Introduction
As initially conceived, the purpose of this chapter was to write about Quot and Hilbert functors and to prove that these are algebraic spaces provided certain technical conditions are satisfied. This material, in the setting of schemes, is covered in Grothendieck’s lectures in the séminair Bourbaki, see [Gr-I], [Gr-II], [Gr-III], [Gr-IV], [Gr-V], and [Gr-VI]. For projective schemes the Quot and Hilbert schemes live inside Grassmannians of spaces of sections of suitable very ample invertible sheaves, and this provides a method of construction for these schemes. Our approach is different: we use Artin’s axioms to prove Quot and Hilb are algebraic spaces.
Upon further consideration, it turned out to be more convenient for the development of theory in the Stacks project, to start the discussion with the stack \(\Cohstack_{X/B}\) of coherent sheaves (with proper support over the base) as introduced in [lieblich_remarks]. For us \(f : X \to B\) is a morphism of algebraic spaces satisfying suitable technical conditions, although this can be generalized (see below). Given modules \(\mathcal{F}\) and \(\mathcal{G}\) on \(X\), under suitably hypotheses, the functor \(T/B \mapsto \Hom_{X_T}(\mathcal{F}_T, \mathcal{G}_T)\) is an algebraic space \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) over \(B\). See Section 08JS. The subfunctor \(\mathit{Isom}(\mathcal{F}, \mathcal{G})\) of isomorphisms is shown to be an algebraic space in Section 08K7. This is used in the next sections to show the diagonal of the stack \(\Cohstack_{X/B}\) is representable. We prove \(\Cohstack_{X/B}\) is an algebraic stack in Section 08KA when \(X \to B\) is flat and in Section 08WB in general. Please see the introduction of this section for pointers to the literature.
Having proved this, it is rather straightforward to prove that \(\Quotfunctor_{\mathcal{F}/X/B}\), \(\Hilbfunctor_{X/B}\), and \(\Picardfunctor_{X/B}\) are algebraic spaces and that \(\Picardstack_{X/B}\) is an algebraic stack. See Sections 09TQ, 0CZX, 0D24, and 0D02.
In the usual manner we deduce that the functor \(\mathit{Mor}_B(Z, X)\) of relative morphisms is an algebraic space (under suitable hypotheses) in Section 0D19.
In Section 0D1D we prove that the stack in groupoids \[\Spacesstack'_{fp, flat, proper}\] parametrizing flat families of proper algebraic spaces satisfies all of Artin’s axioms (including openness of versality) except for formal effectiveness. We’ve chosen the very awkward notation for this stack intentionally, because the reader should be careful in using its properties.
In Section 0D1L we prove that the stack \(\Polarizedstack\) parametrizing flat families of polarized proper algebraic spaces is an algebraic stack. Because of our work on flat families of proper algebraic spaces, this comes down to proving formal effectiveness for polarized schemes which is often known as Grothendieck’s algebraization theorem.
In Section 0D4Y we prove that the stack \(\Curvesstack\) parametrizing families of curves is algebraic.
In Section 0DLB we study moduli of complexes on a proper morphism and we obtain an algebraic stack \(\Complexesstack_{X/B}\). The idea of the statement and the proof are taken from [lieblich-complexes].
What is not in this chapter? There is almost no discussion of the properties the resulting moduli spaces and moduli stacks possess (beyond their algebraicity); to read about this we refer to Moduli Stacks, Section 0DLU. In most of the results discussed, we can generalize the constructions by considering a morphism \(\mathcal{X} \to \mathcal{B}\) of algebraic stacks instead of a morphism \(X \to B\) of algebraic space. We will discuss this (insert future reference here). In the case of Hilbert spaces there is a more general notion of “Hilbert stacks” which we will discuss in a separate chapter, see (insert future reference here).
Conventions
We have intentionally placed this chapter, as well as the chapters “Examples of Stacks”, “Sheaves on Algebraic Stacks”, “Criteria for Representability”, and “Artin’s Axioms” before the general development of the theory of algebraic stacks. The reason for this is that starting with the next chapter (see Properties of Stacks, Section 04XA) we will no longer distinguish between a scheme and the algebraic stack it gives rise to. Thus our language will become more flexible and easier for a human to parse, but also less precise. These first few chapters, including the initial chapter “Algebraic Stacks”, lay the groundwork that later allow us to ignore some of the very technical distinctions between different ways of thinking about algebraic stacks. But especially in the chapters “Artin’s Axioms” and “Criteria of Representability” we need to be very precise about what objects exactly we are working with, as we are trying to show that certain constructions produce algebraic stacks or algebraic spaces.
Unfortunately, this means that some of the notation, conventions and terminology is awkward and may seem backwards to the more experienced reader. We hope the reader will forgive us!
The standing assumption is that all schemes are contained in a big fppf site \(\Sch_{fppf}\). And all rings \(A\) considered have the property that \(\Spec(A)\) is (isomorphic) to an object of this big site.
Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). In this chapter and the following we will write \(X \times_S X\) for the product of \(X\) with itself (in the category of algebraic spaces over \(S\)), instead of \(X \times X\).
The Hom functor
In this section we study the functor of homomorphisms defined below.
Situation
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\), \(\mathcal{G}\) be quasi-coherent \(\mathcal{O}_X\)-modules. For any scheme \(T\) over \(B\) we will denote \(\mathcal{F}_T\) and \(\mathcal{G}_T\) the base changes of \(\mathcal{F}\) and \(\mathcal{G}\) to \(T\), in other words, the pullbacks via the projection morphism \(X_T = X \times_B T \to X\). We consider the functor [08JU]\[\begin{equation} \mathit{Hom}(\mathcal{F}, \mathcal{G}) : (\Sch/B)^{opp} \longrightarrow \textit{Sets},\quad T \longrightarrow \Hom_{\mathcal{O}_{X_T}}(\mathcal{F}_T, \mathcal{G}_T) \end{equation}\]
In Situation 08JT we sometimes think of \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) as a functor \((\Sch/S)^{opp} \to \textit{Sets}\) endowed with a morphism \(\mathit{Hom}(\mathcal{F}, \mathcal{G}) \to B\). Namely, if \(T\) is a scheme over \(S\), then an element of \(\mathit{Hom}(\mathcal{F}, \mathcal{G})(T)\) consists of a pair \((h, u)\), where \(h\) is a morphism \(h : T \to B\) and \(u : \mathcal{F}_T \to \mathcal{G}_T\) is an \(\mathcal{O}_{X_T}\)-module map where \(X_T = T \times_{h, B} X\) and \(\mathcal{F}_T\) and \(\mathcal{G}_T\) are the pullbacks to \(X_T\). In particular, when we say that \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) is an algebraic space, we mean that the corresponding functor \((\Sch/S)^{opp} \to \textit{Sets}\) is an algebraic space.
Lemma
In Situation 08JT the functor \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) satisfies the sheaf property for the fpqc topology.
Proof
Let \(\{T_i \to T\}_{i \in I}\) be an fpqc covering of schemes over \(B\). Set \(X_i = X_{T_i} = X \times_S T_i\) and \(\mathcal{F}_i = u_{T_i}\) and \(\mathcal{G}_i = \mathcal{G}_{T_i}\). Note that \(\{X_i \to X_T\}_{i \in I}\) is an fpqc covering of \(X_T\), see Topologies on Spaces, Lemma 03MR. Thus a family of maps \(u_i : \mathcal{F}_i \to \mathcal{G}_i\) such that \(u_i\) and \(u_j\) restrict to the same map on \(X_{T_i \times_T T_j}\) comes from a unique map \(u : \mathcal{F}_T \to \mathcal{G}_T\) by descent (Descent on Spaces, Proposition 04W8).
Sanity check: \(\mathit{Hom}\) sheaf plays the same role among algebraic spaces over \(S\).
Lemma
In Situation 08JT. Let \(T\) be an algebraic space over \(S\). We have \[\Mor_{\Sh((\Sch/S)_{fppf})}(T, \mathit{Hom}(\mathcal{F}, \mathcal{G})) = \{(h, u) \mid h : T \to B, u : \mathcal{F}_T \to \mathcal{G}_T\}\] where \(\mathcal{F}_T, \mathcal{G}_T\) denote the pullbacks of \(\mathcal{F}\) and \(\mathcal{G}\) to the algebraic space \(X \times_{B, h} T\).
Proof
Choose a scheme \(U\) and a surjective étale morphism \(p : U \to T\). Let \(R = U \times_T U\) with projections \(t, s : R \to U\).
Let \(v : T \to \mathit{Hom}(\mathcal{F}, \mathcal{G})\) be a natural transformation. Then \(v(p)\) corresponds to a pair \((h_U, u_U)\) over \(U\). As \(v\) is a transformation of functors we see that the pullbacks of \((h_U, u_U)\) by \(s\) and \(t\) agree. Since \(T = U/R\) (Spaces, Lemma 0262), we obtain a morphism \(h : T \to B\) such that \(h_U = h \circ p\). Then \(\mathcal{F}_U\) is the pullback of \(\mathcal{F}_T\) to \(X_U\) and similarly for \(\mathcal{G}_U\). Hence \(u_U\) descends to a \(\mathcal{O}_{X_T}\)-module map \(u : \mathcal{F}_T \to \mathcal{G}_T\) by Descent on Spaces, Proposition 04W8.
Conversely, let \((h, u)\) be a pair over \(T\). Then we get a natural transformation \(v : T \to \mathit{Hom}(\mathcal{F}, \mathcal{G})\) by sending a morphism \(a : T' \to T\) where \(T'\) is a scheme to \((h \circ a, a^*u)\). We omit the verification that the construction of this and the previous paragraph are mutually inverse.
Remark
In Situation 08JT let \(B' \to B\) be a morphism of algebraic spaces over \(S\). Set \(X' = X \times_B B'\) and denote \(\mathcal{F}'\), \(\mathcal{G}'\) the pullback of \(\mathcal{F}\), \(\mathcal{G}\) to \(X'\). Then we obtain a functor \(\mathit{Hom}(\mathcal{F}', \mathcal{G}') : (\Sch/B')^{opp} \to \textit{Sets}\) associated to the base change \(f' : X' \to B'\). For a scheme \(T\) over \(B'\) it is clear that we have \[\mathit{Hom}(\mathcal{F}', \mathcal{G}')(T) = \mathit{Hom}(\mathcal{F}, \mathcal{G})(T)\] where on the right hand side we think of \(T\) as a scheme over \(B\) via the composition \(T \to B' \to B\). This trivial remark will occasionally be useful to change the base algebraic space.
Lemma
In Situation 08JT let \(\{X_i \to X\}_{i \in I}\) be an fppf covering and for each \(i, j \in I\) let \(\{X_{ijk} \to X_i \times_X X_j\}\) be an fppf covering. Denote \(\mathcal{F}_i\), resp. \(\mathcal{F}_{ijk}\) the pullback of \(\mathcal{F}\) to \(X_i\), resp. \(X_{ijk}\). Similarly define \(\mathcal{G}_i\) and \(\mathcal{G}_{ijk}\). For every scheme \(T\) over \(B\) the diagram \[\xymatrix{ \mathit{Hom}(\mathcal{F}, \mathcal{G})(T) \ar[r] & \prod\nolimits_i \mathit{Hom}(\mathcal{F}_i, \mathcal{G}_i)(T) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod\nolimits_{i, j, k} \mathit{Hom}(\mathcal{F}_{ijk}, \mathcal{G}_{ijk})(T) }\] presents the first arrow as the equalizer of the other two.
Proof
Let \(u_i : \mathcal{F}_{i, T} \to \mathcal{G}_{i, T}\) be an element in the equalizer of \(\text{pr}_0^*\) and \(\text{pr}_1^*\). Since the base change of an fppf covering is an fppf covering (Topologies on Spaces, Lemma 03Y9) we see that \(\{X_{i, T} \to X_T\}_{i \in I}\) and \(\{X_{ijk, T} \to X_{i, T} \times_{X_T} X_{j, T}\}\) are fppf coverings. Applying Descent on Spaces, Proposition 04W8 we first conclude that \(u_i\) and \(u_j\) restrict to the same morphism over \(X_{i, T} \times_{X_T} X_{j, T}\), whereupon a second application shows that there is a unique morphism \(u : \mathcal{F}_T \to \mathcal{G}_T\) restricting to \(u_i\) for each \(i\). This finishes the proof.
Lemma
In Situation 08JT. If \(\mathcal{F}\) is of finite presentation and \(f\) is quasi-compact and quasi-separated, then \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) is limit preserving.
Proof
Let \(T = \lim_{i \in I} T_i\) be a directed limit of affine \(B\)-schemes. We have to show that \[\mathit{Hom}(\mathcal{F}, \mathcal{G})(T) = \colim \mathit{Hom}(\mathcal{F}, \mathcal{G})(T_i)\] Pick \(0 \in I\). We may replace \(B\) by \(T_0\), \(X\) by \(X_{T_0}\), \(\mathcal{F}\) by \(\mathcal{F}_{T_0}\), \(\mathcal{G}\) by \(\mathcal{G}_{T_0}\), and \(I\) by \(\{i \in I \mid i \geq 0\}\). See Remark 08JW. Thus we may assume \(B = \Spec(R)\) is affine.
When \(B\) is affine, then \(X\) is quasi-compact and quasi-separated. Choose a surjective étale morphism \(U \to X\) where \(U\) is an affine scheme (Properties of Spaces, Lemma 03H6). Since \(X\) is quasi-separated, the scheme \(U \times_X U\) is quasi-compact and we may choose a surjective étale morphism \(V \to U \times_X U\) where \(V\) is an affine scheme. Applying Lemma 08K3 we see that \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) is the equalizer of two maps between \[\mathit{Hom}(\mathcal{F}|_U, \mathcal{G}|_U) \quad\text{and}\quad \mathit{Hom}(\mathcal{F}|_V, \mathcal{G}|_V)\] This reduces us to the case that \(X\) is affine.
In the affine case the statement of the lemma reduces to the following problem: Given a ring map \(R \to A\), two \(A\)-modules \(M\), \(N\) and a directed system of \(R\)-algebras \(C = \colim C_i\). When is it true that the map \[\colim \Hom_{A \otimes_R C_i}(M \otimes_R C_i, N \otimes_R C_i) \longrightarrow \Hom_{A \otimes_R C}(M \otimes_R C, N \otimes_R C)\] is bijective? By Algebra, Lemma 05LI this holds if \(M \otimes_R C\) is of finite presentation over \(A \otimes_R C\), i.e., when \(M\) is of finite presentation over \(A\).
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(i : X' \to X\) be a closed immersion of algebraic spaces over \(B\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module and let \(\mathcal{G}'\) be a quasi-coherent \(\mathcal{O}_{X'}\)-module. Then \[\mathit{Hom}(\mathcal{F}, i_*\mathcal{G}') = \mathit{Hom}(i^*\mathcal{F}, \mathcal{G}')\] as functors on \((\Sch/B)\).
Proof
Let \(g : T \to B\) be a morphism where \(T\) is a scheme. Denote \(i_T : X'_T \to X_T\) the base change of \(i\). Denote \(h : X_T \to X\) and \(h' : X'_T \to X'\) the projections. Observe that \((h')^*i^*\mathcal{F} = i_T^*h^*\mathcal{F}\). As a closed immersion is affine (Morphisms of Spaces, Lemma 07U2) we have \(h^*i_*\mathcal{G} = i_{T, *}(h')^*\mathcal{G}\) by Cohomology of Spaces, Lemma 07U8. Thus we have \[\begin{align*} \mathit{Hom}(\mathcal{F}, i_*\mathcal{G}')(T) & = \Hom_{\mathcal{O}_{X_T}}(h^*\mathcal{F}, h^*i_*\mathcal{G}') \\ & = \Hom_{\mathcal{O}_{X_T}}(h^*\mathcal{F}, i_{T, *}(h')^*\mathcal{G}) \\ & = \Hom_{\mathcal{O}_{X'_T}}(i_T^*h^*\mathcal{F}, (h')^*\mathcal{G}) \\ & = \Hom_{\mathcal{O}_{X'_T}}((h')^*i^*\mathcal{F}, (h')^*\mathcal{G}) \\ & = \mathit{Hom}(i^*\mathcal{F}, \mathcal{G}')(T) \end{align*}\] as desired. The middle equality follows from the adjointness of the functors \(i_{T, *}\) and \(i_T^*\).
Lemma
Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(K\) be a pseudo-coherent object of \(D(\mathcal{O}_B)\).
If for all \(g : T \to B\) in \((\Sch/B)\) the cohomology sheaf \(H^{-1}(Lg^*K)\) is zero, then the functor \[(\Sch/B)^{opp} \longrightarrow \textit{Sets},\quad (g : T \to B) \longmapsto H^0(T, H^0(Lg^*K))\] is an algebraic space affine and of finite presentation over \(B\).
If for all \(g : T \to B\) in \((\Sch/B)\) the cohomology sheaves \(H^i(Lg^*K)\) are zero for \(i < 0\), then \(K\) is perfect, \(K\) locally has tor amplitude in \([0, b]\), and the functor \[(\Sch/B)^{opp} \longrightarrow \textit{Sets},\quad (g : T \to B) \longmapsto H^0(T, Lg^*K)\] is an algebraic space affine and of finite presentation over \(B\).
Proof
Under the assumptions of (2) we have \(H^0(T, Lg^*K) = H^0(T, H^0(Lg^*K))\). Let us prove that the rule \(T \mapsto H^0(T, H^0(Lg^*K))\) satisfies the sheaf property for the fppf topology. To do this assume we have an fppf covering \(\{h_i : T_i \to T\}\) of a scheme \(g : T \to B\) over \(B\). Set \(g_i = g \circ h_i\). Note that since \(h_i\) is flat, we have \(Lh_i^* = h_i^*\) and \(h_i^*\) commutes with taking cohomology. Hence \[H^0(T_i, H^0(Lg_i^*K)) = H^0(T_i, H^0(h_i^*Lg^*K)) = H^0(T, h_i^*H^0(Lg^*K))\] Similarly for the pullback to \(T_i \times_T T_j\). Since \(Lg^*K\) is a pseudo-coherent complex on \(T\) (Cohomology on Sites, Lemma 08H4) the cohomology sheaf \(\mathcal{F} = H^0(Lg^*K)\) is quasi-coherent (Derived Categories of Spaces, Lemma 08JL). Hence by Descent on Spaces, Proposition 04W8 we see that \[H^0(T, \mathcal{F}) = \Ker( \prod H^0(T_i, h_i^*\mathcal{F}) \to \prod H^0(T_i \times_T T_j, (T_i \times_T T_j \to T)^*\mathcal{F}))\] In this way we see that the rules in (1) and (2) satisfy the sheaf property for fppf coverings. This means we may apply Bootstrap, Lemma 04U0 to see it suffices to prove the representability étale locally on \(B\). Moreover, we may check whether the end result is affine and of finite presentation étale locally on \(B\), see Morphisms of Spaces, Lemmas 03WG and 0410. Hence we may assume that \(B\) is an affine scheme.
Assume \(B = \Spec(A)\) is an affine scheme. By the results of Derived Categories of Spaces, Lemmas 08JL, 071Q, and 08HE we deduce that in the rest of the proof we may think of \(K\) as a perfect object of the derived category of complexes of modules on \(B\) in the Zariski topology. By Derived Categories of Schemes, Lemmas 08E5, 06Z0, and 08E7 we can find a pseudo-coherent complex \(M^\bullet\) of \(A\)-modules such that \(K\) is the corresponding object of \(D(\mathcal{O}_B)\). Our assumption on pullbacks implies that \(M^\bullet \otimes^\mathbf{L}_A \kappa(\mathfrak p)\) has vanishing \(H^{-1}\) for all primes \(\mathfrak p \subset A\). By More on Algebra, Lemma 068U we can write \[M^\bullet = \tau_{\geq 0}M^\bullet \oplus \tau_{\leq - 1}M^\bullet\] with \(\tau_{\geq 0}M^\bullet\) perfect with Tor amplitude in \([0, b]\) for some \(b \geq 0\) (here we also have used More on Algebra, Lemmas 066Y and 066N). Note that in case (2) we also see that \(\tau_{\leq - 1}M^\bullet = 0\) in \(D(A)\) whence \(M^\bullet\) and \(K\) are perfect with tor amplitude in \([0, b]\). For any \(B\)-scheme \(g : T \to B\) we have \[H^0(T, H^0(Lg^*K)) = H^0(T, H^0(Lg^*\tau_{\geq 0}K))\] (by the dual of Derived Categories, Lemma 05TC) hence we may replace \(K\) by \(\tau_{\geq 0}K\) and correspondingly \(M^\bullet\) by \(\tau_{\geq 0}M^\bullet\). In other words, we may assume \(M^\bullet\) has tor amplitude in \([0, b]\).
Assume \(M^\bullet\) has tor amplitude in \([0, b]\). We may assume \(M^\bullet\) is a bounded above complex of finite free \(A\)-modules (by our definition of pseudo-coherent complexes, see More on Algebra, Definition 064Q and the discussion following the definition). By More on Algebra, Lemma 0653 we see that \(M = \Coker(M^{- 1} \to M^0)\) is flat. By Algebra, Lemma 00NX we see that \(M\) is finite locally free. Hence \(M^\bullet\) is quasi-isomorphic to \[M \to M^1 \to M^2 \to \ldots \to M^d \to 0 \ldots\] Note that this is a K-flat complex (Cohomology, Lemma 06YD), hence derived pullback of \(K\) via a morphism \(T \to B\) is computed by the complex \[g^*\widetilde{M} \to g^*\widetilde{M^1} \to \ldots\] Thus it suffices to show that the functor \[(g : T \to B) \longmapsto \Ker( \Gamma(T,g^*\widetilde{M}) \to \Gamma(T, g^*(\widetilde{M^1}) )\] is representable by an affine scheme of finite presentation over \(B\).
We may still replace \(B\) by the members of an affine open covering in order to prove this last statement. Hence we may assume that \(M\) is finite free (recall that \(M^1\) is finite free to begin with). Write \(M = A^{\oplus n}\) and \(M^1 = A^{\oplus m}\). Let the map \(M \to M^1\) be given by the \(m \times n\) matrix \((a_{ij})\) with coefficients in \(A\). Then \(\widetilde{M} = \mathcal{O}_B^{\oplus n}\) and \(\widetilde{M^1} = \mathcal{O}_B^{\oplus m}\). Thus the functor above is equal to the functor \[(g : T \to B) \longmapsto \{(f_1, \ldots, f_n) \in \Gamma(T, \mathcal{O}_T) \mid \sum g^\sharp(a_{ij})f_i = 0,\ j = 1, \ldots, m\}\] Clearly this is representable by the affine scheme \[\Spec\left(A[x_1, \ldots, x_n]/(\sum a_{ij}x_i; j = 1, \ldots, m)\right)\] and the lemma has been proved.
The functor \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) is representable in a number of situations. All of our results will be based on the following basic case. The proof of this lemma as given below is in some sense the natural generalization to the proof of [EGA, III, Cor 7.7.8].
Lemma
In Situation 08JT assume that
\(B\) is a Noetherian algebraic space,
\(f\) is locally of finite type and quasi-separated,
\(\mathcal{F}\) is a finite type \(\mathcal{O}_X\)-module, and
\(\mathcal{G}\) is a finite type \(\mathcal{O}_X\)-module, flat over \(B\), with support proper over \(B\).
Then the functor \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) is an algebraic space affine and of finite presentation over \(B\).
Proof
We may replace \(X\) by a quasi-compact open neighbourhood of the support of \(\mathcal{G}\), hence we may assume \(X\) is Noetherian. In this case \(X\) and \(f\) are quasi-compact and quasi-separated. Choose an approximation \(P \to \mathcal{F}\) by a perfect complex \(P\) of the triple \((X, \mathcal{F}, -1)\), see Derived Categories of Spaces, Definition 08HI and Theorem 08HP). Then the induced map \[\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) \longrightarrow \Hom_{D(\mathcal{O}_X)}(P, \mathcal{G})\] is an isomorphism because \(P \to \mathcal{F}\) induces an isomorphism \(H^0(P) \to \mathcal{F}\) and \(H^i(P) = 0\) for \(i > 0\). Moreover, for any morphism \(g : T \to B\) denote \(h : X_T = T \times_B X \to X\) the projection and set \(P_T = Lh^*P\). Then it is equally true that \[\Hom_{\mathcal{O}_{X_T}}(\mathcal{F}_T, \mathcal{G}_T) \longrightarrow \Hom_{D(\mathcal{O}_{X_T})}(P_T, \mathcal{G}_T)\] is an isomorphism, as \(P_T = Lh^*P \to Lh^*\mathcal{F} \to \mathcal{F}_T\) induces an isomorphism \(H^0(P_T) \to \mathcal{F}_T\) (because \(h^*\) is right exact and \(H^i(P) = 0\) for \(i > 0\)). Thus it suffices to prove the result for the functor \[T \longmapsto \Hom_{D(\mathcal{O}_{X_T})}(P_T, \mathcal{G}_T).\] By the Leray spectral sequence (see Cohomology on Sites, Remark 08J6) we have \[\Hom_{D(\mathcal{O}_{X_T})}(P_T, \mathcal{G}_T) = H^0(X_T, R\SheafHom(P_T, \mathcal{G}_T)) = H^0(T, Rf_{T, *}R\SheafHom(P_T, \mathcal{G}_T))\] where \(f_T : X_T \to T\) is the base change of \(f\). By Derived Categories of Spaces, Lemma 08JQ we have \[Rf_{T, *}R\SheafHom(P_T, \mathcal{G}_T) = Lg^*Rf_*R\SheafHom(P, \mathcal{G}).\] By Derived Categories of Spaces, Lemma 0DKK the object \(K = Rf_*R\SheafHom(P, \mathcal{G})\) of \(D(\mathcal{O}_B)\) is perfect. This means we can apply Lemma 08JX as long as we can prove that the cohomology sheaf \(H^i(Lg^*K)\) is \(0\) for all \(i < 0\) and \(g : T \to B\) as above. This is clear from the last displayed formula as the cohomology sheaves of \(Rf_{T, *}R\SheafHom(P_T, \mathcal{G}_T)\) are zero in negative degrees due to the fact that \(R\SheafHom(P_T, \mathcal{G}_T)\) has vanishing cohomology sheaves in negative degrees as \(P_T\) is perfect with vanishing cohomology sheaves in positive degrees.
Here is a cheap consequence of Lemma 08JY.
Proposition
In Situation 08JT assume that
\(f\) is of finite presentation, and
\(\mathcal{G}\) is a finitely presented \(\mathcal{O}_X\)-module, flat over \(B\), with support proper over \(B\).
Then the functor \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) is an algebraic space affine over \(B\). If \(\mathcal{F}\) is of finite presentation, then \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) is of finite presentation over \(B\).
Proof
By Lemma 08JV the functor \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) satisfies the sheaf property for fppf coverings. This mean we may1 apply Bootstrap, Lemma 04SK to check the representability étale locally on \(B\). Moreover, we may check whether the end result is affine or of finite presentation étale locally on \(B\), see Morphisms of Spaces, Lemmas 03WG and 0410. Hence we may assume that \(B\) is an affine scheme.
Assume \(B\) is an affine scheme. As \(f\) is of finite presentation, it follows \(X\) is quasi-compact and quasi-separated. Thus we can write \(\mathcal{F} = \colim \mathcal{F}_i\) as a filtered colimit of \(\mathcal{O}_X\)-modules of finite presentation (Limits of Spaces, Lemma 07V9). It is clear that \[\mathit{Hom}(\mathcal{F}, \mathcal{G}) = \lim \mathit{Hom}(\mathcal{F}_i, \mathcal{G})\] Hence if we can show that each \(\mathit{Hom}(\mathcal{F}_i, \mathcal{G})\) is representable by an affine scheme, then we see that the same thing holds for \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\). Use the material in Limits, Section 01YV and Limits of Spaces, Section 07SE. Thus we may assume that \(\mathcal{F}\) is of finite presentation.
Say \(B = \Spec(R)\). Write \(R = \colim R_i\) with each \(R_i\) a finite type \(\mathbf{Z}\)-algebra. Set \(B_i = \Spec(R_i)\). By the results of Limits of Spaces, Lemmas 07SK and 07V7 we can find an \(i\), a morphism of algebraic spaces \(X_i \to B_i\), and finitely presented \(\mathcal{O}_{X_i}\)-modules \(\mathcal{F}_i\) and \(\mathcal{G}_i\) such that the base change of \((X_i, \mathcal{F}_i, \mathcal{G}_i)\) to \(B\) recovers \((X, \mathcal{F}, \mathcal{G})\). By Limits of Spaces, Lemma 08K0 we may, after increasing \(i\), assume that \(\mathcal{G}_i\) is flat over \(B_i\). By Limits of Spaces, Lemma 08K2 we may similarly assume the scheme theoretic support of \(\mathcal{G}_i\) is proper over \(B_i\). At this point we can apply Lemma 08JY to see that \(H_i = \mathit{Hom}(\mathcal{F}_i, \mathcal{G}_i)\) is an algebraic space affine of finite presentation over \(B_i\). Pulling back to \(B\) (using Remark 08JW) we see that \(H_i \times_{B_i} B = \mathit{Hom}(\mathcal{F}, \mathcal{G})\) and we win.
The Isom functor
In Situation 08JT we can consider the subfunctor \[\mathit{Isom}(\mathcal{F}, \mathcal{G}) \subset \mathit{Hom}(\mathcal{F}, \mathcal{G})\] whose value on a scheme \(T\) over \(B\) is the set of invertible \(\mathcal{O}_{X_T}\)-homomorphisms \(u : \mathcal{F}_T \to \mathcal{G}_T\).
