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Publicly Available Published by De Gruyter August 9, 2014

Openness of versality via coherent functors

Jack Hall

Abstract

We give a proof of openness of versality using coherent functors. As an application, we streamline Artin’s criterion for algebraicity of a stack. We also introduce multi-step obstruction theories, employing them to produce obstruction theories for the stack of coherent sheaves, the Quot functor, and spaces of maps in the presence of non-flatness.

Introduction

In M. Artin’s classic paper on stacks, a criterion for algebraicity is expounded [7, Theorem 5.3]. In the present paper, we take a novel approach to algebraicity, proving an algebraicity criterion for stacks which is easier to apply, more widely applicable, and admitting a substantially simpler proof.

Theorem A

Fix an excellent scheme S and a category X that is fibered in groupoids over the category of S-schemes, Sch/S. Then, X is an algebraic stack that is locally of finite presentation over S, if and only if the following conditions are satisfied.

  1. [Stack]X is a stack over the site (𝐒𝐜𝐡/S)E´t.

  2. [Limit preservation] For any inverse system of affine S-schemes {SpecAj}jJ with limit SpecA, the natural functor

    limjX(SpecAj)X(SpecA)

    is an equivalence of categories.

  3. [Homogeneity] For any diagram of affine S-schemes [SpecBSpecA𝑖SpecA], where i is a nilpotent closed immersion, the natural functor

    X(Spec(B×AA))X(SpecA)×X(SpecA)X(SpecB)

    is an equivalence of categories.

  4. [Effectivity] For any 𝔪-adically complete local noetherian ring (B,𝔪) with an S-scheme structure SpecBS such that the induced morphism Spec(B/𝔪)S is locally of finite type, the natural functor

    X(SpecB)limnX(Spec(B/𝔪n))

    is an equivalence of categories.

  5. [Conditions on automorphisms and deformations] For any affine S-scheme T that is locally of finite type over S and ξX(T), the functors

    AutX/S(ξ,-),DefX/S(ξ,-):𝐐𝐂𝐨𝐡(T)𝐀𝐛

    are coherent.

  6. [Conditions on obstructions] For any affine S-scheme T that is locally of finite type over S and ξX(T), there exist an integer n and a coherent n-step obstruction theory for X at ξ.

Except for conditions (5) and (6), Theorem A is similar to Artin’s criterion [7, Theorem 5.3]. Note, however, that we have fewer conditions, and these conditions are cleaner (e.g. no deformation situations). The conditions of Theorem A are also stable under composition, in the sense of [45].

This paper began with the realization that the homogeneity condition (3), which is stronger than the analogous condition of [7, (S1)], together with conditions (5) and (6), simplifies and broadens the applicability of existing results.

Our usage of the term “coherent” in conditions (5) and (6) of Theorem A is in a different sense than what many readers may be familiar with and is due to M. Auslander [9]. For an affine scheme S, a functor F:𝐐𝐂𝐨𝐡(S)𝐀𝐛 is coherent if there exists a morphism of quasi-coherent 𝒪S-modules 𝒦1𝒦2 such that for all 𝐐𝐂𝐨𝐡(S), there is a natural isomorphism of abelian groups

F()coker(Hom𝒪S(𝒦2,)Hom𝒪S(𝒦1,)).

It is proven in [20] that most functors arising in moduli are coherent.

Relation with other work

The idea of using the Exal functors to simplify M. Artin’s results is due to H. Flenner [15]. Our results and techniques are quite different, however. In particular, H. Flenner [15] does not address the relationship between formal smoothness and formal versality.

Independently, work in the Stacks Project [44, 07T0] has provided a different perspective on Artin’s results. This approach, however, requires that the deformation–obstruction theory is given by a bounded complex. If there are non-flat or non-tame objects in the moduli problem, the existence of such a complex is subtle. Note that while the problems with non-tame stacks can be dealt with by [20, Theorem B], the problems with non-flatness likely needed to be handled by derived algebraic geometry [44, blog:2572].

Using the ideas of B. Töen and G. Vezzosi [46, §1.4], J. Lurie has developed a criterion for algebraicity in the derived context [31, Theorem 3.2.1]. Conditions (5) and (6) of Theorem A are related to Lurie’s requirement of the existence of a cotangent complex. Lurie’s criterion is not applicable to Artin stacks, though it is a future intention [31, Remark 2]. J. Pridham has proved a criterion for Artin stacks [39, Theorem 3.16] which is related to the results of Lurie’s Ph.D. thesis [30, Theorems 7.1.6, 7.5.1] and also exploits a derived analogue of the homogeneity condition (3) in order to simplify Lurie’s conditions.

