Introduction
In this chapter we use the material from the preceding sections to give criteria under which a presheaf of sets on the category of schemes is an algebraic space. Some of this material comes from the work of Artin, see [ArtinI], [ArtinII], [Artin-Theorem-Representability], [Artin-Construction-Techniques], [Artin-Algebraic-Spaces], [Artin-Algebraic-Approximation], [Artin-Implicit-Function], and [ArtinVersal]. However, our method will be to use as much as possible arguments similar to those of the paper by Keel and Mori, see [K-M].
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 \(X \times X\).
Morphisms representable by algebraic spaces
Here we define the notion of one presheaf being relatively representable by algebraic spaces over another, and we prove some properties of this notion.
Definition
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F\), \(G\) be presheaves on \(\Sch_{fppf}/S\). We say a morphism \(a : F \to G\) is representable by algebraic spaces if for every \(U \in \Ob((\Sch/S)_{fppf})\) and any \(\xi : U \to G\) the fiber product \(U \times_{\xi, G} F\) is an algebraic space.
Here is a sanity check.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Then \(f\) is representable by algebraic spaces.
Proof
This is formal. It relies on the fact that the category of algebraic spaces over \(S\) has fibre products, see Spaces, Lemma 02X2.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ G' \times_G F \ar[r] \ar[d]^{a'} & F \ar[d]^a \\ G' \ar[r] & G }\] be a fibre square of presheaves on \((\Sch/S)_{fppf}\). If \(a\) is representable by algebraic spaces so is \(a'\).
Proof
Omitted. Hint: This is formal.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\) be representable by algebraic spaces. If \(G\) is a sheaf, then so is \(F\).
Proof
(Same as the proof of Spaces, Lemma 02WD.) Let \(\{\varphi_i : T_i \to T\}\) be a covering of the site \((\Sch/S)_{fppf}\). Let \(s_i \in F(T_i)\) which satisfy the sheaf condition. Then \(\sigma_i = a(s_i) \in G(T_i)\) satisfy the sheaf condition also. Hence there exists a unique \(\sigma \in G(T)\) such that \(\sigma_i = \sigma|_{T_i}\). By assumption \(F' = h_T \times_{\sigma, G, a} F\) is a sheaf. Note that \((\varphi_i, s_i) \in F'(T_i)\) satisfy the sheaf condition also, and hence come from some unique \((\text{id}_T, s) \in F'(T)\). Clearly \(s\) is the section of \(F\) we are looking for.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\) be representable by algebraic spaces. Then \(\Delta_{F/G} : F \to F \times_G F\) is representable by algebraic spaces.
Proof
(Same as the proof of Spaces, Lemma 05L9.) Let \(U\) be a scheme. Let \(\xi = (\xi_1, \xi_2) \in (F \times_G F)(U)\). Set \(\xi' = a(\xi_1) = a(\xi_2) \in G(U)\). By assumption there exist an algebraic space \(V\) and a morphism \(V \to U\) representing the fibre product \(U \times_{\xi', G} F\). In particular, the elements \(\xi_1, \xi_2\) give morphisms \(f_1, f_2 : U \to V\) over \(U\). Because \(V\) represents the fibre product \(U \times_{\xi', G} F\) and because \(\xi' = a \circ \xi_1 = a \circ \xi_2\) we see that if \(g : U' \to U\) is a morphism then \[g^*\xi_1 = g^*\xi_2 \Leftrightarrow f_1 \circ g = f_2 \circ g.\] In other words, we see that \(U \times_{\xi, F \times_G F} F\) is represented by \(V \times_{\Delta, V \times V, (f_1, f_2)} U\) which is an algebraic space.
The proof of Lemma 02YS below is actually slightly tricky. Namely, we cannot use the argument of the proof of Spaces, Lemma 02WY because we do not yet know that a composition of transformations representable by algebraic spaces is representable by algebraic spaces. In fact, we will use this lemma to prove that statement.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\) be representable by algebraic spaces. If \(G\) is an algebraic space, then so is \(F\).
Proof
We have seen in Lemma 02YR that \(F\) is a sheaf.
Let \(U\) be a scheme and let \(U \to G\) be a surjective étale morphism. In this case \(U \times_G F\) is an algebraic space. Let \(W\) be a scheme and let \(W \to U \times_G F\) be a surjective étale morphism.
First we claim that \(W \to F\) is representable. To see this let \(X\) be a scheme and let \(X \to F\) be a morphism. Then \[W \times_F X = W \times_{U \times_G F} U \times_G F \times_F X = W \times_{U \times_G F} (U \times_G X)\] Since both \(U \times_G F\) and \(G\) are algebraic spaces we see that this is a scheme.
Next, we claim that \(W \to F\) is surjective and étale (this makes sense now that we know it is representable). This follows from the formula above since both \(W \to U \times_G F\) and \(U \to G\) are étale and surjective, hence \(W \times_{U \times_G F} (U \times_G X) \to U \times_G X\) and \(U \times_G X \to X\) are surjective and étale, and the composition of surjective étale morphisms is surjective and étale.
Set \(R = W \times_F W\). By the above \(R\) is a scheme and the projections \(t, s : R \to W\) are étale. It is clear that \(R\) is an equivalence relation, and \(W \to F\) is a surjection of sheaves. Hence \(R\) is an étale equivalence relation and \(F = W/R\). Hence \(F\) is an algebraic space by Spaces, Theorem 02WW.
Lemma
Let \(S\) be a scheme. Let \(a : F \to G\) be a map of presheaves on \((\Sch/S)_{fppf}\). Suppose \(a : F \to G\) is representable by algebraic spaces. If \(X\) is an algebraic space over \(S\), and \(X \to G\) is a map of presheaves then \(X \times_G F\) is an algebraic space.
Proof
By Lemma 03Y0 the transformation \(X \times_G F \to X\) is representable by algebraic spaces. Hence it is an algebraic space by Lemma 02YS.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ F \ar[r]^a & G \ar[r]^b & H }\] be maps of presheaves on \((\Sch/S)_{fppf}\). If \(a\) and \(b\) are representable by algebraic spaces, so is \(b \circ a\).
Proof
Let \(T\) be a scheme over \(S\), and let \(T \to H\) be a morphism. By assumption \(T \times_H G\) is an algebraic space. Hence by Lemma 03XY we see that \(T \times_H F = (T \times_H G) \times_G F\) is an algebraic space as well.
Lemma
Let \(S\) be a scheme. Let \(F_i, G_i : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\), \(i = 1, 2\). Let \(a_i : F_i \to G_i\), \(i = 1, 2\) be representable by algebraic spaces. Then \[a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2\] is a representable by algebraic spaces.
Proof
Write \(a_1 \times a_2\) as the composition \(F_1 \times F_2 \to G_1 \times F_2 \to G_1 \times G_2\). The first arrow is the base change of \(a_1\) by the map \(G_1 \times F_2 \to G_1\), and the second arrow is the base change of \(a_2\) by the map \(G_1 \times G_2 \to G_2\). Hence this lemma is a formal consequence of Lemmas 03Y1 and 03Y0.
Lemma
Let \(S\) be a scheme. Let \(a : F \to G\) and \(b : G \to H\) be transformations of functors \((\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Assume
\(\Delta : G \to G \times_H G\) is representable by algebraic spaces, and
\(b \circ a : F \to H\) is representable by algebraic spaces.
Then \(a\) is representable by algebraic spaces.
Proof
Let \(U\) be a scheme over \(S\) and let \(\xi \in G(U)\). Then \[U \times_{\xi, G, a} F = (U \times_{b(\xi), H, b \circ a} F) \times_{(\xi, a), (G \times_H G), \Delta} G\] Hence the result using Lemma 03XY.
Lemma
Let \(S \in \Ob(\Sch_{fppf})\). Let \(F\) be a presheaf of sets on \((\Sch/S)_{fppf}\). Assume
\(F\) is a sheaf for the Zariski topology on \((\Sch/S)_{fppf}\),
there exists an index set \(I\) and subfunctors \(F_i \subset F\) such that
each \(F_i\) is an fppf sheaf,
each \(F_i \to F\) is representable by algebraic spaces,
\(\coprod F_i \to F\) becomes surjective after fppf sheafification.
Then \(F\) is an fppf sheaf.
Proof
Let \(T \in \Ob((\Sch/S)_{fppf})\) and let \(s \in F(T)\). By (2)(c) there exists an fppf covering \(\{T_j \to T\}\) such that \(s|_{T_j}\) is a section of \(F_{\alpha(j)}\) for some \(\alpha(j) \in I\). Let \(W_j \subset T\) be the image of \(T_j \to T\) which is an open subscheme Morphisms, Lemma 01UA. By (2)(b) we see \(F_{\alpha(j)} \times_{F, s|_{W_j}} W_j \to W_j\) is a monomorphism of algebraic spaces through which \(T_j\) factors. Since \(\{T_j \to W_j\}\) is an fppf covering, we conclude that \(F_{\alpha(j)} \times_{F, s|_{W_j}} W_j = W_j\), in other words \(s|_{W_j} \in F_{\alpha(j)}(W_j)\). Hence we conclude that \(\coprod F_i \to F\) is surjective for the Zariski topology.
Let \(\{T_j \to T\}\) be an fppf covering in \((\Sch/S)_{fppf}\). Let \(s, s' \in F(T)\) with \(s|_{T_j} = s'|_{T_j}\) for all \(j\). We want to show that \(s, s'\) are equal. As \(F\) is a Zariski sheaf by (1) we may work Zariski locally on \(T\). By the result of the previous paragraph we may assume there exist \(i\) such that \(s \in F_i(T)\). Then we see that \(s'|_{T_j}\) is a section of \(F_i\). By (2)(b) we see \(F_{i} \times_{F, s'} T \to T\) is a monomorphism of algebraic spaces through which all of the \(T_j\) factor. Hence we conclude that \(s' \in F_i(T)\). Since \(F_i\) is a sheaf for the fppf topology we conclude that \(s = s'\).
Let \(\{T_j \to T\}\) be an fppf covering in \((\Sch/S)_{fppf}\) and let \(s_j \in F(T_j)\) such that \(s_j|_{T_j \times_T T_{j'}} = s_{j'}|_{T_j \times_T T_{j'}}\). By assumption (2)(c) we may refine the covering and assume that \(s_j \in F_{\alpha(j)}(T_j)\) for some \(\alpha(j) \in I\). Let \(W_j \subset T\) be the image of \(T_j \to T\) which is an open subscheme Morphisms, Lemma 01UA. Then \(\{T_j \to W_j\}\) is an fppf covering. Since \(F_{\alpha(j)}\) is a sub presheaf of \(F\) we see that the two restrictions of \(s_j\) to \(T_j \times_{W_j} T_j\) agree as elements of \(F_{\alpha(j)}(T_j \times_{W_j} T_j)\). Hence, the sheaf condition for \(F_{\alpha(j)}\) implies there exists a \(s'_j \in F_{\alpha(j)}(W_j)\) whose restriction to \(T_j\) is \(s_j\). For a pair of indices \(j\) and \(j'\) the sections \(s'_j|_{W_j \cap W_{j'}}\) and \(s'_{j'}|_{W_j \cap W_{j'}}\) of \(F\) agree by the result of the previous paragraph. This finishes the proof by the fact that \(F\) is a Zariski sheaf.