We sometimes think of \(\mathit{Isom}(\mathcal{F}, \mathcal{G})\) as a functor \((\Sch/S)^{opp} \to \textit{Sets}\) endowed with a morphism \(\mathit{Isom}(\mathcal{F}, \mathcal{G}) \to B\). Namely, if \(T\) is a scheme over \(S\), then an element of \(\mathit{Isom}(\mathcal{F}, \mathcal{G})(T)\) consists of a pair \((h, u)\), where \(h\) is a morphism \(h : T \to B\) and \(u : \mathcal{F}_T \to \mathcal{G}_T\) is an \(\mathcal{O}_{X_T}\)-module isomorphism where \(X_T = T \times_{h, B} X\) and \(\mathcal{F}_T\) and \(\mathcal{G}_T\) are the pullbacks to \(X_T\). In particular, when we say that \(\mathit{Isom}(\mathcal{F}, \mathcal{G})\) is an algebraic space, we mean that the corresponding functor \((\Sch/S)^{opp} \to \textit{Sets}\) is an algebraic space.
Lemma
In Situation 08JT the functor \(\mathit{Isom}(\mathcal{F}, \mathcal{G})\) satisfies the sheaf property for the fpqc topology.
Proof
We have already seen that \(\mathit{Hom}(\mathcal{F}, \mathcal{G})\) satisfies the sheaf property. Hence it remains to show the following: Given an fpqc covering \(\{T_i \to T\}_{i \in I}\) of schemes over \(B\) and an \(\mathcal{O}_{X_T}\)-linear map \(u : \mathcal{F}_T \to \mathcal{G}_T\) such that \(u_{T_i}\) is an isomorphism for all \(i\), then \(u\) is an isomorphism. Since \(\{X_i \to X_T\}_{i \in I}\) is an fpqc covering of \(X_T\), see Topologies on Spaces, Lemma 03MR, this follows from Descent on Spaces, Proposition 04W8.
Sanity check: \(\mathit{Isom}\) sheaf plays the same role among algebraic spaces over \(S\).
Lemma
In Situation 08JT. Let \(T\) be an algebraic space over \(S\). We have \[\Mor_{\Sh((\Sch/S)_{fppf})}(T, \mathit{Isom}(\mathcal{F}, \mathcal{G})) = \{(h, u) \mid h : T \to B, u : \mathcal{F}_T \to \mathcal{G}_T\text{ isomorphism}\}\] where \(\mathcal{F}_T, \mathcal{G}_T\) denote the pullbacks of \(\mathcal{F}\) and \(\mathcal{G}\) to the algebraic space \(X \times_{B, h} T\).
Proof
Observe that the left and right hand side of the equality are subsets of the left and right hand side of the equality in Lemma 0D3S. We omit the verification that these subsets correspond under the identification given in the proof of that lemma.
Proposition
In Situation 08JT assume that
\(f\) is of finite presentation, and
\(\mathcal{F}\) and \(\mathcal{G}\) are finitely presented \(\mathcal{O}_X\)-modules, flat over \(B\), with support proper over \(B\).
Then the functor \(\mathit{Isom}(\mathcal{F}, \mathcal{G})\) is an algebraic space affine of finite presentation over \(B\).
Proof
We will use the abbreviations \(H = \mathit{Hom}(\mathcal{F}, \mathcal{G})\), \(I = \mathit{Hom}(\mathcal{F}, \mathcal{F})\), \(H' = \mathit{Hom}(\mathcal{G}, \mathcal{F})\), and \(I' = \mathit{Hom}(\mathcal{G}, \mathcal{G})\). By Proposition 08K6 the functors \(H\), \(I\), \(H'\), \(I'\) are algebraic spaces and the morphisms \(H \to B\), \(I \to B\), \(H' \to B\), and \(I' \to B\) are affine and of finite presentation. The composition of maps gives a morphism \[c : H' \times_B H \longrightarrow I \times_B I',\quad (u', u) \longmapsto (u \circ u', u' \circ u)\] of algebraic spaces over \(B\). Since \(I \times_B I' \to B\) is separated, the section \(\sigma : B \to I \times_B I'\) corresponding to \((\text{id}_\mathcal{F}, \text{id}_\mathcal{G})\) is a closed immersion (Morphisms of Spaces, Lemma 03KP). Moreover, \(\sigma\) is of finite presentation (Morphisms of Spaces, Lemma 05WT). Hence \[\mathit{Isom}(\mathcal{F}, \mathcal{G}) = (H' \times_B H) \times_{c, I \times_B I', \sigma} B\] is an algebraic space affine of finite presentation over \(B\) as well. Some details omitted.
The stack of coherent sheaves
In this section we prove that the stack of coherent sheaves on \(X/B\) is algebraic under suitable hypotheses. This is a special case of [lieblich_remarks, Theorem 2.1.1] which treats the case of the stack of coherent sheaves on an Artin stack over a base.
Situation
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Assume that \(f\) is of finite presentation. We denote \(\Cohstack_{X/B}\) the category whose objects are triples \((T, g, \mathcal{F})\) where
\(T\) is a scheme over \(S\),
\(g : T \to B\) is a morphism over \(S\), and setting \(X_T = T \times_{g, B} X\)
\(\mathcal{F}\) is a quasi-coherent \(\mathcal{O}_{X_T}\)-module of finite presentation, flat over \(T\), with support proper over \(T\).
A morphism \((T, g, \mathcal{F}) \to (T', g', \mathcal{F}')\) is given by a pair \((h, \varphi)\) where
\(h : T \to T'\) is a morphism of schemes over \(B\) (i.e., \(g' \circ h = g\)), and
\(\varphi : (h')^*\mathcal{F}' \to \mathcal{F}\) is an isomorphism of \(\mathcal{O}_{X_T}\)-modules where \(h' : X_T \to X_{T'}\) is the base change of \(h\).
Thus \(\Cohstack_{X/B}\) is a category and the rule \[p : \Cohstack_{X/B} \longrightarrow (\Sch/S)_{fppf}, \quad (T, g, \mathcal{F}) \longmapsto T\] is a functor. For a scheme \(T\) over \(S\) we denote \(\Cohstack_{X/B, T}\) the fibre category of \(p\) over \(T\). These fibre categories are groupoids.
Lemma
In Situation 08KB the functor \(p : \Cohstack_{X/B} \longrightarrow (\Sch/S)_{fppf}\) is fibred in groupoids.
Proof
We show that \(p\) is fibred in groupoids by checking conditions (1) and (2) of Categories, Definition 003T. Given an object \((T', g', \mathcal{F}')\) of \(\Cohstack_{X/B}\) and a morphism \(h : T \to T'\) of schemes over \(S\) we can set \(g = h \circ g'\) and \(\mathcal{F} = (h')^*\mathcal{F}'\) where \(h' : X_T \to X_{T'}\) is the base change of \(h\). Then it is clear that we obtain a morphism \((T, g, \mathcal{F}) \to (T', g', \mathcal{F}')\) of \(\Cohstack_{X/B}\) lying over \(h\). This proves (1). For (2) suppose we are given morphisms \[(h_1, \varphi_1) : (T_1, g_1, \mathcal{F}_1) \to (T, g, \mathcal{F}) \quad\text{and}\quad (h_2, \varphi_2) : (T_2, g_2, \mathcal{F}_2) \to (T, g, \mathcal{F})\] of \(\Cohstack_{X/B}\) and a morphism \(h : T_1 \to T_2\) such that \(h_2 \circ h = h_1\). Then we can let \(\varphi\) be the composition \[(h')^*\mathcal{F}_2 \xrightarrow{(h')^*\varphi_2^{-1}} (h')^*(h_2)^*\mathcal{F} = (h_1)^*\mathcal{F} \xrightarrow{\varphi_1} \mathcal{F}_1\] to obtain the morphism \((h, \varphi) : (T_1, g_1, \mathcal{F}_1) \to (T_2, g_2, \mathcal{F}_2)\) that witnesses the truth of condition (2).
Lemma
In Situation 08KB. Denote \(\mathcal{X} = \Cohstack_{X/B}\). Then \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces.
Proof
Consider two objects \(x = (T, g, \mathcal{F})\) and \(y = (T, h, \mathcal{G})\) of \(\mathcal{X}\) over a scheme \(T\). We have to show that \(\mathit{Isom}_\mathcal{X}(x, y)\) is an algebraic space over \(T\), see Algebraic Stacks, Lemma 045G. If for \(a : T' \to T\) the restrictions \(x|_{T'}\) and \(y|_{T'}\) are isomorphic in the fibre category \(\mathcal{X}_{T'}\), then \(g \circ a = h \circ a\). Hence there is a transformation of presheaves \[\mathit{Isom}_\mathcal{X}(x, y) \longrightarrow \text{Equalizer}(g, h)\] Since the diagonal of \(B\) is representable (by schemes) this equalizer is a scheme. Thus we may replace \(T\) by this equalizer and the sheaves \(\mathcal{F}\) and \(\mathcal{G}\) by their pullbacks. Thus we may assume \(g = h\). In this case we have \(\mathit{Isom}_\mathcal{X}(x, y) = \mathit{Isom}(\mathcal{F}, \mathcal{G})\) and the result follows from Proposition 08K9.
Lemma
In Situation 08KB the functor \(p : \Cohstack_{X/B} \longrightarrow (\Sch/S)_{fppf}\) is a stack in groupoids.
Proof
To prove that \(\Cohstack_{X/B}\) is a stack in groupoids, we have to show that the presheaves \(\mathit{Isom}\) are sheaves and that descent data are effective. The statement on \(\mathit{Isom}\) follows from Lemma 08W6, see Algebraic Stacks, Lemma 045G. Let us prove the statement on descent data. Suppose that \(\{a_i : T_i \to T\}\) is an fppf covering of schemes over \(S\). Let \((\xi_i, \varphi_{ij})\) be a descent datum for \(\{T_i \to T\}\) with values in \(\Cohstack_{X/B}\). For each \(i\) we can write \(\xi_i = (T_i, g_i, \mathcal{F}_i)\). Denote \(\text{pr}_0 : T_i \times_T T_j \to T_i\) and \(\text{pr}_1 : T_i \times_T T_j \to T_j\) the projections. The condition that \(\xi_i|_{T_i \times_T T_j} = \xi_j|_{T_i \times_T T_j}\) implies in particular that \(g_i \circ \text{pr}_0 = g_j \circ \text{pr}_1\). Thus there exists a unique morphism \(g : T \to B\) such that \(g_i = g \circ a_i\), see Descent on Spaces, Lemma 04P2. Denote \(X_T = T \times_{g, B} X\). Set \(X_i = X_{T_i} = T_i \times_{g_i, B} X = T_i \times_{a_i, T} X_T\) and \[X_{ij} = X_{T_i} \times_{X_T} X_{T_j} = X_i \times_{X_T} X_j\] with projections \(\text{pr}_i\) and \(\text{pr}_j\) to \(X_i\) and \(X_j\). Observe that the pullback of \((T_i, g_i, \mathcal{F}_i)\) by \(\text{pr}_0 : T_i \times_T T_j \to T_i\) is given by \((T_i \times_T T_j, g_i \circ \text{pr}_0, \text{pr}_i^*\mathcal{F}_i)\). Hence a descent datum for \(\{T_i \to T\}\) in \(\Cohstack_{X/B}\) is given by the objects \((T_i, g \circ a_i, \mathcal{F}_i)\) and for each pair \(i, j\) an isomorphism of \(\mathcal{O}_{X_{ij}}\)-modules \[\varphi_{ij} : \text{pr}_i^*\mathcal{F}_i \longrightarrow \text{pr}_j^*\mathcal{F}_j\] satisfying the cocycle condition over (the pullback of \(X\) to) \(T_i \times_T T_j \times_T T_k\). Ok, and now we simply use that \(\{X_i \to X_T\}\) is an fppf covering so that we can view \((\mathcal{F}_i, \varphi_{ij})\) as a descent datum for this covering. By Descent on Spaces, Proposition 04W8 this descent datum is effective and we obtain a quasi-coherent sheaf \(\mathcal{F}\) over \(X_T\) restricting to \(\mathcal{F}_i\) on \(X_i\). By Morphisms of Spaces, Lemma 05VY we see that \(\mathcal{F}\) is flat over \(T\) and Descent on Spaces, Lemma 060V guarantees that \(\mathcal{F}\) is of finite presentation as an \(\mathcal{O}_{X_T}\)-module. Finally, by Descent on Spaces, Lemma 0422 we see that the scheme theoretic support of \(\mathcal{F}\) is proper over \(T\) as we’ve assumed the scheme theoretic support of \(\mathcal{F}_i\) is proper over \(T_i\) (note that taking scheme theoretic support commutes with flat base change by Morphisms of Spaces, Lemma 089C). In this way we obtain our desired object over \(T\).
Remark
In Situation 08KB the rule \((T, g, \mathcal{F}) \mapsto (T, g)\) defines a \(1\)-morphism \[\Cohstack_{X/B} \longrightarrow \mathcal{S}_B\] of stacks in groupoids (see Lemma 08KC, Algebraic Stacks, Section 04SU, and Examples of Stacks, Section 0305). Let \(B' \to B\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{S}_{B'} \to \mathcal{S}_B\) be the associated \(1\)-morphism of stacks fibred in sets. Set \(X' = X \times_B B'\). We obtain a stack in groupoids \(\Cohstack_{X'/B'} \to (\Sch/S)_{fppf}\) associated to the base change \(f' : X' \to B'\). In this situation the diagram \[\vcenter{ \xymatrix{ \Cohstack_{X'/B'} \ar[r] \ar[d] & \Cohstack_{X/B} \ar[d] \\ \mathcal{S}_{B'} \ar[r] & \mathcal{S}_B } } \quad \begin{matrix} \text{or in} \\ \text{another} \\ \text{notation} \end{matrix} \quad \vcenter{ \xymatrix{ \Cohstack_{X'/B'} \ar[r] \ar[d] & \Cohstack_{X/B} \ar[d] \\ \Sch/B' \ar[r] & \Sch/B } }\] is \(2\)-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space.
Lemma
In Situation 08KB assume that \(B \to S\) is locally of finite presentation. Then \(p : \Cohstack_{X/B} \to (\Sch/S)_{fppf}\) is limit preserving (Artin’s Axioms, Definition 07XL).
Proof
Write \(B(T)\) for the discrete category whose objects are the \(S\)-morphisms \(T \to B\). Let \(T = \lim T_i\) be a filtered limit of affine schemes over \(S\). Assigning to an object \((T, h, \mathcal{F})\) of \(\Cohstack_{X/B, T}\) the object \(h\) of \(B(T)\) gives us a commutative diagram of fibre categories \[\xymatrix{ \colim \Cohstack_{X/B, T_i} \ar[r] \ar[d] & \Cohstack_{X/B, T} \ar[d] \\ \colim B(T_i) \ar[r] & B(T) }\] We have to show the top horizontal arrow is an equivalence. Since we have assumed that \(B\) is locally of finite presentation over \(S\) we see from Limits of Spaces, Remark 05N0 that the bottom horizontal arrow is an equivalence. This means that we may assume \(T = \lim T_i\) be a filtered limit of affine schemes over \(B\). Denote \(g_i : T_i \to B\) and \(g : T \to B\) the corresponding morphisms. Set \(X_i = T_i \times_{g_i, B} X\) and \(X_T = T \times_{g, B} X\). Observe that \(X_T = \colim X_i\) and that the algebraic spaces \(X_i\) and \(X_T\) are quasi-separated and quasi-compact (as they are of finite presentation over the affines \(T_i\) and \(T\)). By Limits of Spaces, Lemma 07V7 we see that \[\colim \textit{FP}(X_i) = \textit{FP}(X_T).\] where \(\textit{FP}(W)\) is short hand for the category of finitely presented \(\mathcal{O}_W\)-modules. The results of Limits of Spaces, Lemmas 08K0 and 08K2 tell us the same thing is true if we replace \(\textit{FP}(X_i)\) and \(\textit{FP}(X_T)\) by the full subcategory of objects flat over \(T_i\) and \(T\) with scheme theoretic support proper over \(T_i\) and \(T\). This proves the lemma.
Lemma
In Situation 08KB. Let \[\xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' }\] be a pushout in the category of schemes over \(S\) where \(Z \to Z'\) is a thickening and \(Z \to Y\) is affine, see More on Morphisms, Lemma 07RT. Then the functor on fibre categories \[\Cohstack_{X/B, Y'} \longrightarrow \Cohstack_{X/B, Y} \times_{\Cohstack_{X/B, Z}} \Cohstack_{X/B, Z'}\] is an equivalence.
Proof
Observe that the corresponding map \[B(Y') \longrightarrow B(Y) \times_{B(Z)} B(Z')\] is a bijection, see Pushouts of Spaces, Lemma 07SY. Thus using the commutative diagram \[\xymatrix{ \Cohstack_{X/B, Y'} \ar[r] \ar[d] & \Cohstack_{X/B, Y} \times_{\Cohstack_{X/B, Z}} \Cohstack_{X/B, Z'} \ar[d] \\ B(Y') \ar[r] & B(Y) \times_{B(Z)} B(Z') }\] we see that we may assume that \(Y'\) is a scheme over \(B'\). By Remark 08LP we may replace \(B\) by \(Y'\) and \(X\) by \(X \times_B Y'\). Thus we may assume \(B = Y'\). In this case the statement follows from Pushouts of Spaces, Lemma 08KV.
Lemma
Let \[\xymatrix{ X \ar[d] \ar[r]_i & X' \ar[d] \\ T \ar[r] & T' }\] be a cartesian square of algebraic spaces where \(T \to T'\) is a first order thickening. Let \(\mathcal{F}'\) be an \(\mathcal{O}_{X'}\)-module flat over \(T'\). Set \(\mathcal{F} = i^*\mathcal{F}'\). The following are equivalent
\(\mathcal{F}'\) is a quasi-coherent \(\mathcal{O}_{X'}\)-module of finite presentation,
\(\mathcal{F}'\) is an \(\mathcal{O}_{X'}\)-module of finite presentation,
\(\mathcal{F}\) is a quasi-coherent \(\mathcal{O}_X\)-module of finite presentation,
\(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite presentation,
Proof
Recall that a finitely presented module is quasi-coherent hence the equivalence of (1) and (2) and (3) and (4). The equivalence of (2) and (4) is a special case of Deformation Theory, Lemma 08VU.
Lemma
In Situation 08KB assume that \(S\) is a locally Noetherian scheme and \(B \to S\) is locally of finite presentation. Let \(k\) be a finite type field over \(S\) and let \(x_0 = (\Spec(k), g_0, \mathcal{G}_0)\) be an object of \(\mathcal{X} = \Cohstack_{X/B}\) over \(k\). Then the spaces \(T\mathcal{F}_{\mathcal{X}, k, x_0}\) and \(\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})\) (Artin’s Axioms, Section 07WY) are finite dimensional.
Proof
Observe that by Lemma 08LQ our stack in groupoids \(\mathcal{X}\) satisfies property (RS*) defined in Artin’s Axioms, Section 07Y6. In particular \(\mathcal{X}\) satisfies (RS). Hence all associated predeformation categories are deformation categories (Artin’s Axioms, Lemma 07WU) and the statement makes sense.
In this paragraph we show that we can reduce to the case \(B = \Spec(k)\). Set \(X_0 = \Spec(k) \times_{g_0, B} X\) and denote \(\mathcal{X}_0 = \Cohstack_{X_0/k}\). In Remark 08LP we have seen that \(\mathcal{X}_0\) is the \(2\)-fibre product of \(\mathcal{X}\) with \(\Spec(k)\) over \(B\) as categories fibred in groupoids over \((\Sch/S)_{fppf}\). Thus by Artin’s Axioms, Lemma 07X2 we reduce to proving that \(B\), \(\Spec(k)\), and \(\mathcal{X}_0\) have finite dimensional tangent spaces and infinitesimal automorphism spaces. The tangent space of \(B\) and \(\Spec(k)\) are finite dimensional by Artin’s Axioms, Lemma 07X1 and of course these have vanishing \(\text{Inf}\). Thus it suffices to deal with \(\mathcal{X}_0\).
Let \(k[\epsilon]\) be the dual numbers over \(k\). Let \(\Spec(k[\epsilon]) \to B\) be the composition of \(g_0 : \Spec(k) \to B\) and the morphism \(\Spec(k[\epsilon]) \to \Spec(k)\) coming from the inclusion \(k \to k[\epsilon]\). Set \(X_0 = \Spec(k) \times_B X\) and \(X_\epsilon = \Spec(k[\epsilon]) \times_B X\). Observe that \(X_\epsilon\) is a first order thickening of \(X_0\) flat over the first order thickening \(\Spec(k) \to \Spec(k[\epsilon])\). Unwinding the definitions and using Lemma 08W7 we see that \(T\mathcal{F}_{\mathcal{X}_0, k, x_0}\) is the set of lifts of \(\mathcal{G}_0\) to a flat module on \(X_\epsilon\). By Deformation Theory, Lemma 08VW we conclude that \[T\mathcal{F}_{\mathcal{X}_0, k, x_0} = \Ext^1_{\mathcal{O}_{X_0}}(\mathcal{G}_0, \mathcal{G}_0)\] Here we have used the identification \(\epsilon k[\epsilon] \cong k\) of \(k[\epsilon]\)-modules. Using Deformation Theory, Lemma 08VW once more we see that \[\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \Ext^0_{\mathcal{O}_{X_0}}(\mathcal{G}_0, \mathcal{G}_0)\] These spaces are finite dimensional over \(k\) as \(\mathcal{G}_0\) has support proper over \(\Spec(k)\). Namely, \(X_0\) is of finite presentation over \(\Spec(k)\), hence Noetherian. Since \(\mathcal{G}_0\) is of finite presentation it is a coherent \(\mathcal{O}_{X_0}\)-module. Thus we may apply Derived Categories of Spaces, Lemma 0D0T to conclude the desired finiteness.
Lemma
In Situation 08KB assume that \(S\) is a locally Noetherian scheme and that \(f : X \to B\) is separated. Let \(\mathcal{X} = \Cohstack_{X/B}\). Then the functor Artin’s Axioms, Equation (07X6) is an equivalence.
Proof
Let \(A\) be an \(S\)-algebra which is a complete local Noetherian ring with maximal ideal \(\mathfrak m\) whose residue field \(k\) is of finite type over \(S\). We have to show that the category of objects over \(A\) is equivalent to the category of formal objects over \(A\). Since we know this holds for the category \(\mathcal{S}_B\) fibred in sets associated to \(B\) by Artin’s Axioms, Lemma 07X8, it suffices to prove this for those objects lying over a given morphism \(\Spec(A) \to B\).
Set \(X_A = \Spec(A) \times_B X\) and \(X_n = \Spec(A/\mathfrak m^n) \times_B X\). By Grothendieck’s existence theorem (More on Morphisms of Spaces, Theorem 08BE) we see that the category of coherent modules \(\mathcal{F}\) on \(X_A\) with support proper over \(\Spec(A)\) is equivalent to the category of systems \((\mathcal{F}_n)\) of coherent modules \(\mathcal{F}_n\) on \(X_n\) with support proper over \(\Spec(A/\mathfrak m^n)\). The equivalence sends \(\mathcal{F}\) to the system \((\mathcal{F} \otimes_A A/\mathfrak m^n)\). See discussion in More on Morphisms of Spaces, Remark 08BF. To finish the proof of the lemma, it suffices to show that \(\mathcal{F}\) is flat over \(A\) if and only if all \(\mathcal{F} \otimes_A A/\mathfrak m^n\) are flat over \(A/\mathfrak m^n\). This follows from More on Morphisms of Spaces, Lemma 08VP.
Lemma
In Situation 08KB assume that \(S\) is a locally Noetherian scheme, \(S = B\), and \(f : X \to B\) is flat. Let \(\mathcal{X} = \Cohstack_{X/B}\). Then we have openness of versality for \(\mathcal{X}\) (see Artin’s Axioms, Definition 07XQ).
Proof
This proof is based on the criterion of Artin’s Axioms, Lemma 07YZ. Let \(U \to S\) be of finite type morphism of schemes, \(x\) an object of \(\mathcal{X}\) over \(U\) and \(u_0 \in U\) a finite type point such that \(x\) is versal at \(u_0\). After shrinking \(U\) we may assume that \(u_0\) is a closed point (Morphisms, Lemma 01TA) and \(U = \Spec(A)\) with \(U \to S\) mapping into an affine open \(\Spec(\Lambda)\) of \(S\). Let \(\mathcal{F}\) be the coherent module on \(X_A = \Spec(A) \times_S X\) flat over \(A\) corresponding to the given object \(x\).
According to Deformation Theory, Lemma 08VW we have an isomorphism of functors \[T_x(M) = \Ext^1_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)\] and given any surjection \(A' \to A\) of \(\Lambda\)-algebras with square zero kernel \(I\) we have an obstruction class \[\xi_{A'} \in \Ext^2_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A I)\] This uses that for any \(A' \to A\) as above the base change \(X_{A'} = \Spec(A') \times_B X\) is flat over \(A'\). Moreover, the construction of the obstruction class is functorial in the surjection \(A' \to A\) (for fixed \(A\)) by Deformation Theory, Lemma 0CYE. Apply Derived Categories of Spaces, Lemma 08JR to the computation of the Ext groups \(\Ext^i_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)\) for \(i \leq m\) with \(m = 2\). We find a perfect object \(K \in D(A)\) and functorial isomorphisms \[H^i(K \otimes_A^\mathbf{L} M) \longrightarrow \Ext^i_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)\] for \(i \leq m\) compatible with boundary maps. This object \(K\), together with the displayed identifications above gives us a datum as in Artin’s Axioms, Situation 07YX. Finally, condition (iv) of Artin’s Axioms, Lemma 07YY holds by Deformation Theory, Lemma 08VY. Thus Artin’s Axioms, Lemma 07YZ does indeed apply and the lemma is proved.
Proof
This proof is based on Artin’s Axioms, Lemma 0CYF. Conditions (1), (2), and (3) of that lemma correspond to Lemmas 08W6, 08LQ, and 08KD.
We have constructed an obstruction theory in the chapter on deformation theory. Namely, given an \(S\)-algebra \(A\) and an object \(x\) of \(\Cohstack_{X/B}\) over \(\Spec(A)\) given by \(\mathcal{F}\) on \(X_A\) we set \(\mathcal{O}_x(M) = \Ext^2_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)\) and if \(A' \to A\) is a surjection with kernel \(I\), then as obstruction element we take the element \[o_x(A') = o(\mathcal{F}, \mathcal{F} \otimes_A I, 1) \in \mathcal{O}_x(I) = \Ext^2_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A I)\] of Deformation Theory, Lemma 08VW. All properties of an obstruction theory as defined in Artin’s Axioms, Definition 07YG follow from this lemma except for functoriality of obstruction classes as formulated in condition (ii) of the definition. But as stated in the footnote to assumption (4) of Artin’s Axioms, Lemma 0CYF it suffices to check functoriality of obstruction classes for a fixed \(A\) which follows from Deformation Theory, Lemma 0CYE. Deformation Theory, Lemma 08VW also tells us that \(T_x(M) = \Ext^1_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)\) for any \(A\)-module \(M\).
To finish the proof it suffices to show that \(T_x(\prod M_n) = \prod T_x(M_n)\) and \(\mathcal{O}_x(\prod M_n) = \prod \mathcal{O}_x(M)\). Apply Derived Categories of Spaces, Lemma 08JR to the computation of the Ext groups \(\Ext^i_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)\) for \(i \leq m\) with \(m = 2\). We find a perfect object \(K \in D(A)\) and functorial isomorphisms \[H^i(K \otimes_A^\mathbf{L} M) \longrightarrow \Ext^i_{X_A}(\mathcal{F}, \mathcal{F} \otimes_A M)\] for \(i = 1, 2\). A straightforward argument shows that \[H^i(K \otimes_A^\mathbf{L} \prod M_n) = \prod H^i(K \otimes_A^\mathbf{L} M_n)\] whenever \(K\) is a pseudo-coherent object of \(D(A)\). In fact, this property (for all \(i\)) characterizes pseudo-coherent complexes, see More on Algebra, Lemma 0CYB.
Theorem
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Assume that \(f\) is of finite presentation, separated, and flat2. Then \(\Cohstack_{X/B}\) is an algebraic stack over \(S\).
Proof
Set \(\mathcal{X} = \Cohstack_{X/B}\). We have seen that \(\mathcal{X}\) is a stack in groupoids over \((\Sch/S)_{fppf}\) with diagonal representable by algebraic spaces (Lemmas 08KC and 08W6). Hence it suffices to find a scheme \(W\) and a surjective and smooth morphism \(W \to \mathcal{X}\).
Let \(B'\) be a scheme and let \(B' \to B\) be a surjective étale morphism. Set \(X' = B' \times_B X\) and denote \(f' : X' \to B'\) the projection. Then \(\mathcal{X}' = \Cohstack_{X'/B'}\) is equal to the \(2\)-fibre product of \(\mathcal{X}\) with the category fibred in sets associated to \(B'\) over the category fibred in sets associated to \(B\) (Remark 08LP). By the material in Algebraic Stacks, Section 03YJ the morphism \(\mathcal{X}' \to \mathcal{X}\) is surjective and étale. Hence it suffices to prove the result for \(\mathcal{X}'\). In other words, we may assume \(B\) is a scheme.
