Introduction
The goal of this chapter is to discuss pushouts in the category of algebraic spaces. This can be done with varying assumptions. A fairly general pushout construction is given in [Temkin-Tyomkin]: one of the morphisms is affine and the other is a closed immersion. We discuss a particular case of this in Section 07SW where we assume one of the morphisms is affine and the other is a thickening, a situation that often comes up in deformation theory.
In Sections 0AEP and 0AF2 we discuss diagrams \[\xymatrix{ f^{-1}(X \setminus Z) \ar[r] \ar[d] & Y \ar[d]^f \\ X \setminus Z \ar[r] & X }\] where \(f\) is a quasi-compact and quasi-separated morphism of algebraic spaces, \(Z \to X\) is a closed immersion of finite presentation, the map \(f^{-1}(Z) \to Z\) is an isomorphism, and \(f\) is flat along \(f^{-1}(Z)\). In this situation we glue quasi-coherent modules on \(X \setminus Z\) and \(Y\) (in Section 0AEP) to quasi-coherent modules on \(X\) and we glue algebraic spaces over \(X \setminus Z\) and \(Y\) (in Section 0AF2) to algebraic spaces over \(X\).
In Section 0AGF we discuss how proper birational morphisms of Noetherian algebraic spaces give rise to coequalizer diagrams in algebraic spaces in some sense.
In Section 0F44 we use the construction of elementary distinguished squares in Section 0DVH to prove Nagata’s theorem on compactifications in the setting of algebraic spaces.
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 \(X \times X\).
Colimits of algebraic spaces
We briefly discuss colimits of algebraic spaces. Let \(S\) be a scheme. Let \(\mathcal{I} \to (\Sch/S)_{fppf}\), \(i \mapsto X_i\) be a diagram (see Categories, Section 002D). For each \(i\) we may consider the small étale site \(X_{i, \etale}\) whose objects are schemes étale over \(X_i\), see Properties of Spaces, Section 03EB. For each morphism \(i \to j\) of \(\mathcal{I}\) we have the morphism \(X_i \to X_j\) and hence a pullback functor \(X_{j, \etale} \to X_{i, \etale}\). Hence we obtain a pseudo functor from \(\mathcal{I}^{opp}\) into the \(2\)-category of categories. Denote \[\lim_i X_{i, \etale}\] the \(2\)-limit (see insert future reference here). What does this mean concretely? An object of this limit is a system of étale morphisms \(U_i \to X_i\) over \(\mathcal{I}\) such that for each \(i \to j\) in \(\mathcal{I}\) the diagram \[\xymatrix{ U_i \ar[r] \ar[d] & U_j \ar[d] \\ X_i \ar[r] & X_j }\] is cartesian. Morphisms between objects are defined in the obvious manner. Suppose that \(f_i : X_i \to T\) is a family of morphisms such that for each \(i \to j\) the composition \(X_i \to X_j \to T\) is equal to \(f_i\). Then we get a functor \(T_\etale \to \lim X_{i, \etale}\). With this notation in hand we can formulate our lemma.
Lemma
Let \(S\) be a scheme. Let \(\mathcal{I} \to (\Sch/S)_{fppf}\), \(i \mapsto X_i\) be a diagram of schemes over \(S\) as above. Assume that
\(X = \colim X_i\) exists in the category of schemes,
\(\coprod X_i \to X\) is surjective,
if \(U \to X\) is étale and \(U_i = X_i \times_X U\), then \(U = \colim U_i\) in the category of schemes, and
every object \((U_i \to X_i)\) of \(\lim X_{i, \etale}\) with \(U_i \to X_i\) separated is in the essential image of the functor \(X_\etale \to \lim X_{i, \etale}\).
Then \(X = \colim X_i\) in the category of algebraic spaces over \(S\) also.
Proof
Let \(Z\) be an algebraic space over \(S\). Suppose that \(f_i : X_i \to Z\) is a family of morphisms such that for each \(i \to j\) the composition \(X_i \to X_j \to Z\) is equal to \(f_i\). We have to construct a morphism of algebraic spaces \(f : X \to Z\) such that we can recover \(f_i\) as the composition \(X_i \to X \to Z\). Let \(W \to Z\) be a surjective étale morphism of a scheme to \(Z\). We may assume that \(W\) is a disjoint union of affines and in particular we may assume that \(W \to Z\) is separated. For each \(i\) set \(U_i = W \times_{Z, f_i} X_i\) and denote \(h_i : U_i \to W\) the projection. Then \(U_i \to X_i\) forms an object of \(\lim X_{i, \etale}\) with \(U_i \to X_i\) separated. By assumption (4) we can find an étale morphism \(U \to X\) and (functorial) isomorphisms \(U_i = X_i \times_X U\). By assumption (3) there exists a morphism \(h : U \to W\) such that the compositions \(U_i \to U \to W\) are \(h_i\). Let \(g : U \to Z\) be the composition of \(h\) with the map \(W \to Z\). To finish the proof we have to show that \(g : U \to Z\) descends to a morphism \(X \to Z\). To do this, consider the morphism \((h, h) : U \times_X U \to W \times_S W\). Composing with \(U_i \times_{X_i} U_i \to U \times_X U\) we obtain \((h_i, h_i)\) which factors through \(W \times_Z W\). Since \(U \times_X U\) is the colimit of the schemes \(U_i \times_{X_i} U_i\) by (3) we see that \((h, h)\) factors through \(W \times_Z W\). Hence the two compositions \(U \times_X U \to U \to W \to Z\) are equal. Because each \(U_i \to X_i\) is surjective and assumption (2) we see that \(U \to X\) is surjective. As \(Z\) is a sheaf for the étale topology, we conclude that \(g : U \to Z\) descends to \(f : X \to Z\) as desired.
We can check that a cocone is a colimit (fpqc) locally on the cocone.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(\mathcal{I} \to (\Sch/S)_{fppf}\), \(i \mapsto X_i\) be a diagram of algebraic spaces over \(B\). Let \((X, X_i \to X)\) be a cocone for the diagram in the category of algebraic spaces over \(B\) (Categories, Remark 0G2U). If there exists a fpqc covering \(\{U_a \to X\}_{a \in A}\) such that
for all \(a \in A\) we have \(U_a = \colim X_i \times_X U_a\) in the category of algebraic spaces over \(B\), and
for all \(a, b \in A\) we have \(U_a \times_X U_b = \colim X_i \times_X U_a \times_X U_b\) in the category of algebraic spaces over \(B\),
then \(X = \colim X_i\) in the category of algebraic spaces over \(B\).
Proof
Namely, for an algebraic space \(Y\) over \(B\) a morphism \(X \to Y\) over \(B\) is the same thing as a collection of morphism \(U_a \to Y\) which agree on the overlaps \(U_a \times_X U_b\) for all \(a, b \in A\), see Descent on Spaces, Lemma 04P2.
We are going to find a common partial generalization of Lemmas 07SX and 0GFQ which can in particular be used to reduce a colimit construction to a subcategory of the category of all algebraic spaces.
Let \(S\) be a scheme and let \(B\) be an algebraic space over \(S\). Let \(\mathcal{I}\) be an index category and let \(i \mapsto X_i\) be a diagram in the category of algebraic spaces over \(B\), see Categories, Section 002D. For each \(i\) we may consider the small étale site \(X_{i, spaces, \etale}\) whose objects are algebraic spaces étale over \(X_i\), see Properties of Spaces, Section 03EB. For each morphism \(i \to j\) of \(\mathcal{I}\) we have the morphism \(X_i \to X_j\) and hence a pullback functor \(X_{j, spaces, \etale} \to X_{i, spaces, \etale}\). Hence we obtain a pseudo functor from \(\mathcal{I}^{opp}\) into the \(2\)-category of categories. Denote \[\lim_i X_{i, spaces, \etale}\] the \(2\)-limit (see insert future reference here). What does this mean concretely? An object of this limit is a diagram \(i \mapsto (U_i \to X_i)\) in the category of arrows of algebraic spaces over \(B\) such that for each \(i \to j\) in \(\mathcal{I}\) the diagram \[\xymatrix{ U_i \ar[r] \ar[d] & U_j \ar[d] \\ X_i \ar[r] & X_j }\] is cartesian. Morphisms between objects are defined in the obvious manner. Suppose that \(f_i : X_i \to Z\) is a family of morphisms of algebraic spaces over \(B\) such that for each \(i \to j\) the composition \(X_i \to X_j \to Z\) is equal to \(f_i\). Then we get a functor \(Z_{spaces, \etale} \to \lim X_{i, spaces, \etale}\). With this notation in hand we can formulate our next lemma.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(\mathcal{I} \to (\Sch/S)_{fppf}\), \(i \mapsto X_i\) be a diagram of algebraic spaces over \(B\). Let \((X, X_i \to X)\) be a cocone for the diagram in the category of algebraic spaces over \(B\) (Categories, Remark 0G2U). Assume that
the base change functor \(X_{spaces, \'etale} \to \lim X_{i, spaces, \etale}\), sending \(U\) to \(U_i = X_i \times_X U\) is an equivalence,
given
\(B'\) affine and étale over \(B\),
\(Z\) an affine scheme over \(B'\),
\(U \to X \times_B B'\) an étale morphism of algebraic spaces with \(U\) affine,
\(f_i : U_i \to Z\) a cocone over \(B'\) of the diagram \(i \mapsto U_i = U \times_X X_i\),
there exists a unique morphism \(f : U \to Z\) over \(B'\) such that \(f_i\) equals the composition \(U_i \to U \to Z\).
Then \(X = \colim X_i\) in the category of all algebraic spaces over \(B\).
Proof
In this paragraph we reduce to the case where \(B\) is an affine scheme. Let \(B' \to B\) be an étale morphism of algebraic spaces. Observe that conditions (1) and (2) are preserved if we replace \(B\), \(X_i\), \(X\) by \(B'\), \(X_i \times_B B'\), \(X \times_B B'\). Let \(\{B_a \to B\}_{a \in A}\) be an étale covering with \(B_a\) affine, see Properties of Spaces, Lemma 03FX. For \(a \in A\) denote \(X_a\), \(X_{a, i}\) the base changes of \(X\) and the diagram to \(B_a\). For \(a, b \in A\) denote \(X_{a, b}\) and \(X_{a, b, i}\) the base changes of \(X\) and the diagram to \(B_a \times_B B_b\). By Lemma 0GFQ it suffices to prove that \(X_a = \colim X_{a, i}\) and \(X_{a, b} = \colim X_{a, b, i}\). This reduces us to the case where \(B = B_a\) (an affine scheme) or \(B = B_a \times_B B_b\) (a separated scheme). Repeating the argument once more, we conclude that we may assume \(B\) is an affine scheme (this uses that the intersection of affine opens in a separated scheme is affine).
Assume \(B\) is an affine scheme. Let \(Z\) be an algebraic space over \(B\). We have to show \[\Mor_B(X, Z) \longrightarrow \lim \Mor_B(X_i, Z)\] is a bijection.
Proof of injectivity. Let \(f, g : X \to Z\) be morphisms such that the compositions \(f_i, g_i : X_i \to Z\) are the same for all \(i\). Choose an affine scheme \(Z'\) and an étale morphism \(Z' \to Z\). By Properties of Spaces, Lemma 03FX we know we can cover \(Z\) by such affines. Set \(U = X \times_{f, Z} Z'\) and \(U' = X \times_{g, Z} Z'\) and denote \(p : U \to X\) and \(p' : U' \to X\) the projections. Since \(f_i = g_i\) for all \(i\), we see that \[U_i = X_i \times_{f_i, Z} Z' = X_i \times_{g_i, Z} Z' = U'_i\] compatible with transition morphisms. By (1) there is a unique isomorphism \(\epsilon : U \to U'\) as algebraic spaces over \(X\), i.e., with \(p = p' \circ \epsilon\) which is compatible with the displayed identifications. Choose an étale covering \(\{h_a : U_a \to U\}\) with \(U_a\) affine. By (2) we see that \(f \circ p \circ h_a = g \circ p' \circ \epsilon \circ h_a = g \circ p \circ h_a\). Since \(\{h_a : U_a \to U\}\) is an étale covering we conclude \(f \circ p = g \circ p\). Since the collection of morphisms \(p : U \to X\) we obtain in this manner is an étale covering, we conclude that \(f = g\).
Proof of surjectivity. Let \(f_i : X_i \to Z\) be an element of the right hand side of the displayed arrow in the first paragraph of the proof. It suffices to find an étale covering \(\{U_c \to X\}_{c \in C}\) such that the families \(f_{c, i} \in \lim_i \Mor_B(X_i \times_X U_c, Z)\) come from morphisms \(f_c : U_c \to Z\). Namely, by the uniqueness proved above the morphisms \(f_c\) will agree on \(U_c \times_X U_b\) and hence will descend to give the desired morphism \(f : X \to Z\). To find our covering, we first choose an étale covering \(\{g_a : Z_a \to Z\}_{a \in A}\) where each \(Z_a\) is affine. Then we let \(U_{a, i} = X_i \times_{f_i, Z} Z_a\). By (1) we find \(U_{a, i} = X_i \times_X U_a\) for some algebraic spaces \(U_a\) étale over \(X\). Then we choose étale coverings \(\{U_{a, b} \to U_a\}_{b \in B_a}\) with \(U_{a, b}\) affine and we consider the morphisms \[U_{a, b, i} = X_i \times_X U_{a, b} \to X_i \times_X U_a = X_i \times_{f_i, Z} Z_a \to Z_a\] By (2) we obtain morphisms \(f_{a, b} : U_{a, b} \to Z_a\) compatible with these morphisms. Setting \(C = \coprod_{a \in A} B_a\) and for \(c \in C\) corresponding to \(b \in B_a\) setting \(U_c = U_{a, b}\) and \(f_c = g_a \circ f_{a, b} : U_c \to Z\) we conclude.
Here is an application of these ideas to reduce the general case to the case of separated algebraic spaces.
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(\mathcal{I} \to (\Sch/S)_{fppf}\), \(i \mapsto X_i\) be a diagram of algebraic spaces over \(B\). Assume that
each \(X_i\) is separated over \(B\),
\(X = \colim X_i\) exists in the category of algebraic spaces separated over \(B\),
\(\coprod X_i \to X\) is surjective,
if \(U \to X\) is an étale separated morphism of algebraic spaces and \(U_i = X_i \times_X U\), then \(U = \colim U_i\) in the category of algebraic spaces separated over \(B\), and
every object \((U_i \to X_i)\) of \(\lim X_{i, spaces, \etale}\) with \(U_i \to X_i\) separated is of the form \(U_i = X_i \times_X U\) for some étale separated morphism of algebraic spaces \(U \to X\).
Then \(X = \colim X_i\) in the category of all algebraic spaces over \(B\).
Proof
We encourage the reader to look instead at Lemma 0GHL and its proof.
Let \(Z\) be an algebraic space over \(B\). Suppose that \(f_i : X_i \to Z\) is a family of morphisms such that for each \(i \to j\) the composition \(X_i \to X_j \to Z\) is equal to \(f_i\). We have to construct a morphism of algebraic spaces \(f : X \to Z\) over \(B\) such that we can recover \(f_i\) as the composition \(X_i \to X \to Z\). Let \(W \to Z\) be a surjective étale morphism of a scheme to \(Z\). We may assume that \(W\) is a disjoint union of affines and in particular we may assume that \(W \to Z\) is separated and that \(W\) is separated over \(B\). For each \(i\) set \(U_i = W \times_{Z, f_i} X_i\) and denote \(h_i : U_i \to W\) the projection. Then \(U_i \to X_i\) forms an object of \(\lim X_{i, spaces, \etale}\) with \(U_i \to X_i\) separated. By assumption (5) we can find a separated étale morphism \(U \to X\) of algebraic spaces and (functorial) isomorphisms \(U_i = X_i \times_X U\). By assumption (4) there exists a morphism \(h : U \to W\) over \(B\) such that the compositions \(U_i \to U \to W\) are \(h_i\). Let \(g : U \to Z\) be the composition of \(h\) with the map \(W \to Z\). To finish the proof we have to show that \(g : U \to Z\) descends to a morphism \(X \to Z\). To do this, consider the morphism \((h, h) : U \times_X U \to W \times_S W\). Composing with \(U_i \times_{X_i} U_i \to U \times_X U\) we obtain \((h_i, h_i)\) which factors through \(W \times_Z W\). Since \(U \times_X U\) is the colimit of the algebraic spaces \(U_i \times_{X_i} U_i\) in the category of algebraic spaces separated over \(B\) by (4) we see that \((h, h)\) factors through \(W \times_Z W\). Hence the two compositions \(U \times_X U \to U \to W \to Z\) are equal. Because each \(U_i \to X_i\) is surjective and assumption (2) we see that \(U \to X\) is surjective. As \(Z\) is a sheaf for the étale topology, we conclude that \(g : U \to Z\) descends to \(f : X \to Z\) as desired.
Descending étale sheaves
This section is the analogue for algebraic spaces of Étale Cohomology, Section 0GEX.
In order to conveniently express our results we need some notation. Let \(S\) be a scheme. Let \(\mathcal{U} = \{f_i : X_i \to X\}\) be a family of morphisms of algebraic spaces over \(S\) with fixed target. A descent datum for étale sheaves with respect to \(\mathcal{U}\) is a family \(((\mathcal{F}_i)_{i \in I}, (\varphi_{ij})_{i, j \in I})\) where
\(\mathcal{F}_i\) is in \(\Sh(X_{i, \etale})\), and
\(\varphi_{ij} : \text{pr}_{0, small}^{-1} \mathcal{F}_i \longrightarrow \text{pr}_{1, small}^{-1} \mathcal{F}_j\) is an isomorphism in \(\Sh((X_i \times_X X_j)_\etale)\)
such that the cocycle condition holds: the diagrams \[\xymatrix{ \text{pr}_{0, small}^{-1}\mathcal{F}_i \ar[dr]_{\text{pr}_{02, small}^{-1}\varphi_{ik}} \ar[rr]^{\text{pr}_{01, small}^{-1}\varphi_{ij}} & & \text{pr}_{1, small}^{-1}\mathcal{F}_j \ar[dl]^{\text{pr}_{12, small}^{-1}\varphi_{jk}} \\ & \text{pr}_{2, small}^{-1}\mathcal{F}_k }\] commute in \(\Sh((X_i \times_X X_j \times_X X_k)_\etale)\). There is an obvious notion of morphisms of descent data and we obtain a category of descent data. A descent datum \(((\mathcal{F}_i)_{i \in I}, (\varphi_{ij})_{i, j \in I})\) is called effective if there exist a \(\mathcal{F}\) in \(\Sh(X_\etale)\) and isomorphisms \(\varphi_i : f_{i, small}^{-1} \mathcal{F} \to \mathcal{F}_i\) in \(\Sh(X_{i, \etale})\) compatible with the \(\varphi_{ij}\), i.e., such that \[\varphi_{ij} = \text{pr}_{1, small}^{-1} (\varphi_j) \circ \text{pr}_{0, small}^{-1} (\varphi_i^{-1})\] Another way to say this is the following. Given an object \(\mathcal{F}\) of \(\Sh(X_\etale)\) we obtain the canonical descent datum \((f_{i, small}^{-1}\mathcal{F}_i, c_{ij})\) where \(c_{ij}\) is the canonical isomorphism \[c_{ij} : \text{pr}_{0, small}^{-1} f_{i, small}^{-1}\mathcal{F} \longrightarrow \text{pr}_{1, small}^{-1} f_{j, small}^{-1}\mathcal{F}\] The descent datum \(((\mathcal{F}_i)_{i \in I}, (\varphi_{ij})_{i, j \in I})\) is effective if and only if it is isomorphic to the canonical descent datum associated to some \(\mathcal{F}\) in \(\Sh(X_\etale)\).
