Introduction
The purpose of this chapter is to find criteria guaranteeing that a stack in groupoids over the category of schemes with the fppf topology is an algebraic stack. Historically, this often involved proving that certain functors were representable, see Grothendieck’s lectures [Gr-I], [Gr-II], [Gr-III], [Gr-IV], [Gr-V], and [Gr-VI]. This explains the title of this chapter. Another important source 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].
Some of the notation, conventions and terminology in this chapter is awkward and may seem backwards to the more experienced reader. This is intentional. Please see Quot, Section 05X6 for an explanation.
Conventions
The conventions we use in this chapter are the same as those in the chapter on algebraic stacks, see Algebraic Stacks, Section 026M.
What we already know
The analogue of this chapter for algebraic spaces is the chapter entitled “Bootstrap”, see Bootstrap, Section 046B. That chapter already contains some representability results. Moreover, some of the preliminary material treated there we already have worked out in the chapter on algebraic stacks. Here is a list:
We discuss morphisms of presheaves representable by algebraic spaces in Bootstrap, Section 02YP. In Algebraic Stacks, Section 04SX we discuss the notion of a \(1\)-morphism of categories fibred in groupoids being representable by algebraic spaces.
We discuss properties of morphisms of presheaves representable by algebraic spaces in Bootstrap, Section 046E. In Algebraic Stacks, Section 03YJ we discuss properties of \(1\)-morphisms of categories fibred in groupoids representable by algebraic spaces.
We proved that if \(F\) is a sheaf whose diagonal is representable by algebraic spaces and which has an étale covering by an algebraic space, then \(F\) is an algebraic space, see Bootstrap, Theorem 03Y3. (This is a weak version of the result in the next item on the list.)
We proved that if \(F\) is a sheaf and if 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, then \(F\) is an algebraic space, see Bootstrap, Theorem 04S6.
We have also proved the “smooth” analogue of (05XI) for algebraic stacks: If \(\mathcal{X}\) is a stack in groupoids over \((\Sch/S)_{fppf}\) and if there exists a stack in groupoids \(\mathcal{U}\) over \((\Sch/S)_{fppf}\) which is representable by an algebraic space and a \(1\)-morphism \(u : \mathcal{U} \to \mathcal{X}\) which is representable by algebraic spaces, surjective, and smooth then \(\mathcal{X}\) is an algebraic stack, see Algebraic Stacks, Lemma 05UL.
Our first task now is to prove the analogue of (05XI) for algebraic stacks in general; it is Theorem 06DC.
Morphisms of stacks in groupoids
This section is preliminary and should be skipped on a first reading.
Lemma
Let \(\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(\mathcal{X} \to \mathcal{Z}\) and \(\mathcal{Y} \to \mathcal{Z}\) are representable by algebraic spaces and étale so is \(\mathcal{X} \to \mathcal{Y}\).
Proof
Let \(\mathcal{U}\) be a representable category fibred in groupoids over \(S\). Let \(f : \mathcal{U} \to \mathcal{Y}\) be a \(1\)-morphism. We have to show that \(\mathcal{X} \times_\mathcal{Y} \mathcal{U}\) is representable by an algebraic space and étale over \(\mathcal{U}\). Consider the composition \(h : \mathcal{U} \to \mathcal{Z}\). Then \[\mathcal{X} \times_\mathcal{Z} \mathcal{U} \longrightarrow \mathcal{Y} \times_\mathcal{Z} \mathcal{U}\] is a \(1\)-morphism between categories fibres in groupoids which are both representable by algebraic spaces and both étale over \(\mathcal{U}\). Hence by Properties of Spaces, Lemma 03FV this is represented by an étale morphism of algebraic spaces. Finally, we obtain the result we want as the morphism \(f\) induces a morphism \(\mathcal{U} \to \mathcal{Y} \times_\mathcal{Z} \mathcal{U}\) and we have \[\mathcal{X} \times_\mathcal{Y} \mathcal{U} = (\mathcal{X} \times_\mathcal{Z} \mathcal{U}) \times_{(\mathcal{Y} \times_\mathcal{Z} \mathcal{U})} \mathcal{U}.\]
Lemma
Let \(\mathcal{X}, \mathcal{Y}, \mathcal{Z}\) be stacks in groupoids over \((\Sch/S)_{fppf}\). Suppose that \(\mathcal{X} \to \mathcal{Y}\) and \(\mathcal{Z} \to \mathcal{Y}\) are \(1\)-morphisms. If
\(\mathcal{Y}\), \(\mathcal{Z}\) are representable by algebraic spaces \(Y\), \(Z\) over \(S\),
the associated morphism of algebraic spaces \(Y \to Z\) is surjective, flat and locally of finite presentation, and
\(\mathcal{Y} \times_\mathcal{Z} \mathcal{X}\) is a stack in setoids,
then \(\mathcal{X}\) is a stack in setoids.
Proof
This is a special case of Stacks, Lemma 05W9.
The following lemma is the analogue of Algebraic Stacks, Lemma 05UL and will be superseded by the stronger Theorem 06DC.
Lemma
Let \(S\) be a scheme. Let \(u : \mathcal{U} \to \mathcal{X}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). If
\(\mathcal{U}\) is representable by an algebraic space, and
\(u\) is representable by algebraic spaces, surjective, flat and locally of finite presentation,
then \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) representable by algebraic spaces.
Proof
Given two schemes \(T_1\), \(T_2\) over \(S\) denote \(\mathcal{T}_i = (\Sch/T_i)_{fppf}\) the associated representable fibre categories. Suppose given \(1\)-morphisms \(f_i : \mathcal{T}_i \to \mathcal{X}\). According to Algebraic Stacks, Lemma 045G it suffices to prove that the \(2\)-fibered product \(\mathcal{T}_1 \times_\mathcal{X} \mathcal{T}_2\) is representable by an algebraic space. By Stacks, Lemma 05UJ this is in any case a stack in setoids. Thus \(\mathcal{T}_1 \times_\mathcal{X} \mathcal{T}_2\) corresponds to some sheaf \(F\) on \((\Sch/S)_{fppf}\), see Stacks, Lemma 0432. Let \(U\) be the algebraic space which represents \(\mathcal{U}\). By assumption \[\mathcal{T}_i' = \mathcal{U} \times_{u, \mathcal{X}, f_i} \mathcal{T}_i\] is representable by an algebraic space \(T'_i\) over \(S\). Hence \(\mathcal{T}_1' \times_\mathcal{U} \mathcal{T}_2'\) is representable by the algebraic space \(T'_1 \times_U T'_2\). Consider the commutative diagram \[\xymatrix{ & \mathcal{T}_1 \times_{\mathcal X} \mathcal{T}_2 \ar[rr]\ar'[d][dd] & & \mathcal{T}_1 \ar[dd] \\ \mathcal{T}_1' \times_\mathcal{U} \mathcal{T}_2' \ar[ur]\ar[rr]\ar[dd] & & \mathcal{T}_1' \ar[ur]\ar[dd] \\ & \mathcal{T}_2 \ar'[r][rr] & & \mathcal X \\ \mathcal{T}_2' \ar[rr]\ar[ur] & & \mathcal{U} \ar[ur] }\] In this diagram the bottom square, the right square, the back square, and the front square are \(2\)-fibre products. A formal argument then shows that \(\mathcal{T}_1' \times_\mathcal{U} \mathcal{T}_2' \to \mathcal{T}_1 \times_{\mathcal X} \mathcal{T}_2\) is the “base change” of \(\mathcal{U} \to \mathcal{X}\), more precisely the diagram \[\xymatrix{ \mathcal{T}_1' \times_\mathcal{U} \mathcal{T}_2' \ar[d] \ar[r] & \mathcal{U} \ar[d] \\ \mathcal{T}_1 \times_{\mathcal X} \mathcal{T}_2 \ar[r] & \mathcal{X} }\] is a \(2\)-fibre square. Hence \(T'_1 \times_U T'_2 \to F\) is representable by algebraic spaces, flat, locally of finite presentation and surjective, see Algebraic Stacks, Lemmas 04SY, 0302, 04TC, and 045C. Therefore \(F\) is an algebraic space by Bootstrap, Theorem 04S6 and we win.
Lemma
Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). The following are equivalent
\(\Delta_\Delta : \mathcal{X} \to \mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X}\) is representable by algebraic spaces,
for every \(1\)-morphism \(\mathcal{V} \to \mathcal{X} \times \mathcal{X}\) with \(\mathcal{V}\) representable (by a scheme) the fibre product \(\mathcal{Y} = \mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}} \mathcal{V}\) has diagonal representable by algebraic spaces.
Proof
Although this is a bit of a brain twister, it is completely formal. Namely, recall that \(\mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X} = \mathcal{I}_\mathcal{X}\) is the inertia of \(\mathcal{X}\) and that \(\Delta_\Delta\) is the identity section of \(\mathcal{I}_\mathcal{X}\), see Categories, Section 04Z2. Thus condition (1) says the following: Given a scheme \(V\), an object \(x\) of \(\mathcal{X}\) over \(V\), and a morphism \(\alpha : x \to x\) of \(\mathcal{X}_V\) the condition “\(\alpha = \text{id}_x\)” defines an algebraic space over \(V\). (In other words, there exists a monomorphism of algebraic spaces \(W \to V\) such that a morphism of schemes \(f : T \to V\) factors through \(W\) if and only if \(f^*\alpha = \text{id}_{f^*x}\).)
On the other hand, let \(V\) be a scheme and let \(x, y\) be objects of \(\mathcal{X}\) over \(V\). Then \((x, y)\) define a morphism \(\mathcal{V} = (\Sch/V)_{fppf} \to \mathcal{X} \times \mathcal{X}\). Next, let \(h : V' \to V\) be a morphism of schemes and let \(\alpha : h^*x \to h^*y\) and \(\beta : h^*x \to h^*y\) be morphisms of \(\mathcal{X}_{V'}\). Then \((\alpha, \beta)\) define a morphism \(\mathcal{V}' = (\Sch/V)_{fppf} \to \mathcal{Y} \times \mathcal{Y}\). Condition (2) now says that (with any choices as above) the condition “\(\alpha = \beta\)” defines an algebraic space over \(V\).
To see the equivalence, given \((\alpha, \beta)\) as in (2) we see that (1) implies that “\(\alpha^{-1} \circ \beta = \text{id}_{h^*x}\)” defines an algebraic space. The implication (2) \(\Rightarrow\) (1) follows by taking \(h = \text{id}_V\) and \(\beta = \text{id}_x\).
Limit preserving on objects
Let \(S\) be a scheme. Let \(p : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). We will say that \(p\) is limit preserving on objects if the following condition holds: Given any data consisting of
an affine scheme \(U = \lim_{i \in I} U_i\) which is written as the directed limit of affine schemes \(U_i\) over \(S\),
an object \(y_i\) of \(\mathcal{Y}\) over \(U_i\) for some \(i\),
an object \(x\) of \(\mathcal{X}\) over \(U\), and
an isomorphism \(\gamma : p(x) \to y_i|_U\),
then there exists an \(i' \geq i\), an object \(x_{i'}\) of \(\mathcal{X}\) over \(U_{i'}\), an isomorphism \(\beta : x_{i'}|_U \to x\), and an isomorphism \(\gamma_{i'} : p(x_{i'}) \to y_i|_{U_{i'}}\) such that [06CU]\[\begin{equation} \vcenter{ \xymatrix{ p(x_{i'}|_U) \ar[d]_{p(\beta)} \ar[rr]_{\gamma_{i'}|_U} & & (y_i|_{U_{i'}})|_U \ar@{=}[d] \\ p(x) \ar[rr]^\gamma & & y_i|_U } } \end{equation}\] commutes. In this situation we say that “\((i', x_{i'}, \beta, \gamma_{i'})\) is a solution to the problem posed by our data (1), (2), (3), (4)”. The motivation for this definition comes from Limits of Spaces, Lemma 06BC.
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(p : \mathcal{X} \to \mathcal{Y}\) is limit preserving on objects, then so is the base change \(p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}\) of \(p\) by \(q\).
Proof
This is formal. Let \(U = \lim_{i \in I} U_i\) be the directed limit of affine schemes \(U_i\) over \(S\), let \(z_i\) be an object of \(\mathcal{Z}\) over \(U_i\) for some \(i\), let \(w\) be an object of \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\) over \(U\), and let \(\delta : p'(w) \to z_i|_U\) be an isomorphism. We may write \(w = (U, x, z, \alpha)\) for some object \(x\) of \(\mathcal{X}\) over \(U\) and object \(z\) of \(\mathcal{Z}\) over \(U\) and isomorphism \(\alpha : p(x) \to q(z)\). Note that \(p'(w) = z\) hence \(\delta : z \to z_i|_U\). Set \(y_i = q(z_i)\) and \(\gamma = q(\delta) \circ \alpha : p(x) \to y_i|_U\). As \(p\) is limit preserving on objects there exists an \(i' \geq i\) and an object \(x_{i'}\) of \(\mathcal{X}\) over \(U_{i'}\) as well as isomorphisms \(\beta : x_{i'}|_U \to x\) and \(\gamma_{i'} : p(x_{i'}) \to y_i|_{U_{i'}}\) such that (06CU) commutes. Then we consider the object \(w_{i'} = (U_{i'}, x_{i'}, z_i|_{U_{i'}}, \gamma_{i'})\) of \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\) over \(U_{i'}\) and define isomorphisms \[w_{i'}|_U = (U, x_{i'}|_U, z_i|_U, \gamma_{i'}|_U) \xrightarrow{(\beta, \delta^{-1})} (U, x, z, \alpha) = w\] and \[p'(w_{i'}) = z_i|_{U_{i'}} \xrightarrow{\text{id}} z_i|_{U_{i'}}.\] These combine to give a solution to the problem.
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Y} \to \mathcal{Z}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(p\) and \(q\) are limit preserving on objects, then so is the composition \(q \circ p\).
Proof
This is formal. Let \(U = \lim_{i \in I} U_i\) be the directed limit of affine schemes \(U_i\) over \(S\), let \(z_i\) be an object of \(\mathcal{Z}\) over \(U_i\) for some \(i\), let \(x\) be an object of \(\mathcal{X}\) over \(U\), and let \(\gamma : q(p(x)) \to z_i|_U\) be an isomorphism. As \(q\) is limit preserving on objects there exist an \(i' \geq i\), an object \(y_{i'}\) of \(\mathcal{Y}\) over \(U_{i'}\), an isomorphism \(\beta : y_{i'}|_U \to p(x)\), and an isomorphism \(\gamma_{i'} : q(y_{i'}) \to z_i|_{U_{i'}}\) such that (06CU) is commutative. As \(p\) is limit preserving on objects there exist an \(i'' \geq i'\), an object \(x_{i''}\) of \(\mathcal{X}\) over \(U_{i''}\), an isomorphism \(\beta' : x_{i''}|_U \to x\), and an isomorphism \(\gamma'_{i''} : p(x_{i''}) \to y_{i'}|_{U_{i''}}\) such that (06CU) is commutative. The solution is to take \(x_{i''}\) over \(U_{i''}\) with isomorphism \[q(p(x_{i''})) \xrightarrow{q(\gamma'_{i''})} q(y_{i'})|_{U_{i''}} \xrightarrow{\gamma_{i'}|_{U_{i''}}} z_i|_{U_{i''}}\] and isomorphism \(\beta' : x_{i''}|_U \to x\). We omit the verification that (06CU) is commutative.
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(p\) is representable by algebraic spaces, then the following are equivalent:
\(p\) is limit preserving on objects, and
\(p\) is locally of finite presentation (see Algebraic Stacks, Definition 03YK).