Assume \(B\) is a scheme. In this case we may replace \(S\) by \(B\), see Algebraic Stacks, Section 04X4. Thus we may assume \(S = B\).
Assume \(S = B\). Choose an affine open covering \(S = \bigcup U_i\). Denote \(\mathcal{X}_i\) the restriction of \(\mathcal{X}\) to \((\Sch/U_i)_{fppf}\). If we can find schemes \(W_i\) over \(U_i\) and surjective smooth morphisms \(W_i \to \mathcal{X}_i\), then we set \(W = \coprod W_i\) and we obtain a surjective smooth morphism \(W \to \mathcal{X}\). Thus we may assume \(S = B\) is affine.
Assume \(S = B\) is affine, say \(S = \Spec(\Lambda)\). Write \(\Lambda = \colim \Lambda_i\) as a filtered colimit with each \(\Lambda_i\) of finite type over \(\mathbf{Z}\). For some \(i\) we can find a morphism of algebraic spaces \(X_i \to \Spec(\Lambda_i)\) which is of finite presentation, separated, and flat and whose base change to \(\Lambda\) is \(X\). See Limits of Spaces, Lemmas 07SK, 0851, and 08K0. If we show that \(\Cohstack_{X_i/\Spec(\Lambda_i)}\) is an algebraic stack, then it follows by base change (Remark 08LP and Algebraic Stacks, Section 04X4) that \(\mathcal{X}\) is an algebraic stack. Thus we may assume that \(\Lambda\) is a finite type \(\mathbf{Z}\)-algebra.
Assume \(S = B = \Spec(\Lambda)\) is affine of finite type over \(\mathbf{Z}\). In this case we will verify conditions (1), (2), (3), (4), and (5) of Artin’s Axioms, Lemma 07Y4 to conclude that \(\mathcal{X}\) is an algebraic stack. Note that \(\Lambda\) is a G-ring, see More on Algebra, Proposition 07PX. Hence all local rings of \(S\) are G-rings. Thus (5) holds. By Lemma 08WA we have that \(\mathcal{X}\) satisfies openness of versality, hence (4) holds. To check (2) we have to verify axioms [-1], [0], [1], [2], and [3] of Artin’s Axioms, Section 07XJ. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas 08KC, 08KD, 08LQ, 08W8. Condition (3) follows from Lemma 08W9. Finally, condition (1) is Lemma 08W6. This finishes the proof of the theorem.
The stack of coherent sheaves in the non-flat case
In Theorem 08WC the assumption that \(f : X \to B\) is flat is not necessary. In this section we give a different proof which avoids the flatness assumption and avoids checking openness of versality by using the results in Flatness on Spaces, Section 0CX3 and Artin’s Axioms, Section 0CXR.
For a different approach to this problem the reader may wish to consult [ArtinI] and follow the method discussed in the papers [olsson-starr], [lieblich_remarks], [olsson_proper], [Hall-Rydh], [Hall-Rydh-Hilbert], [rydh_representability]. Some of these papers deal with the more general case of the stack of coherent sheaves on an algebraic stack over an algebraic stack and others deal with similar problems in the case of Hilbert stacks or Quot functors. Our strategy will be to show algebraicity of some cases of Hilbert stacks and Quot functors as a consequence of the algebraicity of the stack of coherent sheaves.
Theorem
Let \(S\) be a scheme. Let \(f : X \to B\) be morphism of algebraic spaces over \(S\). Assume that \(f\) is of finite presentation and separated. Then \(\Cohstack_{X/B}\) is an algebraic stack over \(S\).
Proof
Only the last step of the proof is different from the proof in the flat case, but we repeat all the arguments here to make sure everything works.
Set \(\mathcal{X} = \Cohstack_{X/B}\). We have seen that \(\mathcal{X}\) is a stack in groupoids over \((\Sch/S)_{fppf}\) with diagonal representable by algebraic spaces (Lemmas 08KC and 08W6). Hence it suffices to find a scheme \(W\) and a surjective and smooth morphism \(W \to \mathcal{X}\).
Let \(B'\) be a scheme and let \(B' \to B\) be a surjective étale morphism. Set \(X' = B' \times_B X\) and denote \(f' : X' \to B'\) the projection. Then \(\mathcal{X}' = \Cohstack_{X'/B'}\) is equal to the \(2\)-fibre product of \(\mathcal{X}\) with the category fibred in sets associated to \(B'\) over the category fibred in sets associated to \(B\) (Remark 08LP). By the material in Algebraic Stacks, Section 03YJ the morphism \(\mathcal{X}' \to \mathcal{X}\) is surjective and étale. Hence it suffices to prove the result for \(\mathcal{X}'\). In other words, we may assume \(B\) is a scheme.
Assume \(B\) is a scheme. In this case we may replace \(S\) by \(B\), see Algebraic Stacks, Section 04X4. Thus we may assume \(S = B\).
Assume \(S = B\). Choose an affine open covering \(S = \bigcup U_i\). Denote \(\mathcal{X}_i\) the restriction of \(\mathcal{X}\) to \((\Sch/U_i)_{fppf}\). If we can find schemes \(W_i\) over \(U_i\) and surjective smooth morphisms \(W_i \to \mathcal{X}_i\), then we set \(W = \coprod W_i\) and we obtain a surjective smooth morphism \(W \to \mathcal{X}\). Thus we may assume \(S = B\) is affine.
Assume \(S = B\) is affine, say \(S = \Spec(\Lambda)\). Write \(\Lambda = \colim \Lambda_i\) as a filtered colimit with each \(\Lambda_i\) of finite type over \(\mathbf{Z}\). For some \(i\) we can find a morphism of algebraic spaces \(X_i \to \Spec(\Lambda_i)\) which is separated and of finite presentation and whose base change to \(\Lambda\) is \(X\). See Limits of Spaces, Lemmas 07SK and 0851. If we show that \(\Cohstack_{X_i/\Spec(\Lambda_i)}\) is an algebraic stack, then it follows by base change (Remark 08LP and Algebraic Stacks, Section 04X4) that \(\mathcal{X}\) is an algebraic stack. Thus we may assume that \(\Lambda\) is a finite type \(\mathbf{Z}\)-algebra.
Assume \(S = B = \Spec(\Lambda)\) is affine of finite type over \(\mathbf{Z}\). In this case we will verify conditions (1), (2), (3), (4), and (5) of Artin’s Axioms, Lemma 07Y4 to conclude that \(\mathcal{X}\) is an algebraic stack. Note that \(\Lambda\) is a G-ring, see More on Algebra, Proposition 07PX. Hence all local rings of \(S\) are G-rings. Thus (5) holds. To check (2) we have to verify axioms [-1], [0], [1], [2], and [3] of Artin’s Axioms, Section 07XJ. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas 08KC, 08KD, 08LQ, 08W8. Condition (3) is Lemma 08W9. Condition (1) is Lemma 08W6.
It remains to show condition (4) which is openness of versality. To see this we will use Artin’s Axioms, Lemma 0CXU. We have already seen that \(\mathcal{X}\) has diagonal representable by algebraic spaces, has (RS*), and is limit preserving (see lemmas used above). Hence we only need to see that \(\mathcal{X}\) satisfies the strong formal effectiveness formulated in Artin’s Axioms, Lemma 0CXU. This is Flatness on Spaces, Theorem 0CXB and the proof is complete.
The functor of quotients
In this section we discuss some generalities regarding the functor \(Q_{\mathcal{F}/X/B}\) defined below. The notation \(\Quotfunctor_{\mathcal{F}/X/B}\) is reserved for a subfunctor of \(\text{Q}_{\mathcal{F}/X/B}\). We urge the reader to skip this section on a first reading.
Situation
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. For any scheme \(T\) over \(B\) we will denote \(X_T\) the base change of \(X\) to \(T\) and \(\mathcal{F}_T\) the pullback of \(\mathcal{F}\) via the projection morphism \(X_T = X \times_B T \to X\). Given such a \(T\) we set \[\text{Q}_{\mathcal{F}/X/B}(T) = \left\{ \begin{matrix} \text{quotients }\mathcal{F}_T \to \mathcal{Q}\text{ where } \mathcal{Q}\text{ is a}\\ \text{quasi-coherent } \mathcal{O}_{X_T}\text{-module flat over }T \end{matrix} \right\}\] We identify quotients if they have the same kernel. Suppose that \(T' \to T\) is a morphism of schemes over \(B\) and \(\mathcal{F}_T \to \mathcal{Q}\) is an element of \(\text{Q}_{\mathcal{F}/X/B}(T)\). Then the pullback \(\mathcal{Q}' = (X_{T'} \to X_T)^*\mathcal{Q}\) is a quasi-coherent \(\mathcal{O}_{X_{T'}}\)-module flat over \(T'\) by Morphisms of Spaces, Lemma 05VW. Thus we obtain a functor [082N]\[\begin{equation} \text{Q}_{\mathcal{F}/X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets} \end{equation}\] This is the functor of quotients of \(\mathcal{F}/X/B\). We define a subfunctor [0CZL]\[\begin{equation} \text{Q}^{fp}_{\mathcal{F}/X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets} \end{equation}\] which assigns to \(T\) the subset of \(\text{Q}_{\mathcal{F}/X/B}(T)\) consisting of those quotients \(\mathcal{F}_T \to \mathcal{Q}\) such that \(\mathcal{Q}\) is of finite presentation as an \(\mathcal{O}_{X_T}\)-module. This is a subfunctor by Properties of Spaces, Section 05VR.
In Situation 082M we sometimes think of \(\text{Q}_{\mathcal{F}/X/B}\) as a functor \((\Sch/S)^{opp} \to \textit{Sets}\) endowed with a morphism \(\text{Q}_{\mathcal{F}/X/S} \to B\). Namely, if \(T\) is a scheme over \(S\), then an element of \(\text{Q}_{\mathcal{F}/X/B}(T)\) is a pair \((h, \mathcal{Q})\) where \(h\) a morphism \(h : T \to B\) and \(\mathcal{Q}\) is a \(T\)-flat quotient \(\mathcal{F}_T \to \mathcal{Q}\) of finite presentation on \(X_T = X \times_{B, h} T\). In particular, when we say that \(\text{Q}_{\mathcal{F}/X/S}\) is an algebraic space, we mean that the corresponding functor \((\Sch/S)^{opp} \to \textit{Sets}\) is an algebraic space. Similar remarks apply to \(\text{Q}^{fp}_{\mathcal{F}/X/B}\).
Remark
In Situation 082M let \(B' \to B\) be a morphism of algebraic spaces over \(S\). Set \(X' = X \times_B B'\) and denote \(\mathcal{F}'\) the pullback of \(\mathcal{F}\) to \(X'\). Thus we have the functor \(Q_{\mathcal{F}'/X'/B'}\) on the category of schemes over \(B'\). For a scheme \(T\) over \(B'\) it is clear that we have \[Q_{\mathcal{F}'/X'/B'}(T) = Q_{\mathcal{F}/X/B}(T)\] where on the right hand side we think of \(T\) as a scheme over \(B\) via the composition \(T \to B' \to B\). Similar remarks apply to \(\text{Q}^{fp}_{\mathcal{F}/X/B}\). These trivial remarks will occasionally be useful to change the base algebraic space.
Remark
Let \(S\) be a scheme, \(X\) an algebraic space over \(S\), and \(\mathcal{F}\) a quasi-coherent \(\mathcal{O}_X\)-module. Suppose that \(\{f_i : X_i \to X\}_{i \in I}\) is an fpqc covering and for each \(i, j \in I\) we are given an fpqc covering \(\{X_{ijk} \to X_i \times_X X_j\}\). In this situation we have a bijection \[\left\{ \begin{matrix} \text{quotients }\mathcal{F} \to \mathcal{Q}\text{ where } \\ \mathcal{Q}\text{ is a quasi-coherent }\\ \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{families of quotients }f_i^*\mathcal{F} \to \mathcal{Q}_i \text{ where } \\ \mathcal{Q}_i\text{ is quasi-coherent and } \mathcal{Q}_i\text{ and }\mathcal{Q}_j\\ \text{ restrict to the same quotient on }X_{ijk} \end{matrix} \right\}\] Namely, let \((f_i^*\mathcal{F} \to \mathcal{Q}_i)_{i \in I}\) be an element of the right hand side. Then since \(\{X_{ijk} \to X_i \times_X X_j\}\) is an fpqc covering we see that the pullbacks of \(\mathcal{Q}_i\) and \(\mathcal{Q}_j\) restrict to the same quotient of the pullback of \(\mathcal{F}\) to \(X_i \times_X X_j\) (by fully faithfulness in Descent on Spaces, Proposition 04W8). Hence we obtain a descent datum for quasi-coherent modules with respect to \(\{X_i \to X\}_{i \in I}\). By Descent on Spaces, Proposition 04W8 we find a map of quasi-coherent \(\mathcal{O}_X\)-modules \(\mathcal{F} \to \mathcal{Q}\) whose restriction to \(X_i\) recovers the given maps \(f_i^*\mathcal{F} \to \mathcal{Q}_i\). Since the family of morphisms \(\{X_i \to X\}\) is jointly surjective and flat, for every point \(x \in |X|\) there exists an \(i\) and a point \(x_i \in |X_i|\) mapping to \(x\). Note that the induced map on local rings \(\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X_i, \overline{x_i}}\) is faithfully flat, see Morphisms of Spaces, Section 03MK. Thus we see that \(\mathcal{F} \to \mathcal{Q}\) is surjective.
Lemma
In Situation 082M. The functors \(\text{Q}_{\mathcal{F}/X/B}\) and \(\text{Q}^{fp}_{\mathcal{F}/X/B}\) satisfy the sheaf property for the fpqc topology.
Proof
Let \(\{T_i \to T\}_{i \in I}\) be an fpqc covering of schemes over \(S\). Set \(X_i = X_{T_i} = X \times_S T_i\) and \(\mathcal{F}_i = \mathcal{F}_{T_i}\). Note that \(\{X_i \to X_T\}_{i \in I}\) is an fpqc covering of \(X_T\) (Topologies on Spaces, Lemma 03MR) and that \(X_{T_i \times_T T_{i'}} = X_i \times_{X_T} X_{i'}\). Suppose that \(\mathcal{F}_i \to \mathcal{Q}_i\) is a collection of elements of \(\text{Q}_{\mathcal{F}/X/B}(T_i)\) such that \(\mathcal{Q}_i\) and \(\mathcal{Q}_{i'}\) restrict to the same element of \(\text{Q}_{\mathcal{F}/X/B}(T_i \times_T T_{i'})\). By Remark 08IU we obtain a surjective map of quasi-coherent \(\mathcal{O}_{X_T}\)-modules \(\mathcal{F}_T \to \mathcal{Q}\) whose restriction to \(X_i\) recovers the given quotients. By Morphisms of Spaces, Lemma 05VY we see that \(\mathcal{Q}\) is flat over \(T\). Finally, in the case of \(\text{Q}^{fp}_{\mathcal{F}/X/B}\), i.e., if \(\mathcal{Q}_i\) are of finite presentation, then Descent on Spaces, Lemma 060V guarantees that \(\mathcal{Q}\) is of finite presentation as an \(\mathcal{O}_{X_T}\)-module.
Sanity check: \(\text{Q}_{\mathcal{F}/X/B}\), \(\text{Q}^{fp}_{\mathcal{F}/X/B}\) play the same role among algebraic spaces over \(S\).
Lemma
In Situation 082M. Let \(T\) be an algebraic space over \(S\). We have \[\Mor_{\Sh((\Sch/S)_{fppf})}(T, \text{Q}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}\\ \mathcal{Q}\text{ is quasi-coherent and flat over }T \end{matrix} \right\}\] where \(\mathcal{F}_T\) denotes the pullback of \(\mathcal{F}\) to the algebraic space \(X \times_{B, h} T\). Similarly, we have \[\Mor_{\Sh((\Sch/S)_{fppf})}(T, \text{Q}^{fp}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}\\ \mathcal{Q}\text{ is of finite presentation and flat over }T \end{matrix} \right\}\]
Proof
Choose a scheme \(U\) and a surjective étale morphism \(p : U \to T\). Let \(R = U \times_T U\) with projections \(t, s : R \to U\).
Let \(v : T \to \text{Q}_{\mathcal{F}/X/B}\) be a natural transformation. Then \(v(p)\) corresponds to a pair \((h_U, \mathcal{F}_U \to \mathcal{Q}_U)\) over \(U\). As \(v\) is a transformation of functors we see that the pullbacks of \((h_U, \mathcal{F}_U \to \mathcal{Q}_U)\) by \(s\) and \(t\) agree. Since \(T = U/R\) (Spaces, Lemma 0262), we obtain a morphism \(h : T \to B\) such that \(h_U = h \circ p\). By Descent on Spaces, Proposition 04W8 the quotient \(\mathcal{Q}_U\) descends to a quotient \(\mathcal{F}_T \to \mathcal{Q}\) over \(X_T\). Since \(U \to T\) is surjective and flat, it follows from Morphisms of Spaces, Lemma 05VY that \(\mathcal{Q}\) is flat over \(T\).
Conversely, let \((h, \mathcal{F}_T \to \mathcal{Q})\) be a pair over \(T\). Then we get a natural transformation \(v : T \to \text{Q}_{\mathcal{F}/X/B}\) by sending a morphism \(a : T' \to T\) where \(T'\) is a scheme to \((h \circ a, \mathcal{F}_{T'} \to a^*\mathcal{Q})\). We omit the verification that the construction of this and the previous paragraph are mutually inverse.
In the case of \(\text{Q}^{fp}_{\mathcal{F}/X/B}\) we add: given a morphism \(h : T \to B\), a quasi-coherent sheaf on \(X_T\) is of finite presentation as an \(\mathcal{O}_{X_T}\)-module if and only if the pullback to \(X_U\) is of finite presentation as an \(\mathcal{O}_{X_U}\)-module. This follows from the fact that \(X_U \to X_T\) is surjective and étale and Descent on Spaces, Lemma 060V.
Lemma
In Situation 082M let \(\{X_i \to X\}_{i \in I}\) be an fpqc covering and for each \(i, j \in I\) let \(\{X_{ijk} \to X_i \times_X X_j\}\) be an fpqc covering. Denote \(\mathcal{F}_i\), resp. \(\mathcal{F}_{ijk}\) the pullback of \(\mathcal{F}\) to \(X_i\), resp. \(X_{ijk}\). For every scheme \(T\) over \(B\) the diagram \[\xymatrix{ Q_{\mathcal{F}/X/B}(T) \ar[r] & \prod\nolimits_i Q_{\mathcal{F}_i/X_i/B}(T) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod\nolimits_{i, j, k} Q_{\mathcal{F}_{ijk}/X_{ijk}/B}(T) }\] presents the first arrow as the equalizer of the other two. The same is true for the functor \(\text{Q}^{fp}_{\mathcal{F}/X/B}\).
Proof
Let \(\mathcal{F}_{i, T} \to \mathcal{Q}_i\) be an element in the equalizer of \(\text{pr}_0^*\) and \(\text{pr}_1^*\). By Remark 08IU we obtain a surjection \(\mathcal{F}_T \to \mathcal{Q}\) of quasi-coherent \(\mathcal{O}_{X_T}\)-modules whose restriction to \(X_{i, T}\) recovers \(\mathcal{F}_i \to \mathcal{Q}_i\). By Morphisms of Spaces, Lemma 05VY we see that \(\mathcal{Q}\) is flat over \(T\) as desired. In the case of the functor \(\text{Q}^{fp}_{\mathcal{F}/X/B}\), i.e., if \(\mathcal{Q}_i\) is of finite presentation, then \(\mathcal{Q}\) is of finite presentation too by Descent on Spaces, Lemma 060V.
Lemma
In Situation 082M assume also that (a) \(f\) is quasi-compact and quasi-separated and (b) \(\mathcal{F}\) is of finite presentation. Then the functor \(\text{Q}^{fp}_{\mathcal{F}/X/B}\) is limit preserving in the following sense: If \(T = \lim T_i\) is a directed limit of affine schemes over \(B\), then \(\text{Q}^{fp}_{\mathcal{F}/X/B}(T) = \colim \text{Q}^{fp}_{\mathcal{F}/X/B}(T_i)\).
Proof
Let \(T = \lim T_i\) be as in the statement of the lemma. Choose \(i_0 \in I\) and replace \(I\) by \(\{i \in I \mid i \geq i_0\}\). We may set \(B = S = T_{i_0}\) and we may replace \(X\) by \(X_{T_0}\) and \(\mathcal{F}\) by the pullback to \(X_{T_0}\). Then \(X_T = \lim X_{T_i}\), see Limits of Spaces, Lemma 07SF. Let \(\mathcal{F}_T \to \mathcal{Q}\) be an element of \(\text{Q}^{fp}_{\mathcal{F}/X/B}(T)\). By Limits of Spaces, Lemma 07V7 there exists an \(i\) and a map \(\mathcal{F}_{T_i} \to \mathcal{Q}_i\) of \(\mathcal{O}_{X_{T_i}}\)-modules of finite presentation whose pullback to \(X_T\) is the given quotient map.
We still have to check that, after possibly increasing \(i\), the map \(\mathcal{F}_{T_i} \to \mathcal{Q}_i\) is surjective and \(\mathcal{Q}_i\) is flat over \(T_i\). To do this, choose an affine scheme \(U\) and a surjective étale morphism \(U \to X\) (see Properties of Spaces, Lemma 03H6). We may check surjectivity and flatness over \(T_i\) after pulling back to the étale cover \(U_{T_i} \to X_{T_i}\) (by definition). This reduces us to the case where \(X = \Spec(B_0)\) is an affine scheme of finite presentation over \(B = S = T_0 = \Spec(A_0)\). Writing \(T_i = \Spec(A_i)\), then \(T = \Spec(A)\) with \(A = \colim A_i\) we have reached the following algebra problem. Let \(M_i \to N_i\) be a map of finitely presented \(B_0 \otimes_{A_0} A_i\)-modules such that \(M_i \otimes_{A_i} A \to N_i \otimes_{A_i} A\) is surjective and \(N_i \otimes_{A_i} A\) is flat over \(A\). Show that for some \(i' \geq i\) \(M_i \otimes_{A_i} A_{i'} \to N_i \otimes_{A_i} A_{i'}\) is surjective and \(N_i \otimes_{A_i} A_{i'}\) is flat over \(A\). The first follows from Algebra, Lemma 05LI and the second from Algebra, Lemma 02JO.
Lemma
In Situation 082M. Let \[\xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' }\] be a pushout in the category of schemes over \(B\) where \(Z \to Z'\) is a thickening and \(Z \to Y\) is affine, see More on Morphisms, Lemma 07RT. Then the natural map \[Q_{\mathcal{F}/X/B}(Y') \longrightarrow Q_{\mathcal{F}/X/B}(Y) \times_{Q_{\mathcal{F}/X/B}(Z)} Q_{\mathcal{F}/X/B}(Z')\] is bijective. If \(X \to B\) is locally of finite presentation, then the same thing is true for \(Q^{fp}_{\mathcal{F}/X/B}\).
Proof
Let us construct an inverse map. Namely, suppose we have \(\mathcal{F}_Y \to \mathcal{A}\), \(\mathcal{F}_{Z'} \to \mathcal{B}'\), and an isomorphism \(\mathcal{A}|_{X_Z} \to \mathcal{B}'|_{X_Z}\) compatible with the given surjections. Then we apply Pushouts of Spaces, Lemma 08KV to get a quasi-coherent module \(\mathcal{A}'\) on \(X_{Y'}\) flat over \(Y'\). Since this sheaf is constructed as a fibre product (see proof of cited lemma) there is a canonical map \(\mathcal{F}_{Y'} \to \mathcal{A}'\). That this map is surjective can be seen because it factors as \[\begin{matrix} \mathcal{F}_{Y'} \\ \downarrow \\ (X_Y \to X_{Y'})_*\mathcal{F}_Y \times_{(X_Z \to X_{Y'})_*\mathcal{F}_Z} (X_{Z'} \to X_{Y'})_*\mathcal{F}_{Z'} \\ \downarrow \\ \mathcal{A}' = (X_Y \to X_{Y'})_*\mathcal{A} \times_{(X_Z \to X_{Y'})_*\mathcal{A}|_{X_Z}} (X_{Z'} \to X_{Y'})_*\mathcal{B}' \end{matrix}\] and the first arrow is surjective by More on Algebra, Lemma 08IG and the second by More on Algebra, Lemma 08KJ.
In the case of \(Q^{fp}_{\mathcal{F}/X/B}\) all we have to show is that the construction above produces a finitely presented module. This is explained in More on Algebra, Remark 08KR in the commutative algebra setting. The current case of modules over algebraic spaces follows from this by étale localization.
Remark
In Situation 082M assume that \(\mathcal{F}\) is flat over \(B\). Let \(T \subset T'\) be an first order thickening of schemes over \(B\) with ideal sheaf \(\mathcal{J}\). Then \(X_T \subset X_{T'}\) is a first order thickening of algebraic spaces whose ideal sheaf \(\mathcal{I}\) is a quotient of \(f_T^*\mathcal{J}\). We will think of sheaves on \(X_{T'}\), resp. \(T'\) as sheaves on \(X_T\), resp. \(T\) using the fundamental equivalence described in More on Morphisms of Spaces, Section 05ZJ. Let \[0 \to \mathcal{K} \to \mathcal{F}_T \to \mathcal{Q} \to 0\] define an element \(x\) of \(Q_{\mathcal{F}/X/B}(T)\). Since \(\mathcal{F}_{T'}\) is flat over \(T'\) we have a short exact sequence \[0 \to f_T^*\mathcal{J} \otimes_{\mathcal{O}_{X_T}} \mathcal{F}_T \xrightarrow{i} \mathcal{F}_{T'} \xrightarrow{\pi} \mathcal{F}_T \to 0\] and we have \(f_T^*\mathcal{J} \otimes_{\mathcal{O}_{X_T}} \mathcal{F}_T = \mathcal{I} \otimes_{\mathcal{O}_{X_T}} \mathcal{F}_T\), see Deformation Theory, Lemma 08MQ. Let us use the abbreviation \(f_T^*\mathcal{J} \otimes_{\mathcal{O}_{X_T}} \mathcal{G} = \mathcal{G} \otimes_{\mathcal{O}_T} \mathcal{J}\) for an \(\mathcal{O}_{X_T}\)-module \(\mathcal{G}\). Since \(\mathcal{Q}\) is flat over \(T\), we obtain a short exact sequence \[0 \to \mathcal{K} \otimes_{\mathcal{O}_T} \mathcal{J} \to \mathcal{F}_T \otimes_{\mathcal{O}_T} \mathcal{J} \to \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J} \to \to 0\] Combining the above we obtain an canonical extension \[0 \to \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J} \to \pi^{-1}(\mathcal{K})/i(\mathcal{K} \otimes_{\mathcal{O}_T} \mathcal{J}) \to \mathcal{K} \to 0\] of \(\mathcal{O}_{X_T}\)-modules. This defines a canonical class \[o_x(T') \in \Ext^1_{\mathcal{O}_{X_T}}(\mathcal{K}, \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J})\] If \(o_x(T')\) is zero, then we obtain a splitting of the short exact sequence defining it, in other words, we obtain a \(\mathcal{O}_{X_{T'}}\)-submodule \(\mathcal{K}' \subset \pi^{-1}(\mathcal{K})\) sitting in a short exact sequence \(0 \to \mathcal{K} \otimes_{\mathcal{O}_T} \mathcal{J} \to \mathcal{K}' \to \mathcal{K} \to 0\). Then it follows from the lemma reference above that \(\mathcal{Q}' = \mathcal{F}_{T'}/\mathcal{K}'\) is a lift of \(x\) to an element of \(Q_{\mathcal{F}/X/B}(T')\). Conversely, the reader sees that the existence of a lift implies that \(o_x(T')\) is zero. Moreover, if \(x \in Q_{\mathcal{F}/X/B}^{fp}(T)\), then automatically \(x' \in Q_{\mathcal{F}/X/B}^{fp}(T')\) by Deformation Theory, Lemma 08VU. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi).
Remark
In Situation 082M assume that \(\mathcal{F}\) is flat over \(B\). We continue the discussion of Remark 0CZU. Assume \(o_x(T') = 0\). Then we claim that the set of lifts \(x' \in Q_{\mathcal{F}/X/B}(T')\) is a principal homogeneous space under the group \[\Hom_{\mathcal{O}_{X_T}}(\mathcal{K}, \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J})\] Namely, given any \(\mathcal{F}_{T'} \to \mathcal{Q}'\) flat over \(T'\) lifting the quotient \(\mathcal{Q}\) we obtain a commutative diagram with exact rows and columns \[\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & \mathcal{K} \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{F}_T \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{Q} \otimes \mathcal{J} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}' \ar[r] \ar[d] & \mathcal{F}_{T'} \ar[r] \ar[d] & \mathcal{Q}' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[d] \ar[r] & \mathcal{F}_T \ar[d] \ar[r] & \mathcal{Q} \ar[d] \ar[r] & 0 \\ & 0 & 0 & 0 }\] (to see this use the observations made in the previous remark). Given a map \(\varphi : \mathcal{K} \to \mathcal{Q} \otimes \mathcal{J}\) we can consider the subsheaf \(\mathcal{K}'_\varphi \subset \mathcal{F}_{T'}\) consisting of those local sections \(s\) whose image in \(\mathcal{F}_T\) is a local section \(k\) of \(\mathcal{K}\) and whose image in \(\mathcal{Q}'\) is the local section \(\varphi(k)\) of \(\mathcal{Q} \otimes \mathcal{J}\). Then set \(\mathcal{Q}'_\varphi = \mathcal{F}_{T'}/\mathcal{K}'_\varphi\). Conversely, any second lift of \(x\) corresponds to one of the qotients constructed in this manner. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi).