If the family consists of a single morphism \(\{X \to Y\}\), then we think of a descent datum as a pair \((\mathcal{F}, \varphi)\) where \(\mathcal{F}\) is an object of \(\Sh(X_\etale)\) and \(\varphi\) is an isomorphism \[\text{pr}_{0, small}^{-1} \mathcal{F} \longrightarrow \text{pr}_{1, small}^{-1} \mathcal{F}\] in \(\Sh((X \times_Y X)_\etale)\) such that the cocycle condition holds: \[\xymatrix{ \text{pr}_{0, small}^{-1}\mathcal{F} \ar[dr]_{\text{pr}_{02, small}^{-1}\varphi} \ar[rr]^{\text{pr}_{01, small}^{-1}\varphi} & & \text{pr}_{1, small}^{-1}\mathcal{F} \ar[dl]^{\text{pr}_{12, small}^{-1}\varphi} \\ & \text{pr}_{2, small}^{-1}\mathcal{F} }\] commutes in \(\Sh((X \times_Y X \times_Y X)_\etale)\). There is a notion of morphisms of descent data and effectivity exactly as before.
Lemma
Let \(S\) be a scheme. Let \(\{f_i : X_i \to X\}\) be an étale covering of algebraic spaces. The functor \[\Sh(X_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{f_i : X_i \to X\}\] is an equivalence of categories.
Proof
In Properties of Spaces, Section 03EB we have defined a site \(X_{spaces, \etale}\) whose objects are algebraic spaces étale over \(X\) with étale coverings. Moreover, we have identifications \(\Sh(X_\etale) = \Sh(X_{spaces, \etale})\) compatible with morphisms of algebraic spaces, i.e., compatible with pushforward and pullback. Hence the statement of the lemma follows from the much more general discussion in Sites, Section 04TP.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\{Y_i \to Y\}_{i \in I}\) be an étale covering of algebraic spaces. If for each \(i \in I\) the functor \[\Sh(Y_{i, \etale}) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \times_Y Y_i \to Y_i\}\] is an equivalence of categories and for each \(i, j \in I\) the functor \[\Sh((Y_i \times_Y Y_j)_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt } \{X \times_Y Y_i \times_Y Y_j \to Y_i \times_Y Y_j\}\] is an equivalence of categories, then \[\Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\}\] is an equivalence of categories.
Proof
Formal consequence of Lemma 0GFS and the definitions.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is representable (by schemes) and \(f\) has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation Then \[\Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\}\] is an equivalence of categories.
Proof
Each of the properties of morphisms of algebraic spaces mentioned in the statement of the lemma is preserved by arbitrary base change, see the lists in Spaces, Section 02WE. Thus we can apply Lemma 0GFT to see that we can work étale locally on \(Y\). In this way we reduce to the case where \(Y\) is a scheme; some details omitted. In this case \(X\) is also a scheme and the result follows from Étale Cohomology, Lemma 0GEZ, 0GF0, or 0GF2.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\pi : X' \to X\) be a morphism of algebraic spaces. Assume
\(f \circ \pi\) is representable (by schemes),
\(f \circ \pi\) has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation.
Then \[\Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\}\] is an equivalence of categories.
Proof
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. Then the functor \[\Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\}\] is an equivalence of categories.
Proof
Observe that the base change of a proper surjective morphism is proper and surjective, see Morphisms of Spaces, Lemmas 04WP and 03MH. Hence by Lemma 0GFT we may work étale locally on \(Y\). Hence we reduce to \(Y\) being an affine scheme; some details omitted.
Assume \(Y\) is affine. By Lemma 0GFV it suffices to find a morphism \(X' \to X\) where \(X'\) is a scheme such that \(X' \to Y\) is surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation.
In case \(X \to Y\) is integral and surjective, we can take \(X = X'\) as an integral morphism is representable.
If \(f\) is proper and surjective, then the algebraic space \(X\) is quasi-compact and separated, see Morphisms of Spaces, Section 03HC and Lemma 04ZH. Choose a scheme \(X'\) and a surjective finite morphism \(X' \to X\), see Limits of Spaces, Proposition 09YC. Then \(X' \to Y\) is surjective and proper.
Finally, if \(X \to Y\) is surjective and flat and locally of finite presentation then we can take an affine étale covering \(\{U_i \to X\}\) and set \(X'\) equal to the disjoint \(\coprod U_i\).
Lemma
Let \(S\) be a scheme. Let \(\{f_i : X_i \to X\}\) be an fppf covering of algebraic spaces over \(S\). The functor \[\Sh(X_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{f_i : X_i \to X\}\] is an equivalence of categories.
Proof
We have Lemma 0GFW for the morphism \(f : \coprod X_i \to X\). Then a formal argument shows that descent data for \(f\) are the same thing as descent data for the covering, compare with Descent, Lemma 023X. Details omitted.
Lemma
Let \(S\) be a scheme. Let \(f : Y' \to Y\) be a proper morphism of algebraic spaces over \(S\). Let \(i : Z \to Y\) be a closed immersion. Set \(E = Z \times_Y Y'\). Picture \[\xymatrix{ E \ar[d]_g \ar[r]_j & Y' \ar[d]^f \\ Z \ar[r]^i & Y }\] If \(f\) is an isomorphism over \(Y \setminus Z\), then the functor \[\Sh(Y_\etale) \longrightarrow \Sh(Y'_\etale) \times_{\Sh(E_\etale)} \Sh(Z_\etale)\] is an equivalence of categories.
Proof
Observe that \(X = Y' \coprod Z \to Y\) is a proper surjective morphism. Thus it suffice to construct an equivalence of categories \[\Sh(Y'_\etale) \times_{\Sh(E_\etale)} \Sh(Z_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\}\] compatible with pullback functors from \(Y\) because then we can use Lemma 0GFW to conclude. Thus let \((\mathcal{G}', \mathcal{G}, \alpha)\) be an object of \(\Sh(Y'_\etale) \times_{\Sh(E_\etale)} \Sh(Z_\etale)\) with notation as in Categories, Example 003R. Then we can consider the sheaf \(\mathcal{F}\) on \(X\) defined by taking \(\mathcal{G}'\) on the summand \(Y'\) and \(\mathcal{G}\) on the summand \(Z\). We have \[X \times_Y X = Y' \times_Y Y' \amalg Y' \times_Y Z \amalg Z \times_Y Y' \amalg Z \times_Y Z = Y' \times_Y Y' \amalg E \amalg E \amalg Z\] The isomorphisms of the two pullbacks of \(\mathcal{F}\) to this algebraic space are obvious over the summands \(E\), \(E\), \(Z\). The interesting part of the proof is to find an isomorphism \(\text{pr}_{0, small}^{-1}\mathcal{G}' \to \text{pr}_{1, small}^{-1}\mathcal{G}'\) over \(Y' \times_Y Y'\) satisfying the cocycle condition. However, our assumption that \(Y' \to Y\) is an isomorphism over \(Y \setminus Z\) implies that \[h : Y \coprod E \times_Z E \longrightarrow Y' \times_Y Y'\] is a surjective proper morphism. (It is in fact a finite morphism as it is the disjoint union of two closed immersions.) Hence it suffices to construct an isomorphism of the pullbacks of \(\text{pr}_{0, small}^{-1}\mathcal{G}'\)and \(\text{pr}_{1, small}^{-1}\mathcal{G}'\) by \(h_{small}\) satisfying a certain cocycle condition. For the diagonal, it is clear how to do this. And for the pullback to \(E \times_Z E\) we use that both sheaves pull back to the pullback of \(\mathcal{G}\) by the morphism \(E \times_Z E \to Z\). We omit the details.
Descending étale morphisms of algebraic spaces
In this section we combine the glueing results for étale sheaves given in Section 0GFR with the flexibility of algebraic spaces to get some descent statements for étale morphisms of algebraic spaces.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper surjective morphism of algebraic spaces over \(S\). Any descent datum \((U/X, \varphi)\) relative to \(f\) (Descent on Spaces, Definition 0ADG) with \(U\) étale over \(X\) is effective (Descent on Spaces, Definition 0ADQ). More precisely, there exists an étale morphism \(V \to Y\) of algebraic spaces whose corresponding canonical descent datum is isomorphic to \((U/X, \varphi)\).
Proof
Recall that \(U\) gives rise to a representable sheaf \(\mathcal{F} = h_U\) in \(\Sh(X_{spaces, \etale}) = \Sh(X_\etale)\), see Properties of Spaces, Section 03EB. The descent datum on \(U\) relative to \(f\) exactly gives a descent datum \((\mathcal{F}, \varphi)\) for étale sheaves with respect to \(\{X \to Y\}\). By Lemma 0GFW this descent datum is effective. Let \(\mathcal{G}\) be the corresponding sheaf on \(Y_\etale\). By Properties of Spaces, Lemma 0GF6 we obtain an étale morphism \(V \to Y\) of algebraic spaces corresponding to \(\mathcal{G}\); we omit the verification of the set theoretic condition1. The given isomorphism \(\mathcal{F} \to f_{small}^{-1}\mathcal{G}\) corresponds to an isomorphism \(U \to V \times_Y X\) compatible with the descent datum.
Lemma
Let \(S\) be a scheme. Let \(f : Y' \to Y\) be a proper morphism of algebraic spaces over \(S\). Let \(i : Z \to Y\) be a closed immersion. Set \(E = Z \times_Y Y'\). Picture \[\xymatrix{ E \ar[d]_g \ar[r]_j & Y' \ar[d]^f \\ Z \ar[r]^i & Y }\] If \(f\) is an isomorphism over \(Y \setminus Z\), then the functor \[Y_{spaces, \etale} \longrightarrow Y'_{spaces, \etale} \times_{E_{spaces, \etale}} Z_{spaces, \etale}\] is an equivalence of categories.
Proof
Let \((V' \to Y', W \to Z, \alpha)\) be an object of the right hand side. Recall that \(V'\), resp. \(W\) gives rise to a representable sheaf \(\mathcal{G}' = h_{V'}\) in \(\Sh(Y'_{spaces, \etale}) = \Sh(Y'_\etale)\), resp. \(\mathcal{G} = h_W\) in \(\Sh(Z_{spaces, \etale}) = \Sh(Z_\etale)\), see Properties of Spaces, Section 03EB. The isomorphism \(\alpha : V' \times_{Y'} E \to W \times_Z E\) determines an isomorphism \(j_{small}^{-1}\mathcal{G}' \to g_{small}^{-1}\mathcal{G}\) of sheaves on \(E\). By Lemma 0GFY we obtain a unique sheaf \(\mathcal{F}\) on \(Y\) pulling pack to \(\mathcal{G}'\) and \(\mathcal{G}\) compatibly with the isomorphism. By Properties of Spaces, Lemma 0GF6 we obtain an étale morphism \(V \to Y\) of algebraic spaces corresponding to \(\mathcal{F}\); we omit the verification of the set theoretic condition2. The given isomorphism \(\mathcal{G}' \to f_{small}^{-1}\mathcal{F}\) and \(\mathcal{G} \to i_{small}^{-1}\mathcal{F}\) corresponds to isomorphisms \(V' \to V \times_Y Y'\) and \(W \to V \times_Y Z\) compatible with \(\alpha\) as desired.
Pushouts along thickenings and affine morphisms
This section is analogue of More on Morphisms, Section 07RS.
Lemma
Let \(S\) be a scheme. Let \(X \to X'\) be a thickening of schemes over \(S\) and let \(X \to Y\) be an affine morphism of schemes over \(S\). Let \(Y' = Y \amalg_X X'\) be the pushout in the category of schemes (see More on Morphisms, Lemma 07RT). Then \(Y'\) is also a pushout in the category of algebraic spaces over \(S\).
Proof
This is an immediate consequence of Lemma 07SX and More on Morphisms, Lemmas 07RT, 07RV, and 07RX.
Lemma
Let \(S\) be a scheme. Let \(X \to X'\) be a thickening of algebraic spaces over \(S\) and let \(X \to Y\) be an affine morphism of algebraic spaces over \(S\). Then there exists a pushout \[\xymatrix{ X \ar[r] \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r] & Y \amalg_X X' }\] in the category of algebraic spaces over \(S\). Moreover \(Y' = Y \amalg_X X'\) is a thickening of \(Y\) and \[\mathcal{O}_{Y'} = \mathcal{O}_Y \times_{f_*\mathcal{O}_X} f'_*\mathcal{O}_{X'}\] as sheaves on \(Y_\etale = (Y')_\etale\).
Proof
Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Set \(U = V \times_Y X\). This is a scheme affine over \(V\) with a surjective étale morphism \(U \to X\). By More on Morphisms of Spaces, Lemma 05ZS there exists a \(U' \to X'\) surjective étale with \(U = U' \times_{X'} X\). In particular the morphism of schemes \(U \to U'\) is a thickening too. Apply More on Morphisms, Lemma 07RT to obtain a pushout \(V' = V \amalg_U U'\) in the category of schemes.
We repeat this procedure to construct a pushout \[\xymatrix{ U \times_X U \ar[d] \ar[r] & U' \times_{X'} U' \ar[d] \\ V \times_Y V \ar[r] & R' }\] in the category of schemes. Consider the morphisms \[U \times_X U \to U \to V',\quad U' \times_{X'} U' \to U' \to V',\quad V \times_Y V \to V \to V'\] where we use the first projection in each case. Clearly these glue to give a morphism \(t' : R' \to V'\) which is étale by More on Morphisms, Lemma 07RX. Similarly, we obtain \(s' : R' \to V'\) étale. The morphism \(j' = (t', s') : R' \to V' \times_S V'\) is unramified (as \(t'\) is étale) and a monomorphism when restricted to the closed subscheme \(V \times_Y V \subset R'\). As \(V \times_Y V \subset R'\) is a thickening it follows that \(j'\) is a monomorphism too. Finally, \(j'\) is an equivalence relation as we can use the functoriality of pushouts of schemes to construct a morphism \(c' : R' \times_{s', V', t'} R' \to R'\) (details omitted). At this point we set \(Y' = V'/R'\), see Spaces, Theorem 02WW.
We have morphisms \(X' = U'/U' \times_{X'} U' \to V'/R' = Y'\) and \(Y = V/V \times_Y V \to V'/R' = Y'\). By construction these fit into the commutative diagram \[\xymatrix{ X \ar[r] \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r] & Y' }\] Since \(Y \to Y'\) is a thickening we have \(Y_\etale = (Y')_\etale\), see More on Morphisms of Spaces, Lemma 05ZS. The commutativity of the diagram gives a map of sheaves \[\mathcal{O}_{Y'} \longrightarrow \mathcal{O}_Y \times_{f_*\mathcal{O}_X} f'_*\mathcal{O}_{X'}\] on this set. By More on Morphisms, Lemma 07RT this map is an isomorphism when we restrict to the scheme \(V'\), hence it is an isomorphism.
To finish the proof we show that the diagram above is a pushout in the category of algebraic spaces. To see this, let \(Z\) be an algebraic space and let \(a' : X' \to Z\) and \(b : Y \to Z\) be morphisms of algebraic spaces. By Lemma 07SY we obtain a unique morphism \(h : V' \to Z\) fitting into the commutative diagrams \[\vcenter{ \xymatrix{ U' \ar[d] \ar[r] & V' \ar[d]^h \\ X' \ar[r]^{a'} & Z } } \quad\text{and}\quad \vcenter{ \xymatrix{ V \ar[r] \ar[d] & V' \ar[d]^h \\ Y \ar[r]^b & Z } }\] The uniqueness shows that \(h \circ t' = h \circ s'\). Hence \(h\) factors uniquely as \(V' \to Y' \to Z\) and we win.
In the following lemma we use the fibre product of categories as defined in Categories, Example 003R.
Lemma
Let \(S\) be a base scheme. Let \(X \to X'\) be a thickening of algebraic spaces over \(S\) and let \(X \to Y\) be an affine morphism of algebraic spaces over \(S\). Let \(Y' = Y \amalg_X X'\) be the pushout (see Lemma 07VX). Base change gives a functor \[F : (\textit{Spaces}/Y') \longrightarrow (\textit{Spaces}/Y) \times_{(\textit{Spaces}/Y')} (\textit{Spaces}/X')\] given by \(V' \longmapsto (V' \times_{Y'} Y, V' \times_{Y'} X', 1)\) which sends \((\Sch/Y')\) into \((\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')\). The functor \(F\) has a left adjoint \[G : (\textit{Spaces}/Y) \times_{(\textit{Spaces}/Y')} (\textit{Spaces}/X') \longrightarrow (\textit{Spaces}/Y')\] which sends the triple \((V, U', \varphi)\) to the pushout \(V \amalg_{(V \times_Y X)} U'\) in the category of algebraic spaces over \(S\). The functor \(G\) sends \((\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')\) into \((\Sch/Y')\).
Proof
The proof is completely formal. Since the morphisms \(X \to X'\) and \(X \to Y\) are representable it is clear that \(F\) sends \((\Sch/Y')\) into \((\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')\).
Let us construct \(G\). Let \((V, U', \varphi)\) be an object of the fibre product category. Set \(U = U' \times_{X'} X\). Note that \(U \to U'\) is a thickening. Since \(\varphi : V \times_Y X \to U' \times_{X'} X = U\) is an isomorphism we have a morphism \(U \to V\) over \(X \to Y\) which identifies \(U\) with the fibre product \(X \times_Y V\). In particular \(U \to V\) is affine, see Morphisms of Spaces, Lemma 03WI. Hence we can apply Lemma 07VX to get a pushout \(V' = V \amalg_U U'\). Denote \(V' \to Y'\) the morphism we obtain in virtue of the fact that \(V'\) is a pushout and because we are given morphisms \(V \to Y\) and \(U' \to X'\) agreeing on \(U\) as morphisms into \(Y'\). Setting \(G(V, U', \varphi) = V'\) gives the functor \(G\).
If \((V, U', \varphi)\) is an object of \((\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')\) then \(U = U' \times_{X'} X\) is a scheme too and we can form the pushout \(V' = V \amalg_U U'\) in the category of schemes by More on Morphisms, Lemma 07RT. By Lemma 07SY this is also a pushout in the category of schemes, hence \(G\) sends \((\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')\) into \((\Sch/Y')\).
