Descent and Algebraic Spaces

Introduction

In the chapter on topologies on algebraic spaces (see Topologies on Spaces, Section 03Y5) we introduced étale, fppf, smooth, syntomic and fpqc coverings of algebraic spaces. In this chapter we discuss what kind of structures over algebraic spaces can be descended through such coverings. See for example [Gr-I], [Gr-II], [Gr-III], [Gr-IV], [Gr-V], and [Gr-VI].

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 \(X \times X\).

Descent data for quasi-coherent sheaves

This section is the analogue of Descent, Section 023A for algebraic spaces. It makes sense to read that section first.

Definition

Let \(S\) be a scheme. Let \(\{f_i : X_i \to X\}_{i \in I}\) be a family of morphisms of algebraic spaces over \(S\) with fixed target \(X\).

1. A descent datum \((\mathcal{F}_i, \varphi_{ij})\) for quasi-coherent sheaves with respect to the given family is given by a quasi-coherent sheaf \(\mathcal{F}_i\) on \(X_i\) for each \(i \in I\), an isomorphism of quasi-coherent \(\mathcal{O}_{X_i \times_X X_j}\)-modules \(\varphi_{ij} : \text{pr}_0^*\mathcal{F}_i \to \text{pr}_1^*\mathcal{F}_j\) for each pair \((i, j) \in I^2\) such that for every triple of indices \((i, j, k) \in I^3\) the diagram \[\xymatrix{ \text{pr}_0^*\mathcal{F}_i \ar[rd]_{\text{pr}_{01}^*\varphi_{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi_{ik}} & & \text{pr}_2^*\mathcal{F}_k \\ & \text{pr}_1^*\mathcal{F}_j \ar[ru]_{\text{pr}_{12}^*\varphi_{jk}} & }\] of \(\mathcal{O}_{X_i \times_X X_j \times_X X_k}\)-modules commutes. This is called the cocycle condition.

2. A morphism \(\psi : (\mathcal{F}_i, \varphi_{ij}) \to (\mathcal{F}'_i, \varphi'_{ij})\) of descent data is given by a family \(\psi = (\psi_i)_{i\in I}\) of morphisms of \(\mathcal{O}_{X_i}\)-modules \(\psi_i : \mathcal{F}_i \to \mathcal{F}'_i\) such that all the diagrams \[\xymatrix{ \text{pr}_0^*\mathcal{F}_i \ar[r]_{\varphi_{ij}} \ar[d]_{\text{pr}_0^*\psi_i} & \text{pr}_1^*\mathcal{F}_j \ar[d]^{\text{pr}_1^*\psi_j} \\ \text{pr}_0^*\mathcal{F}'_i \ar[r]^{\varphi'_{ij}} & \text{pr}_1^*\mathcal{F}'_j \\ }\] commute.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) and \(\mathcal{V} = \{V_j \to V\}_{j \in J}\) be families of morphisms of algebraic spaces over \(S\) with fixed targets. Let \((g, \alpha : I \to J, (g_i)) : \mathcal{U} \to \mathcal{V}\) be a morphism of families of maps with fixed target, see Sites, Definition 00VT. Let \((\mathcal{F}_j, \varphi_{jj'})\) be a descent datum for quasi-coherent sheaves with respect to the family \(\{V_j \to V\}_{j \in J}\). Then

1. The system \[\left(g_i^*\mathcal{F}_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}\right)\] is a descent datum with respect to the family \(\{U_i \to U\}_{i \in I}\).

2. This construction is functorial in the descent datum \((\mathcal{F}_j, \varphi_{jj'})\).

3. Given a second morphism \((g', \alpha' : I \to J, (g'_i))\) of families of maps with fixed target with \(g = g'\) there exists a functorial isomorphism of descent data \[(g_i^*\mathcal{F}_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}) \cong ((g'_i)^*\mathcal{F}_{\alpha'(i)}, (g'_i \times g'_{i'})^*\varphi_{\alpha'(i)\alpha'(i')}).\]

Proof

Omitted. Hint: The maps \(g_i^*\mathcal{F}_{\alpha(i)} \to (g'_i)^*\mathcal{F}_{\alpha'(i)}\) which give the isomorphism of descent data in part (3) are the pullbacks of the maps \(\varphi_{\alpha(i)\alpha'(i)}\) by the morphisms \((g_i, g'_i) : U_i \to V_{\alpha(i)} \times_V V_{\alpha'(i)}\).

Let \(g : U \to V\) be a morphism of algebraic spaces. The lemma above tells us that there is a well defined pullback functor between the categories of descent data relative to families of maps with target \(V\) and \(U\) provided there is a morphism between those families of maps which “lives over \(g\)”.

Definition

Let \(S\) be a scheme. Let \(\{U_i \to U\}_{i \in I}\) be a family of morphisms of algebraic spaces over \(S\) with fixed target.

1. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_U\)-module. We call the unique descent on \(\mathcal{F}\) datum with respect to the covering \(\{U \to U\}\) the trivial descent datum.

2. The pullback of the trivial descent datum to \(\{U_i \to U\}\) is called the canonical descent datum. Notation: \((\mathcal{F}|_{U_i}, can)\).

3. A descent datum \((\mathcal{F}_i, \varphi_{ij})\) for quasi-coherent sheaves with respect to the given family is said to be effective if there exists a quasi-coherent sheaf \(\mathcal{F}\) on \(U\) such that \((\mathcal{F}_i, \varphi_{ij})\) is isomorphic to \((\mathcal{F}|_{U_i}, can)\).

Lemma

Let \(S\) be a scheme. Let \(U\) be an algebraic space over \(S\). Let \(\{U_i \to U\}\) be a Zariski covering of \(U\), see Topologies on Spaces, Definition 041G. Any descent datum on quasi-coherent sheaves for the family \(\mathcal{U} = \{U_i \to U\}\) is effective. Moreover, the functor from the category of quasi-coherent \(\mathcal{O}_U\)-modules to the category of descent data with respect to \(\{U_i \to U\}\) is fully faithful.

Proof

Omitted.

Fpqc descent of quasi-coherent sheaves

The main application of flat descent for modules is the corresponding descent statement for quasi-coherent sheaves with respect to fpqc-coverings.

Proposition

Let \(S\) be a scheme. Let \(\{X_i \to X\}\) be an fpqc covering of algebraic spaces over \(S\), see Topologies on Spaces, Definition 03MQ. Any descent datum on quasi-coherent sheaves for \(\{X_i \to X\}\) is effective. Moreover, the functor from the category of quasi-coherent \(\mathcal{O}_X\)-modules to the category of descent data with respect to \(\{X_i \to X\}\) is fully faithful.

Proof

This is more or less a formal consequence of the corresponding result for schemes, see Descent, Proposition 023T. Here is a strategy for a proof:

1. The fact that \(\{X_i \to X\}\) is a refinement of the trivial covering \(\{X \to X\}\) gives, via Lemma 04W4, a functor \(\QCoh(\mathcal{O}_X) \to DD(\{X_i \to X\})\) from the category of quasi-coherent \(\mathcal{O}_X\)-modules to the category of descent data for the given family.

2. In order to prove the proposition we will construct a quasi-inverse functor \(back : DD(\{X_i \to X\}) \to \QCoh(\mathcal{O}_X)\).

3. Applying again Lemma 04W4 we see that there is a functor \(DD(\{X_i \to X\}) \to DD(\{T_j \to X\})\) if \(\{T_j \to X\}\) is a refinement of the given family. Hence in order to construct the functor \(back\) we may assume that each \(X_i\) is a scheme, see Topologies on Spaces, Lemma 0419. This reduces us to the case where all the \(X_i\) are schemes.

4. A quasi-coherent sheaf on \(X\) is by definition a quasi-coherent \(\mathcal{O}_X\)-module on \(X_\etale\). Now for any \(U \in \Ob(X_\etale)\) we get an fppf covering \(\{U_i \times_X X_i \to U\}\) by schemes and a morphism \(g : \{U_i \times_X X_i \to U\} \to \{X_i \to X\}\) of coverings lying over \(U \to X\). Given a descent datum \(\xi = (\mathcal{F}_i, \varphi_{ij})\) we obtain a quasi-coherent \(\mathcal{O}_U\)-module \(\mathcal{F}_{\xi, U}\) corresponding to the pullback \(g^*\xi\) of Lemma 04W4 to the covering of \(U\) and using effectivity for fppf covering of schemes, see Descent, Proposition 023T.

5. Check that \(\xi \mapsto \mathcal{F}_{\xi, U}\) is functorial in \(\xi\). Omitted.

6. Check that \(\xi \mapsto \mathcal{F}_{\xi, U}\) is compatible with morphisms \(U \to U'\) of the site \(X_\etale\), so that the system of sheaves \(\mathcal{F}_{\xi, U}\) corresponds to a quasi-coherent \(\mathcal{F}_\xi\) on \(X_\etale\), see Properties of Spaces, Lemma 03LZ. Details omitted.

7. Check that \(back : \xi \mapsto \mathcal{F}_\xi\) is quasi-inverse to the functor constructed in (1). Omitted.

This finishes the proof.

Quasi-coherent modules and affines

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Recall that \(X_{affine, \etale}\) is the full subcategory of \(X_\etale\) whose objects are affine turned into a site by declaring the coverings to be the standard étale coverings. See Properties of Spaces, Definition 0H01. By Properties of Spaces, Lemma 04JS we have an equivalence of topoi \(g : \Sh(X_{affine, \etale}) \to \Sh(X_\etale)\) whose pullback functor is given by restriction. Recall that \(\mathcal{O}_X\) denotes the structure sheaf on \(X_\etale\). Then we obtain an equivalence [0H03]\[\begin{equation} (\Sh(X_{affine, \etale}), \mathcal{O}_X|_{X_{affine, \etale}}) \longrightarrow (\Sh(X_\etale), \mathcal{O}_X) \end{equation}\] of ringed topoi. We will often write \(\mathcal{O}_X\) in stead of \(\mathcal{O}_X|_{X_{affine, \etale}}\). Having said this we can compare quasi-coherent modules as well.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}_X\)-modules on \(X_{affine, \etale}\). The following are equivalent

1. for every morphism \(U \to U'\) of \(X_{affine, \etale}\) the map \(\mathcal{F}(U') \otimes_{\mathcal{O}_X(U')} \mathcal{O}_X(U) \to \mathcal{F}(U)\) is an isomorphism,

2. \(\mathcal{F}\) is a quasi-coherent module on the ringed site \((X_{affine, \etale}, \mathcal{O}_X)\) in the sense of Modules on Sites, Definition 03DL,

3. \(\mathcal{F}\) corresponds to a quasi-coherent module on \(X\) via the equivalence (0H03),

Proof

Assume (1) holds. To show that \(\mathcal{F}\) is a sheaf, let \(\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}\) be a covering of \(X_{affine, \etale}\). The sheaf condition for \(\mathcal{F}\) and \(\mathcal{U}\), by our assumption on \(\mathcal{F}\), reduces to showing that \[0 \to \mathcal{F}(U) \to \prod \mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{O}_X(U_i) \to \prod \mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{O}_X(U_i \times_U U_j)\] is exact. This is true because \(\mathcal{O}_X(U) \to \prod \mathcal{O}_X(U_i)\) is faithfully flat (by Descent, Lemma 03FI and the fact that coverings in \(X_{affine, \etale}\) are standard étale coverings) and we may apply Descent, Lemma 023M. Next, we show that \(\mathcal{F}\) is quasi-coherent on \(X_{affine, \etale}\). Namely, for \(U\) in \(X_{affine, \etale}\), set \(R = \mathcal{O}_X(U)\) and choose a presentation \[\bigoplus\nolimits_{k \in K} R \longrightarrow \bigoplus\nolimits_{l \in L} R \longrightarrow \mathcal{F}(U) \longrightarrow 0\] by free \(R\)-modules. By property (1) and the right exactness of tensor product we see that for every morphism \(U' \to U\) in \(X_{affine, \etale}\) we obtain a presentation \[\bigoplus\nolimits_{k \in K} \mathcal{O}_X(U') \longrightarrow \bigoplus\nolimits_{l \in L} \mathcal{O}_X(U') \longrightarrow \mathcal{F}(U') \longrightarrow 0\] In other words, we see that the restriction of \(\mathcal{F}\) to the localized category \(X_{affine, etale}/U\) has a presentation \[\bigoplus\nolimits_{k \in K} \mathcal{O}_X|_{X_{affine, \etale}/U} \longrightarrow \bigoplus\nolimits_{l \in L} \mathcal{O}_X|_{X_{affine, \etale}/U} \longrightarrow \mathcal{F}|_{X_{affine, \etale}/U} \longrightarrow 0\] as required to show that \(\mathcal{F}\) is quasi-coherent. With apologies for the horrible notation, this finishes the proof that (1) implies (2).

Since the notion of a quasi-coherent module is intrinsic (Modules on Sites, Lemma 03DM) we see that the equivalence (0H03) induces an equivalence between categories of quasi-coherent modules. Thus we have the equivalence of (2) and (3).

Let us assume (3) and prove (1). Namely, let \(\mathcal{G}\) be a quasi-coherent module on \(X\) corresponding to \(\mathcal{F}\). Let \(h : U \to U' \to X\) be a morphism of \(X_{affine, \etale}\). Denote \(f : U \to X\) and \(f' : U' \to X\) the structure morphisms, so that \(f = f' \circ h\). We have \(\mathcal{F}(U') = \Gamma(U', (f')^*\mathcal{G})\) and \(\mathcal{F}(U) = \Gamma(U, f^*\mathcal{G}) = \Gamma(U, h^*(f')^*\mathcal{G})\). Hence (1) holds by Schemes, Lemma 01I9.