Proof
Assume (2). Let \(U = \lim_{i \in I} U_i\) be the directed limit of affine schemes \(U_i\) over \(S\), let \(y_i\) be an object of \(\mathcal{Y}\) over \(U_i\) for some \(i\), let \(x\) be an object of \(\mathcal{X}\) over \(U\), and let \(\gamma : p(x) \to y_i|_U\) be an isomorphism. Let \(X_{y_i}\) denote an algebraic space over \(U_i\) representing the \(2\)-fibre product \[(\Sch/U_i)_{fppf} \times_{y_i, \mathcal{Y}, p} \mathcal{X}.\] Note that \(\xi = (U, U \to U_i, x, \gamma^{-1})\) defines an object of this \(2\)-fibre product over \(U\). Via the \(2\)-Yoneda lemma \(\xi\) corresponds to a morphism \(f_\xi : U \to X_{y_i}\) over \(U_i\). By Limits of Spaces, Proposition 04AK there exists an \(i' \geq i\) and a morphism \(f_{i'} : U_{i'} \to X_{y_i}\) such that \(f_\xi\) is the composition of \(f_{i'}\) and the projection morphism \(U \to U_{i'}\). Also, the \(2\)-Yoneda lemma tells us that \(f_{i'}\) corresponds to an object \(\xi_{i'} = (U_{i'}, U_{i'} \to U_i, x_{i'}, \alpha)\) of the displayed \(2\)-fibre product over \(U_{i'}\) whose restriction to \(U\) recovers \(\xi\). In particular we obtain an isomorphism \(\gamma : x_{i'}|U \to x\). Note that \(\alpha : y_i|_{U_{i'}} \to p(x_{i'})\). Hence we see that taking \(x_{i'}\), the isomorphism \(\gamma : x_{i'}|U \to x\), and the isomorphism \(\beta = \alpha^{-1} : p(x_{i'}) \to y_i|_{U_{i'}}\) is a solution to the problem.
Assume (1). Choose a scheme \(T\) and a \(1\)-morphism \(y : (\Sch/T)_{fppf} \to \mathcal{Y}\). Let \(X_y\) be an algebraic space over \(T\) representing the \(2\)-fibre product \((\Sch/T)_{fppf} \times_{y, \mathcal{Y}, p} \mathcal{X}\). We have to show that \(X_y \to T\) is locally of finite presentation. To do this we will use the criterion in Limits of Spaces, Remark 05N0. Consider an affine scheme \(U = \lim_{i \in I} U_i\) written as the directed limit of affine schemes over \(T\). Pick any \(i \in I\) and set \(y_i = y|_{U_i}\). Also denote \(i'\) an element of \(I\) which is bigger than or equal to \(i\). By the \(2\)-Yoneda lemma morphisms \(U \to X_y\) over \(T\) correspond bijectively to isomorphism classes of pairs \((x, \alpha)\) where \(x\) is an object of \(\mathcal{X}\) over \(U\) and \(\alpha : y|_U \to p(x)\) is an isomorphism. Of course giving \(\alpha\) is, up to an inverse, the same thing as giving an isomorphism \(\gamma : p(x) \to y_i|_U\). Similarly for morphisms \(U_{i'} \to X_y\) over \(T\). Hence (1) guarantees that the canonical map \[\colim_{i' \geq i} X_y(U_{i'}) \longrightarrow X_y(U)\] is surjective in this situation. It follows from Limits of Spaces, Lemma 0CM6 that \(X_y \to T\) is locally of finite presentation.
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Assume \(p\) is representable by algebraic spaces and an open immersion. Then \(p\) is limit preserving on objects.
Proof
This follows from Lemma 06CX and (via the general principle Algebraic Stacks, Lemma 045F) from the fact that an open immersion of algebraic spaces is locally of finite presentation, see Morphisms of Spaces, Lemma 06CN.
Let \(S\) be a scheme. In the following lemma we need the notion of the size of an algebraic space \(X\) over \(S\). Namely, given a cardinal \(\kappa\) we will say \(X\) has \(\text{size}(X) \leq \kappa\) if and only if there exists a scheme \(U\) with \(\text{size}(U) \leq \kappa\) (see Sets, Section 000H) and a surjective étale morphism \(U \to X\).
Lemma
Let \(S\) be a scheme. Let \(\kappa = \text{size}(T)\) for some \(T \in \Ob((\Sch/S)_{fppf})\). Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\) such that
\(\mathcal{Y} \to (\Sch/S)_{fppf}\) is limit preserving on objects,
for an affine scheme \(V\) locally of finite presentation over \(S\) and \(y \in \Ob(\mathcal{Y}_V)\) the fibre product \((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}\) is representable by an algebraic space of size \(\leq \kappa\)1,
\(\mathcal{X}\) and \(\mathcal{Y}\) are stacks for the Zariski topology.
Then \(f\) is representable by algebraic spaces.
Proof
Let \(V\) be a scheme over \(S\) and \(y \in \mathcal{Y}_V\). We have to prove \((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}\) is representable by an algebraic space.
Case I: \(V\) is affine and maps into an affine open \(\Spec(\Lambda) \subset S\). Then we can write \(V = \lim V_i\) with each \(V_i\) affine and of finite presentation over \(\Spec(\Lambda)\), see Algebra, Lemma 00QN. Then \(y\) comes from an object \(y_i\) over \(V_i\) for some \(i\) by assumption (1). By assumption (3) the fibre product \((\Sch/V_i)_{fppf} \times_{y_i, \mathcal{Y}} \mathcal{X}\) is representable by an algebraic space \(Z_i\). Then \((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}\) is representable by \(Z \times_{V_i} V\).
Case II: \(V\) is general. Choose an affine open covering \(V = \bigcup_{i \in I} V_i\) such that each \(V_i\) maps into an affine open of \(S\). We first claim that \(\mathcal{Z} = (\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}\) is a stack in setoids for the Zariski topology. Namely, it is a stack in groupoids for the Zariski topology by Stacks, Lemma 02ZL. Then suppose that \(z\) is an object of \(\mathcal{Z}\) over a scheme \(T\). Denote \(g : T \to V\) the morphism corresponding to the projection of \(z\) in \((\Sch/V)_{fppf}\). Consider the Zariski sheaf \(\mathit{I} = \mathit{Isom}_{\mathcal{Z}}(z, z)\). By Case I we see that \(\mathit{I}|_{g^{-1}(V_i)} = *\) (the singleton sheaf). Hence \(\mathcal{I} = *\). Thus \(\mathcal{Z}\) is fibred in setoids. To finish the proof we have to show that the Zariski sheaf \(Z : T \mapsto \Ob(\mathcal{Z}_T)/\cong\) is an algebraic space, see Algebraic Stacks, Lemma 02ZX. There is a map \(p : Z \to V\) (transformation of functors) and by Case I we know that \(Z_i = p^{-1}(V_i)\) is an algebraic space. The morphisms \(Z_i \to Z\) are representable by open immersions and \(\coprod Z_i \to Z\) is surjective (in the Zariski topology). Hence \(Z\) is a sheaf for the fppf topology by Bootstrap, Lemma 07WE. Thus Spaces, Lemma 02WR applies and we conclude that \(Z\) is an algebraic space2.
Lemma
Let \(S\) be a scheme. Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(\mathcal{P}\) be a property of morphisms of algebraic spaces as in Algebraic Stacks, Definition 03YK. If
\(f\) is representable by algebraic spaces,
\(\mathcal{Y} \to (\Sch/S)_{fppf}\) is limit preserving on objects,
for an affine scheme \(V\) locally of finite presentation over \(S\) and \(y \in \mathcal{Y}_V\) the resulting morphism of algebraic spaces \(f_y : F_y \to V\), see Algebraic Stacks, Equation (0402), has property \(\mathcal{P}\).
Then \(f\) has property \(\mathcal{P}\).
Proof
Let \(V\) be a scheme over \(S\) and \(y \in \mathcal{Y}_V\). We have to show that \(F_y \to V\) has property \(\mathcal{P}\). Since \(\mathcal{P}\) is fppf local on the base we may assume that \(V\) is an affine scheme which maps into an affine open \(\Spec(\Lambda) \subset S\). Thus we can write \(V = \lim V_i\) with each \(V_i\) affine and of finite presentation over \(\Spec(\Lambda)\), see Algebra, Lemma 00QN. Then \(y\) comes from an object \(y_i\) over \(V_i\) for some \(i\) by assumption (2). By assumption (3) the morphism \(F_{y_i} \to V_i\) has property \(\mathcal{P}\). As \(\mathcal{P}\) is stable under arbitrary base change and since \(F_y = F_{y_i} \times_{V_i} V\) we conclude that \(F_y \to V\) has property \(\mathcal{P}\) as desired.
Formally smooth on objects
Let \(S\) be a scheme. Let \(p : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). We will say that \(p\) is formally smooth on objects if the following condition holds: Given any data consisting of
a first order thickening \(U \subset U'\) of affine schemes over \(S\),
an object \(y'\) of \(\mathcal{Y}\) over \(U'\),
an object \(x\) of \(\mathcal{X}\) over \(U\), and
an isomorphism \(\gamma : p(x) \to y'|_U\),
then there exists an object \(x'\) of \(\mathcal{X}\) over \(U'\) with an isomorphism \(\beta : x'|_U \to x\) and an isomorphism \(\gamma' : p(x') \to y'\) such that [06D0]\[\begin{equation} \vcenter{ \xymatrix{ p(x'|_U) \ar[d]_{p(\beta)} \ar[rr]_{\gamma'|_U} & & y'|_U \ar@{=}[d] \\ p(x) \ar[rr]^\gamma & & y'|_U } } \end{equation}\] commutes. In this situation we say that “\((x', \beta, \gamma')\) is a solution to the problem posed by our data (1), (2), (3), (4)”.
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(p : \mathcal{X} \to \mathcal{Y}\) is formally smooth on objects, then so is the base change \(p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}\) of \(p\) by \(q\).
Proof
This is formal. Let \(U \subset U'\) be a first order thickening of affine schemes over \(S\), let \(z'\) be an object of \(\mathcal{Z}\) over \(U'\), let \(w\) be an object of \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\) over \(U\), and let \(\delta : p'(w) \to z'|_U\) be an isomorphism. We may write \(w = (U, x, z, \alpha)\) for some object \(x\) of \(\mathcal{X}\) over \(U\) and object \(z\) of \(\mathcal{Z}\) over \(U\) and isomorphism \(\alpha : p(x) \to q(z)\). Note that \(p'(w) = z\) hence \(\delta : z \to z|_U\). Set \(y' = q(z')\) and \(\gamma = q(\delta) \circ \alpha : p(x) \to y'|_U\). As \(p\) is formally smooth on objects there exists an object \(x'\) of \(\mathcal{X}\) over \(U'\) as well as isomorphisms \(\beta : x'|_U \to x\) and \(\gamma' : p(x') \to y'\) such that (06D0) commutes. Then we consider the object \(w = (U', x', z', \gamma')\) of \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\) over \(U'\) and define isomorphisms \[w'|_U = (U, x'|_U, z'|_U, \gamma'|_U) \xrightarrow{(\beta, \delta^{-1})} (U, x, z, \alpha) = w\] and \[p'(w') = z' \xrightarrow{\text{id}} z'.\] These combine to give a solution to the problem.
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Y} \to \mathcal{Z}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(p\) and \(q\) are formally smooth on objects, then so is the composition \(q \circ p\).
Proof
This is formal. Let \(U \subset U'\) be a first order thickening of affine schemes over \(S\), let \(z'\) be an object of \(\mathcal{Z}\) over \(U'\), let \(x\) be an object of \(\mathcal{X}\) over \(U\), and let \(\gamma : q(p(x)) \to z'|_U\) be an isomorphism. As \(q\) is formally smooth on objects there exist an object \(y'\) of \(\mathcal{Y}\) over \(U'\), an isomorphism \(\beta : y'|_U \to p(x)\), and an isomorphism \(\gamma' : q(y') \to z'\) such that (06D0) is commutative. As \(p\) is formally smooth on objects there exist an object \(x'\) of \(\mathcal{X}\) over \(U'\), an isomorphism \(\beta' : x'|_U \to x\), and an isomorphism \(\gamma'' : p(x') \to y'\) such that (06D0) is commutative. The solution is to take \(x'\) over \(U'\) with isomorphism \[q(p(x')) \xrightarrow{q(\gamma'')} q(y') \xrightarrow{\gamma'} z'\] and isomorphism \(\beta' : x'|_U \to x\). We omit the verification that (06D0) is commutative.
Note that the class of formally smooth morphisms of algebraic spaces is stable under arbitrary base change and local on the target in the fpqc topology, see More on Morphisms of Spaces, Lemma 061F and 06CS. Hence condition (2) in the lemma below makes sense.
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(p\) is representable by algebraic spaces, then the following are equivalent:
\(p\) is formally smooth on objects, and
\(p\) is formally smooth (see Algebraic Stacks, Definition 03YK).
Proof
Assume (2). Let \(U \subset U'\) be a first order thickening of affine schemes over \(S\), let \(y'\) be an object of \(\mathcal{Y}\) over \(U'\), let \(x\) be an object of \(\mathcal{X}\) over \(U\), and let \(\gamma : p(x) \to y'|_U\) be an isomorphism. Let \(X_{y'}\) denote an algebraic space over \(U'\) representing the \(2\)-fibre product \[(\Sch/U')_{fppf} \times_{y', \mathcal{Y}, p} \mathcal{X}.\] Note that \(\xi = (U, U \to U', x, \gamma^{-1})\) defines an object of this \(2\)-fibre product over \(U\). Via the \(2\)-Yoneda lemma \(\xi\) corresponds to a morphism \(f_\xi : U \to X_{y'}\) over \(U'\). As \(X_{y'} \to U'\) is formally smooth by assumption there exists a morphism \(f' : U' \to X_{y'}\) such that \(f_\xi\) is the composition of \(f'\) and the morphism \(U \to U'\). Also, the \(2\)-Yoneda lemma tells us that \(f'\) corresponds to an object \(\xi' = (U', U' \to U', x', \alpha)\) of the displayed \(2\)-fibre product over \(U'\) whose restriction to \(U\) recovers \(\xi\). In particular we obtain an isomorphism \(\gamma : x'|U \to x\). Note that \(\alpha : y' \to p(x')\). Hence we see that taking \(x'\), the isomorphism \(\gamma : x'|U \to x\), and the isomorphism \(\beta = \alpha^{-1} : p(x') \to y'\) is a solution to the problem.
Assume (1). Choose a scheme \(T\) and a \(1\)-morphism \(y : (\Sch/T)_{fppf} \to \mathcal{Y}\). Let \(X_y\) be an algebraic space over \(T\) representing the \(2\)-fibre product \((\Sch/T)_{fppf} \times_{y, \mathcal{Y}, p} \mathcal{X}\). We have to show that \(X_y \to T\) is formally smooth. Hence it suffices to show that given a first order thickening \(U \subset U'\) of affine schemes over \(T\), then \(X_y(U') \to X_y(U')\) is surjective (morphisms in the category of algebraic spaces over \(T\)). Set \(y' = y|_{U'}\). By the \(2\)-Yoneda lemma morphisms \(U \to X_y\) over \(T\) correspond bijectively to isomorphism classes of pairs \((x, \alpha)\) where \(x\) is an object of \(\mathcal{X}\) over \(U\) and \(\alpha : y|_U \to p(x)\) is an isomorphism. Of course giving \(\alpha\) is, up to an inverse, the same thing as giving an isomorphism \(\gamma : p(x) \to y'|_U\). Similarly for morphisms \(U' \to X_y\) over \(T\). Hence (1) guarantees the surjectivity of \(X_y(U') \to X_y(U')\) in this situation and we win.
Surjective on objects
Let \(S\) be a scheme. Let \(p : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). We will say that \(p\) is surjective on objects if the following condition holds: Given any data consisting of
a field \(k\) over \(S\), and
an object \(y\) of \(\mathcal{Y}\) over \(\Spec(k)\),
then there exists an extension \(K/k\) of fields over \(S\), an object \(x\) of \(\mathcal{X}\) over \(\Spec(K)\) such that \(p(x) \cong y|_{\Spec(K)}\).
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(p : \mathcal{X} \to \mathcal{Y}\) is surjective on objects, then so is the base change \(p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}\) of \(p\) by \(q\).
Proof
This is formal. Let \(z\) be an object of \(\mathcal{Z}\) over a field \(k\). As \(p\) is surjective on objects there exists an extension \(K/k\) and an object \(x\) of \(\mathcal{X}\) over \(K\) and an isomorphism \(\alpha : p(x) \to q(z)|_{\Spec(K)}\). Then \(w = (\Spec(K), x, z|_{\Spec(K)}, \alpha)\) is an object of \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\) over \(K\) with \(p'(w) = z|_{\Spec(K)}\).
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Y} \to \mathcal{Z}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(p\) and \(q\) are surjective on objects, then so is the composition \(q \circ p\).
