complete.
Lemma
In Situation 0GUY, assume \(X\) is Noetherian. Then \(X\) has the resolution property if and only if \(\pi_*\mathcal{I}\) is the quotient of a finite locally free \(\mathcal{O}_X\)-module.
Proof
The module \(\pi_*\mathcal{I}\) is coherent by Cohomology of Spaces, Lemma 07UH. Hence if \(X\) has the resolution property then \(\pi_*\mathcal{I}\) is the quotient of a finite locally free \(\mathcal{O}_X\)-module. Conversely, assume given a surjection \(\mathcal{E} \to \pi_*\mathcal{I}\) for some finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\). Observe that for all \(n \geq 1\) there is a surjection \[\pi_*\mathcal{I} \otimes_{\mathcal{O}_X} \pi_*\mathcal{I}^n \longrightarrow \pi_*\mathcal{I}^{n + 1}\] Hence \(\mathcal{E}^{\otimes n}\) surjects onto \(\pi_*\mathcal{I}^n\) for all \(n \geq 1\). We conclude that \(X\) has the resolution property if we combine this with the result of Lemma 0GUZ.
Lemma
In Situation 0GUY, the algebraic space \(X\) has the resolution property if and only if \(\pi_*\mathcal{I}\) is the quotient of a finite locally free \(\mathcal{O}_X\)-module.
Proof
The pushforward \(\pi_*\mathcal{G}\) of a finite type quasi-coherent \(\mathcal{O}_Y\)-module \(\mathcal{G}\) is a finite type quasi-coherent \(\mathcal{O}_X\)-module by Descent on Spaces, Lemma 060Z. In particular, if \(X\) has the resolution property, then \(\pi_*\mathcal{I}\) is the quotient of a finite locally free \(\mathcal{O}_X\)-module by Derived Categories of Spaces, Definition 0GUS.
Assume that we have a surjection \(\mathcal{E} \to \pi_*\mathcal{I}\) for some finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\). In the rest of the proof we show that \(X\) has the resolution property by reducing to the Noetherian case handled in Lemma 0GV0. We suggest the reader skip the rest of the proof.
A first reduction is that we may view \(X\) as an algebraic space over \(\Spec(\mathbf{Z})\), see Spaces, Definition 03I5. (This doesn’t affect the conditions nor the conclusion of the lemma.)
By Limits of Spaces, Lemma 07VR we can write \(Y = \lim Y_i\) with \(Y_i\) finite and of finite presentation over \(X\) and where the transition maps are closed immersions. Consider the closed subspace \(Z = V(\mathcal{I})\) of \(Y\). Since \(\mathcal{I}\) is of finite type, the morphism \(Z \to Y\) is of finite presentation. Hence we can find an \(i\) and a morphism \(Z_i \to Y_i\) of finite presentation whose base change to \(Y\) is \(Z \to Y\), see Limits of Spaces, Lemma 07SK. For \(i' \geq i\) denote \(Z_{i'} = Z_i \times_{Y_i} Y_{i'}\). After increasing \(i\) we may assume \(Z_i \to Y_i\) is a closed immersion (of finite presentation), see Limits of Spaces, Lemma 0850. Denote \(\mathcal{I}_i \subset \mathcal{O}_{Y_i}\) the ideal sheaf of \(Z_i\) and denote \(\pi_i : Y_i \to X\) the structure morphism. Similarly for \(i' \geq i\). Since \(Z = \lim_{i' \geq i} Z_{i'}\) we have \[\pi_*\mathcal{I} = \colim \pi_{i', *}\mathcal{I}_{i'}\] The transition maps in the system are all surjective as follows from the surjectivity of the maps \(\pi_{i, *}\mathcal{O}_{Y_i} \to \pi_{i', *}\mathcal{O}_{Y_{i'}}\) and the fact that \(Z_{i'} = Z_i \times_{Y_i} Y_{i'}\). By Cohomology of Spaces, Lemma 07U7 for some \(i' \geq i\) the map \(\mathcal{E} \to \pi_*\mathcal{I}\) lifts to a map \(\mathcal{E} \to \pi_{i', *}\mathcal{I}_{i'}\). After increasing \(i'\) this map \(\mathcal{E} \to \pi_{i', *}\mathcal{I}_{i'}\) becomes surjective (since if not the colimit of the cokernels, having surjective transition maps, is nonzero). This reduces us to the case discussed in the next paragraph.