Descent of finiteness properties of modules

This section is the analogue for the case of algebraic spaces of Descent, Section 05AY. The goal is to show that one can check a quasi-coherent module has a certain finiteness conditions by checking on the members of a covering. We will repeatedly use the following proof scheme. Suppose that \(X\) is an algebraic space, and that \(\{X_i \to X\}\) is a fppf (resp. fpqc) covering. Let \(U \to X\) be a surjective étale morphism such that \(U\) is a scheme. Then there exists an fppf (resp. fpqc) covering \(\{Y_j \to X\}\) such that

1. \(\{Y_j \to X\}\) is a refinement of \(\{X_i \to X\}\),

2. each \(Y_j\) is a scheme, and

3. each morphism \(Y_j \to X\) factors though \(U\), and

4. \(\{Y_j \to U\}\) is an fppf (resp. fpqc) covering of \(U\).

Namely, first refine \(\{X_i \to X\}\) by an fppf (resp. fpqc) covering such that each \(X_i\) is a scheme, see Topologies on Spaces, Lemma 042T, resp. Lemma 0419. Then set \(Y_i = U \times_X X_i\). A quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) is of finite type, of finite presentation, etc if and only if the quasi-coherent \(\mathcal{O}_U\)-module \(\mathcal{F}|_U\) is of finite type, of finite presentation, etc. Hence we can use the existence of the refinement \(\{Y_j \to X\}\) to reduce the proof of the following lemmas to the case of schemes. We will indicate this by saying that “the result follows from the case of schemes by étale localization”.

Lemma

Let \(X\) be an algebraic space over a scheme \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is a finite type \(\mathcal{O}_{X_i}\)-module. Then \(\mathcal{F}\) is a finite type \(\mathcal{O}_X\)-module.

Proof

This follows from the case of schemes, see Descent, Lemma 05AZ, by étale localization.

Lemma

Let \(X\) be an algebraic space over a scheme \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is an \(\mathcal{O}_{X_i}\)-module of finite presentation. Then \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite presentation.

Proof

This follows from the case of schemes, see Descent, Lemma 05B0, by étale localization.

Lemma

Let \(X\) be an algebraic space over a scheme \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is a flat \(\mathcal{O}_{X_i}\)-module. Then \(\mathcal{F}\) is a flat \(\mathcal{O}_X\)-module.

Proof

This follows from the case of schemes, see Descent, Lemma 05B1, by étale localization.

Lemma

Let \(X\) be an algebraic space over a scheme \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is a finite locally free \(\mathcal{O}_{X_i}\)-module. Then \(\mathcal{F}\) is a finite locally free \(\mathcal{O}_X\)-module.

Proof

This follows from the case of schemes, see Descent, Lemma 05B2, by étale localization.

The definition of a locally projective quasi-coherent sheaf can be found in Properties of Spaces, Section 060P. It is also proved there that this notion is preserved under pullback.

Lemma

Let \(X\) be an algebraic space over a scheme \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is a locally projective \(\mathcal{O}_{X_i}\)-module. Then \(\mathcal{F}\) is a locally projective \(\mathcal{O}_X\)-module.

Proof

This follows from the case of schemes, see Descent, Lemma 05JZ, by étale localization.

We also add here two results which are related to the results above, but are of a slightly different nature.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume \(f\) is a finite morphism. Then \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite type if and only if \(f_*\mathcal{F}\) is an \(\mathcal{O}_Y\)-module of finite type.

Proof

As \(f\) is finite it is representable. Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Then \(U = V \times_Y X\) is a scheme with a surjective étale morphism towards \(X\) and a finite morphism \(\psi : U \to V\) (the base change of \(f\)). Since \(\psi_*(\mathcal{F}|_U) = f_*\mathcal{F}|_V\) the result of the lemma follows immediately from the schemes version which is Descent, Lemma 05B3.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume \(f\) is finite and of finite presentation. Then \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite presentation if and only if \(f_*\mathcal{F}\) is an \(\mathcal{O}_Y\)-module of finite presentation.

Proof

As \(f\) is finite it is representable. Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Then \(U = V \times_Y X\) is a scheme with a surjective étale morphism towards \(X\) and a finite morphism \(\psi : U \to V\) (the base change of \(f\)). Since \(\psi_*(\mathcal{F}|_U) = f_*\mathcal{F}|_V\) the result of the lemma follows immediately from the schemes version which is Descent, Lemma 05B4.

Fpqc coverings

This section is the analogue of Descent, Section 023P. At the moment we do not know if all of the material for fpqc coverings of schemes holds also for algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(\{f_i : T_i \to T\}_{i \in I}\) be an fpqc covering of algebraic spaces over \(S\). Suppose that for each \(i\) we have an open subspace \(W_i \subset T_i\) such that for all \(i, j \in I\) we have \(\text{pr}_0^{-1}(W_i) = \text{pr}_1^{-1}(W_j)\) as open subspaces of \(T_i \times_T T_j\). Then there exists a unique open subspace \(W \subset T\) such that \(W_i = f_i^{-1}(W)\) for each \(i\).

Proof

By Topologies on Spaces, Lemma 0419 we may assume each \(T_i\) is a scheme. Choose a scheme \(U\) and a surjective étale morphism \(U \to T\). Then \(\{T_i \times_T U \to U\}\) is an fpqc covering of \(U\) and \(T_i \times_T U\) is a scheme for each \(i\). Hence we see that the collection of opens \(W_i \times_T U\) comes from a unique open subscheme \(W' \subset U\) by Descent, Lemma 03N0. As \(U \to X\) is open we can define \(W \subset X\) the Zariski open which is the image of \(W'\), see Properties of Spaces, Section 03BT. We omit the verification that this works, i.e., that \(W_i\) is the inverse image of \(W\) for each \(i\).

Lemma

Let \(S\) be a scheme. Let \(\{T_i \to T\}\) be an fpqc covering of algebraic spaces over \(S\), see Topologies on Spaces, Definition 03MQ. Then given an algebraic space \(B\) over \(S\) the sequence \[\xymatrix{ \Mor_S(T, B) \ar[r] & \prod\nolimits_i \Mor_S(T_i, B) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{i, j} \Mor_S(T_i \times_T T_j, B) }\] is an equalizer diagram. In other words, every representable functor on the category of algebraic spaces over \(S\) satisfies the sheaf condition for fpqc coverings.

Proof

We know this is true if \(\{T_i \to T\}\) is an fpqc covering of schemes, see Properties of Spaces, Proposition 0APL. This is the key fact and we encourage the reader to skip the rest of the proof which is formal. Choose a scheme \(U\) and a surjective étale morphism \(U \to T\). Let \(U_i\) be a scheme and let \(U_i \to T_i \times_T U\) be a surjective étale morphism. Then \(\{U_i \to U\}\) is an fpqc covering. This follows from Topologies on Spaces, Lemmas 03MR and 03MS. By the above we have the result for \(\{U_i \to U\}\).

What this means is the following: Suppose that \(b_i : T_i \to B\) is a family of morphisms with \(b_i \circ \text{pr}_0 = b_j \circ \text{pr}_1\) as morphisms \(T_i \times_T T_j \to B\). Then we let \(a_i : U_i \to B\) be the composition of \(U_i \to T_i\) with \(b_i\). By what was said above we find a unique morphism \(a : U \to B\) such that \(a_i\) is the composition of \(a\) with \(U_i \to U\). The uniqueness guarantees that \(a \circ \text{pr}_0 = a \circ \text{pr}_1\) as morphisms \(U \times_T U \to B\). Then since \(T = U/(U \times_T U)\) as a sheaf, we find that \(a\) comes from a unique morphism \(b : T \to B\). Chasing diagrams we find that \(b\) is the morphism we are looking for.

Descent of finiteness and smoothness properties of morphisms

The following type of lemma is occasionally useful.

Lemma

Let \(S\) be a scheme. Let \(X \to Y \to Z\) be morphism of algebraic spaces. Let \(P\) be one of the following properties of morphisms of algebraic spaces over \(S\): flat, locally finite type, locally finite presentation. Assume that \(X \to Z\) has \(P\) and that \(X \to Y\) is a surjection of sheaves on \((\Sch/S)_{fppf}\). Then \(Y \to Z\) is \(P\).

Proof

Choose a scheme \(W\) and a surjective étale morphism \(W \to Z\). Choose a scheme \(V\) and a surjective étale morphism \(V \to W \times_Z Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to V \times_Y X\). By assumption we can find an fppf covering \(\{V_i \to V\}\) and lifts \(V_i \to X\) of the morphism \(V_i \to Y\). Since \(U \to X\) is surjective étale we see that over the members of the fppf covering \(\{V_i \times_X U \to V\}\) we have lifts into \(U\). Hence \(U \to V\) induces a surjection of sheaves on \((\Sch/S)_{fppf}\). By our definition of what it means to have property \(P\) for a morphism of algebraic spaces (see Morphisms of Spaces, Definition 03ML, Definition 03XF, and Definition 03XP) we see that \(U \to W\) has \(P\) and we have to show \(V \to W\) has \(P\). Thus we reduce the question to the case of morphisms of schemes which is treated in Descent, Lemma 06NB.

A more standard case of the above lemma is the following. (The version with “flat” follows from Morphisms of Spaces, Lemma 05VY.)

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & B }\] be a commutative diagram of morphisms of algebraic spaces over \(S\). Assume that \(f\) is surjective, flat, and locally of finite presentation and assume that \(p\) is locally of finite presentation (resp. locally of finite type). Then \(q\) is locally of finite presentation (resp. locally of finite type).

Proof

Since \(\{X \to Y\}\) is an fppf covering, it induces a surjection of fppf sheaves (Topologies on Spaces, Lemma 0469) and the lemma is a special case of Lemma 06NR. On the other hand, an easier argument is to deduce it from the analogue for schemes. Namely, the problem is étale local on \(B\) and \(Y\) (Morphisms of Spaces, Lemmas 040Y and 0410). Hence we may assume that \(B\) and \(Y\) are affine schemes. Since \(|X| \to |Y|\) is open (Morphisms of Spaces, Lemma 042S), we can choose an affine scheme \(U\) and an étale morphism \(U \to X\) such that the composition \(U \to Y\) is surjective. In this case the result follows from Descent, Lemma 02KL.

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & B }\] be a commutative diagram of morphisms of algebraic spaces over \(S\). Assume that

1. \(f\) is surjective, and syntomic (resp. smooth, resp. étale),

2. \(p\) is syntomic (resp. smooth, resp. étale).

Then \(q\) is syntomic (resp. smooth, resp. étale).

Proof

We deduce this from the analogue for schemes. Namely, the problem is étale local on \(B\) and \(Y\) (Morphisms of Spaces, Lemmas 03ZA, 03ZF, and 03XT). Hence we may assume that \(B\) and \(Y\) are affine schemes. Since \(|X| \to |Y|\) is open (Morphisms of Spaces, Lemma 042S), we can choose an affine scheme \(U\) and an étale morphism \(U \to X\) such that the composition \(U \to Y\) is surjective. In this case the result follows from Descent, Lemma 02KM.

Actually we can strengthen this result as follows.

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & B }\] be a commutative diagram of morphisms of algebraic spaces over \(S\). Assume that

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

2. \(p\) is smooth (resp. étale).

Then \(q\) is smooth (resp. étale).

Proof

We deduce this from the analogue for schemes. Namely, the problem is étale local on \(B\) and \(Y\) (Morphisms of Spaces, Lemmas 03ZF and 03XT). Hence we may assume that \(B\) and \(Y\) are affine schemes. Since \(|X| \to |Y|\) is open (Morphisms of Spaces, Lemma 042S), we can choose an affine scheme \(U\) and an étale morphism \(U \to X\) such that the composition \(U \to Y\) is surjective. In this case the result follows from Descent, Lemma 05B5.

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & B }\] be a commutative diagram of morphisms of algebraic spaces over \(S\). Assume that

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

2. \(p\) is syntomic.

Then both \(q\) and \(f\) are syntomic.

Proof

We deduce this from the analogue for schemes. Namely, the problem is étale local on \(B\) and \(Y\) (Morphisms of Spaces, Lemma 03ZA). Hence we may assume that \(B\) and \(Y\) are affine schemes. Since \(|X| \to |Y|\) is open (Morphisms of Spaces, Lemma 042S), we can choose an affine scheme \(U\) and an étale morphism \(U \to X\) such that the composition \(U \to Y\) is surjective. In this case the result follows from Descent, Lemma 05B7.

Descending properties of spaces

In this section we put some results of the following kind.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(x \in |X|\). If \(f\) is flat at \(x\) and \(X\) is geometrically unibranch at \(x\), then \(Y\) is geometrically unibranch at \(f(x)\).

Proof

Consider the map of étale local rings \(\mathcal{O}_{Y, f(\overline{x})} \to \mathcal{O}_{X, \overline{x}}\). By Morphisms of Spaces, Lemma 04NG this is flat. Hence if \(\mathcal{O}_{X, \overline{x}}\) has a unique minimal prime, so does \(\mathcal{O}_{Y, f(\overline{x})}\) (by going down, see Algebra, Lemma 00HS).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If \(f\) is flat and surjective and \(X\) is reduced, then \(Y\) is reduced.

Proof

Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to X \times_Y V\). As \(f\) is surjective and flat, the morphism of schemes \(U \to V\) is surjective and flat. In this way we reduce the problem to the case of schemes (as reducedness of \(X\) and \(Y\) is defined in terms of reducedness of \(U\) and \(V\), see Properties of Spaces, Section 03E5). The case of schemes is Descent, Lemma 06QM.

Lemma

Let \(f : X \to Y\) be a morphism of algebraic spaces. If \(f\) is locally of finite presentation, flat, and surjective and \(X\) is locally Noetherian, then \(Y\) is locally Noetherian.

Proof

Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to X \times_Y V\). As \(f\) is surjective, flat, and locally of finite presentation the morphism of schemes \(U \to V\) is surjective, flat, and locally of finite presentation. In this way we reduce the problem to the case of schemes (as being locally Noetherian for \(X\) and \(Y\) is defined in terms of being locally Noetherian of \(U\) and \(V\), see Properties of Spaces, Section 03E5). In the case of schemes the result follows from Descent, Lemma 034C.