To prove that the Quot functors for separated Deligne–Mumford stacks are algebraic spaces, M. Olsson and J. Starr [38, Theorem 1.1] did not apply [7, Corollary 5.4], which like [7, Theorem 5.3] is formulated in terms of a single-step obstruction theory. The reason for this is simple: in the presence of non-flatness, it is difficult to formulate a single-step obstruction theory with good properties.

They circumvented this predicament by the use of Artin’s original algebraicity criterion [6, Theorem 5.3]. This earlier algebraicity criterion is not formulated in terms of the existence and properties of a single-step obstruction theory – but in terms of certain explicit lifting problems – making its application more complicated (note that J. Starr [45, Theorem 2.15] has subsequently generalized the criteria of [6, Theorem 5.3] to stacks). To solve these lifting problems, M. Olsson and J. Starr [38, Lemma 2.5] used a 2-step process. This 2-step process is insufficiently functorial to define a multi-step obstruction theory in the sense of this paper. It is, however, closely related, and inspired the multi-step obstruction theories we define.

M. Olsson and J. Starr [38, p. 4077] noted that M. Artin had incorrectly computed the obstruction theory of the Quot functor in the presence of non-flatness [6, (6.4)]. We have also located some other articles in the literature that have not observed the subtlety of deformation theory in the presence of non-flatness (see Sections 89). We would like to emphasize that the impact of this on the main ideas of these articles is small. Indeed, the relevant arguments in these articles are still perfectly valid in the flat case, which covers most cases of interest to geometers. In the non-flat case, the relevant statements in these articles can be shown to hold with the techniques and examples of this article.

By work of M. Olsson [36, Remark 1.7], the conditions of Theorem A are seen to be necessary. The sufficiency of the conditions of Theorem A is demonstrated by the following sequence of observations:

  1. formally versal deformations exist,

  2. algebraizations of formally versal deformations exist, and

  3. formal versality at a point implies smoothness in a neighborhood.

Using the generalizations of M. Artin’s techniques [7] due to B. Conrad and J. de Jong [11, Theorem 1.5], conditions (1)–(4) of Theorem A prove (i) and (ii). The main contribution of this paper is the usage of conditions (3), (5), and (6) of Theorem A to prove (iii).

Note that in our proof of (iii), the techniques of Artin approximation [5] are not used. This is in contrast to M. Artin’s treatments [6, 7], where this technique features prominently. In a paper joint with D. Rydh [21], we illustrate how refinements of the homogeneity condition (3) clarify and simplify M. Artin’s results on versality.

Outline

In Section 1, we discuss the notion of homogeneity. Homogeneity is a generalization of the Schlessinger–Rim criteria [12, Exposé VI]. This section is quite categorical, but it is the only section of the paper that is such. Morally, homogeneity provides a stack X with a linear structure at every point, which we describe in Section 2. To be precise, for any scheme T, together with an object ξX(T), homogeneity produces an additive functor ExalX(ξ,-):𝐐𝐂𝐨𝐡(T)𝐀𝐛 sharply controlling the deformation theory of ξ. The author learnt these ideas from J. Wise (in person) and his paper [48], though they are likely well known to experts, and go back at least as far as the work of H. Flenner [15]. In Section 3, we recall and generalize – to the relative setting – the notion of limit preserving groupoid [7, §1]. The results in Section 3 are similar to those obtained by Lieblich–Osserman [27, §2.4].

In Section 4, we recall the notions of formal versality and formal smoothness. We next recast these notions in terms of vanishing criteria for the functors ExalX(T,-). The central technical result of this paper is Theorem 4.4 – our new proof of (iii).

In Section 5, we briefly review coherent functors. In Section 6, we formalize multi-step obstruction theories. In Section 7, we prove Theorem A.

The remainder of the paper is devoted to applications. In Section 8, we compute a 2-step obstruction theory for the stack of coherent sheaves. Finally, in Section 9, we compute a 2-step obstruction theory for the stack of morphisms between two algebraic stacks.

In Appendix A, we prove that pushouts of algebraic stacks along nilimmersions and affine morphisms exist. This aids in the verification of the homogeneity condition (3) in practice. In Appendix B, we consider left-exact sequences of Picard categories and a resulting 7-term exact sequence. In Appendix C, we state two basic results on local Tor-functors for morphisms of algebraic stacks.

Assumptions, conventions, and notations

If 𝒞 is a category, then denote the opposite category by 𝒞. A fibration of categories Q:𝒞𝒟 has the property that every arrow in the category 𝒟 admits a strongly cartesian lift. For an object d of the category 𝒟, we denote the resulting fiber category by Q(d). It will also be convenient to say that the category 𝒞 is fibered over 𝒟. If the category 𝒞 is fibered over 𝒟 and every arrow in the category 𝒞 is strongly cartesian, then we say that the functor Q is fibered in groupoids. The assumptions guarantee that if the category 𝒞 is fibered in groupoids over 𝒟, then for every object d of the category 𝒟 the fiber category Q(d) is a groupoid.

