Introduction
Algebraic spaces were first introduced by Michael Artin, see [ArtinI], [ArtinII], [Artin-Theorem-Representability], [Artin-Construction-Techniques], [Artin-Algebraic-Spaces], [Artin-Algebraic-Approximation], [Artin-Implicit-Function], and [ArtinVersal]. Some of the foundational material was developed jointly with Knutson, who produced the book [Kn]. Artin defined (see [Artin-Implicit-Function, Definition 1.3]) an algebraic space as a sheaf for the étale topology which is locally in the étale topology representable. In most of Artin’s work the categories of schemes considered are schemes locally of finite type over a fixed excellent Noetherian base.
Our definition is slightly different from Artin’s original definition. Namely, our algebraic spaces are sheaves for the fppf topology whose diagonal is representable and which have an étale “cover” by a scheme. Working with the fppf topology instead of the étale topology is just a technical point and scarcely makes any difference; we will show in Bootstrap, Section 076L that we would have gotten the same category of algebraic spaces if we had worked with the étale topology. In that same chapter we will prove that the condition on the diagonal can in some sense be removed, see Bootstrap, Section 03XV.
After defining algebraic spaces we make some foundational observations. The main result in this chapter is that with our definitions an algebraic space is the same thing as an étale equivalence relation, see the discussion in Section 0261 and Theorem 02WW. The analogue of this theorem in Artin’s setting is [Artin-Implicit-Function, Theorem 1.5], or [Kn, Proposition II.1.7]. In other words, the sheaf defined by an étale equivalence relation has a representable diagonal. It follows that our definition agrees with Artin’s original definition in a broad sense. It also means that one can give examples of algebraic spaces by simply writing down an étale equivalence relation.
In Section 02X3 we introduce various separation axioms on algebraic spaces that we have found in the literature. Finally in Section 02Z0 we give some weird and not so weird examples of algebraic spaces.
General remarks
We work in a suitable big fppf site \(\Sch_{fppf}\) as in Topologies, Definition 021R. So, if not explicitly stated otherwise all schemes will be objects of \(\Sch_{fppf}\). In Section 03FO we discuss what changes if you change the big fppf site.
We will always work relative to a base \(S\) contained in \(\Sch_{fppf}\). And we will then work with the big fppf site \((\Sch/S)_{fppf}\), see Topologies, Definition 021S. The absolute case can be recovered by taking \(S = \Spec(\mathbf{Z})\).
If \(U, T\) are schemes over \(S\), then we denote \(U(T)\) for the set of \(T\)-valued points over \(S\). In a formula: \(U(T) = \Mor_S(T, U)\).
Note that any fpqc covering is a universal effective epimorphism, see Descent, Lemma 023Q. Hence the topology on \(\Sch_{fppf}\) is weaker than the canonical topology and all representable presheaves are sheaves.
Representable morphisms of presheaves
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 a representable transformation of functors, see Categories, Definition 0023. This means that for every \(U \in \Ob((\Sch/S)_{fppf})\) and any \(\xi \in G(U)\) the fiber product \(h_U \times_{\xi, G} F\) is representable. Choose a representing object \(V_\xi\) and an isomorphism \(h_{V_\xi} \to h_U \times_G F\). By the Yoneda lemma, see Categories, Lemma 001P, the projection \(h_{V_\xi} \to h_U \times_G F \to h_U\) comes from a unique morphism of schemes \(a_\xi : V_\xi \to U\). Suggestively we could represent this by the diagram \[\xymatrix{ V_\xi \ar@{~>}[r] \ar[d]_{a_\xi} & h_{V_\xi} \ar[d] \ar[r] & F \ar[d]^a \\ U \ar@{~>}[r] & h_U \ar[r]^\xi & G }\] where the squiggly arrows represent the Yoneda embedding. Here are some lemmas about this notion that work in great generality.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\) and let \(X\), \(Y\) be objects of \((\Sch/S)_{fppf}\). Let \(f : X \to Y\) be a morphism of schemes. Then \[h_f : h_X \longrightarrow h_Y\] is a representable transformation of functors.
Proof
This is formal and relies only on the fact that the category \((\Sch/S)_{fppf}\) has fibre products.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\), \(b : G \to H\) be representable transformations of functors. Then \[b \circ a : F \longrightarrow H\] is a representable transformation of functors.
Proof
This is entirely formal and works in any category.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\) be a representable transformation of functors. Let \(b : H \to G\) be any transformation of functors. Consider the fibre product diagram \[\xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G }\] Then the base change \(a'\) is a representable transformation of functors.
Proof
This is entirely formal and works in any category.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). 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 transformations of functors. Then \[a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2\] is a representable transformation of functors.
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 02WA and 02WB.
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 a representable transformation of functors. If \(G\) is a sheaf, then so is \(F\).
Proof
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 representable presheaf and hence (see remarks in Section 025T) 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 a representable transformation of functors. Then \(\Delta_{F/G} : F \to F \times_G F\) is representable.
Proof
Let \(U \in \Ob((\Sch/S)_{fppf})\). 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 a scheme \(V\) and a morphism \(V \to U\) representing the fibre product \(h_U \times_{\xi', G} F\). 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 \(h_U \times_{\xi', G} F\) 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 \(h_U \times_{\xi, F \times_G F} F\) is represented by \(V \times_{\Delta, V \times V, (f_1, f_2)} U\) which is a scheme.
Lists of useful properties of morphisms of schemes
For ease of reference we list in the following remarks the properties of morphisms which possess some of the properties required of them in later results.
Remark
Here is a list of properties/types of morphisms which are stable under arbitrary base change:
closed, open, and locally closed immersions, see Schemes, Lemma 01JY,
quasi-compact, see Schemes, Lemma 01K5,
universally closed, see Schemes, Definition 01KB,
(quasi-)separated, see Schemes, Lemma 01KU,
monomorphism, see Schemes, Lemma 02YC
surjective, see Morphisms, Lemma 01S1,
universally injective, see Morphisms, Lemma 01S4,
affine, see Morphisms, Lemma 01SD,
quasi-affine, see Morphisms, Lemma 01SO,
(locally) of finite type, see Morphisms, Lemma 01T4,
(locally) quasi-finite, see Morphisms, Lemma 01TM,
(locally) of finite presentation, see Morphisms, Lemma 01TS,
locally of finite type of relative dimension \(d\), see Morphisms, Lemma 02NK,
universally open, see Morphisms, Definition 01U0,
flat, see Morphisms, Lemma 01U9,
syntomic, see Morphisms, Lemma 01UI,
smooth, see Morphisms, Lemma 01VB,
unramified (resp. G-unramified), see Morphisms, Lemma 02GA,
étale, see Morphisms, Lemma 02GO,
proper, see Morphisms, Lemma 01W4,
H-projective, see Morphisms, Lemma 01WF,
(locally) projective, see Morphisms, Lemma 02V6,
finite or integral, see Morphisms, Lemma 01WL,
finite locally free, see Morphisms, Lemma 02KD,
universally submersive, see Morphisms, Lemma 0CES,
universal homeomorphism, see Morphisms, Lemma 0CEU.
Add more as needed.
Remark
Of the properties of morphisms which are stable under base change (as listed in Remark 02WF) the following are also stable under compositions:
closed, open and locally closed immersions, see Schemes, Lemma 02V0,
quasi-compact, see Schemes, Lemma 01K6,
universally closed, see Morphisms, Lemma 01W3,
(quasi-)separated, see Schemes, Lemma 01KU,
monomorphism, see Schemes, Lemma 01L5,
surjective, see Morphisms, Lemma 01S0,
universally injective, see Morphisms, Lemma 02V1,
affine, see Morphisms, Lemma 01SC,
quasi-affine, see Morphisms, Lemma 01SN,
(locally) of finite type, see Morphisms, Lemma 01T3,
(locally) quasi-finite, see Morphisms, Lemma 01TL,
(locally) of finite presentation, see Morphisms, Lemma 01TR,
universally open, see Morphisms, Lemma 02V2,
flat, see Morphisms, Lemma 01U7,
syntomic, see Morphisms, Lemma 01UH,
smooth, see Morphisms, Lemma 01VA,
unramified (resp. G-unramified), see Morphisms, Lemma 02G9,
étale, see Morphisms, Lemma 02GN,
proper, see Morphisms, Lemma 01W3,
H-projective, see Morphisms, Lemma 01WE,
finite or integral, see Morphisms, Lemma 01WK,
finite locally free, see Morphisms, Lemma 02KC,
universally submersive, see Morphisms, Lemma 0CET,
universal homeomorphism, see Morphisms, Lemma 0CEV.
Add more as needed.
Remark
Of the properties mentioned which are stable under base change (as listed in Remark 02WF) the following are also fpqc local on the base (and a fortiori fppf local on the base):
for immersions we have this for
quasi-compact, see Descent, Lemma 02KQ,
universally closed, see Descent, Lemma 02KS,
monomorphism, see Descent, Lemma 02YK,
surjective, see Descent, Lemma 02KV,
universally injective, see Descent, Lemma 02KW,
affine, see Descent, Lemma 02L5,
quasi-affine, see Descent, Lemma 02L7,
(locally) of finite type, see Descent, Lemmas 02KX, and 02KZ,
(locally) quasi-finite, see Descent, Lemma 02VI,
(locally) of finite presentation, see Descent, Lemmas 02KY, and 02L0,
locally of finite type of relative dimension \(d\), see Descent, Lemma 02VJ,
universally open, see Descent, Lemma 02KT,
flat, see Descent, Lemma 02L2,
syntomic, see Descent, Lemma 02VK,
smooth, see Descent, Lemma 02VL,
unramified (resp. G-unramified), see Descent, Lemma 02VM,
étale, see Descent, Lemma 02VN,
proper, see Descent, Lemma 02L1,
finite or integral, see Descent, Lemma 02LA,
finite locally free, see Descent, Lemma 02VO,
universally submersive, see Descent, Lemma 0CEW,
universal homeomorphism, see Descent, Lemma 0CEX.
Note that the property of being an “immersion” may not be fpqc local on the base, but in Descent, Lemma 02YM we proved that it is fppf local on the base.
Properties of representable morphisms of presheaves
Here is the definition that makes this work.
Definition
With \(S\), and \(a : F \to G\) representable as above. Let \(\mathcal{P}\) be a property of morphisms of schemes which
is preserved under any base change, see Schemes, Definition 01JZ, and
is fppf local on the base, see Descent, Definition 02KO.
In this case we say that \(a\) has property \(\mathcal{P}\) if for every \(U \in \Ob((\Sch/S)_{fppf})\) and any \(\xi \in G(U)\) the resulting morphism of schemes \(V_\xi \to U\) 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.
Remark
Consider the property \(\mathcal{P}=\)“surjective”. In this case there could be some ambiguity if we say “let \(F \to G\) be a surjective map”. Namely, we could mean the notion defined in Definition 025V above, or we could mean a surjective map of presheaves, see Sites, Definition 00V6, or, if both \(F\) and \(G\) are sheaves, we could mean a surjective map of sheaves, see Sites, Definition 00WM. If not mentioned otherwise when discussing morphisms of algebraic spaces we will always mean the first. See Lemma 05VM for a case where surjectivity implies surjectivity as a map of sheaves.
Here is a sanity check.
Lemma
Let \(S\), \(X\), \(Y\) be objects of \(\Sch_{fppf}\). Let \(f : X \to Y\) be a morphism of schemes. Let \(\mathcal{P}\) be as in Definition 025V. Then \(h_X \longrightarrow h_Y\) has property \(\mathcal{P}\) if and only if \(f\) has property \(\mathcal{P}\).
Proof
Note that the lemma makes sense by Lemma 02W9. Proof omitted.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(\mathcal{P}\) be a property as in Definition 025V which is stable under composition. Let \(a : F \to G\), \(b : G \to H\) be representable transformations of functors. If \(a\) and \(b\) have property \(\mathcal{P}\) so does \(b \circ a : F \longrightarrow H\).