Lemma

Let \(f : X \to Y\) be a morphism of algebraic spaces. If \(f\) is locally of finite presentation, flat, and surjective and \(X\) is regular, then \(Y\) is regular.

Proof

By Lemma 06MJ we know that \(Y\) is locally Noetherian. Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). It suffices to prove that the local rings of \(V\) are all regular local rings, see Properties, Lemma 02IT. Choose a scheme \(U\) and a surjective étale morphism \(U \to X \times_Y V\). As \(f\) is surjective and flat the morphism of schemes \(U \to V\) is surjective and flat. By assumption \(U\) is a regular scheme in particular all of its local rings are regular (by the lemma above). Hence the lemma follows from Algebra, Lemma 00OF.

Lemma

Let \(f : X \to Y\) be a smooth morphism of algebraic spaces. If \(Y\) is reduced, then \(X\) is reduced. If \(f\) is surjective and \(X\) is reduced, then \(Y\) is reduced.

Proof

Choose a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\) and \(V\) are schemes, the vertical arrows are surjective and étale, and \(U \to X \times_Y V\) is surjective étale. Observe that \(X\) is a reduced algebraic space if and only if \(U\) is a reduced scheme by our definition of reduced algebraic spaces in Properties of Spaces, Section 03E5. Similarly for \(Y\) and \(V\). The morphism \(U \to V\) is a smooth morphism of schemes, see Morphisms of Spaces, Lemma 03ZF. Since being reduced is local for the smooth topology for schemes (Descent, Lemma 034E) we see that \(U\) is reduced if \(V\) is reduced. On the other hand, if \(X \to Y\) is surjective, then \(U \to V\) is surjective and in this case if \(U\) is reduced, then \(V\) is reduced.

Descending properties of morphisms

In this section we introduce the notion of when a property of morphisms of algebraic spaces is local on the target in a topology. Please compare with Descent, Section 02KN.

Definition

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\). Let \(\tau \in \{fpqc, fppf, syntomic, smooth, \etale\}\). We say \(\mathcal{P}\) is \(\tau\) local on the base, or \(\tau\) local on the target, or local on the base for the \(\tau\)-topology if for any \(\tau\)-covering \(\{Y_i \to Y\}_{i \in I}\) of algebraic spaces and any morphism of algebraic spaces \(f : X \to Y\) we have \[f \text{ has }\mathcal{P} \Leftrightarrow \text{each }Y_i \times_Y X \to Y_i\text{ has }\mathcal{P}.\]

To be sure, since isomorphisms are always coverings we see (or require) that property \(\mathcal{P}\) holds for \(X \to Y\) if and only if it holds for any arrow \(X' \to Y'\) isomorphic to \(X \to Y\). If a property is \(\tau\)-local on the target then it is preserved by base changes by morphisms which occur in \(\tau\)-coverings. Here is a formal statement.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{fpqc, fppf, syntomic, smooth, \etale\}\). Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\) which is \(\tau\) local on the target. Let \(f : X \to Y\) have property \(\mathcal{P}\). For any morphism \(Y' \to Y\) which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. étale, the base change \(f' : Y' \times_Y X \to Y'\) of \(f\) has property \(\mathcal{P}\).

Proof

This is true because we can fit \(Y' \to Y\) into a family of morphisms which forms a \(\tau\)-covering.

A simple often used consequence of the above is that if \(f : X \to Y\) has property \(\mathcal{P}\) which is \(\tau\)-local on the target and \(f(X) \subset V\) for some open subspace \(V \subset Y\), then also the induced morphism \(X \to V\) has \(\mathcal{P}\). Proof: The base change \(f\) by \(V \to Y\) gives \(X \to V\).

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{fppf, syntomic, smooth, \etale\}\). Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\) which is \(\tau\) local on the target. For any morphism of algebraic spaces \(f : X \to Y\) over \(S\) there exists a largest open subspace \(W(f) \subset Y\) such that the restriction \(X_{W(f)} \to W(f)\) has \(\mathcal{P}\). Moreover,

1. if \(g : Y' \to Y\) is a morphism of algebraic spaces which is flat and locally of finite presentation, syntomic, smooth, or étale and the base change \(f' : X_{Y'} \to Y'\) has \(\mathcal{P}\), then \(g\) factors through \(W(f)\),

2. if \(g : Y' \to Y\) is flat and locally of finite presentation, syntomic, smooth, or étale, then \(W(f') = g^{-1}(W(f))\), and

3. if \(\{g_i : Y_i \to Y\}\) is a \(\tau\)-covering, then \(g_i^{-1}(W(f)) = W(f_i)\), where \(f_i\) is the base change of \(f\) by \(Y_i \to Y\).

Proof

Consider the union \(W_{set} \subset |Y|\) of the images \(g(|Y'|) \subset |Y|\) of morphisms \(g : Y' \to Y\) with the properties:

1. \(g\) is flat and locally of finite presentation, syntomic, smooth, or étale, and

2. the base change \(Y' \times_{g, Y} X \to Y'\) has property \(\mathcal{P}\).

Since such a morphism \(g\) is open (see Morphisms of Spaces, Lemma 042S) we see that \(W_{set}\) is an open subset of \(|Y|\). Denote \(W \subset Y\) the open subspace whose underlying set of points is \(W_{set}\), see Properties of Spaces, Lemma 03BZ. Since \(\mathcal{P}\) is local in the \(\tau\) topology the restriction \(X_W \to W\) has property \(\mathcal{P}\) because we are given a covering \(\{Y' \to W\}\) of \(W\) such that the pullbacks have \(\mathcal{P}\). This proves the existence and proves that \(W(f)\) has property (1). To see property (2) note that \(W(f') \supset g^{-1}(W(f))\) because \(\mathcal{P}\) is stable under base change by flat and locally of finite presentation, syntomic, smooth, or étale morphisms, see Lemma 06EM. On the other hand, if \(Y'' \subset Y'\) is an open such that \(X_{Y''} \to Y''\) has property \(\mathcal{P}\), then \(Y'' \to Y\) factors through \(W\) by construction, i.e., \(Y'' \subset g^{-1}(W(f))\). This proves (2). Assertion (3) follows from (2) because each morphism \(Y_i \to Y\) is flat and locally of finite presentation, syntomic, smooth, or étale by our definition of a \(\tau\)-covering.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\). Assume

1. if \(X_i \to Y_i\), \(i = 1, 2\) have property \(\mathcal{P}\) so does \(X_1 \amalg X_2 \to Y_1 \amalg Y_2\),

2. a morphism of algebraic spaces \(f : X \to Y\) has property \(\mathcal{P}\) if and only if for every affine scheme \(Z\) and morphism \(Z \to Y\) the base change \(Z \times_Y X \to Z\) of \(f\) has property \(\mathcal{P}\), and

3. for any surjective flat morphism of affine schemes \(Z' \to Z\) over \(S\) and a morphism \(f : X \to Z\) from an algebraic space to \(Z\) we have \[f' : Z' \times_Z X \to Z'\text{ has }\mathcal{P} \Rightarrow f\text{ has }\mathcal{P}.\]

Then \(\mathcal{P}\) is fpqc local on the base.

Proof

If \(\mathcal{P}\) has property (2), then it is automatically stable under any base change. Hence the direct implication in Definition 03YH.

Let \(\{Y_i \to Y\}_{i \in I}\) be an fpqc covering of algebraic spaces over \(S\). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume each base change \(f_i : Y_i \times_Y X \to Y_i\) has property \(\mathcal{P}\). Our goal is to show that \(f\) has \(\mathcal{P}\). Let \(Z\) be an affine scheme, and let \(Z \to Y\) be a morphism. By (2) it suffices to show that the morphism of algebraic spaces \(Z \times_Y X \to Z\) has \(\mathcal{P}\). Since \(\{Y_i \to Y\}_{i \in I}\) is an fpqc covering we know there exists a standard fpqc covering \(\{Z_j \to Z\}_{j = 1, \ldots , n}\) and morphisms \(Z_j \to Y_{i_j}\) over \(Y\) for suitable indices \(i_j \in I\). Since \(f_{i_j}\) has \(\mathcal{P}\) we see that \[Z_j \times_Y X = Z_j \times_{Y_{i_j}} (Y_{i_j} \times_Y X) \longrightarrow Z_j\] has \(\mathcal{P}\) as a base change of \(f_{i_j}\) (see first remark of the proof). Set \(Z' = \coprod_{j = 1, \ldots, n} Z_j\), so that \(Z' \to Z\) is a flat and surjective morphism of affine schemes over \(S\). By (1) we conclude that \(Z' \times_Y X \to Z'\) has property \(\mathcal{P}\). Since this is the base change of the morphism \(Z \times_Y X \to Z\) by the morphism \(Z' \to Z\) we conclude that \(Z \times_Y X \to Z\) has property \(\mathcal{P}\) as desired.

Descending properties of morphisms in the fpqc topology

In this section we find a large number of properties of morphisms of algebraic spaces which are local on the base in the fpqc topology. Please compare with Descent, Section 02YJ for the case of morphisms of schemes.

Lemma

Let \(S\) be a scheme. The property \(\mathcal{P}(f) =\)“\(f\) is quasi-compact” is fpqc local on the base on algebraic spaces over \(S\).

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03KG. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is quasi-compact. We have to show that \(f\) is quasi-compact. To see this, using Morphisms of Spaces, Lemma 03KG again, it is enough to show that for every affine scheme \(Y\) and morphism \(Y \to Z\) the fibre product \(Y \times_Z X\) is quasi-compact. Here is a picture: [041M]\[\begin{equation} \vcenter{ \xymatrix{ Y \times_Z Z' \times_Z X \ar[dd] \ar[rr] \ar[rd] & & Z' \times_Z X \ar'[d][dd]^{f'} \ar[rd] \\ & Y \times_Z X \ar[dd] \ar[rr] & & X \ar[dd]^f \\ Y \times_Z Z' \ar'[r][rr] \ar[rd] & & Z' \ar[rd] \\ & Y \ar[rr] & & Z } } \end{equation}\] Note that all squares are cartesian and the bottom square consists of affine schemes. The assumption that \(f'\) is quasi-compact combined with the fact that \(Y \times_Z Z'\) is affine implies that \(Y \times_Z Z' \times_Z X\) is quasi-compact. Since \[Y \times_Z Z' \times_Z X \longrightarrow Y \times_Z X\] is surjective as a base change of \(Z' \to Z\) we conclude that \(Y \times_Z X\) is quasi-compact, see Morphisms of Spaces, Lemma 040W. This finishes the proof.

Lemma

Let \(S\) be a scheme. The property \(\mathcal{P}(f) =\)“\(f\) is quasi-separated” is fpqc local on the base on algebraic spaces over \(S\).

Proof

A base change of a quasi-separated morphism is quasi-separated, see Morphisms of Spaces, Lemma 03KL. Hence the direct implication in Definition 03YH.

Let \(\{Y_i \to Y\}_{i \in I}\) be an fpqc covering of algebraic spaces over \(S\). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume each base change \(X_i := Y_i \times_Y X \to Y_i\) is quasi-separated. This means that each of the morphisms \[\Delta_i : X_i \longrightarrow X_i \times_{Y_i} X_i = Y_i \times_Y (X \times_Y X)\] is quasi-compact. The base change of a fpqc covering is an fpqc covering, see Topologies on Spaces, Lemma 03MR hence \(\{Y_i \times_Y (X \times_Y X) \to X \times_Y X\}\) is an fpqc covering of algebraic spaces. Moreover, each \(\Delta_i\) is the base change of the morphism \(\Delta : X \to X \times_Y X\). Hence it follows from Lemma 041L that \(\Delta\) is quasi-compact, i.e., \(f\) is quasi-separated.

Lemma

Let \(S\) be a scheme. The property \(\mathcal{P}(f) =\)“\(f\) is universally closed” is fpqc local on the base on algebraic spaces over \(S\).

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03IT. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is universally closed. We have to show that \(f\) is universally closed. To see this, using Morphisms of Spaces, Lemma 03IT again, it is enough to show that for every affine scheme \(Y\) and morphism \(Y \to Z\) the map \(|Y \times_Z X| \to |Y|\) is closed. Consider the cube (041M). The assumption that \(f'\) is universally closed implies that \(|Y \times_Z Z' \times_Z X| \to |Y \times_Z Z'|\) is closed. As \(Y \times_Z Z' \to Y\) is quasi-compact, surjective, and flat as a base change of \(Z' \to Z\) we see the map \(|Y \times_Z Z'| \to |Y|\) is submersive, see Morphisms, Lemma 02JY. Moreover the map \[|Y \times_Z Z' \times_Z X| \longrightarrow |Y \times_Z Z'| \times_{|Y|} |Y \times_Z X|\] is surjective, see Properties of Spaces, Lemma 03H4. It follows by elementary topology that \(|Y \times_Z X| \to |Y|\) is closed.

Lemma

Let \(S\) be a scheme. The property \(\mathcal{P}(f) =\)“\(f\) is universally open” is fpqc local on the base on algebraic spaces over \(S\).

Proof

The proof is the same as the proof of Lemma 041O.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is universally submersive” is fpqc local on the base.

Proof

The proof is the same as the proof of Lemma 041O.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is surjective” is fpqc local on the base.

Proof

Omitted. (Hint: Use Properties of Spaces, Lemma 03H4.)

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is universally injective” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03IT. Let \(Z' \to Z\) be a flat surjective morphism of affine schemes over \(S\) and let \(f : X \to Z\) be a morphism from an algebraic space to \(Z\). Assume that the base change \(f' : X' \to Z'\) is universally injective. Let \(K\) be a field, and let \(a, b : \Spec(K) \to X\) be two morphisms such that \(f \circ a = f \circ b\). As \(Z' \to Z\) is surjective there exists a field extension \(K'/K\) and a morphism \(\Spec(K') \to Z'\) such that the following solid diagram commutes \[\xymatrix{ \Spec(K') \ar[rrd] \ar@{-->}[rd]_{a', b'} \ar[dd] \\ & X' \ar[r] \ar[d] & Z' \ar[d] \\ \Spec(K) \ar[r]^{a, b} & X \ar[r] & Z }\] As the square is cartesian we get the two dotted arrows \(a'\), \(b'\) making the diagram commute. Since \(X' \to Z'\) is universally injective we get \(a' = b'\). This forces \(a = b\) as \(\{\Spec(K') \to \Spec(K)\}\) is an fpqc covering, see Properties of Spaces, Proposition 0APL. Hence \(f\) is universally injective as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a universal homeomorphism” is fpqc local on the base.