Let T be a scheme. Denote by |T| the underlying topological space (with the Zariski topology) and 𝒪T the (Zariski) sheaf of rings on |T|. If t|T|, then let κ(t) denote its residue field. Denote by 𝐐𝐂𝐨𝐡(T) the abelian category of quasi-coherent sheaves on the scheme T. Let 𝐒𝐜𝐡/T denote the category of schemes over T. The big étale site over T will be denoted by (𝐒𝐜𝐡/T)E´t. If T is locally noetherian, then let 𝐂𝐨𝐡(T) denote the abelian category coherent sheaves on T.

Let A be a ring and let M be an A-module. Denote the quasi-coherent 𝒪SpecA-module associated to M by M~. Denote the abelian category of all (resp. coherent) A-modules by 𝐌𝐨𝐝(A) (resp. 𝐂𝐨𝐡(A)).

As in [44], we make no separation assumptions on our algebraic stacks and spaces. As in [37], we use the lisse-étale site for sheaves on algebraic stacks.

Fix a 1-morphism of algebraic stacks f:XY. Given another 1-morphism of algebraic stacks WY we denote the pullback along this 1-morphism by fW:XWW.

A morphism of algebraic S-stacks UV is a locally nilpotent closed immersion if it is a closed immersion defined by a quasi-coherent sheaf of ideals , such that fppf-locally (equivalently, smooth-locally) on V there always exists an integer n such that n=(0).

1 Homogeneity

Schlessinger’s conditions [42], for a functor of artinian rings, are fundamental to infinitesimal deformation theory. Schlessinger’s conditions were generalized to groupoids by R. S. Rim [12, Exposé VI], clarifying infinitesimal deformation theory in the presence of automorphisms. Schlessinger’s and Rim’s conditions are both instances of the notion of homogeneity, which can be traced back to A. Grothendieck [18, no. 195]. A generalization of Rim’s conditions was recently considered by J. Wise [48, §2]. In this section, we will develop a relative formulation of homogeneity for use in this paper.

Throughout this section, we let S be a scheme. An S-groupoid is a pair (X,aX), where X is a category and aX:X𝐒𝐜𝐡/S is a fibration in groupoids. A 1-morphism of S-groupoids Φ:(Y,aY)(Z,aZ) is a functor Φ:YZ that commutes strictly over 𝐒𝐜𝐡/S. We will typically refer to an S-groupoid (X,aX) just as “X”.

Example 1.1

For any S-scheme T, there is a canonical functor

𝐒𝐜𝐡/T𝐒𝐜𝐡/S:(WT)(WTS)

which is faithful. In particular, we may view an S-scheme T as an S-groupoid. Thus a morphism of S-schemes g:UV induces a 1-morphism of S-groupoids

𝐒𝐜𝐡/g:𝐒𝐜𝐡/U𝐒𝐜𝐡/V.

The converse is also true: any 1-morphism of S-groupoids G:𝐒𝐜𝐡/U𝐒𝐜𝐡/V is uniquely isomorphic to a 1-morphism of the form 𝐒𝐜𝐡/g for some morphism of S-schemes g:UV.

Definition 1.2

Fix an S-groupoid X. An X-scheme is a pair (T,σT) consisting of an S-scheme T together with a 1-morphism of S-groupoids σT:𝐒𝐜𝐡/TX. A morphism of X-schemes (f,αf):(U,σU)(V,σV) is given by a morphism of S-schemes f:UV together with a 2-morphism αf:σUσV𝐒𝐜𝐡/f. The collection of all X-schemes forms a 1-category, which we denote as 𝐒𝐜𝐡/X.

For a 1-morphism of S-groupoids Φ:YZ there is an induced functor

𝐒𝐜𝐡/Φ:𝐒𝐜𝐡/Y𝐒𝐜𝐡/Z.

It is readily seen that for an S-groupoid X, the category 𝐒𝐜𝐡/X is also an S-groupoid. The content of the 2-Yoneda Lemma is essentially that the natural 1-morphism of S-groupoids 𝐒𝐜𝐡/XX is an equivalence. An inverse to this equivalence is given by picking a clivage for X.

The principal advantage of working with the fibered category 𝐒𝐜𝐡/X is that it admits a canonical clivage. In practice, this means that given an X-scheme V and a morphism of S-schemes p:UV, the way to make U an X-scheme is already chosen for us: it is the composition

𝐒𝐜𝐡/U𝐒𝐜𝐡/p𝐒𝐜𝐡/VX.