Proof
Note that the lemma makes sense by Lemma 02WA. Proof omitted.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(\mathcal{P}\) be a property as in Definition 025V. Let \(a : F \to G\) be a representable transformation of functors. Let \(b : H \to G\) be any transformation of functors. Consider the fibre product diagram \[\xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G }\] If \(a\) has property \(\mathcal{P}\) then also the base change \(a'\) has property \(\mathcal{P}\).
Proof
Note that the lemma makes sense by Lemma 02WB. Proof omitted.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(\mathcal{P}\) be a property as in Definition 025V. Let \(a : F \to G\) be a representable transformation of functors. Let \(b : H \to G\) be any transformation of functors. Consider the fibre product diagram \[\xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G }\] Assume that \(b\) induces a surjective map of fppf sheaves \(H^\# \to G^\#\). In this case, if \(a'\) has property \(\mathcal{P}\), then also \(a\) has property \(\mathcal{P}\).
Proof
First we remark that by Lemma 02WB the transformation \(a'\) is representable. Let \(U \in \Ob((\Sch/S)_{fppf})\), and let \(\xi \in G(U)\). By assumption there exists an fppf covering \(\{U_i \to U\}_{i \in I}\) and elements \(\xi_i \in H(U_i)\) mapping to \(\xi|_U\) via \(b\). From general category theory it follows that for each \(i\) we have a fibre product diagram \[\xymatrix{ U_i \times_{\xi_i, H, a'} (H \times_{b, G, a} F) \ar[r] \ar[d] & U \times_{\xi, G, a} F \ar[d] \\ U_i \ar[r] & U }\] By assumption the left vertical arrow is a morphism of schemes which has property \(\mathcal{P}\). Since \(\mathcal{P}\) is local in the fppf topology this implies that also the right vertical arrow has property \(\mathcal{P}\) as desired.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). 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 transformations of functors. Let \(\mathcal{P}\) be a property as in Definition 025V 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 02WC. Proof omitted.
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 a representable transformation of functors. Let \(\mathcal{P}\), \(\mathcal{P}'\) be properties as in Definition 025V. Suppose that for any morphism of schemes \(f : X \to Y\) 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, 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 (representable by) a scheme and the morphism \(T' \to T\) is a flat, locally of finite presentation, and surjective. Hence \(\{T' \to T\}\) is an fppf covering such that \(g|_{T'} \in G(T')\) comes from an element of \(F(T')\), namely the map \(T' \to F\). This proves the map is surjective as a map of sheaves, see Sites, Definition 00WM.
Here is a characterization of those functors for which the diagonal is representable.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F\) be a presheaf of sets on \((\Sch/S)_{fppf}\). The following are equivalent:
the diagonal \(F \to F \times F\) is representable,
for \(U \in \Ob((\Sch/S)_{fppf})\) and any \(a \in F(U)\) the map \(a : h_U \to F\) is representable,
for every pair \(U, V \in \Ob((\Sch/S)_{fppf})\) and any \(a \in F(U)\), \(b \in F(V)\) the fibre product \(h_U \times_{a, F, b} h_V\) is representable.
Proof
This is completely formal, see Categories, Lemma 0024. It depends only on the fact that the category \((\Sch/S)_{fppf}\) has products of pairs of objects and fibre products, see Topologies, Lemma 021U.
In the situation of the lemma, for any morphism \(\xi : h_U \to F\) as in the lemma, it makes sense to say that \(\xi\) has property \(\mathcal{P}\), for any property as in Definition 025V. In particular this holds for \(\mathcal{P} =\) “surjective” and \(\mathcal{P} =\) “étale”, see Remark 02WH above. We will use this remark in the definition of algebraic spaces below.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F\) be a presheaf of sets on \((\Sch/S)_{fppf}\). Let \(\mathcal{P}\) be a property as in Definition 025V. If for every \(U, V \in \Ob((\Sch/S)_{fppf})\) and \(a \in F(U)\), \(b \in F(V)\) we have
\(h_U \times_{a, F, b} h_V\) is representable, say by the scheme \(W\), and
the morphism \(W \to U \times_S V\) corresponding to \(h_U \times_{a, F, b} h_V \to h_U \times h_V\) has property \(\mathcal{P}\),
then \(\Delta : F \to F \times F\) is representable and has property \(\mathcal{P}\).
Proof
Observe that \(\Delta\) is representable by Lemma 025W. We can formulate condition (2) as saying that the transformation \(h_U \times_{a, F, b} h_V \to h_{U \times_S V}\) has property \(\mathcal{P}\), see Lemma 02WJ. Consider \(T \in \Ob((\Sch/S)_{fppf})\) and \((a, b) \in (F \times F)(T)\). Observe that we have the commutative diagram \[\xymatrix{ F \times_{\Delta, F \times F, (a, b)} h_T \ar[d] \ar[r] & h_T \ar[d]^{\Delta_{T/S}} \\ h_T \times_{a, F, b} h_T \ar[r] \ar[d] & h_{T \times_S T} \ar[d]^{(a, b)} \\ F \ar[r]^\Delta & F \times F }\] both of whose squares are cartesian. In this way we see that the morphism \(F \times_{F \times F} h_T \to h_T\) is the base change of a morphism having property \(\mathcal{P}\) by \(\Delta_{T/S}\). Since \(\mathcal{P}\) is preserved under base change this finishes the proof.
Algebraic spaces
Here is the definition.
Definition
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). An algebraic space over \(S\) is a presheaf \[F : (\Sch/S)^{opp}_{fppf} \longrightarrow \textit{Sets}\] with the following properties
The presheaf \(F\) is a sheaf.
The diagonal morphism \(F \to F \times F\) is representable.
There exists a scheme \(U \in \Ob((\Sch/S)_{fppf})\) and a map \(h_U \to F\) which is surjective and étale1.
There are two differences with the “usual” definition, for example the definition in Knutson’s book [Kn].
The first is that we require \(F\) to be a sheaf in the fppf topology. One reason for doing this is that many natural examples of algebraic spaces satisfy the sheaf condition for the fppf coverings (and even for fpqc coverings). Also, one of the reasons that algebraic spaces have been so useful is via Michael Artin’s results on algebraic spaces. Built into his method is a condition which guarantees the result is locally of finite presentation over \(S\). Combined it somehow seems to us that the fppf topology is the natural topology to work with. In the end the category of algebraic spaces ends up being the same. See Bootstrap, Section 076L.
The second is that we only require the diagonal map for \(F\) to be representable, whereas in [Kn] it is required that it also be quasi-compact. If \(F = h_U\) for some scheme \(U\) over \(S\) this corresponds to the condition that \(U\) be quasi-separated. Our point of view is to try to prove a certain number of the results that follow only assuming that the diagonal of \(F\) be representable, and simply add an additional hypothesis wherever this is necessary. In any case it has the pleasing consequence that the following lemma is true.
Lemma
A scheme is an algebraic space. More precisely, given a scheme \(T \in \Ob((\Sch/S)_{fppf})\) the representable functor \(h_T\) is an algebraic space.
Proof
The functor \(h_T\) is a sheaf by our remarks in Section 025T. The diagonal \(h_T \to h_T \times h_T = h_{T \times T}\) is representable because \((\Sch/S)_{fppf}\) has fibre products. The identity map \(h_T \to h_T\) is surjective étale.
Definition
Let \(F\), \(F'\) be algebraic spaces over \(S\). A morphism \(f : F \to F'\) of algebraic spaces over \(S\) is a transformation of functors from \(F\) to \(F'\).
The category of algebraic spaces over \(S\) contains the category \((\Sch/S)_{fppf}\) as a full subcategory via the Yoneda embedding \(T/S \mapsto h_T\). From now on we no longer distinguish between a scheme \(T/S\) and the algebraic space it represents. Thus when we say “Let \(f : T \to F\) be a morphism from the scheme \(T\) to the algebraic space \(F\)”, we mean that \(T \in \Ob((\Sch/S)_{fppf})\), that \(F\) is an algebraic space over \(S\), and that \(f : h_T \to F\) is a morphism of algebraic spaces over \(S\).
Fibre products of algebraic spaces
The category of algebraic spaces over \(S\) has both products and fibre products.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G\) be algebraic spaces over \(S\). Then \(F \times G\) is an algebraic space, and is a product in the category of algebraic spaces over \(S\).
Proof
It is clear that \(H = F \times G\) is a sheaf. The diagonal of \(H\) is simply the product of the diagonals of \(F\) and \(G\). Hence it is representable by Lemma 02WC. Finally, if \(U \to F\) and \(V \to G\) are surjective étale morphisms, with \(U, V \in \Ob((\Sch/S)_{fppf})\), then \(U \times V \to F \times G\) is surjective étale by Lemma 02WM.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(H\) be a sheaf on \((\Sch/S)_{fppf}\) whose diagonal is representable. Let \(F, G\) be algebraic spaces over \(S\). Let \(F \to H\), \(G \to H\) be maps of sheaves. Then \(F \times_H G\) is an algebraic space.
Proof
We check the 3 conditions of Definition 025Y. A fibre product of sheaves is a sheaf, hence \(F \times_H G\) is a sheaf. The diagonal of \(F \times_H G\) is the left vertical arrow in \[\xymatrix{ F \times_H G \ar[r] \ar[d]_\Delta & F \times G \ar[d]^{\Delta_F \times \Delta_G} \\ (F \times F) \times_{(H \times H)} (G \times G) \ar[r] & (F \times F) \times (G \times G) }\] which is cartesian. Hence \(\Delta\) is representable as the base change of the morphism on the right which is representable, see Lemmas 02WC and 02WB. Finally, let \(U, V \in \Ob((\Sch/S)_{fppf})\) and \(a : U \to F\), \(b : V \to G\) be surjective and étale. As \(\Delta_H\) is representable, we see that \(U \times_H V\) is a scheme. The morphism \[U \times_H V \longrightarrow F \times_H G\] is surjective and étale as a composition of the base changes \(U \times_H V \to U \times_H G\) and \(U \times_H G \to F \times_H G\) of the étale surjective morphisms \(U \to F\) and \(V \to G\), see Lemmas 02WA and 02WB. This proves the last condition of Definition 025Y holds and we conclude that \(F \times_H G\) is an algebraic space.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F \to H\), \(G \to H\) be morphisms of algebraic spaces over \(S\). Then \(F \times_H G\) is an algebraic space, and is a fibre product in the category of algebraic spaces over \(S\).
Proof
It follows from the stronger Lemma 04T9 that \(F \times_H G\) is an algebraic space. It is clear that \(F \times_H G\) is a fibre product in the category of algebraic spaces over \(S\) since that is a full subcategory of the category of (pre)sheaves of sets on \((\Sch/S)_{fppf}\).
Glueing algebraic spaces
In this section we really start abusing notation and not distinguish between schemes and the spaces they represent.
Lemma
Let \(S \in \Ob(\Sch_{fppf})\). Let \(F\) and \(G\) be sheaves on \((\Sch/S)_{fppf}^{opp}\) and denote \(F \amalg G\) the coproduct in the category of sheaves. The map \(F \to F \amalg G\) is representable by open and closed immersions.
Proof
Let \(U\) be a scheme and let \(\xi \in (F \amalg G)(U)\). Recall the coproduct in the category of sheaves is the sheafification of the coproduct presheaf (Sites, Lemma 00WI). Thus there exists an fppf covering \(\{g_i : U_i \to U\}_{i \in I}\) and a disjoint union decomposition \(I = I' \amalg I''\) such that \(U_i \to U \to F \amalg G\) factors through \(F\), resp. \(G\) if and only if \(i \in I'\), resp. \(i \in I''\). Since \(F\) and \(G\) have empty intersection in \(F \amalg G\) we conclude that \(U_i \times_U U_j\) is empty if \(i \in I'\) and \(j \in I''\). Hence \(U' = \bigcup_{i \in I'} g_i(U_i)\) and \(U'' = \bigcup_{i \in I''} g_i(U_i)\) are disjoint open (Morphisms, Lemma 01UA) subschemes of \(U\) with \(U = U' \amalg U''\). We omit the verification that \(U' = U \times_{F \amalg G} F\).