Proof
This is formal. Let \(z\) be an object of \(\mathcal{Z}\) over a field \(k\). As \(q\) is surjective on objects there exists a field extension \(K/k\) and an object \(y\) of \(\mathcal{Y}\) over \(K\) such that \(q(y) \cong x|_{\Spec(K)}\). As \(p\) is surjective on objects there exists a field extension \(L/K\) and an object \(x\) of \(\mathcal{X}\) over \(L\) such that \(p(x) \cong y|_{\Spec(L)}\). Then the field extension \(L/k\) and the object \(x\) of \(\mathcal{X}\) over \(L\) satisfy \(q(p(x)) \cong z|_{\Spec(L)}\) as desired.
Lemma
Let \(p : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(p\) is representable by algebraic spaces, then the following are equivalent:
\(p\) is surjective on objects, and
\(p\) is surjective (see Algebraic Stacks, Definition 03YK).
Proof
Assume (2). Let \(k\) be a field and let \(y\) be an object of \(\mathcal{Y}\) over \(k\). Let \(X_y\) denote an algebraic space over \(k\) representing the \(2\)-fibre product \[(\Sch/\Spec(k))_{fppf} \times_{y, \mathcal{Y}, p} \mathcal{X}.\] As we’ve assumed that \(p\) is surjective we see that \(X_y\) is not empty. Hence we can find a field extension \(K/k\) and a \(K\)-valued point \(x\) of \(X_y\). Via the \(2\)-Yoneda lemma this corresponds to an object \(x\) of \(\mathcal{X}\) over \(K\) together with an isomorphism \(p(x) \cong y|_{\Spec(K)}\) and we see that (1) holds.
Assume (1). Choose a scheme \(T\) and a \(1\)-morphism \(y : (\Sch/T)_{fppf} \to \mathcal{Y}\). Let \(X_y\) be an algebraic space over \(T\) representing the \(2\)-fibre product \((\Sch/T)_{fppf} \times_{y, \mathcal{Y}, p} \mathcal{X}\). We have to show that \(X_y \to T\) is surjective. By Morphisms of Spaces, Definition 03ME we have to show that \(|X_y| \to |T|\) is surjective. This means exactly that given a field \(k\) over \(T\) and a morphism \(t : \Spec(k) \to T\) there exists a field extension \(K/k\) and a morphism \(x : \Spec(K) \to X_y\) such that \[\xymatrix{ \Spec(K) \ar[d] \ar[r]_x & X_y \ar[d] \\ \Spec(k) \ar[r]^t & T }\] commutes. By the \(2\)-Yoneda lemma this means exactly that we have to find \(k \subset K\) and an object \(x\) of \(\mathcal{X}\) over \(K\) such that \(p(x) \cong t^*y|_{\Spec(K)}\). Hence (1) guarantees that this is the case and we win.
Algebraic morphisms
The following notion is occasionally useful.
Definition
Let \(S\) be a scheme. Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). We say that \(F\) is algebraic if for every scheme \(T\) and every object \(\xi\) of \(\mathcal{Y}\) over \(T\) the \(2\)-fibre product \[(\Sch/T)_{fppf} \times_{\xi, \mathcal{Y}} \mathcal{X}\] is an algebraic stack over \(S\).
With this terminology in place we have the following result that generalizes Algebraic Stacks, Lemma 05UM.
Lemma
Let \(S\) be a scheme. Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). If
\(\mathcal{Y}\) is an algebraic stack, and
\(F\) is algebraic (see above),
then \(\mathcal{X}\) is an algebraic stack.
Proof
By assumption (1) there exists a scheme \(T\) and an object \(\xi\) of \(\mathcal{Y}\) over \(T\) such that the corresponding \(1\)-morphism \(\xi : (\Sch/T)_{fppf} \to \mathcal{Y}\) is smooth an surjective. Then \(\mathcal{U} = (\Sch/T)_{fppf} \times_{\xi, \mathcal{Y}} \mathcal{X}\) is an algebraic stack by assumption (2). Choose a scheme \(U\) and a surjective smooth \(1\)-morphism \((\Sch/U)_{fppf} \to \mathcal{U}\). The projection \(\mathcal{U} \longrightarrow \mathcal{X}\) is, as the base change of the morphism \(\xi : (\Sch/T)_{fppf} \to \mathcal{Y}\), surjective and smooth, see Algebraic Stacks, Lemma 045C. Then the composition \((\Sch/U)_{fppf} \to \mathcal{U} \to \mathcal{X}\) is surjective and smooth as a composition of surjective and smooth morphisms, see Algebraic Stacks, Lemma 045B. Hence \(\mathcal{X}\) is an algebraic stack by Algebraic Stacks, Lemma 05UL.
Lemma
Let \(S\) be a scheme. Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). If \(\mathcal{X}\) is an algebraic stack and \(\Delta : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}\) is representable by algebraic spaces, then \(F\) is algebraic.
Proof
Choose a representable stack in groupoids \(\mathcal{U}\) and a surjective smooth \(1\)-morphism \(\mathcal{U} \to \mathcal{X}\). Let \(T\) be a scheme and let \(\xi\) be an object of \(\mathcal{Y}\) over \(T\). The morphism of \(2\)-fibre products \[(\Sch/T)_{fppf} \times_{\xi, \mathcal{Y}} \mathcal{U} \longrightarrow (\Sch/T)_{fppf} \times_{\xi, \mathcal{Y}} \mathcal{X}\] is representable by algebraic spaces, surjective, and smooth as a base change of \(\mathcal{U} \to \mathcal{X}\), see Algebraic Stacks, Lemmas 0302 and 045C. By our condition on the diagonal of \(\mathcal{Y}\) we see that the source of this morphism is representable by an algebraic space, see Algebraic Stacks, Lemma 045G. Hence the target is an algebraic stack by Algebraic Stacks, Lemma 05UL.
Lemma
Let \(S\) be a scheme. Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). If \(F\) is algebraic and \(\Delta : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}\) is representable by algebraic spaces, then \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces.
Proof
Assume \(F\) is algebraic and \(\Delta : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}\) is representable by algebraic spaces. Take a scheme \(U\) over \(S\) and two objects \(x_1, x_2\) of \(\mathcal{X}\) over \(U\). We have to show that \(\mathit{Isom}(x_1, x_2)\) is an algebraic space over \(U\), see Algebraic Stacks, Lemma 045G. Set \(y_i = F(x_i)\). We have a morphism of sheaves of sets \[f : \mathit{Isom}(x_1, x_2) \to \mathit{Isom}(y_1, y_2)\] and the target is an algebraic space by assumption. Thus it suffices to show that \(f\) is representable by algebraic spaces, see Bootstrap, Lemma 02YS. Thus we can choose a scheme \(V\) over \(U\) and an isomorphism \(\beta : y_{1, V} \to y_{2, V}\) and we have to show the functor \[(\Sch/V)_{fppf} \to \textit{Sets},\quad T/V \mapsto \{\alpha : x_{1, T} \to x_{2, T} \text{ in }\mathcal{X}_T \mid F(\alpha) = \beta|_T\}\] is an algebraic space. Consider the objects \(z_1 = (V, x_{1, V}, \text{id})\) and \(z_2 = (V, x_{2, V}, \beta)\) of \[\mathcal{Z} = (\Sch/V)_{fppf} \times_{y_{1, V}, \mathcal{Y}} \mathcal{X}\] Then it is straightforward to verify that the functor above is equal to \(\mathit{Isom}(z_1, z_2)\) on \((\Sch/V)_{fppf}\). Hence this is an algebraic space by our assumption that \(F\) is algebraic (and the definition of algebraic stacks).
Spaces of sections
Given morphisms \(W \to Z \to U\) we can consider the functor that associates to a scheme \(U'\) over \(U\) the set of sections \(\sigma : Z_{U'} \to W_{U'}\) of the base change \(W_{U'} \to Z_{U'}\) of the morphism \(W \to Z\). In this section we prove some preliminary lemmas on this functor.
Lemma
Let \(Z \to U\) be a finite morphism of schemes. Let \(W\) be an algebraic space and let \(W \to Z\) be a surjective étale morphism. Then there exists a surjective étale morphism \(U' \to U\) and a section \[\sigma : Z_{U'} \to W_{U'}\] of the morphism \(W_{U'} \to Z_{U'}\).
Proof
We may choose a separated scheme \(W'\) and a surjective étale morphism \(W' \to W\). Hence after replacing \(W\) by \(W'\) we may assume that \(W\) is a separated scheme. Write \(f : W \to Z\) and \(\pi : Z \to U\). Note that \(f \circ \pi : W \to U\) is separated as \(W\) is separated (see Schemes, Lemma 01KV). Let \(u \in U\) be a point. Clearly it suffices to find an étale neighbourhood \((U', u')\) of \((U, u)\) such that a section \(\sigma\) exists over \(U'\). Let \(z_1, \ldots, z_r\) be the points of \(Z\) lying above \(u\). For each \(i\) choose a point \(w_i \in W\) which maps to \(z_i\). We may pick an étale neighbourhood \((U', u') \to (U, u)\) such that the conclusions of More on Morphisms, Lemma 02LO hold for both \(Z \to U\) and the points \(z_1, \ldots, z_r\) and \(W \to U\) and the points \(w_1, \ldots, w_r\). Hence, after replacing \((U, u)\) by \((U', u')\) and relabeling, we may assume that all the field extensions \(\kappa(z_i)/\kappa(u)\) and \(\kappa(w_i)/\kappa(u)\) are purely inseparable, and moreover that there exist disjoint union decompositions \[Z = V_1 \amalg \ldots \amalg V_r \amalg A, \quad W = W_1 \amalg \ldots \amalg W_r \amalg B\] by open and closed subschemes with \(z_i \in V_i\), \(w_i \in W_i\) and \(V_i \to U\), \(W_i \to U\) finite. After replacing \(U\) by \(U \setminus \pi(A)\) we may assume that \(A = \emptyset\), i.e., \(Z = V_1 \amalg \ldots \amalg V_r\). After replacing \(W_i\) by \(W_i \cap f^{-1}(V_i)\) and \(B\) by \(B \cup \bigcup W_i \cap f^{-1}(Z \setminus V_i)\) we may assume that \(f\) maps \(W_i\) into \(V_i\). Then \(f_i = f|_{W_i} : W_i \to V_i\) is a morphism of schemes finite over \(U\), hence finite (see Morphisms, Lemma 035D). It is also étale (by assumption), \(f_i^{-1}(\{z_i\}) = w_i\), and induces an isomorphism of residue fields \(\kappa(z_i) = \kappa(w_i)\) (because both are purely inseparable extensions of \(\kappa(u)\) and \(\kappa(w_i)/\kappa(z_i)\) is separable as \(f\) is étale). Hence by Étale Morphisms, Lemma 04DH we see that \(f_i\) is an isomorphism in a neighbourhood \(V_i'\) of \(z_i\). Since \(\pi : Z \to U\) is closed, after shrinking \(U\), we may assume that \(W_i \to V_i\) is an isomorphism. This proves the lemma.
Lemma
Let \(Z \to U\) be a finite locally free morphism of schemes. Let \(W\) be an algebraic space and let \(W \to Z\) be an étale morphism. Then the functor \[F : (\Sch/U)_{fppf}^{opp} \longrightarrow \textit{Sets},\] defined by the rule \[U' \longmapsto F(U') = \{\sigma : Z_{U'} \to W_{U'}\text{ section of }W_{U'} \to Z_{U'}\}\] is an algebraic space and the morphism \(F \to U\) is étale.
Proof
Assume first that \(W \to Z\) is also separated. Let \(U'\) be a scheme over \(U\) and let \(\sigma \in F(U')\). By Morphisms of Spaces, Lemma 03KP the morphism \(\sigma\) is a closed immersion. Moreover, \(\sigma\) is étale by Properties of Spaces, Lemma 03FV. Hence \(\sigma\) is also an open immersion, see Morphisms of Spaces, Lemma 05W5. In other words, \(Z_\sigma = \sigma(Z_{U'}) \subset W_{U'}\) is an open subspace such that the morphism \(Z_\sigma \to Z_{U'}\) is an isomorphism. In particular, the morphism \(Z_\sigma \to U'\) is finite. Hence we obtain a transformation of functors \[F \longrightarrow (W/U)_{fin}, \quad \sigma \longmapsto (U' \to U, Z_\sigma)\] where \((W/U)_{fin}\) is the finite part of the morphism \(W \to U\) introduced in More on Groupoids in Spaces, Section 04PB. It is clear that this transformation of functors is injective (since we can recover \(\sigma\) from \(Z_\sigma\) as the inverse of the isomorphism \(Z_\sigma \to Z_{U'}\)). By More on Groupoids in Spaces, Proposition 04QH we know that \((W/U)_{fin}\) is an algebraic space étale over \(U\). Hence to finish the proof in this case it suffices to show that \(F \to (W/U)_{fin}\) is representable and an open immersion. To see this suppose that we are given a morphism of schemes \(U' \to U\) and an open subspace \(Z' \subset W_{U'}\) such that \(Z' \to U'\) is finite. Then it suffices to show that there exists an open subscheme \(U'' \subset U'\) such that a morphism \(T \to U'\) factors through \(U''\) if and only if \(Z' \times_{U'} T\) maps isomorphically to \(Z \times_{U'} T\). This follows from More on Morphisms of Spaces, Lemma 05XD (here we use that \(Z \to B\) is flat and locally of finite presentation as well as finite). Hence we have proved the lemma in case \(W \to Z\) is separated as well as étale.
In the general case we choose a separated scheme \(W'\) and a surjective étale morphism \(W' \to W\). Note that the morphisms \(W' \to W\) and \(W \to Z\) are separated as their source is separated. Denote \(F'\) the functor associated to \(W' \to Z \to U\) as in the lemma. In the first paragraph of the proof we showed that \(F'\) is representable by an algebraic space étale over \(U\). By Lemma 05XQ the map of functors \(F' \to F\) is surjective for the étale topology on \(\Sch/U\). Moreover, if \(U'\) and \(\sigma : Z_{U'} \to W_{U'}\) define a point \(\xi \in F(U')\), then the fibre product \[F'' = F' \times_{F, \xi} U'\] is the functor on \(\Sch/U'\) associated to the morphisms \[W'_{U'} \times_{W_{U'}, \sigma} Z_{U'} \to Z_{U'} \to U'.\] Since the first morphism is separated as a base change of a separated morphism, we see that \(F''\) is an algebraic space étale over \(U'\) by the result of the first paragraph. It follows that \(F' \to F\) is a surjective étale transformation of functors, which is representable by algebraic spaces. Hence \(F\) is an algebraic space by Bootstrap, Theorem 04S6. Since \(F' \to F\) is an étale surjective morphism of algebraic spaces it follows that \(F \to U\) is étale because \(F' \to U\) is étale.
Relative morphisms
We continue the discussion started in More on Morphisms, Section 0BL0.
Let \(S\) be a scheme. Let \(Z \to B\) and \(X \to B\) be morphisms of algebraic spaces over \(S\). Given a scheme \(T\) we can consider pairs \((a, b)\) where \(a : T \to B\) is a morphism and \(b : T \times_{a, B} Z \to T \times_{a, B} X\) is a morphism over \(T\). Picture [05Y1]\[\begin{equation} \vcenter{ \xymatrix{ T \times_{a, B} Z \ar[rd] \ar[rr]_b & & T \times_{a, B} X \ar[ld] & Z \ar[rd] & & X \ar[ld] \\ & T \ar[rrr]^a & & & B } } \end{equation}\] Of course, we can also think of \(b\) as a morphism \(b : T \times_{a, B} Z \to X\) such that \[\xymatrix{ T \times_{a, B} Z \ar[r] \ar[d] \ar@/^1pc/[rrr]_-b & Z \ar[rd] & & X \ar[ld] \\ T \ar[rr]^a & & B }\] commutes. In this situation we can define a functor [05Y2]\[\begin{equation} \mathit{Mor}_B(Z, X) : (\Sch/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \{(a, b)\text{ as above}\} \end{equation}\] Sometimes we think of this as a functor defined on the category of schemes over \(B\), in which case we drop \(a\) from the notation.