Let us prove that \(G\) is a left adjoint to \(F\). Let \(Z\) be an algebraic space over \(Y'\). We have to show that \[\Mor(V', Z) = \Mor((V, U', \varphi), F(Z))\] where the morphism sets are taking in their respective categories. Let \(g' : V' \to Z\) be a morphism. Denote \(\tilde g\), resp. \(\tilde f'\) the composition of \(g'\) with the morphism \(V \to V'\), resp. \(U' \to V'\). Base change \(\tilde g\), resp. \(\tilde f'\) by \(Y \to Y'\), resp. \(X' \to Y'\) to get a morphism \(g : V \to Z \times_{Y'} Y\), resp. \(f' : U' \to Z \times_{Y'} X'\). Then \((g, f')\) is an element of the right hand side of the equation above (details omitted). Conversely, suppose that \((g, f') : (V, U', \varphi) \to F(Z)\) is an element of the right hand side. We may consider the composition \(\tilde g : V \to Z\), resp. \(\tilde f' : U' \to Z\) of \(g\), resp. \(f\) by \(Z \times_{Y'} X' \to Z\), resp. \(Z \times_{Y'} Y \to Z\). Then \(\tilde g\) and \(\tilde f'\) agree as morphism from \(U\) to \(Z\). By the universal property of pushout, we obtain a morphism \(g' : V' \to Z\), i.e., an element of the left hand side. We omit the verification that these constructions are mutually inverse.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ A \ar[r] \ar[d] & C \ar[d] \ar[r] & E \ar[d] \\ B \ar[r] & D \ar[r] & F }\] be a commutative diagram of algebraic spaces over \(S\). Assume that \(A, B, C, D\) and \(A, B, E, F\) form cartesian squares and that \(B \to D\) is surjective étale. Then \(C, D, E, F\) is a cartesian square.
Proof
This is formal.
Lemma
In the situation of Lemma 07VY the functor \(F \circ G\) is isomorphic to the identity functor.
Proof
We will prove that \(F \circ G\) is isomorphic to the identity by reducing this to the corresponding statement of More on Morphisms, Lemma 07RV.
Choose a scheme \(Y_1\) and a surjective étale morphism \(Y_1 \to Y\). Set \(X_1 = Y_1 \times_Y X\). This is a scheme affine over \(Y_1\) with a surjective étale morphism \(X_1 \to X\). By More on Morphisms of Spaces, Lemma 05ZS there exists a \(X'_1 \to X'\) surjective étale with \(X_1 = X_1' \times_{X'} X\). In particular the morphism of schemes \(X_1 \to X_1'\) is a thickening too. Apply More on Morphisms, Lemma 07RT to obtain a pushout \(Y_1' = Y_1 \amalg_{X_1} X_1'\) in the category of schemes. In the proof of Lemma 07VX we constructed \(Y'\) as a quotient of an étale equivalence relation on \(Y_1'\) such that we get a commutative diagram [07W1]\[\begin{equation} \vcenter{ \xymatrix{ & X \ar[rr] \ar'[d][dd] & & X' \ar[dd] \\ X_1 \ar[rr] \ar[dd] \ar[ru] & & X_1' \ar[dd] \ar[ru] & \\ & Y \ar'[r][rr] & & Y' \\ Y_1 \ar[rr] \ar[ru] & & Y_1' \ar[ru] } } \end{equation}\] where all squares except the front and back squares are cartesian (the front and back squares are pushouts) and the northeast arrows are surjective étale. Denote \(F_1\), \(G_1\) the functors constructed in More on Morphisms, Lemma 07RV for the front square. Then the diagram of categories \[\xymatrix{ (\Sch/Y_1') \ar@<-1ex>[r]_-{F_1} \ar[d] & (\Sch/Y_1) \times_{(\Sch/Y_1')} (\Sch/X_1') \ar[d] \ar@<-1ex>[l]_-{G_1} \\ (\textit{Spaces}/Y') \ar@<-1ex>[r]_-F & (\textit{Spaces}/Y) \times_{(\textit{Spaces}/Y')} (\textit{Spaces}/X') \ar@<-1ex>[l]_-G }\] is commutative by simple considerations regarding base change functors and the agreement of pushouts in schemes with pushouts in spaces of Lemma 07SY.
Let \((V, U', \varphi)\) be an object of \((\textit{Spaces}/Y) \times_{(\textit{Spaces}/Y')} (\textit{Spaces}/X')\). Denote \(U = U' \times_{X'} X\) so that \(G(V, U', \varphi) = V \amalg_U U'\). Choose a scheme \(V_1\) and a surjective étale morphism \(V_1 \to Y_1 \times_Y V\). Set \(U_1 = V_1 \times_Y X\). Then \[U_1 = V_1 \times_Y X \longrightarrow (Y_1 \times_Y V) \times_Y X = X_1 \times_Y V = X_1 \times_X X \times_Y V = X_1 \times_X U\] is surjective étale too. By More on Morphisms of Spaces, Lemma 05ZS there exists a thickening \(U_1 \to U_1'\) and a surjective étale morphism \(U_1' \to X_1' \times_{X'} U'\) whose base change to \(X_1 \times_X U\) is the displayed morphism. At this point \((V_1, U'_1, \varphi_1)\) is an object of \((\Sch/Y_1) \times_{(\Sch/Y_1')} (\Sch/X_1')\). In the proof of Lemma 07VX we constructed \(G(V, U', \varphi) = V \amalg_U U'\) as a quotient of an étale equivalence relation on \(G_1(V_1, U_1', \varphi_1) = V_1 \amalg_{U_1} U_1'\) such that we get a commutative diagram [07W2]\[\begin{equation} \vcenter{ \xymatrix{ & U \ar[rr] \ar'[d][dd] & & U' \ar[dd] \\ U_1 \ar[rr] \ar[dd] \ar[ru] & & U_1' \ar[dd] \ar[ru] & \\ & V \ar'[r][rr] & & G(V, U', \varphi) \\ V_1 \ar[rr] \ar[ru] & & G_1(V_1, U_1', \varphi_1) \ar[ru] } } \end{equation}\] where all squares except the front and back squares are cartesian (the front and back squares are pushouts) and the northeast arrows are surjective étale. In particular \[G_1(V_1, U_1', \varphi_1) \to G(V, U', \varphi)\] is surjective étale.
Finally, we come to the proof of the lemma. We have to show that the adjunction mapping \((V, U', \varphi) \to F(G(V, U', \varphi))\) is an isomorphism. We know \((V_1, U_1', \varphi_1) \to F_1(G_1(V_1, U_1', \varphi_1))\) is an isomorphism by More on Morphisms, Lemma 07RV. Recall that \(F\) and \(F_1\) are given by base change. Using the properties of (07W2) and Lemma 07VZ we see that \(V \to G(V, U', \varphi) \times_{Y'} Y\) and \(U' \to G(V, U', \varphi) \times_{Y'} X'\) are isomorphisms, i.e., \((V, U', \varphi) \to F(G(V, U', \varphi))\) is an isomorphism.
Lemma
Let \(S\) be a base scheme. Let \(X \to X'\) be a thickening of algebraic spaces over \(S\) and let \(X \to Y\) be an affine morphism of algebraic spaces over \(S\). Let \(Y' = Y \amalg_X X'\) be the pushout (see Lemma 07VX). Let \(V' \to Y'\) be a morphism of algebraic spaces over \(S\). Set \(V = Y \times_{Y'} V'\), \(U' = X' \times_{Y'} V'\), and \(U = X \times_{Y'} V'\). There is an equivalence of categories between
quasi-coherent \(\mathcal{O}_{V'}\)-modules flat over \(Y'\), and
the category of triples \((\mathcal{G}, \mathcal{F}', \varphi)\) where
\(\mathcal{G}\) is a quasi-coherent \(\mathcal{O}_V\)-module flat over \(Y\),
\(\mathcal{F}'\) is a quasi-coherent \(\mathcal{O}_{U'}\)-module flat over \(X\), and
\(\varphi : (U \to V)^*\mathcal{G} \to (U \to U')^*\mathcal{F}'\) is an isomorphism of \(\mathcal{O}_U\)-modules.
The equivalence maps \(\mathcal{G}'\) to \(((V \to V')^*\mathcal{G}', (U' \to V')^*\mathcal{G}', can)\). Suppose \(\mathcal{G}'\) corresponds to the triple \((\mathcal{G}, \mathcal{F}', \varphi)\). Then
\(\mathcal{G}'\) is a finite type \(\mathcal{O}_{V'}\)-module if and only if \(\mathcal{G}\) and \(\mathcal{F}'\) are finite type \(\mathcal{O}_Y\) and \(\mathcal{O}_{U'}\)-modules.
if \(V' \to Y'\) is locally of finite presentation, then \(\mathcal{G}'\) is an \(\mathcal{O}_{V'}\)-module of finite presentation if and only if \(\mathcal{G}\) and \(\mathcal{F}'\) are \(\mathcal{O}_Y\) and \(\mathcal{O}_{U'}\)-modules of finite presentation.
Proof
A quasi-inverse functor assigns to the triple \((\mathcal{G}, \mathcal{F}', \varphi)\) the fibre product \[(V \to V')_*\mathcal{G} \times_{(U \to V')_*\mathcal{F}} (U' \to V')_*\mathcal{F}'\] where \(\mathcal{F} = (U \to U')^*\mathcal{F}'\). This works, because on affines étale over \(V'\) and \(Y'\) we recover the equivalence of More on Algebra, Lemma 07RW. Details omitted.
Parts (a) and (b) reduce by étale localization (Properties of Spaces, Section 05VR) to the case where \(V'\) and \(Y'\) are affine in which case the result follows from More on Algebra, Lemmas 08IH and 08KP.
Lemma
In the situation of Lemma 07W0. If \(V' = G(V, U', \varphi)\) for some triple \((V, U', \varphi)\), then
\(V' \to Y'\) is locally of finite type if and only if \(V \to Y\) and \(U' \to X'\) are locally of finite type,
\(V' \to Y'\) is flat if and only if \(V \to Y\) and \(U' \to X'\) are flat,
\(V' \to Y'\) is flat and locally of finite presentation if and only if \(V \to Y\) and \(U' \to X'\) are flat and locally of finite presentation,
\(V' \to Y'\) is smooth if and only if \(V \to Y\) and \(U' \to X'\) are smooth,
\(V' \to Y'\) is étale if and only if \(V \to Y\) and \(U' \to X'\) are étale, and
add more here as needed.
If \(W'\) is flat over \(Y'\), then the adjunction mapping \(G(F(W')) \to W'\) is an isomorphism. Hence \(F\) and \(G\) define mutually quasi-inverse functors between the category of spaces flat over \(Y'\) and the category of triples \((V, U', \varphi)\) with \(V \to Y\) and \(U' \to X'\) flat.
Proof
Choose a diagram (07W1) as in the proof of Lemma 07W0.
Proof of (1) – (5). Let \((V, U', \varphi)\) be an object of \((\textit{Spaces}/Y) \times_{(\textit{Spaces}/Y')} (\textit{Spaces}/X')\). Construct a diagram (07W2) as in the proof of Lemma 07W0. Then the base change of \(G(V, U', \varphi) \to Y'\) to \(Y'_1\) is \(G_1(V_1, U_1', \varphi_1) \to Y_1'\). Hence (1) – (5) follow immediately from the corresponding statements of More on Morphisms, Lemma 07RX for schemes.
Suppose that \(W' \to Y'\) is flat. Choose a scheme \(W'_1\) and a surjective étale morphism \(W'_1 \to Y_1' \times_{Y'} W'\). Observe that \(W'_1 \to W'\) is surjective étale as a composition of surjective étale morphisms. We know that \(G_1(F_1(W_1')) \to W_1'\) is an isomorphism by More on Morphisms, Lemma 07RX applied to \(W'_1\) over \(Y'_1\) and the front of the diagram (with functors \(G_1\) and \(F_1\) as in the proof of Lemma 07W0). Then the construction of \(G(F(W'))\) (as a pushout, i.e., as constructed in Lemma 07VX) shows that \(G_1(F_1(W'_1)) \to G(F(W))\) is surjective étale. Whereupon we conclude that \(G(F(W)) \to W\) is étale, see for example Properties of Spaces, Lemma 03FS. But \(G(F(W)) \to W\) is an isomorphism on underlying reduced algebraic spaces (by construction), hence it is an isomorphism.
Pushouts along closed immersions and integral morphisms
This section is analogue of More on Morphisms, Section 0ECH.
Lemma
In More on Morphisms, Situation 0ECI let \(Y \amalg_Z X\) be the pushout in the category of schemes (More on Morphisms, Proposition 0E25). Then \(Y \amalg_Z X\) is also a pushout in the category of algebraic spaces over \(S\).
Proof
This is a consequence of Lemma 07SX, the proposition mentioned in the lemma and More on Morphisms, Lemmas 0ECK and 0ECL. Conditions (1) and (2) of Lemma 07SX follow immediately. To see (3) and (4) note that an étale morphism is locally quasi-finite and use that the equivalence of categories of More on Morphisms, Lemma 0ECL is constructed using the pushout construction of More on Morphisms, Lemmas 0ECK. Minor details omitted.
Pushouts and derived categories
In this section we discuss the behaviour of the derived category of modules under pushouts.
Lemma
Let \(S\) be a scheme. Consider a pushout \[\xymatrix{ X \ar[r]_i \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r]^j & Y' }\] in the category of algebraic spaces over \(S\) as in Lemma 07VX. Assume \(i\) is a thickening. Then the essential image of the functor3 \[D(\mathcal{O}_{Y'}) \longrightarrow D(\mathcal{O}_Y) \times_{D(\mathcal{O}_X)} D(\mathcal{O}_{X'})\] contains every triple \((M, K', \alpha)\) where \(M \in D(\mathcal{O}_Y)\) and \(K' \in D(\mathcal{O}_{X'})\) are pseudo-coherent.
Proof
Let \((M, K', \alpha)\) be an object of the target of the functor of the lemma. Here \(\alpha : Lf^*M \to Li^*K'\) is an isomorphism which is adjoint to a map \(\beta : M \to Rf_*Li^*K'\). Thus we obtain maps \[Rj_*M \xrightarrow{Rj_*\beta} Rj_*Rf_*Li^*K' = Rf'_*Ri_*Li^*K' \leftarrow Rf'_*K'\] where the arrow pointing left comes from \(K' \to Ri_*Li^*K'\). Choose a distinguished triangle \[M' \to Rj_*M \oplus Rf'_*K' \to Rj_*Rf_*Li^*K' \to M'[1]\] in \(D(\mathcal{O}_{Y'})\). The first arrow defines canonical maps \(Lj^*M' \to M\) and \(L(f')^*M' \to K'\) compatible with \(\alpha\). Thus it suffices to show that the maps \(Lj^*M' \to M\) and \(L(f')^*M' \to K\) are isomorphisms. This we may check étale locally on \(Y'\), hence we may assume \(Y'\) is étale.
Assume \(Y'\) affine and \(M \in D(\mathcal{O}_Y)\) and \(K' \in D(\mathcal{O}_{X'})\) are pseudo-coherent. Say our pushout corresponds to the fibre product \[\xymatrix{ B & B' \ar[l] \\ A \ar[u] & A' \ar[l] \ar[u] }\] of rings where \(B' \to B\) is surjective with locally nilpotent kernel \(I\) (and hence \(A' \to A\) is surjective with locally nilpotent kernel \(I\) as well). The assumption on \(M\) and \(K'\) imply that \(M\) comes from a pseudo-coherent object of \(D(A)\) and \(K'\) comes from a pseudo-coherent object of \(D(B')\), see Derived Categories of Spaces, Lemmas 08JL, 071Q, and 08HE and Derived Categories of Schemes, Lemma 06Z0 and 08E7. Moreover, pushforward and derived pullback agree with the corresponding operations on derived categories of modules, see Derived Categories of Spaces, Remark 08GH and Derived Categories of Schemes, Lemmas 0DJK and 08DW. This reduces us to the statement formulated in the next paragraph. (To be sure these references show the object \(M'\) lies \(D_\QCoh(\mathcal{O}_{Y'})\) as this is a triangulated subcategory of \(D(\mathcal{O}_{Y'})\).)
Given a diagram of rings as above and a triple \((M, K', \alpha)\) where \(M \in D(A)\), \(K' \in D(B')\) are pseudo-coherent and \(\alpha : M \otimes_A^\mathbf{L} B \to K' \otimes_{B'}^\mathbf{L} B\) is an isomorphism suppose we have distinguished triangle \[M' \to M \oplus K' \to K' \otimes_{B'}^\mathbf{L} B \to M'[1]\] in \(D(A')\). Goal: show that the induced maps \(M' \otimes_{A'}^\mathbf{L} A \to M\) and \(M' \otimes_{A'}^\mathbf{L} B' \to K'\) are isomorphisms. To do this, choose a bounded above complex \(E^\bullet\) of finite free \(A\)-modules representing \(M\). Since \((B', I)\) is a henselian pair (More on Algebra, Lemma 0ALI) with \(B = B'/I\) we may apply More on Algebra, Lemma 0BCE to see that there exists a bounded above complex \(P^\bullet\) of free \(B'\)-modules such that \(\alpha\) is represented by an isomorphism \(E^\bullet \otimes_A B \cong P^\bullet \otimes_{B'} B\). Then we can consider the short exact sequence \[0 \to L^\bullet \to E^\bullet \oplus P^\bullet \to P^\bullet \otimes_{B'} B \to 0\] of complexes of \(B'\)-modules. More on Algebra, Lemma 0D2J implies \(L^\bullet\) is a bounded above complex of finite projective \(A'\)-modules (in fact it is rather easy to show directly that \(L^n\) is finite free in our case) and that we have \(L^\bullet \otimes_{A'} A = E^\bullet\) and \(L^\bullet \otimes_{A'} B' = P^\bullet\). The short exact sequence gives a distinguished triangle \[L^\bullet \to M \oplus K' \to K' \otimes_{B'}^\mathbf{L} B \to (L^\bullet)[1]\] in \(D(A')\) (Derived Categories, Section 014Z) which is isomorphic to the given distinguished triangle by general properties of triangulated categories (Derived Categories, Section 05QN). In other words, \(L^\bullet\) represents \(M'\) compatibly with the given maps. Thus the maps \(M' \otimes_{A'}^\mathbf{L} A \to M\) and \(M' \otimes_{A'}^\mathbf{L} B' \to K'\) are isomorphisms because we just saw that the corresponding thing is true for \(L^\bullet\).
Constructing elementary distinguished squares
Elementary distinguished squares were defined in Derived Categories of Spaces, Section 08GL.
Lemma
Let \(S\) be a scheme. Let \((U \subset W, f : V \to W)\) be an elementary distinguished square. Then \[\xymatrix{ U \times_W V \ar[r] \ar[d] & V \ar[d]^f \\ U \ar[r] & W }\] is a pushout in the category of algebraic spaces over \(S\).