Assume \(X\) is an algebraic space over \(\mathbf{Z}\) and that \(Y \to X\) is of finite presentation. By absolute Noetherian approximation we can write \(X = \lim X_i\) as a directed limit, where each \(X_i\) is a quasi-separated algebraic space of finite type over \(\mathbf{Z}\) and the transition morphisms are affine, see Limits of Spaces, Proposition 07SU. Since \(\pi : Y \to X\) is of finite presentation we can find an \(i\) and a morphism \(\pi_i : Y_i \to X_i\) of finite presentation whose base change to \(X\) is \(\pi\), see Limits of Spaces, Lemma 07SK. After increasing \(i\) we may assume \(\pi_i\) is finite, see Limits of Spaces, Lemma 084Z. Next, we may assume there exists a finite locally free \(\mathcal{O}_{X_i}\)-module \(\mathcal{E}_i\) whose pullback to \(X\) is \(\mathcal{E}\), see Limits of Spaces, Lemma 0D2X. We may also assume there is a map \(\mathcal{E}_i \to \pi_{i, *}\mathcal{O}_{Y_i}\) whose pullback to \(X\) is the composition \(\mathcal{E} \to \pi_*\mathcal{I} \to \pi_*\mathcal{O}_Y\), see Limits of Spaces, Lemma 07V7. The cokernel \[\mathcal{E}_i \to \pi_{i, *}\mathcal{O}_{Y_i} \to \mathcal{Q}_i \to 0\] is a coherent \(\mathcal{O}_{Y_i}\)-module whose pullback to \(X\) is the (finitely presented) cokernel \(\mathcal{Q}\) of the map \(\mathcal{E} \to \pi_*\mathcal{O}_Y\). In other words, we have \(\mathcal{Q} = \pi_*(\mathcal{O}_Y/\mathcal{I})\). Consider the map \[\mathcal{E}_i \otimes_{\mathcal{O}_{X_i}} \pi_{i, *} \mathcal{O}_{Y_i} \longrightarrow \pi_{i, *}\mathcal{O}_{Y_i} \otimes_{\mathcal{O}_{X_i}} \pi_{i, *} \mathcal{O}_{Y_i} \to \pi_{i, *} \mathcal{O}_{Y_i} \to \mathcal{Q}_i\] where the second arrow is given by the algebra structure on \(\pi_{i, *}\mathcal{O}_{Y_i}\). The pullback of this map to \(Y\) is zero because the image of \(\mathcal{E} \to \pi_*\mathcal{O}_Y\) is the ideal \(\pi_*\mathcal{I}\). Hence by Limits of Spaces, Lemma 07V7 after increasing \(i\) we may assume the displayed composition is zero. This exactly means that the imag of \(\mathcal{E}_i \to \pi_{i, *}\mathcal{O}_{Y_i}\) is of the form \(\pi_{i, *}\mathcal{I}_i\) for some coherent ideal sheaf \(\mathcal{I}_i \subset \mathcal{O}_{Y_i}\). Since \(\mathcal{E}_i \to \pi_{i, *}\mathcal{O}_{Y_i}\) pulls back to \(\mathcal{E} \to \pi_*\mathcal{O}_Y\) we see that the pullback of \(\mathcal{I}_i\) to \(Y\) generates \(\mathcal{I}\). Denote \(V_i \subset Y_i\) the open subspace whose complement is \(V(\mathcal{I}_i) \subset Y_i\). Then \(V\) is the inverse image of \(V_i\) by the comments above. After increasing \(i\) we may assume that \(V_i\) is affine and that \(\pi_i|_{V_i} : V_i \to X_i\) is étale, see Limits of Spaces, Lemmas 07SQ and 07SL. Having said all of this, we may apply Lemma 0GV0 to conclude that \(X_i\) has the resolution property. Since \(X \to X_i\) is affine we conclude that \(X\) has the resolution property too by Derived Categories of Spaces, Lemma 0GUU.