Properties of maps of presheaves representable by algebraic spaces
Here is the definition that makes this work.
Definition
Let \(S\) be a scheme. Let \(a : F \to G\) be a map of presheaves on \((\Sch/S)_{fppf}\) which is representable by algebraic spaces. Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces which
is preserved under any base change, and
is fppf local on the base, see Descent on Spaces, Definition 03YH.
In this case we say that \(a\) has property \(\mathcal{P}\) if for every scheme \(U\) and \(\xi : U \to G\) the resulting morphism of algebraic spaces \(U \times_G F \to U\) has property \(\mathcal{P}\).
It is important to note that we will only use this definition for properties of morphisms that are stable under base change, and local in the fppf topology on the base. This is not because the definition doesn’t make sense otherwise; rather it is because we may want to give a different definition which is better suited to the property we have in mind.
The definition above applies1 for example to the properties of being “surjective”, “quasi-compact”, “étale”, “smooth”, “flat”, “separated”, “(locally) of finite type”, “(locally) quasi-finite”, “(locally) of finite presentation”, “affine”, “proper”, and “a closed immersion”. In other words, \(a\) is surjective (resp. quasi-compact, étale, smooth, flat, separated, (locally) of finite type, (locally) quasi-finite, (locally) of finite presentation, proper, a closed immersion) if for every scheme \(T\) and map \(\xi : T \to G\) the morphism of algebraic spaces \(T \times_{\xi, G} F \to T\) is surjective (resp. quasi-compact, étale, flat, separated, (locally) of finite type, (locally) quasi-finite, (locally) of finite presentation, proper, a closed immersion).
Next, we check consistency with the already existing notions. By Lemma 03BN any morphism between algebraic spaces over \(S\) is representable by algebraic spaces. And by Morphisms of Spaces, Lemma 03MF (resp. 03KG, 03XT, 03ZF, 03MM, 03KM, 040Y, 040Z, 0410, 03WG, 083R, 03M4) the definition of surjective (resp. quasi-compact, étale, smooth, flat, separated, (locally) of finite type, (locally) quasi-finite, (locally) of finite presentation, affine, proper, closed immersion) above agrees with the already existing definition of morphisms of algebraic spaces.
Some formal lemmas follow.
Lemma
Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property as in Definition 03XZ. Let \[\xymatrix{ G' \times_G F \ar[r] \ar[d]^{a'} & F \ar[d]^a \\ G' \ar[r] & G }\] be a fibre square of presheaves on \((\Sch/S)_{fppf}\). If \(a\) is representable by algebraic spaces and has \(\mathcal{P}\) so does \(a'\).
Proof
Omitted. Hint: This is formal.
Lemma
Let \(S\) be a scheme. Let \(\mathcal{P}\) be a property as in Definition 03XZ, and assume \(\mathcal{P}\) is stable under composition. Let \[\xymatrix{ F \ar[r]^a & G \ar[r]^b & H }\] be maps of presheaves on \((\Sch/S)_{fppf}\). If \(a\), \(b\) are representable by algebraic spaces and has \(\mathcal{P}\) so does \(b \circ a\).
Proof
Omitted. Hint: See Lemma 03Y1 and use stability under composition.
Lemma
Let \(S\) be a scheme. Let \(F_i, G_i : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\), \(i = 1, 2\). Let \(a_i : F_i \to G_i\), \(i = 1, 2\) be representable by algebraic spaces. Let \(\mathcal{P}\) be a property as in Definition 03XZ which is stable under composition. If \(a_1\) and \(a_2\) have property \(\mathcal{P}\) so does \(a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2\).
Proof
Note that the lemma makes sense by Lemma 046D. Proof omitted.
Lemma
Let \(S\) be a scheme. Let \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\) be a transformation of functors representable by algebraic spaces. Let \(\mathcal{P}\), \(\mathcal{P}'\) be properties as in Definition 03XZ. Suppose that for any morphism \(f : X \to Y\) of algebraic spaces over \(S\) we have \(\mathcal{P}(f) \Rightarrow \mathcal{P}'(f)\). If \(a\) has property \(\mathcal{P}\), then \(a\) has property \(\mathcal{P}'\).
Proof
Formal.
Lemma
Let \(S\) be a scheme. Let \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be sheaves. Let \(a : F \to G\) be representable by algebraic spaces, flat, locally of finite presentation, and surjective. Then \(a : F \to G\) is surjective as a map of sheaves.
Proof
Let \(T\) be a scheme over \(S\) and let \(g : T \to G\) be a \(T\)-valued point of \(G\). By assumption \(T' = F \times_G T\) is an algebraic space and the morphism \(T' \to T\) is a flat, locally of finite presentation, and surjective morphism of algebraic spaces. Let \(U \to T'\) be a surjective étale morphism, where \(U\) is a scheme. Then by the definition of flat morphisms of algebraic spaces the morphism of schemes \(U \to T\) is flat. Similarly for “locally of finite presentation”. The morphism \(U \to T\) is surjective also, see Morphisms of Spaces, Lemma 03MF. Hence we see that \(\{U \to T\}\) is an fppf covering such that \(g|_U \in G(U)\) comes from an element of \(F(U)\), namely the map \(U \to T' \to F\). This proves the map is surjective as a map of sheaves, see Sites, Definition 00WM.
Bootstrapping the diagonal
In this section we prove that the diagonal of a sheaf \(F\) on \((\Sch/S)_{fppf}\) is representable as soon as there exists an “fppf cover” of \(F\) by a scheme or by an algebraic space, see Lemma 046K.
Lemma
Let \(S\) be a scheme. If \(F\) is a presheaf on \((\Sch/S)_{fppf}\). The following are equivalent:
\(\Delta_F : F \to F \times F\) is representable by algebraic spaces,
for every scheme \(T\) any map \(T \to F\) is representable by algebraic spaces, and
for every algebraic space \(X\) any map \(X \to F\) is representable by algebraic spaces.
Proof
Assume (1). Let \(X \to F\) be as in (3). Let \(T\) be a scheme, and let \(T \to F\) be a morphism. Then we have \[T \times_F X = (T \times_S X) \times_{F \times F, \Delta} F\] which is an algebraic space by Lemma 03XY and (1). Hence \(X \to F\) is representable, i.e., (3) holds. The implication (3) \(\Rightarrow\) (2) is trivial. Assume (2). Let \(T\) be a scheme, and let \((a, b) : T \to F \times F\) be a morphism. Then \[F \times_{\Delta_F, F \times F} T = (T \times_{a, F, b} T) \times_{T \times T, \Delta_T} T\] which is an algebraic space by assumption. Hence \(\Delta_F\) is representable by algebraic spaces, i.e., (1) holds.
In particular if \(F\) is a presheaf satisfying the equivalent conditions of the lemma, then for any morphism \(X \to F\) where \(X\) is an algebraic space it makes sense to say that \(X \to F\) is surjective (resp. étale, flat, locally of finite presentation) by using Definition 03XZ.
Before we actually do the bootstrap we prove a fun lemma.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ E \ar[r]_a \ar[d]_f & F \ar[d]^g \\ H \ar[r]^b & G }\] be a cartesian diagram of sheaves on \((\Sch/S)_{fppf}\), so \(E = H \times_G F\). If
\(g\) is representable by algebraic spaces, surjective, flat, and locally of finite presentation, and
\(a\) is representable by algebraic spaces, separated, and locally quasi-finite
then \(b\) is representable (by schemes) as well as separated and locally quasi-finite.
Proof
Let \(T\) be a scheme, and let \(T \to G\) be a morphism. We have to show that \(T \times_G H\) is a scheme, and that the morphism \(T \times_G H \to T\) is separated and locally quasi-finite. Thus we may base change the whole diagram to \(T\) and assume that \(G\) is a scheme. In this case \(F\) is an algebraic space. Let \(U\) be a scheme, and let \(U \to F\) be a surjective étale morphism. Then \(U \to F\) is representable, surjective, flat and locally of finite presentation by Morphisms of Spaces, Lemmas 0467 and 0468. By Lemma 03Y1 \(U \to G\) is surjective, flat and locally of finite presentation also. Note that the base change \(E \times_F U \to U\) of \(a\) is still separated and locally quasi-finite (by Lemma 046F). Hence we may replace the upper part of the diagram of the lemma by \(E \times_F U \to U\). In other words, we may assume that \(F \to G\) is a surjective, flat morphism of schemes which is locally of finite presentation. In particular, \(\{F \to G\}\) is an fppf covering of schemes. By Morphisms of Spaces, Proposition 03XX we conclude that \(E\) is a scheme also. By Descent, Lemma 02W5 the fact that \(E = H \times_G F\) means that we get a descent datum on \(E\) relative to the fppf covering \(\{F \to G\}\). By More on Morphisms, Lemma 02W8 this descent datum is effective. By Descent, Lemma 02W5 again this implies that \(H\) is a scheme. By Descent, Lemmas 02KU and 02VI it now follows that \(b\) is separated and locally quasi-finite.
Here is the result that the section title refers to.
Lemma
Let \(S\) be a scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Assume that
the presheaf \(F\) is a sheaf,
there exists an algebraic space \(X\) and a map \(X \to F\) which is representable by algebraic spaces, surjective, flat and locally of finite presentation.
Then \(\Delta_F\) is representable (by schemes).
Proof
Let \(U \to X\) be a surjective étale morphism from a scheme towards \(X\). Then \(U \to X\) is representable, surjective, flat and locally of finite presentation by Morphisms of Spaces, Lemmas 0467 and 0468. By Lemma 046G the composition \(U \to F\) is representable by algebraic spaces, surjective, flat and locally of finite presentation also. Thus we see that \(R = U \times_F U\) is an algebraic space, see Lemma 03XY. The morphism of algebraic spaces \(R \to U \times_S U\) is a monomorphism, hence separated (as the diagonal of a monomorphism is an isomorphism, see Morphisms of Spaces, Lemma 042M). Since \(U \to F\) is locally of finite presentation, both morphisms \(R \to U\) are locally of finite presentation, see Lemma 046F. Hence \(R \to U \times_S U\) is locally of finite type (use Morphisms of Spaces, Lemmas 0464 and 0462). Altogether this means that \(R \to U \times_S U\) is a monomorphism which is locally of finite type, hence a separated and locally quasi-finite morphism, see Morphisms of Spaces, Lemma 0463.