Lemma
Let \(S\) be a scheme. Let \(Z \to B\) and \(X \to B\) be morphisms of algebraic spaces over \(S\). Then
\(\mathit{Mor}_B(Z, X)\) is a sheaf on \((\Sch/S)_{fppf}\).
If \(T\) is an algebraic space over \(S\), then there is a canonical bijection \[\Mor_{\Sh((\Sch/S)_{fppf})}(T, \mathit{Mor}_B(Z, X)) = \{(a, b)\text{ as in }(\href{criteria.html#criteria-equation-hom}{05Y1})\}\]
Proof
Let \(T\) be an algebraic space over \(S\). Let \(\{T_i \to T\}\) be an fppf covering of \(T\) (as in Topologies on Spaces, Section 03Y7). Suppose that \((a_i, b_i) \in \mathit{Mor}_B(Z, X)(T_i)\) such that \((a_i, b_i)|_{T_i \times_T T_j} = (a_j, b_j)|_{T_i \times_T T_j}\) for all \(i, j\). Then by Descent on Spaces, Lemma 04P2 there exists a unique morphism \(a : T \to B\) such that \(a_i\) is the composition of \(T_i \to T\) and \(a\). Then \(\{T_i \times_{a_i, B} Z \to T \times_{a, B} Z\}\) is an fppf covering too and the same lemma implies there exists a unique morphism \(b : T \times_{a, B} Z \to T \times_{a, B} X\) such that \(b_i\) is the composition of \(T_i \times_{a_i, B} Z \to T \times_{a, B} Z\) and \(b\). Hence \((a, b) \in \mathit{Mor}_B(Z, X)(T)\) restricts to \((a_i, b_i)\) over \(T_i\) for all \(i\).
Note that the result of the preceding paragraph in particular implies (1).
Let \(T\) be an algebraic space over \(S\). In order to prove (2) we will construct mutually inverse maps between the displayed sets. In the following when we say “pair” we mean a pair \((a, b)\) fitting into (05Y1).
Let \(v : T \to \mathit{Mor}_B(Z, X)\) be a natural transformation. Choose a scheme \(U\) and a surjective étale morphism \(p : U \to T\). Then \(v(p) \in \mathit{Mor}_B(Z, X)(U)\) corresponds to a pair \((a_U, b_U)\) over \(U\). Let \(R = U \times_T U\) with projections \(t, s : R \to U\). As \(v\) is a transformation of functors we see that the pullbacks of \((a_U, b_U)\) by \(s\) and \(t\) agree. Hence, since \(\{U \to T\}\) is an fppf covering, we may apply the result of the first paragraph that deduce that there exists a unique pair \((a, b)\) over \(T\).
Conversely, let \((a, b)\) be a pair over \(T\). Let \(U \to T\), \(R = U \times_T U\), and \(t, s : R \to U\) be as above. Then the restriction \((a, b)|_U\) gives rise to a transformation of functors \(v : h_U \to \mathit{Mor}_B(Z, X)\) by the Yoneda lemma (Categories, Lemma 001P). As the two pullbacks \(s^*(a, b)|_U\) and \(t^*(a, b)|_U\) are equal, we see that \(v\) coequalizes the two maps \(h_t, h_s : h_R \to h_U\). Since \(T = U/R\) is the fppf quotient sheaf by Spaces, Lemma 0262 and since \(\mathit{Mor}_B(Z, X)\) is an fppf sheaf by (1) we conclude that \(v\) factors through a map \(T \to \mathit{Mor}_B(Z, X)\).
We omit the verification that the two constructions above are mutually inverse.
Lemma
Let \(S\) be a scheme. Let \(Z \to B\), \(X \to B\), and \(B' \to B\) be morphisms of algebraic spaces over \(S\). Set \(Z' = B' \times_B Z\) and \(X' = B' \times_B X\). Then \[\mathit{Mor}_{B'}(Z', X') = B' \times_B \mathit{Mor}_B(Z, X)\] in \(\Sh((\Sch/S)_{fppf})\).
Proof
The equality as functors follows immediately from the definitions. The equality as sheaves follows from this because both sides are sheaves according to Lemma 05Y3 and the fact that a fibre product of sheaves is the same as the corresponding fibre product of pre-sheaves (i.e., functors).
Lemma
Let \(S\) be a scheme. Let \(Z \to B\) and \(X' \to X \to B\) be morphisms of algebraic spaces over \(S\). Assume
\(X' \to X\) is étale, and
\(Z \to B\) is finite locally free.
Then \(\mathit{Mor}_B(Z, X') \to \mathit{Mor}_B(Z, X)\) is representable by algebraic spaces and étale. If \(X' \to X\) is also surjective, then \(\mathit{Mor}_B(Z, X') \to \mathit{Mor}_B(Z, X)\) is surjective.
Proof
Let \(U\) be a scheme and let \(\xi = (a, b)\) be an element of \(\mathit{Mor}_B(Z, X)(U)\). We have to prove that the functor \[h_U \times_{\xi, \mathit{Mor}_B(Z, X)} \mathit{Mor}_B(Z, X')\] is representable by an algebraic space étale over \(U\). Set \(Z_U = U \times_{a, B} Z\) and \(W = Z_U \times_{b, X} X'\). Then \(W \to Z_U \to U\) is as in Lemma 05XR and the sheaf \(F\) defined there is identified with the fibre product displayed above. Hence the first assertion of the lemma. The second assertion follows from this and Lemma 05XQ which guarantees that \(F \to U\) is surjective in the situation above.
Proposition
Let \(S\) be a scheme. Let \(Z \to B\) and \(X \to B\) be morphisms of algebraic spaces over \(S\). If \(Z \to B\) is finite locally free then \(\mathit{Mor}_B(Z, X)\) is an algebraic space.
Proof
Choose a scheme \(B' = \coprod B'_i\) which is a disjoint union of affine schemes \(B'_i\) and an étale surjective morphism \(B' \to B\). We may also assume that \(B'_i \times_B Z\) is the spectrum of a ring which is finite free as a \(\Gamma(B'_i, \mathcal{O}_{B'_i})\)-module. By Lemma 05Y4 and Spaces, Lemma 02WL the morphism \(\mathit{Mor}_{B'}(Z', X') \to \mathit{Mor}_B(Z, X)\) is surjective étale. Hence by Bootstrap, Theorem 04S6 it suffices to prove the proposition when \(B = B'\) is a disjoint union of affine schemes \(B'_i\) so that each \(B'_i \times_B Z\) is finite free over \(B'_i\). Then it actually suffices to prove the result for the restriction to each \(B'_i\). Thus we may assume that \(B\) is affine and that \(\Gamma(Z, \mathcal{O}_Z)\) is a finite free \(\Gamma(B, \mathcal{O}_B)\)-module.
Choose a scheme \(X'\) which is a disjoint union of affine schemes and a surjective étale morphism \(X' \to X\). By Lemma 05Y5 the morphism \(\mathit{Mor}_B(Z, X') \to \mathit{Mor}_B(Z, X)\) is representable by algebraic spaces, étale, and surjective. Hence by Bootstrap, Theorem 04S6 it suffices to prove the proposition when \(X\) is a disjoint union of affine schemes. This reduces us to the case discussed in the next paragraph.
Assume \(X = \coprod_{i \in I} X_i\) is a disjoint union of affine schemes, \(B\) is affine, and that \(\Gamma(Z, \mathcal{O}_Z)\) is a finite free \(\Gamma(B, \mathcal{O}_B)\)-module. For any finite subset \(E \subset I\) set \[F_E = \mathit{Mor}_B(Z, \coprod\nolimits_{i \in E} X_i).\] By More on Morphisms, Lemma 05Y6 we see that \(F_E\) is an algebraic space. Consider the morphism \[\coprod\nolimits_{E \subset I\text{ finite}} F_E \longrightarrow \mathit{Mor}_B(Z, X)\] Each of the morphisms \(F_E \to \mathit{Mor}_B(Z, X)\) is an open immersion, because it is simply the locus parametrizing pairs \((a, b)\) where \(b\) maps into the open subscheme \(\coprod\nolimits_{i \in E} X_i\) of \(X\). Moreover, if \(T\) is quasi-compact, then for any pair \((a, b)\) the image of \(b\) is contained in \(\coprod\nolimits_{i \in E} X_i\) for some \(E \subset I\) finite. Hence the displayed arrow is in fact an open covering and we win3 by Spaces, Lemma 02WR.
Restriction of scalars
Suppose \(X \to Z \to B\) are morphisms of algebraic spaces over \(S\). Given a scheme \(T\) we can consider pairs \((a, b)\) where \(a : T \to B\) is a morphism and \(b : T \times_{a, B} Z \to X\) is a morphism over \(Z\). Picture [05Y9]\[\begin{equation} \vcenter{ \xymatrix{ & X \ar[d] \\ T \times_{a, B} Z \ar[d] \ar[ru]^b \ar[r] & Z \ar[d] \\ T \ar[r]^a & B } } \end{equation}\] In this situation we can define a functor [05YA]\[\begin{equation} \text{Res}_{Z/B}(X) : (\Sch/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \{(a, b)\text{ as above}\} \end{equation}\] Sometimes we think of this as a functor defined on the category of schemes over \(B\), in which case we drop \(a\) from the notation.
Lemma
Let \(S\) be a scheme. Let \(X \to Z \to B\) be morphisms of algebraic spaces over \(S\). Then
\(\text{Res}_{Z/B}(X)\) is a sheaf on \((\Sch/S)_{fppf}\).
If \(T\) is an algebraic space over \(S\), then there is a canonical bijection \[\Mor_{\Sh((\Sch/S)_{fppf})}(T, \text{Res}_{Z/B}(X)) = \{(a, b)\text{ as in }(\href{criteria.html#criteria-equation-pairs}{05Y9})\}\]
Proof
Let \(T\) be an algebraic space over \(S\). Let \(\{T_i \to T\}\) be an fppf covering of \(T\) (as in Topologies on Spaces, Section 03Y7). Suppose that \((a_i, b_i) \in \text{Res}_{Z/B}(X)(T_i)\) such that \((a_i, b_i)|_{T_i \times_T T_j} = (a_j, b_j)|_{T_i \times_T T_j}\) for all \(i, j\). Then by Descent on Spaces, Lemma 04P2 there exists a unique morphism \(a : T \to B\) such that \(a_i\) is the composition of \(T_i \to T\) and \(a\). Then \(\{T_i \times_{a_i, B} Z \to T \times_{a, B} Z\}\) is an fppf covering too and the same lemma implies there exists a unique morphism \(b : T \times_{a, B} Z \to X\) such that \(b_i\) is the composition of \(T_i \times_{a_i, B} Z \to T \times_{a, B} Z\) and \(b\). Hence \((a, b) \in \text{Res}_{Z/B}(X)(T)\) restricts to \((a_i, b_i)\) over \(T_i\) for all \(i\).
Note that the result of the preceding paragraph in particular implies (1).
Let \(T\) be an algebraic space over \(S\). In order to prove (2) we will construct mutually inverse maps between the displayed sets. In the following when we say “pair” we mean a pair \((a, b)\) fitting into (05Y9).
Let \(v : T \to \text{Res}_{Z/B}(X)\) be a natural transformation. Choose a scheme \(U\) and a surjective étale morphism \(p : U \to T\). Then \(v(p) \in \text{Res}_{Z/B}(X)(U)\) corresponds to a pair \((a_U, b_U)\) over \(U\). Let \(R = U \times_T U\) with projections \(t, s : R \to U\). As \(v\) is a transformation of functors we see that the pullbacks of \((a_U, b_U)\) by \(s\) and \(t\) agree. Hence, since \(\{U \to T\}\) is an fppf covering, we may apply the result of the first paragraph that deduce that there exists a unique pair \((a, b)\) over \(T\).
Conversely, let \((a, b)\) be a pair over \(T\). Let \(U \to T\), \(R = U \times_T U\), and \(t, s : R \to U\) be as above. Then the restriction \((a, b)|_U\) gives rise to a transformation of functors \(v : h_U \to \text{Res}_{Z/B}(X)\) by the Yoneda lemma (Categories, Lemma 001P). As the two pullbacks \(s^*(a, b)|_U\) and \(t^*(a, b)|_U\) are equal, we see that \(v\) coequalizes the two maps \(h_t, h_s : h_R \to h_U\). Since \(T = U/R\) is the fppf quotient sheaf by Spaces, Lemma 0262 and since \(\text{Res}_{Z/B}(X)\) is an fppf sheaf by (1) we conclude that \(v\) factors through a map \(T \to \text{Res}_{Z/B}(X)\).
We omit the verification that the two constructions above are mutually inverse.
Of course the sheaf \(\text{Res}_{Z/B}(X)\) comes with a natural transformation of functors \(\text{Res}_{Z/B}(X) \to B\). We will use this without further mention in the following.
Lemma
Let \(S\) be a scheme. Let \(X \to Z \to B\) and \(B' \to B\) be morphisms of algebraic spaces over \(S\). Set \(Z' = B' \times_B Z\) and \(X' = B' \times_B X\). Then \[\text{Res}_{Z'/B'}(X') = B' \times_B \text{Res}_{Z/B}(X)\] in \(\Sh((\Sch/S)_{fppf})\).
Proof
The equality as functors follows immediately from the definitions. The equality as sheaves follows from this because both sides are sheaves according to Lemma 05YB and the fact that a fibre product of sheaves is the same as the corresponding fibre product of pre-sheaves (i.e., functors).
Lemma
Let \(S\) be a scheme. Let \(X' \to X \to Z \to B\) be morphisms of algebraic spaces over \(S\). Assume
\(X' \to X\) is étale, and
\(Z \to B\) is finite locally free.
Then \(\text{Res}_{Z/B}(X') \to \text{Res}_{Z/B}(X)\) is representable by algebraic spaces and étale. If \(X' \to X\) is also surjective, then \(\text{Res}_{Z/B}(X') \to \text{Res}_{Z/B}(X)\) is surjective.
Proof
Let \(U\) be a scheme and let \(\xi = (a, b)\) be an element of \(\text{Res}_{Z/B}(X)(U)\). We have to prove that the functor \[h_U \times_{\xi, \text{Res}_{Z/B}(X)} \text{Res}_{Z/B}(X')\] is representable by an algebraic space étale over \(U\). Set \(Z_U = U \times_{a, B} Z\) and \(W = Z_U \times_{b, X} X'\). Then \(W \to Z_U \to U\) is as in Lemma 05XR and the sheaf \(F\) defined there is identified with the fibre product displayed above. Hence the first assertion of the lemma. The second assertion follows from this and Lemma 05XQ which guarantees that \(F \to U\) is surjective in the situation above.
At this point we can use the lemmas above to prove that \(\text{Res}_{Z/B}(X)\) is an algebraic space whenever \(Z \to B\) is finite locally free in almost exactly the same way as in the proof that \(\mathit{Mor}_B(Z, X)\) is an algebraic spaces, see Proposition 05Y7. Instead we will directly deduce this result from the following lemma and the fact that \(\mathit{Mor}_B(Z, X)\) is an algebraic space.
Lemma
Let \(S\) be a scheme. Let \(X \to Z \to B\) be morphisms of algebraic spaces over \(S\). The following diagram \[\xymatrix{ \mathit{Mor}_B(Z, X) \ar[r] & \mathit{Mor}_B(Z, Z) \\ \text{Res}_{Z/B}(X) \ar[r] \ar[u] & B \ar[u]_{\text{id}_Z} }\] is a cartesian diagram of sheaves on \((\Sch/S)_{fppf}\).
Proof
Omitted. Hint: Exercise in the functorial point of view in algebraic geometry.
Proposition
Let \(S\) be a scheme. Let \(X \to Z \to B\) be morphisms of algebraic spaces over \(S\). If \(Z \to B\) is finite locally free then \(\text{Res}_{Z/B}(X)\) is an algebraic space.
Proof
By Proposition 05Y7 the functors \(\mathit{Mor}_B(Z, X)\) and \(\mathit{Mor}_B(Z, Z)\) are algebraic spaces. Hence this follows from the cartesian diagram of Lemma 05YE and the fact that fibre products of algebraic spaces exist and are given by the fibre product in the underlying category of sheaves of sets (see Spaces, Lemma 04T9).