Lemma
In Situation 082M, let \(T \subset T'\) be a first order thickening of schemes over \(B\) with ideal sheaf \(\mathcal{J}\). Let \[\begin{equation} 0 \longrightarrow \mathcal{K} \longrightarrow \mathcal{F}_T \longrightarrow \mathcal{Q} \longrightarrow 0 \end{equation}\] define an element of \(Q_{\mathcal{F}/X/B}(T)\). Set \[\mathcal{A} = \SheafHom_{\mathcal{O}_{X_T}} (\mathcal{K}, \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J}).\] On the common underlying topological space of \(X_T\) and \(X_{T'}\), let \(\mathcal{L}\) be the sheaf whose sections over an open \(U\) are the lifts of the restriction of (quot-equation-quotient-to-lift) to a quotient of \(\mathcal{F}_{T'}|_U\) flat over \(T'\). Then \(\mathcal{A}\) acts on \(\mathcal{L}\) and this action is simply transitive wherever \(\mathcal{L}\) is nonempty.
If lifts exist locally, then \(\mathcal{L}\) is an \(\mathcal{A}\)-torsor. Its class \[c(\mathcal{Q}) \in H^1(X_T, \mathcal{A})\] vanishes if and only if a global lift exists. If it vanishes, the set of global lifts is a principal homogeneous space under \(H^0(X_T, \mathcal{A})\). In particular, a global lift exists if local lifts exist and \(H^1(X_T, \mathcal{A}) = 0\).
Proof
We identify sheaves on \(X_T\) and \(X_{T'}\) using the equivalence associated to a thickening. Write \(\mathcal{I}\) for the ideal of \(X_T \subset X_{T'}\). Suppose over an open \(U\) we have a lift \[0 \to \mathcal{K}' \to \mathcal{F}_{T'}|_U \to \mathcal{Q}' \to 0.\] Flatness of \(\mathcal{Q}'\) gives canonical identifications \[\mathcal{I}\mathcal{Q}' = \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J} \quad\text{and}\quad \mathcal{K}'/\mathcal{I}\mathcal{K}' = \mathcal{K}|_U;\] see Deformation Theory, Lemma 08MQ.
Given a second lift with quotient \(\mathcal{Q}''\), the composite \(\mathcal{K}' \to \mathcal{F}_{T'}|_U \to \mathcal{Q}''\) has image in \(\mathcal{I}\mathcal{Q}''\) and annihilates \(\mathcal{I}\mathcal{K}'\). It therefore determines a unique map \[\mathcal{K}|_U \longrightarrow (\mathcal{Q} \otimes_{\mathcal{O}_T}\mathcal{J})|_U.\] Conversely, for such a map \(\varphi\), define \(\mathcal{K}'_\varphi \subset \mathcal{F}_{T'}|_U\) to consist of the local sections whose reduction belongs to \(\mathcal{K}\) and whose image in \(\mathcal{Q}'\) is \(\varphi\) of that reduction. Then \(\mathcal{F}_{T'}|_U/\mathcal{K}'_\varphi\) reduces to \(\mathcal{Q}\) and has the same \(\mathcal{I}\)-multiple as \(\mathcal{Q}'\). The flatness criterion in the lemma cited above shows that it is flat over \(T'\). These two constructions are inverse to one another and are compatible with restriction. This proves the assertion about the action.
Quotients and their kernels glue over open coverings, so \(\mathcal{L}\) is a sheaf. If it is locally nonempty, the first part makes it an \(\mathcal{A}\)-torsor. The remaining assertions follow from Cohomology, Lemmas 02FQ and 02FP.
The Quot functor
In this section we prove the Quot functor is an algebraic space.
Situation
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Assume that \(f\) is of finite presentation. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. For any scheme \(T\) over \(B\) we will denote \(X_T\) the base change of \(X\) to \(T\) and \(\mathcal{F}_T\) the pullback of \(\mathcal{F}\) via the projection morphism \(X_T = X \times_S T \to X\). Given such a \(T\) we set \[\Quotfunctor_{\mathcal{F}/X/B}(T) = \left\{ \begin{matrix} \text{quotients }\mathcal{F}_T \to \mathcal{Q}\text{ where } \mathcal{Q}\text{ is a quasi-coherent }\\ \mathcal{O}_{X_T}\text{-module of finite presentation, flat over }T\\ \text{with support proper over }T \end{matrix} \right\}\] By Derived Categories of Spaces, Lemma 0CZJ this is a subfunctor of the functor \(Q^{fp}_{\mathcal{F}/X/B}\) we discussed in Section 082L. Thus we obtain a functor [09TS]\[\begin{equation} \Quotfunctor_{\mathcal{F}/X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets} \end{equation}\] This is the Quot functor associated to \(\mathcal{F}/X/B\).
In Situation 09TR we sometimes think of \(\Quotfunctor_{\mathcal{F}/X/B}\) as a functor \((\Sch/S)^{opp} \to \textit{Sets}\) endowed with a morphism \(\Quotfunctor_{\mathcal{F}/X/B} \to B\). Namely, if \(T\) is a scheme over \(S\), then an element of \(\Quotfunctor_{\mathcal{F}/X/B}(T)\) is a pair \((h, \mathcal{Q})\) where \(h\) is a morphism \(h : T \to B\) and \(Q\) is a finitely presented, \(T\)-flat quotient \(\mathcal{F}_T \to \mathcal{Q}\) on \(X_T = X \times_{B, h} T\) with support proper over \(T\). In particular, when we say that \(\Quotfunctor_{\mathcal{F}/X/B}\) is an algebraic space, we mean that the corresponding functor \((\Sch/S)^{opp} \to \textit{Sets}\) is an algebraic space.
Lemma
In Situation 09TR. The functor \(\Quotfunctor_{\mathcal{F}/X/B}\) satisfies the sheaf property for the fpqc topology.
Proof
In Lemma 082P we have seen that the functor \(\text{Q}^{fp}_{\mathcal{F}/X/S}\) is a sheaf. Recall that for a scheme \(T\) over \(S\) the subset \(\Quotfunctor_{\mathcal{F}/X/S}(T) \subset \text{Q}_{\mathcal{F}/X/S}(T)\) picks out those quotients whose support is proper over \(T\). This defines a subsheaf by the result of Descent on Spaces, Lemma 0422 combined with Morphisms of Spaces, Lemma 089C which shows that taking scheme theoretic support commutes with flat base change.
Sanity check: \(\Quotfunctor_{\mathcal{F}/X/B}\) plays the same role among algebraic spaces over \(S\).
Lemma
In Situation 09TR. Let \(T\) be an algebraic space over \(S\). We have \[\Mor_{\Sh((\Sch/S)_{fppf})}(T, \Quotfunctor_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}\\ \mathcal{Q}\text{ is of finite presentation and}\\ \text{flat over }T\text{ with support proper over }T \end{matrix} \right\}\] where \(\mathcal{F}_T\) denotes the pullback of \(\mathcal{F}\) to the algebraic space \(X \times_{B, h} T\).
Proof
Observe that the left and right hand side of the equality are subsets of the left and right hand side of the second equality in Lemma 0D3U. To see that these subsets correspond under the identification given in the proof of that lemma it suffices to show: given \(h : T \to B\), a surjective étale morphism \(U \to T\), a finite type quasi-coherent \(\mathcal{O}_{X_T}\)-module \(\mathcal{Q}\) the following are equivalent
the scheme theoretic support of \(\mathcal{Q}\) is proper over \(T\), and
the scheme theoretic support of \((X_U \to X_T)^*\mathcal{Q}\) is proper over \(U\).
This follows from Descent on Spaces, Lemma 0422 combined with Morphisms of Spaces, Lemma 089C which shows that taking scheme theoretic support commutes with flat base change.
Proposition
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). If \(f\) is of finite presentation and separated, then \(\Quotfunctor_{\mathcal{F}/X/B}\) is an algebraic space. If \(\mathcal{F}\) is of finite presentation, then \(\Quotfunctor_{\mathcal{F}/X/B} \to B\) is locally of finite presentation.
Proof
By Lemma 09TT we have that \(\Quotfunctor_{\mathcal{F}/X/B}\) is a sheaf in the fppf topology. Let \(\textit{Quot}_{\mathcal{F}/X/B}\) be the stack in groupoids corresponding to \(\Quotfunctor_{\mathcal{F}/X/S}\), see Algebraic Stacks, Section 04SU. By Algebraic Stacks, Proposition 04SZ it suffices to show that \(\textit{Quot}_{\mathcal{F}/X/B}\) is an algebraic stack. Consider the \(1\)-morphism of stacks in groupoids \[\textit{Quot}_{\mathcal{F}/X/S} \longrightarrow \Cohstack_{X/B}\] on \((\Sch/S)_{fppf}\) which associates to the quotient \(\mathcal{F}_T \to \mathcal{Q}\) the module \(\mathcal{Q}\). By Theorem 09DS we know that \(\Cohstack_{X/B}\) is an algebraic stack. By Algebraic Stacks, Lemma 05UM it suffices to show that this \(1\)-morphism is representable by algebraic spaces.
Let \(T\) be a scheme over \(S\) and let the object \((h, \mathcal{G})\) of \(\Cohstack_{X/B}\) over \(T\) correspond to a \(1\)-morphism \(\xi : (\Sch/T)_{fppf} \to \Cohstack_{X/B}\). The \(2\)-fibre product \[\mathcal{Z} = (\Sch/T)_{fppf} \times_{\xi, \Cohstack_{X/B}} \textit{Quot}_{\mathcal{F}/X/S}\] is a stack in setoids, see Stacks, Lemma 05UI. The corresponding sheaf of sets (i.e., functor, see Stacks, Lemmas 05UI and 0430) assigns to a scheme \(T'/T\) the set of surjections \(u : \mathcal{F}_{T'} \to \mathcal{G}_{T'}\) of quasi-coherent modules on \(X_{T'}\). Thus we see that \(\mathcal{Z}\) is representable by an open subspace (by Flatness on Spaces, Lemma 09TP) of the algebraic space \(\mathit{Hom}(\mathcal{F}_T, \mathcal{G})\) from Proposition 08K6.
Lemma
In Situation 09TR, put \(Q = \Quotfunctor_{\mathcal{F}/X/B}\) and assume that \(Q\) is an algebraic space (for example, this holds if \(X \to B\) is separated by Proposition 09TU). Let \(g : T \to Q\) correspond to an exact sequence \[0 \longrightarrow \mathcal{K} \longrightarrow \mathcal{F}_T \longrightarrow \mathcal{G} \longrightarrow 0.\] For every quasi-coherent \(\mathcal{O}_T\)-module \(\mathcal{M}\) there is a canonical isomorphism, functorial in \(\mathcal{M}\), \[\begin{equation} \Hom_{\mathcal{O}_T}(g^*\Omega_{Q/B}, \mathcal{M}) = \Hom_{\mathcal{O}_{X_T}} (\mathcal{K}, \mathcal{G} \otimes_{\mathcal{O}_T} \mathcal{M}). \end{equation}\] In particular, if \(x : \Spec(k) \to Q_b\) is a point, then \[T_{Q_b, x} = \Hom_{\mathcal{O}_{X_k}}(\mathcal{K}, \mathcal{G}).\]
Proof
Let \(T \subset T'\) be the split first order thickening with underlying space \(T\), structure sheaf \(\mathcal{O}_{T'} = \mathcal{O}_T \oplus \mathcal{M}\), and square zero ideal \(\mathcal{M}\). Give \(T'\) its \(B\)-structure through the retraction \(T' \to T\). The set of lifts of \(g\) to a \(B\)-morphism \(T' \to Q\) is a principal homogeneous space under the left hand side of (quot-equation-cotangent-space-quot); this follows from More on Morphisms of Spaces, Lemma 061C. It has a distinguished element obtained by composing \(T' \to T\) with \(g\).
By representability of \(Q\) the same set is the set of lifts of the quotient \(\mathcal{F}_T \to \mathcal{G}\). Finite presentation and properness of the support are unchanged by this nilpotent thickening. Lemma quot-lemma-quotient-lifts-torsor identifies this set, with the same distinguished element, as a principal homogeneous space under the right hand side of (quot-equation-cotangent-space-quot). The difference constructions in the two torsor actions give the displayed canonical isomorphism. Taking \(T = \Spec(k)\) and \(\mathcal{M} = k\) gives the final statement.
Remark
Let \(S\) be a Noetherian scheme all of whose local rings are G-rings. Let \(X\) be an algebraic space over \(S\) whose structure morphism \(f : X \to S\) is of finite presentation and separated. Let \(\mathcal{F}\) be a finitely presented quasi-coherent sheaf on \(X\) flat over \(S\). In this remark we sketch how one can use Artin’s axioms to prove that \(\Quotfunctor_{\mathcal{F}/X/S}\) is an algebraic space locally of finite presentation over \(S\) and avoid using the algebraicity of the stack of coherent sheaves as was done in the proof of Proposition 09TU.
We check the conditions listed in Artin’s Axioms, Proposition 07Y1. Representability of the diagonal of \(\Quotfunctor_{\mathcal{F}/X/S}\) can be seen as follows: suppose we have two quotients \(\mathcal{F}_T \to \mathcal{Q}_i\), \(i = 1, 2\). Denote \(\mathcal{K}_1\) the kernel of the first one. Then we have to show that the locus of \(T\) over which \(u : \mathcal{K}_1 \to \mathcal{Q}_2\) becomes zero is representable. This follows for example from Flatness on Spaces, Lemma 083M or from a discussion of the \(\mathit{Hom}\) sheaf earlier in this chapter. Axioms [0] (sheaf), [1] (limits), [2] (Rim-Schlessinger) follow from Lemmas 09TT, 082Q, and 08IW (plus some extra work to deal with the properness condition). Axiom [3] (finite dimensionality of tangent spaces) follows from the description of the infinitesimal deformations in Remark 0CZV and finiteness of cohomology of coherent sheaves on proper algebraic spaces over fields (Cohomology of Spaces, Lemma 08AR). Axiom [4] (effectiveness of formal objects) follows from Grothendieck’s existence theorem (More on Morphisms of Spaces, Theorem 08BE). As usual, the trickiest to verify is axiom [5] (openness of versality). One can for example use the obstruction theory described in Remark 0CZU and the description of deformations in Remark 0CZV to do this using the criterion in Artin’s Axioms, Lemma 0CYF. Please compare with the second proof of Lemma 08WA.
The Hilbert functor
In this section we prove the Hilb functor is an algebraic space.
Situation
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Assume that \(f\) is of finite presentation. For any scheme \(T\) over \(B\) we will denote \(X_T\) the base change of \(X\) to \(T\). Given such a \(T\) we set \[\Hilbfunctor_{X/B}(T) = \left\{ \begin{matrix} \text{closed subspaces }Z \subset X_T\text{ such that }Z \to T\\ \text{is of finite presentation, flat, and proper} \end{matrix} \right\}\] Since base change preserves the required properties (Spaces, Lemma 02YW and Morphisms of Spaces, Lemmas 03XR, 03MO, and 04WP) we obtain a functor [0CZZ]\[\begin{equation} \Hilbfunctor_{X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets} \end{equation}\] This is the Hilbert functor associated to \(X/B\).
In Situation 0CZY we sometimes think of \(\Hilbfunctor_{X/B}\) as a functor \((\Sch/S)^{opp} \to \textit{Sets}\) endowed with a morphism \(\Hilbfunctor_{X/S} \to B\). Namely, if \(T\) is a scheme over \(S\), then an element of \(\Hilbfunctor_{X/B}(T)\) is a pair \((h, Z)\) where \(h\) is a morphism \(h : T \to B\) and \(Z \subset X_T = X \times_{B, h} T\) is a closed subscheme, flat, proper, and of finite presentation over \(T\). In particular, when we say that \(\Hilbfunctor_{X/B}\) is an algebraic space, we mean that the corresponding functor \((\Sch/S)^{opp} \to \textit{Sets}\) is an algebraic space.
Of course the Hilbert functor is just a special case of the Quot functor.
Lemma
In Situation 0CZY we have \(\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}\).
Proof
Let \(T\) be a scheme over \(B\). Given an element \(Z \in \Hilbfunctor_{X/B}(T)\) we can consider the quotient \(\mathcal{O}_{X_T} \to i_*\mathcal{O}_Z\) where \(i : Z \to X_T\) is the inclusion morphism. Note that \(i_*\mathcal{O}_Z\) is quasi-coherent. Since \(Z \to T\) and \(X_T \to T\) are of finite presentation, we see that \(i\) is of finite presentation (Morphisms of Spaces, Lemma 05WT), hence \(i_*\mathcal{O}_Z\) is an \(\mathcal{O}_{X_T}\)-module of finite presentation (Descent on Spaces, Lemma 0610). Since \(Z \to T\) is proper we see that \(i_*\mathcal{O}_Z\) has support proper over \(T\) (as defined in Derived Categories of Spaces, Section 0CZB). Since \(\mathcal{O}_Z\) is flat over \(T\) and \(i\) is affine, we see that \(i_*\mathcal{O}_Z\) is flat over \(T\) (small argument omitted). Hence \(\mathcal{O}_{X_T} \to i_*\mathcal{O}_Z\) is an element of \(\Quotfunctor_{\mathcal{O}_X/X/B}(T)\).
Conversely, given an element \(\mathcal{O}_{X_T} \to \mathcal{Q}\) of \(\Quotfunctor_{\mathcal{O}_X/X/B}(T)\), we can consider the closed immersion \(i : Z \to X_T\) corresponding to the quasi-coherent ideal sheaf \(\mathcal{I} = \Ker(\mathcal{O}_{X_T} \to \mathcal{Q})\) (Morphisms of Spaces, Lemma 03MB). By construction of \(Z\) we see that \(\mathcal{Q} = i_*\mathcal{O}_Z\). Then we can read the arguments given above backwards to see that \(Z\) defines an element of \(\Hilbfunctor_{X/B}(T)\). For example, \(\mathcal{I}\) is quasi-coherent of finite type (Modules on Sites, Lemma 082T) hence \(i : Z \to X_T\) is of finite presentation (Morphisms of Spaces, Lemma 084Q) hence \(Z \to T\) is of finite presentation (Morphisms of Spaces, Lemma 03XQ). Properness of \(Z \to T\) follows from the discussion in Derived Categories of Spaces, Section 0CZB. Flatness of \(Z \to T\) follows from flatness of \(\mathcal{Q}\) over \(T\).
We omit the (immediate) verification that the two constructions given above are mutually inverse.
Sanity check: \(\Hilbfunctor_{X/B}\) sheaf plays the same role among algebraic spaces over \(S\).
Lemma
In Situation 0CZY. Let \(T\) be an algebraic space over \(S\). We have \[\Mor_{\Sh((\Sch/S)_{fppf})}(T, \Hilbfunctor_{X/B}) = \left\{ \begin{matrix} (h, Z)\text{ where }h : T \to B,\ Z \subset X_T \\ \text{finite presentation, flat, proper over }T \end{matrix} \right\}\] where \(X_T = X \times_{B, h} T\).
Proof
By Lemma 0D00 we have \(\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}\). Thus we can apply Lemma 0D3V to see that the left hand side is bijective with the set of surjections \(\mathcal{O}_{X_T} \to \mathcal{Q}\) which are finitely presented, flat over \(T\), and have support proper over \(T\). Arguing exactly as in the proof of Lemma 0D00 we see that such quotients correspond exactly to the closed immersions \(Z \to X_T\) such that \(Z \to T\) is proper, flat, and of finite presentation.
Proposition
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). If \(f\) is of finite presentation and separated, then \(\Hilbfunctor_{X/B}\) is an algebraic space locally of finite presentation over \(B\).
Proof
The Picard stack
The Picard stack for a morphism of algebraic spaces was introduced in Examples of Stacks, Section 0372. We will deduce it is an open substack of the stack of coherent sheaves (in good cases) from the following lemma.
Lemma
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\) which is flat, of finite presentation, and proper. The natural map \[\Picardstack_{X/B} \longrightarrow \Cohstack_{X/B}\] is representable by open immersions.
Proof
Observe that the map simply sends a triple \((T, g, \mathcal{L})\) as in Examples of Stacks, Section 0372 to the same triple \((T, g, \mathcal{L})\) but where now we view this as a triple of the kind described in Situation 08KB. This works because the invertible \(\mathcal{O}_{X_T}\)-module \(\mathcal{L}\) is certainly a finitely presented \(\mathcal{O}_{X_T}\)-module, it is flat over \(T\) because \(X_T \to T\) is flat, and the support is proper over \(T\) as \(X_T \to T\) is proper (Morphisms of Spaces, Lemmas 03MO and 04WP). Thus the statement makes sense.
Having said this, it is clear that the content of the lemma is the following: given an object \((T, g, \mathcal{F})\) of \(\Cohstack_{X/B}\) there is an open subscheme \(U \subset T\) such that for a morphism of schemes \(T' \to T\) the following are equivalent
\(T' \to T\) factors through \(U\),
the pullback \(\mathcal{F}_{T'}\) of \(\mathcal{F}\) by \(X_{T'} \to X_T\) is invertible.
Let \(W \subset |X_T|\) be the set of points \(x \in |X_T|\) such that \(\mathcal{F}\) is locally free in a neighbourhood of \(x\). By More on Morphisms of Spaces, Lemma 0CZT. \(W\) is open and formation of \(W\) commutes with arbitrary base change. Clearly, if \(T' \to T\) satisfies (b), then \(|X_{T'}| \to |X_T|\) maps into \(W\). Hence we may replace \(T\) by the open \(T \setminus f_T(|X_T| \setminus W)\) in order to construct \(U\). After doing so we reach the situation where \(\mathcal{F}\) is finite locally free. In this case we get a disjoint union decomposition \(X_T = X_0 \amalg X_1 \amalg X_2 \amalg \ldots\) into open and closed subspaces such that the restriction of \(\mathcal{F}\) is locally free of rank \(i\) on \(X_i\). Then clearly \[U = T \setminus f_T(|X_0| \cup |X_2| \cup |X_3| \cup \ldots )\] works. (Note that if we assume that \(T\) is quasi-compact, then \(X_T\) is quasi-compact hence only a finite number of \(X_i\) are nonempty and so \(U\) is indeed open.)
Proposition
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). If \(f\) is flat, of finite presentation, and proper, then \(\Picardstack_{X/B}\) is an algebraic stack.
Proof
Immediate consequence of Lemma 0D03, Algebraic Stacks, Lemma 05UM and either Theorem 08WC or Theorem 09DS
The Picard functor
In this section we revisit the Picard functor discussed in Picard Schemes of Curves, Section 0B9K. The discussion will be more general as we want to study the Picard functor of a morphism of algebraic spaces as in the section on the Picard stack, see Section 0D02.
Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). An invertible sheaf on \(X\) is an invertible \(\mathcal{O}_X\)-module on \(X_\etale\), see Modules on Sites, Definition 0409. The group of isomorphism classes of invertible modules is denoted \(\Pic(X)\), see Modules on Sites, Definition 040C. Given a morphism \(f : X \to Y\) of algebraic spaces over \(S\) pullback defines a group homomorphism \(\Pic(Y) \to \Pic(X)\). The assignment \(X \leadsto \Pic(X)\) is a contravariant functor from the category of schemes to the category of abelian groups. This functor is not representable, but it turns out that a relative variant of this construction sometimes is representable.
Situation
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). We define \[\Picardfunctor_{X/B} : (\Sch/B)^{opp} \longrightarrow \textit{Sets}\] as the fppf sheafification of the functor which to a scheme \(T\) over \(B\) associates the group \(\Pic(X_T)\).
In Situation 0D25 we sometimes think of \(\Picardfunctor_{X/B}\) as a functor \((\Sch/S)^{opp} \to \textit{Sets}\) endowed with a morphism \(\Picardfunctor_{X/B} \to B\). In this point of view, we define \(\Picardfunctor_{X/B}\) to be the fppf sheafification of the functor \[T/S \longmapsto \{(h, \mathcal{L}) \mid h : T \to B,\ \mathcal{L} \in \Pic(X \times_{B, h} T)\}\] In particular, when we say that \(\Picardfunctor_{X/B}\) is an algebraic space, we mean that the corresponding functor \((\Sch/S)^{opp} \to \textit{Sets}\) is an algebraic space.
An often used remark is that if \(T\) is a scheme over \(B\), then \(\Picardfunctor_{X_T/T}\) is the restriction of \(\Picardfunctor_{X/B}\) to \((\Sch/T)_{fppf}\).
Lemma
In Situation 0D25 the functor \(\Picardfunctor_{X/B}\) is the sheafification of the functor \(T \mapsto \Ob(\Picardstack_{X/B, T})/\cong\).
Proof
Since the fibre category \(\Picardstack_{X/B, T}\) of the Picard stack \(\Picardstack_{X/B}\) over \(T\) is the category of invertible sheaves on \(X_T\) (see Section 0D02 and Examples of Stacks, Section 0372) this is immediate from the definitions.
It turns out to be nontrivial to see what the value of \(\Picardfunctor_{X/B}\) is on schemes \(T\) over \(B\). Here is a lemma that helps with this task.
Lemma
In Situation 0D25. If \(\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}\) is an isomorphism for all schemes \(T\) over \(B\), then \[0 \to \Pic(T) \to \Pic(X_T) \to \Picardfunctor_{X/B}(T)\] is an exact sequence for all \(T\).
Proof
We may replace \(B\) by \(T\) and \(X\) by \(X_T\) and assume that \(B = T\) to simplify the notation. Let \(\mathcal{N}\) be an invertible \(\mathcal{O}_B\)-module. If \(f^*\mathcal{N} \cong \mathcal{O}_X\), then we see that \(f_*f^*\mathcal{N} \cong f_*\mathcal{O}_X \cong \mathcal{O}_B\) by assumption. Since \(\mathcal{N}\) is locally trivial, we see that the canonical map \(\mathcal{N} \to f_*f^*\mathcal{N}\) is locally an isomorphism (because \(\mathcal{O}_B \to f_*f^*\mathcal{O}_B\) is an isomorphism by assumption). Hence we conclude that \(\mathcal{N} \to f_*f^*\mathcal{N} \to \mathcal{O}_B\) is an isomorphism and we see that \(\mathcal{N}\) is trivial. This proves the first arrow is injective.
Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module which is in the kernel of \(\Pic(X) \to \Picardfunctor_{X/B}(B)\). Then there exists an fppf covering \(\{B_i \to B\}\) such that \(\mathcal{L}\) pulls back to the trivial invertible sheaf on \(X_{B_i}\). Choose a trivializing section \(s_i\). Then \(\text{pr}_0^*s_i\) and \(\text{pr}_1^*s_j\) are both trivialising sections of \(\mathcal{L}\) over \(X_{B_i \times_B B_j}\) and hence differ by a multiplicative unit \[f_{ij} \in \Gamma(X_{B_i \times_B B_j}, \mathcal{O}_{X_{B_i \times_B B_j}}^*) = \Gamma(B_i \times_B B_j, \mathcal{O}_{B_i \times_B B_j}^*)\] (equality by our assumption on pushforward of structure sheaves). Of course these elements satisfy the cocycle condition on \(B_i \times_B B_j \times_B B_k\), hence they define a descent datum on invertible sheaves for the fppf covering \(\{B_i \to B\}\). By Descent, Proposition 023T there is an invertible \(\mathcal{O}_B\)-module \(\mathcal{N}\) with trivializations over \(B_i\) whose associated descent datum is \(\{f_{ij}\}\). (The proposition applies because \(B\) is a scheme by the replacement performed at the start of the proof.) Then \(f^*\mathcal{N} \cong \mathcal{L}\) as the functor from descent data to modules is fully faithful.
Proposition
Let \(\Lambda\) be a Noetherian ring, let \(S = \Spec(\Lambda)\), and let \(f : X \to S\) be a flat morphism of schemes. Assume that \[\mathcal{O}_T \longrightarrow f_{T, *}\mathcal{O}_{X_T}\] is an isomorphism for every scheme \(T\) of finite type over \(S\). Denote by \(\mathcal{A}_\Lambda\) the category of \(\Lambda\)-algebras which have finite length as \(\Lambda\)-modules. Then the functor \[F : \mathcal{A}_\Lambda \longrightarrow \textit{Sets}, \qquad A \longmapsto \Picardfunctor_{X/S}(\Spec(A))\] is strictly prorepresentable.
For \(A \in \mathcal{A}_\Lambda\), put \(A_\epsilon = A[\epsilon]/(\epsilon^2)\). If \(\xi \in F(A)\), then the fibre \[F(A_\epsilon, \xi) = \{\xi' \in F(A_\epsilon) \mid \xi'|_A = \xi\}\] is canonically isomorphic to \[H^1(X_A, \mathcal{O}_{X_A}).\]
Proof
The functor \(F\) commutes with finite products. We verify the two local equalizer tests of Formal Deformation Theory, Theorem formal-defos-theorem-grothendieck-levelt-prorepresentability.
Let \(A \to B\) be a map in \(\mathcal{A}_\Lambda\) with \(A\) local. If \(B\) is a nonzero finite free \(A\)-module, then \(\Spec(B) \to \Spec(A)\) is an fppf cover. The required equalizer is therefore the sheaf condition in the definition of \(\Picardfunctor_{X/S}\).