Now we are ready to prove that \(\Delta_F\) is representable. Let \(T\) be a scheme, and let \((a, b) : T \to F \times F\) be a morphism. Set \[T' = (U \times_S U) \times_{F \times F} T.\] Note that \(U \times_S U \to F \times F\) is representable by algebraic spaces, surjective, flat and locally of finite presentation by Lemma 046H. Hence \(T'\) is an algebraic space, and the projection morphism \(T' \to T\) is surjective, flat, and locally of finite presentation. Consider \(Z = T \times_{F \times F} F\) (this is a sheaf) and \[Z' = T' \times_{U \times_S U} R = T' \times_T Z.\] We see that \(Z'\) is an algebraic space, and \(Z' \to T'\) is separated and locally quasi-finite by the discussion in the first paragraph of the proof which showed that \(R\) is an algebraic space and that the morphism \(R \to U \times_S U\) has those properties. Hence we may apply Lemma 046J to the diagram \[\xymatrix{ Z' \ar[r] \ar[d] & T' \ar[d] \\ Z \ar[r] & T }\] and we conclude.
Here is a variant of the result above.
Lemma
Let \(S\) be a scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Let \(X\) be a scheme and let \(X \to F\) be representable by algebraic spaces and locally quasi-finite. Then \(X \to F\) is representable (by schemes).
Proof
Let \(T\) be a scheme and let \(T \to F\) be a morphism. We have to show that the algebraic space \(X \times_F T\) is representable by a scheme. Consider the morphism \[X \times_F T \longrightarrow X \times_{\Spec(\mathbf{Z})} T\] Since \(X \times_F T \to T\) is locally quasi-finite, so is the displayed arrow (Morphisms of Spaces, Lemma 03XN). On the other hand, the displayed arrow is a monomorphism and hence separated (Morphisms of Spaces, Lemma 042N). Thus \(X \times_F T\) is a scheme by Morphisms of Spaces, Proposition 03XX.
Bootstrap
We warn the reader right away that the result of this section will be superseded by the stronger Theorem 04S6. On the other hand, the theorem in this section is quite a bit easier to prove and still provides quite a bit of insight into how things work, especially for those readers mainly interested in Deligne-Mumford stacks.
In Spaces, Section 025X we defined an algebraic space as a sheaf in the fppf topology whose diagonal is representable, and such that there exist a surjective étale morphism from a scheme towards it. In this section we show that a sheaf in the fppf topology whose diagonal is representable by algebraic spaces and which has an étale surjective covering by an algebraic space is also an algebraic space. In other words, the category of algebraic spaces is an enlargement of the category of schemes by those fppf sheaves \(F\) which have a representable diagonal and an étale covering by a scheme. The result of this section says that doing the same process again starting with the category of algebraic spaces, does not lead to yet another category.
Another motivation for the material in this section is that it will guarantee later that a Deligne-Mumford stack whose inertia stack is trivial is equivalent to an algebraic space, see Algebraic Stacks, Lemma 045H.
Here is the main result of this section (as we mentioned above this will be superseded by the stronger Theorem 04S6).
Theorem
Let \(S\) be a scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Assume that
the presheaf \(F\) is a sheaf,
the diagonal morphism \(F \to F \times F\) is representable by algebraic spaces, and
there exists an algebraic space \(X\) and a map \(X \to F\) which is surjective, and étale.
or assume that
the presheaf \(F\) is a sheaf, and
there exists an algebraic space \(X\) and a map \(X \to F\) which is representable by algebraic spaces, surjective, and étale.
Then \(F\) is an algebraic space.
Proof
We will use the remarks directly below Definition 03XZ without further mention.
Assume (1), (2), and (3) and let \(X \to F\) be as in (3). By Lemma 03Y2 the morphism \(X \to F\) is representable by algebraic spaces. Thus we see that (a) and (b) hold.
Assume (a) and (b) and let \(X \to F\) be as in (b). Let \(U \to X\) be a surjective étale morphism from a scheme towards \(X\). By Lemma 03Y1 the transformation \(U \to F\) is representable by algebraic spaces, surjective, and étale. Hence to prove that \(F\) is an algebraic space boils down to proving that \(\Delta_F\) is representable (Spaces, Definition 025Y). This follows immediately from Lemma 046K. On the other hand we can circumvent this lemma and show directly \(F\) is an algebraic space as in the next paragraph.
Namely, let \(U\) be a scheme and let \(U \to F\) be representable by algebraic spaces, surjective, and étale. Consider the fibre product \(R = U \times_F U\). Both projections \(R \to U\) are representable by algebraic spaces, surjective, and étale (Lemma 046F). In particular \(R\) is an algebraic space by Lemma 02YS. The morphism of algebraic spaces \(R \to U \times_S U\) is a monomorphism, hence separated (as the diagonal of a monomorphism is an isomorphism). Since \(R \to U\) is étale, we see that \(R \to U\) is locally quasi-finite, see Morphisms of Spaces, Lemma 03XU. We conclude that also \(R \to U \times_S U\) is locally quasi-finite by Morphisms of Spaces, Lemma 03XN. Hence Morphisms of Spaces, Proposition 03XX applies and \(R\) is a scheme. By Lemma 04S1 the map \(U \to F\) is a surjection of sheaves. Thus \(F = U/R\). We conclude that \(F\) is an algebraic space by Spaces, Theorem 02WW.
Finding opens
First we prove a lemma which is a slight improvement and generalization of Spaces, Lemma 02WU to quotient sheaves associated to groupoids.
Lemma
Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(g : U' \to U\) be a morphism. Assume
the composition \[\xymatrix{ U' \times_{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^h & R \ar[r]_s & U }\] has an open image \(W \subset U\), and
the resulting map \(h : U' \times_{g, U, t} R \to W\) defines a surjection of sheaves in the fppf topology.
Let \(R' = R|_{U'}\) be the restriction of \(R\) to \(U'\). Then the map of quotient sheaves \[U'/R' \to U/R\] in the fppf topology is representable, and is an open immersion.
Proof
Note that \(W\) is an \(R\)-invariant open subscheme of \(U\). This is true because the set of points of \(W\) is the set of points of \(U\) which are equivalent in the sense of Groupoids, Lemma 022Q to a point of \(g(U') \subset U\) (the lemma applies as \(j : R \to U \times_S U\) is a pre-equivalence relation by Groupoids, Lemma 0232). Also \(g : U' \to U\) factors through \(W\). Let \(R|_W\) be the restriction of \(R\) to \(W\). Then it follows that \(R'\) is also the restriction of \(R|_W\) to \(U'\). Hence we can factor the map of sheaves of the lemma as \[U'/R' \longrightarrow W/R|_W \longrightarrow U/R\] By Groupoids, Lemma 02VH we see that the first arrow is an isomorphism of sheaves. Hence it suffices to show the lemma in case \(g\) is the immersion of an \(R\)-invariant open into \(U\).
Assume \(U' \subset U\) is an \(R\)-invariant open and \(g\) is the inclusion morphism. Set \(F = U/R\) and \(F' = U'/R'\). By Groupoids, Lemma 045Z or 02VH the map \(F' \to F\) is injective. Let \(\xi \in F(T)\). We have to show that \(T \times_{\xi, F} F'\) is representable by an open subscheme of \(T\). There exists an fppf covering \(\{f_i : T_i \to T\}\) such that \(\xi|_{T_i}\) is the image via \(U \to U/R\) of a morphism \(a_i : T_i \to U\). Set \(V_i = a_i^{-1}(U')\). We claim that \(V_i \times_T T_j = T_i \times_T V_j\) as open subschemes of \(T_i \times_T T_j\).
As \(a_i \circ \text{pr}_0\) and \(a_j \circ \text{pr}_1\) are morphisms \(T_i \times_T T_j \to U\) which both map to the section \(\xi|_{T_i \times_T T_j} \in F(T_i \times_T T_j)\) we can find an fppf covering \(\{f_{ijk} : T_{ijk} \to T_i \times_T T_j\}\) and morphisms \(r_{ijk} : T_{ijk} \to R\) such that \[a_i \circ \text{pr}_0 \circ f_{ijk} = s \circ r_{ijk}, \quad a_j \circ \text{pr}_1 \circ f_{ijk} = t \circ r_{ijk},\] see Groupoids, Lemma 045Y. Since \(U'\) is \(R\)-invariant we have \(s^{-1}(U') = t^{-1}(U')\) and hence \(f_{ijk}^{-1}(V_i \times_T T_j) = f_{ijk}^{-1}(T_i \times_T V_j)\). As \(\{f_{ijk}\}\) is surjective this implies the claim above. Hence by Descent, Lemma 03N0 there exists an open subscheme \(V \subset T\) such that \(f_i^{-1}(V) = V_i\). We claim that \(V\) represents \(T \times_{\xi, F} F'\).
As a first step, we will show that \(\xi|_V\) lies in \(F'(V) \subset F(V)\). Namely, the family of morphisms \(\{V_i \to V\}\) is an fppf covering, and by construction we have \(\xi|_{V_i} \in F'(V_i)\). Hence by the sheaf property of \(F'\) we get \(\xi|_V \in F'(V)\). Finally, let \(T' \to T\) be a morphism of schemes and that \(\xi|_{T'} \in F'(T')\). To finish the proof we have to show that \(T' \to T\) factors through \(V\). We can find a fppf covering \(\{T'_j \to T'\}_{j \in J}\) and morphisms \(b_j : T'_j \to U'\) such that \(\xi|_{T'_j}\) is the image via \(U' \to U/R\) of \(b_j\). Clearly, it is enough to show that the compositions \(T'_j \to T\) factor through \(V\). Hence we may assume that \(\xi|_{T'}\) is the image of a morphism \(b : T' \to U'\). Now, it is enough to show that \(T'\times_T T_i \to T_i\) factors through \(V_i\). Over the scheme \(T' \times_T T_i\) the restriction of \(\xi\) is the image of two elements of \((U/R)(T' \times_T T_i)\), namely \(a_i \circ \text{pr}_1\), and \(b \circ \text{pr}_0\), the second of which factors through the \(R\)-invariant open \(U'\). Hence by Groupoids, Lemma 045Y there exists a covering \(\{h_k : Z_k \to T' \times_T T_i\}\) and morphisms \(r_k : Z_k \to R\) such that \(a_i \circ \text{pr}_1 \circ h_k = s \circ r_k\) and \(b \circ \text{pr}_0 \circ h_k = t \circ r_k\). As \(U'\) is an \(R\)-invariant open the fact that \(b\) has image in \(U'\) then implies that each \(a_i \circ \text{pr}_1 \circ h_k\) has image in \(U'\). It follows from this that \(T' \times_T T_i \to T_i\) has image in \(V_i\) by definition of \(V_i\) which concludes the proof.
Slicing equivalence relations
In this section we explain how to “improve” a given equivalence relation by slicing. This is not a kind of “étale slicing” that you may be used to but a much coarser kind of slicing.