Proof
Observe that \(U \amalg V \to W\) is a surjective étale morphism. The fibre product \[(U \amalg V) \times_W (U \amalg V)\] is the disjoint union of four pieces, namely \(U = U \times_W U\), \(U \times_W V\), \(V \times_W U\), and \(V \times_W V\). There is a surjective étale morphism \[V \amalg (U \times_W V) \times_U (U \times_W V) \longrightarrow V \times_W V\] because \(f\) induces an isomorphism over \(W \setminus U\) (part of the definition of being an elementary distinguished square). Let \(B\) be an algebraic space over \(S\) and let \(g : V \to B\) and \(h : U \to B\) be morphisms over \(S\) which agree after restricting to \(U \times_W V\). Then the description of \((U \amalg V) \times_W (U \amalg V)\) given above shows that \(h \amalg g : U \amalg V \to B\) equalizes the two projections. Since \(B\) is a sheaf for the étale topology we obtain a unique factorization of \(h \amalg g\) through \(W\) as desired.
Lemma
Let \(S\) be a scheme. Let \(V\), \(U\) be algebraic spaces over \(S\). Let \(V' \subset V\) be an open subspace and let \(f' : V' \to U\) be a separated étale morphism of algebraic spaces over \(S\). Then there exists a pushout \[\xymatrix{ V' \ar[r] \ar[d] & V \ar[d]^f \\ U \ar[r] & W }\] in the category of algebraic spaces over \(S\) and moreover \((U \subset W, f : V \to W)\) is an elementary distinguished square.
Proof
We are going to construct \(W\) as the quotient of an étale equivalence relation \(R\) on \(U \amalg V\). Such a quotient is an algebraic space for example by Bootstrap, Theorem 04S6. Moreover, the proof of Lemma 0DVI tells us to take \[R = U \amalg V' \amalg V' \amalg V \amalg (V' \times_U V' \setminus \Delta_{V'/U}(V'))\] Since we assumed \(V' \to U\) is separated, the image of \(\Delta_{V'/U}\) is closed and hence the complement is an open subspace. The morphism \(j : R \to (U \amalg V) \times_S (U \amalg V)\) is given by \[u,\ v',\ v',\ v,\ (v'_1, v'_2) \mapsto (u, u),\ (f'(v'), v'),\ (v', f'(v')),\ (v, v),\ (v'_1, v'_2)\] with obvious notation. It is immediately verified that this is a monomorphism, an equivalence relation, and that the induced morphisms \(s, t : R \to U \amalg V\) are étale. Let \(W = (U \amalg V)/R\) be the quotient algebraic space. We obtain a commutative diagram as in the statement of the lemma. To finish the proof it suffices to show that this diagram is an elementary distinguished square, since then Lemma 0DVI implies that it is a pushout. Thus we have to show that \(U \to W\) is open and that \(f\) is étale and is an isomorphism over \(W \setminus U\). This follows from the choice of \(R\); we omit the details.
Formal glueing of quasi-coherent modules
This section is the analogue of More on Algebra, Section 05E5. In the case of morphisms of schemes, the result can be found in the paper by Joyet [Joyet]; this is a good place to start reading. For a discussion of applications to descent problems for stacks, see the paper by Moret-Bailly [MB]. In the case of an affine morphism of schemes there is a statement in the appendix of the paper [Ferrand-Raynaud] but one needs to add the hypothesis that the closed subscheme is cut out by a finitely generated ideal (as in the paper by Joyet) since otherwise the result does not hold. A generalization of this material to (higher) derived categories with potential applications to nonflat situations can be found in [Bhatt-Algebraize, Section 5].
We start with a lemma on abelian sheaves supported on closed subsets.
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). Let \(Z \subset X\) closed subspace such that \(f^{-1}Z \to Z\) is integral and universally injective. Let \(\overline{y}\) be a geometric point of \(Y\) and \(\overline{x} = f(\overline{y})\). We have \[(Rf_*Q)_{\overline{x}} = Q_{\overline{y}}\] in \(D(\textit{Ab})\) for any object \(Q\) of \(D(Y_\etale)\) supported on \(|f^{-1}Z|\).
Proof
Consider the commutative diagram of algebraic spaces \[\xymatrix{ f^{-1}Z \ar[r]_{i'} \ar[d]_{f'} & Y \ar[d]_f \\ Z \ar[r]^i & X }\] By Cohomology of Spaces, Lemma 0AEI we can write \(Q = Ri'_*K'\) for some object \(K'\) of \(D(f^{-1}Z_\etale)\). By Morphisms of Spaces, Lemma 0AEH we have \(K' = (f')^{-1}K\) with \(K = Rf'_*K'\). Then we have \(Rf_*Q = Rf_*Ri'_*K' = Ri_*Rf'_*K' = Ri_*K\). Let \(\overline{z}\) be the geometric point of \(Z\) corresponding to \(\overline{x}\) and let \(\overline{z}'\) be the geometric point of \(f^{-1}Z\) corresponding to \(\overline{y}\). We obtain the result of the lemma as follows \[Q_{\overline{y}} = (Ri'_*K')_{\overline{y}} = K'_{\overline{z}'} = (f')^{-1}K_{\overline{z}'} = K_{\overline{z}} = Ri_*K_{\overline{x}} = Rf_*Q_{\overline{x}}\] The middle equality holds because of the description of the stalk of a pullback given in Properties of Spaces, Lemma 04K2.
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). Let \(Z \subset X\) closed subspace such that \(f^{-1}Z \to Z\) is integral and universally injective. Let \(\overline{y}\) be a geometric point of \(Y\) and \(\overline{x} = f(\overline{y})\). Let \(\mathcal{G}\) be an abelian sheaf on \(Y\). Then the map of two term complexes \[\left(f_*\mathcal{G}_{\overline{x}} \to (f \circ j')_*(\mathcal{G}|_V)_{\overline{x}}\right) \longrightarrow \left(\mathcal{G}_{\overline{y}} \to j'_*(\mathcal{G}|_V)_{\overline{y}}\right)\] induces an isomorphism on kernels and an injection on cokernels. Here \(V = Y \setminus f^{-1}Z\) and \(j' : V \to Y\) is the inclusion.
Proof
Choose a distinguished triangle \[\mathcal{G} \to Rj'_*\mathcal{G}|_V \to Q \to \mathcal{G}[1]\] n \(D(Y_\etale)\). The cohomology sheaves of \(Q\) are supported on \(|f^{-1}Z|\). We apply \(Rf_*\) and we obtain \[Rf_*\mathcal{G} \to Rf_*Rj'_*\mathcal{G}|_V \to Rf_*Q \to Rf_*\mathcal{G}[1]\] Taking stalks at \(\overline{x}\) we obtain an exact sequence \[0 \to (R^{-1}f_*Q)_{\overline{x}} \to f_*\mathcal{G}_{\overline{x}} \to (f \circ j')_*(\mathcal{G}|_V)_{\overline{x}} \to (R^0f_*Q)_{\overline{x}}\] We can compare this with the exact sequence \[0 \to H^{-1}(Q)_{\overline{y}} \to \mathcal{G}_{\overline{y}} \to j'_*(\mathcal{G}|_V)_{\overline{y}} \to H^0(Q)_{\overline{y}}\] Thus we see that the lemma follows because \(Q_{\overline{y}} = Rf_*Q_{\overline{x}}\) by Lemma 0AEQ.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(f : Y \to X\) be a quasi-compact and quasi-separated morphism. Let \(\overline{x}\) be a geometric point of \(X\) and let \(\Spec(\mathcal{O}_{X, \overline{x}}) \to X\) be the canonical morphism. For a quasi-coherent module \(\mathcal{G}\) on \(Y\) we have \[f_*\mathcal{G}_{\overline{x}} = \Gamma(Y \times_X \Spec(\mathcal{O}_{X, \overline{x}}), p^*\mathcal{F})\] where \(p : Y \times_X \Spec(\mathcal{O}_{X, \overline{x}}) \to Y\) is the projection.
Proof
Observe that \(f_*\mathcal{G}_{\overline{x}} = \Gamma(\Spec(\mathcal{O}_{X, \overline{x}}), h^*f_*\mathcal{G})\) where \(h : \Spec(\mathcal{O}_{X, \overline{x}}) \to X\). Hence the result is true because \(h\) is flat so that Cohomology of Spaces, Lemma 073K applies.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(i : Z \to X\) be a closed immersion of finite presentation. Let \(Q \in D_\QCoh(\mathcal{O}_X)\) be supported on \(|Z|\). Let \(\overline{x}\) be a geometric point of \(X\) and let \(I_{\overline{x}} \subset \mathcal{O}_{X, \overline{x}}\) be the stalk of the ideal sheaf of \(Z\). Then the cohomology modules \(H^n(Q_{\overline{x}})\) are \(I_{\overline{x}}\)-power torsion (see More on Algebra, Definition 05E6).
Proof
Choose an affine scheme \(U\) and an étale morphism \(U \to X\) such that \(\overline{x}\) lifts to a geometric point \(\overline{u}\) of \(U\). Then we can replace \(X\) by \(U\), \(Z\) by \(U \times_X Z\), \(Q\) by the restriction \(Q|_U\), and \(\overline{x}\) by \(\overline{u}\). Thus we may assume that \(X = \Spec(A)\) is affine. Let \(I \subset A\) be the ideal defining \(Z\). Since \(i : Z \to X\) is of finite presentation, the ideal \(I = (f_1, \ldots, f_r)\) is finitely generated. The object \(Q\) comes from a complex of \(A\)-modules \(M^\bullet\), see Derived Categories of Spaces, Lemma 071Q and Derived Categories of Schemes, Lemma 06Z0. Since the cohomology sheaves of \(Q\) are supported on \(Z\) we see that the localization \(M^\bullet_f\) is acyclic for each \(f \in I\). Take \(x \in H^p(M^\bullet)\). By the above we can find \(n_i\) such that \(f_i^{n_i} x = 0\) in \(H^p(M^\bullet)\) for each \(i\). Then with \(n = \sum n_i\) we see that \(I^n\) annihilates \(x\). Thus \(H^p(M^\bullet)\) is \(I\)-power torsion. Since the ring map \(A \to \mathcal{O}_{X, \overline{x}}\) is flat and since \(I_{\overline{x}} = I\mathcal{O}_{X, \overline{x}}\) we conclude.
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). Let \(Z \subset X\) be a closed subspace. Assume \(f^{-1}Z \to Z\) is an isomorphism and that \(f\) is flat in every point of \(f^{-1}Z\). For any \(Q\) in \(D_\QCoh(\mathcal{O}_Y)\) supported on \(|f^{-1}Z|\) we have \(Lf^*Rf_*Q = Q\).
Proof
We show the canonical map \(Lf^*Rf_*Q \to Q\) is an isomorphism by checking on stalks at \(\overline{y}\). If \(\overline{y}\) is not in \(f^{-1}Z\), then both sides are zero and the result is true. Assume the image \(\overline{x}\) of \(\overline{y}\) is in \(Z\). By Lemma 0AEQ we have \(Rf_*Q_{\overline{x}} = Q_{\overline{y}}\) and since \(f\) is flat at \(\overline{y}\) we see that \[(Lf^*Rf_*Q)_{\overline{y}} = (Rf_*Q)_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}} = Q_{\overline{y}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}}\] Thus we have to check that the canonical map \[Q_{\overline{y}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}} \longrightarrow Q_{\overline{y}}\] is an isomorphism in the derived category. Let \(I_{\overline{x}} \subset \mathcal{O}_{X, \overline{x}}\) be the stalk of the ideal sheaf defining \(Z\). Since \(Z \to X\) is locally of finite presentation this ideal is finitely generated and the cohomology groups of \(Q_{\overline{y}}\) are \(I_{\overline{y}} = I_{\overline{x}}\mathcal{O}_{Y, \overline{y}}\)-power torsion by Lemma 0AET applied to \(Q\) on \(Y\). It follows that they are also \(I_{\overline{x}}\)-power torsion. The ring map \(\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Y, \overline{y}}\) is flat and induces an isomorphism after dividing by \(I_{\overline{x}}\) and \(I_{\overline{y}}\) because we assumed that \(f^{-1}Z \to Z\) is an isomorphism. Hence we see that the cohomology modules of \(Q_{\overline{y}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}}\) are equal to the cohomology modules of \(Q_{\overline{y}}\) by More on Algebra, Lemma 05E9 which finishes the proof.
Situation
Here \(S\) is a base scheme, \(f : Y \to X\) is a quasi-compact and quasi-separated morphism of algebraic spaces over \(S\), and \(Z \to X\) is a closed immersion of finite presentation. We assume that \(f^{-1}(Z) \to Z\) is an isomorphism and that \(f\) is flat in every point \(x \in |f^{-1}Z|\). We set \(U = X \setminus Z\) and \(V = Y \setminus f^{-1}(Z)\). Picture \[\xymatrix{ V \ar[r]_{j'} \ar[d]_{f|_V} & Y \ar[d]^f \\ U \ar[r]^j & X }\]
In Situation 0AEV we define \(\textit{QCoh}(Y \to X, Z)\) as the category of triples \((\mathcal{H}, \mathcal{G}, \varphi)\) where \(\mathcal{H}\) is a quasi-coherent sheaf of \(\mathcal{O}_U\)-modules, \(\mathcal{G}\) is a quasi-coherent sheaf of \(\mathcal{O}_Y\)-modules, and \(\varphi : f^*\mathcal{H} \to \mathcal{G}|_V\) is an isomorphism of \(\mathcal{O}_V\)-modules. There is a canonical functor [0AEW]\[\begin{equation} \QCoh(\mathcal{O}_X) \longrightarrow \textit{QCoh}(Y \to X, Z) \end{equation}\] which maps \(\mathcal{F}\) to the system \((\mathcal{F}|_U, f^*\mathcal{F}, can)\). By analogy with the proof given in the affine case, we construct a functor in the opposite direction. To an object \((\mathcal{H}, \mathcal{G}, \varphi)\) we assign the \(\mathcal{O}_X\)-module [0AEX]\[\begin{equation} \Ker(j_*\mathcal{H} \oplus f_*\mathcal{G} \to (f \circ j')_*\mathcal{G}|_V) \end{equation}\] Observe that \(j\) and \(j'\) are quasi-compact morphisms as \(Z \to X\) is of finite presentation. Hence \(f_*\), \(j_*\), and \((f \circ j')_*\) transform quasi-coherent modules into quasi-coherent modules (Morphisms of Spaces, Lemma 03M9). Thus the module (0AEX) is quasi-coherent.
Lemma
In Situation 0AEV. The functor (0AEX) is right adjoint to the functor (0AEW).
Proof
This follows easily from the adjointness of \(f^*\) to \(f_*\) and \(j^*\) to \(j_*\). Details omitted.
Lemma
In Situation 0AEV. Let \(X' \to X\) be a flat morphism of algebraic spaces. Set \(Z' = X' \times_X Z\) and \(Y' = X' \times_X Y\). The pullbacks \(\QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_{X'})\) and \(\QCoh(Y \to X, Z) \to \QCoh(Y' \to X', Z')\) are compatible with the functors (0AEX) and 0AEW).
Proof
This is true because pullback commutes with pullback and because flat pullback commutes with pushforward along quasi-compact and quasi-separated morphisms, see Cohomology of Spaces, Lemma 073K.
Proposition
In Situation 0AEV the functor (0AEW) is an equivalence with quasi-inverse given by (0AEX).
Proof
We first treat the special case where \(X\) and \(Y\) are affine schemes and where the morphism \(f\) is flat. Say \(X = \Spec(R)\) and \(Y = \Spec(S)\). Then \(f\) corresponds to a flat ring map \(R \to S\). Moreover, \(Z \subset X\) is cut out by a finitely generated ideal \(I \subset R\). Choose generators \(f_1, \ldots, f_t \in I\). By the description of quasi-coherent modules in terms of modules (Schemes, Section 01I6), we see that the category \(\textit{QCoh}(Y \to X, Z)\) is canonically equivalent to the category \(\text{Glue}(R \to S, f_1, \ldots, f_t)\) of More on Algebra, Remark 05EL such that the functors (0AEW) and (0AEX) correspond to the functors \(\text{Can}\) and \(H^0\). Hence the result follows from More on Algebra, Proposition 05ER in this case.
We return to the general case. Let \(\mathcal{F}\) be a quasi-coherent module on \(X\). We will show that \[\alpha : \mathcal{F} \longrightarrow \Ker\left(j_*\mathcal{F}|_U \oplus f_*f^*\mathcal{F} \to (f \circ j')_*f^*\mathcal{F}|_V\right)\] is an isomorphism. Let \((\mathcal{H}, \mathcal{G}, \varphi)\) be an object of \(\QCoh(Y \to X, Z)\). We will show that \[\beta : f^*\Ker\left( j_*\mathcal{H} \oplus f_*\mathcal{G} \to (f \circ j')_*\mathcal{G}|_V \right) \longrightarrow \mathcal{G}\] and \[\gamma : j^*\Ker\left( j_*\mathcal{H} \oplus f_*\mathcal{G} \to (f \circ j')_*\mathcal{G}|_V \right) \longrightarrow \mathcal{H}\] are isomorphisms. To see these statements are true it suffices to look at stalks. Let \(\overline{y}\) be a geometric point of \(Y\) mapping to the geometric point \(\overline{x}\) of \(X\).
Fix an object \((\mathcal{H}, \mathcal{G}, \varphi)\) of \(\QCoh(Y \to X, Z)\). By Lemma 0AER and a diagram chase (omitted) the canonical map \[\Ker(j_*\mathcal{H} \oplus f_*\mathcal{G} \to (f \circ j')_*\mathcal{G}|_V)_{\overline{x}} \longrightarrow \Ker( j_*\mathcal{H}_{\overline{x}} \oplus \mathcal{G}_{\overline{y}} \to j'_*\mathcal{G}_{\overline{y}} )\] is an isomorphism.
In particular, if \(\overline{y}\) is a geometric point of \(V\), then we see that \(j'_*\mathcal{G}_{\overline{y}} = \mathcal{G}_{\overline{y}}\) and hence that this kernel is equal to \(\mathcal{H}_{\overline{x}}\). This easily implies that \(\alpha_{\overline{x}}\), \(\beta_{\overline{x}}\), and \(\beta_{\overline{y}}\) are isomorphisms in this case.
Next, assume that \(\overline{y}\) is a point of \(f^{-1}Z\). Let \(I_{\overline{x}} \subset \mathcal{O}_{X, \overline{x}}\), resp. \(I_{\overline{y}} \subset \mathcal{O}_{Y, \overline{y}}\) be the stalk of the ideal cutting out \(Z\), resp. \(f^{-1}Z\). Then \(I_{\overline{x}}\) is a finitely generated ideal, \(I_{\overline{y}} = I_{\overline{x}}\mathcal{O}_{Y, \overline{y}}\), and \(\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Y, \overline{y}}\) is a flat local homomorphism inducing an isomorphism \(\mathcal{O}_{X, \overline{x}}/I_{\overline{x}} = \mathcal{O}_{Y, \overline{y}}/I_{\overline{y}}\). At this point we can bootstrap using the diagram of categories \[\xymatrix{ \QCoh(\mathcal{O}_X) \ar[r]_-{(\href{spaces-pushouts.html#spaces-pushouts-equation-formal-glueing-modules}{0AEW})} \ar[d] & \QCoh(Y \to X, Z) \ar[d] \ar@/_2pc/[l]^{(\href{spaces-pushouts.html#spaces-pushouts-equation-reverse}{0AEX})} \\ \text{Mod}_{\mathcal{O}_{X, \overline{x}}} \ar[r]^-{\text{Can}} & \text{Glue}(\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Y, \overline{y}}, f_1, \ldots, f_t) \ar@/^2pc/[l]_{H^0} }\] Namely, as in the first paragraph of the proof we identify \[\text{Glue}(\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Y, \overline{y}}, f_1, \ldots, f_t) = \QCoh(\Spec(\mathcal{O}_{Y, \overline{y}}) \to \Spec(\mathcal{O}_{X, \overline{x}}), V(I_{\overline{x}}))\] The right vertical functor is given by pullback, and it is clear that the inner square is commutative. Our computation of the stalk of the kernel in the third paragraph of the proof combined with Lemma 0AES implies that the outer square (using the curved arrows) commutes. Thus we conclude using the case of a flat morphism of affine schemes which we handled in the first paragraph of the proof.