Lemma
Let \(S \in \Ob(\Sch_{fppf})\). Let \(U \in \Ob((\Sch/S)_{fppf})\). Given a set \(I\) and sheaves \(F_i\) on \(\Ob((\Sch/S)_{fppf})\), if \(U \cong \coprod_{i\in I} F_i\) as sheaves, then each \(F_i\) is representable by an open and closed subscheme \(U_i\) and \(U \cong \coprod U_i\) as schemes.
Proof
By Lemma 0F15 the map \(F_i \to U\) is representable by open and closed immersions. Hence \(F_i\) is representable by an open and closed subscheme \(U_i\) of \(U\). We have \(U = \coprod U_i\) because we have \(U \cong \coprod F_i\) as sheaves and we can test the equality on points.
Lemma
Let \(S \in \Ob(\Sch_{fppf})\). Let \(F\) be an algebraic space over \(S\). Given a set \(I\) and sheaves \(F_i\) on \(\Ob((\Sch/S)_{fppf})\), if \(F \cong \coprod_{i\in I} F_i\) as sheaves, then each \(F_i\) is an algebraic space over \(S\).
Proof
The representability of \(F \to F \times F\) implies that each diagonal morphism \(F_i \to F_i \times F_i\) is representable (immediate from the definitions and the fact that \(F \times_{(F \times F)} (F_i \times F_i) = F_i\)). Choose a scheme \(U\) in \((\Sch/S)_{fppf}\) and a surjective étale morphism \(U \to F\) (this exist by hypothesis). The base change \(U \times_F F_i \to F_i\) is surjective and étale by Lemma 02WL. On the other hand, \(U \times_F F_i\) is a scheme by Lemma 0F15. Thus we have verified all the conditions in Definition 025Y and \(F_i\) is an algebraic space.
The condition on the size of \(I\) and the \(F_i\) in the following lemma may be ignored by those not worried about set theoretic questions.
Lemma
Let \(S \in \Ob(\Sch_{fppf})\). Suppose given a set \(I\) and algebraic spaces \(F_i\), \(i \in I\). Then \(F = \coprod_{i \in I} F_i\) is an algebraic space provided \(I\), and the \(F_i\) are not too “large”: for example if we can choose surjective étale morphisms \(U_i \to F_i\) such that \(\coprod_{i \in I} U_i\) is isomorphic to an object of \((\Sch/S)_{fppf}\), then \(F\) is an algebraic space.
Proof
By construction \(F\) is a sheaf. We omit the verification that the diagonal morphism of \(F\) is representable. Finally, if \(U\) is an object of \((\Sch/S)_{fppf}\) isomorphic to \(\coprod_{i \in I} U_i\) then it is straightforward to verify that the resulting map \(U \to \coprod F_i\) is surjective and étale.
Here is the analogue of Schemes, Lemma 01JJ.
Lemma
Let \(S \in \Ob(\Sch_{fppf})\). Let \(F\) be a presheaf of sets on \((\Sch/S)_{fppf}\). Assume
\(F\) is a sheaf,
there exists an index set \(I\) and subfunctors \(F_i \subset F\) such that
Then \(F\) is an algebraic space.
Proof
Let \(T\) be an object of \((\Sch/S)_{fppf}\). Let \(T \to F\) be a morphism. By assumption (2)(b) and (2)(c) the fibre product \(F_i \times_F T\) is representable by an open subscheme \(V_i \subset T\). It follows that \((\coprod F_i) \times_F T\) is represented by the scheme \(\coprod V_i\) over \(T\). By assumption (2)(d) there exists an fppf covering \(\{T_j \to T\}_{j \in J}\) such that \(T_j \to T \to F\) factors through \(F_i\), \(i = i(j)\). Hence \(T_j \to T\) factors through the open subscheme \(V_{i(j)} \subset T\). Since \(\{T_j \to T\}\) is jointly surjective, it follows that \(T = \bigcup V_i\) is an open covering. In particular, the transformation of functors \(\coprod F_i \to F\) is representable and surjective in the sense of Definition 025V (see Remark 02YN for a discussion).
Next, let \(T' \to F\) be a second morphism from an object in \((\Sch/S)_{fppf}\). Write as above \(T' = \bigcup V'_i\) with \(V'_i = T' \times_F F_i\). To show that the diagonal \(F \to F \times F\) is representable we have to show that \(G = T \times_F T'\) is representable, see Lemma 025W. Consider the subfunctors \(G_i = G \times_F F_i\). Note that \(G_i = V_i \times_{F_i} V'_i\), and hence is representable as \(F_i\) is an algebraic space. By the above the \(G_i\) form a Zariski covering of \(G\). Hence by Schemes, Lemma 01JJ we see \(G\) is representable.
Choose a scheme \(U \in \Ob((\Sch/S)_{fppf})\) and a surjective étale morphism \(U \to \coprod F_i\) (this exists by hypothesis). We may write \(U = \coprod U_i\) with \(U_i\) the inverse image of \(F_i\), see Lemma 02WO. We claim that \(U \to F\) is surjective and étale. Surjectivity follows as \(\coprod F_i \to F\) is surjective (see first paragraph of the proof) by applying Lemma 02WK. Consider the fibre product \(U \times_F T\) where \(T \to F\) is as above. We have to show that \(U \times_F T \to T\) is étale. Since \(U \times_F T = \coprod U_i \times_F T\) it suffices to show each \(U_i \times_F T \to T\) is étale. Since \(U_i \times_F T = U_i \times_{F_i} V_i\) this follows from the fact that \(U_i \to F_i\) is étale and \(V_i \to T\) is an open immersion (and Morphisms, Lemmas 02GP and 02GN).
Presentations of algebraic spaces
Given an algebraic space we can find a “presentation” of it.
Lemma
Let \(F\) be an algebraic space over \(S\). Let \(f : U \to F\) be a surjective étale morphism from a scheme to \(F\). Set \(R = U \times_F U\). Then
\(j : R \to U \times_S U\) defines an equivalence relation on \(U\) over \(S\) (see Groupoids, Definition 022P).
the morphisms \(s, t : R \to U\) are étale, and
the diagram \[\xymatrix{ R \ar@<1ex>[r] \ar@<-1ex>[r] & U \ar[r] & F }\] is a coequalizer diagram in \(\Sh((\Sch/S)_{fppf})\).
Proof
Let \(T/S\) be an object of \((\Sch/S)_{fppf}\). Then \(R(T) = \{(a, b) \in U(T) \times U(T) \mid f \circ a = f \circ b\}\) which defines an equivalence relation on \(U(T)\). The morphisms \(s, t : R \to U\) are étale because the morphism \(U \to F\) is étale.
To prove (3) we first show that \(U \to F\) is a surjection of sheaves, see Sites, Definition 00WM. Let \(\xi \in F(T)\) with \(T\) as above. Let \(V = T \times_{\xi, F, f}U\). By assumption \(V\) is a scheme and \(V \to T\) is surjective étale. Hence \(\{V \to T\}\) is a covering for the fppf topology. Since \(\xi|_V\) factors through \(U\) by construction we conclude \(U \to F\) is surjective. Surjectivity implies that \(F\) is the coequalizer of the diagram by Sites, Lemma 086K.
This lemma suggests the following definitions.
Definition
Let \(S\) be a scheme. Let \(U\) be a scheme over \(S\). An étale equivalence relation on \(U\) over \(S\) is an equivalence relation \(j : R \to U \times_S U\) such that \(s, t : R \to U\) are étale morphisms of schemes.
Definition
Let \(F\) be an algebraic space over \(S\). A presentation of \(F\) is given by a scheme \(U\) over \(S\) and an étale equivalence relation \(R\) on \(U\) over \(S\), and a surjective étale morphism \(U \to F\) such that \(R = U \times_F U\).
Equivalently we could ask for the existence of an isomorphism \[U/R \cong F\] where the quotient \(U/R\) is as defined in Groupoids, Section 02VE. To construct algebraic spaces we will study the converse question, namely, for which equivalence relations the quotient sheaf \(U/R\) is an algebraic space. It will finally turn out this is always the case if \(R\) is an étale equivalence relation on \(U\) over \(S\), see Theorem 02WW.
Algebraic spaces and equivalence relations
Suppose given a scheme \(U\) over \(S\) and an étale equivalence relation \(R\) on \(U\) over \(S\). We would like to show this defines an algebraic space. We will produce a series of lemmas that prove the quotient sheaf \(U/R\) (see Groupoids, Definition 02VG) has all the properties required of it in Definition 025Y.
Lemma
Let \(S\) be a scheme. Let \(U\) be a scheme over \(S\). Let \(j = (s, t) : R \to U \times_S U\) be an étale equivalence relation on \(U\) over \(S\). Let \(U' \to U\) be an étale morphism. Let \(R'\) be the restriction of \(R\) to \(U'\), see Groupoids, Definition 02V9. Then \(j' : R' \to U' \times_S U'\) is an étale equivalence relation also.
Proof
It is clear from the description of \(s', t'\) in Groupoids, Lemma 02VB that \(s' , t' : R' \to U'\) are étale as compositions of base changes of étale morphisms (see Morphisms, Lemma 02GO and 02GN).
We will often use the following lemma to find open subspaces of algebraic spaces. A slight improvement (with more general hypotheses) of this lemma is Bootstrap, Lemma 046M.
Lemma
Let \(S\) be a scheme. Let \(U\) be a scheme over \(S\). Let \(j = (s, t) : R \to U \times_S U\) be a pre-relation. Let \(g : U' \to U\) be a morphism. Assume
\(j\) is an equivalence relation,
\(s, t : R \to U\) are surjective, flat and locally of finite presentation,
\(g\) is flat and locally of finite presentation.
Let \(R' = R|_{U'}\) be the restriction of \(R\) to \(U'\). Then \(U'/R' \to U/R\) is representable, and is an open immersion.
Proof
By Groupoids, Lemma 02V8 the morphism \(j' = (s', t') : R' \to U' \times_S U'\) defines an equivalence relation. Since \(g\) is flat and locally of finite presentation we see that \(g\) is universally open as well (Morphisms, Lemma 01UA). For the same reason \(s, t\) are universally open as well. Let \(W^1 = g(U') \subset U\), and let \(W = t(s^{-1}(W^1))\). Then \(W^1\) and \(W\) are open in \(U\). Moreover, as \(j\) is an equivalence relation we have \(t(s^{-1}(W)) = W\) (see Groupoids, Lemma 03LO for example).
By Groupoids, Lemma 045Z the map of sheaves \(F' = U'/R' \to F = U/R\) is injective. Let \(a : T \to F\) be a morphism from a scheme into \(U/R\). We have to show that \(T \times_F F'\) is representable by an open subscheme of \(T\).
The morphism \(a\) is given by the following data: an fppf covering \(\{\varphi_j : T_j \to T\}_{j \in J}\) of \(T\) and morphisms \(a_j : T_j \to U\) such that the maps \[a_j \times a_{j'} : T_j \times_T T_{j'} \longrightarrow U \times_S U\] factor through \(j : R \to U \times_S U\) via some (unique) maps \(r_{jj'} : T_j \times_T T_{j'} \to R\). The system \((a_j)\) corresponds to \(a\) in the sense that the diagrams \[\xymatrix{ T_j \ar[r]_{a_j} \ar[d] & U \ar[d] \\ T \ar[r]^a & F }\] commute.