Lemma
Let \(S\) be a scheme. Let \(j : R \to U \times_S U\) be an equivalence relation on schemes over \(S\). Assume \(s, t : R \to U\) are flat and locally of finite presentation. Then there exists an equivalence relation \(j' : R' \to U'\times_S U'\) on schemes over \(S\), and an isomorphism \[U'/R' \longrightarrow U/R\] induced by a morphism \(U' \to U\) which maps \(R'\) into \(R\) such that \(s', t' : R \to U\) are flat, locally of finite presentation and locally quasi-finite.
Proof
We will prove this lemma in several steps. We will use without further mention that an equivalence relation gives rise to a groupoid scheme and that the restriction of an equivalence relation is an equivalence relation, see Groupoids, Lemmas 02V8, 0233, and 02VD.
Step 1: We may assume that \(s, t : R \to U\) are locally of finite presentation and Cohen-Macaulay morphisms. Namely, as in More on Groupoids, Lemma 0460 let \(g : U' \to U\) be the open subscheme such that \(t^{-1}(U') \subset R\) is the maximal open over which \(s : R \to U\) is Cohen-Macaulay, and denote \(R'\) the restriction of \(R\) to \(U'\). By the lemma cited above we see that \[\xymatrix{ t^{-1}(U') \ar@{=}[r] & U' \times_{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^h & R \ar[r]_s & U }\] is surjective. Since \(h\) is flat and locally of finite presentation, we see that \(\{h\}\) is a fppf covering. Hence by Groupoids, Lemma 02VH we see that \(U'/R' \to U/R\) is an isomorphism. By the construction of \(U'\) we see that \(s', t'\) are Cohen-Macaulay and locally of finite presentation.
Step 2. Assume \(s, t\) are Cohen-Macaulay and locally of finite presentation. Let \(u \in U\) be a point of finite type. By More on Groupoids, Lemma 04N0 there exists an affine scheme \(U'\) and a morphism \(g : U' \to U\) such that
\(g\) is an immersion,
\(u \in U'\),
\(g\) is locally of finite presentation,
\(h\) is flat, locally of finite presentation and locally quasi-finite, and
the morphisms \(s', t' : R' \to U'\) are flat, locally of finite presentation and locally quasi-finite.
Here we have used the notation introduced in More on Groupoids, Situation 04MY.
Step 3. For each point \(u \in U\) which is of finite type choose a \(g_u : U'_u \to U\) as in Step 2 and denote \(R'_u\) the restriction of \(R\) to \(U'_u\). Denote \(h_u = s \circ \text{pr}_1 : U'_u \times_{g_u, U, t} R \to U\). Set \(U' = \coprod_{u \in U} U'_u\), and \(g = \coprod g_u\). Let \(R'\) be the restriction of \(R\) to \(U'\) as above. We claim that the pair \((U', g)\) works2. Note that \[\begin{align*} R' = & \coprod\nolimits_{u_1, u_2 \in U} (U'_{u_1} \times_{g_{u_1}, U, t} R) \times_R (R \times_{s, U, g_{u_2}} U'_{u_2}) \\ = & \coprod\nolimits_{u_1, u_2 \in U} (U'_{u_1} \times_{g_{u_1}, U, t} R) \times_{h_{u_1}, U, g_{u_2}} U'_{u_2} \end{align*}\] Hence the projection \(s' : R' \to U' = \coprod U'_{u_2}\) is flat, locally of finite presentation and locally quasi-finite as a base change of \(\coprod h_{u_1}\). Finally, by construction the morphism \(h : U' \times_{g, U, t} R \to U\) is equal to \(\coprod h_u\) hence its image contains all points of finite type of \(U\). Since each \(h_u\) is flat and locally of finite presentation we conclude that \(h\) is flat and locally of finite presentation. In particular, the image of \(h\) is open (see Morphisms, Lemma 01UA) and since the set of points of finite type is dense (see Morphisms, Lemma 02J4) we conclude that the image of \(h\) is \(U\). This implies that \(\{h\}\) is an fppf covering. By Groupoids, Lemma 02VH this means that \(U'/R' \to U/R\) is an isomorphism. This finishes the proof of the lemma.
Quotient by a subgroupoid
We need one more lemma before we can do our final bootstrap. Let us discuss what is going on in terms of “plain” groupoids before embarking on the scheme theoretic version.
Let \(\mathcal{C}\) be a groupoid, see Categories, Definition 0018. As discussed in Groupoids, Section 0230 this corresponds to a quintuple \((\text{Ob}, \text{Arrows}, s, t, c)\). Suppose we are given a subset \(P \subset \text{Arrows}\) such that \((\text{Ob}, P, s|_P, t|_P, c|_P)\) is also a groupoid and such that there are no nontrivial automorphisms in \(P\). Then we can construct the quotient groupoid \((\overline{\text{Ob}}, \overline{\text{Arrows}}, \overline{s}, \overline{t}, \overline{c})\) as follows:
\(\overline{\text{Ob}} = \text{Ob}/P\) is the set of \(P\)-isomorphism classes,
\(\overline{\text{Arrows}} = P\backslash \text{Arrows}/P\) is the set of arrows in \(\mathcal{C}\) up to pre-composing and post-composing by arrows of \(P\),
the source and target maps \(\overline{s}, \overline{t} : P\backslash \text{Arrows}/P \to \text{Ob}/P\) are induced by \(s, t\),
composition is defined by the rule \(\overline{c}(\overline{a}, \overline{b}) = \overline{c(a, b)}\) which is well defined.
In fact, it turns out that the original groupoid \((\text{Ob}, \text{Arrows}, s, t, c)\) is canonically isomorphic to the restriction (see discussion in Groupoids, Section 02VA) of the groupoid \((\overline{\text{Ob}}, \overline{\text{Arrows}}, \overline{s}, \overline{t}, \overline{c})\) via the quotient map \(g : \text{Ob} \to \overline{\text{Ob}}\). Recall that this means that \[\text{Arrows} = \text{Ob} \times_{g, \overline{\text{Ob}}, \overline{t}} \overline{\text{Arrows}} \times_{\overline{s}, \overline{\text{Ob}}, g} \text{Ob}\] which holds as \(P\) has no nontrivial automorphisms. We omit the details.
The following lemma holds in much greater generality, but this is the version we use in the proof of the final bootstrap (after which we can more easily prove the more general versions of this lemma).
Lemma
Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(P \to R\) be monomorphism of schemes. Assume that
\((U, P, s|_P, t|_P, c|_{P \times_{s, U, t}P})\) is a groupoid scheme,
\(s|_P, t|_P : P \to U\) are finite locally free,
\(j|_P : P \to U \times_S U\) is a monomorphism.
\(U\) is affine, and
\(j : R \to U \times_S U\) is separated and locally quasi-finite,
Then \(U/P\) is representable by an affine scheme \(\overline{U}\), the quotient morphism \(U \to \overline{U}\) is finite locally free, and \(P = U \times_{\overline{U}} U\). Moreover, \(R\) is the restriction of a groupoid scheme \((\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})\) on \(\overline{U}\) via the quotient morphism \(U \to \overline{U}\).
Proof
Conditions (1), (2), (3), and (4) and Groupoids, Proposition 03BM imply the affine scheme \(\overline{U}\) representing \(U/P\) exists, the morphism \(U \to \overline{U}\) is finite locally free, and \(P = U \times_{\overline{U}} U\). The identification \(P = U \times_{\overline{U}} U\) is such that \(t|_P = \text{pr}_0\) and \(s|_P = \text{pr}_1\), and such that composition is equal to \(\text{pr}_{02} : U \times_{\overline{U}} U \times_{\overline{U}} U \to U \times_{\overline{U}} U\). A product of finite locally free morphisms is finite locally free (see Spaces, Lemma 02WM and Morphisms, Lemmas 02KD and 02KC). To get \(\overline{R}\) we are going to descend the scheme \(R\) via the finite locally free morphism \(U \times_S U \to \overline{U} \times_S \overline{U}\). Namely, note that \[(U \times_S U) \times_{(\overline{U} \times_S \overline{U})} (U \times_S U) = P \times_S P\] by the above. Thus giving a descent datum (see Descent, Definition 023V) for \(R / U \times_S U / \overline{U} \times_S \overline{U}\) consists of an isomorphism \[\varphi : R \times_{(U \times_S U), t \times t} (P \times_S P) \longrightarrow (P \times_S P) \times_{s \times s, (U \times_S U)} R\] over \(P \times_S P\) satisfying a cocycle condition. We define \(\varphi\) on \(T\)-valued points by the rule \[\varphi : (r, (p, p')) \longmapsto ((p, p'), p^{-1} \circ r \circ p')\] where the composition is taken in the groupoid category \((U(T), R(T), s, t, c)\). This makes sense because for \((r, (p, p'))\) to be a \(T\)-valued point of the source of \(\varphi\) it needs to be the case that \(t(r) = t(p)\) and \(s(r) = t(p')\). Note that this map is an isomorphism with inverse given by \(((p, p'), r') \mapsto (p \circ r' \circ (p')^{-1}, (p, p'))\). To check the cocycle condition we have to verify that \(\varphi_{02} = \varphi_{12} \circ \varphi_{01}\) as maps over \[(U \times_S U) \times_{(\overline{U} \times_S \overline{U})} (U \times_S U) \times_{(\overline{U} \times_S \overline{U})} (U \times_S U) = (P \times_S P) \times_{s \times s, (U \times_S U), t \times t} (P \times_S P)\] By explicit calculation we see that \[\begin{matrix} \varphi_{02} & (r, (p_1, p_1'), (p_2, p_2')) & \mapsto & ((p_1, p_1'), (p_2, p_2'), (p_1 \circ p_2)^{-1} \circ r \circ (p_1' \circ p_2')) \\ \varphi_{01} & (r, (p_1, p_1'), (p_2, p_2')) & \mapsto & ((p_1, p_1'), p_1^{-1} \circ r \circ p_1', (p_2, p_2')) \\ \varphi_{12} & ((p_1, p_1'), r, (p_2, p_2')) & \mapsto & ((p_1, p_1'), (p_2, p_2'), p_2^{-1} \circ r \circ p_2') \end{matrix}\] (with obvious notation) which implies what we want. As \(j\) is separated and locally quasi-finite by (5) we may apply More on Morphisms, Lemma 02W8 to get a scheme \(\overline{R} \to \overline{U} \times_S \overline{U}\) and an isomorphism \[R \to \overline{R} \times_{(\overline{U} \times_S \overline{U})} (U \times_S U)\] which identifies the descent datum \(\varphi\) with the canonical descent datum on \(\overline{R} \times_{(\overline{U} \times_S \overline{U})} (U \times_S U)\), see Descent, Definition 023Z.