Lemma
In Situation 0AEV the functor \(Rf_*\) induces an equivalence between \(D_{\QCoh, |f^{-1}Z|}(\mathcal{O}_Y)\) and \(D_{\QCoh, |Z|}(\mathcal{O}_X)\) with quasi-inverse given by \(Lf^*\).
Proof
Since \(f\) is quasi-compact and quasi-separated we see that \(Rf_*\) defines a functor from \(D_{\QCoh, |f^{-1}Z|}(\mathcal{O}_Y)\) to \(D_{\QCoh, |Z|}(\mathcal{O}_X)\), see Derived Categories of Spaces, Lemma 08FA. By Derived Categories of Spaces, Lemma 08F4 we see that \(Lf^*\) maps \(D_{\QCoh, |Z|}(\mathcal{O}_X)\) into \(D_{\QCoh, |f^{-1}Z|}(\mathcal{O}_Y)\). In Lemma 0AEU we have seen that \(Lf^*Rf_*Q = Q\) for \(Q\) in \(D_{\QCoh, |f^{-1}Z|}(\mathcal{O}_Y)\). By the dual of Derived Categories, Lemma 09J1 to finish the proof it suffices to show that \(Lf^*K = 0\) implies \(K = 0\) for \(K\) in \(D_{\QCoh, |Z|}(\mathcal{O}_X)\). This follows from the fact that \(f\) is flat at all points of \(f^{-1}Z\) and the fact that \(f^{-1}Z \to Z\) is surjective.
Lemma
In Situation 0AEV there exists an fpqc covering \(\{X_i \to X\}_{i \in I}\) refining the family \(\{U \to X, Y \to X\}\).
Proof
For the definition and general properties of fpqc coverings we refer to Topologies, Section 022A. In particular, we can first choose an étale covering \(\{X_i \to X\}\) with \(X_i\) affine and by base changing \(Y\), \(Z\), and \(U\) to each \(X_i\) we reduce to the case where \(X\) is affine. In this case \(U\) is quasi-compact and hence a finite union \(U = U_1 \cup \ldots \cup U_n\) of affine opens. Then \(Z\) is quasi-compact hence also \(f^{-1}Z\) is quasi-compact. Thus we can choose an affine scheme \(W\) and an étale morphism \(h : W \to Y\) such that \(h^{-1}f^{-1}Z \to f^{-1}Z\) is surjective. Say \(W = \Spec(B)\) and \(h^{-1}f^{-1}Z = V(J)\) where \(J \subset B\) is an ideal of finite type. By Pro-étale Cohomology, Lemma 096V there exists a localization \(B \to B'\) such that points of \(\Spec(B')\) correspond exactly to points of \(W = \Spec(B)\) specializing to \(h^{-1}f^{-1}Z = V(J)\). It follows that the composition \(\Spec(B') \to \Spec(B) = W \to Y \to X\) is flat as by assumption \(f : Y \to X\) is flat at all the points of \(f^{-1}Z\). Then \(\{\Spec(B') \to X, U_1 \to X, \ldots, U_n \to X\}\) is an fpqc covering by Topologies, Lemma 03L7.
Formal glueing of algebraic spaces
In Situation 0AEV we consider the category \(\textit{Spaces}(Y \to X, Z)\) of commutative diagrams of algebraic spaces over \(S\) of the form \[\xymatrix{ U' \ar[d] & V' \ar[l] \ar[d] \ar[r] & Y' \ar[d] \\ U & V \ar[l] \ar[r] & Y }\] where both squares are cartesian. There is a canonical functor [0AF3]\[\begin{equation} \textit{Spaces}/X \longrightarrow \textit{Spaces}(Y \to X, Z) \end{equation}\] which maps \(X' \to X\) to the morphisms \(U \times_X X' \leftarrow V \times_X X' \rightarrow Y \times_X X'\).
Lemma
In Situation 0AEV the functor (0AF3) restricts to an equivalence
from the category of algebraic spaces affine over \(X\) to the full subcategory of \(\textit{Spaces}(Y \to X, Z)\) consisting of \((U' \leftarrow V' \rightarrow Y')\) with \(U' \to U\), \(V' \to V\), and \(Y' \to Y\) affine,
from the category of closed immersions \(X' \to X\) to the full subcategory of \(\textit{Spaces}(Y \to X, Z)\) consisting of \((U' \leftarrow V' \rightarrow Y')\) with \(U' \to U\), \(V' \to V\), and \(Y' \to Y\) closed immersions, and
same statement as in (2) for finite morphisms.
Proof
The category of algebraic spaces affine over \(X\) is equivalent to the category of quasi-coherent sheaves \(\mathcal{A}\) of \(\mathcal{O}_X\)-algebras. The full subcategory of \(\textit{Spaces}(Y \to X, Z)\) consisting of \((U' \leftarrow V' \rightarrow Y')\) with \(U' \to U\), \(V' \to V\), and \(Y' \to Y\) affine is equivalent to the category of algebra objects of \(\QCoh(Y \to X, Z)\). In both cases this follows from Morphisms of Spaces, Lemma 081V with quasi-inverse given by the relative spectrum construction (Morphisms of Spaces, Definition 081W) which commutes with arbitrary base change. Thus part (1) of the lemma follows from Proposition 0AF0.
Fully faithfulness in part (2) follows from part (1). For essential surjectivity, we reduce by part (1) to proving that \(X' \to X\) is a closed immersion if and only if both \(U \times_X X' \to U\) and \(Y \times_X X' \to Y\) are closed immersions. By Lemma 0AF1 \(\{U \to X, Y \to X\}\) can be refined by an fpqc covering. Hence the result follows from Descent on Spaces, Lemma 0420.
For (3) use the argument proving (2) and Descent on Spaces, Lemma 0426.
Lemma
Proof
By a formal argument with base change, this reduces to the following question: A morphism \(a : X' \to X\) of algebraic spaces such that \(U \times_X X' \to U\) and \(Y \times_X X' \to Y\) are isomorphisms, is an isomorphism. The family \(\{U \to X, Y \to X\}\) can be refined by an fpqc covering by Lemma 0AF1. Hence the result follows from Descent on Spaces, Lemma 041Y.
Lemma
In Situation 0AEV the functor (0AF3) is fully faithful on algebraic spaces separated over \(X\). More precisely, it induces a bijection \[\Mor_X(X'_1, X'_2) \longrightarrow \Mor_{\textit{Spaces}(Y \to X, Z)}(F(X'_1), F(X'_2))\] whenever \(X'_2 \to X\) is separated.
Proof
Since \(X'_2 \to X\) is separated, the graph \(i : X'_1 \to X'_1 \times_X X'_2\) of a morphism \(X'_1 \to X'_2\) over \(X\) is a closed immersion, see Morphisms of Spaces, Lemma 03KO. Moreover a closed immersion \(i : T \to X'_1 \times_X X'_2\) is the graph of a morphism if and only if \(\text{pr}_1 \circ i\) is an isomorphism. The same is true for
the graph of a morphism \(U \times_X X'_1 \to U \times_X X'_2\) over \(U\),
the graph of a morphism \(V \times_X X'_1 \to V \times_X X'_2\) over \(V\), and
the graph of a morphism \(Y \times_X X'_1 \to Y \times_X X'_2\) over \(Y\).
Moreover, if morphisms as in (1), (2), (3) fit together to form a morphism in the category \(\textit{Spaces}(Y \to X, Z)\), then these graphs fit together to give an object of \(\textit{Spaces}(Y \times_X (X'_1 \times_X X'_2) \to X'_1 \times_X X'_2, Z \times_X (X'_1 \times_X X'_2))\) whose triple of morphisms are closed immersions. The proof is finished by applying Lemmas 0AF4 and 0AF5.
Glueing and the Beauville-Laszlo theorem
Let \(R \to R'\) be a ring homomorphism and let \(f \in R\) be an element such that \[0 \to R \to R_f \oplus R' \to R'_f \to 0\] is a short exact sequence. This implies that \(R/f^nR \cong R'/f^nR'\) for all \(n\) and \((R \to R', f)\) is a glueing pair in the sense of More on Algebra, Section 0BNI. Set \(X = \Spec(R)\), \(U = \Spec(R_f)\), \(X' = \Spec(R')\) and \(U' = \Spec(R'_f)\). Picture \[\xymatrix{ U' \ar[r] \ar[d] & X' \ar[d] \\ U \ar[r] & X }\] In this situation we can consider the category \(\textit{Spaces}(U \leftarrow U' \to X')\) whose objects are commutative diagrams \[\xymatrix{ V \ar[d] & V' \ar[l] \ar[d] \ar[r] & Y' \ar[d] \\ U & U' \ar[l] \ar[r] & X' }\] of algebraic spaces with both squares cartesian and whose morphism are defined in the obvious manner. An object of this category will be denoted \((V, V', Y')\) with arrows suppressed from the notation. There is a functor [0F9N]\[\begin{equation} \textit{Spaces}/X \longrightarrow \textit{Spaces}(U \leftarrow U' \to X') \end{equation}\] given by base change: \(Y \mapsto (U \times_X Y, U' \times_X Y, X' \times_X Y)\).
We have seen in More on Algebra, Section 0BNI that not every \(R\)-module \(M\) can be recovered from its gluing data. Similarly, the functor (0F9N) won’t be fully faithful on the category of all spaces over \(X\). In order to single out a suitable subcategory of algebraic spaces over \(X\) we need a lemma.
Lemma
Let \((R \to R', f)\) be a glueing pair, see above. Let \(Y\) be an algebraic space over \(X\). The following are equivalent
there exists an étale covering \(\{Y_i \to Y\}_{i \in I}\) with \(Y_i\) affine and \(\Gamma(Y_i, \mathcal{O}_{Y_i})\) glueable as an \(R\)-module,
for every étale morphism \(W \to Y\) with \(W\) affine \(\Gamma(W, \mathcal{O}_W)\) is a glueable \(R\)-module.
Proof
It is immediate that (2) implies (1). Assume \(\{Y_i \to Y\}\) is as in (1) and let \(W \to Y\) be as in (2). Then \(\{Y_i \times_Y W \to W\}_{i \in I}\) is an étale covering, which we may refine by an étale covering \(\{W_j \to W\}_{j = 1, \ldots, m}\) with \(W_j\) affine (Topologies, Lemma 0218). Thus to finish the proof it suffices to show the following three algebraic statements:
if \(R \to A \to B\) are ring maps with \(A \to B\) étale and \(A\) glueable as an \(R\)-module, then \(B\) is glueable as an \(R\)-module,
finite products of glueable \(R\)-modules are glueable,
if \(R \to A \to B\) are ring maps with \(A \to B\) faithfully étale and \(B\) glueable as an \(R\)-module, then \(A\) is glueable as an \(R\)-module.
Namely, the first of these will imply that \(\Gamma(W_j, \mathcal{O}_{W_j})\) is a glueable \(R\)-module, the second will imply that \(\prod \Gamma(W_j, \mathcal{O}_{W_j})\) is a glueable \(R\)-module, and the third will imply that \(\Gamma(W, \mathcal{O}_W)\) is a glueable \(R\)-module.
Consider an étale \(R\)-algebra homomorphism \(A \to B\). Set \(A' = A \otimes_R R'\) and \(B' = B \otimes_R R' = A' \otimes_A B\). Statements (1) and (3) then follow from the following facts: (a) \(A\), resp. \(B\) is glueable if and only if the sequence \[0 \to A \to A_f \oplus A' \to A'_f \to 0, \quad\text{resp.}\quad 0 \to B \to B_f \oplus B' \to B'_f \to 0,\] is exact, (b) the second sequence is equal to the functor \(- \otimes_A B\) applied to the first and (c) (faithful) flatness of \(A \to B\). We omit the proof of (2).
Let \((R \to R', f)\) be a glueing pair, see above. We will say an algebraic space \(Y\) over \(X = \Spec(R)\) is glueable for \((R \to R', f)\) if the equivalent conditions of Lemma 0F9P are satisfied.
Lemma
Let \((R \to R', f)\) be a glueing pair, see above. The functor (0F9N) restricts to an equivalence between the category of affine \(Y/X\) which are glueable for \((R \to R', f)\) and the full subcategory of objects \((V, V', Y')\) of \(\textit{Spaces}(U \leftarrow U' \to X')\) with \(V\), \(V'\), \(Y'\) affine.
Proof
Let \((V, V', Y')\) be an object of \(\textit{Spaces}(U \leftarrow U' \to X')\) with \(V\), \(V'\), \(Y'\) affine. Write \(V = \Spec(A_1)\) and \(Y' = \Spec(A')\). By our definition of the category \(\textit{Spaces}(U \leftarrow U' \to X')\) we find that \(V'\) is the spectrum of \(A_1 \otimes_{R_f} R'_f = A_1 \otimes_R R'\) and the spectrum of \(A'_f\). Hence we get an isomorphism \(\varphi : A'_f \to A_1 \otimes_R R'\) of \(R'_f\)-algebras. By More on Algebra, Theorem 0BP2 there exists a unique glueable \(R\)-module \(A\) and isomorphisms \(A_f \to A_1\) and \(A \otimes_R R' \to A'\) of modules compatible with \(\varphi\). Since the sequence \[0 \to A \to A_1 \oplus A' \to A'_f \to 0\] is short exact, the multiplications on \(A_1\) and \(A'\) define a unique \(R\)-algebra structure on \(A\) such that the maps \(A \to A_1\) and \(A \to A'\) are ring homomorphisms. We omit the verification that this construction defines a quasi-inverse to the functor (0F9N) restricted to the subcategories mentioned in the statement of the lemma.
Lemma
Let \(P\) be one of the following properties of morphisms: “finite”, “closed immersion”, “flat”, “finite type”, “flat and finite presentation”, “étale”. Under the equivalence of Lemma 0F9Q the morphisms having \(P\) correspond to morphisms of triples whose components have \(P\).
Proof
Let \(P'\) be one of the following properties of homomorphisms of rings: “finite”, “surjective”, “flat”, “finite type”, “flat and of finite presentation”, “étale”. Translated into algebra, the statement means the following: If \(A \to B\) is an \(R\)-algebra homomorphism and \(A\) and \(B\) are glueable for \((R \to R', f)\), then \(A_f \to B_f\) and \(A \otimes_R R' \to B \otimes_R R'\) have \(P'\) if and only if \(A \to B\) has \(P'\).
By More on Algebra, Lemmas 0BNN and 0BP7 the algebraic statement is true for \(P'\) equal to “finite” or “flat”.
If \(A_f \to B_f\) and \(A \otimes_R R' \to B \otimes_R R'\) are surjective, then \(N = B/A\) is an \(R\)-module with \(N_f = 0\) and \(N \otimes_R R' = 0\) and hence vanishes by More on Algebra, Lemma 0BNL. Thus \(A \to B\) is surjective.
If \(A_f \to B_f\) and \(A \otimes_R R' \to B \otimes_R R'\) are finite type, then we can choose an \(A\)-algebra homomorphism \(A[x_1, \ldots, x_n] \to B\) such that \(A_f[x_1, \ldots, x_n] \to B_f\) and \((A \otimes_R R')[x_1, \ldots, x_n] \to B \otimes_R R'\) are surjective (small detail omitted). We conclude that \(A[x_1, \ldots, x_n] \to B\) is surjective by the previous result. Thus \(A \to B\) is of finite type.
If \(A_f \to B_f\) and \(A \otimes_R R' \to B \otimes_R R'\) are flat and of finite presentation, then we know that \(A \to B\) is flat and of finite type by what we have already shown. Choose a surjection \(A[x_1, \ldots, x_n] \to B\) and denote \(I\) the kernel. By flatness of \(B\) over \(A\) we see that \(I_f\) is the kernel of \(A_f[x_1, \ldots, x_n] \to B_f\) and \(I \otimes_R R'\) is the kernel of \(A \otimes_R R'[x_1, \ldots, x_n] \to B \otimes_R R'\). Thus \(I_f\) is a finite \(A_f[x_1, \ldots, x_n]\)-module and \(I \otimes_R R'\) is a finite \((A \otimes_R R')[x_1, \ldots, x_n]\)-module. By More on Algebra, Lemma 0BNN applied to \(I\) viewed as a module over \(A[x_1, \ldots, x_n]\) we conclude that \(I\) is a finitely generated ideal and we conclude \(A \to B\) is flat and of finite presentation.
If \(A_f \to B_f\) and \(A \otimes_R R' \to B \otimes_R R'\) are étale, then we know that \(A \to B\) is flat and of finite presentation by what we have already shown. Since the fibres of \(\Spec(B) \to \Spec(A)\) are isomorphic to fibres of \(\Spec(B_f) \to \Spec(A_f)\) or \(\Spec(B/fB) \to \Spec(A/fA)\), we conclude that \(A \to B\) is unramified, see Morphisms, Lemmas 02G7 and 02G8. We conclude that \(A \to B\) is étale by Morphisms, Lemma 02GV for example.
Lemma
Let \((R \to R', f)\) be a glueing pair, see above. The functor (0F9N) is faithful on the full subcategory of algebraic spaces \(Y/X\) glueable for \((R \to R', f)\).
Proof
Let \(f, g : Y \to Z\) be two morphisms of algebraic spaces over \(X\) with \(Y\) and \(Z\) glueable for \((R \to R', f)\) such that \(f\) and \(g\) are mapped to the same morphism in the category \(\textit{Spaces}(U \leftarrow U' \to X')\). We have to show the equalizer \(E \to Y\) of \(f\) and \(g\) is an isomorphism. Working étale locally on \(Y\) we may assume \(Y\) is an affine scheme. Then \(E\) is a scheme and the morphism \(E \to Y\) is a monomorphism and locally quasi-finite, see Morphisms of Spaces, Lemma 03HK. Moreover, the base change of \(E \to Y\) to \(U\) and to \(X'\) is an isomorphism. As \(Y\) is the disjoint union of the affine open \(V = U \times_X Y\) and the affine closed \(V(f) \times_X Y\), we conclude \(E\) is the disjoint union of their isomorphic inverse images. It follows in particular that \(E\) is quasi-compact. By Zariski’s main theorem (More on Morphisms, Lemma 05K0) we conclude that \(E\) is quasi-affine. Set \(B = \Gamma(E, \mathcal{O}_E)\) and \(A = \Gamma(Y, \mathcal{O}_Y)\) so that we have an \(R\)-algebra homomorphism \(A \to B\). Since \(E \to Y\) becomes an isomorphism after base change to \(U\) and \(X'\) we obtain ring maps \(B \to A_f\) and \(B \to A \otimes_R R'\) agreeing as maps into \(A \otimes_R R'_f\). Since \(A\) is glueable for \((R \to R', f)\) we get a ring map \(B \to A\) which is left inverse to the map \(A \to B\). The corresponding morphism \(Y = \Spec(A) \to \Spec(B)\) maps into the open subscheme \(E \subset \Spec(B)\) pointwise because this is true after base change to \(U\) and \(X'\). Hence we get a morphism \(Y \to E\) over \(Y\). Since \(E \to Y\) is a monomorhism we conclude \(Y \to E\) is an isomorphism as desired.