It is for this reason that working with 𝐒𝐜𝐡/X greatly simplifies proofs and definitions. Calculations, however, are typically easier to perform in X.

Fix a class P of morphisms of S-schemes and an S-groupoid X. Then, a morphism of X-schemes p:UV is P if the underlying morphism of S-schemes is P. The following squares will feature frequently and prominently throughout the article.

Definition 1.3

Fix a scheme S, a class P of morphisms of S-schemes, and an S-groupoid X. A P-nil pair over X is a pair

(V𝑝T,V𝑗V),

where p and j are morphisms of X-schemes, p is P, and j is a locally nilpotent closed immersion. A P-nil square over X is a commutative diagram of X-schemes:

(1.1)

(1.1)

where the pair (V𝑝T,V𝑗V) is P-nil over X. A P-nil square over X is cocartesian if it is cocartesian in the category of X-schemes. A P-nil square over X is geometric if p is affine, i is a locally nilpotent closed immersion, and there is a natural isomorphism

𝒪Ti*𝒪T×p*j*𝒪Vp*𝒪V.

The following definition is a trivial generalization of the ideas of M. Olsson [34, Appendix A], J. Starr [45, §2], and J. Wise [48, §2].

Definition 1.4

Definition 1.4 (P-homogeneity)

Fix a scheme S and a class P of morphisms of S-schemes. A 1-morphism of S-groupoids Φ:YZ is P-homogeneous if the following two conditions are satisfied.

  1. A P-nil square over Y is cocartesian if and only if the induced P-nil square over Z is cocartesian.

  2. If a P-nil pair over Y can be completed to a cocartesian P-nil square over Z, then it can be completed to a P-nil square over Y.

An S-groupoid X is P-homogeneous if its structure 1-morphism is P-homogeneous.

For homogeneity, we will be interested in the following classes of morphisms:

  1. 𝐍𝐢𝐥 – locally nilpotent closed immersions,

  2. 𝐂𝐥 – closed immersions,

  3. 𝐫𝐍𝐢𝐥 – morphisms VT such that there exists (V0V)𝐍𝐢𝐥 with the composition (V0VT)𝐍𝐢𝐥,

  4. 𝐫𝐂𝐥 – morphisms VT such that there exists (V0V)𝐍𝐢𝐥 with the composition (V0VT)𝐂𝐥,

  5. 𝐀𝐟𝐟 – affine morphisms.

By [17, IV.18.12.11], universal homeomorphisms of schemes are integral, thus affine. Hence, there is a containment of classes of morphisms of S-schemes:

In [21, Appendix A], it is shown that if X is limit preserving, in the sense of [7, §1], and a stack for the Zariski topology, then 𝐫𝐂𝐥-homogeneity is equivalent to the condition (S1) of [7, (2.3)]. To assuage any fears of circularity, we would like to emphasize that this result will not be used in this paper.

J. Wise [48, Proposition 2.1] has shown that every algebraic stack is 𝐀𝐟𝐟-homogeneous. In Appendix A, we generalize results of D. Ferrand [14] and obtain techniques to prove that many “geometric” moduli problems are 𝐀𝐟𝐟-homogeneous.

The following definition is a convenient computational tool. A 1-morphism of S-groupoids Φ:YZ is formally étale if for every Z-scheme V and every locally nilpotent closed immersion of Z-schemes VV, then every Y-scheme structure on V that is compatible with its Z-scheme structure under Φ lifts uniquely to a compatible Y-scheme structure on V. That is, there is always a unique solution to the following lifting problem:

Note that if Y is a stack for the étale topology, then it suffices to verify the above lifting property étale-locally on V. Indeed, the uniqueness in the definition of formally étale guarantees the cocycle condition necessary to perform the descent. Also, if Y and Z are schemes, then the induced 1-morphism of S-groupoids 𝐒𝐜𝐡/Y𝐒𝐜𝐡/Z is formally étale if and only if the morphism of schemes YZ is formally étale [17, IV4.17.1.1].

The following lemma provides several methods to prove that a 1-morphism of S-groupoids is P-homogeneous, at least in the situation where P𝐀𝐟𝐟.

Lemma 1.5

Fix a scheme S, a 1-morphism of S-groupoids Φ:YZ, and a class PAff of morphisms of S-schemes.

  1. Every cocartesian P-nil square over Y is geometric. In particular, if Z satisfies (HP1), then every cocartesian P-nil square over Y is cocartesian over Z.

  2. Let (V𝑝T,V𝑗V) be a P-nil pair over Y that may be completed to a cocartesian P-nil square over Z as in (1.1). If Φ is P-homogeneous, then this cocartesian P-nil square over Z lifts uniquely to a P-nil square over Y that is simultaneously cocartesian and geometric.