Lemma
Let \(S\) be a scheme. Let \(X = \lim X_i\) be a limit of a direct system of quasi-compact and quasi-separated algebraic spaces over \(S\) with affine transition morphisms. Then \(X\) has the resolution property if and only if \(X_i\) has the resolution properties for some \(i\).
Proof
If \(X_i\) has the resolution property, then \(X\) does by Derived Categories of Spaces, Lemma 0GUU. Assume \(X\) has the resolution property. Choose \(i \in I\). We may choose an affine scheme \(V_i\) and a surjective étale morphism \(V_i \to X_i\) (Properties of Spaces, Lemma 03H6). We may choose an embedding \(j : V_i \to Y_i\) with \(Y_i\) finite and finitely presented over \(X_i\) (Lemma 0874). We may choose a finite type quasi-coherent ideal \(\mathcal{I}_i \subset \mathcal{O}_{Y_i}\) such that \(V_i = Y_i \setminus V(\mathcal{I}_i)\) (Limits of Spaces, Lemma 0855). Denote \(V \to Y \to X\) the base changes of \(V_i \to Y_i \to X_i\) to \(X\). Denote \(\mathcal{I} \subset \mathcal{O}_Y\) the pullback of the ideal \(\mathcal{I}_i\). By the easy direction of Lemma 0GV1 there exists a finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\) and a surjection \(\mathcal{E} \to \pi_*\mathcal{I}\). Note that since \(\pi_i : Y_i \to X_i\) is finite and of finite presentation we also have that \(\pi : Y \to X\) is finite and of finite presentation and that the \(\mathcal{O}_{X_i}\)-modules \(\pi_{i, *}\mathcal{O}_{Y_i}\) and \(\pi_{i, *}(\mathcal{O}_{Y_i}/\mathcal{I}_i)\) are of finite presentation and pullback to \(X\) to give \(\pi_*\mathcal{O}_Y\) and \(\pi_*(\mathcal{O}_Y/\mathcal{I})\). Thus by Limits of Spaces, Lemma 07V7 after increasing \(i\) we can find a finite locally free \(\mathcal{O}_{X_i}\)-module \(\mathcal{E}_i\) and a map \(\mathcal{E}_i \to \pi_{i, *}\mathcal{O}_{Y_i}\) whose base change to \(X\) recovers the composition \(\mathcal{E} \to \pi_*\mathcal{I} \to \pi_*\mathcal{O}_Y\). The pullbacks of the finitely presented \(\mathcal{O}_{X_i}\)-modules \(\Coker(\mathcal{E}_i \to \pi_{i, *}\mathcal{O}_{Y_i})\) and \(\pi_{i, *}(\mathcal{O}_{Y_i}/\mathcal{I}_i)\) to \(X\) agree as quotients of \(\pi_*\mathcal{O}_Y\). Hence by Limits of Spaces, Lemma 07V7 we may assume that these agree, in other words that the image of \(\mathcal{E}_i \to \pi_{i, *}\mathcal{O}_{X_i}\) is equal to \(\pi_{i, *}\mathcal{I}_i\). Then we conclude that \(X_i\) has the resolution property by Lemma 0GV1.
Lemma
Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space with the resolution property. Then \(X\) has affine diagonal over \(\mathbf{Z}\) (as in Properties of Spaces, Definition 03BS).