Lemma
Let \((R \to R', f)\) be a glueing pair, see above. The functor (0F9N) is fully faithful on the full subcategory of algebraic spaces \(Y/X\) which are (a) glueable for \((R \to R', f)\) and (b) have affine diagonal \(Y \to Y \times_X Y\).
Proof
Let \(Y, Z\) be two algebraic spaces over \(X\) which are both glueable for \((R \to R', f)\) and assume the diagonal of \(Z\) is affine. Let \(a : U \times_X Y \to U \times_X Z\) over \(U\) and \(b : X' \times_X Y \to X' \times_X Z\) over \(X'\) be two morphisms of algebraic spaces which induce the same morphism \(c : U' \times_X Y \to U' \times_X Z\) over \(U'\). We want to construct a morphism \(f : Y \to Z\) over \(X\) which produces the morphisms \(a\), \(b\) on base change to \(U\), \(X'\). By the faithfulness of Lemma 0F9S, it suffices to construct the morphism \(f\) étale locally on \(Y\) (details omitted). Thus we may and do assume \(Y\) is affine.
Let \(y \in |Y|\) be a point. If \(y\) maps into the open \(U \subset X\), then \(U \times_X Y\) is an open of \(Y\) on which the morphism \(f\) is defined (we can just take \(a\)). Thus we may assume \(y\) maps into the closed subset \(V(f)\) of \(X\). Since \(R/fR = R'/fR'\) there is a unique point \(y' \in |X' \times_X Y|\) mapping to \(y\). Denote \(z' = b(y') \in |X' \times_X Z|\) and \(z \in |Z|\) the images of \(y'\). Choose an étale neighbourhood \((W, w) \to (Z, z)\) with \(W\) affine. Observe that \[(U \times_X W) \times_{U \times_X Z, a} (U \times_X Y),\quad (U' \times_X W) \times_{U' \times_X Z, c} (U' \times_X Y),\] and \[(X' \times_X W) \times_{X' \times_X Z, b} (X' \times_X Y)\] form an object of \(\textit{Spaces}(U \leftarrow U' \to X')\) with affine parts (this is where we use that \(Z\) has affine diagonal). Hence by Lemma 0F9Q there exists a unique affine scheme \(V\) glueable for \((R \to R', f)\) such that \[(U \times_X V, U' \times_X V, X' \times_X V)\] is the triple displayed above. By fully faithfulness for the affine case (Lemma 0F9Q) we get a unique morphisms \(V \to W\) and \(V \to Y\) agreeing with the first and second projection morphisms over \(U\) and \(X'\) in the construction above. By Lemma 0F9R the morphism \(V \to Y\) is étale. To finish the proof, it suffices to show that there is a point \(v \in |V|\) mapping to \(y\) (because then \(f\) is defined on an étale neighbourhood of \(y\), namely \(V\)). There is a unique point \(w' \in |X' \times_X W|\) mapping to \(w\). By uniqueness \(w'\) is mapped to \(z'\) under the map \(|X' \times_X W| \to |X' \times_X Z|\). Then we consider the cartesian diagram \[\xymatrix{ X' \times_X V \ar[r] \ar[d] & X' \times_X W \ar[d] \\ X' \times_X Y \ar[r] & X' \times_X Z }\] to see that there is a point \(v' \in |X' \times_X V|\) mapping to \(y'\) and \(w'\), see Properties of Spaces, Lemma 03H4. Of course the image \(v\) of \(v'\) in \(|V|\) maps to \(y\) and the proof is complete.
Lemma
Let \((R \to R', f)\) be a glueing pair, see above. Any object \((V, V', Y')\) of \(\textit{Spaces}(U \leftarrow U' \to X')\) with \(V\), \(V'\), \(Y'\) quasi-affine is isomorphic to the image under the functor (0F9N) of a separated algebraic space \(Y\) over \(X\).
Proof
Choose \(n'\), \(T' \to Y'\) and \(n_1\), \(T_1 \to V\) as in Properties, Lemma 0F82. Picture \[\xymatrix{ & & T_1 \times_V V' \times_Y T' \ar[ld] \ar[rd] \\ T_1 \ar[d] & T_1 \times_V V' \ar[l] \ar[dr] & & V' \times_{Y'} T' \ar[r] \ar[dl] & T' \ar[d] \\ V & & V' \ar[rr] \ar[ll] & & Y' }\] Observe that \(T_1 \times_V V'\) and \(V' \times_{Y'} T'\) are affine (namely the morphisms \(V' \to V\) and \(V' \to Y'\) are affine as base changes of the affine morphisms \(U' \to U\) and \(U' \to X'\)). By construction we see that \[\mathbf{A}^{n'}_{T_1 \times_V V'} \cong T_1 \times_V V' \times_{Y'} T' \cong \mathbf{A}^{n_1}_{V' \times_{Y'} T'}\] In other words, the affine schemes \(\mathbf{A}^{n'}_{T_1}\) and \(\mathbf{A}^{n_1}_{T'}\) are part of a triple making an affine object of \(\textit{Spaces}(U \leftarrow U' \to X')\). By Lemma 0F9Q there exists a morphism of affine schemes \(T \to X\) and isomorphisms \(U \times_X T \cong \mathbf{A}^{n'}_{T_1}\) and \(X' \times_X T \cong \mathbf{A}^{n_1}_{T'}\) compatible with the isomorphisms displayed above. These isomorphisms produce morphisms \[U \times_X T \longrightarrow V \quad\text{and}\quad X' \times_X T \longrightarrow Y'\] satisfying the property of Properties, Lemma 0F82 with \(n = n' + n_1\) and moreover define a morphism from the triple \((U \times_X T, U' \times_X T, X' \times_X T)\) to our triple \((V, V', Y')\) in the category \(\textit{Spaces}(U \leftarrow U' \to X')\).
By Lemma 0F9Q there is an affine scheme \(W\) whose image in \(\textit{Spaces}(U \leftarrow U' \to X')\) is isomorphic to the triple \[((U \times_X T) \times_V (U \times_X T), (U' \times_X T) \times_{V'} (U' \times_X T), (X' \times_X T) \times_{Y'} (X' \times_X T))\] By fully faithfulness of this construction, we obtain two maps \(p_0, p_1 : W \to T\) whose base changes to \(U, U', X'\) are the projection morphisms. By Lemma 0F9R the morphisms \(p_0, p_1\) are flat and of finite presentation and the morphism \((p_0, p_1) : W \to T \times_X T\) is a closed immersion. In fact, \(W \to T \times_X T\) is an equivalence relation: by the lemmas used above we may check symmetry, reflexivity, and transitivity after base change to \(U\) and \(X'\), where these are obvious (details omitted). Thus the quotient sheaf \[Y = T/W\] is an algebraic space for example by Bootstrap, Theorem 04S6. Since it is clear that \(Y/X\) is sent to the triple \((V, V', Y')\). The base change of the diagonal \(\Delta : Y \to Y \times_X Y\) by the quasi-compact surjective flat morphism \(T \times_X T \to Y \times_X Y\) is the closed immersion \(W \to T \times_X T\). Thus \(\Delta\) is a closed immersion by Descent on Spaces, Lemma 0420. Thus the algebraic space \(Y\) is separated and the proof is complete.
Coequalizers and glueing
Let \(X\) be a Noetherian algebraic space and \(Z \to X\) a closed subspace. Let \(X' \to X\) be the blowing up in \(Z\). In this section we show that \(X\) can be recovered from \(X'\), \(Z_n\) and glueing data where \(Z_n\) is the \(n\)th infinitesimal neighbourhood of \(Z\) in \(X\).
Lemma
Let \(S\) be a scheme. Let \[g : Y \longrightarrow X\] be a morphism of algebraic spaces over \(S\). Assume \(X\) is locally Noetherian, and \(g\) is proper. Let \(R = Y \times_X Y\) with projection morphisms \(t, s : R \to Y\). There exists a coequalizer \(X'\) of \(s, t : R \to Y\) in the category of algebraic spaces over \(S\). Moreover
The morphism \(X' \to X\) is finite.
The morphism \(Y \to X'\) is proper.
The morphism \(Y \to X'\) is surjective.
The morphism \(X' \to X\) is universally injective.
If \(g\) is surjective, the morphism \(X' \to X\) is a universal homeomorphism.
Proof
Denote \(h : R \to X\) denote the composition of either \(s\) or \(t\) with \(g\). Then \(h\) is proper by Morphisms of Spaces, Lemmas 04WP and 04XY. The sheaves \[g_*\mathcal{O}_Y \quad\text{and}\quad h_*\mathcal{O}_R\] are coherent \(\mathcal{O}_X\)-algebras by Cohomology of Spaces, Lemma 08AR. The \(X\)-morphisms \(s\), \(t\) induce \(\mathcal{O}_X\)-algebra maps \(s^\sharp, t^\sharp\) from the first to the second. Set \[\mathcal{A} = \text{Equalizer}\left(s^\sharp, t^\sharp : g_*\mathcal{O}_Y \longrightarrow h_*\mathcal{O}_R\right)\] Then \(\mathcal{A}\) is a coherent \(\mathcal{O}_X\)-algebra and we can define \[X' = \underline{\Spec}_X(\mathcal{A})\] as in Morphisms of Spaces, Definition 081W. By Morphisms of Spaces, Remark 081X and functoriality of the \(\underline{\Spec}\) construction there is a factorization \[Y \longrightarrow X' \longrightarrow X\] and the morphism \(g' : Y \to X'\) equalizes \(s\) and \(t\).
Before we show that \(X'\) is the coequalizer of \(s\) and \(t\), we show that \(Y \to X'\) and \(X' \to X\) have the desired properties. Since \(\mathcal{A}\) is a coherent \(\mathcal{O}_X\)-module it is clear that \(X' \to X\) is a finite morphism of algebraic spaces. This proves (1). The morphism \(Y \to X'\) is proper by Morphisms of Spaces, Lemma 04NX. This proves (2). Denote \(Y \to Y' \to X\) with \(Y' = \underline{\Spec}_X(g_*\mathcal{O}_Y)\) the Stein factorization of \(g\), see More on Morphisms of Spaces, Theorem 0A1B. Of course we obtain morphisms \(Y \to Y' \to X' \to X\) fitting with the morphisms studied above. Since \(\mathcal{O}_{X'} \subset g_*\mathcal{O}_Y\) is a finite extension we see that \(Y' \to X'\) is finite and surjective. Some details omitted; hint: use Algebra, Lemma 00GQ and reduce to the affine case by étale localization. Since \(Y \to Y'\) is surjective (with geometrically connected fibres) we conclude that \(Y \to X'\) is surjective. This proves (3). To show that \(X' \to X\) is universally injective, we have to show that \(X' \to X' \times_X X'\) is surjective, see Morphisms of Spaces, Definition 03MV and Lemma 040X. Since \(Y \to X'\) is surjective (see above) and since base changes and compositions of surjective morphisms are surjective by Morphisms of Spaces, Lemmas 03MH and 03MG we see that \(Y \times_X Y \to X' \times_X X'\) is surjective. However, since \(Y \to X'\) equalizes \(s\) and \(t\), we see that \(Y \times_X Y \to X' \times_X X'\) factors through \(X' \to X' \times_X X'\) and we conclude this latter map is surjective. This proves (4). Finally, if \(g\) is surjective, then since \(g\) factors through \(X' \to X\) we see that \(X' \to X\) is surjective. Since a surjective, universally injective, finite morphism is a universal homeomorphism (because it is universally bijective and universally closed), this proves (5).
In the rest of the proof we show that \(Y \to X'\) is the coequalizer of \(s\) and \(t\) in the category of algebraic spaces over \(S\). Observe that \(X'\) is locally Noetherian (Morphisms of Spaces, Lemma 04ZK). Moreover, observe that \(Y \times_{X'} Y \to Y \times_X Y\) is an isomorphism as \(Y \to X'\) equalizes \(s\) and \(t\) (this is a categorical statement). Hence in order to prove the statement that \(Y \to X'\) is the coequalizer of \(s\) and \(t\), we may and do assume \(X = X'\). In other words, \(\mathcal{O}_X\) is the equalizer of the maps \(s^\sharp, t^\sharp : g_*\mathcal{O}_Y \to h_*\mathcal{O}_R\).
Let \(X_1 \to X\) be a flat morphism of algebraic spaces over \(S\) with \(X_1\) locally Noetherian. Denote \(g_1 : Y_1 \to X_1\), \(h_1 : R_1 \to X_1\) and \(s_1, t_1 : R_1 \to Y_1\) the base changes of \(g, h, s, t\) to \(X_1\). Of course \(g_1\) is proper and \(R_1 = Y_1 \times_{X_1} Y_1\). Since we have flat base change for pushforward of quasi-coherent modules, Cohomology of Spaces, Lemma 073K, we see that \(\mathcal{O}_{X_1}\) is the equalizer of the maps \(s_1^\sharp, t_1^\sharp : g_{1, *}\mathcal{O}_{Y_1} \to h_{1, *}\mathcal{O}_{R_1}\). Hence all the assumptions we have are preserved by this base change.
At this point we are going to check conditions (1) and (2) of Lemma 0GHL. Condition (1) follows from Lemma 0GG0 and the fact that \(g\) is proper and surjective (because \(X = X'\)). To check condition (2), by the remarks on base change above, we reduce to the statement discussed and proved in the next paragraph.
Assume \(S = \Spec(A)\) is an affine scheme, \(X = X'\) is an affine scheme, and \(Z\) is an affine scheme over \(S\). We have to show that \[\Mor_S(X, Z) \longrightarrow \text{Equalizer}(s, t : \Mor_S(Y, Z) \to \Mor_S(R, Z))\] is bijective. However, this is clear from the fact that \(X = X'\) which implies \(\mathcal{O}_X\) is the equalizer of the maps \(s^\sharp, t^\sharp : g_*\mathcal{O}_Y \to h_*\mathcal{O}_R\) which in turn implies \[\Gamma(X, \mathcal{O}_X) = \text{Equalizer}\left( s^\sharp, t^\sharp : \Gamma(Y, \mathcal{O}_Y) \to \Gamma(R, \mathcal{O}_R) \right)\] Namely, we have \[\Mor_S(X, Z) = \Hom_A(\Gamma(Z, \mathcal{O}_Z), \Gamma(X, \mathcal{O}_X))\] and similarly for \(Y\) and \(R\), see Properties of Spaces, Lemma 05Z1.
We will work in the following situation.
Situation
Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). Let \(Z \to X\) be a closed immersion and let \(U \subset X\) be the complementary open subspace. Finally, let \(f : X' \to X\) be a proper morphism of algebraic spaces such that \(f^{-1}(U) \to U\) is an isomorphism.
Lemma
In Situation 0AGH let \(Y = X' \amalg Z\) and \(R = Y \times_X Y\) with projections \(t, s : R \to Y\). There exists a coequalizer \(X_1\) of \(s, t : R \to Y\) in the category of algebraic spaces over \(S\). The morphism \(X_1 \to X\) is a finite universal homeomorphism, an isomorphism over \(U\), and \(Z \to X\) lifts to \(X_1\).
Proof
Existence of \(X_1\) and the fact that \(X_1 \to X\) is a finite universal homeomorphism is a special case of Lemma 0AGG. The formation of \(X_1\) commutes with étale localization on \(X\) (see proof of Lemma 0AGG). Thus the morphism \(X_1 \to X\) is an isomorphism over \(U\). It is immediate from the construction that \(Z \to X\) lifts to \(X_1\).
In Situation 0AGH for \(n \geq 1\) let \(Z_n \subset X\) be the \(n\)th order infinitesimal neighbourhood of \(Z\) in \(X\), i.e., the closed subscheme defined by the \(n\)th power of the sheaf of ideals cutting out \(Z\). Consider \(Y_n = X' \amalg Z_n\) and \(R_n = Y_n \times_X Y_n\) and the coequalizer \[\xymatrix{ R_n \ar@<1ex>[r] \ar@<-1ex>[r] & Y_n \ar[r] & X_n \ar[r] & X }\] as in Lemma 0AGI. The maps \(Y_n \to Y_{n + 1}\) and \(R_n \to R_{n + 1}\) induce morphisms [0AGJ]\[\begin{equation} X_1 \to X_2 \to X_3 \to \ldots \to X \end{equation}\] Each of these morphisms is a universal homeomorphism as the morphisms \(X_n \to X\) are universal homeomorphisms.
Lemma
In Situation 0AGH assume \(X\) quasi-compact. In (0AGJ) for all \(n\) large enough, there exists an \(m\) such that \(X_n \to X_{n + m}\) factors through a closed immersion \(X \to X_{n + m}\).
Proof
Let’s look a bit more closely at the construction of \(X_n\) and how it changes as we increase \(n\). We have \(X_n = \underline{\Spec}(\mathcal{A}_n)\) where \(\mathcal{A}_n\) is the equalizer of \(s_n^\sharp\) and \(t_n^\sharp\) going from \(g_{n , *}\mathcal{O}_{Y_n}\) to \(h_{n, *}\mathcal{O}_{R_n}\). Here \(g_n : Y_n = X' \amalg Z_n \to X\) and \(h_n : R_n = Y_n \times_X Y_n \to X\) are the given morphisms. Let \(\mathcal{I} \subset \mathcal{O}_X\) be the coherent sheaf of ideals corresponding to \(Z\). Then \[g_{n, *}\mathcal{O}_{Y_n} = f_*\mathcal{O}_{X'} \times \mathcal{O}_X/\mathcal{I}^n\] Similarly, we have a decomposition \[R_n = X' \times_X X' \amalg X' \times_X Z_n \amalg Z_n \times_X X' \amalg Z_n \times_X Z_n\] As \(Z_n \to X\) is a monomorphism, we see that \(X' \times_X Z_n = Z_n \times_X X'\) and that this identification is compatible with the two morphisms to \(X\), with the two morphisms to \(X'\), and with the two morphisms to \(Z_n\). Denote \(f_n : X' \times_X Z_n \to X\) the morphism to \(X\). Denote \[\mathcal{A} = \text{Equalizer}( \xymatrix{ f_*\mathcal{O}_{X'} \ar@<1ex>[r] \ar@<-1ex>[r] & (f \times f)_*\mathcal{O}_{X' \times_X X'} } )\] By the remarks above we find that \[\mathcal{A}_n = \text{Equalizer}( \xymatrix{ \mathcal{A} \times \mathcal{O}_X/\mathcal{I}^n \ar@<1ex>[r] \ar@<-1ex>[r] & f_{n, *}\mathcal{O}_{X' \times_X Z_n} } )\] We have canonical maps \[\mathcal{O}_X \to \ldots \to \mathcal{A}_3 \to \mathcal{A}_2 \to \mathcal{A}_1\] of coherent \(\mathcal{O}_X\)-algebras. The statement of the lemma means that for \(n\) large enough there exists an \(m \geq 0\) such that the image of \(\mathcal{A}_{n + m} \to \mathcal{A}_n\) is isomorphic to \(\mathcal{O}_X\). This we may check étale locally on \(X\). Hence by Properties of Spaces, Lemma 03H6 we may assume \(X\) is an affine Noetherian scheme.