Proof

This can be proved in exactly the same manner as Lemma 041O. Alternatively, one can use that a map of topological spaces is a homeomorphism if and only if it is injective, surjective, and open. Thus a universal homeomorphism is the same thing as a surjective, universally injective, and universally open morphism. See Morphisms of Spaces, Lemma 03MH and Morphisms of Spaces, Definitions 03MV, 03ME, 03Z2, 05Z5. Thus the lemma follows from Lemmas 041Q, 041R, and 041P.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is locally of finite type” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 040Y. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is locally of finite type. We have to show that \(f\) is locally of finite type. Let \(U\) be a scheme and let \(U \to X\) be surjective and étale. By Morphisms of Spaces, Lemma 040Y again, it is enough to show that \(U \to Z\) is locally of finite type. Since \(f'\) is locally of finite type, and since \(Z' \times_Z U\) is a scheme étale over \(Z' \times_Z X\) we conclude (by the same lemma again) that \(Z' \times_Z U \to Z'\) is locally of finite type. As \(\{Z' \to Z\}\) is an fpqc covering we conclude that \(U \to Z\) is locally of finite type by Descent, Lemma 02KX as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is locally of finite presentation” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 0410. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is locally of finite presentation. We have to show that \(f\) is locally of finite presentation. Let \(U\) be a scheme and let \(U \to X\) be surjective and étale. By Morphisms of Spaces, Lemma 0410 again, it is enough to show that \(U \to Z\) is locally of finite presentation. Since \(f'\) is locally of finite presentation, and since \(Z' \times_Z U\) is a scheme étale over \(Z' \times_Z X\) we conclude (by the same lemma again) that \(Z' \times_Z U \to Z'\) is locally of finite presentation. As \(\{Z' \to Z\}\) is an fpqc covering we conclude that \(U \to Z\) is locally of finite presentation by Descent, Lemma 02KY as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is of finite type” is fpqc local on the base.

Proof

Combine Lemmas 041L and 041S.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is of finite presentation” is fpqc local on the base.

Proof

Combine Lemmas 041L, 041N and 041T.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is flat” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03MM. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is flat. We have to show that \(f\) is flat. Let \(U\) be a scheme and let \(U \to X\) be surjective and étale. By Morphisms of Spaces, Lemma 03MM again, it is enough to show that \(U \to Z\) is flat. Since \(f'\) is flat, and since \(Z' \times_Z U\) is a scheme étale over \(Z' \times_Z X\) we conclude (by the same lemma again) that \(Z' \times_Z U \to Z'\) is flat. As \(\{Z' \to Z\}\) is an fpqc covering we conclude that \(U \to Z\) is flat by Descent, Lemma 02L2 as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is an open immersion” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03M4. Consider a cartesian diagram \[\xymatrix{ X' \ar[r] \ar[d] & X \ar[d] \\ Z' \ar[r] & Z }\] of algebraic spaces over \(S\) where \(Z' \to Z\) is a surjective flat morphism of affine schemes, and \(X' \to Z'\) is an open immersion. We have to show that \(X \to Z\) is an open immersion. Note that \(|X'| \subset |Z'|\) corresponds to an open subscheme \(U' \subset Z'\) (isomorphic to \(X'\)) with the property that \(\text{pr}_0^{-1}(U') = \text{pr}_1^{-1}(U')\) as open subschemes of \(Z' \times_Z Z'\). Hence there exists an open subscheme \(U \subset Z\) such that \(X' = (Z' \to Z)^{-1}(U)\), see Descent, Lemma 03N0. By Properties of Spaces, Proposition 0APL we see that \(X\) satisfies the sheaf condition for the fpqc topology. Now we have the fpqc covering \(\mathcal{U} = \{U' \to U\}\) and the element \(U' \to X' \to X \in \check{H}^0(\mathcal{U}, X)\). By the sheaf condition we obtain a morphism \(U \to X\) such that \[\xymatrix{ U' \ar[r] \ar[d]^{\cong} \ar@/_3ex/[dd] & U \ar[d] \ar@/^3ex/[dd] \\ X' \ar[r] \ar[d] & X \ar[d] \\ Z' \ar[r] & Z }\] is commutative. On the other hand, we know that for any scheme \(T\) over \(S\) and \(T\)-valued point \(T \to X\) the composition \(T \to X \to Z\) is a morphism such that \(Z' \times_Z T \to Z'\) factors through \(U'\). Clearly this means that \(T \to Z\) factors through \(U\). In other words the map of sheaves \(U \to X\) is bijective and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is an isomorphism” is fpqc local on the base.

Proof

Combine Lemmas 041Q and 041X.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is affine” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03WG. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is affine. Let \(X'\) be a scheme representing \(Z' \times_Z X\). We obtain a canonical isomorphism \[\varphi : X' \times_Z Z' \longrightarrow Z' \times_Z X'\] since both schemes represent the algebraic space \(Z' \times_Z Z' \times_Z X\). This is a descent datum for \(X'/Z'/Z\), see Descent, Definition 023V (verification omitted, compare with Descent, Lemma 02W5). Since \(X' \to Z'\) is affine this descent datum is effective, see Descent, Lemma 0245. Thus there exists a scheme \(Y \to Z\) over \(Z\) and an isomorphism \(\psi : Z' \times_Z Y \to X'\) compatible with descent data. Of course \(Y \to Z\) is affine (by construction or by Descent, Lemma 02L5). Note that \(\mathcal{Y} = \{Z' \times_Z Y \to Y\}\) is a fpqc covering, and interpreting \(\psi\) as an element of \(X(Z' \times_Z Y)\) we see that \(\psi \in \check{H}^0(\mathcal{Y}, X)\). By the sheaf condition for \(X\) with respect to this covering (see Properties of Spaces, Proposition 0APL) we obtain a morphism \(Y \to X\). By construction the base change of this to \(Z'\) is an isomorphism, hence an isomorphism by Lemma 041Y. This proves that \(X\) is representable by an affine scheme and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a closed immersion” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03M4. Consider a cartesian diagram \[\xymatrix{ X' \ar[r] \ar[d] & X \ar[d] \\ Z' \ar[r] & Z }\] of algebraic spaces over \(S\) where \(Z' \to Z\) is a surjective flat morphism of affine schemes, and \(X' \to Z'\) is a closed immersion. We have to show that \(X \to Z\) is a closed immersion. The morphism \(X' \to Z'\) is affine. Hence by Lemma 041Z we see that \(X\) is a scheme and \(X \to Z\) is affine. It follows from Descent, Lemma 02L6 that \(X \to Z\) is a closed immersion as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is separated” is fpqc local on the base.

Proof

A base change of a separated morphism is separated, see Morphisms of Spaces, Lemma 03KL. Hence the direct implication in Definition 03YH.

Let \(\{Y_i \to Y\}_{i \in I}\) be an fpqc covering of algebraic spaces over \(S\). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume each base change \(X_i := Y_i \times_Y X \to Y_i\) is separated. This means that each of the morphisms \[\Delta_i : X_i \longrightarrow X_i \times_{Y_i} X_i = Y_i \times_Y (X \times_Y X)\] is a closed immersion. The base change of a fpqc covering is an fpqc covering, see Topologies on Spaces, Lemma 03MR hence \(\{Y_i \times_Y (X \times_Y X) \to X \times_Y X\}\) is an fpqc covering of algebraic spaces. Moreover, each \(\Delta_i\) is the base change of the morphism \(\Delta : X \to X \times_Y X\). Hence it follows from Lemma 0420 that \(\Delta\) is a closed immersion, i.e., \(f\) is separated.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is proper” is fpqc local on the base.

Proof

The lemma follows by combining Lemmas 041O, 0421 and 041U.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is quasi-affine” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03WM. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is quasi-affine. Let \(X'\) be a scheme representing \(Z' \times_Z X\). We obtain a canonical isomorphism \[\varphi : X' \times_Z Z' \longrightarrow Z' \times_Z X'\] since both schemes represent the algebraic space \(Z' \times_Z Z' \times_Z X\). This is a descent datum for \(X'/Z'/Z\), see Descent, Definition 023V (verification omitted, compare with Descent, Lemma 02W5). Since \(X' \to Z'\) is quasi-affine this descent datum is effective, see Descent, Lemma 0247. Thus there exists a scheme \(Y \to Z\) over \(Z\) and an isomorphism \(\psi : Z' \times_Z Y \to X'\) compatible with descent data. Of course \(Y \to Z\) is quasi-affine (by construction or by Descent, Lemma 02L7). Note that \(\mathcal{Y} = \{Z' \times_Z Y \to Y\}\) is a fpqc covering, and interpreting \(\psi\) as an element of \(X(Z' \times_Z Y)\) we see that \(\psi \in \check{H}^0(\mathcal{Y}, X)\). By the sheaf condition for \(X\) (see Properties of Spaces, Proposition 0APL) we obtain a morphism \(Y \to X\). By construction the base change of this to \(Z'\) is an isomorphism, hence an isomorphism by Lemma 041Y. This proves that \(X\) is representable by a quasi-affine scheme and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a quasi-compact immersion” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemmas 03M4 and 03KG. Consider a cartesian diagram \[\xymatrix{ X' \ar[r] \ar[d] & X \ar[d] \\ Z' \ar[r] & Z }\] of algebraic spaces over \(S\) where \(Z' \to Z\) is a surjective flat morphism of affine schemes, and \(X' \to Z'\) is a quasi-compact immersion. We have to show that \(X \to Z\) is a closed immersion. The morphism \(X' \to Z'\) is quasi-affine. Hence by Lemma 0423 we see that \(X\) is a scheme and \(X \to Z\) is quasi-affine. It follows from Descent, Lemma 02L8 that \(X \to Z\) is a quasi-compact immersion as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is integral” is fpqc local on the base.

Proof

An integral morphism is the same thing as an affine, universally closed morphism. See Morphisms of Spaces, Lemma 0415. Hence the lemma follows on combining Lemmas 041O and 041Z.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is finite” is fpqc local on the base.

Proof

An finite morphism is the same thing as an integral, morphism which is locally of finite type. See Morphisms of Spaces, Lemma 0414. Hence the lemma follows on combining Lemmas 041S and 0425.

Lemma

The properties \(\mathcal{P}(f) =\)“\(f\) is locally quasi-finite” and \(\mathcal{P}(f) =\)“\(f\) is quasi-finite” are fpqc local on the base.

Proof

We have already seen that “quasi-compact” is fpqc local on the base, see Lemma 041L. Hence it is enough to prove the lemma for “locally quasi-finite”. We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 040Z. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is locally quasi-finite. We have to show that \(f\) is locally quasi-finite. Let \(U\) be a scheme and let \(U \to X\) be surjective and étale. By Morphisms of Spaces, Lemma 040Z again, it is enough to show that \(U \to Z\) is locally quasi-finite. Since \(f'\) is locally quasi-finite, and since \(Z' \times_Z U\) is a scheme étale over \(Z' \times_Z X\) we conclude (by the same lemma again) that \(Z' \times_Z U \to Z'\) is locally quasi-finite. As \(\{Z' \to Z\}\) is an fpqc covering we conclude that \(U \to Z\) is locally quasi-finite by Descent, Lemma 02VI as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is syntomic” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03ZA. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is syntomic. We have to show that \(f\) is syntomic. Let \(U\) be a scheme and let \(U \to X\) be surjective and étale. By Morphisms of Spaces, Lemma 03ZA again, it is enough to show that \(U \to Z\) is syntomic. Since \(f'\) is syntomic, and since \(Z' \times_Z U\) is a scheme étale over \(Z' \times_Z X\) we conclude (by the same lemma again) that \(Z' \times_Z U \to Z'\) is syntomic. As \(\{Z' \to Z\}\) is an fpqc covering we conclude that \(U \to Z\) is syntomic by Descent, Lemma 02VK as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is smooth” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03ZF. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is smooth. We have to show that \(f\) is smooth. Let \(U\) be a scheme and let \(U \to X\) be surjective and étale. By Morphisms of Spaces, Lemma 03ZF again, it is enough to show that \(U \to Z\) is smooth. Since \(f'\) is smooth, and since \(Z' \times_Z U\) is a scheme étale over \(Z' \times_Z X\) we conclude (by the same lemma again) that \(Z' \times_Z U \to Z'\) is smooth. As \(\{Z' \to Z\}\) is an fpqc covering we conclude that \(U \to Z\) is smooth by Descent, Lemma 02VL as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is unramified” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03ZK. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is unramified. We have to show that \(f\) is unramified. Let \(U\) be a scheme and let \(U \to X\) be surjective and étale. By Morphisms of Spaces, Lemma 03ZK again, it is enough to show that \(U \to Z\) is unramified. Since \(f'\) is unramified, and since \(Z' \times_Z U\) is a scheme étale over \(Z' \times_Z X\) we conclude (by the same lemma again) that \(Z' \times_Z U \to Z'\) is unramified. As \(\{Z' \to Z\}\) is an fpqc covering we conclude that \(U \to Z\) is unramified by Descent, Lemma 02VM as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is étale” is fpqc local on the base.