Suppose that \(A \to B\) is injective and \(B/A\) has length \(1\). If \(\mathfrak m\) is the maximal ideal of \(A\), then \(\mathfrak mB = \mathfrak m\) by Formal Deformation Theory, Lemma formal-defos-lemma-length-one-cech-equalizer. Put \[k = A/\mathfrak m, \qquad C = B/\mathfrak m.\] Thus \(A = B \times_C k\) and \(k \to C\) is faithfully flat. We recall why invertible modules have effective descent in this situation. If \(U = \Spec(R)\) is an affine open of \(X_A\), then \(R\) is flat over \(A\) and tensoring the cartesian square of rings with \(R\) gives \[R = (R \otimes_A B) \mathbin{\times}_{R \otimes_A C} (R \otimes_A k).\] Reduce the descent datum on an invertible module over \(U_B\) modulo \(\mathfrak m\). Faithfully flat descent along \(k \to C\) gives an invertible module over \(U_k\). More on Algebra, Lemmas 0D2H and 0D2I patch the two modules to a finite flat module over \(U\); it has rank \(1\) after pullback to \(U_B\) and \(U_k\), whose images cover \(U\), and hence is invertible. Full faithfulness in Descent, Lemma descent-lemma-equalizer-flat-module-descent makes these local descents unique and hence glues them on overlaps. This is the scheme version of the argument in Descent, Lemma descent-lemma-artinian-small-effective-descent-flat-modules.
It remains to pass from the Picard groupoid to the fppf sheaf of its isomorphism classes. For every finite type \(S\)-scheme \(D\) and every invertible module \(\mathcal{L}\) on \(X_D\), the hypothesis gives \[\operatorname{Aut}(\mathcal{L}) = \Gamma(X_D, \mathcal{O}_{X_D}^*) = \Gamma(D, \mathcal{O}_D^*).\] Hence the automorphisms in the preceding descent argument are exactly the automorphisms pulled back from the base. Applying the same effective descent argument to invertible modules on the bases shows that quotienting by these automorphisms commutes with the equalizer. Consequently \[F(A) \longrightarrow F(B) \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} F(B \otimes_A B)\] is an equalizer. The cited prorepresentability criterion now proves the first assertion.
For the tangent assertion, \(X_A \subset X_{A_\epsilon}\) is a first order thickening with ideal \(\mathcal{O}_{X_A}\epsilon\). More on Morphisms, Lemma 0C6R gives the corresponding exact sequence of units, first cohomology, and Picard groups. Our hypothesis identifies the map on global units with the surjection \[A_\epsilon^* \longrightarrow A^*.\] It follows that the kernel of restriction on Picard classes is canonically \[H^1(X_A, \mathcal{O}_{X_A}).\] Translation by the distinguished lift of \(\xi\) obtained by base change identifies the fibre in the statement with this kernel. The construction is fppf local on \(\Spec(A)\) and therefore applies to the sheafified Picard functor.
Remark
The parenthetical reduction of the global-functions hypothesis in the cited source has a correction: the phrase “if \(f\) is proper” must read “if \(f\) is proper and separable”. The proposition above uses the base-change hypothesis itself, so this correction does not enter its proof.
Lemma
In Situation 0D25 let \(\sigma : B \to X\) be a section. Assume that \(\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}\) is an isomorphism for all \(T\) over \(B\). Then \[0 \to \Pic(T) \to \Pic(X_T) \to \Picardfunctor_{X/B}(T) \to 0\] is a split exact sequence with splitting given by \(\sigma_T^* : \Pic(X_T) \to \Pic(T)\).
Proof
Denote \(K(T) = \Ker(\sigma_T^* : \Pic(X_T) \to \Pic(T))\). Since \(\sigma\) is a section of \(f\) we see that \(\Pic(X_T)\) is the direct sum of \(\Pic(T)\) and \(K(T)\). Thus by Lemma 0D27 we see that \(K(T) \subset \Picardfunctor_{X/B}(T)\) for all \(T\). Moreover, it is clear from the construction that \(\Picardfunctor_{X/B}\) is the sheafification of the presheaf \(K\). To finish the proof it suffices to show that \(K\) satisfies the sheaf condition for fppf coverings which we do in the next paragraph.
Let \(\{T_i \to T\}\) be an fppf covering. Let \(\mathcal{L}_i\) be elements of \(K(T_i)\) which map to the same elements of \(K(T_i \times_T T_j)\) for all \(i\) and \(j\). Choose an isomorphism \(\alpha_i : \mathcal{O}_{T_i} \to \sigma_{T_i}^*\mathcal{L}_i\) for all \(i\). Choose an isomorphism \[\varphi_{ij} : \mathcal{L}_i|_{X_{T_i \times_T T_j}} \longrightarrow \mathcal{L}_j|_{X_{T_i \times_T T_j}}\] If the map \[\alpha_j|_{T_i \times_T T_j} \circ \sigma_{T_i \times_T T_j}^*\varphi_{ij} \circ \alpha_i|_{T_i \times_T T_j} : \mathcal{O}_{T_i \times_T T_j} \to \mathcal{O}_{T_i \times_T T_j}\] is not equal to multiplication by \(1\) but some \(u_{ij}\), then we can scale \(\varphi_{ij}\) by \(u_{ij}^{-1}\) to correct this. Having done this, consider the self map \[\varphi_{ki}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{jk}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{ij}|_{X_{T_i \times_T T_j \times_T T_k}} \quad\text{on}\quad \mathcal{L}_i|_{X_{T_i \times_T T_j \times_T T_k}}\] which is given by multiplication by some section \(f_{ijk}\) of the structure sheaf of \(X_{T_i \times_T T_j \times_T T_k}\). By our choice of \(\varphi_{ij}\) we see that the pullback of this map by \(\sigma\) is equal to multiplication by \(1\). By our assumption on functions on \(X\), we see that \(f_{ijk} = 1\). Thus we obtain a descent datum for the fppf covering \(\{X_{T_i} \to X_T\}\). By Descent on Spaces, Proposition 04W8 there is an invertible \(\mathcal{O}_{X_T}\)-module \(\mathcal{L}\) and an isomorphism \(\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}\) whose pullback to \(X_{T_i}\) recovers \((\mathcal{L}_i, \alpha_i)\) (small detail omitted). Thus \(\mathcal{L}\) defines an object of \(K(T)\) as desired.
Lemma
In Situation 0D25, assume that \(\mathcal{O}_U \to f_{U, *}\mathcal{O}_{X_U}\) is an isomorphism for every scheme \(U\) over \(B\). Let \(T \to B\) be a scheme and let \(T' \to T\) be a surjective fppf morphism such that \(X_{T'} \to T'\) has a section. Set \(T'' = T' \times_T T'\). Then the diagram \[\Picardfunctor_{X/B}(T) \longrightarrow \Pic(X_{T'})/\Pic(T') \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} \Pic(X_{T''})/\Pic(T'')\] is an equalizer. In particular, every element of \(\Picardfunctor_{X/B}(T)\) is represented after pullback to \(T'\) by an invertible module on \(X_{T'}\). If \(X_T\) is a scheme and \(X_T \to T\) is a surjective fppf morphism, then one may take \(T' = X_T\); the required section after base change is the diagonal.
Proof
Since \(\Picardfunctor_{X/B}\) is an fppf sheaf, the diagram \[\Picardfunctor_{X/B}(T) \longrightarrow \Picardfunctor_{X/B}(T') \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} \Picardfunctor_{X/B}(T'')\] is an equalizer. The section over \(T'\) and either of its pullbacks to \(T''\) allow us to apply Lemma 0D28 over \(T'\) and \(T''\). This identifies the last two terms with the two quotients in the statement. These identifications are compatible with pullback because they are induced by the canonical maps from Picard groups to the Picard functor.
In Situation 0D25 let \(\sigma : B \to X\) be a section. We denote \(\Picardstack_{X/B, \sigma}\) the category defined as follows:
An object is a quadruple \((T, h, \mathcal{L}, \alpha)\), where \((T, h, \mathcal{L})\) is an object of \(\Picardstack_{X/B}\) over \(T\) and \(\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}\) is an isomorphism.
A morphism \((g, \varphi) : (T, h, \mathcal{L}, \alpha) \to (T', h', \mathcal{L}', \alpha')\) is given by a morphism of schemes \(g : T \to T'\) with \(h = h' \circ g\) and an isomorphism \(\varphi : (g')^*\mathcal{L}' \to \mathcal{L}\) such that \(\sigma_T^*\varphi \circ g^*\alpha' = \alpha\). Here \(g' : X_T \to X_{T'}\) is the base change of \(g\).
There is a natural faithful forgetful functor \[\Picardstack_{X/B, \sigma} \longrightarrow \Picardstack_{X/B}\] In this way we view \(\Picardstack_{X/B, \sigma}\) as a category over \((\Sch/S)_{fppf}\).
Lemma
In Situation 0D25 let \(\sigma : B \to X\) be a section. Then \(\Picardstack_{X/B, \sigma}\) as defined above is a stack in groupoids over \((\Sch/S)_{fppf}\).
Proof
We already know that \(\Picardstack_{X/B}\) is a stack in groupoids over \((\Sch/S)_{fppf}\) by Examples of Stacks, Lemma 04WN. Let us show descent for objects for \(\Picardstack_{X/B, \sigma}\). Let \(\{T_i \to T\}\) be an fppf covering and let \(\xi_i = (T_i, h_i, \mathcal{L}_i, \alpha_i)\) be an object of \(\Picardstack_{X/B, \sigma}\) lying over \(T_i\), and let \(\varphi_{ij} : \text{pr}_0^*\xi_i \to \text{pr}_1^*\xi_j\) be a descent datum. Applying the result for \(\Picardstack_{X/B}\) we see that we may assume we have an object \((T, h, \mathcal{L})\) of \(\Picardstack_{X/B}\) over \(T\) which pulls back to \(\xi_i\) for all \(i\). Then we get \[\alpha_i : \mathcal{O}_{T_i} \to \sigma_{T_i}^*\mathcal{L}_i = (T_i \to T)^*\sigma_T^*\mathcal{L}\] Since the maps \(\varphi_{ij}\) are compatible with the \(\alpha_i\) we see that \(\alpha_i\) and \(\alpha_j\) pullback to the same map on \(T_i \times_T T_j\). By descent of quasi-coherent sheaves (Descent, Proposition 023T), we see that the \(\alpha_i\) are the restriction of a single map \(\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}\) as desired. We omit the proof of descent for morphisms.
Lemma
In Situation 0D25 let \(\sigma : B \to X\) be a section. The morphism \(\Picardstack_{X/B, \sigma} \to \Picardstack_{X/B}\) is representable, surjective, and smooth.
Proof
Let \(T\) be a scheme and let \((\Sch/T)_{fppf} \to \Picardstack_{X/B}\) be given by the object \(\xi = (T, h, \mathcal{L})\) of \(\Picardstack_{X/B}\) over \(T\). We have to show that \[(\Sch/T)_{fppf} \times_{\xi, \Picardstack_{X/B}} \Picardstack_{X/B, \sigma}\] is representable by a scheme \(V\) and that the corresponding morphism \(V \to T\) is surjective and smooth. See Algebraic Stacks, Sections 04ST, 04SX, and 03YJ. The forgetful functor \(\Picardstack_{X/B, \sigma} \to \Picardstack_{X/B}\) is faithful on fibre categories and for \(T'/T\) the set of isomorphism classes is the set of isomorphisms \[\alpha' : \mathcal{O}_{T'} \longrightarrow (T' \to T)^*\sigma_T^*\mathcal{L}\] See Algebraic Stacks, Lemma 02ZY. We know this functor is representable by an affine scheme \(U\) of finite presentation over \(T\) by Proposition 08K9 (applied to \(\text{id} : T \to T\) and \(\mathcal{O}_T\) and \(\sigma^*\mathcal{L}\)). Working Zariski locally on \(T\) we may assume that \(\sigma_T^*\mathcal{L}\) is isomorphic to \(\mathcal{O}_T\) and then we see that our functor is representable by \(\mathbf{G}_m \times T\) over \(T\). Hence \(U \to T\) Zariski locally on \(T\) looks like the projection \(\mathbf{G}_m \times T \to T\) which is indeed smooth and surjective.
Lemma
In Situation 0D25 let \(\sigma : B \to X\) be a section. If \(\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}\) is an isomorphism for all \(T\) over \(B\), then \(\Picardstack_{X/B, \sigma} \to (\Sch/S)_{fppf}\) is fibred in setoids with set of isomorphism classes over \(T\) given by \[\coprod\nolimits_{h : T \to B} \Ker(\sigma_T^* : \Pic(X \times_{B, h} T) \to \Pic(T))\]
Proof
If \(\xi = (T, h, \mathcal{L}, \alpha)\) is an object of \(\Picardstack_{X/B, \sigma}\) over \(T\), then an automorphism \(\varphi\) of \(\xi\) is given by multiplication with an invertible global section \(u\) of the structure sheaf of \(X_T\) such that moreover \(\sigma_T^*u = 1\). Then \(u = 1\) by our assumption that \(\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}\) is an isomorphism. Hence \(\Picardstack_{X/B, \sigma}\) is fibred in setoids over \((\Sch/S)_{fppf}\). Given \(T\) and \(h : T \to B\) the set of isomorphism classes of pairs \((\mathcal{L}, \alpha)\) is the same as the set of isomorphism classes of \(\mathcal{L}\) with \(\sigma_T^*\mathcal{L} \cong \mathcal{O}_T\) (isomorphism not specified). This is clear because any two choices of \(\alpha\) differ by a global unit on \(T\) and this is the same thing as a global unit on \(X_T\).
Proposition
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Assume that
\(f\) is flat, of finite presentation, and proper, and
\(\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}\) is an isomorphism for all schemes \(T\) over \(B\).
Then \(\Picardfunctor_{X/B}\) is an algebraic space.
In the situation of the proposition the algebraic stack \(\Picardstack_{X/B}\) is a gerbe over the algebraic space \(\Picardfunctor_{X/B}\). After developing the general theory of gerbes, this provides a shorter proof of the proposition (but using more general theory).
Proof
There exists a surjective, flat, finitely presented morphism \(B' \to B\) of algebraic spaces such that the base change \(X' = X \times_B B'\) over \(B'\) has a section: namely, we can take \(B' = X\). Observe that \(\Picardfunctor_{X'/B'} = B' \times_B \Picardfunctor_{X/B}\). Hence \(\Picardfunctor_{X'/B'} \to \Picardfunctor_{X/B}\) is representable by algebraic spaces, surjective, flat, and finitely presented. Hence, if we can show that \(\Picardfunctor_{X'/B'}\) is an algebraic space, then it follows that \(\Picardfunctor_{X/B}\) is an algebraic space by Bootstrap, Theorem 04S6. In this way we reduce to the case described in the next paragraph.
In addition to the assumptions of the proposition, assume that we have a section \(\sigma : B \to X\). By Proposition 0D04 we see that \(\Picardstack_{X/B}\) is an algebraic stack. By Lemma 0D2A and Algebraic Stacks, Lemma 05UM we see that \(\Picardstack_{X/B, \sigma}\) is an algebraic stack. By Lemma 0D2B and Algebraic Stacks, Lemma 02ZX we see that \(T \mapsto \Ker(\sigma_T^* : \Pic(X_T) \to \Pic(T))\) is an algebraic space. By Lemma 0D28 this functor is the same as \(\Picardfunctor_{X/B}\).
Lemma
With assumptions and notation as in Proposition 0D2C. Then the diagonal \(\Picardfunctor_{X/B} \to \Picardfunctor_{X/B} \times_B \Picardfunctor_{X/B}\) is representable by immersions. In other words, \(\Picardfunctor_{X/B} \to B\) is locally separated.
Proof
Let \(T\) be a scheme over \(B\) and let \(s, t \in \Picardfunctor_{X/B}(T)\). We want to show that there exists a locally closed subscheme \(Z \subset T\) such that \(s|_Z = t|_Z\) and such that a morphism \(T' \to T\) factors through \(Z\) if and only if \(s|_{T'} = t|_{T'}\).
We first reduce the general problem to the case where \(s\) and \(t\) come from invertible modules on \(X_T\). We suggest the reader skip this step. Choose an fppf covering \(\{T_i \to T\}_{i \in I}\) such that \(s|_{T_i}\) and \(t|_{T_i}\) come from \(\Pic(X_{T_i})\) for all \(i\). Suppose that we can show the result for all the pairs \(s|_{T_i}, t|_{T_i}\). Then we obtain locally closed subschemes \(Z_i \subset T_i\) with the desired universal property. It follows that \(Z_i\) and \(Z_j\) have the same scheme theoretic inverse image in \(T_i \times_T T_j\). This determines a descend datum on \(Z_i/T_i\). Since \(Z_i \to T_i\) is locally quasi-finite, it follows from More on Morphisms, Lemma 02W8 that we obtain a locally quasi-finite morphism \(Z \to T\) recovering \(Z_i \to T_i\) by base change. Then \(Z \to T\) is an immersion by Descent, Lemma 02YM. Finally, because \(\Picardfunctor_{X/B}\) is an fppf sheaf, we conclude that \(s|_Z = t|_Z\) and that \(Z\) satisfies the universal property mentioned above.
Assume \(s\) and \(t\) come from invertible modules \(\mathcal{V}\), \(\mathcal{W}\) on \(X_T\). Set \(\mathcal{L} = \mathcal{V} \otimes \mathcal{W}^{\otimes -1}\) We are looking for a locally closed subscheme \(Z\) of \(T\) such that \(T' \to T\) factors through \(Z\) if and only if \(\mathcal{L}_{X_{T'}}\) is the pullback of an invertible sheaf on \(T'\), see Lemma 0D27. Hence the existence of \(Z\) follows from More on Morphisms of Spaces, Lemma 0D23.
Relative morphisms
We continue the discussion from Criteria for Representability, Section 05Y0. In that section, starting with a scheme \(S\) and morphisms of algebraic spaces \(Z \to B\) and \(X \to B\) over \(S\) we constructed a functor \[\mathit{Mor}_B(Z, X) : (\Sch/B)^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \{f : Z_T \to X_T\}\] We sometimes think of \(\mathit{Mor}_B(Z, X)\) as a functor \((\Sch/S)^{opp} \to \textit{Sets}\) endowed with a morphism \(\mathit{Mor}_B(Z, X) \to B\). Namely, if \(T\) is a scheme over \(S\), then an element of \(\mathit{Mor}_B(Z, X)(T)\) is a pair \((f, h)\) where \(h\) is a morphism \(h : T \to B\) and \(f : Z \times_{B, h} T \to X \times_{B, h} T\) is a morphism of algebraic spaces over \(T\). In particular, when we say that \(\mathit{Mor}_B(Z, X)\) is an algebraic space, we mean that the corresponding functor \((\Sch/S)^{opp} \to \textit{Sets}\) is an algebraic space.
Lemma
Let \(S\) be a scheme. Consider morphisms of algebraic spaces \(Z \to B\) and \(X \to B\) over \(S\). If \(X \to B\) is separated and \(Z \to B\) is of finite presentation, flat, and proper, then there is a natural injective transformation of functors \[\mathit{Mor}_B(Z, X) \longrightarrow \Hilbfunctor_{Z \times_B X/B}\] which maps a morphism \(f : Z_T \to X_T\) to its graph.
Proof
Given a scheme \(T\) over \(B\) and a morphism \(f_T : Z_T \to X_T\) over \(T\), the graph of \(f\) is the morphism \(\Gamma_f = (\text{id}, f) : Z_T \to Z_T \times_T X_T = (Z \times_B X)_T\). Recall that being separated, flat, proper, or finite presentation are properties of morphisms of algebraic spaces which are stable under base change (Morphisms of Spaces, Lemmas 03KL, 03MO, 04WP, and 03XR). Hence \(\Gamma_f\) is a closed immersion by Morphisms of Spaces, Lemma 03KO. Moreover, \(\Gamma_f(Z_T)\) is flat, proper, and of finite presentation over \(T\). Thus \(\Gamma_f(Z_T)\) defines an element of \(\Hilbfunctor_{Z \times_B X/B}(T)\). To show the transformation is injective it suffices to show that two morphisms with the same graph are the same. This is true because if \(Y \subset (Z \times_B X)_T\) is the graph of a morphism \(f\), then we can recover \(f\) by using the inverse of \(\text{pr}_1|_Y : Y \to Z_T\) composed with \(\text{pr}_2|_Y\).
Lemma
Assumption and notation as in Lemma 0D1A. The transformation \(\mathit{Mor}_B(Z, X) \longrightarrow \Hilbfunctor_{Z \times_B X/B}\) is representable by open immersions.
Proof
Let \(T\) be a scheme over \(B\) and let \(Y \subset (Z \times_B X)_T\) be an element of \(\Hilbfunctor_{Z \times_B X/B}(T)\). Then we see that \(Y\) is the graph of a morphism \(Z_T \to X_T\) over \(T\) if and only if \(k = \text{pr}_1|_Y : Y \to Z_T\) is an isomorphism. By More on Morphisms of Spaces, Lemma 05XD there exists an open subscheme \(V \subset T\) such that for any morphism of schemes \(T' \to T\) we have \(k_{T'} : Y_{T'} \to Z_{T'}\) is an isomorphism if and only if \(T' \to T\) factors through \(V\). This proves the lemma.
Proposition
Let \(S\) be a scheme. Let \(Z \to B\) and \(X \to B\) be morphisms of algebraic spaces over \(S\). Assume \(X \to B\) is of finite presentation and separated and \(Z \to B\) is of finite presentation, flat, and proper. Then \(\mathit{Mor}_B(Z, X)\) is an algebraic space locally of finite presentation over \(B\).
Proof
The stack of algebraic spaces
This section continuous the discussion started in Examples of Stacks, Sections 04SP, 04UC, and 04UH. Working over \(\mathbf{Z}\), the discussion therein shows that we have a stack in groupoids \[p'_{ft} : \Spacesstack'_{ft} \longrightarrow \Sch_{fppf}\] parametrizing (nonflat) families of finite type algebraic spaces. More precisely, an object3 of \(\Spacesstack'_{ft}\) is a finite type morphism \(X \to S\) from an algebraic space \(X\) to a scheme \(S\) and a morphism \((X' \to S') \to (X \to S)\) is given by a pair \((f, g)\) where \(f : X' \to X\) is a morphism of algebraic spaces and \(g : S' \to S\) is a morphism of schemes which fit into a commutative diagram \[\xymatrix{ X' \ar[d] \ar[r]_f & X \ar[d] \\ S' \ar[r]^g & S }\] inducing an isomorphism \(X' \to S' \times_S X\), in other words, the diagram is cartesian in the category of algebraic spaces. The functor \(p'_{ft}\) sends \((X \to S)\) to \(S\) and sends \((f, g)\) to \(g\). We define a full subcategory \[\Spacesstack'_{fp, flat, proper} \subset \Spacesstack'_{ft}\] consisting of objects \(X \to S\) of \(\Spacesstack'_{ft}\) such that \(X \to S\) is of finite presentation, flat, and proper. We denote \[p'_{fp, flat, proper} : \Spacesstack'_{fp, flat, proper} \longrightarrow \Sch_{fppf}\] the restriction of the functor \(p'_{ft}\) to the indicated subcategory. We first review the results already obtained in the references listed above, and then we start adding further results.
Lemma
The category \(\Spacesstack'_{ft}\) is fibred in groupoids over \(\Sch_{fppf}\). The same is true for \(\Spacesstack'_{fp, flat, proper}\).
Proof
We have seen this in Examples of Stacks, Section 04UH for the case of \(\Spacesstack'_{ft}\) and this easily implies the result for the other case. However, let us also prove this directly by checking conditions (1) and (2) of Categories, Definition 003T.
Condition (1). Let \(X \to S\) be an object of \(\Spacesstack'_{ft}\) and let \(S' \to S\) be a morphism of schemes. Then we set \(X' = S' \times_S X\). Note that \(X' \to S'\) is of finite type by Morphisms of Spaces, Lemma 03XH. to obtain a morphism \((X' \to S') \to (X \to S)\) lying over \(S' \to S\). Argue similarly for the other case using Morphisms of Spaces, Lemmas 03XR, 03MO, and 04WP.
Condition (2). Consider morphisms \((f, g) : (X' \to S') \to (X \to S)\) and \((a, b) : (Y \to T) \to (X \to S)\) of \(\Spacesstack'_{ft}\). Given a morphism \(h : T \to S'\) with \(g \circ h = b\) we have to show there is a unique morphism \((k, h) : (Y \to T) \to (X' \to S')\) of \(\Spacesstack'_{ft}\) such that \((f, g) \circ (k, h) = (a, b)\). This is clear from the fact that \(X' = S' \times_S X\). The same therefore works for any full subcategory of \(\Spacesstack'_{ft}\) satisfying (1).
Lemma
The diagonal \[\Delta : \Spacesstack'_{fp, flat, proper} \longrightarrow \Spacesstack'_{fp, flat, proper} \times \Spacesstack'_{fp, flat, proper}\] is representable by algebraic spaces.
Proof
We will use criterion (2) of Algebraic Stacks, Lemma 045G. Let \(S\) be a scheme and let \(X\) and \(Y\) be algebraic spaces of finite presentation over \(S\), flat over \(S\), and proper over \(S\). We have to show that the functor \[\mathit{Isom}_S(X, Y) : (\Sch/S)_{fppf} \longrightarrow \textit{Sets}, \quad T \longmapsto \{f : X_T \to Y_T \text{ isomorphism}\}\] is an algebraic space. An elementary argument shows that \(\mathit{Isom}_S(X, Y)\) sits in a fibre product \[\xymatrix{ \mathit{Isom}_S(X, Y) \ar[r] \ar[d] & S \ar[d]_{(\text{id}, \text{id})} \\ \mathit{Mor}_S(X, Y) \times \mathit{Mor}_S(Y, X) \ar[r] & \mathit{Mor}_S(X, X) \times \mathit{Mor}_S(Y, Y) }\] The bottom arrow sends \((\varphi, \psi)\) to \((\psi \circ \varphi, \varphi \circ \psi)\). By Proposition 0D1C the functors on the bottom row are algebraic spaces over \(S\). Hence the result follows from the fact that the category of algebraic spaces over \(S\) has fibre products.
Lemma
The category \(\Spacesstack'_{ft}\) is a stack in groupoids over \(\Sch_{fppf}\). The same is true for \(\Spacesstack'_{fp, flat, proper}\).
Proof
The reason this lemma holds is the slogan: any fppf descent datum for algebraic spaces is effective, see Bootstrap, Section 04SJ. More precisely, the lemma for \(\Spacesstack'_{ft}\) follows from Examples of Stacks, Lemma 04UD as we saw in Examples of Stacks, Section 04UH. However, let us review the proof. We need to check conditions (1), (2), and (3) of Stacks, Definition 02ZI.
Property (1) we have seen in Lemma 0D1E.
Property (2) follows from Lemma 0D1F in the case of \(\Spacesstack'_{fp, flat, proper}\). In the case of \(\Spacesstack'_{ft}\) it follows from Examples of Stacks, Lemma 04UA (and this is really the “correct” reference).
Condition (3) for \(\Spacesstack'_{ft}\) is checked as follows. Suppose given
an fppf covering \(\{U_i \to U\}_{i \in I}\) in \(\Sch_{fppf}\),
for each \(i \in I\) an algebraic space \(X_i\) of finite type over \(U_i\), and
for each \(i, j \in I\) an isomorphism \(\varphi_{ij} : X_i \times_U U_j \to U_i \times_U X_j\) of algebraic spaces over \(U_i \times_U U_j\) satisfying the cocycle condition over \(U_i \times_U U_j \times_U U_k\).
We have to show there exists an algebraic space \(X\) of finite type over \(U\) and isomorphisms \(X_{U_i} \cong X_i\) over \(U_i\) recovering the isomorphisms \(\varphi_{ij}\). This follows from Bootstrap, Lemma 0ADV part (2). By Descent on Spaces, Lemma 041U we see that \(X \to U\) is of finite type. In the case of \(\Spacesstack'_{fp, flat, proper}\) one additionally uses Descent on Spaces, Lemma 041V, 041W, and 0422 in the last step.
Sanity check: the stacks \(\Spacesstack'_{ft}\) and \(\Spacesstack'_{fp, flat, proper}\) play the same role among algebraic spaces.
Lemma
Let \(T\) be an algebraic space over \(\mathbf{Z}\). Let \(\mathcal{S}_T\) denote the corresponding algebraic stack (Algebraic Stacks, Sections 04SU, 02ZV, and 03YR). We have an equivalence of categories \[\left\{ \begin{matrix} \text{morphisms of algebraic spaces }\\ X \to T\text{ of finite type} \end{matrix} \right\} \longrightarrow \Mor_{\textit{Cat}/\Sch_{fppf}}(\mathcal{S}_T, \Spacesstack'_{ft})\] and an equivalence of categories \[\left\{ \begin{matrix} \text{morphisms of algebraic spaces }X \to T\\ \text{of finite presentation, flat, and proper} \end{matrix} \right\} \longrightarrow \Mor_{\textit{Cat}/\Sch_{fppf}}(\mathcal{S}_T, \Spacesstack'_{fp, flat, proper})\]
Proof
We are going to deduce this lemma from the fact that it holds for schemes (essentially by construction of the stacks) and the fact that fppf descent data for algebraic spaces over algerbaic spaces are effective. We strongly encourage the reader to skip the proof.
The construction from left to right in either arrow is straightforward: given \(X \to T\) of finite type the functor \(\mathcal{S}_T \to \Spacesstack'_{ft}\) assigns to \(U/T\) the base change \(X_U \to U\). We will explain how to construct a quasi-inverse.