Consider the open subsets \(W_j = a_j^{-1}(W) \subset T_j\). Since \(t(s^{-1}(W)) = W\) we see that \[W_j \times_T T_{j'} = r_{jj'}^{-1}(t^{-1}(W)) = r_{jj'}^{-1}(s^{-1}(W)) = T_j \times_T W_{j'}.\] By Descent, Lemma 03N0 this means there exists an open \(W_T \subset T\) such that \(\varphi_j^{-1}(W_T) = W_j\) for all \(j \in J\). We claim that \(W_T \to T\) represents \(T \times_F F' \to T\).
First, let us show that \(W_T \to T \to F\) is an element of \(F'(W_T)\). Since \(\{W_j \to W_T\}_{j \in J}\) is an fppf covering of \(W_T\), it is enough to show that each \(W_j \to U \to F\) is an element of \(F'(W_j)\) (as \(F'\) is a sheaf for the fppf topology). Consider the commutative diagram \[\xymatrix{ W'_j \ar[rr] \ar[dd] \ar[rd] & & U' \ar[d]^g \\ & s^{-1}(W^1) \ar[r]_s \ar[d]^t & W^1 \ar[d] \\ W_j \ar[r]^{a_j|_{W_j}} & W \ar[r] & F }\] where \(W'_j = W_j \times_W s^{-1}(W^1) \times_{W^1} U'\). Since \(t\) and \(g\) are surjective, flat and locally of finite presentation, so is \(W'_j \to W_j\). Hence the restriction of the element \(W_j \to U \to F\) to \(W'_j\) is an element of \(F'\) as desired.
Suppose that \(f : T' \to T\) is a morphism of schemes such that \(a|_{T'} \in F'(T')\). We have to show that \(f\) factors through the open \(W_T\). Since \(\{T' \times_T T_j \to T'\}\) is an fppf covering of \(T'\) it is enough to show each \(T' \times_T T_j \to T\) factors through \(W_T\). Hence we may assume \(f\) factors as \(\varphi_j \circ f_j : T' \to T_j \to T\) for some \(j\). In this case the condition \(a|_{T'} \in F'(T')\) means that there exists some fppf covering \(\{\psi_i : T'_i \to T'\}_{i \in I}\) and some morphisms \(b_i : T'_i \to U'\) such that \[\xymatrix{ T'_i \ar[r]_{b_i} \ar[d]_{f_j \circ \psi_i} & U' \ar[r]_g & U \ar[d] \\ T_j \ar[r]^{a_j} & U \ar[r] & F }\] is commutative. This commutativity means that there exists a morphism \(r'_i : T'_i \to R\) such that \(t \circ r'_i = a_j \circ f_j \circ \psi_i\), and \(s \circ r'_i = g \circ b_i\). This implies that \(\Im(f_j \circ \psi_i) \subset W_j\) and we win.
The following lemma is not completely trivial although it looks like it should be trivial.
Lemma
Let \(S\) be a scheme. Let \(U\) be a scheme over \(S\). Let \(j = (s, t) : R \to U \times_S U\) be an étale equivalence relation on \(U\) over \(S\). If the quotient \(U/R\) is an algebraic space, then \(U \to U/R\) is étale and surjective. Hence \((U, R, U \to U/R)\) is a presentation of the algebraic space \(U/R\).
Proof
Denote \(c : U \to U/R\) the morphism in question. Let \(T\) be a scheme and let \(a : T \to U/R\) be a morphism. We have to show that the morphism (of schemes) \(\pi : T \times_{a, U/R, c} U \to T\) is étale and surjective. The morphism \(a\) corresponds to an fppf covering \(\{\varphi_i : T_i \to T\}\) and morphisms \(a_i : T_i \to U\) such that \(a_i \times a_{i'} : T_i \times_T T_{i'} \to U \times_S U\) factors through \(R\), and such that \(c \circ a_i = a \circ \varphi_i\). Hence \[T_i \times_{\varphi_i, T} T \times_{a, U/R, c} U = T_i \times_{c \circ a_i, U/R, c} U = T_i \times_{a_i, U} U \times_{c, U/R, c} U = T_i \times_{a_i, U, t} R.\] Since \(t\) is étale and surjective we conclude that the base change of \(\pi\) to \(T_i\) is surjective and étale. Since the property of being surjective and étale is local on the base in the fpqc topology (see Remark 02WH) we win.
Lemma
Let \(S\) be a scheme. Let \(U\) be a scheme over \(S\). Let \(j = (s, t) : R \to U \times_S U\) be an étale equivalence relation on \(U\) over \(S\). Assume that \(U\) is affine. Then the quotient \(F = U/R\) is an algebraic space, and \(U \to F\) is étale and surjective.
Proof
Since \(j : R \to U \times_S U\) is a monomorphism we see that \(j\) is separated (see Schemes, Lemma 01L4). Since \(U\) is affine we see that \(U \times_S U\) (which comes equipped with a monomorphism into the affine scheme \(U \times U\)) is separated. Hence we see that \(R\) is separated. In particular the morphisms \(s, t\) are separated as well as étale.
Since the composition \(R \to U \times_S U \to U\) is locally of finite type we conclude that \(j\) is locally of finite type (see Morphisms, Lemma 01T8). As \(j\) is also a monomorphism it has finite fibres and we see that \(j\) is locally quasi-finite by Morphisms, Lemma 02NG. Altogether we see that \(j\) is separated and locally quasi-finite.
Our first step is to show that the quotient map \(c : U \to F\) is representable. Consider a scheme \(T\) and a morphism \(a : T \to F\). We have to show that the sheaf \(G = T \times_{a, F, c} U\) is representable. As seen in the proofs of Lemmas 02WU and 02WV there exists an fppf covering \(\{\varphi_i : T_i \to T\}_{i \in I}\) and morphisms \(a_i : T_i \to U\) such that \(a_i \times a_{i'} : T_i \times_T T_{i'} \to U \times_S U\) factors through \(R\), and such that \(c \circ a_i = a \circ \varphi_i\). As in the proof of Lemma 02WV we see that \[\begin{eqnarray*} T_i \times_{\varphi_i, T} G & = & T_i \times_{\varphi_i, T} T \times_{a, U/R, c} U \\ & = & T_i \times_{c \circ a_i, U/R, c} U \\ & = & T_i \times_{a_i, U} U \times_{c, U/R, c} U \\ & = & T_i \times_{a_i, U, t} R \end{eqnarray*}\] Since \(t\) is separated and étale, and in particular separated and locally quasi-finite (by Morphisms, Lemmas 02V5 and 02GV) we see that the restriction of \(G\) to each \(T_i\) is representable by a morphism of schemes \(X_i \to T_i\) which is separated and locally quasi-finite. By Descent, Lemma 02W5 we obtain a descent datum \((X_i, \varphi_{ii'})\) relative to the fppf-covering \(\{T_i \to T\}\). Since each \(X_i \to T_i\) is separated and locally quasi-finite we see by More on Morphisms, Lemma 02W8 that this descent datum is effective. Hence by Descent, Lemma 02W5 (2) we conclude that \(G\) is representable as desired.
The second step of the proof is to show that \(U \to F\) is surjective and étale. This is clear from the above since in the first step above we saw that \(G = T \times_{a, F, c} U\) is a scheme over \(T\) which base changes to schemes \(X_i \to T_i\) which are surjective and étale. Thus \(G \to T\) is surjective and étale (see Remark 02WH). Alternatively one can reread the proof of Lemma 02WV in the current situation.
The third and final step is to show that the diagonal map \(F \to F \times F\) is representable. We first observe that the diagram \[\xymatrix{ R \ar[r] \ar[d]_j & F \ar[d]^\Delta \\ U \times_S U \ar[r] & F \times F }\] is a fibre product square. By Lemma 02WC the morphism \(U \times_S U \to F \times F\) is representable (note that \(h_U \times h_U = h_{U \times_S U}\)). Moreover, by Lemma 02WM the morphism \(U \times_S U \to F \times F\) is surjective and étale (note also that étale and surjective occur in the lists of Remarks 02WH and 02WG). It follows either from Lemma 02WB and the diagram above, or by writing \(R \to F\) as \(R \to U \to F\) and Lemmas 02W9 and 02WA that \(R \to F\) is representable as well. Let \(T\) be a scheme and let \(a : T \to F \times F\) be a morphism. We have to show that \(G = T \times_{a, F \times F, \Delta} F\) is representable. By what was said above the morphism (of schemes) \[T' = (U \times_S U) \times_{F \times F, a} T \longrightarrow T\] is surjective and étale. Hence \(\{T' \to T\}\) is an étale covering of \(T\). Note also that \[T' \times_T G = T' \times_{U \times_S U, j} R\] as can be seen contemplating the following cube \[\xymatrix{ & R \ar[rr] \ar[dd] & & F \ar[dd] \\ T' \times_T G \ar[rr] \ar[dd] \ar[ru] & & G \ar[dd] \ar[ru] & \\ & U \times_S U \ar'[r][rr] & & F \times F \\ T' \ar[rr] \ar[ru] & & T \ar[ru] }\] Hence we see that the restriction of \(G\) to \(T'\) is representable by a scheme \(X\), and moreover that the morphism \(X \to T'\) is a base change of the morphism \(j\). Hence \(X \to T'\) is separated and locally quasi-finite (see second paragraph of the proof). By Descent, Lemma 02W5 we obtain a descent datum \((X, \varphi)\) relative to the fppf-covering \(\{T' \to T\}\). Since \(X \to T'\) is separated and locally quasi-finite we see by More on Morphisms, Lemma 02W8 that this descent datum is effective. Hence by Descent, Lemma 02W5 (2) we conclude that \(G\) is representable as desired.
Theorem
Let \(S\) be a scheme. Let \(U\) be a scheme over \(S\). Let \(j = (s, t) : R \to U \times_S U\) be an étale equivalence relation on \(U\) over \(S\). Then the quotient \(U/R\) is an algebraic space, and \(U \to U/R\) is étale and surjective, in other words \((U, R, U \to U/R)\) is a presentation of \(U/R\).
Proof
By Lemma 02WV it suffices to prove that \(U/R\) is an algebraic space. Let \(U' \to U\) be a surjective, étale morphism. Then \(\{U' \to U\}\) is in particular an fppf covering. Let \(R'\) be the restriction of \(R\) to \(U'\), see Groupoids, Definition 02V9. According to Groupoids, Lemma 02VH we see that \(U/R \cong U'/R'\). By Lemma 02WT \(R'\) is an étale equivalence relation on \(U'\). Thus we may replace \(U\) by \(U'\).
We apply the previous remark to \(U' = \coprod U_i\), where \(U = \bigcup U_i\) is an affine open covering of \(U\). Hence we may and do assume that \(U = \coprod U_i\) where each \(U_i\) is an affine scheme.
Consider the restriction \(R_i\) of \(R\) to \(U_i\). By Lemma 02WT this is an étale equivalence relation. Set \(F_i = U_i/R_i\) and \(F = U/R\). It is clear that \(\coprod F_i \to F\) is surjective. By Lemma 02WU each \(F_i \to F\) is representable, and an open immersion. By Lemma 0265 applied to \((U_i, R_i)\) we see that \(F_i\) is an algebraic space. Then by Lemma 02WV we see that \(U_i \to F_i\) is étale and surjective. From Lemma 02WQ it follows that \(\coprod F_i\) is an algebraic space. Finally, we have verified all hypotheses of Lemma 02WR and it follows that \(F = U/R\) is an algebraic space.
Algebraic spaces, retrofitted
We start building our arsenal of lemmas dealing with algebraic spaces. The first result says that in Definition 025Y we can weaken the condition on the diagonal as follows.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F\) be a sheaf on \((\Sch/S)_{fppf}\) such that there exists \(U \in \Ob((\Sch/S)_{fppf})\) and a map \(U \to F\) which is representable, surjective, and étale. Then \(F\) is an algebraic space.