Since \(U \times_S U \to \overline{U} \times_S \overline{U}\) is finite locally free we conclude that \(R \to \overline{R}\) is finite locally free as a base change. Hence \(R \to \overline{R}\) is surjective as a map of sheaves on \((\Sch/S)_{fppf}\). Our choice of \(\varphi\) implies that given \(T\)-valued points \(r, r' \in R(T)\) these have the same image in \(\overline{R}\) if and only if \(p^{-1} \circ r \circ p'\) for some \(p, p' \in P(T)\). Thus \(\overline{R}\) represents the sheaf \[T \longmapsto \overline{R(T)} = P(T)\backslash R(T)/P(T)\] with notation as in the discussion preceding the lemma. Hence we can define the groupoid structure on \((\overline{U} = U/P, \overline{R} = P\backslash R/P)\) exactly as in the discussion of the “plain” groupoid case. It follows from this that \((U, R, s, t, c)\) is the pullback of this groupoid structure via the morphism \(U \to \overline{U}\). This concludes the proof.
Final bootstrap
The following result goes quite a bit beyond the earlier results.
Theorem
Let \(S\) be a scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Any one of the following conditions implies that \(F\) is an algebraic space:
\(F = U/R\) where \((U, R, s, t, c)\) is a groupoid in algebraic spaces over \(S\) such that \(s, t\) are flat and locally of finite presentation, and \(j = (t, s) : R \to U \times_S U\) is an equivalence relation,
\(F = U/R\) where \((U, R, s, t, c)\) is a groupoid scheme over \(S\) such that \(s, t\) are flat and locally of finite presentation, and \(j = (t, s) : R \to U \times_S U\) is an equivalence relation,
\(F\) is a sheaf and there exists an algebraic space \(U\) and a morphism \(U \to F\) which is representable by algebraic spaces, surjective, flat and locally of finite presentation,
\(F\) is a sheaf and there exists a scheme \(U\) and a morphism \(U \to F\) which is representable by algebraic spaces or schemes, surjective, flat and locally of finite presentation,
\(F\) is a sheaf, \(\Delta_F\) is representable by algebraic spaces, and there exists an algebraic space \(U\) and a morphism \(U \to F\) which is surjective, flat, and locally of finite presentation, or
\(F\) is a sheaf, \(\Delta_F\) is representable, and there exists a scheme \(U\) and a morphism \(U \to F\) which is surjective, flat, and locally of finite presentation.
Proof
Trivial observations: (6) is a special case of (5) and (4) is a special case of (3). We first prove that cases (5) and (3) reduce to case (1). Namely, by bootstrapping the diagonal Lemma 046K we see that (3) implies (5). In case (5) we set \(R = U \times_F U\) which is an algebraic space by assumption. Moreover, by assumption both projections \(s, t : R \to U\) are surjective, flat and locally of finite presentation. The map \(j : R \to U \times_S U\) is clearly an equivalence relation. By Lemma 04S1 the map \(U \to F\) is a surjection of sheaves. Thus \(F = U/R\) which reduces us to case (1).
Next, we show that (1) reduces to (2). Namely, let \((U, R, s, t, c)\) be a groupoid in algebraic spaces over \(S\) such that \(s, t\) are flat and locally of finite presentation, and \(j = (t, s) : R \to U \times_S U\) is an equivalence relation. Choose a scheme \(U'\) and a surjective étale morphism \(U' \to U\). Let \(R' = R|_{U'}\) be the restriction of \(R\) to \(U'\). By Groupoids in Spaces, Lemma 046P we see that \(U/R = U'/R'\). Since \(s', t' : R' \to U'\) are also flat and locally of finite presentation (see More on Groupoids in Spaces, Lemma 04RP) this reduces us to the case where \(U\) is a scheme. As \(j\) is an equivalence relation we see that \(j\) is a monomorphism. As \(s : R \to U\) is locally of finite presentation we see that \(j : R \to U \times_S U\) is locally of finite type, see Morphisms of Spaces, Lemma 0462. By Morphisms of Spaces, Lemma 0463 we see that \(j\) is locally quasi-finite and separated. Hence if \(U\) is a scheme, then \(R\) is a scheme by Morphisms of Spaces, Proposition 03XX. Thus we reduce to proving the theorem in case (2).
Assume \(F = U/R\) where \((U, R, s, t, c)\) is a groupoid scheme over \(S\) such that \(s, t\) are flat and locally of finite presentation, and \(j = (t, s) : R \to U \times_S U\) is an equivalence relation. By Lemma 0489 we reduce to that case where \(s, t\) are flat, locally of finite presentation, and locally quasi-finite. Let \(U = \bigcup_{i \in I} U_i\) be an affine open covering (with index set \(I\) of cardinality \(\leq\) than the size of \(U\) to avoid set theoretic problems later – most readers can safely ignore this remark). Let \((U_i, R_i, s_i, t_i, c_i)\) be the restriction of \(R\) to \(U_i\). It is clear that \(s_i, t_i\) are still flat, locally of finite presentation, and locally quasi-finite as \(R_i\) is the open subscheme \(s^{-1}(U_i) \cap t^{-1}(U_i)\) of \(R\) and \(s_i, t_i\) are the restrictions of \(s, t\) to this open. By Lemma 046M (or the simpler Spaces, Lemma 02WU) the map \(U_i/R_i \to U/R\) is representable by open immersions. Hence if we can show that \(F_i = U_i/R_i\) is an algebraic space, then \(\coprod_{i \in I} F_i\) is an algebraic space by Spaces, Lemma 02WQ. As \(U = \bigcup U_i\) is an open covering it is clear that \(\coprod F_i \to F\) is surjective. Thus it follows that \(U/R\) is an algebraic space, by Spaces, Lemma 02WR. In this way we reduce to the case where \(U\) is affine and \(s, t\) are flat, locally of finite presentation, and locally quasi-finite and \(j\) is an equivalence.
Assume \((U, R, s, t, c)\) is a groupoid scheme over \(S\), with \(U\) affine, such that \(s, t\) are flat, locally of finite presentation, and locally quasi-finite, and \(j\) is an equivalence relation. Choose \(u \in U\). We apply More on Groupoids in Spaces, Lemma 04S0 to \(u \in U, R, s, t, c\). We obtain an affine scheme \(U'\), an étale morphism \(g : U' \to U\), a point \(u' \in U'\) with \(\kappa(u) = \kappa(u')\) such that the restriction \(R' = R|_{U'}\) is quasi-split over \(u'\). Note that the image \(g(U')\) is open as \(g\) is étale and contains \(u\). Hence, repeatedly applying the lemma, we can find finitely many points \(u_i \in U\), \(i = 1, \ldots, n\), affine schemes \(U'_i\), étale morphisms \(g_i : U_i' \to U\), points \(u'_i \in U'_i\) with \(g(u'_i) = u_i\) such that (a) each restriction \(R'_i\) is quasi-split over some point in \(U'_i\) and (b) \(U = \bigcup_{i = 1, \ldots, n} g_i(U'_i)\). Now we rerun the last part of the argument in the preceding paragraph: Using Lemma 046M (or the simpler Spaces, Lemma 02WU) the map \(U'_i/R'_i \to U/R\) is representable by open immersions. If we can show that \(F_i = U'_i/R'_i\) is an algebraic space, then \(\coprod_{i \in I} F_i\) is an algebraic space by Spaces, Lemma 02WQ. As \(\{g_i : U'_i \to U\}\) is an étale covering it is clear that \(\coprod F_i \to F\) is surjective. Thus it follows that \(U/R\) is an algebraic space, by Spaces, Lemma 02WR. In this way we reduce to the case where \(U\) is affine and \(s, t\) are flat, locally of finite presentation, and locally quasi-finite, \(j\) is an equivalence, and \(R\) is quasi-split over \(u\) for some \(u \in U\).
Assume \((U, R, s, t, c)\) is a groupoid scheme over \(S\), with \(U\) affine, \(u \in U\) such that \(s, t\) are flat, locally of finite presentation, and locally quasi-finite and \(j = (t, s) : R \to U \times_S U\) is an equivalence relation and \(R\) is quasi-split over \(u\). Let \(P \subset R\) be a quasi-splitting of \(R\) over \(u\). By Lemma 04S4 we see that \((U, R, s, t, c)\) is the restriction of a groupoid \((\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})\) by a surjective finite locally free morphism \(U \to \overline{U}\) such that \(P = U \times_{\overline{U}} U\). Note that \(s\) admits a factorization \[R = U \times_{\overline{U}, \overline{t}} \overline{R} \times_{\overline{s}, \overline{U}} U \xrightarrow{\text{pr}_{23}} \overline{R} \times_{\overline{s}, \overline{U}} U \xrightarrow{\text{pr}_2} U\] The map \(\text{pr}_2\) is the base change of \(\overline{s}\), and the map \(\text{pr}_{23}\) is a base change of the surjective finite locally free map \(U \to \overline{U}\). Since \(s\) is flat, locally of finite presentation, and locally quasi-finite and since \(\text{pr}_{23}\) is surjective finite locally free (as a base change of such), we conclude that \(\text{pr}_2\) is flat, locally of finite presentation, and locally quasi-finite by Descent, Lemmas 036K and 036N and Morphisms, Lemma 0GWS. Since \(\text{pr}_2\) is the base change of the morphism \(\overline{s}\) by \(U \to \overline{U}\) and \(\{U \to \overline{U}\}\) is an fppf covering we conclude \(\overline{s}\) is flat, locally of finite presentation, and locally quasi-finite, see Descent, Lemmas 02L2, 02KY, and 02VI. The same goes for \(\overline{t}\). Consider the commutative diagram \[\xymatrix{ U \times_{\overline{U}} U \ar@{=}[r] \ar[rd] & P \ar[r] \ar[d] & R \ar[d] \\ & \overline{U} \ar[r]^{\overline{e}} & \overline{R} }\] It is a general fact about restrictions that the outer four corners form a cartesian diagram. By the equality we see the inner square is cartesian. Since \(P\) is open in \(R\) (by definition of a quasi-splitting) we conclude that \(\overline{e}\) is an open immersion by Descent, Lemma 02L3. An application of Groupoids, Lemma 045Z shows that \(U/R = \overline{U}/\overline{R}\). Hence we have reduced to the case where \((U, R, s, t, c)\) is a groupoid scheme over \(S\), with \(U\) affine, \(u \in U\) such that \(s, t\) are flat, locally of finite presentation, and locally quasi-finite and \(j = (t, s) : R \to U \times_S U\) is an equivalence relation and \(e : U \to R\) is an open immersion!
But of course, if \(e\) is an open immersion and \(s, t\) are flat and locally of finite presentation then the morphisms \(t, s\) are étale. For example you can see this by applying More on Groupoids, Lemma 04R9 which shows that \(\Omega_{R/U} = 0\) which in turn implies that \(s, t : R \to U\) is G-unramified (see Morphisms, Lemma 02G5), which in turn implies that \(s, t\) are étale (see Morphisms, Lemma 02GV). And if \(s, t\) are étale then finally \(U/R\) is an algebraic space by Spaces, Theorem 02WW.