Proposition
Let \(S\) be a scheme. Let \(X \to Z \to B\) be morphisms of algebraic spaces over \(S\), and assume that \(Z \to B\) is finite locally free. Then
if \(X \to Z\) is locally of finite presentation, then \(\text{Res}_{Z/B}(X) \to B\) is locally of finite presentation;
if \(X \to Z\) is smooth, then \(\text{Res}_{Z/B}(X) \to B\) is smooth;
if \(B\), \(Z\), and \(X\) are schemes and \(X \to Z\) is quasi-projective, then \(\text{Res}_{Z/B}(X)\) is a scheme.
Proof
The restriction of scalars is an algebraic space by Proposition 05YF.
Suppose \(X \to Z\) is locally of finite presentation. Let \(T = \lim T_i\) be a directed inverse limit of affine schemes over \(B\). Since \(Z \to B\) is finite, all the schemes \(Z_{T_i}\) and \(Z_T\) are affine and \(Z_T = \lim Z_{T_i}\). Since \(X \to Z\) is locally of finite presentation, Limits of Spaces, Proposition 04AK gives \[\Mor_Z(Z_T, X) = \colim_i \Mor_Z(Z_{T_i}, X).\] This says exactly that \(\text{Res}_{Z/B}(X) \to B\) is limit preserving. Another application of the proposition just cited proves (1).
Suppose \(X \to Z\) is smooth. Part (1) shows that the restriction of scalars is locally of finite presentation. Let \(T \subset T'\) be a first order thickening of affine schemes over \(B\) and let \(T \to \text{Res}_{Z/B}(X)\) correspond to a \(Z\)-morphism \(Z_T \to X\). The spaces \(Z_T\) and \(Z_{T'}\) are affine schemes because they are finite over affine schemes. The morphism \(X \to Z\) is formally smooth by More on Morphisms of Spaces, Lemma 04AM; hence the map \(Z_T \to X\) extends to a \(Z\)-morphism \(Z_{T'} \to X\). Thus the restriction of scalars is formally smooth. The same lemma, together with local finite presentation, proves (2).
Finally, assertion (3) is local on \(B\), so we may assume that \(B\) is affine. The finite morphism \(Z \to B\) is projective by Morphisms, Lemma 0B3I. Hence \(X \to B\) is quasi-projective by Morphisms, Lemma 0C4M. More on Morphisms, Lemma 0BL4 shows that both \(\mathit{Mor}_B(Z, X)\) and \(\mathit{Mor}_B(Z, Z)\) are schemes. The cartesian diagram of Lemma 05YE now expresses \(\text{Res}_{Z/B}(X)\) as a fibre product of schemes. This proves (3).
Example
Let \(T \to S\) be a smooth separated morphism of schemes. Denote by \(\mathcal{I}\) the ideal of the diagonal in \(T \times_S T\), and for \(n \geq 0\) let \(T_n\) be the closed subscheme cut out by \(\mathcal{I}^{n + 1}\). Write \(p_1, p_2 : T_n \to T\) for the two projections. Then \(p_1\) is finite locally free. For a scheme \(X\) over \(T\) set \[\mathit{Germ}^n_{T/S}(X) = \text{Res}_{T_n/T}(T_n \times_{p_2, T} X),\] where \(T_n \to T\) in the subscript is \(p_1\). This is an algebraic space over \(T\); it is a scheme if \(X \to T\) is quasi-projective, and it is smooth over \(T\) if \(X \to T\) is smooth.
Proof
The diagonal is a regular immersion by Divisors, Lemma 067R, hence a quasi-regular immersion by Lemma 063K. Its conormal sheaf is finite locally free, and its conormal algebra is the symmetric algebra by Divisors, Lemma 063M. Thus the successive quotients in the filtration \[\mathcal{O}_{T_n} \supset \mathcal{I}/\mathcal{I}^{n + 1} \supset \cdots \supset \mathcal{I}^n/\mathcal{I}^{n + 1} \supset 0\] are finite locally free \(\mathcal{O}_T\)-modules. It follows that \(\mathcal{O}_{T_n}\) is finite locally free over \(\mathcal{O}_T\). Locally on \(T\), the scheme \(T_n\) is affine because it is a finite order thickening of the affine diagonal; see More on Morphisms, Lemma 06AD. Hence \(p_1 : T_n \to T\) is finite locally free.
The remaining assertions follow from Propositions 05YF and criteria-proposition-restriction-of-scalars-properties, applied to the base change \(T_n \times_{p_2, T} X \to T_n\) of \(X \to T\).
Finite Hilbert stacks
In this section we prove some results concerning the finite Hilbert stacks \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) introduced in Examples of Stacks, Section 05WA.
Lemma
Consider a \(2\)-commutative diagram \[\xymatrix{ \mathcal{X}' \ar[r]_G \ar[d]_{F'} & \mathcal{X} \ar[d]^F \\ \mathcal{Y}' \ar[r]^H & \mathcal{Y} }\] of stacks in groupoids over \((\Sch/S)_{fppf}\) with a given \(2\)-isomorphism \(\gamma : H \circ F' \to F \circ G\). In this situation we obtain a canonical \(1\)-morphism \(\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\). This morphism is compatible with the forgetful \(1\)-morphisms of Examples of Stacks, Equation (05WD).
Proof
We map the object \((U, Z, y', x', \alpha')\) to the object \((U, Z, H(y'), G(x'), \gamma \star \text{id}_H \star \alpha')\) where \(\star\) denotes horizontal composition of \(2\)-morphisms, see Categories, Definition 003E. To a morphism \((f, g, b, a) : (U_1, Z_1, y_1', x_1', \alpha_1') \to (U_2, Z_2, y_2', x_2', \alpha_2')\) we assign \((f, g, H(b), G(a))\). We omit the verification that this defines a functor between categories over \((\Sch/S)_{fppf}\).
Lemma
In the situation of Lemma 05XN assume that the given square is \(2\)-cartesian. Then the diagram \[\xymatrix{ \mathcal{H}_d(\mathcal{X}'/\mathcal{Y}') \ar[r] \ar[d] & \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y} }\] is \(2\)-cartesian.
Proof
We get a \(2\)-commutative diagram by Lemma 05XN and hence we get a \(1\)-morphism (i.e., a functor) \[\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}') \longrightarrow \mathcal{Y}' \times_\mathcal{Y} \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\] We indicate why this functor is essentially surjective. Namely, an object of the category on the right hand side is given by a scheme \(U\) over \(S\), an object \(y'\) of \(\mathcal{Y}'_U\), an object \((U, Z, y, x, \alpha)\) of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) over \(U\) and an isomorphism \(H(y') \to y\) in \(\mathcal{Y}_U\). The assumption means exactly that there exists an object \(x'\) of \(\mathcal{X}'_Z\) such that there exist isomorphisms \(G(x') \cong x\) and \(\alpha' : y'|_Z \to F'(x')\) compatible with \(\alpha\). Then we see that \((U, Z, y', x', \alpha')\) is an object of \(\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}')\) over \(U\). Details omitted.
Lemma
In the situation of Lemma 05XN assume
\(\mathcal{Y}' = \mathcal{Y}\) and \(H = \text{id}_\mathcal{Y}\),
\(G\) is representable by algebraic spaces and étale.
Then \(\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is representable by algebraic spaces and étale. If \(G\) is also surjective, then \(\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is surjective.
Proof
Let \(U\) be a scheme and let \(\xi = (U, Z, y, x, \alpha)\) be an object of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) over \(U\). We have to prove that the \(2\)-fibre product [05XT]\[\begin{equation} (\Sch/U)_{fppf} \times_{\xi, \mathcal{H}_d(\mathcal{X}/\mathcal{Y})} \mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \end{equation}\] is representable by an algebraic space étale over \(U\). An object of this over \(U'\) corresponds to an object \(x'\) in the fibre category of \(\mathcal{X}'\) over \(Z_{U'}\) such that \(G(x') \cong x|_{Z_{U'}}\). By assumption the \(2\)-fibre product \[(\Sch/Z)_{fppf} \times_{x, \mathcal{X}} \mathcal{X}'\] is representable by an algebraic space \(W\) such that the projection \(W \to Z\) is étale. Then (05XT) is representable by the algebraic space \(F\) parametrizing sections of \(W \to Z\) over \(U\) introduced in Lemma 05XR. Since \(F \to U\) is étale we conclude that \(\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is representable by algebraic spaces and étale. Finally, if \(\mathcal{X}' \to \mathcal{X}\) is surjective also, then \(W \to Z\) is surjective, and hence \(F \to U\) is surjective by Lemma 05XQ. Thus in this case \(\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is also surjective.
Lemma
In the situation of Lemma 05XN. Assume that \(G\), \(H\) are representable by algebraic spaces and étale. Then \(\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is representable by algebraic spaces and étale. If also \(H\) is surjective and the induced functor \(\mathcal{X}' \to \mathcal{Y}' \times_\mathcal{Y} \mathcal{X}\) is surjective, then \(\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is surjective.
Proof
Set \(\mathcal{X}'' = \mathcal{Y}' \times_\mathcal{Y} \mathcal{X}\). By Lemma 05XK the \(1\)-morphism \(\mathcal{X}' \to \mathcal{X}''\) is representable by algebraic spaces and étale (in particular the condition in the second statement of the lemma that \(\mathcal{X}' \to \mathcal{X}''\) be surjective makes sense). We obtain a \(2\)-commutative diagram \[\xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X}'' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y}' \ar[r] & \mathcal{Y} }\] It follows from Lemma 05XP that \(\mathcal{H}_d(\mathcal{X}''/\mathcal{Y}')\) is the base change of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) by \(\mathcal{Y}' \to \mathcal{Y}\). In particular we see that \(\mathcal{H}_d(\mathcal{X}''/\mathcal{Y}') \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is representable by algebraic spaces and étale, see Algebraic Stacks, Lemma 045C. Moreover, it is also surjective if \(H\) is. Hence if we can show that the result holds for the left square in the diagram, then we’re done. In this way we reduce to the case where \(\mathcal{Y}' = \mathcal{Y}\) which is the content of Lemma 05YG.
Lemma
Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). Assume that \(\Delta : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}\) is representable by algebraic spaces. Then \[\mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{H}_d(\mathcal{X}) \times \mathcal{Y}\] see Examples of Stacks, Equation (05WD) is representable by algebraic spaces.
Proof
Let \(U\) be a scheme and let \(\xi = (U, Z, p, x, 1)\) be an object of \(\mathcal{H}_d(\mathcal{X}) = \mathcal{H}_d(\mathcal{X}/S)\) over \(U\). Here \(p\) is just the structure morphism of \(U\). The fifth component \(1\) exists and is unique since everything is over \(S\). Also, let \(y\) be an object of \(\mathcal{Y}\) over \(U\). We have to show the \(2\)-fibre product [05YI]\[\begin{equation} (\Sch/U)_{fppf} \times_{\xi \times y, \mathcal{H}_d(\mathcal{X}) \times \mathcal{Y}} \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \end{equation}\] is representable by an algebraic space. To explain why this is so we introduce \[I = \mathit{Isom}_\mathcal{Y}(y|_Z, F(x))\] which is an algebraic space over \(Z\) by assumption. Let \(a : U' \to U\) be a scheme over \(U\). What does it mean to give an object of the fibre category of (05YI) over \(U'\)? Well, it means that we have an object \(\xi' = (U', Z', y', x', \alpha')\) of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) over \(U'\) and isomorphisms \((U', Z', p', x', 1) \cong (U, Z, p, x, 1)|_{U'}\) and \(y' \cong y|_{U'}\). Thus \(\xi'\) is isomorphic to \((U', U' \times_{a, U} Z, a^*y, x|_{U' \times_{a, U} Z}, \alpha)\) for some morphism \[\alpha : a^*y|_{U' \times_{a, U} Z} \longrightarrow F(x|_{U' \times_{a, U} Z})\] in the fibre category of \(\mathcal{Y}\) over \(U' \times_{a, U} Z\). Hence we can view \(\alpha\) as a morphism \(b : U' \times_{a, U} Z \to I\). In this way we see that (05YI) is representable by \(\text{Res}_{Z/U}(I)\) which is an algebraic space by Proposition 05YF.
The following lemma is a (partial) generalization of Lemma 05YG.
Lemma
Let \(F : \mathcal{X} \to \mathcal{Y}\) and \(G : \mathcal{X}' \to \mathcal{X}\) be \(1\)-morphisms of stacks in groupoids over \((\Sch/S)_{fppf}\). If \(G\) is representable by algebraic spaces, then the \(1\)-morphism \[\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \longrightarrow \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\] is representable by algebraic spaces.
Proof
Let \(U\) be a scheme and let \(\xi = (U, Z, y, x, \alpha)\) be an object of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) over \(U\). We have to prove that the \(2\)-fibre product [05YK]\[\begin{equation} (\Sch/U)_{fppf} \times_{\xi, \mathcal{H}_d(\mathcal{X}/\mathcal{Y})} \mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \end{equation}\] is representable by an algebraic space over \(U\). An object of this over \(a : U' \to U\) corresponds to an object \(x'\) of \(\mathcal{X}'\) over \(U' \times_{a, U} Z\) such that \(G(x') \cong x|_{U' \times_{a, U} Z}\). By assumption the \(2\)-fibre product \[(\Sch/Z)_{fppf} \times_{x, \mathcal{X}} \mathcal{X}'\] is representable by an algebraic space \(W\) over \(Z\). It follows that (05YK) is representable by \(\text{Res}_{Z/U}(W)\), which is an algebraic space by Proposition 05YF.
Lemma
Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). Assume \(F\) is representable by algebraic spaces and locally of finite presentation. Then \[p : \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}\] is limit preserving on objects.
Proof
This means we have to show the following: Given
an affine scheme \(U = \lim_i U_i\) which is written as the directed limit of affine schemes \(U_i\) over \(S\),
an object \(y_i\) of \(\mathcal{Y}\) over \(U_i\) for some \(i\), and
an object \(\Xi = (U, Z, y, x, \alpha)\) of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) over \(U\) such that \(y = y_i|_U\),
then there exists an \(i' \geq i\) and an object \(\Xi_{i'} = (U_{i'}, Z_{i'}, y_{i'}, x_{i'}, \alpha_{i'})\) of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) over \(U_{i'}\) with \(\Xi_{i'}|_U = \Xi\) and \(y_{i'} = y_i|_{U_{i'}}\). Namely, the last two equalities will take care of the commutativity of (06CU).
Let \(X_{y_i} \to U_i\) be an algebraic space representing the \(2\)-fibre product \[(\Sch/U_i)_{fppf} \times_{y_i, \mathcal{Y}, F} \mathcal{X}.\] Note that \(X_{y_i} \to U_i\) is locally of finite presentation by our assumption on \(F\). Write \(\Xi\). It is clear that \(\xi = (Z, Z \to U_i, x, \alpha)\) is an object of the \(2\)-fibre product displayed above, hence \(\xi\) gives rise to a morphism \(f_\xi : Z \to X_{y_i}\) of algebraic spaces over \(U_i\) (since \(X_{y_i}\) is the functor of isomorphisms classes of objects of \((\Sch/U_i)_{fppf} \times_{y, \mathcal{Y}, F} \mathcal{X}\), see Algebraic Stacks, Lemma 02ZX). By Limits, Lemmas 01ZM and 06AC there exists an \(i' \geq i\) and a finite locally free morphism \(Z_{i'} \to U_{i'}\) of degree \(d\) whose base change to \(U\) is \(Z\). By Limits of Spaces, Proposition 04AK we may, after replacing \(i'\) by a bigger index, assume there exists a morphism \(f_{i'} : Z_{i'} \to X_{y_i}\) such that \[\xymatrix{ Z \ar[d] \ar[r] \ar@/^3ex/[rr]^{f_\xi} & Z_{i'} \ar[d] \ar[r]_{f_{i'}} & X_{y_i} \ar[d] \\ U \ar[r] & U_{i'} \ar[r] & U_i }\] is commutative. We set \(\Xi_{i'} = (U_{i'}, Z_{i'}, y_{i'}, x_{i'}, \alpha_{i'})\) where
\(y_{i'}\) is the object of \(\mathcal{Y}\) over \(U_{i'}\) which is the pullback of \(y_i\) to \(U_{i'}\),
\(x_{i'}\) is the object of \(\mathcal{X}\) over \(Z_{i'}\) corresponding via the \(2\)-Yoneda lemma to the \(1\)-morphism \[(\Sch/Z_{i'})_{fppf} \to \mathcal{S}_{X_{y_i}} \to (\Sch/U_i)_{fppf} \times_{y_i, \mathcal{Y}, F} \mathcal{X} \to \mathcal{X}\] where the middle arrow is the equivalence which defines \(X_{y_i}\) (notation as in Algebraic Stacks, Sections 02ZV and 04SU).