Proof
Set \(R = U \times_F U\). This is a scheme as \(U \to F\) is assumed representable. The projections \(s, t : R \to U\) are étale 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 02WW the quotient sheaf \(F' = U/R\) is an algebraic space and \(U \to F'\) is surjective and étale. Again since \(R = U \times_F U\) 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 sheaves by Lemma 05VM. Thus \(F' \to F\) is also surjective and we conclude \(F' = F\) is an algebraic space.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(G\) be an algebraic space over \(S\), let \(F\) be a sheaf on \((\Sch/S)_{fppf}\), and let \(G \to F\) be a representable transformation of functors which is surjective and étale. Then \(F\) is an algebraic space.
Proof
Pick a scheme \(U\) and a surjective étale morphism \(U \to G\). Since \(G\) is an algebraic space \(U \to G\) is representable. Hence the composition \(U \to G \to F\) is representable, surjective, and étale. See Lemmas 02WA and 02WK. Thus \(F\) is an algebraic space by Lemma 0BGQ.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F\) be an algebraic space over \(S\). Let \(G \to F\) be a representable transformation of functors. Then \(G\) is an algebraic space.
Proof
By Lemma 02WD we see that \(G\) is a sheaf. The diagram \[\xymatrix{ G \times_F G \ar[r] \ar[d] & F \ar[d]^{\Delta_F} \\ G \times G \ar[r] & F \times F }\] is cartesian. Hence we see that \(G \times_F G \to G \times G\) is representable by Lemma 02WB. By Lemma 05L9 we see that \(G \to G \times_F G\) is representable. Hence \(\Delta_G : G \to G \times G\) is representable as a composition of representable transformations, see Lemma 02WA. Finally, let \(U\) be an object of \((\Sch/S)_{fppf}\) and let \(U \to F\) be surjective and étale. By assumption \(U \times_F G\) is representable by a scheme \(U'\). By Lemma 02WL the morphism \(U' \to G\) is surjective and étale. This verifies the final condition of Definition 025Y and we win.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F\), \(G\) be algebraic spaces over \(S\). Let \(G \to F\) be a representable morphism. Let \(U \in \Ob((\Sch/S)_{fppf})\), and \(q : U \to F\) surjective and étale. Set \(V = G \times_F U\). Finally, let \(\mathcal{P}\) be a property of morphisms of schemes as in Definition 025V. Then \(G \to F\) has property \(\mathcal{P}\) if and only if \(V \to U\) has property \(\mathcal{P}\).
Proof
(This lemma follows from Lemmas 02WL and 03KD, but we give a direct proof here also.) It is clear from the definitions that if \(G \to F\) has property \(\mathcal{P}\), then \(V \to U\) has property \(\mathcal{P}\). Conversely, assume \(V \to U\) has property \(\mathcal{P}\). Let \(T \to F\) be a morphism from a scheme to \(F\). Let \(T' = T \times_F G\) which is a scheme since \(G \to F\) is representable. We have to show that \(T' \to T\) has property \(\mathcal{P}\). Consider the commutative diagram of schemes \[\xymatrix{ V \ar[d] & T \times_F V \ar[d] \ar[l] \ar[r] & T \times_F G \ar[d] \ar@{=}[r] & T' \\ U & T \times_F U \ar[l] \ar[r] & T }\] where both squares are fibre product squares. Hence we conclude the middle arrow has property \(\mathcal{P}\) as a base change of \(V \to U\). Finally, \(\{T \times_F U \to T\}\) is a fppf covering as it is surjective étale, and hence we conclude that \(T' \to T\) has property \(\mathcal{P}\) as it is local on the base in the fppf topology.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(G \to F\) be a transformation of presheaves on \((\Sch/S)_{fppf}\). Let \(\mathcal{P}\) be a property of morphisms of schemes. Assume
\(\mathcal{P}\) is preserved under any base change, fppf local on the base, and morphisms of type \(\mathcal{P}\) satisfy descent for fppf coverings, see Descent, Definition 02W2,
\(G\) is a sheaf,
\(F\) is an algebraic space,
there exists a \(U \in \Ob((\Sch/S)_{fppf})\) and a surjective étale morphism \(U \to F\) such that \(V = G \times_F U\) is representable, and
\(V \to U\) has \(\mathcal{P}\).
Then \(G\) is an algebraic space, \(G \to F\) is representable and has property \(\mathcal{P}\).
Proof
Let \(T\) be a scheme and let \(T \to F\) be a morphism. Then \(U \times_F T \to T\) is surjective étale, hence \(\{U \times_F T \to T\}\) is a covering for the étale topology. Consider \[W = G \times_F (U \times_F T) = V \times_F T = V \times_U (U \times_F T).\] It is a scheme since \(F\) is an algebraic space. The morphism \(W \to U \times_F T\) has property \(\mathcal{P}\) since it is a base change of \(V \to U\). There is an isomorphism \[\begin{align*} W \times_T (U \times_F T) & = (G \times_F (U \times_F T)) \times_T (U \times_F T) \\ & = (U \times_F T) \times_T (G \times_F (U \times_F T)) \\ & = (U \times_F T) \times_T W \end{align*}\] over \((U \times_F T) \times_T (U \times_F T)\). The middle equality maps \(((g, (u_1, t)), (u_2, t))\) to \(((u_1, t), (g, (u_2, t)))\). This defines a descent datum for \(W/U \times_F T/T\), see Descent, Definition 023V. This follows from Descent, Lemma 02W5. Namely we have a sheaf \(G \times_F T\), whose base change to \(U \times_F T\) is represented by \(W\) and the isomorphism above is the one from the proof of Descent, Lemma 02W5. By assumption on \(\mathcal{P}\) the descent datum above is representable. Hence by the last statement of Descent, Lemma 02W5 we see that \(G \times_F T\) is representable. This proves that \(G \to F\) is a representable transformation of functors.
As \(G \to F\) is representable, we see that \(G\) is an algebraic space by Lemma 02WY. The fact that \(G \to F\) has property \(\mathcal{P}\) now follows from Lemma 02WZ.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G\) be algebraic spaces over \(S\). Let \(a : F \to G\) be a morphism. Given any \(V \in \Ob((\Sch/S)_{fppf})\) and a surjective étale morphism \(q : V \to G\) there exists a \(U \in \Ob((\Sch/S)_{fppf})\) and a commutative diagram \[\xymatrix{ U \ar[d]_p \ar[r]_\alpha & V \ar[d]^q \\ F \ar[r]^a & G }\] with \(p\) surjective and étale.
Proof
First choose \(W \in \Ob((\Sch/S)_{fppf})\) with surjective étale morphism \(W \to F\). Next, put \(U = W \times_G V\). Since \(G\) is an algebraic space we see that \(U\) is isomorphic to an object of \((\Sch/S)_{fppf}\). As \(q\) is surjective étale, we see that \(U \to W\) is surjective étale (see Lemma 02WL). Thus \(U \to F\) is surjective étale as a composition of surjective étale morphisms (see Lemma 02WK).
Immersions and Zariski coverings of algebraic spaces
At this point an interesting phenomenon occurs. We have already defined the notion of an open immersion of algebraic spaces (through Definition 025V) but we have yet to define the notion of a point2. Thus the Zariski topology of an algebraic space has already been defined, but there is no space yet!
Perhaps superfluously we formally introduce immersions as follows.
Definition
Let \(S \in \Ob(\Sch_{fppf})\) be a scheme. Let \(F\) be an algebraic space over \(S\).
A morphism of algebraic spaces over \(S\) is called an open immersion if it is representable, and an open immersion in the sense of Definition 025V.
An open subspace of \(F\) is a subfunctor \(F' \subset F\) such that \(F'\) is an algebraic space and \(F' \to F\) is an open immersion.
A morphism of algebraic spaces over \(S\) is called a closed immersion if it is representable, and a closed immersion in the sense of Definition 025V.
A closed subspace of \(F\) is a subfunctor \(F' \subset F\) such that \(F'\) is an algebraic space and \(F' \to F\) is a closed immersion.
A morphism of algebraic spaces over \(S\) is called an immersion if it is representable, and an immersion in the sense of Definition 025V.
A locally closed subspace of \(F\) is a subfunctor \(F' \subset F\) such that \(F'\) is an algebraic space and \(F' \to F\) is an immersion.
We note that these definitions make sense since an immersion is in particular a monomorphism (see Schemes, Lemma 01L7 and Lemma 02YO), and hence the image of an immersion \(G \to F\) of algebraic spaces is a subfunctor \(F' \subset F\) which is (canonically) isomorphic to \(G\). Thus some of the discussion of Schemes, Section 01IM carries over to the setting of algebraic spaces.
Lemma
Let \(S \in \Ob(\Sch_{fppf})\) be a scheme. A composition of (closed, resp. open) immersions of algebraic spaces over \(S\) is a (closed, resp. open) immersion of algebraic spaces over \(S\).
Proof
See Lemma 02WK and Remarks 02WH (see very last line of that remark) and 02WG.
Lemma
Let \(S \in \Ob(\Sch_{fppf})\) be a scheme. A base change of a (closed, resp. open) immersion of algebraic spaces over \(S\) is a (closed, resp. open) immersion of algebraic spaces over \(S\).
Proof
See Lemma 02WL and Remark 02WH (see very last line of that remark).
Lemma
Let \(S \in \Ob(\Sch_{fppf})\) be a scheme. Let \(F\) be an algebraic space over \(S\). Let \(F_1\), \(F_2\) be locally closed subspaces of \(F\). If \(F_1 \subset F_2\) as subfunctors of \(F\), then \(F_1\) is a locally closed subspace of \(F_2\). Similarly for closed and open subspaces.
Proof
Let \(T \to F_2\) be a morphism with \(T\) a scheme. Since \(F_2 \to F\) is a monomorphism, we see that \(T \times_{F_2} F_1 = T \times_F F_1\). The lemma follows formally from this.
Let us formally define the notion of a Zariski open covering of algebraic spaces. Note that in Lemma 02WR we have already encountered such open coverings as a method for constructing algebraic spaces.
Definition
Let \(S \in \Ob(\Sch_{fppf})\) be a scheme. Let \(F\) be an algebraic space over \(S\). A Zariski covering \(\{F_i \subset F\}_{i \in I}\) of \(F\) is given by a set \(I\) and a collection of open subspaces \(F_i \subset F\) such that \(\coprod F_i \to F\) is a surjective map of sheaves.
Note that if \(T\) is a schemes, and \(a : T \to F\) is a morphism, then each of the fibre products \(T \times_F F_i\) is identified with an open subscheme \(T_i \subset T\). The final condition of the definition signifies exactly that \(T = \bigcup_{i \in I} T_i\).
It is clear that the collection \(F_{Zar}\) of open subspaces of \(F\) is a set (as \((\Sch/S)_{fppf}\) is a site, hence a set). Moreover, we can turn \(F_{Zar}\) into a category by letting the morphisms be inclusions of subfunctors (which are automatically open immersions by Lemma 02YX). Finally, Definition 02YY provides the notion of a Zariski covering \(\{F_i \to F'\}_{i \in I}\) in the category \(F_{Zar}\). Hence, just as in the case of a topological space (see Sites, Example 00VJ) by suitably choosing a set of coverings we may obtain a Zariski site of the algebraic space \(F\).
Definition
Let \(S \in \Ob(\Sch_{fppf})\) be a scheme. Let \(F\) be an algebraic space over \(S\). A small Zariski site \(F_{Zar}\) of an algebraic space \(F\) is one of the sites described above.
Hence this gives a notion of what it means for something to be true Zariski locally on an algebraic space, which is how we will use this notion. In general the Zariski topology is not fine enough for our purposes. For example we can consider the category of Zariski sheaves on an algebraic space. It will turn out that this is not the correct thing to consider, even for quasi-coherent sheaves. One only gets the desired result when using the étale or fppf site of \(F\) to define quasi-coherent sheaves.