Proof

We will use Lemma 041J to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 03XT. Let \(Z' \to Z\) be a surjective flat morphism of affine schemes over \(S\). Let \(f : X \to Z\) be a morphism of algebraic spaces, and assume that the base change \(f' : Z' \times_Z X \to Z'\) is étale. We have to show that \(f\) is étale. Let \(U\) be a scheme and let \(U \to X\) be surjective and étale. By Morphisms of Spaces, Lemma 03XT again, it is enough to show that \(U \to Z\) is étale. Since \(f'\) is étale, and since \(Z' \times_Z U\) is a scheme étale over \(Z' \times_Z X\) we conclude (by the same lemma again) that \(Z' \times_Z U \to Z'\) is étale. As \(\{Z' \to Z\}\) is an fpqc covering we conclude that \(U \to Z\) is étale by Descent, Lemma 02VN as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is finite locally free” is fpqc local on the base.

Proof

Being finite locally free is equivalent to being finite, flat and locally of finite presentation (Morphisms of Spaces, Lemma 0416). Hence this follows from Lemmas 0426, 041W, and 041T.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a monomorphism” is fpqc local on the base.

Proof

Let \(f : X \to Y\) be a morphism of algebraic spaces. Let \(\{Y_i \to Y\}\) be an fpqc covering, and assume each of the base changes \(f_i : X_i \to Y_i\) of \(f\) is a monomorphism. We have to show that \(f\) is a monomorphism.

First proof. Note that \(f\) is a monomorphism if and only if \(\Delta : X \to X \times_Y X\) is an isomorphism. By applying this to \(f_i\) we see that each of the morphisms \[\Delta_i : X_i \longrightarrow X_i \times_{Y_i} X_i = Y_i \times_Y (X \times_Y X)\] is an isomorphism. The base change of an fpqc covering is an fpqc covering, see Topologies on Spaces, Lemma 03MR hence \(\{Y_i \times_Y (X \times_Y X) \to X \times_Y X\}\) is an fpqc covering of algebraic spaces. Moreover, each \(\Delta_i\) is the base change of the morphism \(\Delta : X \to X \times_Y X\). Hence it follows from Lemma 041Y that \(\Delta\) is an isomorphism, i.e., \(f\) is a monomorphism.

Second proof. Let \(V\) be a scheme, and let \(V \to Y\) be a surjective étale morphism. If we can show that \(V \times_Y X \to V\) is a monomorphism, then it follows that \(X \to Y\) is a monomorphism. Namely, given any cartesian diagram of sheaves \[\vcenter{ \xymatrix{ \mathcal{F} \ar[r]_a \ar[d]_b & \mathcal{G} \ar[d]^c \\ \mathcal{H} \ar[r]^d & \mathcal{I} } } \quad \quad \mathcal{F} = \mathcal{H} \times_\mathcal{I} \mathcal{G}\] if \(c\) is a surjection of sheaves, and \(a\) is injective, then also \(d\) is injective. This reduces the problem to the case where \(Y\) is a scheme. Moreover, in this case we may assume that the algebraic spaces \(Y_i\) are schemes also, since we can always refine the covering to place ourselves in this situation, see Topologies on Spaces, Lemma 0419.

Assume \(\{Y_i \to Y\}\) is an fpqc covering of schemes. Let \(a, b : T \to X\) be two morphisms such that \(f \circ a = f \circ b\). We have to show that \(a = b\). Since \(f_i\) is a monomorphism we see that \(a_i = b_i\), where \(a_i, b_i : Y_i \times_Y T \to X_i\) are the base changes. In particular the compositions \(Y_i \times_Y T \to T \to X\) are equal. Since \(\{Y_i \times_Y T \to T\}\) is an fpqc covering we deduce that \(a = b\) from Properties of Spaces, Proposition 0APL.

Descending properties of morphisms in the fppf topology

In this section we find some properties of morphisms of algebraic spaces for which we could not (yet) show they are local on the base in the fpqc topology which, however, are local on the base in the fppf topology.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is an immersion” is fppf local on the base.

Proof

Let \(f : X \to Y\) be a morphism of algebraic spaces. Let \(\{Y_i \to Y\}_{i \in I}\) be an fppf covering of \(Y\). Let \(f_i : X_i \to Y_i\) be the base change of \(f\).

If \(f\) is an immersion, then each \(f_i\) is an immersion by Spaces, Lemma 02YW. This proves the direct implication in Definition 03YH.

Conversely, assume each \(f_i\) is an immersion. By Morphisms of Spaces, Lemma 042R this implies each \(f_i\) is separated. By Morphisms of Spaces, Lemma 03XM this implies each \(f_i\) is locally quasi-finite. Hence we see that \(f\) is locally quasi-finite and separated, by applying Lemmas 0421 and 0427. By Morphisms of Spaces, Lemma 0418 this implies that \(f\) is representable!

By Morphisms of Spaces, Lemma 03M4 it suffices to show that for every scheme \(Z\) and morphism \(Z \to Y\) the base change \(Z \times_Y X \to Z\) is an immersion. By Topologies on Spaces, Lemma 042T we can find an fppf covering \(\{Z_i \to Z\}\) by schemes which refines the pullback of the covering \(\{Y_i \to Y\}\) to \(Z\). Hence we see that \(Z \times_Y X \to Z\) (which is a morphism of schemes according to the result of the preceding paragraph) becomes an immersion after pulling back to the members of an fppf (by schemes) of \(Z\). Hence \(Z \times_Y X \to Z\) is an immersion by the result for schemes, see Descent, Lemma 02YM.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is locally separated” is fppf local on the base.

Proof

A base change of a locally separated morphism is locally separated, see Morphisms of Spaces, Lemma 03KL. Hence the direct implication in Definition 03YH.

Let \(\{Y_i \to Y\}_{i \in I}\) be an fppf covering of algebraic spaces over \(S\). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume each base change \(X_i := Y_i \times_Y X \to Y_i\) is locally separated. This means that each of the morphisms \[\Delta_i : X_i \longrightarrow X_i \times_{Y_i} X_i = Y_i \times_Y (X \times_Y X)\] is an immersion. The base change of a fppf covering is an fppf covering, see Topologies on Spaces, Lemma 03Y9 hence \(\{Y_i \times_Y (X \times_Y X) \to X \times_Y X\}\) is an fppf covering of algebraic spaces. Moreover, each \(\Delta_i\) is the base change of the morphism \(\Delta : X \to X \times_Y X\). Hence it follows from Lemma 042U that \(\Delta\) is an immersion, i.e., \(f\) is locally separated.

Application of descent of properties of morphisms

This section is the analogue of Descent, Section 02LB.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(\{g_i : Y_i \to Y\}_{i \in I}\) be an fpqc covering. Let \(f_i : X_i \to Y_i\) be the base change of \(f\) and let \(\mathcal{L}_i\) be the pullback of \(\mathcal{L}\) to \(X_i\). The following are equivalent

1. \(\mathcal{L}\) is ample on \(X/Y\), and

2. \(\mathcal{L}_i\) is ample on \(X_i/Y_i\) for every \(i \in I\).

Proof

The implication (1) \(\Rightarrow\) (2) follows from Divisors on Spaces, Lemma 0D33. Assume (2). To check \(\mathcal{L}\) is ample on \(X/Y\) we may work étale locally on \(Y\), see Divisors on Spaces, Lemma 0D36. Thus we may assume that \(Y\) is a scheme and then we may in turn assume each \(Y_i\) is a scheme too, see Topologies on Spaces, Lemma 0419. In other words, we may assume that \(\{Y_i \to Y\}\) is an fpqc covering of schemes.

By Divisors on Spaces, Lemma 0D34 we see that \(X_i \to Y_i\) is representable (i.e., \(X_i\) is a scheme), quasi-compact, and separated. Hence \(f\) is quasi-compact and separated by Lemmas 041L and 0421. This means that \(\mathcal{A} = \bigoplus_{d \geq 0} f_*\mathcal{L}^{\otimes d}\) is a quasi-coherent graded \(\mathcal{O}_Y\)-algebra (Morphisms of Spaces, Lemma 03M9). Moreover, the formation of \(\mathcal{A}\) commutes with flat base change by Cohomology of Spaces, Lemma 073K. In particular, if we set \(\mathcal{A}_i = \bigoplus_{d \geq 0} f_{i, *}\mathcal{L}_i^{\otimes d}\) then we have \(\mathcal{A}_i = g_i^*\mathcal{A}\). It follows that the natural maps \(\psi_d : f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d}\) of \(\mathcal{O}_X\) pullback to give the natural maps \(\psi_{i, d} : f_i^*(\mathcal{A}_i)_d \to \mathcal{L}_i^{\otimes d}\) of \(\mathcal{O}_{X_i}\)-modules. Since \(\mathcal{L}_i\) is ample on \(X_i/Y_i\) we see that for any point \(x_i \in X_i\), there exists a \(d \geq 1\) such that \(f_i^*(\mathcal{A}_i)_d \to \mathcal{L}_i^{\otimes d}\) is surjective on stalks at \(x_i\). This follows either directly from the definition of a relatively ample module or from Morphisms, Lemma 01VJ. If \(x \in |X|\), then we can choose an \(i\) and an \(x_i \in X_i\) mapping to \(x\). Since \(\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X_i, \overline{x}_i}\) is flat hence faithfully flat, we conclude that for every \(x \in |X|\) there exists a \(d \geq 1\) such that \(f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d}\) is surjective on stalks at \(x\). This implies that the open subset \(U(\psi) \subset X\) of Divisors on Spaces, Lemma 0D2Z corresponding to the map \(\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}\) of graded \(\mathcal{O}_X\)-algebras is equal to \(X\). Consider the corresponding morphism \[r_{\mathcal{L}, \psi} : X \longrightarrow \underline{\text{Proj}}_Y(\mathcal{A})\] It is clear from the above that the base change of \(r_{\mathcal{L}, \psi}\) to \(Y_i\) is the morphism \(r_{\mathcal{L}_i, \psi_i}\) which is an open immersion by Morphisms, Lemma 01VJ. Hence \(r_{\mathcal{L}, \psi}\) is an open immersion by Lemma 041X. Hence \(X\) is a scheme and we conclude \(\mathcal{L}\) is ample on \(X/Y\) by Morphisms, Lemma 01VJ.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper morphism of algebraic spaces over \(S\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. There exists an open subspace \(V \subset Y\) characterized by the following property: A morphism \(Y' \to Y\) of algebraic spaces factors through \(V\) if and only if the pullback \(\mathcal{L}'\) of \(\mathcal{L}\) to \(X' = Y' \times_Y X\) is ample on \(X'/Y'\) (as in Divisors on Spaces, Definition 0D31).

Proof

Suppose that the lemma holds whenever \(Y\) is a scheme. Let \(U\) be a scheme and let \(U \to Y\) be a surjective étale morphism. Let \(R = U \times_Y U\) with projections \(t, s : R \to U\). Denote \(X_U = U \times_Y X\) and \(\mathcal{L}_U\) the pullback. Then we get an open subscheme \(V' \subset U\) as in the lemma for \((X_U \to U, \mathcal{L}_U)\). By the functorial characterization we see that \(s^{-1}(V') = t^{-1}(V')\). Thus there is an open subspace \(V \subset Y\) such that \(V'\) is the inverse image of \(V\) in \(U\). In particular \(V' \to V\) is surjective étale and we conclude that \(\mathcal{L}_V\) is ample on \(X_V/V\) (Divisors on Spaces, Lemma 0D36). Now, if \(Y' \to Y\) is a morphism such that \(\mathcal{L}'\) is ample on \(X'/Y'\), then \(U \times_Y Y' \to Y'\) must factor through \(V'\) and we conclude that \(Y' \to Y\) factors through \(V\). Hence \(V \subset Y\) is as in the statement of the lemma. In this way we reduce to the case dealt with in the next paragraph.

Assume \(Y\) is a scheme. Since the question is local on \(Y\) we may assume \(Y\) is an affine scheme. We will show the following:

1. If \(\Spec(k) \to Y\) is a morphism such that \(\mathcal{L}_k\) is ample on \(X_k/k\), then there is an open neighbourhood \(V \subset Y\) of the image of \(\Spec(k) \to Y\) such that \(\mathcal{L}_V\) is ample on \(X_V/V\).

It is clear that (A) implies the truth of the lemma.

Let \(X \to Y\), \(\mathcal{L}\), \(\Spec(k) \to Y\) be as in (A). By Lemma 0D3C we may assume that \(k = \kappa(y)\) is the residue field of a point \(y\) of \(Y\).

As \(Y\) is affine we can find a directed set \(I\) and an inverse system of morphisms \(X_i \to Y_i\) of algebraic spaces with \(Y_i\) of finite presentation over \(\mathbf{Z}\), with affine transition morphisms \(X_i \to X_{i'}\) and \(Y_i \to Y_{i'}\), with \(X_i \to Y_i\) proper and of finite presentation, and such that \(X \to Y = \lim (X_i \to Y_i)\). See Limits of Spaces, Lemma 0A0X. After shrinking \(I\) we may assume \(Y_i\) is an (affine) scheme for all \(i\), see Limits of Spaces, Lemma 07SQ. After shrinking \(I\) we can assume we have a compatible system of invertible \(\mathcal{O}_{X_i}\)-modules \(\mathcal{L}_i\) pulling back to \(\mathcal{L}\), see Limits of Spaces, Lemma 0D2X. Let \(y_i \in Y_i\) be the image of \(y\). Then \(\kappa(y) = \colim \kappa(y_i)\). Hence \(X_y = \lim X_{i, y_i}\) and after shrinking \(I\) we may assume \(X_{i, y_i}\) is a scheme for all \(i\), see Limits of Spaces, Lemma 07SR. Hence for some \(i\) we have \(\mathcal{L}_{i, y_i}\) is ample on \(X_{i, y_i}\) by Limits, Lemma 09MT. By Divisors on Spaces, Lemma 0D3A we find an open neighbourhood \(V_i \subset Y_i\) of \(y_i\) such that \(\mathcal{L}_i\) restricted to \(f_i^{-1}(V_i)\) is ample relative to \(V_i\). Letting \(V \subset Y\) be the inverse image of \(V_i\) finishes the proof (hints: use Morphisms, Lemma 0893 and the fact that \(X \to Y \times_{Y_i} X_i\) is affine and the fact that the pullback of an ample invertible sheaf by an affine morphism is ample by Morphisms, Lemma 0892).