\(\alpha_{i'} : y_{i'}|_{Z_{i'}} \to F(x_{i'})\) is the isomorphism coming from the \(2\)-commutativity of the diagram \[\xymatrix{ (\Sch/Z_{i'})_{fppf} \ar[r] \ar[rd] & (\Sch/U_i)_{fppf} \times_{y_i, \mathcal{Y}, F} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d]^F \\ & (\Sch/U_{i'})_{fppf} \ar[r] & \mathcal{Y} }\]
Recall that \(f_\xi : Z \to X_{y_i}\) was the morphism corresponding to the object \(\xi = (Z, Z \to U_i, x, \alpha)\) of \((\Sch/U_i)_{fppf} \times_{y_i, \mathcal{Y}, F} \mathcal{X}\) over \(Z\). By construction \(f_{i'}\) is the morphism corresponding to the object \(\xi_{i'} = (Z_{i'}, Z_{i'} \to U_i, x_{i'}, \alpha_{i'})\). As \(f_\xi = f_{i'} \circ (Z \to Z_{i'})\) we see that the object \(\xi_{i'} = (Z_{i'}, Z_{i'} \to U_i, x_{i'}, \alpha_{i'})\) pulls back to \(\xi\) over \(Z\). Thus \(x_{i'}\) pulls back to \(x\) and \(\alpha_{i'}\) pulls back to \(\alpha\). This means that \(\Xi_{i'}\) pulls back to \(\Xi\) over \(U\) and we win.
The finite Hilbert stack of a point
Let \(d \geq 1\) be an integer. In Examples of Stacks, Definition 05WC we defined a stack in groupoids \(\mathcal{H}_d\). In this section we prove that \(\mathcal{H}_d\) is an algebraic stack. We will throughout assume that \(S = \Spec(\mathbf{Z})\). The general case will follow from this by base change. Recall that the fibre category of \(\mathcal{H}_d\) over a scheme \(T\) is the category of finite locally free morphisms \(\pi : Z \to T\) of degree \(d\). Instead of classifying these directly we first study the quasi-coherent sheaves of algebras \(\pi_*\mathcal{O}_Z\).
Let \(R\) be a ring. Let us temporarily make the following definition: A free \(d\)-dimensional algebra over \(R\) is given by a commutative \(R\)-algebra structure \(m\) on \(R^{\oplus d}\) such that \(e_1 = (1, 0, \ldots, 0)\) is a unit4. We think of \(m\) as an \(R\)-linear map \[m : R^{\oplus d} \otimes_R R^{\oplus d} \longrightarrow R^{\oplus d}\] such that \(m(e_1, x) = m(x, e_1) = x\) and such that \(m\) defines a commutative and associative ring structure. If we write \(m(e_i, e_j) = \sum a_{ij}^ke_k\) then we see this boils down to the conditions \[\left\{ \begin{matrix} \sum_l a_{ij}^la_{lk}^m = \sum_l a_{il}^ma_{jk}^l & \forall i, j, k, m \\ a_{ij}^k = a_{ji}^k & \forall i, j, k \\ a_{i1}^j = \delta_{ij} & \forall i, j \end{matrix} \right.\] where \(\delta_{ij}\) is the Kronecker \(\delta\)-function. OK, so let’s define \[R_{univ} = \mathbf{Z}[a_{ij}^k]/J\] where the ideal \(J\) is the ideal generated by the relations displayed above. Denote \[m_{univ} : R_{univ}^{\oplus d} \otimes_{R_{univ}} R_{univ}^{\oplus d} \longrightarrow R_{univ}^{\oplus d}\] the free \(d\)-dimensional algebra \(m\) over \(R_{univ}\) whose structure constants are the classes of \(a_{ij}^k\) modulo \(J\). Then it is clear that given any free \(d\)-dimensional algebra \(m\) over a ring \(R\) there exists a unique \(\mathbf{Z}\)-algebra homomorphism \(\psi : R_{univ} \to R\) such that \(\psi_*m_{univ} = m\) (this means that \(m\) is what you get by applying the base change functor \(- \otimes_{R_{univ}} R\) to \(m_{univ}\)). In other words, setting \(X = \Spec(R_{univ})\) we obtain a canonical identification \[X(T) = \{\text{free }d\text{-dimensional algebras }m\text{ over }R\}\] for varying \(T = \Spec(R)\). By Zariski localization we obtain the following seemingly more general identification [05YM]\[\begin{equation} X(T) = \{\text{free }d\text{-dimensional algebras } m\text{ over }\Gamma(T, \mathcal{O}_T)\} \end{equation}\] for any scheme \(T\).
Next we talk a little bit about isomorphisms of free \(d\)-dimensional \(R\)-algebras. Namely, suppose that \(m\), \(m'\) are two free \(d\)-dimensional algebras over a ring \(R\). An isomorphism from \(m\) to \(m'\) is given by an invertible \(R\)-linear map \[\varphi : R^{\oplus d} \longrightarrow R^{\oplus d}\] such that \(\varphi(e_1) = e_1\) and such that \[m \circ \varphi \otimes \varphi = \varphi \circ m'.\] Note that we can compose these so that the collection of free \(d\)-dimensional algebras over \(R\) becomes a category. In this way we obtain a functor [05YN]\[\begin{equation} FA_d : \Sch_{fppf}^{opp} \longrightarrow \textit{Groupoids} \end{equation}\] from the category of schemes to groupoids: to a scheme \(T\) we associate the set of free \(d\)-dimensional algebras over \(\Gamma(T, \mathcal{O}_T)\) endowed with the structure of a category using the notion of isomorphisms just defined.
The above suggests we consider the functor \(G\) in groups which associates to any scheme \(T\) the group \[G(T) = \{g \in \text{GL}_d(\Gamma(T, \mathcal{O}_T)) \mid g(e_1) = e_1\}\] It is clear that \(G \subset \text{GL}_d\) (see Groupoids, Example 022W) is the closed subgroup scheme cut out by the equations \(x_{11} = 1\) and \(x_{i1} = 0\) for \(i > 1\). Hence \(G\) is a smooth affine group scheme over \(\Spec(\mathbf{Z})\). Consider the action \[a : G \times_{\Spec(\mathbf{Z})} X \longrightarrow X\] which associates to a \(T\)-valued point \((g, m)\) with \(T = \Spec(R)\) on the left hand side the free \(d\)-dimensional algebra over \(R\) given by \[a(g, m) = g^{-1} \circ m \circ g \otimes g.\] Note that this means that \(g\) defines an isomorphism \(m \to a(g, m)\) of \(d\)-dimensional free \(R\)-algebras. We omit the verification that \(a\) indeed defines an action of the group scheme \(G\) on the scheme \(X\).
Lemma
The functor in groupoids \(FA_d\) defined in (05YN) is isomorphic (!) to the functor in groupoids which associates to a scheme \(T\) the category with
set of objects is \(X(T)\),
set of morphisms is \(G(T) \times X(T)\),
\(s : G(T) \times X(T) \to X(T)\) is the projection map,
\(t : G(T) \times X(T) \to X(T)\) is \(a(T)\), and
composition \(G(T) \times X(T) \times_{s, X(T), t} G(T) \times X(T) \to G(T) \times X(T)\) is given by \(((g, m), (g', m')) \mapsto (gg', m')\).
Proof
We have seen the rule on objects in (05YM). We have also seen above that \(g \in G(T)\) can be viewed as a morphism from \(m\) to \(a(g, m)\) for any free \(d\)-dimensional algebra \(m\). Conversely, any morphism \(m \to m'\) is given by an invertible linear map \(\varphi\) which corresponds to an element \(g \in G(T)\) such that \(m' = a(g, m)\).
In fact the groupoid \((X, G \times X, s, t, c)\) described in the lemma above is the groupoid associated to the action \(a : G \times X \to X\) as defined in Groupoids, Lemma 0234. Since \(G\) is smooth over \(\Spec(\mathbf{Z})\) we see that the two morphisms \(s, t : G \times X \to X\) are smooth: by symmetry it suffices to prove that one of them is, and \(s\) is the base change of \(G \to \Spec(\mathbf{Z})\). Hence \((G \times X, X, s, t, c)\) is a smooth groupoid scheme, and the quotient stack \([X/G]\) is an algebraic stack by Algebraic Stacks, Theorem 04TK.
Proposition
The stack \(\mathcal{H}_d\) is equivalent to the quotient stack \([X/G]\) described above. In particular \(\mathcal{H}_d\) is an algebraic stack.
Proof
Note that by Groupoids in Spaces, Definition 044Q the quotient stack \([X/G]\) is the stackification of the category fibred in groupoids associated to the “presheaf in groupoids” which associates to a scheme \(T\) the groupoid \[(X(T), G(T) \times X(T), s, t, c).\] Since this “presheaf in groupoids” is isomorphic to \(FA_d\) by Lemma 05YP it suffices to prove that the \(\mathcal{H}_d\) is the stackification of (the category fibred in groupoids associated to the “presheaf in groupoids”) \(FA_d\). To do this we first define a functor \[\Spec : FA_d \longrightarrow \mathcal{H}_d\] Recall that the fibre category of \(\mathcal{H}_d\) over a scheme \(T\) is the category of finite locally free morphisms \(Z \to T\) of degree \(d\). Thus given a scheme \(T\) and a free \(d\)-dimensional \(\Gamma(T, \mathcal{O}_T)\)-algebra \(m\) we may assign to this the object \[Z = \underline{\Spec}_T(\mathcal{A})\] of \(\mathcal{H}_{d, T}\) where \(\mathcal{A} = \mathcal{O}_T^{\oplus d}\) endowed with a \(\mathcal{O}_T\)-algebra structure via \(m\). Moreover, if \(m'\) is a second such free \(d\)-dimensional \(\Gamma(T, \mathcal{O}_T)\)-algebra and if \(\varphi : m \to m'\) is an isomorphism of these, then the induced \(\mathcal{O}_T\)-linear map \(\varphi : \mathcal{O}_T^{\oplus d} \to \mathcal{O}_T^{\oplus d}\) induces an isomorphism \[\varphi : \mathcal{A}' \longrightarrow \mathcal{A}\] of quasi-coherent \(\mathcal{O}_T\)-algebras. Hence \[\underline{\Spec}_T(\varphi) : \underline{\Spec}_T(\mathcal{A}) \longrightarrow \underline{\Spec}_T(\mathcal{A}')\] is a morphism in the fibre category \(\mathcal{H}_{d, T}\). We omit the verification that this construction is compatible with base change so we get indeed a functor \(\Spec : FA_d \to \mathcal{H}_d\) as claimed above.
To show that \(\Spec : FA_d \to \mathcal{H}_d\) induces an equivalence between the stackification of \(FA_d\) and \(\mathcal{H}_d\) it suffices to check that
\(\mathit{Isom}(m, m') = \mathit{Isom}(\Spec(m), \Spec(m'))\) for any \(m, m' \in FA_d(T)\).
for any scheme \(T\) and any object \(Z \to T\) of \(\mathcal{H}_{d, T}\) there exists a covering \(\{T_i \to T\}\) such that \(Z|_{T_i}\) is isomorphic to \(\Spec(m)\) for some \(m \in FA_d(T_i)\), and
see Stacks, Lemma 02ZP. The first statement follows from the observation that any isomorphism \[\underline{\Spec}_T(\mathcal{A}) \longrightarrow \underline{\Spec}_T(\mathcal{A}')\] is necessarily given by a global invertible matrix \(g\) when \(\mathcal{A} = \mathcal{A}' = \mathcal{O}_T^{\oplus d}\) as modules. To prove the second statement let \(\pi : Z \to T\) be a finite locally free morphism of degree \(d\). Then \(\mathcal{A}\) is a locally free sheaf \(\mathcal{O}_T\)-modules of rank \(d\). Consider the element \(1 \in \Gamma(T, \mathcal{A})\). This element is nonzero in \(\mathcal{A} \otimes_{\mathcal{O}_{T, t}} \kappa(t)\) for every \(t \in T\) since the scheme \(Z_t = \Spec(\mathcal{A} \otimes_{\mathcal{O}_{T, t}} \kappa(t))\) is nonempty being of degree \(d > 0\) over \(\kappa(t)\). Thus \(1 : \mathcal{O}_T \to \mathcal{A}\) can locally be used as the first basis element (for example you can use Algebra, Lemma 00O0 parts (1) and (2) to see this). Thus, after localizing on \(T\) we may assume that there exists an isomorphism \(\varphi : \mathcal{A} \to \mathcal{O}_T^{\oplus d}\) such that \(1 \in \Gamma(\mathcal{A})\) corresponds to the first basis element. In this situation the multiplication map \(\mathcal{A} \otimes_{\mathcal{O}_T} \mathcal{A} \to \mathcal{A}\) translates via \(\varphi\) into a free \(d\)-dimensional algebra \(m\) over \(\Gamma(T, \mathcal{O}_T)\). This finishes the proof.
Finite Hilbert stacks of spaces
The finite Hilbert stack of an algebraic space is an algebraic stack.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Then \(\mathcal{H}_d(X)\) is an algebraic stack.
Proof
The \(1\)-morphism \[\mathcal{H}_d(X) \longrightarrow \mathcal{H}_d\] is representable by algebraic spaces according to Lemma 05YJ. The stack \(\mathcal{H}_d\) is an algebraic stack according to Proposition 05YQ. Hence \(\mathcal{H}_d(X)\) is an algebraic stack by Algebraic Stacks, Lemma 05UM.
This lemma allows us to bootstrap.
Lemma
Let \(S\) be a scheme. Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\) such that
\(\mathcal{X}\) is representable by an algebraic space, and
\(F\) is representable by algebraic spaces, surjective, flat, and locally of finite presentation.
Then \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is an algebraic stack.
Proof
Choose a representable stack in groupoids \(\mathcal{U}\) over \(S\) and a \(1\)-morphism \(f : \mathcal{U} \to \mathcal{H}_d(\mathcal{X})\) which is representable by algebraic spaces, smooth, and surjective. This is possible because \(\mathcal{H}_d(\mathcal{X})\) is an algebraic stack by Lemma 05YS. Consider the \(2\)-fibre product \[\mathcal{W} = \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \times_{\mathcal{H}_d(\mathcal{X}), f} \mathcal{U}.\] Since \(\mathcal{U}\) is representable (in particular a stack in setoids) it follows from Examples of Stacks, Lemma 05XV and Stacks, Lemma 05UI that \(\mathcal{W}\) is a stack in setoids. The \(1\)-morphism \(\mathcal{W} \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is representable by algebraic spaces, smooth, and surjective as a base change of the morphism \(f\) (see Algebraic Stacks, Lemmas 0302 and 045C). Thus, if we can show that \(\mathcal{W}\) is representable by an algebraic space, then the lemma follows from Algebraic Stacks, Lemma 05UL.
The diagonal of \(\mathcal{Y}\) is representable by algebraic spaces according to Lemma 05XW. We may apply Lemma 05YH to see that the \(1\)-morphism \[\mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{H}_d(\mathcal{X}) \times \mathcal{Y}\] is representable by algebraic spaces. Consider the \(2\)-fibre product \[\mathcal{V} = \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \times_{(\mathcal{H}_d(\mathcal{X}) \times \mathcal{Y}), f \times F} (\mathcal{U} \times \mathcal{X}).\] The projection morphism \(\mathcal{V} \to \mathcal{U} \times \mathcal{X}\) is representable by algebraic spaces as a base change of the last displayed morphism. Hence \(\mathcal{V}\) is an algebraic space (see Bootstrap, Lemma 02YS or Algebraic Stacks, Lemma 0300). The \(1\)-morphism \(\mathcal{V} \to \mathcal{U}\) fits into the following \(2\)-cartesian diagram \[\xymatrix{ \mathcal{V} \ar[d] \ar[r] & \mathcal{X} \ar[d]^F \\ \mathcal{W} \ar[r] & \mathcal{Y} }\] because \[\mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \times_{(\mathcal{H}_d(\mathcal{X}) \times \mathcal{Y}), f \times F} (\mathcal{U} \times \mathcal{X}) = (\mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \times_{\mathcal{H}_d(\mathcal{X}), f} \mathcal{U}) \times_{\mathcal{Y}, F} \mathcal{X}.\] Hence \(\mathcal{V} \to \mathcal{W}\) is representable by algebraic spaces, surjective, flat, and locally of finite presentation as a base change of \(F\). It follows that the same thing is true for the corresponding sheaves of sets associated to \(\mathcal{V}\) and \(\mathcal{W}\), see Algebraic Stacks, Lemma 04TC. Thus we conclude that the sheaf associated to \(\mathcal{W}\) is an algebraic space by Bootstrap, Theorem 04S6.