  3. Let W be a P-homogeneous S-groupoid. Then every P-nil pair (V𝑝T,T𝑗V) over W can be completed to a P-nil square over W that is simultaneously cocartesian and geometric. In particular, P-nil squares over W are cocartesian if and only if they are geometric.

  4. If W is an S-groupoid that is a stack for the Zariski topology, then W is P-homogeneous if and only if for every P-nil pair (SpecASpecB,SpecASpecA) over S, the naturally induced functor

    W(Spec(B×AA))W(SpecB)×W(SpecA)W(SpecA)

    is an equivalence of categories.

  5. Let Ψ:WY be a 1 -morphism of S-groupoids. If Φ is P-homogeneous, then Ψ is P-homogeneous if and only if ΦΨ is P-homogeneous.

  6. If Z is P-homogeneous, then the 1 -morphism Φ is P-homogeneous if and only if for every Z-scheme W, the W-groupoid Y×Z(𝐒𝐜𝐡/W) is P-homogeneous.

  7. If Z and Φ are P-homogeneous, then for every P-homogeneous 1 -morphism of S-groupoids Ψ:WZ, the 1 -morphism Y×ZWW is P-homogeneous.

  8. If Z and Φ are P-homogeneous, then the diagonal 1 -morphism ΔΦ:YY×ZY is P-homogeneous.

  9. If Z is P-homogeneous and Φ is formally étale, then Φ and Y are P-homogeneous.

Proof.

For (1), fix a cocartesian P-nil square over Y as in (1.1). By [14, Théorème 7.1], the induced P-nil pair (V𝑝T,V𝑗V) over Y may be completed to the following cocartesian P-nil square over S which is geometric:

The universal properties produce a unique S-morphism t:T~T. The morphism t promotes T~ to a Y-scheme and it follows from the universal property defining T that there is a uniquely induced Y-morphism u:TT~ such that tu=IdT. The universal property defining T~ in the category of S-schemes shows that ut=IdT~. Thus u is an isomorphism over Y and the result follows.

For (2), by (HP2), it follows that there is a P-nil square over Y,

The P-nil square over Y above induces a P-nil square over Z. Since the P-nil square over Z as in (1.1) is cocartesian, it follows that there is a uniquely induced Z-morphism TT′′ that is compatible with the data. Since T′′ is a Y-scheme, T inherits the structure of a Y-scheme. It follows that the cocartesian P-nil square over Z as in (1.1) lifts to a P-nil square over Y and, by (HP1), it is cocartesian and thus the lifting is unique. That the resulting square is geometric follows from (1).

The claims (3) and (5) both follow from (1) and (2).

The claims (4) and (6) both follow from (1), (2), and (3).

The claim (7) follows from (6).

For (8), by (7) and (5), we know that Y×ZY is P-homogeneous. The result now follows from (5) applied to YY×ZYY.

For (9), by (5), it is sufficient to prove that Φ is P-homogeneous. Since Φ is formally étale, we may use (1) to deduce that Φ satisfies (HP1). By (3), every cocartesian P-nil square over Z is geometric. Since Φ is formally étale, it follows that Φ satisfies (HP2). ∎

2 Extensions

The results of this section are well known to experts, being similar to those obtained by H. Flenner [15] and J. Wise [48, §2.3].

Fix a scheme S and an S-groupoid X. An X-extension is a square zero closed immersion of X-schemes i:TT. An obligatory observation is that the i-1𝒪T-module ker(i-1𝒪T𝒪T) is canonically a quasi-coherent 𝒪T-module. If the X-scheme T is affine, so is the scheme T; see [17, I.5.1.9]. A morphism of X-extensions

(i1:T1T1)(i2:T2T2)

is a commutative diagram of X-schemes

In a natural way, the collection of X-extensions forms a category, which we denote as 𝐄𝐱𝐚𝐥X. There is a natural functor 𝐄𝐱𝐚𝐥X𝐒𝐜𝐡/X:(i:TT)T.

We denote by 𝐄𝐱𝐚𝐥X(T) the fiber of the category 𝐄𝐱𝐚𝐥X over the X-scheme T. An X-extension of T is an object of 𝐄𝐱𝐚𝐥X(T). There is a natural functor

𝐄𝐱𝐚𝐥X(T)𝐐𝐂𝐨𝐡(T),(i:TT)ker(i-1𝒪T𝒪T).

We denote by 𝐄𝐱𝐚𝐥X(T,I) the fiber category of 𝐄𝐱𝐚𝐥X(T) over the quasi-coherent 𝒪T-module I. An X-extension of T by I is an object of 𝐄𝐱𝐚𝐥X(T,I).