Separation conditions on algebraic spaces
A separation condition on an algebraic space \(F\) is a condition on the diagonal morphism \(F \to F \times F\). Let us first list the properties the diagonal has automatically. Since the diagonal is representable by definition the following lemma makes sense (through Definition 025V).
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F\) be an algebraic space over \(S\). Let \(\Delta : F \to F \times F\) be the diagonal morphism. Then
\(\Delta\) is locally of finite type,
\(\Delta\) is a monomorphism,
\(\Delta\) is separated, and
\(\Delta\) is locally quasi-finite.
Proof
Let \(F = U/R\) be a presentation of \(F\). As in the proof of Lemma 0265 the diagram \[\xymatrix{ R \ar[r] \ar[d]_j & F \ar[d]^\Delta \\ U \times_S U \ar[r] & F \times F }\] is cartesian. Hence according to Lemma 02WZ it suffices to show that \(j\) has the properties listed in the lemma. (Note that each of the properties (1) – (4) occur in the lists of Remarks 02WF and 02WH.) Since \(j\) is an equivalence relation it is a monomorphism. Hence it is separated by Schemes, Lemma 01L4. As \(R\) is an étale equivalence relation we see that \(s, t : R \to U\) are étale. Hence \(s, t\) are locally of finite type. Then it follows from Morphisms, Lemma 01T8 that \(j\) is locally of finite type. Finally, as it is a monomorphism its fibres are finite. Thus we conclude that it is locally quasi-finite by Morphisms, Lemma 02NG.
Here are some common types of separation conditions, relative to the base scheme \(S\). There is also an absolute notion of these conditions which we will discuss in Properties of Spaces, Section 03BR. Moreover, we will discuss separation conditions for a morphism of algebraic spaces in Morphisms of Spaces, Section 03HJ.
Definition
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F\) be an algebraic space over \(S\). Let \(\Delta : F \to F \times F\) be the diagonal morphism.
We say \(F\) is separated over \(S\) if \(\Delta\) is a closed immersion.
We say \(F\) is locally separated over \(S\)3 if \(\Delta\) is an immersion.
We say \(F\) is quasi-separated over \(S\) if \(\Delta\) is quasi-compact.
We say \(F\) is Zariski locally quasi-separated over \(S\)4 if there exists a Zariski covering \(F = \bigcup_{i \in I} F_i\) such that each \(F_i\) is quasi-separated.
Note that if the diagonal is quasi-compact (when \(F\) is separated or quasi-separated) then the diagonal is actually quasi-finite and separated, hence quasi-affine (by More on Morphisms, Lemma 02LR).
Examples of algebraic spaces
In this section we construct some examples of algebraic spaces. Some of these were suggested by B. Conrad. Since we do not yet have a lot of theory at our disposal the discussion is a bit awkward in some places.
Example
Let \(k\) be a field of characteristic \(\not = 2\). Let \(U = \mathbf{A}^1_k\). Set \[j : R = \Delta \amalg \Gamma \longrightarrow U \times_k U\] where \(\Delta = \{(x, x) \mid x \in \mathbf{A}^1_k\}\) and \(\Gamma = \{(x, -x) \mid x \in \mathbf{A}^1_k, x \not = 0\}\). It is clear that \(s, t : R \to U\) are étale, and hence \(j\) is an étale equivalence relation. The quotient \(X = U/R\) is an algebraic space by Theorem 02WW. Since \(R\) is quasi-compact we see that \(X\) is quasi-separated. On the other hand, \(X\) is not locally separated because the morphism \(j\) is not an immersion.
Example
Let \(k\) be a field. Let \(k'/k\) be a degree \(2\) Galois extension with \(\text{Gal}(k'/k) = \{1, \sigma\}\). Let \(S = \Spec(k[x])\) and \(U = \Spec(k'[x])\). Note that \[U \times_S U = \Spec((k' \otimes_k k')[x]) = \Delta(U) \amalg \Delta'(U)\] where \(\Delta' = (1, \sigma) : U \to U \times_S U\). Take \[R = \Delta(U) \amalg \Delta'(U \setminus \{0_U\})\] where \(0_U \in U\) denotes the \(k'\)-rational point whose \(x\)-coordinate is zero. It is easy to see that \(R\) is an étale equivalence relation on \(U\) over \(S\) and hence \(X = U/R\) is an algebraic space by Theorem 02WW. Here are some properties of \(X\) (some of which will not make sense until later):
\(X \to S\) is an isomorphism over \(S \setminus \{0_S\}\),
the morphism \(X \to S\) is étale (see Properties of Spaces, Definition 03FR)
the fibre \(0_X\) of \(X \to S\) over \(0_S\) is isomorphic to \(\Spec(k') = 0_U\),
\(X\) is not a scheme because if it were, then \(\mathcal{O}_{X, 0_X}\) would be a local domain \((\mathcal{O}, \mathfrak m, \kappa)\) with fraction field \(k(x)\), with \(x \in \mathfrak m\) and residue field \(\kappa = k'\) which is impossible,
\(X\) is not separated, but it is locally separated and quasi-separated,
there exists a surjective, finite, étale morphism \(S' \to S\) such that the base change \(X' = S' \times_S X\) is a scheme (namely, if we base change to \(S' = \Spec(k'[x])\) then \(U\) splits into two copies of \(S'\) and \(X'\) becomes isomorphic to the affine line with \(0\) doubled, see Schemes, Example 01JD), and
if we think of \(X\) as a finite type algebraic space over \(\Spec(k)\), then similarly the base change \(X_{k'}\) is a scheme but \(X\) is not a scheme.
In particular, this gives an example of a descent datum for schemes relative to the covering \(\{\Spec(k') \to \Spec(k)\}\) which is not effective.
See also Examples, Lemma 08KF, which shows that descent data need not be effective even for a projective morphism of schemes. That example gives a smooth separated algebraic space of dimension 3 over \({\mathbf C}\) which is not a scheme.
We will use the following lemma as a convenient way to construct algebraic spaces as quotients of schemes by free group actions.
Lemma
Let \(U \to S\) be a morphism of \(\Sch_{fppf}\). Let \(G\) be an abstract group. Let \(G \to \text{Aut}_S(U)\) be a group homomorphism. Assume
if \(u \in U\) is a point, and \(g(u) = u\) for some non-identity element \(g \in G\), then \(g\) induces a nontrivial automorphism of \(\kappa(u)\).
Then \[j : R = \coprod\nolimits_{g \in G} U \longrightarrow U \times_S U, \quad (g, x) \longmapsto (g(x), x)\] is an étale equivalence relation and hence \[F = U/R\] is an algebraic space by Theorem 02WW.
Proof
In the statement of the lemma the symbol \(\text{Aut}_S(U)\) denotes the group of automorphisms of \(U\) over \(S\). Assume \((*)\) holds. Let us show that \[j : R = \coprod\nolimits_{g \in G} U \longrightarrow U \times_S U, \quad (g, x) \longmapsto (g(x), x)\] is a monomorphism. This signifies that if \(T\) is a nonempty scheme, and \(h : T \to U\) is a \(T\)-valued point such that \(g \circ h = g' \circ h\) then \(g = g'\). Suppose \(T \not = \emptyset\), \(h : T \to U\) and \(g \circ h = g' \circ h\). Let \(t \in T\). Consider the composition \(\Spec(\kappa(t)) \to \Spec(\kappa(h(t))) \to U\). Then we conclude that \(g^{-1} \circ g'\) fixes \(u = h(t)\) and acts as the identity on its residue field. Hence \(g = g'\) by \((*)\).
Thus if \((*)\) holds we see that \(j\) is a relation (see Groupoids, Definition 022P). Moreover, it is an equivalence relation since on \(T\)-valued points for a connected scheme \(T\) we see that \(R(T) = G \times U(T) \to U(T) \times U(T)\) (recall that we always work over \(S\)). Moreover, the morphisms \(s, t : R \to U\) are étale since \(R\) is a disjoint product of copies of \(U\). This proves that \(j : R \to U \times_S U\) is an étale equivalence relation.
Given a scheme \(U\) and an action of a group \(G\) on \(U\) we say the action of \(G\) on \(U\) is free if condition \((*)\) of Lemma 02Z2 holds. This is equivalent to the notion of a free action of the constant group scheme \(G_S\) on \(U\) as defined in Groupoids, Definition 07S1. The lemma can be interpreted as saying that quotients of schemes by free actions of groups exist in the category of algebraic spaces.
Definition
Notation \(U \to S\), \(G\), \(R\) as in Lemma 02Z2. If the action of \(G\) on \(U\) satisfies \((*)\) we say \(G\) acts freely on the scheme \(U\). In this case the algebraic space \(U/R\) is denoted \(U/G\) and is called the quotient of \(U\) by \(G\).
This notation is consistent with the notation \(U/G\) introduced in Groupoids, Definition 02VG. We will later make sense of the quotient as an algebraic stack without any assumptions on the action whatsoever; when we do this we will use the notation \([U/G]\). Before we discuss the examples we prove some more lemmas to facilitate the discussion. Here is a lemma discussing the various separation conditions for this quotient when \(G\) is finite.
Lemma
Notation and assumptions as in Lemma 02Z2. Assume \(G\) is finite. Then
if \(U \to S\) is quasi-separated, then \(U/G\) is quasi-separated over \(S\), and
if \(U \to S\) is separated, then \(U/G\) is separated over \(S\).
Proof
In the proof of Lemma 02X4 we saw that it suffices to prove the corresponding properties for the morphism \(j : R \to U \times_S U\). If \(U \to S\) is quasi-separated, then for every affine open \(V \subset U\) which maps into an affine of \(S\) the opens \(g(V) \cap V\) are quasi-compact. It follows that \(j\) is quasi-compact. If \(U \to S\) is separated, the diagonal \(\Delta_{U/S}\) is a closed immersion. Hence \(j : R \to U \times_S U\) is a finite coproduct of closed immersions with disjoint images. Hence \(j\) is a closed immersion.
Lemma
Notation and assumptions as in Lemma 02Z2. If \(\Spec(k) \to U/G\) is a morphism, then there exist
a finite Galois extension \(k'/k\),
a finite subgroup \(H \subset G\),
an isomorphism \(H \to \text{Gal}(k'/k)\), and
an \(H\)-equivariant morphism \(\Spec(k') \to U\).
Conversely, such data determine a morphism \(\Spec(k) \to U/G\).
Proof
Consider the fibre product \(V = \Spec(k) \times_{U/G} U\). Here is a diagram \[\xymatrix{ V \ar[r] \ar[d] & U \ar[d] \\ \Spec(k) \ar[r] & U/G }\] Then \(V\) is a nonempty scheme étale over \(\Spec(k)\) and hence is a disjoint union \(V = \coprod_{i \in I} \Spec(k_i)\) of spectra of fields \(k_i\) finite separable over \(k\) (Morphisms, Lemma 02GL). We have \[\begin{align*} V \times_{\Spec(k)} V & = (\Spec(k) \times_{U/G} U) \times_{\Spec(k)}(\Spec(k) \times_{U/G} U) \\ & = \Spec(k) \times_{U/G} U \times_{U/G} U \\ & = \Spec(k) \times_{U/G} U \times G \\ & = V \times G \end{align*}\] The action of \(G\) on \(U\) induces an action of \(a : G \times V \to V\). The displayed equality means that \(G \times V \to V \times_{\Spec(k)} V\), \((g, v) \mapsto (a(g, v), v)\) is an isomorphism. In particular we see that for every \(i\) we have an isomorphism \(H_i \times \Spec(k_i) \to \Spec(k_i \otimes_k k_i)\) where \(H_i \subset G\) is the subgroup of elements fixing \(i \in I\). Thus \(H_i\) is finite and is the Galois group of \(k_i/k\). We omit the converse construction.