If \(T\) is a scheme, then there is a quasi-inverse by the \(2\)-Yoneda lemma, see Categories, Lemma 004B. Let \(p : U \to T\) be a surjective étale morphism where \(U\) is a scheme. Let \(R = U \times_T U\) with projections \(s, t : R \to U\). Observe that we obtain morphisms \[\xymatrix{ \mathcal{S}_{U \times_T U \times_T U} \ar@<2ex>[r] \ar[r] \ar@<-2ex>[r] & \mathcal{S}_R \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{S}_U \ar[r] & \mathcal{S}_T }\] satisfying various compatibilities (on the nose).
Let \(G : \mathcal{S}_T \to \Spacesstack'_{ft}\) be a functor over \(\Sch_{fppf}\). The restriction of \(G\) to \(\mathcal{S}_U\) via the map displayed above corresponds to a finite type morphism \(X_U \to U\) of algebraic spaces via the \(2\)-Yoneda lemma. Since \(p \circ s = p \circ t\) we see that \(R \times_{s, U} X_U\) and \(R \times_{t, U} X_U\) both correspond to the restriction of \(G\) to \(\mathcal{S}_R\). Thus we obtain a canonical isomorphism \(\varphi : X_U \times_{U, t} R \to R \times_{s, U} X_U\) over \(R\). This isomorphism satisfies the cocycle condition by the various compatibilities of the diagram given above. Thus a descent datum which is effective by Bootstrap, Lemma 0ADV part (2). In other words, we obtain an object \(X \to T\) of the right hand side category. We omit checking the construction \(G \leadsto X\) is functorial and that it is quasi-inverse to the other construction. In the case of \(\Spacesstack'_{fp, flat, proper}\) one additionally uses Descent on Spaces, Lemma 041V, 041W, and 0422 in the last step to see that \(X \to T\) is of finite presentation, flat, and proper.
Remark
Let \(B\) be an algebraic space over \(\Spec(\mathbf{Z})\). Let \(B\textit{-Spaces}'_{ft}\) be the category consisting of pairs \((X \to S, h : S \to B)\) where \(X \to S\) is an object of \(\Spacesstack'_{ft}\) and \(h : S \to B\) is a morphism. A morphism \((X' \to S', h') \to (X \to S, h)\) in \(B\textit{-Spaces}'_{ft}\) is a morphism \((f, g)\) in \(\Spacesstack'_{ft}\) such that \(h \circ g = h'\). In this situation the diagram \[\xymatrix{ B\textit{-Spaces}'_{ft} \ar[r] \ar[d] & \Spacesstack'_{ft} \ar[d] \\ (\Sch/B)_{fppf} \ar[r] & \Sch_{fppf} }\] is \(2\)-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case \(\Spacesstack'_{ft}\) to the case of families over a given base algebraic space. Of course, a similar construction works for \(B\textit{-Spaces}'_{fp, flat, proper}\)
Lemma
The stack \(p'_{fp, flat, proper} : \Spacesstack'_{fp, flat, proper} \to \Sch_{fppf}\) is limit preserving (Artin’s Axioms, Definition 07XL).
Proof
Let \(T = \lim T_i\) be the limits of a directed inverse system of affine schemes. By Limits of Spaces, Lemma 07SK the category of algebraic spaces of finite presentation over \(T\) is the colimit of the categories of algebraic spaces of finite presentation over \(T_i\). To finish the proof use that flatness and properness descends through the limit, see Limits of Spaces, Lemmas 08K0 and 08K1.
Lemma
Let \[\xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' }\] be a pushout in the category of schemes where \(T \to T'\) is a thickening and \(T \to S\) is affine, see More on Morphisms, Lemma 07RT. Then the functor on fibre categories \[\begin{matrix} \Spacesstack'_{fp, flat, proper, S'} \\ \downarrow \\ \Spacesstack'_{fp, flat, proper, S} \times_{\Spacesstack'_{fp, flat, proper, T}} \Spacesstack'_{fp, flat, proper, T'} \end{matrix}\] is an equivalence.
Proof
The functor is an equivalence if we drop “proper” from the list of conditions and replace “of finite presentation” by “locally of finite presentation”, see Pushouts of Spaces, Lemma 07W3. Thus it suffices to show that given a morphism \(X' \to S'\) of an algebraic space to \(S'\) which is flat and locally of finite presentation, then \(X' \to S'\) is proper if and only if \(S \times_{S'} X' \to S\) and \(T' \times_{S'} X' \to T'\) are proper. One implication follows from the fact that properness is preserved under base change (Morphisms of Spaces, Lemma 04WP) and the other from the fact that properness of \(S \times_{S'} X' \to S\) implies properness of \(X' \to S'\) by More on Morphisms of Spaces, Lemma 09ZZ.
Lemma
Let \(k\) be a field and let \(x = (X \to \Spec(k))\) be an object of \(\mathcal{X} = \Spacesstack'_{fp, flat, proper}\) over \(\Spec(k)\).
If \(k\) is of finite type over \(\mathbf{Z}\), then the vector spaces \(T\mathcal{F}_{\mathcal{X}, k, x}\) and \(\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x})\) (see Artin’s Axioms, Section 07WY) are finite dimensional, and
in general the vector spaces \(T_x(k)\) and \(\text{Inf}_x(k)\) (see Artin’s Axioms, Section 07Y6) are finite dimensional.
Proof
The discussion in Artin’s Axioms, Section 07WY only applies to fields of finite type over the base scheme \(\Spec(\mathbf{Z})\). Our stack satisfies (RS*) by Lemma 0D1J and we may apply Artin’s Axioms, Lemma 07Y9 to get the vector spaces \(T_x(k)\) and \(\text{Inf}_x(k)\) mentioned in (2). Moreover, in the finite type case these spaces agree with the ones mentioned in (1) by Artin’s Axioms, Remark 0D18. With this out of the way we can start the proof. Observe that the first order thickening \(\Spec(k) \to \Spec(k[\epsilon]) = \Spec(k[k])\) has conormal module \(k\). Hence the formula in Deformation Theory, Lemma 0D17 describing infinitesimal deformations of \(X\) and infinitesimal automorphisms of \(X\) become \[T_x(k) = \Ext^1_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) \quad\text{and}\quad \text{Inf}_x(k) = \Ext^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X)\] By More on Morphisms of Spaces, Lemma 0D0Z and the fact that \(X\) is Noetherian, we see that \(\NL_{X/k}\) has coherent cohomology sheaves zero except in degrees \(0\) and \(-1\). By Derived Categories of Spaces, Lemma 0D0T the displayed \(\Ext\)-groups are finite \(k\)-vector spaces and the proof is complete.
Beware that openness of versality (as proved in the next lemma) is a bit strange because our stack does not satisfy formal effectiveness, see Examples, Section 0D1Q. Later we will apply the openness of versality to suitable substacks of \(\Spacesstack'_{fp, flat, proper}\) which do satisfy formal effectiveness to conclude that these stacks are algebraic.
Lemma
The stack in groupoids \(\mathcal{X} = \Spacesstack'_{fp, flat, proper}\) satisfies openness of versality over \(\Spec(\mathbf{Z})\). Similarly, after base change (Remark 0D1H) openness of versality holds over any Noetherian base scheme \(S\).
Proof
For the “usual” proof of this fact, please see the discussion in the remark following this proof. We will prove this using Artin’s Axioms, Lemma 0CXU. We have already seen that \(\mathcal{X}\) has diagonal representable by algebraic spaces, has (RS*), and is limit preserving, see Lemmas 0D1F, 0D1J, and 0D1I. Hence we only need to see that \(\mathcal{X}\) satisfies the strong formal effectiveness formulated in Artin’s Axioms, Lemma 0CXU.
Let \((R_n)\) be an inverse system of rings such that \(R_n \to R_m\) is surjective with square zero kernel for all \(n \geq m\). Let \(X_n \to \Spec(R_n)\) be a finitely presented, flat, proper morphism where \(X_n\) is an algebraic space and let \(X_{n + 1} \to X_n\) be a morphism over \(\Spec(R_{n + 1})\) inducing an isomorphism \(X_n = X_{n + 1} \times_{\Spec(R_{n + 1})} \Spec(R_n)\). We have to find a flat, proper, finitely presented morphism \(X \to \Spec(\lim R_n)\) whose source is an algebraic space such that \(X_n\) is the base change of \(X\) for all \(n\).
Let \(I_n = \Ker(R_n \to R_1)\). We may think of \((X_1 \subset X_n) \to (\Spec(R_1) \subset \Spec(R_n))\) as a morphism of first order thickenings. (Please read some of the material on thickenings of algebraic spaces in More on Morphisms of Spaces, Section 05ZJ before continuing.) The structure sheaf of \(X_n\) is an extension \[0 \to \mathcal{O}_{X_1} \otimes_{R_1} I_n \to \mathcal{O}_{X_n} \to \mathcal{O}_{X_1} \to 0\] over \(0 \to I_n \to R_n \to R_1\), see More on Morphisms of Spaces, Lemma 06BH. Let’s consider the extension \[0 \to \lim \mathcal{O}_{X_1} \otimes_{R_1} I_n \to \lim \mathcal{O}_{X_n} \to \mathcal{O}_{X_1} \to 0\] over \(0 \to \lim I_n \to \lim R_n \to R_1 \to 0\). The displayed sequence is exact as the \(R^1\lim\) of the system of kernels is zero by Derived Categories of Spaces, Lemma 0D3E. Observe that the map \[\mathcal{O}_{X_1} \otimes_{R_1} \lim I_n \longrightarrow \lim \mathcal{O}_{X_1} \otimes_{R_1} I_n\] induces an isomorphism upon applying the functor \(DQ_X\), see Derived Categories of Spaces, Lemma 0D3F. Hence we obtain a unique extension \[0 \to \mathcal{O}_{X_1} \otimes_{R_1} \lim I_n \to \mathcal{O}' \to \mathcal{O}_{X_1} \to 0\] over \(0 \to \lim I_n \to \lim R_n \to R_1 \to 0\) by the equivalence of categories of Deformation Theory, Lemma 0D3Q. The sheaf \(\mathcal{O}'\) determines a first order thickening of algebraic spaces \(X_1 \subset X\) over \(\Spec(R_1) \subset \Spec(\lim R_n)\) by More on Morphisms of Spaces, Lemma 05ZT. Observe that \(X \to \Spec(\lim R_n)\) is flat by the already used More on Morphisms of Spaces, Lemma 06BH. By More on Morphisms of Spaces, Lemma 0CG6 we see that \(X \to \Spec(\lim R_n)\) is proper and of finite presentation. This finishes the proof.
Remark
Lemma 0D3X can also be shown using either Artin’s Axioms, Lemma 07YZ (as in the first proof of Lemma 08WA), or using an obstruction theory as in Artin’s Axioms, Lemma 0CYF (as in the second proof of Lemma 08WA). In both cases one uses the deformation and obstruction theory developed in Cotangent, Section 08V3 to translate the needed properties of deformations and obstructions into \(\Ext\)-groups to which Derived Categories of Spaces, Lemma 08JR can be applied. The second method (using an obstruction theory and therefore using the full cotangent complex) is perhaps the “standard” method used in most references.
The stack of polarized proper schemes
To study the stack of polarized proper schemes it suffices to work over \(\mathbf{Z}\) as we can later pullback to any scheme or algebraic space we want (see Remark 0D1N).
Situation
We define a category \(\Polarizedstack\) as follows. Objects are pairs \((X \to S, \mathcal{L})\) where
\(X \to S\) is a morphism of schemes which is proper, flat, and of finite presentation, and
\(\mathcal{L}\) is an invertible \(\mathcal{O}_X\)-module which is relatively ample on \(X/S\) (Morphisms, Definition 01VH).
A morphism \((X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})\) between objects is given by a triple \((f, g, \varphi)\) where \(f : X' \to X\) and \(g : S' \to S\) are morphisms of schemes which fit into a commutative diagram \[\xymatrix{ X' \ar[d] \ar[r]_f & X \ar[d] \\ S' \ar[r]^g & S }\] inducing an isomorphism \(X' \to S' \times_S X\), in other words, the diagram is cartesian, and \(\varphi : f^*\mathcal{L} \to \mathcal{L}'\) is an isomorphism. Composition is defined in the obvious manner (see Examples of Stacks, Sections 04SP and 03YL). The forgetful functor \[p : \Polarizedstack \longrightarrow \Sch_{fppf},\quad (X \to S, \mathcal{L}) \longmapsto S\] is how we view \(\Polarizedstack\) as a category over \(\Sch_{fppf}\) (see Section 05X6 for notation).
In the previous section we have done a substantial amount of work on the stack \(\Spacesstack'_{fp, flat, proper}\) of finitely presented, flat, proper algebraic spaces. To use this material we consider the forgetful functor [0D3Y]\[\begin{equation} \Polarizedstack \longrightarrow \Spacesstack'_{fp, flat, proper},\quad (X \to S, \mathcal{L}) \longmapsto (X \to S) \end{equation}\] This functor will be a useful tool in what follows. Observe that if \((X \to S)\) is in the essential image of (0D3Y), then \(X\) and \(S\) are schemes.
Lemma
The category \(\Polarizedstack\) is fibred in groupoids over \(\Spacesstack'_{fp, flat, proper}\). The category \(\Polarizedstack\) is fibred in groupoids over \(\Sch_{fppf}\).
Proof
We check conditions (1) and (2) of Categories, Definition 003T.
Condition (1). Let \((X \to S, \mathcal{L})\) be an object of \(\Polarizedstack\) and let \((X' \to S') \to (X \to S)\) be a morphism of \(\Spacesstack'_{fp, flat, proper}\). Then we let \(\mathcal{L}'\) be the pullback of \(\mathcal{L}\) to \(X'\). Observe that \(X, S, S'\) are schemes, hence \(X'\) is a scheme as well (as the fibre product of schemes). Then \(\mathcal{L}'\) is ample on \(X'/S'\) by Morphisms, Lemma 0893. In this way we obtain a morphism \((X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})\) lying over \((X' \to S') \to (X \to S)\).
Condition (2). Consider morphisms \((f, g, \varphi) : (X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})\) and \((a, b, \psi) : (Y \to T, \mathcal{N}) \to (X \to S, \mathcal{L})\) of \(\Polarizedstack\). Given a morphism \((k, h) : (Y \to T) \to (X' \to S')\) of \(\Spacesstack'_{fp, flat, proper}\) with \((f, g) \circ (k, h) = (a, b)\) we have to show there is a unique morphism \((k, h, \chi) : (Y \to T, \mathcal{N}) \to (X' \to S', \mathcal{L}')\) of \(\Polarizedstack\) such that \((f, g, \varphi) \circ (k, h, \chi) = (a, b, \psi)\). We can just take \[\chi = \psi \circ (k^*\varphi)^{-1}\] This proves condition (2). A composition of functors defining fibred categories defines a fibred category, see Categories, Lemma 09WV. This we see that \(\Polarizedstack\) is fibred in groupoids over \(\Sch_{fppf}\) (strictly speaking we should check the fibre categories are groupoids and apply Categories, Lemma 003V).
Lemma
The category \(\Polarizedstack\) is a stack in groupoids over \(\Spacesstack'_{fp, flat, proper}\) (endowed with the inherited topology, see Stacks, Definition 06NV). The category \(\Polarizedstack\) is a stack in groupoids over \(\Sch_{fppf}\).
Proof
We prove \(\Polarizedstack\) is a stack in groupoids over \(\Spacesstack'_{fp, flat, proper}\) by checking conditions (1), (2), and (3) of Stacks, Definition 02ZI. We have already seen (1) in Lemma 0D3Z.
A covering of \(\Spacesstack'_{fp, flat, proper}\) comes about in the following manner: Let \(X \to S\) be an object of \(\Spacesstack'_{fp, flat, proper}\). Suppose that \(\{S_i \to S\}_{i \in I}\) is a covering of \(\Sch_{fppf}\). Set \(X_i = S_i \times_S X\). Then \(\{(X_i \to S_i) \to (X \to S)\}_{i \in I}\) is a covering of \(\Spacesstack'_{fp, flat, proper}\) and every covering of \(\Spacesstack'_{fp, flat, proper}\) is isomorphic to one of these. Set \(S_{ij} = S_i \times_S S_j\) and \(X_{ij} = S_{ij} \times_S X\) so that \((X_{ij} \to S_{ij}) = (X_i \to S_i) \times_{(X \to S)} (X_j \to S_j)\). Next, suppose that \(\mathcal{L}, \mathcal{N}\) are ample invertible sheaves on \(X/S\) so that \((X \to S, \mathcal{L})\) and \((X \to S, \mathcal{N})\) are two objects of \(\Polarizedstack\) over the object \((X \to S)\). To check descent for morphisms, we assume we have morphisms \((\text{id}, \text{id}, \varphi_i)\) from \((X_i \to S_i, \mathcal{L}|_{X_i})\) to \((X_i \to S_i, \mathcal{N}|_{X_i})\) whose base changes to morphisms from \((X_{ij} \to S_{ij}, \mathcal{L}|_{X_{ij}})\) to \((X_{ij} \to S_{ij}, \mathcal{N}|_{X_{ij}})\) agree. Then \(\varphi_i : \mathcal{L}|_{X_i} \to \mathcal{N}|_{X_i}\) are isomorphisms of invertible modules over \(X_i\) such that \(\varphi_i\) and \(\varphi_j\) restrict to the same isomorphisms over \(X_{ij}\). By descent for quasi-coherent sheaves (Descent on Spaces, Proposition 04W8) we obtain a unique isomorphism \(\varphi : \mathcal{L} \to \mathcal{N}\) whose restriction to \(X_i\) recovers \(\varphi_i\).
Decent for objects is proved in exactly the same manner. Namely, suppose that \(\{(X_i \to S_i) \to (X \to S)\}_{i \in I}\) is a covering of \(\Spacesstack'_{fp, flat, proper}\) as above. Suppose we have objects \((X_i \to S_i, \mathcal{L}_i)\) of \(\Polarizedstack\) lying over \((X_i \to S_i)\) and a descent datum \[(\text{id}, \text{id}, \varphi_{ij}) : (X_{ij} \to S_{ij}, \mathcal{L}_i|_{X_{ij}}) \to (X_{ij} \to S_{ij}, \mathcal{L}_j|_{X_{ij}})\] satisfying the obvious cocycle condition over \((X_{ijk} \to S_{ijk})\) for every triple of indices. Then by descent for quasi-coherent sheaves (Descent on Spaces, Proposition 04W8) we obtain a unique invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) and isomorphisms \(\mathcal{L}|_{X_i} \to \mathcal{L}_i\) recovering the descent datum \(\varphi_{ij}\). To show that \((X \to S, \mathcal{L})\) is an object of \(\Polarizedstack\) we have to prove that \(\mathcal{L}\) is ample. This follows from Descent on Spaces, Lemma 0D3C.
Since we already have seen that \(\Spacesstack'_{fp, flat, proper}\) is a stack in groupoids over \(\Sch_{fppf}\) (Lemma 0D1G) it now follows formally that \(\Polarizedstack\) is a stack in groupoids over \(\Sch_{fppf}\). See Stacks, Lemma 09WX.
Sanity check: the stack \(\Polarizedstack\) plays the same role among algebraic spaces.
Lemma
Let \(T\) be an algebraic space over \(\mathbf{Z}\). Let \(\mathcal{S}_T\) denote the corresponding algebraic stack (Algebraic Stacks, Sections 04SU, 02ZV, and 03YR). We have an equivalence of categories \[\left\{ \begin{matrix} (X \to T, \mathcal{L})\text{ where }X \to T\text{ is a morphism}\\ \text{of algebraic spaces, is proper, flat, and of}\\ \text{finite presentation and }\mathcal{L}\text{ ample on }X/T \end{matrix} \right\} \longrightarrow \Mor_{\textit{Cat}/\Sch_{fppf}}(\mathcal{S}_T, \Polarizedstack)\]
Proof
Omitted. Hints: Argue exactly as in the proof of Lemma 0E93 and use Descent on Spaces, Proposition 04W8 to descent the invertible sheaf in the construction of the quasi-inverse functor. The relative ampleness property descends by Descent on Spaces, Lemma 0D3C.
Remark
Let \(B\) be an algebraic space over \(\Spec(\mathbf{Z})\). Let \(B\textit{-Polarized}\) be the category consisting of triples \((X \to S, \mathcal{L}, h : S \to B)\) where \((X \to S, \mathcal{L})\) is an object of \(\Polarizedstack\) and \(h : S \to B\) is a morphism. A morphism \((X' \to S', \mathcal{L}', h') \to (X \to S, \mathcal{L}, h)\) in \(B\textit{-Polarized}\) is a morphism \((f, g, \varphi)\) in \(\Polarizedstack\) such that \(h \circ g = h'\). In this situation the diagram \[\xymatrix{ B\textit{-Polarized} \ar[r] \ar[d] & \Polarizedstack \ar[d] \\ (\Sch/B)_{fppf} \ar[r] & \Sch_{fppf} }\] is \(2\)-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case \(\Polarizedstack\) to the case of families over a given base algebraic space.
Lemma
The functor (0D3Y) defines a \(1\)-morphism \[\Polarizedstack \to \Spacesstack'_{fp, flat, proper}\] of stacks in groupoids over \(\Sch_{fppf}\) which is algebraic in the sense of Criteria for Representability, Definition 06CF.
Proof
By Lemmas 0D1G and 0D40 the statement makes sense. To prove it, we choose a scheme \(S\) and an object \(\xi = (X \to S)\) of \(\Spacesstack'_{fp, flat, proper}\) over \(S\). We have to show that \[\mathcal{X} = (\Sch/S)_{fppf} \times_{\xi, \Spacesstack'_{fp, flat, proper}} \Polarizedstack\] is an algebraic stack over \(S\). Observe that an object of \(\mathcal{X}\) is given by a pair \((T/S, \mathcal{L})\) where \(T\) is a scheme over \(S\) and \(\mathcal{L}\) is an invertible \(\mathcal{O}_{X_T}\)-module which is ample on \(X_T/T\). Morphisms are defined in the obvious manner. In particular, we see immediately that we have an inclusion \[\mathcal{X} \subset \Picardstack_{X/S}\] of categories over \((\Sch/S)_{fppf}\), inducing equality on morphism sets. Since \(\Picardstack_{X/S}\) is an algebraic stack by Proposition 0D04 it suffices to show that the inclusion above is representable by open immersions. This is exactly the content of Descent on Spaces, Lemma 0D3D.
Lemma
The diagonal \[\Delta : \Polarizedstack \longrightarrow \Polarizedstack \times \Polarizedstack\] is representable by algebraic spaces.
Proof
This is a formal consequence of Lemmas 0D41 and 0D1F. See Criteria for Representability, Lemma 0D3R.
Lemma
The stack in groupoids \(\Polarizedstack\) is limit preserving (Artin’s Axioms, Definition 07XL).
Proof
Let \(I\) be a directed set and let \((A_i, \varphi_{ii'})\) be a system of rings over \(I\). Set \(S = \Spec(A)\) and \(S_i = \Spec(A_i)\). We have to show that on fibre categories we have \[\Polarizedstack_S = \colim \Polarizedstack_{S_i}\] We know that the category of schemes of finite presentation over \(S\) is the colimit of the category of schemes of finite presentation over \(S_i\), see Limits, Lemma 01ZM. Moreover, given \(X_i \to S_i\) of finite presentation, with limit \(X \to S\), then the category of invertible \(\mathcal{O}_X\)-modules \(\mathcal{L}\) is the colimit of the categories of invertible \(\mathcal{O}_{X_i}\)-modules \(\mathcal{L}_i\), see Limits, Lemma 01ZR and 0B8W. If \(X \to S\) is proper and flat, then for sufficiently large \(i\) the morphism \(X_i \to S_i\) is proper and flat too, see Limits, Lemmas 081F and 04AI. Finally, if \(\mathcal{L}\) is ample on \(X\) then \(\mathcal{L}_i\) is ample on \(X_i\) for \(i\) sufficiently large, see Limits, Lemma 09MT. Putting everything together finishes the proof.
Lemma
In Situation 08KB. Let \[\xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' }\] be a pushout in the category of schemes where \(T \to T'\) is a thickening and \(T \to S\) is affine, see More on Morphisms, Lemma 07RT. Then the functor on fibre categories \[\Polarizedstack_{S'} \longrightarrow \Polarizedstack_S \times_{\Polarizedstack_T} \Polarizedstack_{T'}\] is an equivalence.
Proof
By More on Morphisms, Lemma 07RX there is an equivalence \[\textit{flat-lfp}_{S'} \longrightarrow \textit{flat-lfp}_S \times_{\textit{flat-lfp}_T} \textit{flat-lfp}_{T'}\] where \(\textit{flat-lfp}_S\) signifies the category of schemes flat and locally of finite presentation over \(S\). Let \(X'/S'\) on the left hand side correspond to the triple \((X/S, Y'/T', \varphi)\) on the right hand side. Set \(Y = T \times_{T'} Y'\) which is isomorphic with \(T \times_S X\) via \(\varphi\). Then More on Morphisms, Lemma 08KU shows that we have an equivalence \[\textit{QCoh-flat}_{X'/S'} \longrightarrow \textit{QCoh-flat}_{X/S} \times_{\textit{QCoh-flat}_{Y/T}} \textit{QCoh-flat}_{Y'/T'}\] where \(\textit{QCoh-flat}_{X/S}\) signifies the category of quasi-coherent \(\mathcal{O}_X\)-modules flat over \(S\). Since \(X \to S\), \(Y \to T\), \(X' \to S'\), \(Y' \to T'\) are flat, this will in particular apply to invertible modules to give an equivalence of categories \[\textit{Pic}(X') \longrightarrow \textit{Pic}(X) \times_{\textit{Pic}(Y)} \textit{Pic}(Y')\] where \(\textit{Pic}(X)\) signifies the category of invertible \(\mathcal{O}_X\)-modules. There is a small point here: one has to show that if an object \(\mathcal{F}'\) of \(\textit{QCoh-flat}_{X'/S'}\) pulls back to invertible modules on \(X\) and \(Y'\), then \(\mathcal{F}'\) is an invertible \(\mathcal{O}_{X'}\)-module. It follows from the cited lemma that \(\mathcal{F}'\) is an \(\mathcal{O}_{X'}\)-module of finite presentation. By More on Morphisms, Lemma 080Q it suffices to check the restriction of \(\mathcal{F}'\) to fibres of \(X' \to S'\) is invertible. But the fibres of \(X' \to S'\) are the same as the fibres of \(X \to S\) and hence these restrictions are invertible.
Having said the above we obtain an equivalence of categories if we drop the assumption (for the category of objects over \(S\)) that \(X \to S\) be proper and the assumption that \(\mathcal{L}\) be ample. Now it is clear that if \(X' \to S'\) is proper, then \(X \to S\) and \(Y' \to T'\) are proper (Morphisms, Lemma 01W4). Conversely, if \(X \to S\) and \(Y' \to T'\) are proper, then \(X' \to S'\) is proper by More on Morphisms, Lemma 09ZW. Similarly, if \(\mathcal{L}'\) is ample on \(X'/S'\), then \(\mathcal{L}'|_X\) is ample on \(X/S\) and \(\mathcal{L}'|_{Y'}\) is ample on \(Y'/T'\) (Morphisms, Lemma 0893). Finally, if \(\mathcal{L}'|_X\) is ample on \(X/S\) and \(\mathcal{L}'|_{Y'}\) is ample on \(Y'/T'\), then \(\mathcal{L}'\) is ample on \(X'/S'\) by More on Morphisms, Lemma 0D2R.
Lemma
Let \(k\) be a field and let \(x = (X \to \Spec(k), \mathcal{L})\) be an object of \(\mathcal{X} = \Polarizedstack\) over \(\Spec(k)\).
If \(k\) is of finite type over \(\mathbf{Z}\), then the vector spaces \(T\mathcal{F}_{\mathcal{X}, k, x}\) and \(\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x})\) (see Artin’s Axioms, Section 07WY) are finite dimensional, and
in general the vector spaces \(T_x(k)\) and \(\text{Inf}_x(k)\) (see Artin’s Axioms, Section 07Y6) are finite dimensional.
Proof
The discussion in Artin’s Axioms, Section 07WY only applies to fields of finite type over the base scheme \(\Spec(\mathbf{Z})\). Our stack satisfies (RS*) by Lemma 0D44 and we may apply Artin’s Axioms, Lemma 07Y9 to get the vector spaces \(T_x(k)\) and \(\text{Inf}_x(k)\) mentioned in (2). Moreover, in the finite type case these spaces agree with the ones mentioned in part (1) by Artin’s Axioms, Remark 0D18. With this out of the way we can start the proof.
One proof is to use an argument as in the proof of Lemma 0D1K; this would require us to develop a deformation theory for pairs consisting of a scheme and a quasi-coherent module. Another proof would be the use the result from Lemma 0D1K, the algebraicity of \(\Polarizedstack \to \Spacesstack'_{fp, flat, proper}\), and a computation of the deformation space of an invertible module. However, what we will do instead is to translate the question into a deformation question on graded \(k\)-algebras and deduce the result that way.