Properties of morphisms local on the source

In this section we define what it means for a property of morphisms of algebraic spaces to be local on the source. Please compare with Descent, Section 036F.

Definition

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\). Let \(\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] \etale\}\). We say \(\mathcal{P}\) is \(\tau\) local on the source, or local on the source for the \(\tau\)-topology if for any morphism \(f : X \to Y\) of algebraic spaces over \(S\), and any \(\tau\)-covering \(\{X_i \to X\}_{i \in I}\) of algebraic spaces we have \[f \text{ has }\mathcal{P} \Leftrightarrow \text{each }X_i \to Y\text{ has }\mathcal{P}.\]

To be sure, since isomorphisms are always coverings we see (or require) that property \(\mathcal{P}\) holds for \(X \to Y\) if and only if it holds for any arrow \(X' \to Y'\) isomorphic to \(X \to Y\). If a property is \(\tau\)-local on the source then it is preserved by precomposing with morphisms which occur in \(\tau\)-coverings. Here is a formal statement.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] \etale\}\). Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\) which is \(\tau\) local on the source. Let \(f : X \to Y\) have property \(\mathcal{P}\). For any morphism \(a : X' \to X\) which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. smooth, resp. étale, the composition \(f \circ a : X' \to Y\) has property \(\mathcal{P}\).

Proof

This is true because we can fit \(X' \to X\) into a family of morphisms which forms a \(\tau\)-covering.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] \etale\}\). Suppose that \(\mathcal{P}\) is a property of morphisms of schemes over \(S\) which is étale local on the source-and-target. Denote \(\mathcal{P}_{spaces}\) the corresponding property of morphisms of algebraic spaces over \(S\), see Morphisms of Spaces, Definition 04RD. If \(\mathcal{P}\) is local on the source for the \(\tau\)-topology, then \(\mathcal{P}_{spaces}\) is local on the source for the \(\tau\)-topology.

Proof

Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\{X_i \to X\}_{i \in I}\) be a \(\tau\)-covering of algebraic spaces. Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to X \times_Y V\). For each \(i\) choose a scheme \(U_i\) and a surjective étale morphism \(U_i \to X_i \times_X U\).

Note that \(\{X_i \times_X U \to U\}_{i \in I}\) is a \(\tau\)-covering. Note that each \(\{U_i \to X_i \times_X U\}\) is an étale covering, hence a \(\tau\)-covering. Hence \(\{U_i \to U\}_{i \in I}\) is a \(\tau\)-covering of algebraic spaces over \(S\). But since \(U\) and each \(U_i\) is a scheme we see that \(\{U_i \to U\}_{i \in I}\) is a \(\tau\)-covering of schemes over \(S\).

Now we have \[\begin{align*} f \text{ has }\mathcal{P}_{spaces} & \Leftrightarrow U \to V \text{ has }\mathcal{P} \\ & \Leftrightarrow \text{each }U_i \to V \text{ has }\mathcal{P} \\ & \Leftrightarrow \text{each }X_i \to Y\text{ has }\mathcal{P}_{spaces}. \end{align*}\] the first and last equivalence by the definition of \(\mathcal{P}_{spaces}\) the middle equivalence because we assumed \(\mathcal{P}\) is local on the source in the \(\tau\)-topology.

Properties of morphisms local in the fpqc topology on the source

Here are some properties of morphisms that are fpqc local on the source.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is flat” is fpqc local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 03ML and Descent, Lemma 036K.

Properties of morphisms local in the fppf topology on the source

Here are some properties of morphisms that are fppf local on the source.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is locally of finite presentation” is fppf local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 03XP and Descent, Lemma 036N.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is locally of finite type” is fppf local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 03XF and Descent, Lemma 036O.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is open” is fppf local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 03Z2 and Descent, Lemma 036P.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is universally open” is fppf local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 03Z2 and Descent, Lemma 036Q.

Properties of morphisms local in the syntomic topology on the source

Here are some properties of morphisms that are syntomic local on the source.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is syntomic” is syntomic local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 03Z7 and Descent, Lemma 036S.

Properties of morphisms local in the smooth topology on the source

Here are some properties of morphisms that are smooth local on the source.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is smooth” is smooth local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 03ZC and Descent, Lemma 036U.

Properties of morphisms local in the étale topology on the source

Here are some properties of morphisms that are étale local on the source.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is étale” is étale local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 04RH and Descent, Lemma 036W.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is locally quasi-finite” is étale local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 03XJ and Descent, Lemma 03X4.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is unramified” is étale local on the source.

Proof

Follows from Lemma 06ER using Morphisms of Spaces, Definition 03ZH and Descent, Lemma 03YV.

Properties of morphisms smooth local on source-and-target

Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces. There is an intuitive meaning to the phrase “\(\mathcal{P}\) is smooth local on the source and target”. However, it turns out that this notion is not the same as asking \(\mathcal{P}\) to be both smooth local on the source and smooth local on the target. We have discussed a similar phenomenon (for the étale topology and the category of schemes) in great detail in Descent, Section 04QW (for a quick overview take a look at Descent, Remark 04R4). However, there is an important difference between the case of the smooth and the étale topology. To see this difference we encourage the reader to ponder the difference between Descent, Lemma 04R0 and Lemma 06F9 as well as the difference between Descent, Lemma 04R1 and Lemma 06FA. Namely, in the étale setting the choice of the étale “covering” of the target is immaterial, whereas in the smooth setting it is not.

Definition

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\). We say \(\mathcal{P}\) is smooth local on source-and-target if

1. (stable under precomposing with smooth maps) if \(f : X \to Y\) is smooth and \(g : Y \to Z\) has \(\mathcal{P}\), then \(g \circ f\) has \(\mathcal{P}\),

2. (stable under smooth base change) if \(f : X \to Y\) has \(\mathcal{P}\) and \(Y' \to Y\) is smooth, then the base change \(f' : Y' \times_Y X \to Y'\) has \(\mathcal{P}\), and

3. (locality) given a morphism \(f : X \to Y\) the following are equivalent

   1. \(f\) has \(\mathcal{P}\),

   2. for every \(x \in |X|\) there exists a commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with smooth vertical arrows and \(u \in |U|\) with \(a(u) = x\) such that \(h\) has \(\mathcal{P}\).

The above serves as our definition. In the lemmas below we will show that this is equivalent to \(\mathcal{P}\) being smooth local on the target, smooth local on the source, and stable under post-composing by smooth morphisms.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\) which is smooth local on source-and-target. Then

1. \(\mathcal{P}\) is smooth local on the source,

2. \(\mathcal{P}\) is smooth local on the target,

3. \(\mathcal{P}\) is stable under postcomposing with smooth morphisms: if \(f : X \to Y\) has \(\mathcal{P}\) and \(g : Y \to Z\) is smooth, then \(g \circ f\) has \(\mathcal{P}\).

Proof

We write everything out completely.

Proof of (1). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\{X_i \to X\}_{i \in I}\) be a smooth covering of \(X\). If each composition \(h_i : X_i \to Y\) has \(\mathcal{P}\), then for each \(|x| \in X\) we can find an \(i \in I\) and a point \(x_i \in |X_i|\) mapping to \(x\). Then \((X_i, x_i) \to (X, x)\) is a smooth morphism of pairs, and \(\text{id}_Y : Y \to Y\) is a smooth morphism, and \(h_i\) is as in part (3) of Definition 06F8. Thus we see that \(f\) has \(\mathcal{P}\). Conversely, if \(f\) has \(\mathcal{P}\) then each \(X_i \to Y\) has \(\mathcal{P}\) by Definition 06F8 part (1).

Proof of (2). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\{Y_i \to Y\}_{i \in I}\) be a smooth covering of \(Y\). Write \(X_i = Y_i \times_Y X\) and \(h_i : X_i \to Y_i\) for the base change of \(f\). If each \(h_i : X_i \to Y_i\) has \(\mathcal{P}\), then for each \(x \in |X|\) we pick an \(i \in I\) and a point \(x_i \in |X_i|\) mapping to \(x\). Then \((X_i, x_i) \to (X, x)\) is a smooth morphism of pairs, \(Y_i \to Y\) is smooth, and \(h_i\) is as in part (3) of Definition 06F8. Thus we see that \(f\) has \(\mathcal{P}\). Conversely, if \(f\) has \(\mathcal{P}\), then each \(X_i \to Y_i\) has \(\mathcal{P}\) by Definition 06F8 part (2).

Proof of (3). Assume \(f : X \to Y\) has \(\mathcal{P}\) and \(g : Y \to Z\) is smooth. For every \(x \in |X|\) we can think of \((X, x) \to (X, x)\) as a smooth morphism of pairs, \(Y \to Z\) is a smooth morphism, and \(h = f\) is as in part (3) of Definition 06F8. Thus we see that \(g \circ f\) has \(\mathcal{P}\).

The following lemma is the analogue of Morphisms, Lemma 01SU.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\) which is smooth local on source-and-target. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:

1. \(f\) has property \(\mathcal{P}\),

2. for every \(x \in |X|\) there exists a smooth morphism of pairs \(a : (U, u) \to (X, x)\), a smooth morphism \(b : V \to Y\), and a morphism \(h : U \to V\) such that \(f \circ a = b \circ h\) and \(h\) has \(\mathcal{P}\),

3. for some commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with \(a\), \(b\) smooth and \(a\) surjective the morphism \(h\) has \(\mathcal{P}\),

4. for any commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with \(b\) smooth and \(U \to X \times_Y V\) smooth the morphism \(h\) has \(\mathcal{P}\),

5. there exists a smooth covering \(\{Y_i \to Y\}_{i \in I}\) such that each base change \(Y_i \times_Y X \to Y_i\) has \(\mathcal{P}\),

6. there exists a smooth covering \(\{X_i \to X\}_{i \in I}\) such that each composition \(X_i \to Y\) has \(\mathcal{P}\),

7. there exists a smooth covering \(\{Y_i \to Y\}_{i \in I}\) and for each \(i \in I\) a smooth covering \(\{X_{ij} \to Y_i \times_Y X\}_{j \in J_i}\) such that each morphism \(X_{ij} \to Y_i\) has \(\mathcal{P}\).

Proof

The equivalence of (a) and (b) is part of Definition 06F8. The equivalence of (a) and (e) is Lemma 06F9 part (2). The equivalence of (a) and (f) is Lemma 06F9 part (1). As (a) is now equivalent to (e) and (f) it follows that (a) equivalent to (g).

It is clear that (c) implies (b). If (b) holds, then for any \(x \in |X|\) we can choose a smooth morphism of pairs \(a_x : (U_x, u_x) \to (X, x)\), a smooth morphism \(b_x : V_x \to Y\), and a morphism \(h_x : U_x \to V_x\) such that \(f \circ a_x = b_x \circ h_x\) and \(h_x\) has \(\mathcal{P}\). Then \(h = \coprod h_x : \coprod U_x \to \coprod V_x\) with \(a = \coprod a_x\) and \(b = \coprod b_x\) is a diagram as in (c). (Note that \(h\) has property \(\mathcal{P}\) as \(\{V_x \to \coprod V_x\}\) is a smooth covering and \(\mathcal{P}\) is smooth local on the target.) Thus (b) is equivalent to (c).

Now we know that (a), (b), (c), (e), (f), and (g) are equivalent. Suppose (a) holds. Let \(U, V, a, b, h\) be as in (d). Then \(X \times_Y V \to V\) has \(\mathcal{P}\) as \(\mathcal{P}\) is stable under smooth base change, whence \(U \to V\) has \(\mathcal{P}\) as \(\mathcal{P}\) is stable under precomposing with smooth morphisms. Conversely, if (d) holds, then setting \(U = X\) and \(V = Y\) we see that \(f\) has \(\mathcal{P}\).

Lemma

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\). Assume

1. \(\mathcal{P}\) is smooth local on the source,

2. \(\mathcal{P}\) is smooth local on the target, and

3. \(\mathcal{P}\) is stable under postcomposing with smooth morphisms: if \(f : X \to Y\) has \(\mathcal{P}\) and \(Y \to Z\) is a smooth morphism then \(X \to Z\) has \(\mathcal{P}\).

Then \(\mathcal{P}\) is smooth local on the source-and-target.

Proof

Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces which satisfies conditions (1), (2) and (3) of the lemma. By Lemma 06EQ we see that \(\mathcal{P}\) is stable under precomposing with smooth morphisms. By Lemma 06EM we see that \(\mathcal{P}\) is stable under smooth base change. Hence it suffices to prove part (3) of Definition 06F8 holds.

More precisely, suppose that \(f : X \to Y\) is a morphism of algebraic spaces over \(S\) which satisfies Definition 06F8 part (3)(b). In other words, for every \(x \in X\) there exists a smooth morphism \(a_x : U_x \to X\), a point \(u_x \in |U_x|\) mapping to \(x\), a smooth morphism \(b_x : V_x \to Y\), and a morphism \(h_x : U_x \to V_x\) such that \(f \circ a_x = b_x \circ h_x\) and \(h_x\) has \(\mathcal{P}\). The proof of the lemma is complete once we show that \(f\) has \(\mathcal{P}\). Set \(U = \coprod U_x\), \(a = \coprod a_x\), \(V = \coprod V_x\), \(b = \coprod b_x\), and \(h = \coprod h_x\). We obtain a commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with \(a\), \(b\) smooth, \(a\) surjective. Note that \(h\) has \(\mathcal{P}\) as each \(h_x\) does and \(\mathcal{P}\) is smooth local on the target. Because \(a\) is surjective and \(\mathcal{P}\) is smooth local on the source, it suffices to prove that \(b \circ h\) has \(\mathcal{P}\). This follows as we assumed that \(\mathcal{P}\) is stable under postcomposing with a smooth morphism and as \(b\) is smooth.