LCI locus in the Hilbert stack
Please consult Examples of Stacks, Section 05WA for notation. Fix a \(1\)-morphism \(F : \mathcal{X} \longrightarrow \mathcal{Y}\) of stacks in groupoids over \((\Sch/S)_{fppf}\). Assume that \(F\) is representable by algebraic spaces. Fix \(d \geq 1\). Consider an object \((U, Z, y, x, \alpha)\) of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\). There is an induced \(1\)-morphism \[(\Sch/Z)_{fppf} \longrightarrow (\Sch/U)_{fppf} \times_{y, \mathcal{Y}, F} \mathcal{X}\] (by the universal property of \(2\)-fibre products) which is representable by a morphism of algebraic spaces over \(U\). Namely, since \(F\) is representable by algebraic spaces, we may choose an algebraic space \(X_y\) over \(U\) which represents the \(2\)-fibre product \((\Sch/U)_{fppf} \times_{y, \mathcal{Y}, F} \mathcal{X}\). Since \(\alpha : y|_Z \to F(x)\) is an isomorphism we see that \(\xi = (Z, Z \to U, x, \alpha)\) is an object of the \(2\)-fibre product \((\Sch/U)_{fppf} \times_{y, \mathcal{Y}, F} \mathcal{X}\) over \(Z\). Hence \(\xi\) gives rise to a morphism \(x_\alpha : Z \to X_y\) of algebraic spaces over \(U\) as \(X_y\) is the functor of isomorphisms classes of objects of \((\Sch/U)_{fppf} \times_{y, \mathcal{Y}, F} \mathcal{X}\), see Algebraic Stacks, Lemma 02ZX. Here is a picture [06CK]\[\begin{equation} \vcenter{ \xymatrix{ Z \ar[r]_{x_\alpha} \ar[rd] & X_y \ar[d] \\ & U } } \quad\quad \vcenter{ \xymatrix{ (\Sch/Z)_{fppf} \ar[rd] \ar[r]_-{x, \alpha} & (\Sch/U)_{fppf} \times_{y, \mathcal{Y}, F} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d]^F \\ & (\Sch/U)_{fppf} \ar[r]^y & \mathcal{Y} } } \end{equation}\] We remark that if \((f, g, b, a) : (U, Z, y, x, \alpha) \to (U', Z', y', x', \alpha')\) is a morphism between objects of \(\mathcal{H}_d\), then the morphism \(x'_{\alpha'} : Z' \to X'_{y'}\) is the base change of the morphism \(x_\alpha\) by the morphism \(g : U' \to U\) (details omitted).
Now assume moreover that \(F\) is flat and locally of finite presentation. In this situation we define a full subcategory \[\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \subset \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\] consisting of those objects \((U, Z, y, x, \alpha)\) of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) such that the corresponding morphism \(x_\alpha : Z \to X_y\) is unramified and a local complete intersection morphism (see Morphisms of Spaces, Definition 03ZH and More on Morphisms of Spaces, Definition 06C4 for definitions).
Lemma
Let \(S\) be a scheme. Fix a \(1\)-morphism \(F : \mathcal{X} \longrightarrow \mathcal{Y}\) of stacks in groupoids over \((\Sch/S)_{fppf}\). Assume \(F\) is representable by algebraic spaces, flat, and locally of finite presentation. Then \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) is a stack in groupoids and the inclusion functor \[\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\] is representable and an open immersion.
Proof
Let \(\Xi = (U, Z, y, x, \alpha)\) be an object of \(\mathcal{H}_d\). It follows from the remark following (06CK) that the pullback of \(\Xi\) by \(U' \to U\) belongs to \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) if and only if the base change of \(x_\alpha\) is unramified and a local complete intersection morphism. Note that \(Z \to U\) is finite locally free (hence flat, locally of finite presentation and universally closed) and that \(X_y \to U\) is flat and locally of finite presentation by our assumption on \(F\). Then More on Morphisms of Spaces, Lemmas 05X8 and 06CE imply exists an open subscheme \(W \subset U\) such that a morphism \(U' \to U\) factors through \(W\) if and only if the base change of \(x_\alpha\) via \(U' \to U\) is unramified and a local complete intersection morphism. This implies that \[(\Sch/U)_{fppf} \times_{\Xi, \mathcal{H}_d(\mathcal{X}/\mathcal{Y})} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\] is representable by \(W\). Hence the final statement of the lemma holds. The first statement (that \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) is a stack in groupoids) follows from this and Algebraic Stacks, Lemma 05UN.
Local complete intersection morphisms are “locally unobstructed”. This holds in much greater generality than the special case that we need in this chapter here.
Lemma
Let \(U \subset U'\) be a first order thickening of affine schemes. Let \(X'\) be an algebraic space flat over \(U'\). Set \(X = U \times_{U'} X'\). Let \(Z \to U\) be finite locally free of degree \(d\). Finally, let \(f : Z \to X\) be unramified and a local complete intersection morphism. Then there exists a commutative diagram \[\xymatrix{ (Z \subset Z') \ar[rd] \ar[rr]_{(f, f')} & & (X \subset X') \ar[ld] \\ & (U \subset U') }\] of algebraic spaces over \(U'\) such that \(Z' \to U'\) is finite locally free of degree \(d\) and \(Z = U \times_{U'} Z'\).
Proof
By More on Morphisms of Spaces, Lemma 06CB the conormal sheaf \(\mathcal{C}_{Z/X}\) of the unramified morphism \(Z \to X\) is a finite locally free \(\mathcal{O}_Z\)-module and by More on Morphisms of Spaces, Lemma 06CC we have an exact sequence \[0 \to i^*\mathcal{C}_{X/X'} \to \mathcal{C}_{Z/X'} \to \mathcal{C}_{Z/X} \to 0\] of conormal sheaves. Since \(Z\) is affine this sequence is split. Choose a splitting \[\mathcal{C}_{Z/X'} = i^*\mathcal{C}_{X/X'} \oplus \mathcal{C}_{Z/X}\] Let \(Z \subset Z''\) be the universal first order thickening of \(Z\) over \(X'\) (see More on Morphisms of Spaces, Section 0601). Denote \(\mathcal{I} \subset \mathcal{O}_{Z''}\) the quasi-coherent sheaf of ideals corresponding to \(Z \subset Z''\). By definition we have \(\mathcal{C}_{Z/X'}\) is \(\mathcal{I}\) viewed as a sheaf on \(Z\). Hence the splitting above determines a splitting \[\mathcal{I} = i^*\mathcal{C}_{X/X'} \oplus \mathcal{C}_{Z/X}\] Let \(Z' \subset Z''\) be the closed subscheme cut out by \(\mathcal{C}_{Z/X} \subset \mathcal{I}\) viewed as a quasi-coherent sheaf of ideals on \(Z''\). It is clear that \(Z'\) is a first order thickening of \(Z\) and that we obtain a commutative diagram of first order thickenings as in the statement of the lemma.
Since \(X' \to U'\) is flat and since \(X = U \times_{U'} X'\) we see that \(\mathcal{C}_{X/X'}\) is the pullback of \(\mathcal{C}_{U/U'}\) to \(X\), see More on Morphisms of Spaces, Lemma 06BH. Note that by construction \(\mathcal{C}_{Z/Z'} = i^*\mathcal{C}_{X/X'}\) hence we conclude that \(\mathcal{C}_{Z/Z'}\) is isomorphic to the pullback of \(\mathcal{C}_{U/U'}\) to \(Z\). Applying More on Morphisms of Spaces, Lemma 06BH once again (or its analogue for schemes, see More on Morphisms, Lemma 063Y) we conclude that \(Z' \to U'\) is flat and that \(Z = U \times_{U'} Z'\). Finally, More on Morphisms, Lemma 06AG shows that \(Z' \to U'\) is finite locally free of degree \(d\).
Lemma
Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). Assume \(F\) is representable by algebraic spaces, flat, and locally of finite presentation. Then \[p : \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}\] is formally smooth on objects.
Proof
We have to show the following: Given
an object \((U, Z, y, x, \alpha)\) of \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) over an affine scheme \(U\),
a first order thickening \(U \subset U'\), and
an object \(y'\) of \(\mathcal{Y}\) over \(U'\) such that \(y'|_U = y\),
then there exists an object \((U', Z', y', x', \alpha')\) of \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) over \(U'\) with \(Z = U \times_{U'} Z'\), with \(x = x'|_Z\), and with \(\alpha = \alpha'|_U\). Namely, the last two equalities will take care of the commutativity of (06D0).
Consider the morphism \(x_\alpha : Z \to X_y\) constructed in Equation (06CK). Denote similarly \(X'_{y'}\) the algebraic space over \(U'\) representing the \(2\)-fibre product \((\Sch/U')_{fppf} \times_{y', \mathcal{Y}, F} \mathcal{X}\). By assumption the morphism \(X'_{y'} \to U'\) is flat (and locally of finite presentation). As \(y'|_U = y\) we see that \(X_y = U \times_{U'} X'_{y'}\). Hence we may apply Lemma 06D8 to find \(Z' \to U'\) finite locally free of degree \(d\) with \(Z = U \times_{U'} Z'\) and with \(Z' \to X'_{y'}\) extending \(x_\alpha\). By construction the morphism \(Z' \to X'_{y'}\) corresponds to a pair \((x', \alpha')\). It is clear that \((U', Z', y', x', \alpha')\) is an object of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) over \(U'\) with \(Z = U \times_{U'} Z'\), with \(x = x'|_Z\), and with \(\alpha = \alpha'|_U\). As we’ve seen in Lemma 06CL that \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \subset \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\) is an “open substack” it follows that \((U', Z', y', x', \alpha')\) is an object of \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) as desired.
Lemma
Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). Assume \(F\) is representable by algebraic spaces, flat, surjective, and locally of finite presentation. Then \[\coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{Y}\] is surjective on objects.
Proof
It suffices to prove the following: For any field \(k\) and object \(y\) of \(\mathcal{Y}\) over \(\Spec(k)\) there exists an integer \(d \geq 1\) and an object \((U, Z, y, x, \alpha)\) of \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) with \(U = \Spec(k)\). Namely, in this case we see that \(p\) is surjective on objects in the strong sense that an extension of the field is not needed.
Denote \(X_y\) the algebraic space over \(U = \Spec(k)\) representing the \(2\)-fibre product \((\Sch/U')_{fppf} \times_{y', \mathcal{Y}, F} \mathcal{X}\). By assumption the morphism \(X_y \to \Spec(k)\) is surjective and locally of finite presentation (and flat). In particular \(X_y\) is nonempty. Choose a nonempty affine scheme \(V\) and an étale morphism \(V \to X_y\). Note that \(V \to \Spec(k)\) is (flat), surjective, and locally of finite presentation (by Morphisms of Spaces, Definition 03XP). Pick a closed point \(v \in V\) where \(V \to \Spec(k)\) is Cohen-Macaulay (i.e., \(V\) is Cohen-Macaulay at \(v\)), see More on Morphisms, Lemma 045U. Applying More on Morphisms, Lemma 0570 we find a regular immersion \(Z \to V\) with \(Z = \{v\}\). This implies \(Z \to V\) is a closed immersion. Moreover, it follows that \(Z \to \Spec(k)\) is finite (for example by Algebra, Lemma 00PJ). Hence \(Z \to \Spec(k)\) is finite locally free of some degree \(d\). Now \(Z \to X_y\) is unramified as the composition of a closed immersion followed by an étale morphism (see Morphisms of Spaces, Lemmas 03ZI, 06CR, and 06CQ). Finally, \(Z \to X_y\) is a local complete intersection morphism as a composition of a regular immersion of schemes and an étale morphism of algebraic spaces (see More on Morphisms, Lemma 069L and Morphisms of Spaces, Lemmas 04XX and 06CP and More on Morphisms of Spaces, Lemmas 06C9 and 06C8). The morphism \(Z \to X_y\) corresponds to an object \(x\) of \(\mathcal{X}\) over \(Z\) together with an isomorphism \(\alpha : y|_Z \to F(x)\). We obtain an object \((U, Z, y, x, \alpha)\) of \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y})\). By what was said above about the morphism \(Z \to X_y\) we see that it actually is an object of the subcategory \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) and we win.
Bootstrapping algebraic stacks
The following theorem is one of the main results of this chapter.
Theorem
Let \(S\) be a scheme. Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of stacks in groupoids over \((\Sch/S)_{fppf}\). If
\(\mathcal{X}\) is representable by an algebraic space, and
\(F\) is representable by algebraic spaces, surjective, flat and locally of finite presentation,
then \(\mathcal{Y}\) is an algebraic stack.
Proof
By Lemma 05XW we see that the diagonal of \(\mathcal{Y}\) is representable by algebraic spaces. Hence we only need to verify the existence of a \(1\)-morphism \(f : \mathcal{V} \to \mathcal{Y}\) of stacks in groupoids over \((\Sch/S)_{fppf}\) with \(\mathcal{V}\) representable and \(f\) surjective and smooth. By Lemma 06CI we know that \[\coprod\nolimits_{d \geq 1} \mathcal{H}_d(\mathcal{X}/\mathcal{Y})\] is an algebraic stack. It follows from Lemma 06CL and Algebraic Stacks, Lemma 05UN that \[\coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\] is an algebraic stack as well. Choose a representable stack in groupoids \(\mathcal{V}\) over \((\Sch/S)_{fppf}\) and a surjective and smooth \(1\)-morphism \[\mathcal{V} \longrightarrow \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}).\] We claim that the composition \[\mathcal{V} \longrightarrow \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{Y}\] is smooth and surjective which finishes the proof of the theorem. In fact, the smoothness will be a consequence of Lemmas 06CH and 06D9 and the surjectivity a consequence of Lemma 06DA. We spell out the details in the following paragraph.
By construction \(\mathcal{V} \to \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) is representable by algebraic spaces, surjective, and smooth (and hence also locally of finite presentation and formally smooth by the general principle Algebraic Stacks, Lemma 045F and More on Morphisms of Spaces, Lemma 04AM). Applying Lemmas 06CX, 06D3, and 06D7 we see that \(\mathcal{V} \to \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})\) is limit preserving on objects, formally smooth on objects, and surjective on objects. The \(1\)-morphism \(\coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}\) is
limit preserving on objects: this is Lemma 06CH for \(\mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}\) and we combine it with Lemmas 06CL, 06CY, and 06CW to get it for \(\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}\),
formally smooth on objects by Lemma 06D9, and
surjective on objects by Lemma 06DA.
Using Lemmas 06CW, 06D2, and 06D6 we conclude that the composition \(\mathcal{V} \to \mathcal{Y}\) is limit preserving on objects, formally smooth on objects, and surjective on objects. Using Lemmas 06CX, 06D3, and 06D7 we see that \(\mathcal{V} \to \mathcal{Y}\) is locally of finite presentation, formally smooth, and surjective. Finally, using (via the general principle Algebraic Stacks, Lemma 045F) the infinitesimal lifting criterion (More on Morphisms of Spaces, Lemma 04AM) we see that \(\mathcal{V} \to \mathcal{Y}\) is smooth and we win.
Applications
Our first task is to show that the quotient stack \([U/R]\) associated to a “flat and locally finitely presented groupoid” is an algebraic stack. See Groupoids in Spaces, Definition 044Q for the definition of the quotient stack. The following lemma is preliminary and is the analogue of Algebraic Stacks, Lemma 04X0.
Lemma
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \((U, R, s, t, c)\) be a groupoid in algebraic spaces over \(S\). Assume \(s, t\) are flat and locally of finite presentation. Then the morphism \(\mathcal{S}_U \to [U/R]\) is flat, locally of finite presentation, and surjective.