Let \(\mathcal{C}_k\) be the category of Artinian local \(k\)-algebras \(A\) with residue field \(k\). We get a predeformation category \(p : \mathcal{F} \to \mathcal{C}_k\) from our object \(x\) of \(\mathcal{X}\) over \(k\), see Artin’s Axioms, Section 07T2. Thus \(\mathcal{F}(A)\) is the category of triples \((X_A, \mathcal{L}_A, \alpha)\), where \((X_A, \mathcal{L}_A)\) is an object of \(\Polarizedstack\) over \(A\) and \(\alpha\) is an isomorphism \((X_A, \mathcal{L}_A) \times_{\Spec(A)} \Spec(k) \cong (X, \mathcal{L})\). On the other hand, let \(q : \mathcal{G} \to \mathcal{C}_k\) be the category cofibred in groupoids defined in Deformation Problems, Example 0D3L. Choose \(d_0 \gg 0\) (we’ll see below how large). Let \(P\) be the graded \(k\)-algebra \[P = k \oplus \bigoplus\nolimits_{d \geq d_0} H^0(X, \mathcal{L}^{\otimes d})\] Then \(y = (k, P)\) is an object of \(\mathcal{G}(k)\). Let \(\mathcal{G}_y\) be the predeformation category of Formal Deformation Theory, Remark 06GU. Given \((X_A, \mathcal{F}_A, \alpha)\) as above we set \[Q = A \oplus \bigoplus\nolimits_{d \geq d_0} H^0(X_A, \mathcal{L}_A^{\otimes d})\] The isomorphism \(\alpha\) induces a map \(\beta : Q \to P\). By deformation theory of projective schemes (More on Morphisms, Lemma 0D4F) we obtain a \(1\)-morphism \[\mathcal{F} \longrightarrow \mathcal{G}_y,\quad (X_A, \mathcal{F}_A, \alpha) \longmapsto (Q, \beta : Q \to P)\] of categories cofibred in groupoids over \(\mathcal{C}_k\). In fact, this functor is an equivalence with quasi-inverse given by \(Q \mapsto \underline{\text{Proj}}_A(Q)\). Namely, the scheme \(X_A = \underline{\text{Proj}}_A(Q)\) is flat over \(A\) by Divisors, Lemma 0D4C. Set \(\mathcal{L}_A = \mathcal{O}_{X_A}(1)\); this is flat over \(A\) by the same lemma. We get an isomorphism \((X_A, \mathcal{L}_A) \times_{\Spec(A)} \Spec(k) = (X, \mathcal{L})\) from \(\beta\). Then we can deduce all the desired properties of the pair \((X_A, \mathcal{L}_A)\) from the corresponding properties of \((X, \mathcal{L})\) using the techniques in More on Morphisms, Sections 0CF2 and 063X. Some details omitted.
In conclusion, we see that \(T\mathcal{F} = T\mathcal{G}_y = T_y\mathcal{G}\) and \(\text{Inf}(\mathcal{F}) = \text{Inf}_y(\mathcal{G})\). These vector spaces are finite dimensional by Deformation Problems, Lemma 0DVZ and the proof is complete.
Lemma
Let \((R_n)\) be an inverse system of rings with surjective transition maps whose kernels are locally nilpotent. Set \(R = \lim R_n\). Set \(S_n = \Spec(R_n)\) and \(S = \Spec(R)\). Consider a commutative diagram \[\xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots }\] of schemes with cartesian squares. Suppose given \((\mathcal{L}_n, \varphi_n)\) where each \(\mathcal{L}_n\) is an invertible sheaf on \(X_n\) and \(\varphi_n : i_n^*\mathcal{L}_{n + 1} \to \mathcal{L}_n\) is an isomorphism. If
\(X_n \to S_n\) is proper, flat, of finite presentation, and
\(\mathcal{L}_1\) is ample on \(X_1\)
then there exists a morphism of schemes \(X \to S\) proper, flat, and of finite presentation and an ample invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) and isomorphisms \(X_n \cong X \times_S S_n\) and \(\mathcal{L}_n \cong \mathcal{L}|_{X_n}\) compatible with the morphisms \(i_n\) and \(\varphi_n\).
Proof
Choose \(d_0\) for \(X_1 \to S_1\) and \(\mathcal{L}_1\) as in More on Morphisms, Lemma 0D4F. For any \(n \geq 1\) set \[A_n = R_n \oplus \bigoplus\nolimits_{d \geq d_0} H^0(X_n, \mathcal{L}_n^{\otimes d})\] By the lemma each \(A_n\) is a finitely presented graded \(R_n\)-algebra whose homogeneous parts \((A_n)_d\) are finite projective \(R_n\)-modules such that \(X_n = \text{Proj}(A_n)\) and \(\mathcal{L}_n = \mathcal{O}_{\text{Proj}(A_n)}(1)\). The lemma also guarantees that the maps \[A_1 \leftarrow A_2 \leftarrow A_3 \leftarrow \ldots\] induce isomorphisms \(A_n = A_m \otimes_{R_m} R_n\) for \(n \leq m\). We set \[B = \bigoplus\nolimits_{d \geq 0} B_d \quad\text{with}\quad B_d = \lim_n (A_n)_d\] By More on Algebra, Lemma 0D4B we see that \(B_d\) is a finite projective \(R\)-module and that \(B \otimes_R R_n = A_n\). Thus the scheme \[X = \text{Proj}(B) \quad\text{and}\quad \mathcal{L} = \mathcal{O}_X(1)\] is flat over \(S\) and \(\mathcal{L}\) is a quasi-coherent \(\mathcal{O}_X\)-module flat over \(S\), see Divisors, Lemma 0D4C. Because formation of Proj commutes with base change (Constructions, Lemma 01N2) we obtain canonical isomorphisms \[X \times_S S_n = X_n \quad\text{and}\quad \mathcal{L}|_{X_n} \cong \mathcal{L}_n\] compatible with the transition maps of the system. Thus we may think of \(X_1 \subset X\) as a closed subscheme. Below we will show that \(B\) is of finite presentation over \(R\). By Divisors, Lemmas 0800 and 0D4D this implies that \(X \to S\) is of finite presentation and proper and that \(\mathcal{L} = \mathcal{O}_X(1)\) is of finite presentation as an \(\mathcal{O}_X\)-module. Since the restriction of \(\mathcal{L}\) to the base change \(X_1 \to S_1\) is invertible, we see from More on Morphisms, Lemma 0CZR that \(\mathcal{L}\) is invertible on an open neighbourhood of \(X_1\) in \(X\). Since \(X \to S\) is closed and since \(\Ker(R \to R_1)\) is contained in the Jacobson radical (More on Algebra, Lemma 0CT7) we see that any open neighbourhood of \(X_1\) in \(X\) is equal to \(X\). Thus \(\mathcal{L}\) is invertible. Finally, the set of points in \(S\) where \(\mathcal{L}\) is ample on the fibre is open in \(S\) (More on Morphisms, Lemma 0D2S) and contains \(S_1\) hence equals \(S\). Thus \(X \to S\) and \(\mathcal{L}\) have all the properties required of them in the statement of the lemma.
We prove the claim above. Choose a presentation \(A_1 = R_1[X_1, \ldots, X_s]/(F_1, \ldots, F_t)\) where \(X_i\) are variables having degrees \(d_i\) and \(F_j\) are homogeneous polynomials in \(X_i\) of degree \(e_j\). Then we can choose a map \[\Psi : R[X_1, \ldots, X_s] \longrightarrow B\] lifting the map \(R_1[X_1, \ldots, X_s] \to A_1\). Since each \(B_d\) is finite projective over \(R\) we conclude from Nakayama’s lemma (Algebra, Lemma 00DV using again that \(\Ker(R \to R_1)\) is contained in the Jacobson radical of \(R\)) that \(\Psi\) is surjective. Since \(- \otimes_R R_1\) is right exact we can find \(G_1, \ldots, G_t \in \Ker(\Psi)\) mapping to \(F_1, \ldots, F_t\) in \(R_1[X_1, \ldots, X_s]\). Observe that \(\Ker(\Psi)_d\) is a finite projective \(R\)-module for all \(d \geq 0\) as the kernel of the surjection \(R[X_1, \ldots, X_s]_d \to B_d\) of finite projective \(R\)-modules. We conclude from Nakayama’s lemma once more that \(\Ker(\Psi)\) is generated by \(G_1, \ldots, G_t\).
Lemma
Consider the stack \(\Polarizedstack\) over the base scheme \(\Spec(\mathbf{Z})\). Then every formal object is effective.
Proof
For definitions of the notions in the lemma, please see Artin’s Axioms, Section 07X3. From the definitions we see the lemma follows immediately from the more general Lemma 0D4T.
Lemma
The stack in groupoids \(\Polarizedstack\) satisfies openness of versality over \(\Spec(\mathbf{Z})\). Similarly, after base change (Remark 0D1N) openness of versality holds over any Noetherian base scheme \(S\).
Proof
This follows from Artin’s Axioms, Lemma 0CXU and Lemmas 0D42, 0D44, 0D43, and 0D4T. For the “usual” proof of this fact, please see the discussion in the remark following this proof.
Remark
Lemma 0D4V can also be shown using an obstruction theory as in Artin’s Axioms, Lemma 0CYF (as in the second proof of Lemma 08WA). To do this one has to generalize the deformation and obstruction theory developed in Cotangent, Section 08V3 to the case of pairs of algebraic spaces and quasi-coherent modules. Another possibility is to use that the \(1\)-morphism \(\Polarizedstack \to \Spacesstack'_{fp, flat, proper}\) is algebraic (Lemma 0D41) and the fact that we know openness of versality for the target (Lemma 0D3X and Remark 0D1P).
Theorem
The stack \(\Polarizedstack\) (Situation 0D1M) is algebraic. In fact, for any algebraic space \(B\) the stack \(B\textit{-Polarized}\) (Remark 0D1N) is algebraic.
Proof
The absolute case follows from Artin’s Axioms, Lemma 07Y4 and Lemmas 0D42, 0D44, 0D43, 0D4U, and 0D4V. The case over \(B\) follows from this, the description of \(B\textit{-Polarized}\) as a \(2\)-fibre product in Remark 0D1N, and the fact that algebraic stacks have \(2\)-fibre products, see Algebraic Stacks, Lemma 04T2.
The stack of curves
In this section we prove the stack of curves is algebraic. For a further discussion of moduli of curves, we refer the reader to Moduli of Curves, Section 0DMH.
A curve in the Stacks project is a variety of dimension \(1\). However, when we speak of families of curves, we often allow the fibres to be reducible and/or nonreduced. In this section, the stack of curves will “parametrize proper schemes of dimension \(\leq 1\)”. However, it turns out that in order to get the correct notion of a family we need to allow the total space of our family to be an algebraic space. This leads to the following definition.
Situation
We define a category \(\Curvesstack\) as follows:
Objects are families of curves. More precisely, an object is a morphism \(f : X \to S\) where the base \(S\) is a scheme, the total space \(X\) is an algebraic space, and \(f\) is flat, proper, of finite presentation, and has relative dimension \(\leq 1\) (Morphisms of Spaces, Definition 06LR).
A morphism \((X' \to S') \to (X \to S)\) between objects is given by a pair \((f, g)\) where \(f : X' \to X\) is a morphism of algebraic spaces and \(g : S' \to S\) is a morphism of schemes which fit into a commutative diagram \[\xymatrix{ X' \ar[d] \ar[r]_f & X \ar[d] \\ S' \ar[r]^g & S }\] inducing an isomorphism \(X' \to S' \times_S X\), in other words, the diagram is cartesian.
The forgetful functor \[p : \Curvesstack \longrightarrow \Sch_{fppf},\quad (X \to S) \longmapsto S\] is how we view \(\Curvesstack\) as a category over \(\Sch_{fppf}\) (see Section 05X6 for notation).
It follows from Spaces over Fields, Lemma 0ADD and more generally More on Morphisms of Spaces, Lemma 0AE7 that if \(S\) is the spectrum of a field, or an Artinian local ring, or a Noetherian complete local ring, then for any family of curves \(X \to S\) the total space \(X\) is a scheme. On the other hand, there are families of curves over \(\mathbf{A}^1_k\) where the total space is not a scheme, see Examples, Section 0D5D.
It is clear that [0D50]\[\begin{equation} \Curvesstack \subset \Spacesstack'_{fp, flat, proper} \end{equation}\] and that an object \(X \to S\) of \(\Spacesstack'_{fp, flat, proper}\) is in \(\Curvesstack\) if and only if \(X \to S\) has relative dimension \(\leq 1\). We will use this to verify Artin’s axioms for \(\Curvesstack\).
Lemma
The category \(\Curvesstack\) is fibred in groupoids over \(\Sch_{fppf}\).
Proof
Using the embedding (0D50), the description of the image, and the corresponding fact for \(\Spacesstack'_{fp, flat, proper}\) (Lemma 0D1E) this reduces to the following statement: Given a morphism \[\xymatrix{ X' \ar[r] \ar[d] & X \ar[d] \\ S' \ar[r] & S }\] in \(\Spacesstack'_{fp, flat, proper}\) (recall that this implies in particular the diagram is cartesian) if \(X \to S\) has relative dimension \(\leq 1\), then \(X' \to S'\) has relative dimension \(\leq 1\). This follows from Morphisms of Spaces, Lemma 04NS.
Lemma
The category \(\Curvesstack\) is a stack in groupoids over \(\Sch_{fppf}\).
Proof
Using the embedding (0D50), the description of the image, and the corresponding fact for \(\Spacesstack'_{fp, flat, proper}\) (Lemma 0D1G) this reduces to the following statement: Given an object \(X \to S\) of \(\Spacesstack'_{fp, flat, proper}\) and an fppf covering \(\{S_i \to S\}_{i \in I}\) the following are equivalent:
\(X \to S\) has relative dimension \(\leq 1\), and
for each \(i\) the base change \(X_i \to S_i\) has relative dimension \(\leq 1\).
This follows from Morphisms of Spaces, Lemma 04NS.
Lemma
The diagonal \[\Delta : \Curvesstack \longrightarrow \Curvesstack \times \Curvesstack\] is representable by algebraic spaces.
Proof
This is immediate from the fully faithful embedding (0D50) and the corresponding fact for \(\Spacesstack'_{fp, flat, proper}\) (Lemma 0D1F).
Remark
Let \(B\) be an algebraic space over \(\Spec(\mathbf{Z})\). Let \(B\text{-}\Curvesstack\) be the category consisting of pairs \((X \to S, h : S \to B)\) where \(X \to S\) is an object of \(\Curvesstack\) and \(h : S \to B\) is a morphism. A morphism \((X' \to S', h') \to (X \to S, h)\) in \(B\text{-}\Curvesstack\) is a morphism \((f, g)\) in \(\Curvesstack\) such that \(h \circ g = h'\). In this situation the diagram \[\xymatrix{ B\text{-}\Curvesstack \ar[r] \ar[d] & \Curvesstack \ar[d] \\ (\Sch/B)_{fppf} \ar[r] & \Sch_{fppf} }\] is \(2\)-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case \(\Curvesstack\) to the case of families of curves over a given base algebraic space.
Lemma
The stack \(\Curvesstack \to \Sch_{fppf}\) is limit preserving (Artin’s Axioms, Definition 07XL).
Proof
Using the embedding (0D50), the description of the image, and the corresponding fact for \(\Spacesstack'_{fp, flat, proper}\) (Lemma 0D1I) this reduces to the following statement: Let \(T = \lim T_i\) be the limits of a directed inverse system of affine schemes. Let \(i \in I\) and let \(X_i \to T_i\) be an object of \(\Spacesstack'_{fp, flat, proper}\) over \(T_i\). Assume that \(T \times_{T_i} X_i \to T\) has relative dimension \(\leq 1\). Then for some \(i' \geq i\) the morphism \(T_{i'} \times_{T_i} X_i \to T_i\) has relative dimension \(\leq 1\). This follows from Limits of Spaces, Lemma 0D4K.
Lemma
Let \[\xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' }\] be a pushout in the category of schemes where \(T \to T'\) is a thickening and \(T \to S\) is affine, see More on Morphisms, Lemma 07RT. Then the functor on fibre categories \[\Curvesstack_{S'} \longrightarrow \Curvesstack_S \times_{\Curvesstack_T} \Curvesstack_{T'}\] is an equivalence.
Proof
Using the embedding (0D50), the description of the image, and the corresponding fact for \(\Spacesstack'_{fp, flat, proper}\) (Lemma 0D1J) this reduces to the following statement: given a morphism \(X' \to S'\) of an algebraic space to \(S'\) which is of finite presentation, flat, proper then \(X' \to S'\) has relative dimension \(\leq 1\) if and only if \(S \times_{S'} X' \to S\) and \(T' \times_{S'} X' \to T'\) have relative dimension \(\leq 1\). One implication follows from the fact that having relative dimension \(\leq 1\) is preserved under base change (Morphisms of Spaces, Lemma 04NS). The other follows from the fact that having relative dimension \(\leq 1\) is checked on the fibres and that the fibres of \(X' \to S'\) (over points of the scheme \(S'\)) are the same as the fibres of \(S \times_{S'} X' \to S\) since \(S \to S'\) is a thickening by More on Morphisms, Lemma 07RT.
Lemma
Let \(k\) be a field and let \(x = (X \to \Spec(k))\) be an object of \(\mathcal{X} = \Curvesstack\) over \(\Spec(k)\).
If \(k\) is of finite type over \(\mathbf{Z}\), then the vector spaces \(T\mathcal{F}_{\mathcal{X}, k, x}\) and \(\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x})\) (see Artin’s Axioms, Section 07WY) are finite dimensional, and
in general the vector spaces \(T_x(k)\) and \(\text{Inf}_x(k)\) (see Artin’s Axioms, Section 07Y6) are finite dimensional.
Proof
This is immediate from the fully faithful embedding (0D50) and the corresponding fact for \(\Spacesstack'_{fp, flat, proper}\) (Lemma 0D1K).
Lemma
Consider the stack \(\Curvesstack\) over the base scheme \(\Spec(\mathbf{Z})\). Then every formal object is effective.
Proof
For definitions of the notions in the lemma, please see Artin’s Axioms, Section 07X3. Let \((A, \mathfrak m, \kappa)\) be a Noetherian complete local ring. Let \((X_n \to \Spec(A/\mathfrak m^n))\) be a formal object of \(\Curvesstack\) over \(A\). By More on Morphisms of Spaces, Lemma 0E7R there exists a projective morphism \(X \to \Spec(A)\) and a compatible system of ismomorphisms \(X \times_{\Spec(A)} \Spec(A/\mathfrak m^n) \cong X_n\). By More on Morphisms, Lemma 0D4G we see that \(X \to \Spec(A)\) is flat. By More on Morphisms, Lemma 0D4J we see that \(X \to \Spec(A)\) has relative dimension \(\leq 1\). This proves the lemma.
Lemma
The stack in groupoids \(\mathcal{X} = \Curvesstack\) satisfies openness of versality over \(\Spec(\mathbf{Z})\). Similarly, after base change (Remark 0D54) openness of versality holds over any Noetherian base scheme \(S\).
Proof
This is immediate from the fully faithful embedding (0D50) and the corresponding fact for \(\Spacesstack'_{fp, flat, proper}\) (Lemma 0D3X).
Theorem
The stack \(\Curvesstack\) (Situation 0D4Z) is algebraic. In fact, for any algebraic space \(B\) the stack \(B\text{-}\Curvesstack\) (Remark 0D54) is algebraic.
Proof
The absolute case follows from Artin’s Axioms, Lemma 07Y4 and Lemmas 0D53, 0D56, 0D55, 0D58, and 0D59. The case over \(B\) follows from this, the description of \(B\text{-}\Curvesstack\) as a \(2\)-fibre product in Remark 0D54, and the fact that algebraic stacks have \(2\)-fibre products, see Algebraic Stacks, Lemma 04T2.
Lemma
The \(1\)-morphism (0D50) \[\Curvesstack \longrightarrow \Spacesstack'_{fp, flat, proper}\] is representable by open and closed immersions.
Proof
Since (0D50) is a fully faithful embedding of categories it suffices to show the following: given an object \(X \to S\) of \(\Spacesstack'_{fp, flat, proper}\) there exists an open and closed subscheme \(U \subset S\) such that a morphism \(S' \to S\) factors through \(U\) if and only if the base change \(X' \to S'\) of \(X \to S\) has relative dimension \(\leq 1\). This follows immediately from More on Morphisms of Spaces, Lemma 0D4R.
Remark
Consider the \(2\)-fibre product \[\xymatrix{ \Curvesstack \times_{\Spacesstack'_{fp, flat, proper}} \Polarizedstack \ar[r] \ar[d] & \Polarizedstack \ar[d] \\ \Curvesstack \ar[r] & \Spacesstack'_{fp, flat, proper} }\] This fibre product parametrized polarized curves, i.e., families of curves endowed with a relatively ample invertible sheaf. It turns out that the left vertical arrow \[\textit{PolarizedCurves} \longrightarrow \Curvesstack\] is algebraic, smooth, and surjective. Namely, this \(1\)-morphism is algebraic (as base change of the arrow in Lemma 0D41), every point is in the image, and there are no obstructions to deforming invertible sheaves on curves (see proof of Lemma 0D58). This gives another approach to the algebraicity of \(\Curvesstack\). Namely, by Lemma 0D5B we see that \(\textit{PolarizedCurves}\) is an open and closed substack of the algebraic stack \(\Polarizedstack\) and any stack in groupoids which is the target of a smooth algebraic morphism from an algebraic stack is an algebraic stack.
Moduli of complexes on a proper morphism
The title and the material of this section are taken from [lieblich-complexes]. Let \(S\) be a scheme and let \(f : X \to B\) be a proper, flat, finitely presented morphism of algebraic spaces. We will prove that there is an algebraic stack \[\Complexesstack_{X/B}\] parametrizing “families” of objects of \(D^b_{\textit{Coh}}\) of the fibres with vanishing negative self-exts. More precisely a family is given by a relatively perfect object of the derived category of the total space; this somewhat technical notion is studied in More on Morphisms of Spaces, Section 0DKM.
Already if \(X\) is a proper algebraic space over a field \(k\) we obtain a very interesting algebraic stack. Namely, there is an embedding \[\Cohstack_{X/k} \longrightarrow \Complexesstack_{X/k}\] since for any \(\mathcal{O}\)-module \(\mathcal{F}\) (on any ringed topos) we have \(\Ext^i_\mathcal{O}(\mathcal{F}, \mathcal{F}) = 0\) for \(i < 0\). Although this certainly shows our stack is nonempty, the true motivation for the study of \(\Complexesstack_{X/k}\) is that there are often objects of the derived category \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) with vanishing negative self-exts and nonvanishing cohomology sheaves in more than one degree. For example, \(X\) could be derived equivalent to another proper algebraic space \(Y\) over \(k\), i.e., we have a \(k\)-linear equivalence \[F : D^b_{\textit{Coh}}(\mathcal{O}_Y) \longrightarrow D^b_{\textit{Coh}}(\mathcal{O}_X)\] There are cases where this happens and \(F\) is not given by an isomorphism between \(X\) and \(Y\); for example in the case of an abelian variety and its dual. In this situation \(F\) induces an isomorphism of algebraic stacks \[\Complexesstack_{Y/k} \longrightarrow \Complexesstack_{X/k}\] (insert future reference here) and in particular the stack of coherent sheaves on \(Y\) maps into the stack of complexes on \(X\). Turning this around, if we can understand well enough the geometry of \(\Complexesstack_{X/k}\), then we can try to use this to study all possible derived equivalent \(Y\).
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is proper, flat, and of finite presentation. Let \(K, E \in D(\mathcal{O}_X)\). Assume \(K\) is pseudo-coherent and \(E\) is \(Y\)-perfect (More on Morphisms of Spaces, Definition 0DKN). For a field \(k\) and a morphism \(y : \Spec(k) \to Y\) denote \(K_y\), \(E_y\) the pullback to the fibre \(X_y\).
There is an open \(W \subset Y\) characterized by the property \[y \in |W| \Leftrightarrow \Ext^i_{\mathcal{O}_{X_y}}(K_y, E_y) = 0 \text{ for }i < 0.\]
For any morphism \(V \to Y\) factoring through \(W\) we have \[\Ext^i_{\mathcal{O}_{X_V}}(K_V, E_V) = 0 \quad\text{for}\quad i < 0\] where \(X_V\) is the base change of \(X\) and \(K_V\) and \(E_V\) are the derived pullbacks of \(K\) and \(E\) to \(X_V\).
The functor \(V \mapsto \Hom_{\mathcal{O}_{X_V}}(K_V, E_V)\) is a sheaf on \((\textit{Spaces}/W)_{fppf}\) representable by an algebraic space affine and of finite presentation over \(W\).
Proof
For any morphism \(V \to Y\) the complex \(K_V\) is pseudo-coherent (Cohomology on Sites, Lemma 08H4) and \(E_V\) is \(V\)-perfect (More on Morphisms of Spaces, Lemma 0DKT). Another observation is that given \(y : \Spec(k) \to Y\) and a field extension \(k'/k\) with \(y' : \Spec(k') \to Y\) the induced morphism, we have \[\Ext^i_{\mathcal{O}_{X_{y'}}}(K_{y'}, E_{y'}) = \Ext^i_{\mathcal{O}_{X_y}}(K_y, E_y) \otimes_k k'\] by Derived Categories of Schemes, Lemma 0AA7. Thus the vanishing in (1) is really a property of the induced point \(y \in |Y|\). We will use these two observations without further mention in the proof.
Assume first \(Y\) is an affine scheme. Then we may apply More on Morphisms of Spaces, Lemma 0DKY and find a pseudo-coherent \(L \in D(\mathcal{O}_Y)\) which “universally computes” \(Rf_*R\SheafHom(K, E)\) in the sense described in that lemma. Unwinding the definitions, we obtain for a point \(y \in Y\) the equality \[\Ext^i_{\kappa(y)}(L \otimes_{\mathcal{O}_Y}^\mathbf{L} \kappa(y), \kappa(y)) = \Ext^i_{\mathcal{O}_{X_y}}(K_y, E_y)\] We conclude that \[H^i(L \otimes_{\mathcal{O}_Y}^\mathbf{L} \kappa(y)) = 0 \text{ for } i > 0 \Leftrightarrow \Ext^i_{\mathcal{O}_{X_y}}(K_y, E_y) = 0 \text{ for }i < 0.\] By Derived Categories of Schemes, Lemma 0BDI the set \(W\) of \(y \in Y\) where this happens defines an open of \(Y\). This open \(W\) then satisfies the requirement in (1) for all morphisms from spectra of fields, by the “universality” of \(L\).
Let’s go back to \(Y\) a general algebraic space. Choose an étale covering \(\{V_i \to Y\}\) by affine schemes \(V_i\). Then we see that the subset \(W \subset |Y|\) pulls back to the corresponding subset \(W_i \subset |V_i|\) for \(X_{V_i}\), \(K_{V_i}\), \(E_{V_i}\). By the previous paragraph we find that \(W_i\) is open, hence \(W\) is open. This proves (1) in general. Moreover, parts (2) and (3) are entirely formulated in terms of the category \(\textit{Spaces}/W\) and the restrictions \(X_W\), \(K_W\), \(E_W\). This reduces us to the case \(W = Y\).
Assume \(W = Y\). We claim that for any algebraic space \(V\) over \(Y\) we have \(Rf_{V, *}R\SheafHom(K_V, E_V)\) has vanishing cohomology sheaves in degrees \(< 0\). This will prove (2) because \[\Ext^i_{\mathcal{O}_{X_V}}(K_V, E_V) = H^i(X_V, R\SheafHom(K_V, E_V)) = H^i(V, Rf_{V, *}R\SheafHom(K_V, E_V))\] by Cohomology on Sites, Lemmas 08JA and 0D6H and the vanishing of the cohomology sheaves implies the cohomology group \(H^i\) is zero for \(i < 0\) by Derived Categories, Lemma 05TC.
To prove the claim, we may work étale locally on \(V\). In particular, we may assume \(Y\) is affine and \(W = Y\). Let \(L \in D(\mathcal{O}_Y)\) be as in the second paragraph of the proof. For an algebraic space \(V\) over \(Y\) denote \(L_V\) the derived pullback of \(L\) to \(V\). (An important feature we will use is that \(L\) “works” for all algebraic spaces \(V\) over \(Y\) and not just affine \(V\).) As \(W = Y\) we have \(H^i(L) = 0\) for \(i > 0\) (use More on Algebra, Lemma 0BCC to go from fibres to stalks). Hence \(H^i(L_V) = 0\) for \(i > 0\). The property defining \(L\) is that \[Rf_{V, *}R\SheafHom(K_V, E_V) = R\SheafHom(L_V, \mathcal{O}_V)\] Since \(L_V\) sits in degrees \(\leq 0\), we conclude that \(R\SheafHom(L_V, \mathcal{O}_V)\) sits in degrees \(\geq 0\) thereby proving the claim. This finishes the proof of (2).
Assume \(W = Y\) but make no assumptions on the algebraic space \(Y\). Since we have (2), we see from Simplicial Spaces, Lemma 0DL9 that the functor \(F\) given by \(F(V) = \Hom_{\mathcal{O}_{X_V}}(K_V, E_V)\) is a sheaf4 on \((\textit{Spaces}/Y)_{fppf}\). Thus to prove that \(F\) is an algebraic space and that \(F \to Y\) is affine and of finite presentation, we may work étale locally on \(Y\); see Bootstrap, Lemma 04U0 and Morphisms of Spaces, Lemmas 03WG and 0410. We conclude that it suffices to prove \(F\) is an affine algebraic space of finite presentation over \(Y\) when \(Y\) is an affine scheme. In this case we go back to our pseudo-coherent complex \(L \in D(\mathcal{O}_Y)\). Since \(H^i(L) = 0\) for \(i > 0\), we can represent \(L\) by a complex of the form \[\ldots \to \mathcal{O}_Y^{\oplus m_1} \to \mathcal{O}_Y^{\oplus m_0} \to 0 \to \ldots\] with the last term in degree \(0\), see More on Algebra, Lemma 064U. Combining the two displayed formulas earlier in the proof we find that \[F(V) = \Ker( \Hom_V(\mathcal{O}_V^{\oplus m_0}, \mathcal{O}_V) \to \Hom_V(\mathcal{O}_V^{\oplus m_1}, \mathcal{O}_V) )\] In other words, there is a fibre product diagram \[\xymatrix{ F \ar[d] \ar[r] & Y \ar[d]^0 \\ \mathbf{A}_Y^{m_0} \ar[r] & \mathbf{A}_Y^{m_1} }\] which proves what we want.