Remark

Using Lemma 06FB and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are smooth local on the source-and-target. In each case we list the lemma which implies the property is smooth local on the source and the lemma which implies the property is smooth local on the target. In each case the third assumption of Lemma 06FB is trivial to check, and we omit it. Here is the list:

1. flat, see Lemmas 06ET and 041W,

2. locally of finite presentation, see Lemmas 06EV and 041T,

3. locally finite type, see Lemmas 06EW and 041S,

4. universally open, see Lemmas 06EY and 041P,

5. syntomic, see Lemmas 06F0 and 0428,

6. smooth, see Lemmas 06F2 and 0429,

7. add more here as needed.

Properties of morphisms étale-smooth local on source-and-target

This section is the analogue of Section 06F7 for properties of morphisms which are étale local on the source and smooth local on the target. We give this property a ridiculously long name in order to avoid using it too much.

Definition

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\). We say \(\mathcal{P}\) is étale-smooth local on source-and-target if

1. (stable under precomposing with étale maps) if \(f : X \to Y\) is étale and \(g : Y \to Z\) has \(\mathcal{P}\), then \(g \circ f\) has \(\mathcal{P}\),

2. (stable under smooth base change) if \(f : X \to Y\) has \(\mathcal{P}\) and \(Y' \to Y\) is smooth, then the base change \(f' : Y' \times_Y X \to Y'\) has \(\mathcal{P}\), and

3. (locality) given a morphism \(f : X \to Y\) the following are equivalent

   1. \(f\) has \(\mathcal{P}\),

   2. for every \(x \in |X|\) there exists a commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with \(b\) smooth and \(U \to X \times_Y V\) étale and \(u \in |U|\) with \(a(u) = x\) such that \(h\) has \(\mathcal{P}\).

The above serves as our definition. In the lemmas below we will show that this is equivalent to \(\mathcal{P}\) being étale local on the target, smooth local on the source, and stable under post-composing by étale morphisms.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\) which is étale-smooth local on source-and-target. Then

1. \(\mathcal{P}\) is étale local on the source,

2. \(\mathcal{P}\) is smooth local on the target,

3. \(\mathcal{P}\) is stable under postcomposing with étale morphisms: if \(f : X \to Y\) has \(\mathcal{P}\) and \(g : Y \to Z\) is étale, then \(g \circ f\) has \(\mathcal{P}\), and

4. \(\mathcal{P}\) has a permanence property: given \(f : X \to Y\) and \(g : Y \to Z\) étale such that \(g \circ f\) has \(\mathcal{P}\), then \(f\) has \(\mathcal{P}\).

Proof

We write everything out completely.

Proof of (1). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\{X_i \to X\}_{i \in I}\) be an étale covering of \(X\). If each composition \(h_i : X_i \to Y\) has \(\mathcal{P}\), then for each \(|x| \in X\) we can find an \(i \in I\) and a point \(x_i \in |X_i|\) mapping to \(x\). Then \((X_i, x_i) \to (X, x)\) is an étale morphism of pairs, and \(\text{id}_Y : Y \to Y\) is a smooth morphism, and \(h_i\) is as in part (3) of Definition 0CFZ. Thus we see that \(f\) has \(\mathcal{P}\). Conversely, if \(f\) has \(\mathcal{P}\) then each \(X_i \to Y\) has \(\mathcal{P}\) by Definition 0CFZ part (1).

Proof of (2). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\{Y_i \to Y\}_{i \in I}\) be a smooth covering of \(Y\). Write \(X_i = Y_i \times_Y X\) and \(h_i : X_i \to Y_i\) for the base change of \(f\). If each \(h_i : X_i \to Y_i\) has \(\mathcal{P}\), then for each \(x \in |X|\) we pick an \(i \in I\) and a point \(x_i \in |X_i|\) mapping to \(x\). Then \(X_i \to X \times_Y Y_i\) is an étale morphism (because it is an isomorphism), \(Y_i \to Y\) is smooth, and \(h_i\) is as in part (3) of Definition 06F8. Thus we see that \(f\) has \(\mathcal{P}\). Conversely, if \(f\) has \(\mathcal{P}\), then each \(X_i \to Y_i\) has \(\mathcal{P}\) by Definition 06F8 part (2).

Proof of (3). Assume \(f : X \to Y\) has \(\mathcal{P}\) and \(g : Y \to Z\) is étale. The morphism \(X \to Y \times_Z X\) is étale as a morphism between algebraic spaces étale over \(X\) ( Properties of Spaces, Lemma 03FV). Also \(Y \to Z\) is étale hence a smooth morphism. Thus the diagram \[\xymatrix{ X \ar[d] \ar[r]_f & Y \ar[d] \\ X \ar[r]^{g \circ f} & Z }\] works for every \(x \in |X|\) in part (3) of Definition 06F8 and we conclude that \(g \circ f\) has \(\mathcal{P}\).

Proof of (4). Let \(f : X \to Y\) be a morphism and \(g : Y \to Z\) étale such that \(g \circ f\) has \(\mathcal{P}\). Then by Definition 0CFZ part (2) we see that \(\text{pr}_Y : Y \times_Z X \to Y\) has \(\mathcal{P}\). But the morphism \((f, 1) : X \to Y \times_Z X\) is étale as a section to the étale projection \(\text{pr}_X : Y \times_Z X \to X\), see Morphisms of Spaces, Lemma 05W3. Hence \(f = \text{pr}_Y \circ (f, 1)\) has \(\mathcal{P}\) by Definition 0CFZ part (1).

Lemma

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\) which is etale-smooth local on source-and-target. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:

1. \(f\) has property \(\mathcal{P}\),

2. for every \(x \in |X|\) there exists a smooth morphism \(b : V \to Y\), an étale morphism \(a : U \to V \times_Y X\), and a point \(u \in |U|\) mapping to \(x\) such that \(U \to V\) has \(\mathcal{P}\),

3. for some commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with \(b\) smooth, \(U \to V \times_Y X\) étale, and \(a\) surjective the morphism \(h\) has \(\mathcal{P}\),

4. for any commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with \(b\) smooth and \(U \to X \times_Y V\) étale, the morphism \(h\) has \(\mathcal{P}\),

5. there exists a smooth covering \(\{Y_i \to Y\}_{i \in I}\) such that each base change \(Y_i \times_Y X \to Y_i\) has \(\mathcal{P}\),

6. there exists an étale covering \(\{X_i \to X\}_{i \in I}\) such that each composition \(X_i \to Y\) has \(\mathcal{P}\),

7. there exists a smooth covering \(\{Y_i \to Y\}_{i \in I}\) and for each \(i \in I\) an étale covering \(\{X_{ij} \to Y_i \times_Y X\}_{j \in J_i}\) such that each morphism \(X_{ij} \to Y_i\) has \(\mathcal{P}\).

Proof

The equivalence of (a) and (b) is part of Definition 0CFZ. The equivalence of (a) and (e) is Lemma 0CG0 part (2). The equivalence of (a) and (f) is Lemma 0CG0 part (1). As (a) is now equivalent to (e) and (f) it follows that (a) equivalent to (g).

It is clear that (c) implies (b). If (b) holds, then for any \(x \in |X|\) we can choose a smooth morphism a smooth morphism \(b_x : V_x \to Y\), an étale morphism \(U_x \to V_x \times_Y X\), and \(u_x \in |U_x|\) mapping to \(x\) such that \(U_x \to V_x\) has \(\mathcal{P}\). Then \(h = \coprod h_x : \coprod U_x \to \coprod V_x\) with \(a = \coprod a_x\) and \(b = \coprod b_x\) is a diagram as in (c). (Note that \(h\) has property \(\mathcal{P}\) as \(\{V_x \to \coprod V_x\}\) is a smooth covering and \(\mathcal{P}\) is smooth local on the target.) Thus (b) is equivalent to (c).

Now we know that (a), (b), (c), (e), (f), and (g) are equivalent. Suppose (a) holds. Let \(U, V, a, b, h\) be as in (d). Then \(X \times_Y V \to V\) has \(\mathcal{P}\) as \(\mathcal{P}\) is stable under smooth base change, whence \(U \to V\) has \(\mathcal{P}\) as \(\mathcal{P}\) is stable under precomposing with étale morphisms. Conversely, if (d) holds, then setting \(U = X\) and \(V = Y\) we see that \(f\) has \(\mathcal{P}\).

Lemma

Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces over \(S\). Assume

1. \(\mathcal{P}\) is étale local on the source,

2. \(\mathcal{P}\) is smooth local on the target, and

3. \(\mathcal{P}\) is stable under postcomposing with open immersions: if \(f : X \to Y\) has \(\mathcal{P}\) and \(Y \subset Z\) is an open embedding then \(X \to Z\) has \(\mathcal{P}\).

Then \(\mathcal{P}\) is étale-smooth local on the source-and-target.

Proof

Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces which satisfies conditions (1), (2) and (3) of the lemma. By Lemma 06EQ we see that \(\mathcal{P}\) is stable under precomposing with étale morphisms. By Lemma 06EM we see that \(\mathcal{P}\) is stable under smooth base change. Hence it suffices to prove part (3) of Definition 06F8 holds.

More precisely, suppose that \(f : X \to Y\) is a morphism of algebraic spaces over \(S\) which satisfies Definition 06F8 part (3)(b). In other words, for every \(x \in X\) there exists a smooth morphism \(b_x : V_x \to Y\), an étale morphism \(U_x \to V_x \times_Y X\), and a point \(u_x \in |U_x|\) mapping to \(x\) such that \(h_x : U_x \to V_x\) has \(\mathcal{P}\). The proof of the lemma is complete once we show that \(f\) has \(\mathcal{P}\).

Let \(a_x : U_x \to X\) be the composition \(U_x \to V_x \times_Y X \to X\). Set \(U = \coprod U_x\), \(a = \coprod a_x\), \(V = \coprod V_x\), \(b = \coprod b_x\), and \(h = \coprod h_x\). We obtain a commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with \(b\) smooth, \(U \to V \times_Y X\) étale, \(a\) surjective. Note that \(h\) has \(\mathcal{P}\) as each \(h_x\) does and \(\mathcal{P}\) is smooth local on the target. In the next paragraph we prove that we may assume \(U, V, X, Y\) are schemes; we encourage the reader to skip it.

Let \(X, Y, U, V, a, b, f, h\) be as in the previous paragraph. We have to show \(f\) has \(\mathcal{P}\). Let \(X' \to X\) be a surjective étale morphism with \(X_i\) a scheme. Set \(U' = X' \times_X U\). Then \(U' \to X'\) is surjective and \(U' \to X' \times_Y V\) is étale. Since \(\mathcal{P}\) is étale local on the source, we see that \(U' \to V\) has \(\mathcal{P}\) and that it suffices to show that \(X' \to Y\) has \(\mathcal{P}\). In other words, we may assume that \(X\) is a scheme. Next, choose a surjective étale morphism \(Y' \to Y\) with \(Y'\) a scheme. Set \(V' = V \times_Y Y'\), \(X' = X \times_Y Y'\), and \(U' = U \times_Y Y'\). Then \(U' \to X'\) is surjective and \(U' \to X' \times_{Y'} V'\) is étale. Since \(\mathcal{P}\) is smooth local on the target, we see that \(U' \to V'\) has \(\mathcal{P}\) and that it suffices to prove \(X' \to Y'\) has \(\mathcal{P}\). Thus we may assume both \(X\) and \(Y\) are schemes. Choose a surjective étale morphism \(V' \to V\) with \(V'\) a scheme. Set \(U' = U \times_V V'\). Then \(U' \to X\) is surjective and \(U' \to X \times_Y V'\) is étale. Since \(\mathcal{P}\) is smooth local on the source, we see that \(U' \to V'\) has \(\mathcal{P}\). Thus we may replace \(U, V\) by \(U', V'\) and assume \(X, Y, V\) are schemes. Finally, we replace \(U\) by a scheme surjective étale over \(U\) and we see that we may assume \(U, V, X, Y\) are all schemes.

If \(U, V, X, Y\) are schemes, then \(f\) has \(\mathcal{P}\) by Descent, Lemma 0CF1.

Remark

Using Lemma 0CG2 and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are smooth local on the source-and-target. In each case we list the lemma which implies the property is etale local on the source and the lemma which implies the property is smooth local on the target. In each case the third assumption of Lemma 0CG2 is trivial to check, and we omit it. Here is the list:

1. étale, see Lemmas 06F4 and 042B,

2. locally quasi-finite, see Lemmas 06F5 and 0427,

3. unramified, see Lemmas 06F6 and 042A, and

4. add more here as needed.

Of course any property listed in Remark 06FC is a fortiori an example that could be listed here.

Descent data for spaces over spaces

This section is the analogue of Descent, Section 023U for algebraic spaces. Most of the arguments in this section are formal relying only on the definition of a descent datum.

Definition

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

1. Let \(V \to Y\) be a morphism of algebraic spaces. A descent datum for \(V/Y/X\) is an isomorphism \(\varphi : V \times_X Y \to Y \times_X V\) of algebraic spaces over \(Y \times_X Y\) satisfying the cocycle condition that the diagram \[\xymatrix{ V \times_X Y \times_X Y \ar[rd]^{\varphi_{01}} \ar[rr]_{\varphi_{02}} & & Y \times_X Y \times_X V\\ & Y \times_X V \times_X Y \ar[ru]^{\varphi_{12}} }\] commutes (with obvious notation).

2. We also say that the pair \((V/Y, \varphi)\) is a descent datum relative to \(Y \to X\).

3. A morphism \(f : (V/Y, \varphi) \to (V'/Y, \varphi')\) of descent data relative to \(Y \to X\) is a morphism \(f : V \to V'\) of algebraic spaces over \(Y\) such that the diagram \[\xymatrix{ V \times_X Y \ar[r]_{\varphi} \ar[d]_{f \times \text{id}_Y} & Y \times_X V \ar[d]^{\text{id}_Y \times f} \\ V' \times_X Y \ar[r]^{\varphi'} & Y \times_X V' }\] commutes.