Proof
We could prove this as in the case of schemes, but instead we will deduce the lemma from the case of schemes. First, we may and do assume \(S = \Spec(\mathbf{Z})\). Next, we choose a scheme \(Y\) and a surjective integral morphism \(f : Y \to X\), see Decent Spaces, Lemma 09YB. Then \(f\) is affine, hence \(Y\) has the resolution property by Derived Categories of Spaces, Lemma 0GUU. Hence by the case of schemes, the scheme \(Y\) has affine diagonal, see Derived Categories of Schemes, Lemma 0F8C. Next, we consider the commutative diagram \[\xymatrix{ Y \ar[d] \ar[rr]_{\Delta_Y} & & Y \times_{\mathbf{Z}} Y \ar[d] \\ X \ar[rr]^{\Delta_X} & & X \times_{\mathbf{Z}} X }\] Observe that the right vertical arrow is integral, in particular affine. Let \(W \to X \times_{\mathbf{Z}} X\) be a morphism with \(W\) affine. Then we see that \[Y \times_{X \times_{\mathbf{Z}} X} W = Y \times_{\Delta_Y, Y \times_{\mathbf{Z}} Y} (Y \times_{\mathbf{Z}} Y) \times_{X \times_{\mathbf{Z}} X} W\] is affine. On the other hand, \(Y \to X\) is integral and surjective hence \[Y \times_{X \times_{\mathbf{Z}} X} W \longrightarrow X \times_{X \times_{\mathbf{Z}} X} W\] is integral surjective as the base change of \(Y \to X\) to \(W\). We conclude that the target of this arrow is affine by Limits of Spaces, Proposition 07VT. It follows that \(\Delta_X\) is affine as desired.
Blowing up and the resolution property
We prove that the resolution property is satisfied after a blowing up.
Lemma
Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Assume that \(|X|\) has finitely many irreducible components. There exists a dense quasi-compact open \(U \subset X\) and a \(U\)-admissible blowing up \(X' \to X\) such that the algebraic space \(X'\) has the resolution property.
Proof
By Limits of Spaces, Lemma 0GUN there exists a surjective, finite, and finitely presented morphism \(f : Y \to X\) where \(Y\) is a scheme and a quasi-compact dense open \(U \subset X\) such that \(f^{-1}(U) \to U\) is finite étale. By More on Morphisms, Lemma 0GTT there is a quasi-compact dense open \(V \subset Y\) and a \(V\)-admissible blowing up \(Y' \to Y\) such that \(Y'\) has an ample family of invertible modules. After shrinking \(U\) we may assume that \(f^{-1}(U) \subset V\) (details omitted). Hence \(f' : Y' \to X\) is finite étale over \(U\) and in particular, the morphism \((f')^{-1}(U) \to U\) is finite locally free. By Lemma 0B4K there is a \(U\)-admissible blowing up \(X' \to X\) such that the strict transform \(Y''\) of \(Y'\) is finite locally free over \(X'\). Picture \[\xymatrix{ Y'' \ar[d] \ar[r]_g & Y' \ar[r] & Y \ar[d] \\ X' \ar[rr] & & X }\] Since \(g : Y'' \to Y'\) is a blowing up (Divisors on Spaces, Lemma 0864) in the inverse image of the center of \(X' \to X\), we see that \(g : Y'' \to Y'\) is projective and that there exists some \(g\)-ample invertible module on \(Y''\). Hence by More on Morphisms, Lemma 0GTP we see that \(Y''\) has an ample family of invertible modules. Hence \(Y''\) has the resolution property, see Derived Categories of Schemes, Lemma 0GMM. We conclude that \(X'\) has the resolution property by Derived Categories of Spaces, Lemma 0GUV.
Lemma
Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). There exists a \(t \geq 0\) and closed subspaces \[X \supset Z_0 \supset Z_1 \supset \ldots \supset Z_t = \emptyset\] such that \(Z_i \to X\) is of finite presentation, \(Z_0 \subset X\) is a thickening, and for each \(i = 0, \ldots t - 1\) there exists a \((Z_i \setminus Z_{i - 1})\)-admissible blowing up \(Z'_i \to Z_i\) such that \(Z'_i\) has the resolution property.