Since \(X_n \to X\) is an isomorphism over \(U\) we see that the kernel of \(\mathcal{O}_X \to \mathcal{A}_n\) is supported on \(|Z|\). Since \(X\) is Noetherian, the sequence of kernels \(\mathcal{J}_n = \Ker(\mathcal{O}_X \to \mathcal{A}_n)\) stabilizes (Cohomology of Spaces, Lemma 07UJ). Say \(\mathcal{J}_{n_0} = \mathcal{J}_{n_0 + 1} = \ldots = \mathcal{J}\). By Cohomology of Spaces, Lemma 07UK we find that \(\mathcal{I}^t \mathcal{J} = 0\) for some \(t \geq 0\). On the other hand, there is an \(\mathcal{O}_X\)-algebra map \(\mathcal{A}_n \to \mathcal{O}_X/\mathcal{I}^n\) and hence \(\mathcal{J} \subset \mathcal{I}^n\) for all \(n\). By Artin-Rees (Cohomology of Spaces, Lemma 07UL) we find that \(\mathcal{J} \cap \mathcal{I}^n \subset \mathcal{I}^{n - c}\mathcal{J}\) for some \(c \geq 0\) and all \(n \gg 0\). We conclude that \(\mathcal{J} = 0\).
Pick \(n \geq n_0\) as in the previous paragraph. Then \(\mathcal{O}_X \to \mathcal{A}_n\) is injective. Hence it now suffices to find \(m \geq 0\) such that the image of \(\mathcal{A}_{n + m} \to \mathcal{A}_n\) is equal to the image of \(\mathcal{O}_X\). Observe that \(\mathcal{A}_n\) sits in a short exact sequence \[0 \to \Ker(\mathcal{A} \to f_{n, *}\mathcal{O}_{X' \times_X Z_n}) \to \mathcal{A}_n \to \mathcal{O}_X/\mathcal{I}^n \to 0\] and similarly for \(\mathcal{A}_{n + m}\). Hence it suffices to show \[\Ker(\mathcal{A} \to f_{n + m, *}\mathcal{O}_{X' \times_X Z_{n + m}}) \subset \Im(\mathcal{I}^n \to \mathcal{A})\] for some \(m \geq 0\). To do this we may work étale locally on \(X\) and since \(X\) is Noetherian we may assume that \(X\) is a Noetherian affine scheme. Say \(X = \Spec(R)\) and \(\mathcal{I}\) corresponds to the ideal \(I \subset R\). Let \(\mathcal{A} = \widetilde{A}\) for a finite \(R\)-algebra \(A\). Let \(f_*\mathcal{O}_{X'} = \widetilde{B}\) for a finite \(R\)-algebra \(B\). Then \(R \to A \subset B\) and these maps become isomorphisms on inverting any element of \(I\).
Note that \(f_{n, *}\mathcal{O}_{X' \times_X Z_n}\) is equal to \(f_*(\mathcal{O}_{X'}/I^n\mathcal{O}_{X'})\) in the notation used in Cohomology of Spaces, Section 08AU. By Cohomology of Spaces, Lemma 08AY we see that there exists a \(c \geq 0\) such that \[\Ker(B \to \Gamma(X, f_*(\mathcal{O}_{X'}/I^{n + m + c}\mathcal{O}_{X'}))\] is contained in \(I^{n + m}B\). On the other hand, as \(R \to B\) is finite and an isomorphism after inverting any element of \(I\) we see that \(I^{n + m}B \subset \Im(I^n \to B)\) for \(m\) large enough (can be chosen independent of \(n\)). This finishes the proof as \(A \subset B\).
Remark
The meaning of Lemma 0AGK is the system \(X_1 \to X_2 \to X_3 \to \ldots\) is essentially constant with value \(X\). See Categories, Definition 05PU.
Compactifications
This section is the analogue of More on Flatness, Section 0F3T. The theorem in this section is the main theorem in [CLO].
Let \(B\) be a quasi-compact and quasi-separated algebraic space over some base scheme \(S\). We will say an algebraic space \(X\) over \(B\) has a compactification over \(B\) or is compactifyable over \(B\) if there exists a quasi-compact open immersion \(X \to \overline{X}\) into an algebraic space \(\overline{X}\) proper over \(B\). If \(X\) has a compactification over \(B\), then \(X \to B\) is separated and of finite type. The main theorem of this section is that the converse is true as well.
Lemma
Let \(S\) be a scheme. Let \(X \to Y\) be a morphism of algebraic spaces over \(S\). If \((U \subset X, f : V \to X)\) is an elementary distinguished square such that \(U \to Y\) and \(V \to Y\) are separated and \(U \times_X V \to U \times_Y V\) is closed, then \(X \to Y\) is separated.
Proof
We have to check that \(\Delta : X \to X \times_Y X\) is a closed immersion. There is an étale covering of \(X \times_Y X\) given by the four parts \(U \times_Y U\), \(U \times_Y V\), \(V \times_Y U\), and \(V \times_Y V\). Observe that \((U \times_Y U) \times_{(X \times_Y X), \Delta} X = U\), \((U \times_Y V) \times_{(X \times_Y X), \Delta} X = U \times_X V\), \((V \times_Y U) \times_{(X \times_Y X), \Delta} X = V \times_X U\), and \((V \times_Y V) \times_{(X \times_Y X), \Delta} X = V\). Thus the assumptions of the lemma exactly tell us that \(\Delta\) is a closed immersion.
Lemma
Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(U \subset X\) be a quasi-compact open.
If \(Z_1, Z_2 \subset X\) are closed subspaces of finite presentation such that \(Z_1 \cap Z_2 \cap U = \emptyset\), then there exists a \(U\)-admissible blowing up \(X' \to X\) such that the strict transforms of \(Z_1\) and \(Z_2\) are disjoint.
If \(T_1, T_2 \subset |U|\) are disjoint constructible closed subsets, then there is a \(U\)-admissible blowing up \(X' \to X\) such that the closures of \(T_1\) and \(T_2\) are disjoint.
Proof
Proof of (1). The assumption that \(Z_i \to X\) is of finite presentation signifies that the quasi-coherent ideal sheaf \(\mathcal{I}_i\) of \(Z_i\) is of finite type, see Morphisms of Spaces, Lemma 084Q. Denote \(Z \subset X\) the closed subspace cut out by the product \(\mathcal{I}_1 \mathcal{I}_2\). Observe that \(Z \cap U\) is the disjoint union of \(Z_1 \cap U\) and \(Z_2 \cap U\). By Divisors on Spaces, Lemma 086F there is a \(U \cap Z\)-admissible blowup \(Z' \to Z\) such that the strict transforms of \(Z_1\) and \(Z_2\) are disjoint. Denote \(Y \subset Z\) the center of this blowing up. Then \(Y \to X\) is a closed immersion of finite presentation as the composition of \(Y \to Z\) and \(Z \to X\) (Divisors on Spaces, Definition 086B and Morphisms of Spaces, Lemma 03XQ). Thus the blowing up \(X' \to X\) of \(Y\) is a \(U\)-admissible blowing up. By general properties of strict transforms, the strict transform of \(Z_1, Z_2\) with respect to \(X' \to X\) is the same as the strict transform of \(Z_1, Z_2\) with respect to \(Z' \to Z\), see Divisors on Spaces, Lemma 0864. Thus (1) is proved.
Proof of (2). By Limits of Spaces, Lemma 0855 there exists a finite type quasi-coherent sheaf of ideals \(\mathcal{J}_i \subset \mathcal{O}_U\) such that \(T_i = V(\mathcal{J}_i)\) (set theoretically). By Limits of Spaces, Lemma 0853 there exists a finite type quasi-coherent sheaf of ideals \(\mathcal{I}_i \subset \mathcal{O}_X\) whose restriction to \(U\) is \(\mathcal{J}_i\). Apply the result of part (1) to the closed subspaces \(Z_i = V(\mathcal{I}_i)\) to conclude.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper morphism of quasi-compact and quasi-separated algebraic spaces over \(S\). Let \(V \subset Y\) be a quasi-compact open and \(U = f^{-1}(V)\). Let \(T \subset |V|\) be a closed subset such that \(f|_U : U \to V\) is an isomorphism over an open neighbourhood of \(T\) in \(V\). Then there exists a \(V\)-admissible blowing up \(Y' \to Y\) such that the strict transform \(f' : X' \to Y'\) of \(f\) is an isomorphism over an open neighbourhood of the closure of \(T\) in \(|Y'|\).
Proof
Let \(T' \subset |V|\) be the complement of the maximal open over which \(f|_U\) is an isomorphism. Then \(T', T\) are closed in \(|V|\) and \(T \cap T' = \emptyset\). Since \(|V|\) is a spectral topological space (Properties of Spaces, Lemma 0A4G) we can find constructible closed subsets \(T_c, T'_c\) of \(|V|\) with \(T \subset T_c\), \(T' \subset T'_c\) such that \(T_c \cap T'_c = \emptyset\) (choose a quasi-compact open \(W\) of \(|V|\) containing \(T'\) not meeting \(T\) and set \(T_c = |V| \setminus W\), then choose a quasi-compact open \(W'\) of \(|V|\) containing \(T_c\) not meeting \(T'\) and set \(T'_c = |V| \setminus W'\)). By Lemma 0F46 we may, after replacing \(Y\) by a \(V\)-admissible blowing up, assume that \(T_c\) and \(T'_c\) have disjoint closures in \(|Y|\). Let \(Y_0\) be the open subspace of \(Y\) corresponding to the open \(|Y| \setminus \overline{T}'_c\) and set \(V_0 = V \cap Y_0\), \(U_0 = U \times_V V_0\), and \(X_0 = X \times_Y Y_0\). Since \(U_0 \to V_0\) is an isomorphism, we can find a \(V_0\)-admissible blowing up \(Y'_0 \to Y_0\) such that the strict transform \(X'_0\) of \(X_0\) maps isomorphically to \(Y'_0\), see More on Morphisms of Spaces, Lemma 087F. By Divisors on Spaces, Lemma 086D there exists a \(V\)-admissible blow up \(Y' \to Y\) whose restriction to \(Y_0\) is \(Y'_0 \to Y_0\). If \(f' : X' \to Y'\) denotes the strict transform of \(f\), then we see what we want is true because \(f'\) restricts to an isomorphism over \(Y'_0\).
Lemma
Let \(S\) be a scheme. Consider a diagram \[\xymatrix{ X \ar[d]_f & U \ar[l] \ar[d]_{f|_U} & A \ar[d] \ar[l] \\ Y & V \ar[l] & B \ar[l] }\] of quasi-compact and quasi-separated algebraic spaces over \(S\). Assume
\(f\) is proper,
\(V\) is a quasi-compact open of \(Y\), \(U = f^{-1}(V)\),
\(B \subset V\) and \(A \subset U\) are closed subspaces,
\(f|_A : A \to B\) is an isomorphism, and \(f\) is étale at every point of \(A\).
Then there exists a \(V\)-admissible blowing up \(Y' \to Y\) such that the strict transform \(f' : X' \to Y'\) satisfies: for every geometric point \(\overline{a}\) of the closure of \(|A|\) in \(|X'|\) there exists a quotient \(\mathcal{O}_{X', \overline{a}} \to \mathcal{O}\) such that \(\mathcal{O}_{Y', f'(\overline{a})} \to \mathcal{O}\) is finite flat.
As you can see from the proof, more is true, but the statement is already long enough and this will be sufficient later on.
Proof
Let \(T' \subset |U|\) be the complement of the maximal open on which \(f|_U\) is étale. Then \(T'\) is closed in \(|U|\) and disjoint from \(|A|\). Since \(|U|\) is a spectral topological space (Properties of Spaces, Lemma 0A4G) we can find constructible closed subsets \(T_c, T'_c\) of \(|U|\) with \(|A| \subset T_c\), \(T' \subset T'_c\) such that \(T_c \cap T'_c = \emptyset\) (see proof of Lemma 0F47). By Lemma 0F46 there is a \(U\)-admissible blowing up \(X_1 \to X\) such that \(T_c\) and \(T'_c\) have disjoint closures in \(|X_1|\). Let \(X_{1, 0}\) be the open subspace of \(X_1\) corresponding to the open \(|X_1| \setminus \overline{T}'_c\) and set \(U_0 = U \cap X_{1, 0}\). Observe that the scheme theoretic image \(\overline{A}_1 \subset X_1\) of \(A\) is contained in \(X_{1, 0}\) by construction.
After replacing \(Y\) by a \(V\)-admissible blowing up and taking strict transforms, we may assume \(X_{1, 0} \to Y\) is flat, quasi-finite, and of finite presentation, see More on Morphisms of Spaces, Lemmas 087E and 0878. Consider the commutative diagram \[\vcenter{ \xymatrix{ X_1 \ar[rr] \ar[rd] & & X \ar[ld] \\ & Y } } \quad\text{and the diagram}\quad \vcenter{ \xymatrix{ \overline{A}_1 \ar[rr] \ar[rd] & & \overline{A} \ar[ld] \\ & \overline{B} } }\] of scheme theoretic images. The morphism \(\overline{A}_1 \to \overline{A}\) is surjective because it is proper and hence the scheme theoretic image of \(\overline{A}_1 \to \overline{A}\) must be equal to \(\overline{A}\) and then we can use Morphisms of Spaces, Lemma 0AGD. The statement on étale local rings follows by choosing a lift of the geometric point \(\overline{a}\) to a geometric point \(\overline{a}_1\) of \(\overline{A}_1\) and setting \(\mathcal{O} = \mathcal{O}_{X_1, \overline{a}_1}\). Namely, since \(X_1 \to Y\) is flat and quasi-finite on \(X_{1, 0} \supset \overline{A}_1\), the map \(\mathcal{O}_{Y', f'(\overline{a})} \to \mathcal{O}_{X_1, \overline{a}_1}\) is finite flat, see Algebra, Lemmas 05WR and 04GG.
Lemma
Let \(S\) be a scheme. Let \(X \to B\) and \(Y \to B\) be morphisms of algebraic spaces over \(S\). Let \(U \subset X\) be an open subspace. Let \(V \to X \times_B Y\) be a quasi-compact morphism whose composition with the first projection maps into \(U\). Let \(Z \subset X \times_B Y\) be the scheme theoretic image of \(V \to X \times_B Y\). Let \(X' \to X\) be a \(U\)-admissible blowup. Then the scheme theoretic image of \(V \to X' \times_B Y\) is the strict transform of \(Z\) with respect to the blowing up.
Proof
Denote \(Z' \to Z\) the strict transform. The morphism \(Z' \to X'\) induces a morphism \(Z' \to X' \times_B Y\) which is a closed immersion (as \(Z'\) is a closed subspace of \(X' \times_X Z\) by definition). Thus to finish the proof it suffices to show that the scheme theoretic image \(Z''\) of \(V \to Z'\) is \(Z'\). Observe that \(Z'' \subset Z'\) is a closed subspace such that \(V \to Z'\) factors through \(Z''\). Since both \(V \to X \times_B Y\) and \(V \to X' \times_B Y\) are quasi-compact (for the latter this follows from Morphisms of Spaces, Lemma 03KS and the fact that \(X' \times_B Y \to X \times_B Y\) is separated as a base change of a proper morphism), by Morphisms of Spaces, Lemma 082Z we see that \(Z \cap (U \times_B Y) = Z'' \cap (U \times_B Y)\). Thus the inclusion morphism \(Z'' \to Z'\) is an isomorphism away from the exceptional divisor \(E\) of \(Z' \to Z\). However, the structure sheaf of \(Z'\) does not have any nonzero sections supported on \(E\) (by definition of strict transforms) and we conclude that the surjection \(\mathcal{O}_{Z'} \to \mathcal{O}_{Z''}\) must be an isomorphism.
Lemma
Let \(S\) be a scheme. Let \(B\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(U\) be an algebraic space of finite type and separated over \(B\). Let \(V \to U\) be an étale morphism. If \(V\) has a compactification \(V \subset Y\) over \(B\), then there exists a \(V\)-admissible blowing up \(Y' \to Y\) and an open \(V \subset V' \subset Y'\) such that \(V \to U\) extends to a proper morphism \(V' \to U\).
Proof
Consider the scheme theoretic image \(Z \subset Y \times_B U\) of the “diagonal” morphism \(V \to Y \times_B U\). If we replace \(Y\) by a \(V\)-admissible blowing up, then \(Z\) is replaced by the strict transform with respect to this blowing up, see Lemma 0F49. Hence by More on Morphisms of Spaces, Lemma 087F we may assume \(Z \to Y\) is an open immersion. If \(V' \subset Y\) denotes the image, then we see that the induced morphism \(V' \to U\) is proper because the projection \(Y \times_B U \to U\) is proper and \(V' \cong Z\) is a closed subspace of \(Y \times_B U\).
The following lemma is formulated for finite type separated algebraic spaces over a finite type algebraic space over \(\mathbf{Z}\). The version for quasi-compact and quasi-separated algebraic spaces is true as well (with essentially the same proof), but will be trivially implied by the main theorem in this section. We strongly urge the reader to read the proof of this lemma in the case of schemes first.
Lemma
Let \(B\) be an algebraic space of finite type over \(\mathbf{Z}\). Let \(U\) be an algebraic space of finite type and separated over \(B\). Let \((U_2 \subset U, f : U_1 \to U)\) be an elementary distinguished square. Assume \(U_1\) and \(U_2\) have compactifications over \(B\) and \(U_1 \times_U U_2 \to U\) has dense image. Then \(U\) has a compactification over \(B\).