Proof
Let \(T\) be a scheme and let \(x : (\Sch/T)_{fppf} \to [U/R]\) be a \(1\)-morphism. We have to show that the projection \[\mathcal{S}_U \times_{[U/R]} (\Sch/T)_{fppf} \longrightarrow (\Sch/T)_{fppf}\] is surjective, flat, and locally of finite presentation. We already know that the left hand side is representable by an algebraic space \(F\), see Algebraic Stacks, Lemmas 04WZ and 045G. Hence we have to show the corresponding morphism \(F \to T\) of algebraic spaces is surjective, locally of finite presentation, and flat. Since we are working with properties of morphisms of algebraic spaces which are local on the target in the fppf topology we may check this fppf locally on \(T\). By construction, there exists an fppf covering \(\{T_i \to T\}\) of \(T\) such that \(x|_{(\Sch/T_i)_{fppf}}\) comes from a morphism \(x_i : T_i \to U\). (Note that \(F \times_T T_i\) represents the \(2\)-fibre product \(\mathcal{S}_U \times_{[U/R]} (\Sch/T_i)_{fppf}\) so everything is compatible with the base change via \(T_i \to T\).) Hence we may assume that \(x\) comes from \(x : T \to U\). In this case we see that \[\mathcal{S}_U \times_{[U/R]} (\Sch/T)_{fppf} = (\mathcal{S}_U \times_{[U/R]} \mathcal{S}_U) \times_{\mathcal{S}_U} (\Sch/T)_{fppf} = \mathcal{S}_R \times_{\mathcal{S}_U} (\Sch/T)_{fppf}\] The first equality by Categories, Lemma 02XD and the second equality by Groupoids in Spaces, Lemma 04M9. Clearly the last \(2\)-fibre product is represented by the algebraic space \(F = R \times_{s, U, x} T\) and the projection \(R \times_{s, U, x} T \to T\) is flat and locally of finite presentation as the base change of the flat locally finitely presented morphism of algebraic spaces \(s : R \to U\). It is also surjective as \(s\) has a section (namely the identity \(e : U \to R\) of the groupoid). This proves the lemma.
Here is the first main result of this section.
Theorem
Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \((U, R, s, t, c)\) be a groupoid in algebraic spaces over \(S\). Assume \(s, t\) are flat and locally of finite presentation. Then the quotient stack \([U/R]\) is an algebraic stack over \(S\).
Proof
We check the two conditions of Theorem 06DC for the morphism \[(\Sch/U)_{fppf} \longrightarrow [U/R].\] The first is trivial (as \(U\) is an algebraic space). The second is Lemma 06FH.
When is a quotient stack algebraic?
In Groupoids in Spaces, Section 044O we have defined the quotient stack \([U/R]\) associated to a groupoid \((U, R, s, t, c)\) in algebraic spaces. Note that \([U/R]\) is a stack in groupoids whose diagonal is representable by algebraic spaces (see Bootstrap, Lemma 04TB and Algebraic Stacks, Lemma 045G) and such that there exists an algebraic space \(U\) and a \(1\)-morphism \((\Sch/U)_{fppf} \to [U/R]\) which is an “fppf surjection” in the sense that it induces a map on presheaves of isomorphism classes of objects which becomes surjective after sheafification. However, it is not the case that \([U/R]\) is an algebraic stack in general. This is not a contradiction with Theorem 06DC as the \(1\)-morphism \((\Sch/U)_{fppf} \to [U/R]\) may not be flat and locally of finite presentation.
The easiest way to make examples of non-algebraic quotient stacks is to look at quotients of the form \([S/G]\) where \(S\) is a scheme and \(G\) is a group scheme over \(S\) acting trivially on \(S\). Namely, we will see below (Lemma 06PL) that if \([S/G]\) is algebraic, then \(G \to S\) has to be flat and locally of finite presentation. An explicit example can be found in Examples, Section 04AG.
Lemma
Let \(S\) be a scheme and let \(B\) be an algebraic space over \(S\). Let \((U, R, s, t, c)\) be a groupoid in algebraic spaces over \(B\). The quotient stack \([U/R]\) is an algebraic stack if and only if there exists a morphism of algebraic spaces \(g : U' \to U\) such that
the composition \(U' \times_{g, U, t} R \to R \xrightarrow{s} U\) is a surjection of sheaves, and
the morphisms \(s', t' : R' \to U'\) are flat and locally of finite presentation where \((U', R', s', t', c')\) is the restriction of \((U, R, s, t, c)\) via \(g\).
Proof
First, assume that \(g : U' \to U\) satisfies (1) and (2). Property (1) implies that \([U'/R'] \to [U/R]\) is an equivalence, see Groupoids in Spaces, Lemma 046T. By Theorem 06FI the quotient stack \([U'/R']\) is an algebraic stack. Hence \([U/R]\) is an algebraic stack too, see Algebraic Stacks, Lemma 03YQ.
Conversely, assume that \([U/R]\) is an algebraic stack. We may choose a scheme \(W\) and a surjective smooth \(1\)-morphism \[f : (\Sch/W)_{fppf} \longrightarrow [U/R].\] By the \(2\)-Yoneda lemma (Algebraic Stacks, Section 04SS) this corresponds to an object \(\xi\) of \([U/R]\) over \(W\). By the description of \([U/R]\) in Groupoids in Spaces, Lemma 044X we can find a surjective, flat, locally finitely presented morphism \(b : U' \to W\) of schemes such that \(\xi' = b^*\xi\) corresponds to a morphism \(g : U' \to U\). Note that the \(1\)-morphism \[f' : (\Sch/U')_{fppf} \longrightarrow [U/R].\] corresponding to \(\xi'\) is surjective, flat, and locally of finite presentation, see Algebraic Stacks, Lemma 045B. Hence \((\Sch/U')_{fppf} \times_{[U/R]} (\Sch/U')_{fppf}\) which is represented by the algebraic space \[\mathit{Isom}_{[U/R]}(\text{pr}_0^*\xi', \text{pr}_1^*\xi') = (U' \times_S U') \times_{(g \circ \text{pr}_0, g \circ \text{pr}_1), U \times_S U} R = R'\] (see Groupoids in Spaces, Lemma 044V for the first equality; the second is the definition of restriction) is flat and locally of finite presentation over \(U'\) via both \(s'\) and \(t'\) (by base change, see Algebraic Stacks, Lemma 045C). By this description of \(R'\) and by Algebraic Stacks, Lemma 04T4 we obtain a canonical fully faithful \(1\)-morphism \([U'/R'] \to [U/R]\). This \(1\)-morphism is essentially surjective because \(f'\) is flat, locally of finite presentation, and surjective (see Stacks, Lemma 046N); another way to prove this is to use Algebraic Stacks, Remark 04WY. Finally, we can use Groupoids in Spaces, Lemma 046T to conclude that the composition \(U' \times_{g, U, t} R \to R \xrightarrow{s} U\) is a surjection of sheaves.
Lemma
Let \(S\) be a scheme and let \(B\) be an algebraic space over \(S\). Let \(G\) be a group algebraic space over \(B\). Let \(X\) be an algebraic space over \(B\) and let \(a : G \times_B X \to X\) be an action of \(G\) on \(X\) over \(B\). The quotient stack \([X/G]\) is an algebraic stack if and only if there exists a morphism of algebraic spaces \(\varphi : X' \to X\) such that
\(G \times_B X' \to X\), \((g, x') \mapsto a(g, \varphi(x'))\) is a surjection of sheaves, and
the two projections \(X'' \to X'\) of the algebraic space \(X''\) given by the rule \[T \longmapsto \{(x'_1, g, x'_2) \in (X' \times_B G \times_B X')(T) \mid \varphi(x'_1) = a(g, \varphi(x'_2))\}\] are flat and locally of finite presentation.
Proof
This lemma is a special case of Lemma 06PJ. Namely, the quotient stack \([X/G]\) is by Groupoids in Spaces, Definition 044Q equal to the quotient stack \([X/G \times_B X]\) of the groupoid in algebraic spaces \((X, G \times_B X, s, t, c)\) associated to the group action in Groupoids in Spaces, Lemma 0444. There is one small observation that is needed to get condition (1). Namely, the morphism \(s : G \times_B X \to X\) is the second projection and the morphism \(t : G \times_B X \to X\) is the action morphism \(a\). Hence the morphism \(h : U' \times_{g, U, t} R \to R \xrightarrow{s} U\) from Lemma 06PJ corresponds to the morphism \[X' \times_{\varphi, X, a} (G \times_B X) \xrightarrow{\text{pr}_1} X\] in the current setting. However, because of the symmetry given by the inverse of \(G\) this morphism is isomorphic to the morphism \[(G \times_B X) \times_{\text{pr}_1, X, \varphi} X' \xrightarrow{a} X\] of the statement of the lemma. Details omitted.
Lemma
Let \(S\) be a scheme and let \(B\) be an algebraic space over \(S\). Let \(G\) be a group algebraic space over \(B\). Endow \(B\) with the trivial action of \(G\). Then the quotient stack \([B/G]\) is an algebraic stack if and only if \(G\) is flat and locally of finite presentation over \(B\).
Proof
If \(G\) is flat and locally of finite presentation over \(B\), then \([B/G]\) is an algebraic stack by Theorem 06FI.
Conversely, assume that \([B/G]\) is an algebraic stack. By Lemma 06PK and because the action is trivial, we see there exists an algebraic space \(B'\) and a morphism \(B' \to B\) such that (1) \(B' \to B\) is a surjection of sheaves and (2) the projections \[B' \times_B G \times_B B' \to B'\] are flat and locally of finite presentation. Note that the base change \(B' \times_B G \times_B B' \to G \times_B B'\) of \(B' \to B\) is a surjection of sheaves also. Thus it follows from Descent on Spaces, Lemma 06NR that the projection \(G \times_B B' \to B'\) is flat and locally of finite presentation. By (1) we can find an fppf covering \(\{B_i \to B\}\) such that \(B_i \to B\) factors through \(B' \to B\). Hence \(G \times_B B_i \to B_i\) is flat and locally of finite presentation by base change. By Descent on Spaces, Lemmas 041W and 041T we conclude that \(G \to B\) is flat and locally of finite presentation.
Later we will see that the quotient stack of a smooth \(S\)-space by a group algebraic space \(G\) is smooth, even when \(G\) is not smooth (Morphisms of Stacks, Lemma 0DLS).
Algebraic stacks in the étale topology
Let \(S\) be a scheme. Instead of working with stacks in groupoids over the big fppf site \((\Sch/S)_{fppf}\) we could work with stacks in groupoids over the big étale site \((\Sch/S)_\etale\). All of the material in Algebraic Stacks, Sections 02ZQ, 04SS, 04ST, 04SU, 02ZV, 04SX, 03YJ, and 0304 makes sense for categories fibred in groupoids over \((\Sch/S)_\etale\). Thus we get a second notion of an algebraic stack by working in the étale topology. This notion is (a priori) weaker than the notion introduced in Algebraic Stacks, Definition 026O since a stack in the fppf topology is certainly a stack 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 Sheaves on Stacks, Section 06TN and Topologies, Remark 03FF). Let \(S\) be an object of \(\Sch_\alpha\). Let \[p : \mathcal{X} \to \Sch_\alpha/S\] be a category fibred in groupoids with the following properties:
\(\mathcal{X}\) is a stack in groupoids over \((\Sch/S)_\etale\),
the diagonal \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces5, and
there exists \(U \in \Ob(\Sch_\alpha/S)\) and a \(1\)-morphism \((\Sch/U)_\etale \to \mathcal{X}\) which is surjective and smooth.
Then \(\mathcal{X}\) is an algebraic stack in the sense of Algebraic Stacks, Definition 026O.
Proof
Note that properties (2) and (3) of the lemma and the corresponding properties (2) and (3) of Algebraic Stacks, Definition 026O are independent of the topology. This is true because these properties involve only the notion of a \(2\)-fibre product of categories fibred in groupoids, \(1\)- and \(2\)-morphisms of categories fibred in groupoids, the notion of a \(1\)-morphism of categories fibred in groupoids representable by algebraic spaces, and what it means for such a \(1\)-morphism to be surjective and smooth. Thus all we have to prove is that an étale stack in groupoids \(\mathcal{X}\) with properties (2) and (3) is also an fppf stack in groupoids.
Using (2) let \(R\) be an algebraic space representing \[(\Sch_\alpha/U) \times_\mathcal{X} (\Sch_\alpha/U)\] By (3) the projections \(s, t : R \to U\) are smooth. Exactly as in the proof of Algebraic Stacks, Lemma 04T4 there exists a groupoid in spaces \((U, R, s, t, c)\) and a canonical fully faithful \(1\)-morphism \([U/R]_\etale \to \mathcal{X}\) where \([U/R]_\etale\) is the étale stackification of presheaf in groupoids \[T \longmapsto (U(T), R(T), s(T), t(T), c(T))\] Claim: If \(V \to T\) is a surjective smooth morphism from an algebraic space \(V\) to a scheme \(T\), then there exists an étale covering \(\{T_i \to T\}\) refining the covering \(\{V \to T\}\). This follows from More on Morphisms, Lemma 055V or the more general Sheaves on Stacks, Lemma 06XH. Using the claim and arguing exactly as in Algebraic Stacks, Lemma 04T5 it follows that \([U/R]_\etale \to \mathcal{X}\) is an equivalence.
Next, let \([U/R]\) denote the quotient stack in the fppf topology which is an algebraic stack by Algebraic Stacks, Theorem 04TK. Thus we have \(1\)-morphisms \[U \to [U/R]_\etale \to [U/R].\] Both \(U \to [U/R]_\etale \cong \mathcal{X}\) and \(U \to [U/R]\) are surjective and smooth (the first by assumption and the second by the theorem) and in both cases the fibre product \(U \times_\mathcal{X} U\) and \(U \times_{[U/R]} U\) is representable by \(R\). Hence the \(1\)-morphism \([U/R]_\etale \to [U/R]\) is fully faithful (since morphisms in the quotient stacks are given by morphisms into \(R\), see Groupoids in Spaces, Section 04MB).
Finally, for any scheme \(T\) and morphism \(t : T \to [U/R]\) the fibre product \(V = T \times_{U/R} U\) is an algebraic space surjective and smooth over \(T\). By the claim above there exists an étale covering \(\{T_i \to T\}_{i \in I}\) and morphisms \(T_i \to V\) over \(T\). This proves that the object \(t\) of \([U/R]\) over \(T\) comes étale locally from \(U\). We conclude that \([U/R]_\etale \to [U/R]\) is an equivalence of stacks in groupoids over \((\Sch/S)_\etale\) by Stacks, Lemma 046N. This concludes the proof.
The condition on size can be dropped by those ignoring set theoretic issues.↩︎
To see that the set theoretic condition of that lemma is satisfied we argue as follows: First choose the open covering such that \(|I| \leq \text{size}(V)\). Next, choose schemes \(U_i\) of size \(\leq \max(\kappa, \text{size}(V))\) and surjective étale morphisms \(U_i \to Z_i\); we can do this by assumption (2) and Sets, Lemma 04T6 (details omitted). Then Sets, Lemma 000R implies that \(\coprod U_i\) is an object of \((\Sch/S)_{fppf}\). Hence \(\coprod Z_i\) is an algebraic space by Spaces, Lemma 02WQ.↩︎
Modulo some set theoretic arguments. Namely, we have to show that \(\coprod F_E\) is an algebraic space. This follows because \(|I| \leq \text{size}(X)\) and \(\text{size}(F_E) \leq \text{size}(X)\) as follows from the explicit description of \(F_E\) in the proof of More on Morphisms, Lemma 05Y6. Some details omitted.↩︎
It may be better to think of this as a pair consisting of a multiplication map \(m : R^{\oplus d} \otimes_R R^{\oplus d} \to R^{\oplus d}\) and a ring map \(\psi : R \to R^{\oplus d}\) satisfying a bunch of axioms.↩︎
Here we can either mean sheaves in the étale topology whose diagonal is representable and which have an étale surjective covering by a scheme or algebraic spaces as defined in Algebraic Spaces, Definition 025Y. Namely, by Bootstrap, Lemma 076M there is no difference.↩︎