It follows from this lemma for example that if \(k'/k\) is a finite Galois extension, then \(\Spec(k')/\text{Gal}(k'/k) \cong \Spec(k)\). What happens if the extension is infinite? Here is an example.
Example
Let \(S = \Spec(\mathbf{Q})\). Let \(U = \Spec(\overline{\mathbf{Q}})\). Let \(G = \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\) with obvious action on \(U\). Then by construction property \((*)\) of Lemma 02Z2 holds and we obtain an algebraic space \[X = \Spec(\overline{\mathbf{Q}})/G \longrightarrow S = \Spec(\mathbf{Q}).\] Of course this is totally ridiculous as an approximation of \(S\)! Namely, by the Artin-Schreier theorem, see [JacobsonIII, Theorem 17, page 316], the only finite subgroups of \(\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\) are \(\{1\}\) and the conjugates of the order two group \(\text{Gal}(\overline{\mathbf{Q}}/\overline{\mathbf{Q}} \cap \mathbf{R})\). Hence, if \(\Spec(k) \to X\) is a morphism with \(k\) algebraic over \(\mathbf{Q}\), then it follows from Lemma 02Z5 and the theorem just mentioned that either \(k\) is \(\overline{\mathbf{Q}}\) or isomorphic to \(\overline{\mathbf{Q}} \cap \mathbf{R}\).
What is wrong with the example above is that the Galois group comes equipped with a topology, and this should somehow be part of any construction of a quotient of \(\Spec(\overline{\mathbf{Q}})\). The following example is much more reasonable in my opinion and may actually occur in “nature”.
Example
Let \(k\) be a field of characteristic zero. Let \(U = \mathbf{A}^1_k\) and let \(G = \mathbf{Z}\). As action we take \(n(x) = x + n\), i.e., the action of \(\mathbf{Z}\) on the affine line by translation. The only fixed point is the generic point and it is clearly the case that \(\mathbf{Z}\) injects into the automorphism group of the field \(k(x)\). (This is where we use the characteristic zero assumption.) Consider the morphism \[\gamma : \Spec(k(x)) \longrightarrow X = \mathbf{A}^1_k/\mathbf{Z}\] of the generic point of the affine line into the quotient. We claim that this morphism does not factor through any monomorphism \(\Spec(L) \to X\) of the spectrum of a field to \(X\). (Contrary to what happens for schemes, see Schemes, Section 01J5.) In fact, since \(\mathbf{Z}\) does not have any nontrivial finite subgroups we see from Lemma 02Z5 that for any such factorization \(k(x) = L\). Finally, \(\gamma\) is not a monomorphism since \[\Spec(k(x)) \times_{\gamma, X, \gamma} \Spec(k(x)) \cong \Spec(k(x)) \times \mathbf{Z}.\]
This example suggests that in order to define points of an algebraic space \(X\) we should consider equivalence classes of morphisms from spectra of fields into \(X\) and not the set of monomorphisms from spectra of fields.
We finish with a truly awful example.
Example
Let \(k\) be a field. Let \(A = \prod_{n \in \mathbf{N}} k\) be the infinite product. Set \(U = \Spec(A)\) seen as a scheme over \(S = \Spec(k)\). Note that the projection maps \(\text{pr}_n : A \to k\) define open and closed immersions \(f_n : S \to U\). Set \[R = U \amalg \coprod\nolimits_{(n, m) \in \mathbf{N}^2, \ n \not = m} S\] with morphism \(j\) equal to \(\Delta_{U/S}\) on the component \(U\) and \(j = (f_n, f_m)\) on the component \(S\) corresponding to \((n, m)\). It is clear from the remark above that \(s, t\) are étale. It is also clear that \(j\) is an equivalence relation. Hence we obtain an algebraic space \[X = U/R.\] To see what this means we specialize to the case where the field \(k\) is finite with \(q\) elements. Let us first discuss the topological space \(|U|\) associated to the scheme \(U\) a little bit. All elements of \(A\) satisfy \(x^q = x\). Hence every residue field of \(A\) is isomorphic to \(k\), and all points of \(U\) are closed. But the topology on \(U\) isn’t the discrete topology. Let \(u_n \in |U|\) be the point corresponding to \(f_n\). As mentioned above the points \(u_n\) are the open points (and hence isolated). This implies there have to be other points since we know \(U\) is quasi-compact, see Algebra, Lemma 00E8 (hence not equal to an infinite discrete set). Another way to see this is because the (proper) ideal \[I = \{x = (x_n) \in A \mid \text{all but a finite number of }x_n\text{ are zero}\}\] is contained in a maximal ideal. Note also that every element \(x\) of \(A\) is of the form \(x = ue\) where \(u\) is a unit and \(e\) is an idempotent. Hence a basis for the topology of \(A\) consists of open and closed subsets (see Algebra, Lemma 00EC.) So the topology on \(|U|\) is totally disconnected, but nontrivial. Finally, note that \(\{u_n\}\) is dense in \(|U|\).
We will later define a topological space \(|X|\) associated to \(X\), see Properties of Spaces, Section 03BT. What can we say about \(|X|\)? It turns out that the map \(|U| \to |X|\) is surjective and continuous. All the points \(u_n\) map to the same point \(x_0\) of \(|X|\), and none of the other points get identified. Since \(\{u_n\}\) is dense in \(|U|\) we conclude that the closure of \(x_0\) in \(|X|\) is \(|X|\). In other words \(|X|\) is irreducible and \(x_0\) is a generic point of \(|X|\). This seems bizarre since also \(x_0\) is the image of a section \(S \to X\) of the structure morphism \(X \to S\) (and in the case of schemes this would imply it was a closed point, see Morphisms, Lemma 01TE).
Whatever you think is actually going on in this example, it certainly shows that some care has to be exercised when defining irreducible components, connectedness, etc of algebraic spaces.
Change of big site
In this section we briefly discuss what happens when we change big sites. The upshot is that we can always enlarge the big site at will, hence we may assume any set of schemes we want to consider is contained in the big fppf site over which we consider our algebraic space. Here is a precise statement of the result.
Lemma
Suppose given big sites \(\Sch_{fppf}\) and \(\Sch'_{fppf}\). Assume that \(\Sch_{fppf}\) is contained in \(\Sch'_{fppf}\), see Topologies, Section 022I. Let \(S\) be an object of \(\Sch_{fppf}\). Let \[\begin{align*} g : \Sh((\Sch/S)_{fppf}) \longrightarrow \Sh((\Sch'/S)_{fppf}), \\ f : \Sh((\Sch'/S)_{fppf}) \longrightarrow \Sh((\Sch/S)_{fppf}) \end{align*}\] be the morphisms of topoi of Topologies, Lemma 022K. Let \(F\) be a sheaf of sets on \((\Sch/S)_{fppf}\). Then
if \(F\) is representable by a scheme \(X \in \Ob((\Sch/S)_{fppf})\) over \(S\), then \(f^{-1}F\) is representable too, in fact it is representable by the same scheme \(X\), now viewed as an object of \((\Sch'/S)_{fppf}\), and
if \(F\) is an algebraic space over \(S\), then \(f^{-1}F\) is an algebraic space over \(S\) also.
Proof
Let \(X \in \Ob((\Sch/S)_{fppf})\). Let us write \(h_X\) for the representable sheaf on \((\Sch/S)_{fppf}\) associated to \(X\), and \(h'_X\) for the representable sheaf on \((\Sch'/S)_{fppf}\) associated to \(X\). By the description of \(f^{-1}\) in Topologies, Section 022I we see that \(f^{-1}h_X = h'_X\). This proves (1).
Next, suppose that \(F\) is an algebraic space over \(S\). By Lemma 0262 this means that \(F = h_U/h_R\) for some étale equivalence relation \(R \to U \times_S U\) in \((\Sch/S)_{fppf}\). Since \(f^{-1}\) is an exact functor we conclude that \(f^{-1}F = h'_U/h'_R\). Hence \(f^{-1}F\) is an algebraic space over \(S\) by Theorem 02WW.
Note that this lemma is purely set theoretical and has virtually no content. Moreover, it is not true (in general) that the restriction of an algebraic space over the bigger site is an algebraic space over the smaller site (simply by reasons of cardinality). Hence we can only ever use a simple lemma of this kind to enlarge the base category and never to shrink it.
Lemma
Suppose \(\Sch_{fppf}\) is contained in \(\Sch'_{fppf}\). Let \(S\) be an object of \(\Sch_{fppf}\). Denote \(\textit{Spaces}/S\) the category of algebraic spaces over \(S\) defined using \(\Sch_{fppf}\). Similarly, denote \(\textit{Spaces}'/S\) the category of algebraic spaces over \(S\) defined using \(\Sch'_{fppf}\). The construction of Lemma 03FP defines a fully faithful functor \[\textit{Spaces}/S \longrightarrow \textit{Spaces}'/S\] whose essential image consists of those \(X' \in \Ob(\textit{Spaces}'/S)\) such that there exist \(U, R \in \Ob((\Sch/S)_{fppf})\)5 and morphisms \[U \longrightarrow X' \quad\text{and}\quad R \longrightarrow U \times_{X'} U\] in \(\Sh((\Sch'/S)_{fppf})\) which are surjective as maps of sheaves (for example if the displayed morphisms are surjective and étale).
Proof
In Sites, Lemma 00XU we have seen that the functor \(f^{-1} : \Sh((\Sch/S)_{fppf}) \to \Sh((\Sch'/S)_{fppf})\) is fully faithful (see discussion in Topologies, Section 022I). Hence we see that the displayed functor of the lemma is fully faithful.
Suppose that \(X' \in \Ob(\textit{Spaces}'/S)\) such that there exists \(U \in \Ob((\Sch/S)_{fppf})\) and a map \(U \to X'\) in \(\Sh((\Sch'/S)_{fppf})\) which is surjective as a map of sheaves. Let \(U' \to X'\) be a surjective étale morphism with \(U' \in \Ob((\Sch'/S)_{fppf})\). Let \(\kappa = \text{size}(U)\), see Sets, Section 000H. Then \(U\) has an affine open covering \(U = \bigcup_{i \in I} U_i\) with \(|I| \leq \kappa\). Observe that \(U' \times_{X'} U \to U\) is étale and surjective. For each \(i\) we can pick a quasi-compact open \(U'_i \subset U'\) such that \(U'_i \times_{X'} U_i \to U_i\) is surjective (because the scheme \(U' \times_{X'} U_i\) is the union of the Zariski opens \(W \times_{X'} U_i\) for \(W \subset U'\) affine and because \(U' \times_{X'} U_i \to U_i\) is étale hence open). Then \(\coprod_{i \in I} U'_i \to X\) is surjective étale because of our assumption that \(U \to X\) and hence \(\coprod U_i \to X\) is a surjection of sheaves (details omitted). Because \(U'_i \times_{X'} U \to U'_i\) is a surjection of sheaves and because \(U'_i\) is quasi-compact, we can find a quasi-compact open \(W_i \subset U'_i \times_{X'} U\) such that \(W_i \to U'_i\) is surjective as a map of sheaves (details omitted). Then \(W_i \to U\) is étale and we conclude that \(\text{size}(W_i) \leq \text{size}(U)\), see Sets, Lemma 04T7. By Sets, Lemma 0AHK we conclude that \(\text{size}(U'_i) \leq \text{size}(U)\). Hence \(\coprod_{i \in I} U'_i\) is isomorphic to an object of \((\Sch/S)_{fppf}\) by Sets, Lemma 000Q.