Proof
In this paragraph we use absolute Noetherian approximation to reduce to the case of algebraic spaces of finite presentation over \(\Spec(\mathbf{Z})\). We may view \(X\) as an algebraic space over \(\Spec(\mathbf{Z})\), see Spaces, Definition 03I5 and Properties of Spaces, Definition 03BS. Thus we may apply Limits of Spaces, Proposition 07SU. It follows that we can find an affine morphism \(X \to X_0\) with \(X_0\) of finite presentation over \(\mathbf{Z}\). If we can prove the lemma for \(X_0\), then we can pull back the stratification and the centers of the blowing ups to \(X\) and get the result for \(X\); this uses that the resolution property goes up along affine morphisms (Derived Categories of Spaces, Lemma 0GUU) and that the strict transform of an affine morphism is affine – details omitted. This reduces us to the case discussed in the next paragraph.
Assume \(X\) is of finite presentation over \(\mathbf{Z}\). Then \(X\) is Noetherian and \(|X|\) is a Noetherian topological space (with finitely many irreducible components) of finite dimension. Hence we may use induction on \(\dim(|X|)\). By Lemma 0GV5 there exists a dense open \(U \subset X\) and a \(U\)-admissible blowing up \(X' \to X\) such that \(X'\) has the resolution property. Set \(Z_0 = X\) and let \(Z_1 \subset X\) be the reduced closed subspace with \(|Z_1| = |X| \setminus |U|\). By induction we find an integer \(t \geq 0\) and a filtration \[Z_1 \supset Z_{1, 0} \supset Z_{1, 1} \supset \ldots \supset Z_{1, t} = \emptyset\] by closed subspaces, where \(Z_{1, 0} \to Z_1\) is a thickening and there exist \((Z_{1, i} \setminus Z_{1, i + 1})\)-admissible blowing ups \(Z'_{1, i} \to Z_{1, i}\) such that \(Z'_{1, i}\) has the resolution property. Since \(Z_1\) is reduced, we have \(Z_1 = Z_{1, 0}\). Hence we can set \(Z_i = Z_{1, i - 1}\) and \(Z'_i = Z'_{1, i - 1}\) for \(i \geq 1\) and the lemma is proved.
This is just one possible definition that one can make here. Another slightly weaker condition would be to require that the dotted arrow exists fppf locally on \(T'\). This weaker notion has in some sense better formal properties.↩︎
Because we may check closedness of the diagonal \(X \to X \times_Y X\) over the four open parts \(X'_i \times_Y X'_j\) of \(X \times_Y X\) where it is clear.↩︎
Here is the argument in more detail. Choose a surjective étale morphism \(W' \to B'\) with \(W'\) a scheme. Choose a surjective étale morphism \(W \to B \times_{B'} W'\) with \(W\) a scheme. Choose a surjective étale morphism \(U' \to X' \times_{B'} W'\) with \(U'\) a scheme. Choose a surjective étale morphism \(V' \to Y' \times_{B'} W'\) with \(V'\) a scheme. Observe that \(U' \times_{W'} V' \to X' \times_{B'} Y'\) is surjective étale. Choose a surjective étale morphism \(T' \to Z' \times_{X' \times_{B'} Y'} U' \times_{W'} V'\) with \(T'\) a scheme. Denote \(U\) and \(V\) the base changes of \(U'\) and \(V'\) to \(W\). Then the lemma says that \(X \times_B Y\) and \(Z'\) are Tor independent over \(X' \times_{B'} Y'\) as algebraic spaces if and only if \(U \times_W V\) and \(T'\) are Tor independent over \(U' \times_{W'} V'\) as schemes. Thus it suffices to prove the lemma for the square with corners \(T', U', V', W'\) and base change by \(W \to W'\). The flatness of \(Y' \to B'\) and \(Z' \to X'\) implies flatness of \(V' \to W'\) and \(T' \to U'\).↩︎