Applications
As a first application we obtain the following fundamental fact: \[\fbox{A sheaf which is fppf locally an algebraic space is an algebraic space.}\] This is the content of the following lemma. Note that assumption (2) is equivalent to the condition that \(F|_{(\Sch/S_i)_{fppf}}\) is an algebraic space, see Spaces, Lemma 04SH. Assumption (3) is a set theoretic condition which may be ignored by those not worried about set theoretic questions.
Lemma
Let \(S\) be a scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Let \(\{S_i \to S\}_{i \in I}\) be a covering of \((\Sch/S)_{fppf}\). Assume that
\(F\) is a sheaf,
each \(F_i = h_{S_i} \times F\) is an algebraic space, and
\(\coprod_{i \in I} F_i\) is an algebraic space (see Spaces, Lemma 02WQ).
Then \(F\) is an algebraic space.
Proof
Consider the morphism \(\coprod F_i \to F\). This is the base change of \(\coprod S_i \to S\) via \(F \to S\). Hence it is representable, locally of finite presentation, flat and surjective by our definition of an fppf covering and Lemma 046F. Thus Theorem 04S6 applies to show that \(F\) is an algebraic space.
Here is a special case of Lemma 04SK where we do not need to worry about set theoretical issues.
Lemma
Let \(S\) be a scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Let \(\{S_i \to S\}_{i \in I}\) be a covering of \((\Sch/S)_{fppf}\). Assume that
\(F\) is a sheaf,
each \(F_i = h_{S_i} \times F\) is an algebraic space, and
the morphisms \(F_i \to S_i\) are of finite type.
Then \(F\) is an algebraic space.
Proof
We will use Lemma 04SK above. To do this we will show that the assumption that \(F_i\) is of finite type over \(S_i\) to prove that the set theoretic condition in the lemma is satisfied (after perhaps refining the given covering of \(S\) a bit). We suggest the reader skip the rest of the proof.
If \(S'_i \to S_i\) is a morphism of schemes then \[h_{S'_i} \times F = h_{S'_i} \times_{h_{S_i}} h_{S_i} \times F = h_{S'_i} \times_{h_{S_i}} F_i\] is an algebraic space of finite type over \(S'_i\), see Spaces, Lemma 02X2 and Morphisms of Spaces, Lemma 03XH. Thus we may refine the given covering. After doing this we may assume: (a) each \(S_i\) is affine, and (b) the cardinality of \(I\) is at most the cardinality of the set of points of \(S\). (Since to cover all of \(S\) it is enough that each point is in the image of \(S_i \to S\) for some \(i\).)
Since each \(S_i\) is affine and each \(F_i\) of finite type over \(S_i\) we conclude that \(F_i\) is quasi-compact. Hence by Properties of Spaces, Lemma 03H6 we can find an affine \(U_i \in \Ob((\Sch/S)_{fppf})\) and a surjective étale morphism \(U_i \to F_i\). The fact that \(F_i \to S_i\) is locally of finite type then implies that \(U_i \to S_i\) is locally of finite type, and in particular \(U_i \to S\) is locally of finite type. By Sets, Lemma 04T7 we conclude that \(\text{size}(U_i) \leq \text{size}(S)\). Since also \(|I| \leq \text{size}(S)\) we conclude that \(\coprod_{i \in I} U_i\) is isomorphic to an object of \((\Sch/S)_{fppf}\) by Sets, Lemma 000Q and the construction of \(\Sch\). This implies that \(\coprod F_i\) is an algebraic space by Spaces, Lemma 02WQ and we win.
As a second application we obtain \[\fbox{Any fppf descent datum for algebraic spaces is effective.}\] This holds modulo set theoretical difficulties; as an example result we offer the following lemma.
Lemma
Let \(S\) be a scheme. Let \(\{X_i \to X\}_{i \in I}\) be an fppf covering of algebraic spaces over \(S\).
Proof
Proof of (1). By Descent on Spaces, Lemma 0ADT this translates into the statement that an fppf sheaf \(F\) endowed with a map \(F \to X\) is an algebraic space provided that each \(F \times_X X_i\) is an algebraic space. The restriction on the cardinality of \(I\) implies that coproducts of algebraic spaces indexed by \(I\) are algebraic spaces, see Spaces, Lemma 02WQ and Sets, Lemma 000R. The morphism \[\coprod F \times_X X_i \longrightarrow F\] is representable by algebraic spaces (as the base change of \(\coprod X_i \to X\), see Lemma 03Y0), and surjective, flat, and locally of finite presentation (as the base change of \(\coprod X_i \to X\), see Lemma 046F). Hence part (1) follows from Theorem 04S6.
Proof of (2). First we apply Descent on Spaces, Lemma 0ADT to obtain an fppf sheaf \(F\) endowed with a map \(F \to X\) such that \(F \times_X X_i = Y_i\) for all \(i \in I\). Our goal is to show that \(F\) is an algebraic space. Choose a scheme \(U\) and a surjective étale morphism \(U \to X\). Then \(F' = U \times_X F \to F\) is representable, surjective, and étale as the base change of \(U \to X\). By Theorem 04S6 it suffices to show that \(F' = U \times_X F\) is an algebraic space. We may choose an fppf covering \(\{U_j \to U\}_{j \in J}\) where \(U_j\) is a scheme refining the fppf covering \(\{X_i \times_X U \to U\}_{i \in I}\), see Topologies on Spaces, Lemma 042T. Thus we get a map \(a : J \to I\) and for each \(j\) a morphism \(U_j \to X_{a(j)}\) over \(X\). Then we see that \(U_j \times_U F' = U_j \times_{X_{a(j)}} Y_{a(j)}\) is of finite type over \(U_j\). Hence \(F'\) is an algebraic space by Lemma 04U0.
Here is a different type of application.
Lemma
Let \(S\) be a scheme. Let \(a : F \to G\) and \(b : G \to H\) be transformations of functors \((\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Assume
\(F, G, H\) are sheaves,
\(a : F \to G\) is representable by algebraic spaces, flat, locally of finite presentation, and surjective, and
\(b \circ a : F \to H\) is representable by algebraic spaces.
Then \(b\) is representable by algebraic spaces.
Proof
Let \(U\) be a scheme over \(S\) and let \(\xi \in H(U)\). We have to show that \(U \times_{\xi, H} G\) is an algebraic space. On the other hand, we know that \(U \times_{\xi, H} F\) is an algebraic space and that \(U \times_{\xi, H} F \to U \times_{\xi, H} G\) is representable by algebraic spaces, flat, locally of finite presentation, and surjective as a base change of the morphism \(a\) (see Lemma 046F). Thus the result follows from Theorem 04S6.
Lemma
Assume \(B \to S\) and \((U, R, s, t, c)\) are as in Groupoids in Spaces, Definition 044Q (1). For any scheme \(T\) over \(S\) and objects \(x, y\) of \([U/R]\) over \(T\) the sheaf \(\mathit{Isom}(x, y)\) on \((\Sch/T)_{fppf}\) is an algebraic space.
Proof
By Groupoids in Spaces, Lemma 044W there exists an fppf covering \(\{T_i \to T\}_{i \in I}\) such that \(\mathit{Isom}(x, y)|_{(\Sch/T_i)_{fppf}}\) is an algebraic space for each \(i\). By Spaces, Lemma 04SH this means that each \(F_i = h_{S_i} \times \mathit{Isom}(x, y)\) is an algebraic space. Thus to prove the lemma we only have to verify the set theoretic condition that \(\coprod F_i\) is an algebraic space of Lemma 04SK above to conclude. To do this we use Spaces, Lemma 02WQ which requires showing that \(I\) and the \(F_i\) are not “too large”. We suggest the reader skip the rest of the proof.
Choose \(U' \in \Ob(\Sch/S)_{fppf}\) and a surjective étale morphism \(U' \to U\). Let \(R'\) be the restriction of \(R\) to \(U'\). Since \([U/R] = [U'/R']\) we may, after replacing \(U\) by \(U'\), assume that \(U\) is a scheme. (This step is here so that the fibre products below are over a scheme.)
Note that if we refine the covering \(\{T_i \to T\}\) then it remains true that each \(F_i\) is an algebraic space. Hence we may assume that each \(T_i\) is affine. Since \(T_i \to T\) is locally of finite presentation, this then implies that \(\text{size}(T_i) \leq \text{size}(T)\), see Sets, Lemma 04T7. We may also assume that the cardinality of the index set \(I\) is at most the cardinality of the set of points of \(T\) since to get a covering it suffices to check that each point of \(T\) is in the image. Hence \(|I| \leq \text{size}(T)\). Choose \(W \in \Ob((\Sch/S)_{fppf})\) and a surjective étale morphism \(W \to R\). Note that in the proof of Groupoids in Spaces, Lemma 044W we showed that \(F_i\) is representable by \(T_i \times_{(y_i, x_i), U \times_B U} R\) for some \(x_i, y_i : T_i \to U\). Hence now we see that \(V_i = T_i \times_{(y_i, x_i), U \times_B U} W\) is a scheme which comes with an étale surjection \(V_i \to F_i\). By Sets, Lemma 04T6 we see that \[\text{size}(V_i) \leq \max\{\text{size}(T_i), \text{size}(W)\} \leq \max\{\text{size}(T), \text{size}(W)\}\] Hence, by Sets, Lemma 000Q we conclude that \[\text{size}(\coprod\nolimits_{i \in I} V_i) \leq \max\{|I|, \text{size}(T), \text{size}(W)\}.\] Hence we conclude by our construction of \(\Sch\) that \(\coprod_{i \in I} V_i\) is isomorphic to an object \(V\) of \((\Sch/S)_{fppf}\). This verifies the hypothesis of Spaces, Lemma 02WQ and we win.
Lemma
Let \(S\) be a scheme. Consider an algebraic space \(F\) of the form \(F = U/R\) where \((U, R, s, t, c)\) is a groupoid in algebraic spaces over \(S\) such that \(s, t\) are flat and locally of finite presentation, and \(j = (t, s) : R \to U \times_S U\) is an equivalence relation. Then \(U \to F\) is surjective, flat, and locally of finite presentation.
Proof
This is almost but not quite a triviality. Namely, by Groupoids in Spaces, Lemma 046O and the fact that \(j\) is a monomorphism we see that \(R = U \times_F U\). Choose a scheme \(W\) and a surjective étale morphism \(W \to F\). As \(U \to F\) is a surjection of sheaves we can find an fppf covering \(\{W_i \to W\}\) and maps \(W_i \to U\) lifting the morphisms \(W_i \to F\). Then we see that \[W_i \times_F U = W_i \times_U U \times_F U = W_i \times_{U, t} R\] and the projection \(W_i \times_F U \to W_i\) is the base change of \(t : R \to U\) hence flat and locally of finite presentation, see Morphisms of Spaces, Lemmas 03MO and 03XR. Hence by Descent on Spaces, Lemmas 041W and 041T we see that \(U \to F\) is flat and locally of finite presentation. It is surjective by Spaces, Remark 02YN.