A morphism (TT1)(TT2) in 𝐄𝐱𝐚𝐥X(T,I) induces a commutative diagram of sheaves of rings on the topological space |T|:

The Snake Lemma implies that the morphism of S-schemes T1T2 is an isomorphism. Thus the category 𝐄𝐱𝐚𝐥X(T,I) is a groupoid.

Example 2.1

If X is an algebraic stack, T is an X-scheme, and I is a quasi-coherent 𝒪T-module, then the groupoid 𝐄𝐱𝐚𝐥X(T,I) is equivalent to the Picard category associated to the complex τ0(𝖱Hom𝒪T(τ-1LT/X,I)[1]), where τ-1LT/X is the truncated cotangent complex of [25, Chapter 17]. For a proof of this, see [36, §2.22 and Theorem A.7]. For background material on Picard categories see [8, XVIII.1.4].

Example 2.1 motivates many of the results in this section. The following is a triviality that we record here for future reference.

Lemma 2.2

Fix a scheme S, a formally étale 1-morphism of S-groupoids XY, an X-scheme T, and a quasi-coherent OT-module I. Then, the natural functor

𝐄𝐱𝐚𝐥X(T,I)𝐄𝐱𝐚𝐥Y(T,I)

is an equivalence of categories.

Fix a scheme W and a quasi-coherent 𝒪W-module J. Then, the quasi-coherent 𝒪W-module 𝒪WJ is readily seen to be an 𝒪W-algebra. Indeed, for an open subset U|W| and (w,j), (w,j)Γ(U,𝒪W), let

(w,j)(w,j)=(ww,wj+wj),

which makes 𝒪WJ a sheaf of rings. The natural map 𝒪W𝒪WJ, w(w,0) canonically defines an 𝒪W-algebra, which we denote as 𝒪W[J]. Let W[J] be the W-scheme Spec¯W(𝒪W[J]). Corresponding to the natural surjection of 𝒪W-algebras 𝒪W[J]𝒪W, there is a canonical W-extension of W by J, which we call the trivialW-extension of W by J and denote as (iW,J:WW[J]). In particular, the structure morphism rW,J:W[J]W is a retraction of the morphism iW,J:WW[J].

For a morphism of X-schemes q:UV, let RetX(U/V) be the set of X-retractions to the morphism q:UV. That is,

RetX(U/V)={rHom𝐒𝐜𝐡/X(V,U):rq=IdU}.
Lemma 2.3

Fix a scheme S, an S-groupoid X, an X-scheme T, a quasi-coherent T-module I, and an X-extension (i:TT) of T by I. Then, there is a natural bijection:

Hom𝐄𝐱𝐚𝐥X(T,I)((i:TT),(iT,I:TT[I]))RetX(T/T).

Proof.

For a morphism of X-extensions (TT)(TT[I]), the composition TT[I]rT,IT defines an X-retraction to i. This assignment is bijective. ∎

With some homogeneity assumptions, we are able to prove something meaningful.

Proposition 2.4

Fix a scheme S, an S-groupoid X, and an X-scheme T. Then the functor ExalX(T)QCoh(T) is a fibration in groupoids. If the S-groupoid X is Nil-homogeneous, then ExalX(T,I) is a Picard category for all IQCoh(T).

Proof.

Fix a morphism α:JI in 𝐐𝐂𝐨𝐡(T). This corresponds to a morphism of quasi-coherent 𝒪T-modules α:IJ. Also, fix an X-extension (i:TTI) of T by I. On the topological space |T| we obtain a commutative diagram of sheaves of abelian groups with exact rows

where

𝒪TIIJ=coker(Ii(i,-α(i))𝒪TIJ).

It is easily verified that the sheaf of abelian groups 𝒪TJ=𝒪TIIJ is a sheaf of rings and that the homomorphism α~ is a ring homomorphism. The subsheaf J𝒪TJ defines a square zero sheaf of ideals and as J is quasi-coherent, one immediately concludes that the ringed space (|T|,𝒪TJ) is an S-scheme, TJ, and that we have defined an S-extension (iα:TTJ) of T by J. The morphism of S-schemes TJTI promotes the S-extension iα to an X-extension of T by J. It is immediate that the resulting arrow iαi in 𝐄𝐱𝐚𝐥X(T) is strongly cartesian over the arrow α:JI in 𝐐𝐂𝐨𝐡(T), and we deduce the first claim.

For the second claim, the fibration 𝐄𝐱𝐚𝐥X(T)𝐐𝐂𝐨𝐡(T) induces for each M and N𝐐𝐂𝐨𝐡(T) a functor

πM,N:𝐄𝐱𝐚𝐥X(T,M×N)𝐄𝐱𝐚𝐥X(T,M)×𝐄𝐱𝐚𝐥X(T,N).