Remark

Let \(S\) be a scheme. Let \(Y \to X\) be a morphism of algebraic spaces over \(S\). Let \((V/Y, \varphi)\) be a descent datum relative to \(Y \to X\). We may think of the isomorphism \(\varphi\) as an isomorphism \[(Y \times_X Y) \times_{\text{pr}_0, Y} V \longrightarrow (Y \times_X Y) \times_{\text{pr}_1, Y} V\] of algebraic spaces over \(Y \times_X Y\). So loosely speaking one may think of \(\varphi\) as a map \(\varphi : \text{pr}_0^*V \to \text{pr}_1^*V\)¹. The cocycle condition then says that \(\text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi\). In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.

Here is the definition in case you have a family of morphisms with fixed target.

Definition

Let \(S\) be a scheme. Let \(\{X_i \to X\}_{i \in I}\) be a family of morphisms of algebraic spaces over \(S\) with fixed target \(X\).

1. A descent datum \((V_i, \varphi_{ij})\) relative to the family \(\{X_i \to X\}\) is given by an algebraic space \(V_i\) over \(X_i\) for each \(i \in I\), an isomorphism \(\varphi_{ij} : V_i \times_X X_j \to X_i \times_X V_j\) of algebraic spaces over \(X_i \times_X X_j\) for each pair \((i, j) \in I^2\) such that for every triple of indices \((i, j, k) \in I^3\) the diagram \[\xymatrix{ V_i \times_X X_j \times_X X_k \ar[rd]^{\text{pr}_{01}^*\varphi_{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi_{ik}} & & X_i \times_X X_j \times_X V_k\\ & X_i \times_X V_j \times_X X_k \ar[ru]^{\text{pr}_{12}^*\varphi_{jk}} }\] of algebraic spaces over \(X_i \times_X X_j \times_X X_k\) commutes (with obvious notation).

2. A morphism \(\psi : (V_i, \varphi_{ij}) \to (V'_i, \varphi'_{ij})\) of descent data is given by a family \(\psi = (\psi_i)_{i \in I}\) of morphisms \(\psi_i : V_i \to V'_i\) of algebraic spaces over \(X_i\) such that all the diagrams \[\xymatrix{ V_i \times_X X_j \ar[r]_{\varphi_{ij}} \ar[d]_{\psi_i \times \text{id}} & X_i \times_X V_j \ar[d]^{\text{id} \times \psi_j} \\ V'_i \times_X X_j \ar[r]^{\varphi'_{ij}} & X_i \times_X V'_j }\] commute.

Remark

Let \(S\) be a scheme. Let \(\{X_i \to X\}_{i \in I}\) be a family of morphisms of algebraic spaces over \(S\) with fixed target \(X\). Let \((V_i, \varphi_{ij})\) be a descent datum relative to \(\{X_i \to X\}\). We may think of the isomorphisms \(\varphi_{ij}\) as isomorphisms \[(X_i \times_X X_j) \times_{\text{pr}_0, X_i} V_i \longrightarrow (X_i \times_X X_j) \times_{\text{pr}_1, X_j} V_j\] of algebraic spaces over \(X_i \times_X X_j\). So loosely speaking one may think of \(\varphi_{ij}\) as an isomorphism \(\text{pr}_0^*V_i \to \text{pr}_1^*V_j\) over \(X_i \times_X X_j\). The cocycle condition then says that \(\text{pr}_{02}^*\varphi_{ik} = \text{pr}_{12}^*\varphi_{jk} \circ \text{pr}_{01}^*\varphi_{ij}\). In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.

The reason we will usually work with the version of a family consisting of a single morphism is the following lemma.

Lemma

Let \(S\) be a scheme. Let \(\{X_i \to X\}_{i \in I}\) be a family of morphisms of algebraic spaces over \(S\) with fixed target \(X\). Set \(Y = \coprod_{i \in I} X_i\). There is a canonical equivalence of categories \[\begin{matrix} \text{category of descent data } \\ \text{relative to the family } \{X_i \to X\}_{i \in I} \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } Y/X \end{matrix}\] which maps \((V_i, \varphi_{ij})\) to \((V, \varphi)\) with \(V = \coprod_{i\in I} V_i\) and \(\varphi = \coprod \varphi_{ij}\).

Proof

Observe that \(Y \times_X Y = \coprod_{ij} X_i \times_X X_j\) and similarly for higher fibre products. Giving a morphism \(V \to Y\) is exactly the same as giving a family \(V_i \to X_i\). And giving a descent datum \(\varphi\) is exactly the same as giving a family \(\varphi_{ij}\).

Lemma

Pullback of descent data. Let \(S\) be a scheme.

1. Let \[\xymatrix{ Y' \ar[r]_f \ar[d]_{a'} & Y \ar[d]^a \\ X' \ar[r]^h & X }\] be a commutative diagram of algebraic spaces over \(S\). The construction \[(V \to Y, \varphi) \longmapsto f^*(V \to Y, \varphi) = (V' \to Y', \varphi')\] where \(V' = Y' \times_Y V\) and where \(\varphi'\) is defined as the composition \[\xymatrix{ V' \times_{X'} Y' \ar@{=}[r] & (Y' \times_Y V) \times_{X'} Y' \ar@{=}[r] & (Y' \times_{X'} Y') \times_{Y \times_X Y} (V \times_X Y) \ar[d]^{\text{id} \times \varphi} \\ Y' \times_{X'} V' \ar@{=}[r] & Y' \times_{X'} (Y' \times_Y V) & (Y' \times_X Y') \times_{Y \times_X Y} (Y \times_X V) \ar@{=}[l] }\] defines a functor from the category of descent data relative to \(Y \to X\) to the category of descent data relative to \(Y' \to X'\).

2. Given two morphisms \(f_i : Y' \to Y\), \(i = 0, 1\) making the diagram commute the functors \(f_0^*\) and \(f_1^*\) are canonically isomorphic.

Proof

We omit the proof of (1), but we remark that the morphism \(\varphi'\) is the morphism \((f \times f)^*\varphi\) in the notation introduced in Remark 0ADH. For (2) we indicate which morphism \(f_0^*V \to f_1^*V\) gives the functorial isomorphism. Namely, since \(f_0\) and \(f_1\) both fit into the commutative diagram we see there is a unique morphism \(r : Y' \to Y \times_X Y\) with \(f_i = \text{pr}_i \circ r\). Then we take \[\begin{eqnarray*} f_0^*V & = & Y' \times_{f_0, Y} V \\ & = & Y' \times_{\text{pr}_0 \circ r, Y} V \\ & = & Y' \times_{r, Y \times_X Y} (Y \times_X Y) \times_{\text{pr}_0, Y} V \\ & \xrightarrow{\varphi} & Y' \times_{r, Y \times_X Y} (Y \times_X Y) \times_{\text{pr}_1, Y} V \\ & = & Y' \times_{\text{pr}_1 \circ r, Y} V \\ & = & Y' \times_{f_1, Y} V \\ & = & f_1^*V \end{eqnarray*}\] We omit the verification that this works.

Definition

With \(S, X, X', Y, Y', f, a, a', h\) as in Lemma 0ADL the functor \[(V, \varphi) \longmapsto f^*(V, \varphi)\] constructed in that lemma is called the pullback functor on descent data.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{U}' = \{X'_i \to X'\}_{i \in I'}\) and \(\mathcal{U} = \{X_j \to X\}_{i \in I}\) be families of morphisms with fixed target. Let \(\alpha : I' \to I\), \(g : X' \to X\) and \(g_i : X'_i \to X_{\alpha(i)}\) be a morphism of families of maps with fixed target, see Sites, Definition 00VT.

1. Let \((V_i, \varphi_{ij})\) be a descent datum relative to the family \(\mathcal{U}\). The system \[\left( g_i^*V_{\alpha(i)}, (g_i \times g_j)^*\varphi_{\alpha(i) \alpha(j)} \right)\] (with notation as in Remark 0ADJ) is a descent datum relative to \(\mathcal{U}'\).

2. This construction defines a functor between the category of descent data relative to \(\mathcal{U}\) and the category of descent data relative to \(\mathcal{U}'\).

3. Given a second \(\beta : I' \to I\), \(h : X' \to X\) and \(h'_i : X'_i \to X_{\beta(i)}\) morphism of families of maps with fixed target, then if \(g = h\) the two resulting functors between descent data are canonically isomorphic.

4. These functors agree, via Lemma 0ADK, with the pullback functors constructed in Lemma 0ADL.

Proof

This follows from Lemma 0ADL via the correspondence of Lemma 0ADK.

Definition

With \(\mathcal{U}' = \{X'_i \to X'\}_{i \in I'}\), \(\mathcal{U} = \{X_i \to X\}_{i \in I}\), \(\alpha : I' \to I\), \(g : X' \to X\), and \(g_i : X'_i \to X_{\alpha(i)}\) as in Lemma 0ADN the functor \[(V_i, \varphi_{ij}) \longmapsto (g_i^*V_{\alpha(i)}, (g_i \times g_j)^*\varphi_{\alpha(i) \alpha(j)})\] constructed in that lemma is called the pullback functor on descent data.

If \(\mathcal{U}\) and \(\mathcal{U}'\) have the same target \(X\), and if \(\mathcal{U}'\) refines \(\mathcal{U}\) (see Sites, Definition 00VT) but no explicit pair \((\alpha, g_i)\) is given, then we can still talk about the pullback functor since we have seen in Lemma 0ADN that the choice of the pair does not matter (up to a canonical isomorphism).

Definition

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

1. Given an algebraic space \(U\) over \(X\) we have the trivial descent datum of \(U\) relative to \(\text{id} : X \to X\), namely the identity morphism on \(U\).

2. By Lemma 0ADL we get a canonical descent datum on \(Y \times_X U\) relative to \(Y \to X\) by pulling back the trivial descent datum via \(f\). We often denote \((Y \times_X U, can)\) this descent datum.

3. A descent datum \((V, \varphi)\) relative to \(Y/X\) is called effective if \((V, \varphi)\) is isomorphic to the canonical descent datum \((Y \times_X U, can)\) for some algebraic space \(U\) over \(X\).

Thus being effective means there exists an algebraic space \(U\) over \(X\) and an isomorphism \(\psi : V \to Y \times_X U\) over \(Y\) such that \(\varphi\) is equal to the composition \[V \times_X Y \xrightarrow{\psi \times \text{id}_Y} Y \times_X U \times_S Y = Y \times_X Y \times_X U \xrightarrow{\text{id}_Y \times \psi^{-1}} Y \times_X V\] There is a slight problem here which is that this definition (in spirit) conflicts with the definition given in Descent, Definition 023Z in case \(Y\) and \(X\) are schemes. However, it will always be clear from context which version we mean.

Definition

Let \(S\) be a scheme. Let \(\{X_i \to X\}\) be a family of morphisms of algebraic spaces over \(S\) with fixed target \(X\).

1. Given an algebraic space \(U\) over \(X\) we have a canonical descent datum on the family of algebraic spaces \(X_i \times_X U\) by pulling back the trivial descent datum for \(U\) relative to \(\{\text{id} : S \to S\}\). We denote this descent datum \((X_i \times_X U, can)\).

2. A descent datum \((V_i, \varphi_{ij})\) relative to \(\{X_i \to S\}\) is called effective if there exists an algebraic space \(U\) over \(X\) such that \((V_i, \varphi_{ij})\) is isomorphic to \((X_i \times_X U, can)\).

Descent data in terms of sheaves

This section is the analogue of Descent, Section 02W4. It is slightly different as algebraic spaces are already sheaves.

Lemma

Let \(S\) be a scheme. Let \(\{X_i \to X\}_{i \in I}\) be an fppf covering of algebraic spaces over \(S\) (Topologies on Spaces, Definition 03Y8). There is an equivalence of categories \[\left\{ \begin{matrix} \text{descent data }(V_i, \varphi_{ij})\\ \text{relative to }\{X_i \to X\} \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{sheaves }F\text{ on }(\Sch/S)_{fppf}\text{ endowed}\\ \text{with a map }F \to X\text{ such that each}\\ X_i \times_X F\text{ is an algebraic space} \end{matrix} \right\}.\] Moreover,

1. the algebraic space \(X_i \times_X F\) on the right hand side corresponds to \(V_i\) on the left hand side, and

2. the sheaf \(F\) is an algebraic space² if and only if the corresponding descent datum \((X_i, \varphi_{ij})\) is effective.

Proof

Let us construct the functor from right to left. Let \(F \to X\) be a map of sheaves on \((\Sch/S)_{fppf}\) such that each \(V_i = X_i \times_X F\) is an algebraic space. We have the projection \(V_i \to X_i\). Then both \(V_i \times_X X_j\) and \(X_i \times_X V_j\) represent the sheaf \(X_i \times_X F \times_X X_j\) and hence we obtain an isomorphism \[\varphi_{ii'} : V_i \times_X X_j \to X_i \times_X V_j\] It is straightforward to see that the maps \(\varphi_{ij}\) are morphisms over \(X_i \times_X X_j\) and satisfy the cocycle condition. The functor from right to left is given by this construction \(F \mapsto (V_i, \varphi_{ij})\).

Let us construct a functor from left to right. The isomorphisms \(\varphi_{ij}\) give isomorphisms \[\varphi_{ij} : V_i \times_X X_j \longrightarrow X_i \times_X V_j\] over \(X_i \times X_j\). Set \(F\) equal to the coequalizer in the following diagram \[\xymatrix{ \coprod_{i, i'} V_i \times_X X_j \ar@<1ex>[rr]^-{\text{pr}_0} \ar@<-1ex>[rr]_-{\text{pr}_1 \circ \varphi_{ij}} & & \coprod_i V_i \ar[r] & F }\] The cocycle condition guarantees that \(F\) comes with a map \(F \to X\) and that \(X_i \times_X F\) is isomorphic to \(V_i\). The functor from left to right is given by this construction \((V_i, \varphi_{ij}) \mapsto F\).

We omit the verification that these constructions are mutually quasi-inverse functors. The final statements (1) and (2) follow from the constructions.