Lemma
Let \(S\) be a scheme. Let \(X \to B\) be a morphism of algebraic spaces over \(S\). Let \(G\) be a group algebraic space over \(B\) and let \(a : G \times_B X \to X\) be an action of \(G\) on \(X\) over \(B\). If
\(a\) is a free action, and
\(G \to B\) is flat and locally of finite presentation,
then \(X/G\) (see Groupoids in Spaces, Definition 044J) is an algebraic space, the morphism \(X \to X/G\) is surjective, flat, and locally of finite presentation, and \(X\) is an fppf \(G\)-torsor over \(X/G\).
Proof
The fact that \(X/G\) is an algebraic space is immediate from Theorem 04S6 and the definitions. Namely, \(X/G = X/R\) where \(R = G \times_B X\). The morphisms \(s, t : G \times_B X \to X\) are flat and locally of finite presentation (clear for \(s\) as a base change of \(G \to B\) and by symmetry using the inverse it follows for \(t\)) and the morphism \(j : G \times_B X \to X \times_B X\) is a monomorphism by Groupoids in Spaces, Lemma 06P9 as the action is free. The morphism \(X \to X/G\) is surjective, flat, and locally of finite presentation by Lemma 06PG. To see that \(X \to X/G\) is an fppf \(G\)-torsor (Groupoids in Spaces, Definition 04TY) we have to show that \(G \times_S X \to X \times_{X/G} X\) is an isomorphism and that \(X \to X/G\) fppf locally has sections. The second part is clear from the properties of \(X \to X/G\) already shown. The map \(G \times_S X \to X \times_{X/G} X\) is injective (as a map of fppf sheaves) as the action is free. Finally, the map is also surjective as a map of sheaves by Groupoids in Spaces, Lemma 046O. This finishes the proof.
Lemma
Let \(k\) be a field. Let \(G\) be a commutative group scheme locally of finite type over \(k\). Let \(G^0 \subset G\) be the subgroup scheme of Groupoids, Proposition 0B7R. The fppf quotient \[\pi_0(G) = G/G^0\] is an étale commutative group scheme over \(k\). The morphism \(G \to \pi_0(G)\) is an fppf \(G^0\)-torsor. If \(G\) is quasi-compact, then \(\pi_0(G)\) is finite étale over \(k\).
Proof
The scheme \(G^0\) is quasi-compact and locally of finite type over \(k\). Thus it is of finite type, and hence of finite presentation, over \(k\). It is flat over \(k\). Translation gives a free action of \(G^0\) on \(G\). Lemma 06PH shows that the quotient is an algebraic space and that \(G \to \pi_0(G)\) is an fppf \(G^0\)-torsor. The quotient sheaf is a commutative group sheaf, so the representing algebraic space is a commutative group algebraic space.
Let \(\overline{k}\) be an algebraic closure of \(k\). Formation of \(G^0\) commutes with extension of the ground field. Indeed, \(G^0 \to G\) is a flat closed immersion of finite presentation, hence an open and closed immersion by Morphisms, Lemma 0819, and \(G^0\) is geometrically irreducible. Thus \((G^0)_{\overline{k}}\) is the connected component of the identity of \(G_{\overline{k}}\). Every connected component of \(G_{\overline{k}}\) is open and contains an \(\overline{k}\)-rational point. Translation by such a point identifies that component with \((G^0)_{\overline{k}}\). Since formation of the quotient sheaf commutes with base change, we obtain \[\pi_0(G)_{\overline{k}} = \coprod_{C \in \pi_0(|G_{\overline{k}}|)} \Spec(\overline{k}).\] Consequently \(\pi_0(G)_{\overline{k}}\) is étale over \(\overline{k}\). By Descent on Spaces, Lemma 042B, the algebraic space \(\pi_0(G)\) is étale over \(k\). In particular it is locally quasi-finite, and hence it is a scheme by Algebraic Spaces over Fields, Lemma 06S0.
If \(G\) is quasi-compact, then so is its quotient \(\pi_0(G)\). The description in Algebraic Spaces over Fields, Lemma 06S0 now shows that \(\pi_0(G)\) is finite over \(k\).
Lemma
Let \(k\) be a field and let \(N\) be an étale commutative group scheme over \(k\). Set \(p = 1\) if \(k\) has characteristic zero and set \(p = \text{char}(k)\) otherwise. For \(n \geq 1\), write \(N[n] = \Ker([n] : N \to N)\) and set \[\begin{align*} N_{\mathrm{tors}} & = \bigcup_{n \geq 1} N[n], \\ N_{p'} & = \bigcup_{(n,p)=1} N[n], \\ N_p & = \bigcup_{h \geq 0} N[p^h]. \end{align*}\] These are open and closed subgroup schemes of \(N\), and \[N_{p'} \cap N_p = e, \qquad N_{p'}N_p = N_{\mathrm{tors}}.\] If \(G\) is as in Lemma bootstrap-lemma-component-group-scheme-over-field and \(q : G \to \pi_0(G)\) is the quotient, then \[\begin{align*} G^\tau & = q^{-1}(\pi_0(G)_{\mathrm{tors}}), \\ G^\sigma & = q^{-1}(\pi_0(G)_{p'}), \\ G^\rho & = q^{-1}(\pi_0(G)_p) \end{align*}\] are open and closed subgroup schemes of \(G\). They satisfy \[G^\sigma \cap G^\rho = G^0, \qquad G^\sigma G^\rho = G^\tau.\]
Proof
An étale scheme over a field is a disjoint union of spectra of finite separable field extensions, see Algebraic Spaces over Fields, Lemma 06S0. Hence a union of connected components of \(N\) is open and closed. Each of the three subfunctors displayed in the statement is a union of connected components and is stable under the group operations. It is therefore an open and closed subgroup scheme.
After extending the ground field to a separable closure, the assertions about intersections and products reduce to assertions about an abelian group. A torsion element has a unique sum decomposition into its prime-to-\(p\) part and its \(p\)-primary part. This also covers characteristic zero, where \(p=1\) and the \(p\)-primary subgroup is the identity. Thus the two equalities for \(N\) hold, and they descend to \(k\). Taking inverse images under \(q\) proves all the assertions for \(G\).
Remark
When a group scheme \(G\) represents a Picard functor, the étale group scheme \(\pi_0(G)\) of Lemma bootstrap-lemma-component-group-scheme-over-field is often called the reduced Néron–Severi scheme. If \((G^0)_{\mathrm{red}}\) is a subgroup scheme of \(G\), Lemma 06PH also represents \(G/(G^0)_{\mathrm{red}}\) by a commutative group algebraic space and makes \(G \to G/(G^0)_{\mathrm{red}}\) an fppf torsor.
Lemma
Let \(\{S_i \to S\}_{i \in I}\) be a covering of \((\Sch/S)_{fppf}\). Let \(G\) be a group algebraic space over \(S\), and denote \(G_i = G_{S_i}\) the base changes. Suppose given
for each \(i \in I\) an fppf \(G_i\)-torsor \(X_i\) over \(S_i\), and
for each \(i, j \in I\) a \(G_{S_i \times_S S_j}\)-equivariant isomorphism \(\varphi_{ij} : X_i \times_S S_j \to S_i \times_S X_j\) satisfying the cocycle condition over every \(S_i \times_S S_j \times_S S_j\).
Then there exists an fppf \(G\)-torsor \(X\) over \(S\) whose base change to \(S_i\) is isomorphic to \(X_i\) such that we recover the descent datum \(\varphi_{ij}\).
Proof
We may think of \(X_i\) as a sheaf on \((\Sch/S_i)_{fppf}\), see Spaces, Section 03I3. By Sites, Section 04TP the descent datum \((X_i, \varphi_{ij})\) is effective in the sense that there exists a unique sheaf \(X\) on \((\Sch/S)_{fppf}\) which recovers the algebraic spaces \(X_i\) after restricting back to \((\Sch/S_i)_{fppf}\). Hence we see that \(X_i = h_{S_i} \times X\). By Lemma 04SK we see that \(X\) is an algebraic space, modulo verifying that \(\coprod X_i\) is an algebraic space which we do at the end of the proof. By the equivalence of categories in Sites, Lemma 04TS the action maps \(G_i \times_{S_i} X_i \to X_i\) glue to give a map \(a : G \times_S X \to X\). Now we have to show that \(a\) is an action and that \(X\) is a pseudo-torsor, and fppf locally trivial (see Groupoids in Spaces, Definition 04TY). These may be checked fppf locally, and hence follow from the corresponding properties of the actions \(G_i \times_{S_i} X_i \to X_i\). Hence the lemma is true.
We suggest the reader skip the rest of the proof, which is purely set theoretical. Pick coverings \(\{S_{ij} \to S_j\}_{j \in J_i}\) of \((\Sch/S)_{fppf}\) which trivialize the \(G_i\) torsors \(X_i\) (possible by assumption, and Topologies, Lemma 03WX part (1)). Then \(\{S_{ij} \to S\}_{i \in I, j \in J_i}\) is a covering of \((\Sch/S)_{fppf}\) and hence we may assume that each \(X_i\) is the trivial torsor! Of course we may also refine the covering further, hence we may assume that each \(S_i\) is affine and that the index set \(I\) has cardinality bounded by the cardinality of the set of points of \(S\). Choose \(U \in \Ob((\Sch/S)_{fppf})\) and a surjective étale morphism \(U \to G\). Then we see that \(U_i = U \times_S S_i\) comes with an étale surjective morphism to \(X_i \cong G_i\). By Sets, Lemma 04T6 we see \(\text{size}(U_i) \leq \max\{\text{size}(U), \text{size}(S_i)\}\). By Sets, Lemma 04T7 we have \(\text{size}(S_i) \leq \text{size}(S)\). Hence we see that \(\text{size}(U_i) \leq \max\{\text{size}(U), \text{size}(S)\}\) for all \(i \in I\). Together with the bound on \(|I|\) we found above we conclude from Sets, Lemma 000Q that \(\text{size}(\coprod U_i) \leq \max\{\text{size}(U), \text{size}(S)\}\). Hence Spaces, Lemma 02WQ applies to show that \(\coprod X_i\) is an algebraic space which is what we had to prove.
Algebraic spaces in the étale topology
Let \(S\) be a scheme. Instead of working with sheaves over the big fppf site \((\Sch/S)_{fppf}\) we could work with sheaves over the big étale site \((\Sch/S)_\etale\). All of the material in Algebraic Spaces, Sections 025U and 02WI makes sense for sheaves over \((\Sch/S)_\etale\). Thus we get a second notion of algebraic spaces by working in the étale topology. This notion is (a priori) weaker then the notion introduced in Algebraic Spaces, Definition 025Y since a sheaf in the fppf topology is certainly a sheaf in the étale topology. However, the notions are equivalent as is shown by the following lemma.