Note that this functor is not unique, but for any other choice of such a functor πM,N, there is a unique natural isomorphism of functors πM,NπM,N. This renders the Picard category structure on 𝐄𝐱𝐚𝐥X(T,I) as essentially unique, and on the level of isomorphism classes of objects, the abelian group structure is unique.

By [19, §1.2], it is sufficient to show that the functor πM,N is an equivalence, which we show using the arguments of [17, 0IV.18.3]. For the essential surjectivity, we fix X-extensions (iM:TTM) and (iN:TTN) of T by M and N, respectively. By Lemma 1.5(3), there is a geometric 𝐍𝐢𝐥-nil square over X

In particular, the resulting closed immersion i:TT defines an X-extension of T by M×N. Moreover, it is plain to see that πM,N(i)(iM,iN). The full faithfulness of the functor πM,N follows from a similar argument. ∎

Denote the set of isomorphism classes of the category 𝐄𝐱𝐚𝐥X(T,I) by ExalX(T,I). By Proposition 2.4, if X is 𝐍𝐢𝐥-homogeneous, then there are additive functors

DerX(T,-):𝐐𝐂𝐨𝐡(T)𝐀𝐛,IAut𝐄𝐱𝐚𝐥X(T,I)(iT,I)

and

ExalX(T,-):𝐐𝐂𝐨𝐡(T)𝐀𝐛,IExalX(T,I).

We note that the 0-object of the abelian group DerX(T,I) corresponds to the identity automorphism and the 0-object of the group ExalX(T,I) corresponds to the isomorphism class containing the trivial X-extension of T by I, (iT,I:TT[I]). With a stronger homogeneity assumption, there is an important exact sequence.

Corollary 2.5

Fix a scheme S, an rNil-homogeneous S-groupoid X, and an X-scheme T. Then for every short exact sequence of quasi-coherent OT-modules,

0K𝑘M𝑐C0,

there is a natural 6-term exact sequence of abelian groups:

Proof.

If X is algebraic, then the exact sequence is a trivial consequence of [36, Theorem 1.1]. In general, the result can be recovered from [48, Proposition 2.3 (iv)], where it was shown that the fibered category 𝐄𝐱𝐚𝐥X(T)𝐐𝐂𝐨𝐡(T) is additive and left-exact, in the sense of [19]. We will follow a similar route, but instead employ the results of Appendix B.

By Proposition 2.4, the morphisms k and c induce functors

k*:𝐄𝐱𝐚𝐥X(T,K)𝐄𝐱𝐚𝐥X(T,M)andc*:𝐄𝐱𝐚𝐥X(T,M)𝐄𝐱𝐚𝐥X(T,C).

By Lemma B.1, it remains to prove that the following sequence of Picard categories is exact:

iT,K𝐄𝐱𝐚𝐥X(T,K)k*𝐄𝐱𝐚𝐥X(T,M)c*𝐄𝐱𝐚𝐥X(T,C).

Since ck=0, it follows that there is a naturally induced 2-morphism δ:c*k*0iT,C0. Hence, there is a naturally induced morphism of Picard categories

𝐄𝐱𝐚𝐥X(T,K)𝐄𝐱𝐚𝐥X(T,M)×c*,𝐄𝐱𝐚𝐥X(T,C),0iT,C.

It now remains to exhibit a quasi-inverse to the above functor. By Lemma 2.3, we may view an object of the right-hand side as being given by a pair (i:TTM,r), where r is a retraction of the X-extension of T by C, c*i:TTC. Note that since c is surjective with kernel K, the X-morphism TCTM defines an X-extension of TC by K. In particular, we have an 𝐫𝐍𝐢𝐥-nil pair

(TCTM,TC𝑟T)

over X. Since X is 𝐫𝐍𝐢𝐥-homogeneous, Lemma 1.5(3) implies that the 𝐫𝐍𝐢𝐥-nil pair over X in question can be completed to a cocartesian 𝐫𝐍𝐢𝐥-nil square over X which is geometric. In particular, the resulting morphism j:TT is an X-extension of T by K. Since j is defined by a universal property, we have defined a functor from the right-hand side above to 𝐄𝐱𝐚𝐥X(T,K). The claim follows. ∎

Further strengthening our homogeneity assumption, we obtain more structure.

Corollary 2.6

Fix a scheme S, an Aff-homogeneous S-groupoid X, and an X-scheme T. For all affine and étale morphisms p:VT and quasi-coherent OV-modules M, there is an equivalence of Picard categories:

𝐄𝐱𝐚𝐥X(V,M)𝐄𝐱𝐚𝐥X(T,p*M).