Now let \(X'\), \(U \to X'\) and \(R \to U \times_{X'} U\) be as in the statement of the lemma. In the previous paragraph we have seen that we can find \(U' \in \Ob((\Sch/S)_{fppf})\) and a surjective étale morphism \(U' \to X'\) in \(\Sh((\Sch'/S)_{fppf})\). Then \(U' \times_{X'} U \to U'\) is a surjection of sheaves, i.e., we can find an fppf covering \(\{U'_i \to U'\}\) such that \(U'_i \to U'\) factors through \(U' \times_{X'} U \to U'\). By Sets, Lemma 0AHL we can find \(\tilde U \to U'\) which is surjective, flat, and locally of finite presentation, with \(\text{size}(\tilde U) \leq \text{size}(U')\), such that \(\tilde U \to U'\) factors through \(U' \times_{X'} U \to U'\). Then we consider \[\xymatrix{ U' \times_{X'} U' \ar[d] & \tilde U \times_{X'} \tilde U \ar[l] \ar[d] \ar[r] & U \times_{X'} U \ar[d] \\ U' \times_S U' & \tilde U \times_S \tilde U \ar[l] \ar[r] & U \times_S U }\] The squares are cartesian. We know the objects of the bottom row are represented by objects of \((\Sch/S)_{fppf}\). By the result of the argument of the previous paragraph, the same is true for \(U \times_{X'} U\) (as we have the surjection of sheaves \(R \to U \times_{X'} U\) by assumption). Since \((\Sch/S)_{fppf}\) is closed under fibre products (by construction), we see that \(\tilde U \times_{X'} \tilde U\) is represented by an object of \((\Sch/S)_{fppf}\). Finally, the map \(\tilde U \times_{X'} \tilde U \to U' \times_{X'} U'\) is a surjection of fppf sheaves as \(\tilde U \to U'\) is so. Thus we can once more apply the result of the previous paragraph to conclude that \(R' = U' \times_{X'} U'\) is represented by an object of \((\Sch/S)_{fppf}\). At this point Lemma 0262 and Theorem 02WW imply that \(X = h_{U'}/h_{R'}\) is an object of \(\textit{Spaces}/S\) such that \(f^{-1}X \cong X'\) as desired.
Change of base scheme
In this section we briefly discuss what happens when we change base schemes. The upshot is that given a morphism \(S \to S'\) of base schemes, any algebraic space over \(S\) can be viewed as an algebraic space over \(S'\). And, given an algebraic space \(F'\) over \(S'\) there is a base change \(F'_S\) which is an algebraic space over \(S\). We explain only what happens in case \(S \to S'\) is a morphism of the big fppf site under consideration, if only \(S\) or \(S'\) is contained in the big site, then one first enlarges the big site as in Section 03FO.
Lemma
Suppose given a big site \(\Sch_{fppf}\). Let \(g : S \to S'\) be morphism of \(\Sch_{fppf}\). Let \(j : (\Sch/S)_{fppf} \to (\Sch/S')_{fppf}\) be the corresponding localization functor. Let \(F\) be a sheaf of sets on \((\Sch/S)_{fppf}\). Then
for a scheme \(T'\) over \(S'\) we have \(j_!F(T'/S') = \coprod\nolimits_{\varphi : T' \to S} F(T' \xrightarrow{\varphi} S),\)
if \(F\) is representable by a scheme \(X \in \Ob((\Sch/S)_{fppf})\), then \(j_!F\) is representable by \(j(X)\) which is \(X\) viewed as a scheme over \(S'\), and
if \(F\) is an algebraic space over \(S\), then \(j_!F\) is an algebraic space over \(S'\), and if \(F = U/R\) is a presentation, then \(j_!F = j(U)/j(R)\) is a presentation.
Let \(F'\) be a sheaf of sets on \((\Sch/S')_{fppf}\). Then
for a scheme \(T\) over \(S\) we have \(j^{-1}F'(T/S) = F'(T/S')\),
if \(F'\) is representable by a scheme \(X' \in \Ob((\Sch/S')_{fppf})\), then \(j^{-1}F'\) is representable, namely by \(X'_S = S \times_{S'} X'\), and
if \(F'\) is an algebraic space, then \(j^{-1}F'\) is an algebraic space, and if \(F' = U'/R'\) is a presentation, then \(j^{-1}F' = U'_S/R'_S\) is a presentation.
Proof
The functors \(j_!\), \(j_*\) and \(j^{-1}\) are defined in Sites, Lemma 03EH where it is also shown that \(j = j_{S/S'}\) is the localization of \((\Sch/S')_{fppf}\) at the object \(S/S'\). Hence all of the material on localization functors is available for \(j\). The formula in (1) is Sites, Lemma 03HT. By definition \(j_!\) is the left adjoint to restriction \(j^{-1}\), hence \(j_!\) is right exact. By Sites, Lemma 04BB it also commutes with fibre products and equalizers. By Sites, Lemma 03HU we see that \(j_!h_X = h_{j(X)}\) hence (2) holds. If \(F\) is an algebraic space over \(S\), then we can write \(F = U/R\) (Lemma 0262) and we get \[j_!F = j(U)/j(R)\] because \(j_!\) being right exact commutes with coequalizers, and moreover \(j(R) = j(U) \times_{j_!F} j(U)\) as \(j_!\) commutes with fibre products. Since the morphisms \(j(s), j(t) : j(R) \to j(U)\) are simply the morphisms \(s, t : R \to U\) (but viewed as morphisms of schemes over \(S'\)), they are still étale. Thus \((j(U), j(R), s, t)\) is an étale equivalence relation. Hence by Theorem 02WW we conclude that \(j_!F\) is an algebraic space.
Proof of (4), (5), and (6). The description of \(j^{-1}\) is in Sites, Section 00XZ. The restriction of the representable sheaf associated to \(X'/S'\) is the representable sheaf associated to \(X'_S = S \times_{S'} Y'\) by Sites, Lemma 03CE. The restriction functor \(j^{-1}\) is exact, hence \(j^{-1}F' = U'_S/R'_S\). Again by exactness the sheaf \(R'_S\) is still an equivalence relation on \(U'_S\). Finally the two maps \(R'_S \to U'_S\) are étale as base changes of the étale morphisms \(R' \to U'\). Hence \(j^{-1}F' = U'_S/R'_S\) is an algebraic space by Theorem 02WW and we win.
Note how the presentation \(j_!F = j(U)/j(R)\) is just the presentation of \(F\) but viewed as a presentation by schemes over \(S'\). Hence the following definition makes sense.
Definition
Let \(\Sch_{fppf}\) be a big fppf site. Let \(S \to S'\) be a morphism of this site.
If \(F'\) is an algebraic space over \(S'\), then the base change of \(F'\) to \(S\) is the algebraic space \(j^{-1}F'\) described in Lemma 03I4. We denote it \(F'_S\).
If \(F\) is an algebraic space over \(S\), then \(F\) viewed as an algebraic space over \(S'\) is the algebraic space \(j_!F\) over \(S'\) described in Lemma 03I4. We often simply denote this \(F\); if not then we will write \(j_!F\).
The algebraic space \(j_!F\) comes equipped with a canonical morphism \(j_!F \to S\) of algebraic spaces over \(S'\). This is true simply because the sheaf \(j_!F\) maps to \(h_S\) (see for example the explicit description in Lemma 03I4). In fact, in Sites, Lemma 00Y1 we have seen that the category of sheaves on \((\Sch/S)_{fppf}\) is equivalent to the category of pairs \((\mathcal{F}', \mathcal{F}' \to h_S)\) consisting of a sheaf on \((\Sch/S')_{fppf}\) and a map of sheaves \(\mathcal{F}' \to h_S\). The equivalence assigns to the sheaf \(\mathcal{F}\) the pair \((j_!\mathcal{F}, j_!\mathcal{F} \to h_S)\). This, combined with the above, leads to the following result for categories of algebraic spaces.
Lemma
Let \(\Sch_{fppf}\) be a big fppf site. Let \(S \to S'\) be a morphism of this site. The construction above give an equivalence of categories \[\left\{ \begin{matrix} \text{category of algebraic}\\ \text{spaces over }S \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{category of pairs }(F', F' \to S)\text{ consisting}\\ \text{of an algebraic space }F'\text{ over }S'\text{ and a}\\ \text{morphism }F' \to S\text{ of algebraic spaces over }S' \end{matrix} \right\}\]
Proof
Let \(F\) be an algebraic space over \(S\). The functor from left to right assigns the pair \((j_!F, j_!F \to S)\) to \(F\) which is an object of the right hand side by Lemma 03I4. Since this defines an equivalence of categories of sheaves by Sites, Lemma 00Y1 to finish the proof it suffices to show: if \(F\) is a sheaf and \(j_!F\) is an algebraic space, then \(F\) is an algebraic space. To do this, write \(j_!F = U'/R'\) as in Lemma 0262 with \(U', R' \in \Ob((\Sch/S')_{fppf})\). Then the compositions \(U' \to j_!F \to S\) and \(R' \to j_!F \to S\) are morphisms of schemes over \(S'\). Denote \(U, R\) the corresponding objects of \((\Sch/S)_{fppf}\). The two morphisms \(R' \to U'\) are morphisms over \(S\) and hence correspond to morphisms \(R \to U\). Since these are simply the same morphisms (but viewed over \(S\)) we see that we get an étale equivalence relation over \(S\). As \(j_!\) defines an equivalence of categories of sheaves (see reference above) we see that \(F = U/R\) and by Theorem 02WW we see that \(F\) is an algebraic space.
The following lemma is a slight rephrasing of the above.
Lemma
Let \(\Sch_{fppf}\) be a big fppf site. Let \(S \to S'\) be a morphism of this site. Let \(F'\) be a sheaf on \((\Sch/S')_{fppf}\). The following are equivalent:
The restriction \(F'|_{(\Sch/S)_{fppf}}\) is an algebraic space over \(S\), and
the sheaf \(h_S \times F'\) is an algebraic space over \(S'\).
Proof
The restriction and the product match under the equivalence of categories of Sites, Lemma 00Y1 so that Lemma 04SG above gives the result.
We finish this section with a lemma on a compatibility.
Lemma
Let \(\Sch_{fppf}\) be a big fppf site. Let \(S \to S'\) be a morphism of this site. Let \(F\) be an algebraic space over \(S\). Let \(T\) be a scheme over \(S\) and let \(f : T \to F\) be a morphism over \(S\). Let \(f' : T' \to F'\) be the morphism over \(S'\) we get from \(f\) by applying the equivalence of categories described in Lemma 04SG. For any property \(\mathcal{P}\) as in Definition 025V we have \(\mathcal{P}(f') \Leftrightarrow \mathcal{P}(f)\).
Proof
Suppose that \(U\) is a scheme over \(S\), and \(U \to F\) is a surjective étale morphism. Denote \(U'\) the scheme \(U\) viewed as a scheme over \(S'\). In Lemma 03I4 we have seen that \(U' \to F'\) is surjective étale. Since \[j(T \times_{f, F} U) = T' \times_{f', F'} U'\] the morphism of schemes \(T \times_{f, F} U \to U\) is identified with the morphism of schemes \(T' \times_{f', F'} U' \to U'\). It is the same morphism, just viewed over different base schemes. Hence the lemma follows from Lemma 02WZ.
We will associate a topological space to an algebraic space in Properties of Spaces, Section 03BT, and its opens will correspond exactly to the open subspaces defined below.↩︎
In the literature this often refers to quasi-separated and locally separated algebraic spaces.↩︎
This definition was suggested by B. Conrad.↩︎
Requiring the existence of \(R\) is necessary because of our choice of the function \(Bound\) in Sets, Equation (046U). The size of the fibre product \(U \times_{X'} U\) can grow faster than \(Bound\) in terms of the size of \(U\). We can illustrate this by setting \(S = \Spec(A)\), \(U = \Spec(A[x_i, i \in I])\) and \(R = \coprod_{(\lambda_i) \in A^I} \Spec(A[x_i, y_i]/(x_i - \lambda_i y_i))\). In this case the size of \(R\) grows like \(\kappa^\kappa\) where \(\kappa\) is the size of \(U\).↩︎