Proof
Choose a compactification \(U_i \subset X_i\) over \(B\) for \(i = 1, 2\). We may assume \(U_i\) is scheme theoretically dense in \(X_i\). We may assume there is an open \(V_i \subset X_i\) and a proper morphism \(\psi_i : V_i \to U\) extending \(U_i \to U\), see Lemma 0F4A. Picture \[\xymatrix{ U_i \ar[r] \ar[d] & V_i \ar[r] \ar[dl]^{\psi_i} & X_i \\ U }\] Denote \(Z_1 \subset U\) the reduced closed subspace corresponding to the closed subset \(|U| \setminus |U_2|\). Recall that \(f^{-1}Z_1\) is a closed subspace of \(U_1\) mapping isomorphically to \(Z_1\). Denote \(Z_2 \subset U\) the reduced closed subspace corresponding to the closed subset \(|U| \setminus \Im(|f|) = |U_2| \setminus \Im(|U_1 \times_U U_2| \to |U_2|)\). Thus we have \[U = U_2 \amalg Z_1 = Z_2 \amalg \Im(f) = Z_2 \amalg \Im(U_1 \times_U U_2 \to U_2) \amalg Z_1\] set theoretically. Denote \(Z_{i, i} \subset V_i\) the inverse image of \(Z_i\) under \(\psi_i\). Observe that \(\psi_2\) is an isomorphism over an open neighbourhood of \(Z_2\). Observe that \(Z_{1, 1} = \psi_1^{-1}Z_1 = f^{-1}Z_1 \amalg T\) for some closed subspace \(T \subset V_1\) disjoint from \(f^{-1}Z_1\) and furthermore \(\psi_1\) is étale along \(f^{-1}Z_1\). Denote \(Z_{i, j} \subset V_i\) the inverse image of \(Z_j\) under \(\psi_i\). Observe that \(\psi_i : Z_{i, j} \to Z_j\) is a proper morphism. Since \(Z_i\) and \(Z_j\) are disjoint closed subspaces of \(U\), we see that \(Z_{i, i}\) and \(Z_{i, j}\) are disjoint closed subspaces of \(V_i\).
Denote \(\overline{Z}_{i, i}\) and \(\overline{Z}_{i, j}\) the scheme theoretic images of \(Z_{i, i}\) and \(Z_{i, j}\) in \(X_i\). We recall that \(|Z_{i, j}|\) is dense in \(|\overline{Z}_{i, j}|\), see Morphisms of Spaces, Lemma 088G. After replacing \(X_i\) by a \(V_i\)-admissible blowup we may assume that \(\overline{Z}_{i, i}\) and \(\overline{Z}_{i, j}\) are disjoint, see Lemma 0F46. We assume this holds for both \(X_1\) and \(X_2\). Observe that this property is preserved if we replace \(X_i\) by a further \(V_i\)-admissible blowup. Hence we may replace \(X_1\) by another \(V_1\)-admissible blowup and assume \(|\overline{Z}_{1, 1}|\) is the disjoint union of the closures of \(|T|\) and \(|f^{-1}Z_1|\) in \(|X_1|\).
Set \(V_{12} = V_1 \times_U V_2\). We have an immersion \(V_{12} \to X_1 \times_B X_2\) which is the composition of the closed immersion \(V_{12} = V_1 \times_U V_2 \to V_1 \times_B V_2\) (Morphisms of Spaces, Lemma 03KN) and the open immersion \(V_1 \times_B V_2 \to X_1 \times_B X_2\). Let \(X_{12} \subset X_1 \times_B X_2\) be the scheme theoretic image of \(V_{12} \to X_1 \times_B X_2\). The projection morphisms \[p_1 : X_{12} \to X_1 \quad\text{and}\quad p_2 : X_{12} \to X_2\] are proper as \(X_1\) and \(X_2\) are proper over \(B\). If we replace \(X_1\) by a \(V_1\)-admissible blowing up, then \(X_{12}\) is replaced by the strict transform with respect to this blowing up, see Lemma 0F49.
Denote \(\psi : V_{12} \to U\) the compositions \(\psi = \psi_1 \circ p_1|_{V_{12}} = \psi_2 \circ p_2|_{V_{12}}\). Consider the closed subspace \[Z_{12, 2} = (p_1|_{V_{12}})^{-1}Z_{1, 2} = (p_2|_{V_{12}})^{-1}Z_{2, 2} = \psi^{-1}Z_2 \subset V_{12}\] The morphism \(p_1|_{V_{12}} : V_{12} \to V_1\) is an isomorphism over an open neighbourhood of \(Z_{1, 2}\) because \(\psi_2 : V_2 \to U\) is an isomorphism over an open neighbourhood of \(Z_2\) and \(V_{12} = V_1 \times_U V_2\). By Lemma 0F47 there exists a \(V_1\)-admissible blowing up \(X_1' \to X_1\) such that the strict transform \(p'_1 : X'_{12} \to X'_1\) of \(p_1\) is an isomorphism over an open neighbourhood of the closure of \(|Z_{1, 2}|\) in \(|X'_1|\). After replacing \(X_1\) by \(X'_1\) and \(X_{12}\) by \(X'_{12}\) we may assume that \(p_1\) is an isomorphism over an open neighbourhood of \(|\overline{Z}_{1, 2}|\).
The result of the previous paragraph tells us that \[X_{12} \cap (\overline{Z}_{1, 2} \times_B \overline{Z}_{2, 1}) = \emptyset\] where the intersection taken in \(X_1 \times_B X_2\). Namely, the inverse image \(p_1^{-1}\overline{Z}_{1, 2}\) in \(X_{12}\) maps isomorphically to \(\overline{Z}_{1, 2}\). In particular, we see that \(|Z_{12, 2}|\) is dense in \(|p_1^{-1}\overline{Z}_{1, 2}|\). Thus \(p_2\) maps \(|p_1^{-1}\overline{Z}_{1, 2}|\) into \(|\overline{Z}_{2, 2}|\). Since \(|\overline{Z}_{2, 2}| \cap |\overline{Z}_{2, 1}| = \emptyset\) we conclude.
It turns out that we need to do one additional blowing up before we can conclude the argument. Namely, let \(V_2 \subset W_2 \subset X_2\) be the open subspace with underlying topological space \[|W_2| = |V_2| \cup (|X_2| \setminus |\overline{Z}_{2, 1}|) = |X_2| \setminus \left(|\overline{Z}_{2, 1}| \setminus |Z_{2, 1}|\right)\] Since \(p_2(p_1^{-1}\overline{Z}_{1, 2})\) is contained in \(W_2\) (see above) we see that replacing \(X_2\) by a \(W_2\)-admissible blowup and \(X_{21}\) by the corresponding strict transform will preserve the property of \(p_1\) being an isomorphism over an open neighbourhood of \(\overline{Z}_{1, 2}\). Since \(\overline{Z}_{2, 1} \cap W_2 = \overline{Z}_{2, 1} \cap V_2 = Z_{2, 1}\) we see that \(Z_{2, 1}\) is a closed subspace of \(W_2\) and \(V_2\). Observe that \(V_{12} = V_1 \times_U V_2 = p_1^{-1}(V_1) = p_2^{-1}(V_2)\) as open subspaces of \(X_{12}\) as it is the largest open subspace of \(X_{12}\) over which the morphism \(\psi : V_{12} \to U\) extends; details omitted4. We have the following equalities of closed subspaces of \(V_{12}\): \[p_2^{-1}Z_{2, 1} = p_2^{-1} \psi_2^{-1} Z_1 = p_1^{-1} \psi_1^{-1} Z_1= p_1^{-1}Z_{1, 1} = p_1^{-1}f^{-1}Z_1 \amalg p_1^{-1}T\] Here and below we use the slight abuse of notation of writing \(p_2\) in stead of the restriction of \(p_2\) to \(V_{12}\), etc. Since \(p_2^{-1}(Z_{2, 1})\) is a closed subspace of \(p_2^{-1}(W_2)\) as \(Z_{2, 1}\) is a closed subspace of \(W_2\) we conclude that also \(p_1^{-1}f^{-1}Z_1\) is a closed subspace of \(p_2^{-1}(W_2)\). Finally, the morphism \(p_2 : X_{12} \to X_2\) is étale at points of \(p_1^{-1}f^{-1}Z_1\) as \(\psi_1\) is étale along \(f^{-1}Z_1\) and \(V_{12} = V_1 \times_U V_2\). Thus we may apply Lemma 0F48 to the morphism \(p_2 : X_{12} \to X_2\), the open \(W_2\), the closed subspace \(Z_{2, 1} \subset W_2\), and the closed subspace \(p_1^{-1}f^{-1}Z_1 \subset p_2^{-1}(W_2)\). Hence after replacing \(X_2\) by a \(W_2\)-admissible blowup and \(X_{12}\) by the corresponding strict transform, we obtain for every geometric point \(\overline{y}\) of the closure of \(|p_1^{-1}f^{-1}Z_1|\) a local ring map \(\mathcal{O}_{X_{12}, \overline{y}} \to \mathcal{O}\) such that \(\mathcal{O}_{X_2, p_2(\overline{y})} \to \mathcal{O}\) is finite flat.
Consider the algebraic space \[W_2 = U \coprod\nolimits_{U_2} (X_2 \setminus \overline{Z}_{2, 1}),\] and with \(T \subset V_1\) as in the first paragraph the algebraic space \[W_1 = U \coprod\nolimits_{U_1} (X_1 \setminus \overline{Z}_{1, 2} \cup \overline{T}),\] obtained by pushout, see Lemma 0DVJ. Let us apply Lemma 0F45 to see that \(W_i \to B\) is separated. First, \(U \to B\) and \(X_i \to B\) are separated. Let us check the quasi-compact immersion \(U_i \to U \times_B (X_i \setminus \overline{Z}_{i, j})\) is closed using the valuative criterion, see Morphisms of Spaces, Lemma 03KA. Choose a valuation ring \(A\) over \(B\) with fraction field \(K\) and compatible morphisms \((u, x_i) : \Spec(A) \to U \times_B X_i\) and \(u_i : \Spec(K) \to U_i\). Since \(\psi_i\) is proper, we can find a unique \(v_i : \Spec(A) \to V_i\) compatible with \(u\) and \(u_i\). Since \(X_i\) is proper over \(B\) we see that \(x_i = v_i\). If \(v_i\) does not factor through \(U_i \subset V_i\), then we conclude that \(x_i\) maps the closed point of \(\Spec(A)\) into \(Z_{i, j}\) or \(T\) when \(i = 1\). This finishes the proof because we removed \(\overline{Z}_{i, j}\) and \(\overline{T}\) in the construction of \(W_i\).
On the other hand, for any valuation ring \(A\) over \(B\) with fraction field \(K\) and any morphism \[\gamma : \Spec(K) \to \Im(U_1 \times_U U_2 \to U)\] over \(B\), we claim that after replacing \(A\) by an extension of valuation rings, there is an \(i\) and an extension of \(\gamma\) to a morphism \(h_i : \Spec(A) \to W_i\). Namely, we first extend \(\gamma\) to a morphism \(g_2 : \Spec(A) \to X_2\) using the valuative criterion of properness. If the image of \(g_2\) does not meet \(\overline{Z}_{2, 1}\), then we obtain our morphism into \(W_2\). Otherwise, denote \(\overline{z} \in \overline{Z}_{2, 1}\) a geometric point lying over the image of the closed point under \(g_2\). We may lift this to a geometric point \(\overline{y}\) of \(X_{12}\) in the closure of \(|p_1^{-1}f^{-1}Z_1|\) because the map of spaces \(|p_1^{-1}f^{-1}Z_1| \to |\overline{Z}_{2, 1}|\) is closed with image containing the dense open \(|Z_{2, 1}|\). After replacing \(A\) by its strict henselization (More on Algebra, Lemma 0ASK) we get the following diagram \[\xymatrix{ A \ar@{..>}[rr] & & A' \\ \mathcal{O}_{X_2, \overline{z}} \ar[r] \ar[u] & \mathcal{O}_{X_{12}, \overline{y}} \ar[r] & \mathcal{O} \ar@{..>}[u] }\] where \(\mathcal{O}_{X_{12}, \overline{y}} \to \mathcal{O}\) is the map we found in the 5th paragraph of the proof. Since the horizontal composition is finite and flat we can find an extension of valuation rings \(A'/A\) and dotted arrow making the diagram commute. After replacing \(A\) by \(A'\) this means that we obtain a lift \(g_{12} : \Spec(A) \to X_{12}\) whose closed point maps into the closure of \(|p_1^{-1}f^{-1}Z_1|\). Then \(g_1 = p_1 \circ g_{12} : \Spec(A) \to X_1\) is a morphism whose closed point maps into the closure of \(|f^{-1}Z_1|\). Since the closure of \(|f^{-1}Z_1|\) is disjoint from the closure of \(|T|\) and contained in \(|\overline{Z}_{1, 1}|\) which is disjoint from \(|\overline{Z}_{1, 2}|\) we conclude that \(g_1\) defines a morphism \(h_1 : \Spec(A) \to W_1\) as desired.
Consider a diagram \[\xymatrix{ W_1' \ar[d] \ar[r] & W & W_2' \ar[l] \ar[d] \\ W_1 & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & W_2 }\] as in More on Morphisms of Spaces, Lemma 088Q. By the previous paragraph for every solid diagram \[\xymatrix{ \Spec(K) \ar[r]_\gamma \ar[d] & W \ar[d] \\ \Spec(A) \ar@{..>}[ru] \ar[r] & B }\] where \(\Im(\gamma) \subset \Im(U_1 \times_U U_2 \to U)\) there is an \(i\) and an extension \(h_i : \Spec(A) \to W_i\) of \(\gamma\) after possibly replacing \(A\) by an extension of valuation rings. Using the valuative criterion of properness for \(W'_i \to W_i\), we can then lift \(h_i\) to \(h'_i : \Spec(A) \to W'_i\). Hence the dotted arrow in the diagram exists after possibly extending \(A\). Since \(W\) is separated over \(B\), we see that the choice of extension isn’t needed and the arrow is unique as well, see Morphisms of Spaces, Lemmas 0A3W and 03KU. Then finally the existence of the dotted arrow implies that \(W \to B\) is universally closed by Morphisms of Spaces, Lemma 089G. As \(W \to B\) is already of finite type and separated, we win.
Lemma
Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(U \subset X\) be a proper dense open subspace. Then there exists an affine scheme \(V\) and an étale morphism \(V \to X\) such that
the open subspace \(W = U \cup \Im(V \to X)\) is strictly larger than \(U\),
\((U \subset W, V \to W)\) is a distinguished square, and
\(U \times_W V \to U\) has dense image.
Proof
Choose a stratification \[\emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X\] and morphisms \(f_p : V_p \to U_p\) as in Decent Spaces, Lemma 07ST. Let \(p\) be the smallest integer such that \(U_p \not \subset U\) (this is possible as \(U \not = X\)). Choose an affine open \(V \subset V_p\) such that the étale morphism \(f_p|_V : V \to X\) does not factor through \(U\). Consider the open \(W = U \cup \Im(V \to X)\) and the reduced closed subspace \(Z \subset W\) with \(|Z| = |W| \setminus |U|\). Then \(f^{-1}Z \to Z\) is an isomorphism because we have the corresponding property for the morphism \(f_p\), see the lemma cited above. Thus \((U \subset W, f : V \to W)\) is a distinguished square. It may not be true that the open \(I = \Im(U \times_W V \to U)\) is dense in \(U\). The algebraic space \(U' \subset U\) whose underlying set is \(|U| \setminus \overline{|I|}\) is Noetherian and hence we can find a dense open subscheme \(U'' \subset U'\), see for example Properties of Spaces, Proposition 06NH. Then we can find a dense open affine \(U''' \subset U''\), see Properties, Lemmas 0BA8 and 01ZV. After we replace \(f\) by \(V \amalg U''' \to X\) everything is clear.
Theorem
Let \(S\) be a scheme. Let \(B\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(X \to B\) be a separated, finite type morphism. Then \(X\) has a compactification over \(B\).
Proof
We first reduce to the Noetherian case. We strongly urge the reader to skip this paragraph. First, we may replace \(S\) by \(\Spec(\mathbf{Z})\). See Spaces, Section 03I3 and Properties of Spaces, Definition 03BS. There exists a closed immersion \(X \to X'\) with \(X' \to B\) of finite presentation and separated. See Limits of Spaces, Proposition 0873. If we find a compactification of \(X'\) over \(B\), then taking the scheme theoretic closure of \(X\) in this will give a compactification of \(X\) over \(B\). Thus we may assume \(X \to B\) is separated and of finite presentation. We may write \(B = \lim B_i\) as a directed limit of a system of Noetherian algebraic spaces of finite type over \(\Spec(\mathbf{Z})\) with affine transition morphisms. See Limits of Spaces, Proposition 07SU. We can choose an \(i\) and a morphism \(X_i \to B_i\) of finite presentation whose base change to \(B\) is \(X \to B\), see Limits of Spaces, Lemma 07SK. After increasing \(i\) we may assume \(X_i \to B_i\) is separated, see Limits of Spaces, Lemma 0851. If we can find a compactification of \(X_i\) over \(B_i\), then the base change of this to \(B\) will be a compactification of \(X\) over \(B\). This reduces us to the case discussed in the next paragraph.
Assume \(B\) is of finite type over \(\mathbf{Z}\) in addition to being quasi-compact and quasi-separated. Let \(U \to X\) be an étale morphism of algebraic spaces such that \(U\) has a compactification \(Y\) over \(\Spec(\mathbf{Z})\). The morphism \[U \longrightarrow B \times_{\Spec(\mathbf{Z})} Y\] is separated and quasi-finite by Morphisms of Spaces, Lemma 0463 (the displayed morphism factors into an immersion hence is a monomorphism). Hence by Zariski’s main theorem (More on Morphisms of Spaces, Lemma 082K) there is an open immersion of \(U\) into an algebraic space \(Y'\) finite over \(B \times_{\Spec(\mathbf{Z})} Y\). Then \(Y' \to B\) is proper as the composition \(Y' \to B \times_{\Spec(\mathbf{Z})} Y \to B\) of two proper morphisms (use Morphisms of Spaces, Lemmas 04NZ, 04XY, and 04WP). We conclude that \(U\) has a compactification over \(B\).
There is a dense open subspace \(U \subset X\) which is a scheme. (Properties of Spaces, Proposition 06NH). In fact, we may choose \(U\) to be an affine scheme (Properties, Lemmas 0BA8 and 01ZV). Thus \(U\) has a compactification over \(\Spec(\mathbf{Z})\); this is easily shown directly but also follows from the theorem for schemes, see More on Flatness, Theorem 0F41. By the previous paragraph \(U\) has a compactification over \(B\). By Noetherian induction we can find a maximal dense open subspace \(U \subset X\) which has a compactification over \(B\). We will show that the assumption that \(U \not = X\) leads to a contradiction. Namely, by Lemma 0F4C we can find a strictly larger open \(U \subset W \subset X\) and a distinguished square \((U \subset W, f : V \to W)\) with \(V\) affine and \(U \times_W V\) dense image in \(U\). Since \(V\) is affine, as before it has a compactification over \(B\). Hence Lemma 0F4B applies to show that \(W\) has a compactification over \(B\) which is the desired contradiction.
It follows from the fact that \(\mathcal{F}\) satisfies the corresponding condition.↩︎
It follows from the fact that \(\mathcal{G}\) and \(\mathcal{G}'\) satisfies the corresponding condition.↩︎
All functors given by derived pullback.↩︎
Namely, \(V_1 \times_U V_2\) is proper over \(U\) so if \(\psi\) extends to a larger open of \(X_{12}\), then \(V_1 \times_U V_2\) would be closed in this open by Morphisms of Spaces, Lemma 04NX. Then we get equality as \(V_{12} \subset X_{12}\) is dense.↩︎