Lemma
Denote the common underlying category of \(\Sch_{fppf}\) and \(\Sch_\etale\) by \(\Sch_\alpha\) (see Topologies, Remark 03FF). Let \(S\) be an object of \(\Sch_\alpha\). Let \[F : (\Sch_\alpha/S)^{opp} \longrightarrow \textit{Sets}\] be a presheaf with the following properties:
\(F\) is a sheaf for the étale topology,
the diagonal \(\Delta : F \to F \times F\) is representable, and
there exists \(U \in \Ob(\Sch_\alpha/S)\) and \(U \to F\) which is surjective and étale.
Then \(F\) is an algebraic space in the sense of Algebraic Spaces, Definition 025Y.
Proof
Note that properties (2) and (3) of the lemma and the corresponding properties (2) and (3) of Algebraic Spaces, Definition 025Y are independent of the topology. This is true because these properties involve only the notion of a fibre product of presheaves, maps of presheaves, the notion of a representable transformation of functors, and what it means for such a transformation to be surjective and étale. Thus all we have to prove is that an étale sheaf \(F\) with properties (2) and (3) is also an fppf sheaf.
To do this, let \(R = U \times_F U\). By (2) the presheaf \(R\) is representable by a scheme and by (3) the projections \(R \to U\) are étale. Thus \(j : R \to U \times_S U\) is an étale equivalence relation. Moreover \(U \to F\) identifies \(F\) as the quotient of \(U\) by \(R\) for the étale topology: (a) if \(T \to F\) is a morphism, then \(\{T \times_F U \to T\}\) is an étale covering, hence \(U \to F\) is a surjection of sheaves for the étale topology, (b) if \(a, b : T \to U\) map to the same section of \(F\), then \((a, b) : T \to R\) hence \(a\) and \(b\) have the same image in the quotient of \(U\) by \(R\) for the étale topology. Next, let \(U/R\) denote the quotient sheaf in the fppf topology which is an algebraic space by Spaces, Theorem 02WW. Thus we have morphisms (transformations of functors) \[U \to F \to U/R.\] By the aforementioned Spaces, Theorem 02WW the composition is representable, surjective, and étale. Hence for any scheme \(T\) and morphism \(T \to U/R\) the fibre product \(V = T \times_{U/R} U\) is a scheme surjective and étale over \(T\). In other words, \(\{V \to U\}\) is an étale covering. This proves that \(U \to U/R\) is surjective as a map of sheaves in the étale topology. It follows that \(F \to U/R\) is surjective as a map of sheaves in the étale topology. On the other hand, the map \(F \to U/R\) is injective (as a map of presheaves) since \(R = U \times_{U/R} U\) again by Spaces, Theorem 02WW. It follows that \(F \to U/R\) is an isomorphism of étale sheaves, see Sites, Lemma 00WN which concludes the proof.
There is also an analogue of Spaces, Lemma 0BGQ.
Lemma
Denote the common underlying category of \(\Sch_{fppf}\) and \(\Sch_\etale\) by \(\Sch_\alpha\) (see Topologies, Remark 03FF). Let \(S\) be an object of \(\Sch_\alpha\). Let \[F : (\Sch_\alpha/S)^{opp} \longrightarrow \textit{Sets}\] be a presheaf with the following properties:
\(F\) is a sheaf for the étale topology,
there exists an algebraic space \(U\) over \(S\) and a map \(U \to F\) which is representable by algebraic spaces, surjective, and étale.
Then \(F\) is an algebraic space in the sense of Algebraic Spaces, Definition 025Y.
Proof
Set \(R = U \times_F U\). This is an algebraic space as \(U \to F\) is assumed representable by algebraic spaces. The projections \(s, t : R \to U\) are étale morphisms of algebraic spaces as \(U \to F\) is assumed étale. The map \(j = (t, s) : R \to U \times_S U\) is a monomorphism and an equivalence relation as \(R = U \times_F U\). By Theorem 04S6 the fppf quotient sheaf \(F' = U/R\) is an algebraic space. The morphism \(U \to F'\) is surjective, flat, and locally of finite presentation by Lemma 06PG. The map \(R \to U \times_{F'} U\) is surjective as a map of fppf sheaves by Groupoids in Spaces, Lemma 046O and since \(j\) is a monomorphism it is an isomorphism. Hence the base change of \(U \to F'\) by \(U \to F'\) is étale, and we conclude that \(U \to F'\) is étale by Descent on Spaces, Lemma 042B. Thus \(U \to F'\) is surjective as a map of étale sheaves. This means that \(F'\) is equal to the quotient sheaf \(U/R\) in the étale topology (small check omitted). Hence we obtain a canonical factorization \(U \to F' \to F\) and \(F' \to F\) is an injective map of sheaves. On the other hand, \(U \to F\) is surjective as a map of étale sheaves and hence so is \(F' \to F\). This means that \(F' = F\) and the proof is complete.
In fact, it suffices to have a smooth cover by a scheme and it suffices to assume the diagonal is representable by algebraic spaces.
Lemma
Denote the common underlying category of \(\Sch_{fppf}\) and \(\Sch_\etale\) by \(\Sch_\alpha\) (see Topologies, Remark 03FF). Let \(S\) be an object of \(\Sch_\alpha\). \[F : (\Sch_\alpha/S)^{opp} \longrightarrow \textit{Sets}\] be a presheaf with the following properties:
\(F\) is a sheaf for the étale topology,
the diagonal \(\Delta : F \to F \times F\) is representable by algebraic spaces, and
there exists \(U \in \Ob(\Sch_\alpha/S)\) and \(U \to F\) which is surjective and smooth.
Then \(F\) is an algebraic space in the sense of Algebraic Spaces, Definition 025Y.
Proof
The proof mirrors the proof of Lemma 076M. Let \(R = U \times_F U\). By (2) the presheaf \(R\) is an algebraic space and by (3) the projections \(R \to U\) are smooth and surjective. Denote \((U, R, s, t, c)\) the groupoid associated to the equivalence relation \(j : R \to U \times_S U\) (see Groupoids in Spaces, Lemma 043Y). By Theorem 04S6 we see that \(X = U/R\) (quotient in the fppf-topology) is an algebraic space. Using that the smooth topology and the étale topology have the same sheaves (by More on Morphisms, Lemma 055V) we see the map \(U \to F\) identifies \(F\) as the quotient of \(U\) by \(R\) for the smooth topology (details omitted). Thus we have morphisms (transformations of functors) \[U \to F \to X.\] By Lemma 06PG we see that \(U \to X\) is surjective, flat and locally of finite presentation. By Groupoids in Spaces, Lemma 046O (and the fact that \(j\) is a monomorphism) we have \(R = U \times_X U\). By Descent on Spaces, Lemma 0429 we conclude that \(U \to X\) is smooth and surjective (as the projections \(R \to U\) are smooth and surjective and \(\{U \to X\}\) is an fppf covering). Hence for any scheme \(T\) and morphism \(T \to X\) the fibre product \(T \times_X U\) is an algebraic space surjective and smooth over \(T\). Choose a scheme \(V\) and a surjective étale morphism \(V \to T \times_X U\). Then \(\{V \to T\}\) is a smooth covering such that \(V \to T \to X\) lifts to a morphism \(V \to U\). This proves that \(U \to X\) is surjective as a map of sheaves in the smooth topology. It follows that \(F \to X\) is surjective as a map of sheaves in the smooth topology. On the other hand, the map \(F \to X\) is injective (as a map of presheaves) since \(R = U \times_X U\). It follows that \(F \to X\) is an isomorphism of smooth (\(=\) étale) sheaves, see Sites, Lemma 00WN which concludes the proof.
Finally, here is the analogue of Spaces, Lemma 0BGQ with a smooth morphism covering the space.
Lemma
Denote the common underlying category of \(\Sch_{fppf}\) and \(\Sch_\etale\) by \(\Sch_\alpha\) (see Topologies, Remark 03FF). Let \(S\) be an object of \(\Sch_\alpha\). Let \[F : (\Sch_\alpha/S)^{opp} \longrightarrow \textit{Sets}\] be a presheaf with the following properties:
\(F\) is a sheaf for the étale topology,
there exists an algebraic space \(U\) over \(S\) and a map \(U \to F\) which is representable by algebraic spaces, surjective, and smooth.
Then \(F\) is an algebraic space in the sense of Algebraic Spaces, Definition 025Y.
Proof
The proof is identical to the proof of Lemma 0BH4. Set \(R = U \times_F U\). This is an algebraic space as \(U \to F\) is assumed representable by algebraic spaces. The projections \(s, t : R \to U\) are smooth morphisms of algebraic spaces as \(U \to F\) is assumed smooth. The map \(j = (t, s) : R \to U \times_S U\) is a monomorphism and an equivalence relation as \(R = U \times_F U\). By Theorem 04S6 the fppf quotient sheaf \(F' = U/R\) is an algebraic space. The morphism \(U \to F'\) is surjective, flat, and locally of finite presentation by Lemma 06PG. The map \(R \to U \times_{F'} U\) is surjective as a map of fppf sheaves by Groupoids in Spaces, Lemma 046O and since \(j\) is a monomorphism it is an isomorphism. Hence the base change of \(U \to F'\) by \(U \to F'\) is smooth, and we conclude that \(U \to F'\) is smooth by Descent on Spaces, Lemma 0429. Thus \(U \to F'\) is surjective as a map of étale sheaves (as the smooth topology is equal to the étale topology by More on Morphisms, Lemma 055V). This means that \(F'\) is equal to the quotient sheaf \(U/R\) in the étale topology (small check omitted). Hence we obtain a canonical factorization \(U \to F' \to F\) and \(F' \to F\) is an injective map of sheaves. On the other hand, \(U \to F\) is surjective as a map of étale sheaves (as the smooth topology is the same as the étale topology) and hence so is \(F' \to F\). This means that \(F' = F\) and the proof is complete.
Being preserved under base change holds by Morphisms of Spaces, Lemmas 03MH, 03HF, 0466, 03ZE, 03MO, 03KL, 03XH, 03XL, 03XR, 03WI, 04WP, and Spaces, Lemma 02YW. Being fppf local on the base holds by Descent on Spaces, Lemmas 041Q, 041L, 042B, 0429, 041W, 0421, 041U, 0427, 041T, 041Z, 0422, and 0420.↩︎
Here we should check that \(U'\) is not too large, i.e., that it is isomorphic to an object of the category \(\Sch_{fppf}\), see Section 046C. This is a purely set theoretical matter; let us use the notion of size of a scheme introduced in Sets, Section 000H. Note that each \(U'_u\) has size at most the size of \(U\) and that the cardinality of the index set is at most the cardinality of \(|U|\) which is bounded by the size of \(U\). Hence \(U'\) is isomorphic to an object of \(\Sch_{fppf}\) by Sets, Lemma 000R part (6).↩︎
The restriction on countablility can be ignored by those who do not care about set theoretical issues. We can allow larger index sets here if we can bound the size of the algebraic spaces which we are descending. See for example Lemma 04U0.↩︎