Lemma
Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is proper, flat, and of finite presentation. Let \(E \in D(\mathcal{O}_X)\). Assume
\(E\) is \(S\)-perfect (More on Morphisms of Spaces, Definition 0DKN), and
for every point \(s \in S\) we have \[\Ext^i_{\mathcal{O}_{X_s}}(E_s, E_s) = 0 \quad\text{for}\quad i < 0\] where \(E_s\) is the pullback to the fibre \(X_s\).
Then
(1) and (2) are preserved by arbitrary base change \(V \to Y\),
\(\Ext^i_{\mathcal{O}_{X_V}}(E_V, E_V) = 0\) for \(i < 0\) and all \(V\) over \(Y\),
\(V \mapsto \Hom_{\mathcal{O}_{X_V}}(E_V, E_V)\) is representable by an algebraic space affine and of finite presentation over \(Y\).
Here \(X_V\) is the base change of \(X\) and \(E_V\) is the derived pullback of \(E\) to \(X_V\).
Proof
Immediate consequence of Lemma 0DLC.
Situation
Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is proper, flat, and of finite presentation. We denote \(\Complexesstack_{X/B}\) the category whose objects are triples \((T, g, E)\) where
\(T\) is a scheme over \(S\),
\(g : T \to B\) is a morphism over \(S\), and setting \(X_T = T \times_{g, B} X\)
\(E\) is an object of \(D(\mathcal{O}_{X_T})\) satisfying conditions (1) and (2) of Lemma 0DLD.
A morphism \((T, g, E) \to (T', g', E')\) is given by a pair \((h, \varphi)\) where
\(h : T \to T'\) is a morphism of schemes over \(B\) (i.e., \(g' \circ h = g\)), and
\(\varphi : L(h')^*E' \to E\) is an isomorphism of \(D(\mathcal{O}_{X_T})\) where \(h' : X_T \to X_{T'}\) is the base change of \(h\).
Thus \(\Complexesstack_{X/B}\) is a category and the rule \[p : \Complexesstack_{X/B} \longrightarrow (\Sch/S)_{fppf}, \quad (T, g, E) \longmapsto T\] is a functor. For a scheme \(T\) over \(S\) we denote \(\Complexesstack_{X/B, T}\) the fibre category of \(p\) over \(T\). These fibre categories are groupoids.
Lemma
In Situation 0DLE the functor \(p : \Complexesstack_{X/B} \longrightarrow (\Sch/S)_{fppf}\) is fibred in groupoids.
Proof
We show that \(p\) is fibred in groupoids by checking conditions (1) and (2) of Categories, Definition 003T. Given an object \((T', g', E')\) of \(\Complexesstack_{X/B}\) and a morphism \(h : T \to T'\) of schemes over \(S\) we can set \(g = h \circ g'\) and \(E = L(h')^*E'\) where \(h' : X_T \to X_{T'}\) is the base change of \(h\). Then it is clear that we obtain a morphism \((T, g, E) \to (T', g', E')\) of \(\Complexesstack_{X/B}\) lying over \(h\). This proves (1). For (2) suppose we are given morphisms \[(h_1, \varphi_1) : (T_1, g_1, E_1) \to (T, g, E) \quad\text{and}\quad (h_2, \varphi_2) : (T_2, g_2, E_2) \to (T, g, E)\] of \(\Complexesstack_{X/B}\) and a morphism \(h : T_1 \to T_2\) such that \(h_2 \circ h = h_1\). Then we can let \(\varphi\) be the composition \[L(h')^*E_2 \xrightarrow{L(h')^*\varphi_2^{-1}} L(h')^*L(h_2)^*E = L(h_1)^*E \xrightarrow{\varphi_1} E_1\] to obtain the morphism \((h, \varphi) : (T_1, g_1, E_1) \to (T_2, g_2, E_2)\) that witnesses the truth of condition (2).
Lemma
In Situation 0DLE. Denote \(\mathcal{X} = \Complexesstack_{X/B}\). Then \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces.
Proof
Consider two objects \(x = (T, g, E)\) and \(y = (T, g', E')\) of \(\mathcal{X}\) over a scheme \(T\). We have to show that \(\mathit{Isom}_\mathcal{X}(x, y)\) is an algebraic space over \(T\), see Algebraic Stacks, Lemma 045G. If for \(h : T' \to T\) the restrictions \(x|_{T'}\) and \(y|_{T'}\) are isomorphic in the fibre category \(\mathcal{X}_{T'}\), then \(g \circ h = g' \circ h\). Hence there is a transformation of presheaves \[\mathit{Isom}_\mathcal{X}(x, y) \longrightarrow \text{Equalizer}(g, g')\] Since the diagonal of \(B\) is representable (by schemes) this equalizer is a scheme. Thus we may replace \(T\) by this equalizer and \(E\) and \(E'\) by their pullbacks. Thus we may assume \(g = g'\).
Assume \(g = g'\). After replacing \(B\) by \(T\) and \(X\) by \(X_T\) we arrive at the following problem. Given \(E, E' \in D(\mathcal{O}_X)\) satisfying conditions (1), (2) of Lemma 0DLD we have to show that \(\mathit{Isom}(E, E')\) is an algebraic space. Here \(\mathit{Isom}(E, E')\) is the functor \[(\Sch/B)^{opp} \to \textit{Sets},\quad T \mapsto \{\varphi : E_T \to E'_T \text{ isomorphism in }D(\mathcal{O}_{X_T})\}\] where \(E_T\) and \(E'_T\) are the derived pullbacks of \(E\) and \(E'\) to \(X_T\). Now, let \(W \subset B\), resp. \(W' \subset B\) be the open subspace of \(B\) associated to \(E, E'\), resp. to \(E', E\) by Lemma 0DLC. Clearly, if there exists an isomorphism \(E_T \to E'_T\) as in the definition of \(\mathit{Isom}(E, E')\), then we see that \(T \to B\) factors into both \(W\) and \(W'\) (because we have condition (1) for \(E\) and \(E'\) and we’ll obviously have \(E_t \cong E'_t\) so no nonzero maps \(E_t[i] \to E_t\) or \(E'_t[i] \to E_t\) over the fibre \(X_t\) for \(i > 0\). Thus we may replace \(B\) by the open \(W \cap W'\). In this case the functor \(H = \SheafHom(E, E')\) \[(\Sch/B)^{opp} \to \textit{Sets},\quad T \mapsto \Hom_{\mathcal{O}_{X_T}}(E_T, E'_T)\] is an algebraic space affine and of finite presentation over \(B\) by Lemma 0DLC. The same is true for \(H' = \SheafHom(E', E)\), \(I = \SheafHom(E, E)\), and \(I' = \SheafHom(E', E')\). Therefore we can repeat the argument of the proof of Proposition 08K9 to see that \[\mathit{Isom}(E, E') = (H' \times_B H) \times_{c, I \times_B I', \sigma} B\] for some morphisms \(c\) and \(\sigma\). Thus \(\mathit{Isom}(E, E')\) is an algebraic space.
Lemma
In Situation 0DLE the functor \(p : \Complexesstack_{X/B} \longrightarrow (\Sch/S)_{fppf}\) is a stack in groupoids.
Proof
To prove that \(\Complexesstack_{X/B}\) is a stack in groupoids, we have to show that the presheaves \(\mathit{Isom}\) are sheaves and that descent data are effective. The statement on \(\mathit{Isom}\) follows from Lemma 0DLG, see Algebraic Stacks, Lemma 045G. Let us prove the statement on descent data.
Suppose that \(\{a_i : T_i \to T\}\) is an fppf covering of schemes over \(S\). Let \((\xi_i, \varphi_{ij})\) be a descent datum for \(\{T_i \to T\}\) with values in \(\Complexesstack_{X/B}\). For each \(i\) we can write \(\xi_i = (T_i, g_i, E_i)\). Denote \(\text{pr}_0 : T_i \times_T T_j \to T_i\) and \(\text{pr}_1 : T_i \times_T T_j \to T_j\) the projections. The condition that \(\xi_i|_{T_i \times_T T_j} \cong \xi_j|_{T_i \times_T T_j}\) implies in particular that \(g_i \circ \text{pr}_0 = g_j \circ \text{pr}_1\). Thus there exists a unique morphism \(g : T \to B\) such that \(g_i = g \circ a_i\), see Descent on Spaces, Lemma 04P2. Denote \(X_T = T \times_{g, B} X\). Set \(X_i = X_{T_i} = T_i \times_{g_i, B} X = T_i \times_{a_i, T} X_T\) and \[X_{ij} = X_{T_i} \times_{X_T} X_{T_j} = X_i \times_{X_T} X_j\] with projections \(\text{pr}_i\) and \(\text{pr}_j\) to \(X_i\) and \(X_j\). Observe that the pullback of \((T_i, g_i, E_i)\) by \(\text{pr}_0 : T_i \times_T T_j \to T_i\) is given by \((T_i \times_T T_j, g_i \circ \text{pr}_0, L\text{pr}_i^*E_i)\). Hence a descent datum for \(\{T_i \to T\}\) in \(\Complexesstack_{X/B}\) is given by the objects \((T_i, g \circ a_i, E_i)\) and for each pair \(i, j\) an isomorphism in \(D\mathcal{O}_{X_{ij}})\) \[\varphi_{ij} : L\text{pr}_i^*E_i \longrightarrow L\text{pr}_j^*E_j\] satisfying the cocycle condition over the pullback of \(X\) to \(T_i \times_T T_j \times_T T_k\). Using the vanishing of negative Exts provided by (b) of Lemma 0DLD, we may apply Simplicial Spaces, Lemma 0DLA to obtain descent5 for these complexes. In other words, we find there exists an object \(E\) in \(D_\QCoh(\mathcal{O}_{X_T})\) restricting to \(E_i\) on \(X_{T_i}\) compatible with \(\varphi_{ij}\). Recall that being \(T\)-perfect signifies being pseudo-coherent and having locally finite tor dimension over \(f^{-1}\mathcal{O}_T\). Thus \(E\) is \(T\)-perfect by an application of More on Morphisms of Spaces, Lemmas 0DL1 and 0DL2. Finally, we have to check condition (2) from Lemma 0DLD for \(E\). This immediately follows from the description of the open \(W\) in Lemma 0DLC and the fact that (2) holds for \(E_i\) on \(X_{T_i}/T_i\).
Remark
In Situation 0DLE the rule \((T, g, E) \mapsto (T, g)\) defines a \(1\)-morphism \[\Complexesstack_{X/B} \longrightarrow \mathcal{S}_B\] of stacks in groupoids (see Lemma 0DLH, Algebraic Stacks, Section 04SU, and Examples of Stacks, Section 0305). Let \(B' \to B\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{S}_{B'} \to \mathcal{S}_B\) be the associated \(1\)-morphism of stacks fibred in sets. Set \(X' = X \times_B B'\). We obtain a stack in groupoids \(\Complexesstack_{X'/B'} \to (\Sch/S)_{fppf}\) associated to the base change \(f' : X' \to B'\). In this situation the diagram \[\vcenter{ \xymatrix{ \Complexesstack_{X'/B'} \ar[r] \ar[d] & \Complexesstack_{X/B} \ar[d] \\ \mathcal{S}_{B'} \ar[r] & \mathcal{S}_B } } \quad \begin{matrix} \text{or in} \\ \text{another} \\ \text{notation} \end{matrix} \quad \vcenter{ \xymatrix{ \Complexesstack_{X'/B'} \ar[r] \ar[d] & \Complexesstack_{X/B} \ar[d] \\ \Sch/B' \ar[r] & \Sch/B } }\] is \(2\)-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space.
Lemma
In Situation 0DLE assume that \(B \to S\) is locally of finite presentation. Then \(p : \Complexesstack_{X/B} \to (\Sch/S)_{fppf}\) is limit preserving (Artin’s Axioms, Definition 07XL).
Proof
Write \(B(T)\) for the discrete category whose objects are the \(S\)-morphisms \(T \to B\). Let \(T = \lim T_i\) be a filtered limit of affine schemes over \(S\). Assigning to an object \((T, h, E)\) of \(\Complexesstack_{X/B, T}\) the object \(h\) of \(B(T)\) gives us a commutative diagram of fibre categories \[\xymatrix{ \colim \Complexesstack_{X/B, T_i} \ar[r] \ar[d] & \Complexesstack_{X/B, T} \ar[d] \\ \colim B(T_i) \ar[r] & B(T) }\] We have to show the top horizontal arrow is an equivalence. Since we have assume that \(B\) is locally of finite presentation over \(S\) we see from Limits of Spaces, Remark 05N0 that the bottom horizontal arrow is an equivalence. This means that we may assume \(T = \lim T_i\) be a filtered limit of affine schemes over \(B\). Denote \(g_i : T_i \to B\) and \(g : T \to B\) the corresponding morphisms. Set \(X_i = T_i \times_{g_i, B} X\) and \(X_T = T \times_{g, B} X\). Observe that \(X_T = \colim X_i\). By More on Morphisms of Spaces, Lemma 0DKW the category of \(T\)-perfect objects of \(D(\mathcal{O}_{X_T})\) is the colimit of the categories of \(T_i\)-perfect objects of \(D(\mathcal{O}_{X_{T_i}})\). Thus all we have to prove is that given an \(T_i\)-perfect object \(E_i\) of \(D(\mathcal{O}_{X_{T_i}})\) such that the derived pullback \(E\) of \(E_i\) to \(X_T\) satisfies condition (2) of Lemma 0DLD, then after increasing \(i\) we have that \(E_i\) satisfies condition (2) of Lemma 0DLD. Let \(W \subset |T_i|\) be the open constructed in Lemma 0DLC for \(E_i\) and \(E_i\). By assumption on \(E\) we find that \(T \to T_i\) factors through \(T\). Hence there is an \(i' \geq i\) such that \(T_{i'} \to T_i\) factors through \(W\), see Limits, Lemma 05F4 Then \(i'\) works by construction of \(W\).
Lemma
In Situation 0DLE. Let \[\xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' }\] be a pushout in the category of schemes over \(S\) where \(Z \to Z'\) is a finite order thickening and \(Z \to Y\) is affine, see More on Morphisms, Lemma 07RT. Then the functor on fibre categories \[\Complexesstack_{X/B, Y'} \longrightarrow \Complexesstack_{X/B, Y} \times_{\Complexesstack_{X/B, Z}} \Complexesstack_{X/B, Z'}\] is an equivalence.
Proof
Observe that the corresponding map \[B(Y') \longrightarrow B(Y) \times_{B(Z)} B(Z')\] is a bijection, see Pushouts of Spaces, Lemma 07SY. Thus using the commutative diagram \[\xymatrix{ \Complexesstack_{X/B, Y'} \ar[r] \ar[d] & \Complexesstack_{X/B, Y} \times_{\Complexesstack_{X/B, Z}} \Complexesstack_{X/B, Z'} \ar[d] \\ B(Y') \ar[r] & B(Y) \times_{B(Z)} B(Z') }\] we see that we may assume that \(Y'\) is a scheme over \(B'\). By Remark 0DLI we may replace \(B\) by \(Y'\) and \(X\) by \(X \times_B Y'\). Thus we may assume \(B = Y'\).
Assume \(B = Y'\). We first prove fully faithfulness of our functor. To do this, let \(\xi_1, \xi_2\) be two objects of \(\Complexesstack_{X/B}\) over \(Y'\). Then we have to show that \[\mathit{Isom}(\xi_1, \xi_2)(Y') \longrightarrow \mathit{Isom}(\xi_1, \xi_2)(Y) \times_{\mathit{Isom}(\xi_1, \xi_2)(Z)} \mathit{Isom}(\xi_1, \xi_2)(Z')\] is bijective. However, we already know that \(\mathit{Isom}(\xi_1, \xi_2)\) is an algebraic space over \(B = Y'\). Thus this bijectivity follows from Artin’s Axioms, Lemma 07WN (or the aforementioned Pushouts of Spaces, Lemma 07SY).
Essential surjectivity. Let \((E_Y, E_{Z'}, \alpha)\) be a triple, where \(E_Y \in D(\mathcal{O}_Y)\) and \(E_{Z'} \in D(\mathcal{O}_{X_{Z'}})\) are objects such that \((Y, Y \to B, E_Y)\) is an object of \(\Complexesstack_{X/B}\) over \(Y\), such that \((Z', Z' \to B, E_{Z'})\) is an object of \(\Complexesstack_{X/B}\) over \(Z'\), and \(\alpha : L(X_Z \to X_Y)^*E_Y \to L(X_Z \to X_{Z'})^*E_{Z'}\) is an isomorphism in \(D(\mathcal{O}_{Z'})\). That is to say \[((Y, Y \to B, E_Y), (Z', Z' \to B, E_{Z'}), \alpha)\] is an object of the target of the arrow of our lemma. Observe that the diagram \[\xymatrix{ X_Z \ar[r] \ar[d] & X_{Z'} \ar[d] \\ X_Y \ar[r] & X_{Y'} }\] is a pushout with \(X_Z \to X_Y\) affine and \(X_Z \to X_{Z'}\) a thickening (see Pushouts of Spaces, Lemma 07W3). Hence by Pushouts of Spaces, Lemma 0DL7 we find an object \(E_{Y'} \in D(\mathcal{O}_{X_{Y'}})\) together with isomorphisms \(L(X_Y \to X_{Y'})^*E_{Y'} \to E_Y\) and \(L(X_{Z'} \to X_{Y'})^*E_{Y'} \to E_Z\) compatible with \(\alpha\). Clearly, if we show that \(E_{Y'}\) is \(Y'\)-perfect, then we are done, because property (2) of Lemma 0DLD is a property on points (and \(Y\) and \(Y'\) have the same points). This follows from More on Morphisms of Spaces, Lemma 0DL4.
Lemma
In Situation 0DLE assume that \(S\) is a locally Noetherian scheme and \(B \to S\) is locally of finite presentation. Let \(k\) be a finite type field over \(S\) and let \(x_0 = (\Spec(k), g_0, E_0)\) be an object of \(\mathcal{X} = \Complexesstack_{X/B}\) over \(k\). Then the spaces \(T\mathcal{F}_{\mathcal{X}, k, x_0}\) and \(\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})\) (Artin’s Axioms, Section 07WY) are finite dimensional.
Proof
Observe that by Lemma 0DLK our stack in groupoids \(\mathcal{X}\) satisfies property (RS*) defined in Artin’s Axioms, Section 0CXN. In particular \(\mathcal{X}\) satisfies (RS). Hence all associated predeformation categories are deformation categories (Artin’s Axioms, Lemma 07WU) and the statement makes sense.
In this paragraph we show that we can reduce to the case \(B = \Spec(k)\). Set \(X_0 = \Spec(k) \times_{g_0, B} X\) and denote \(\mathcal{X}_0 = \Complexesstack_{X_0/k}\). In Remark 0DLI we have seen that \(\mathcal{X}_0\) is the \(2\)-fibre product of \(\mathcal{X}\) with \(\Spec(k)\) over \(B\) as categories fibred in groupoids over \((\Sch/S)_{fppf}\). Thus by Artin’s Axioms, Lemma 07X2 we reduce to proving that \(B\), \(\Spec(k)\), and \(\mathcal{X}_0\) have finite dimensional tangent spaces and infinitesimal automorphism spaces. The tangent space of \(B\) and \(\Spec(k)\) are finite dimensional by Artin’s Axioms, Lemma 07X1 and of course these have vanishing \(\text{Inf}\). Thus it suffices to deal with \(\mathcal{X}_0\).
Let \(k[\epsilon]\) be the dual numbers over \(k\). Let \(\Spec(k[\epsilon]) \to B\) be the composition of \(g_0 : \Spec(k) \to B\) and the morphism \(\Spec(k[\epsilon]) \to \Spec(k)\) coming from the inclusion \(k \to k[\epsilon]\). Set \(X_0 = \Spec(k) \times_B X\) and \(X_\epsilon = \Spec(k[\epsilon]) \times_B X\). Observe that \(X_\epsilon\) is a first order thickening of \(X_0\) flat over the first order thickening \(\Spec(k) \to \Spec(k[\epsilon])\). Observe that \(X_0\) and \(X_\epsilon\) give rise to canonically equivalent small étale topoi, see More on Morphisms of Spaces, Section 05ZJ. By More on Morphisms of Spaces, Lemma 0DL4 we see that \(T\mathcal{F}_{\mathcal{X}_0, k, x_0}\) is the set of isomorphism classes of lifts of \(E_0\) to \(X_\epsilon\) in the sense of Deformation Theory, Lemma 0DIZ. We conclude that \[T\mathcal{F}_{\mathcal{X}_0, k, x_0} = \Ext^1_{\mathcal{O}_{X_0}}(E_0, E_0)\] Here we have used the identification \(\epsilon k[\epsilon] \cong k\) of \(k[\epsilon]\)-modules. Using Deformation Theory, Lemma 0DIZ once more we see that there is a surjection \[\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) \leftarrow \Ext^0_{\mathcal{O}_{X_0}}(E_0, E_0)\] of \(k\)-vector spaces. As \(E_0\) is pseudo-coherent it lies in \(D^-_{\textit{Coh}}(\mathcal{O}_{X_0})\) by Derived Categories of Spaces, Lemma 08IK. Since \(E_0\) locally has finite tor dimension and \(X_0\) is quasi-compact we see \(E_0 \in D^b_{\textit{Coh}}(\mathcal{O}_{X_0})\). Thus the \(\Ext\)s above are finite dimensional \(k\)-vector spaces by Derived Categories of Spaces, Lemma 0D0T.
Lemma
In Situation 0DLE assume \(B = S\) is locally Noetherian. Then strong formal effectiveness in the sense of Artin’s Axioms, Remark 0CXT holds for \(p : \Complexesstack_{X/S} \to (\Sch/S)_{fppf}\).
Proof
Let \((R_n)\) be an inverse system of \(S\)-algebras with surjective transition maps whose kernels are locally nilpotent. Set \(R = \lim R_n\). Let \((\xi_n)\) be a system of objects of \(\Complexesstack_{X/B}\) lying over \((\Spec(R_n))\). We have to show \((\xi_n)\) is effective, i.e., there exists an object \(\xi\) of \(\Complexesstack_{X/B}\) lying over \(\Spec(R)\).
Write \(X_R = \Spec(R) \times_S X\) and \(X_n = \Spec(R_n) \times_S X\). Of course \(X_n\) is the base change of \(X_R\) by \(R \to R_n\). Since \(S = B\), we see that \(\xi_n\) corresponds simply to an \(R_n\)-perfect object \(E_n \in D(\mathcal{O}_{X_n})\) satisfying condition (2) of Lemma 0DLD. In particular \(E_n\) is pseudo-coherent. The isomorphisms \(\xi_{n + 1}|_{\Spec(R_n)} \cong \xi_n\) correspond to isomorphisms \(L(X_n \to X_{n + 1})^*E_{n + 1} \to E_n\). Therefore by Flatness on Spaces, Theorem 0DIQ we find a pseudo-coherent object \(E\) of \(D(\mathcal{O}_{X_R})\) with \(E_n\) equal to the derived pullback of \(E\) for all \(n\) compatible with the transition isomorphisms.
Observe that \((R, \Ker(R \to R_1))\) is a henselian pair, see More on Algebra, Lemma 0CT7. In particular, \(\Ker(R \to R_1)\) is contained in the Jacobson radical of \(R\). Then we may apply More on Morphisms of Spaces, Lemma 0DL5 to see that \(E\) is \(R\)-perfect.
Finally, we have to check condition (2) of Lemma 0DLD. By Lemma 0DLC the set of points \(t\) of \(\Spec(R)\) where the negative self-exts of \(E_t\) vanish is an open. Since this condition is true in \(V(\Ker(R \to R_1))\) and since \(\Ker(R \to R_1)\) is contained in the Jacobson radical of \(R\) we conclude it holds for all points.
Theorem
Let \(S\) be a scheme. Let \(f : X \to B\) be morphism of algebraic spaces over \(S\). Assume that \(f\) is proper, flat, and of finite presentation. Then \(\Complexesstack_{X/B}\) is an algebraic stack over \(S\).
Proof
Set \(\mathcal{X} = \Complexesstack_{X/B}\). We have seen that \(\mathcal{X}\) is a stack in groupoids over \((\Sch/S)_{fppf}\) with diagonal representable by algebraic spaces (Lemmas 0DLH and 0DLG). Hence it suffices to find a scheme \(W\) and a surjective and smooth morphism \(W \to \mathcal{X}\).
Let \(B'\) be a scheme and let \(B' \to B\) be a surjective étale morphism. Set \(X' = B' \times_B X\) and denote \(f' : X' \to B'\) the projection. Then \(\mathcal{X}' = \Complexesstack_{X'/B'}\) is equal to the \(2\)-fibre product of \(\mathcal{X}\) with the category fibred in sets associated to \(B'\) over the category fibred in sets associated to \(B\) (Remark 0DLI). By the material in Algebraic Stacks, Section 03YJ the morphism \(\mathcal{X}' \to \mathcal{X}\) is surjective and étale. Hence it suffices to prove the result for \(\mathcal{X}'\). In other words, we may assume \(B\) is a scheme.
Assume \(B\) is a scheme. In this case we may replace \(S\) by \(B\), see Algebraic Stacks, Section 04X4. Thus we may assume \(S = B\).
Assume \(S = B\). Choose an affine open covering \(S = \bigcup U_i\). Denote \(\mathcal{X}_i\) the restriction of \(\mathcal{X}\) to \((\Sch/U_i)_{fppf}\). If we can find schemes \(W_i\) over \(U_i\) and surjective smooth morphisms \(W_i \to \mathcal{X}_i\), then we set \(W = \coprod W_i\) and we obtain a surjective smooth morphism \(W \to \mathcal{X}\). Thus we may assume \(S = B\) is affine.
Assume \(S = B\) is affine, say \(S = \Spec(\Lambda)\). Write \(\Lambda = \colim \Lambda_i\) as a filtered colimit with each \(\Lambda_i\) of finite type over \(\mathbf{Z}\). For some \(i\) we can find a morphism of algebraic spaces \(X_i \to \Spec(\Lambda_i)\) which is proper, flat, of finite presentation and whose base change to \(\Lambda\) is \(X\). See Limits of Spaces, Lemmas 07SK, 08K0, and 08K1. If we show that \(\Complexesstack_{X_i/\Spec(\Lambda_i)}\) is an algebraic stack, then it follows by base change (Remark 0DLI and Algebraic Stacks, Section 04X4) that \(\mathcal{X}\) is an algebraic stack. Thus we may assume that \(\Lambda\) is a finite type \(\mathbf{Z}\)-algebra.
Assume \(S = B = \Spec(\Lambda)\) is affine of finite type over \(\mathbf{Z}\). In this case we will verify conditions (1), (2), (3), (4), and (5) of Artin’s Axioms, Lemma 07Y4 to conclude that \(\mathcal{X}\) is an algebraic stack. Note that \(\Lambda\) is a G-ring, see More on Algebra, Proposition 07PX. Hence all local rings of \(S\) are G-rings. Thus (5) holds. To check (2) we have to verify axioms [-1], [0], [1], [2], and [3] of Artin’s Axioms, Section 07XJ. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas 0DLH, 0DLJ, 0DLK, 0DLL. Condition (3) follows from Lemma 0DLM. Condition (1) is Lemma 0DLG.
It remains to show condition (4) which is openness of versality. To see this we will use Artin’s Axioms, Lemma 0CXU. We have already seen that \(\mathcal{X}\) has diagonal representable by algebraic spaces, has (RS*), and is limit preserving (see lemmas used above). Hence we only need to see that \(\mathcal{X}\) satisfies the strong formal effectiveness formulated in Artin’s Axioms, Lemma 0CXU. This follows from Lemma 0DLM and the proof is complete.
We omit the verification of the set theoretical condition (3) of the referenced lemma.↩︎
We always perform a replacement as in Examples of Stacks, Lemma 04UE.↩︎
To check the sheaf property for a covering \(\{V_i \to V\}_{i \in I}\) first consider the Čech fppf hypercovering \(a : V_\bullet \to V\) with \(V_n = \coprod_{i_0 \ldots i_n} V_{i_0} \times_V \ldots \times_V V_{i_n}\) and then set \(U_\bullet = V_\bullet \times_{a, V} X_V\). Then \(U_\bullet \to X_V\) is an fppf hypercovering to which we may apply Simplicial Spaces, Lemma 0DL9.↩︎
To check this, first consider the Čech fppf hypercovering \(a : T_\bullet \to T\) with \(T_n = \coprod_{i_0 \ldots i_n} T_{i_0} \times_T \ldots \times_T T_{i_n}\) and then set \(U_\bullet = T_\bullet \times_{a, T} X_T\). Then \(U_\bullet \to X_T\) is an fppf hypercovering to which we may apply Simplicial Spaces, Lemma 0DLA.↩︎