Proof.

Let e:WT be an étale morphism. If TT is an X-extension of T by K, then there exists a unique X-extension WW of W by e*K together with an étale morphism WT such that W×TTW and the second projection coincides with e:WT; see [17, IV.18.1.2]. This describes a functor e*:𝐄𝐱𝐚𝐥X(T,K)𝐄𝐱𝐚𝐥X(W,e*K). Taking K=p*M and e=p, we obtain a functor 𝐄𝐱𝐚𝐥X(T,p*M)𝐄𝐱𝐚𝐥X(V,p*p*M). By Proposition 2.4, corresponding to the 𝒪V-module homomorphism p*p*MM, there is an induced functor 𝐄𝐱𝐚𝐥X(V,p*p*M)𝐄𝐱𝐚𝐥X(V,M). Composing these two functors produces a functor 𝐄𝐱𝐚𝐥X(T,p*M)𝐄𝐱𝐚𝐥X(V,M).

Also, since p is affine, 𝐀𝐟𝐟-homogeneity implies that there is a functor

p*:𝐄𝐱𝐚𝐥X(V,M)𝐄𝐱𝐚𝐥X(T,p*M).

Indeed, for any X-extension (j:VV) of V by M, the 𝐀𝐟𝐟-homogeneity of X and Lemma 1.5(3) provide a cocartesian 𝐀𝐟𝐟-nil square over X as in (1.1) which is geometric. In particular, the X-morphism (i:TT) defines an X-extension of T by p*M. The functors 𝐄𝐱𝐚𝐥X(T,p*M)𝐄𝐱𝐚𝐥X(V,M) are clearly quasi-inverse. ∎

3 Limit preservation

In this section we prove that the functors defined in Section 2, MDerX(T,M) and MExalX(T,M), frequently preserve direct limits. We also relativize the notion of limit preserving S-groupoid [7, §1].

Definition 3.1

Let S be a scheme and let Φ:YZ be a 1-morphism of S-groupoids. The 1-morphism Φ is limit preserving if for every inverse system of quasi-compact and quasi-separated Z-schemes with affine transition maps {Tj}jJ and every Y-scheme T, such that as a Z-scheme T is an inverse limit of {Tj}jJ, then the following two conditions hold.

  1. There exist j0J and a Y-scheme structure on Tj0 such that the induced diagram of Y-schemes {Tj}jj0 has inverse limit T.

  2. Let j1J and let Tj1 have two Y-scheme structures such that both of the induced diagrams of Y-schemes {Tj}jj1 have inverse limit T. Then for all jj1, the two Y-scheme structures on Tj are isomorphic.

An S-groupoid X is limit preserving if its structure morphism to 𝐒𝐜𝐡/S is so. Similarly, an X-scheme T is limit preserving if its structure 1-morphism 𝐒𝐜𝐡/TX is so.

Analogous to Lemma 1.5, we have the following easily verified lemma.

Lemma 3.2

Fix a scheme S and a 1-morphism of S-groupoids Φ:YZ.

  1. If Z is a Zariski stack, then it is limit preserving if and only if for every inverse system of affine S-schemes {SpecAj}jJ with limit SpecA, the natural functor

    limjZ(SpecAj)Z(SpecA)

    is an equivalence.

  2. If Z is an algebraic stack, then it is limit preserving if and only if it is locally of finite presentation over S.

  3. If Φ is limit preserving, then for every other limit preserving 1 -morphism WY the composition WZ is limit preserving.

  4. The 1 -morphism Φ is limit preserving if and only if for every Z-scheme T the T-groupoid Y×Z𝐒𝐜𝐡/T is limit preserving.

  5. If Φ is limit preserving, then for every 1 -morphism of S-groupoids WZ, the 1 -morphism Y×ZWW is limit preserving.

  6. If Φ is limit preserving, then the diagonal 1 -morphism ΔΦ:YY×ZY is limit preserving.

Proof.

The claims (1), (3), (4), and (5) are all obvious, thus their proofs are omitted. Claim (2) follows from (1) and [25, Propositions 4.15, 4.18]. For claim (6), combine (4) and (LP2) (note that for a morphism of schemes, this just says that if a morphism is locally of finite presentation, then so too is its diagonal [17, IV.1.4.3.1]). ∎

Example 3.3

Fix a scheme S and a limit preserving S-groupoid X. Then, an X-scheme is limit preserving if and only if it is locally of finite presentation over S.

We now have the main result of this section.

Proposition 3.4

Fix a scheme S, a Nil-homogeneous S-groupoid X, and a quasi-compact, quasi-separated, limit preserving X-scheme T.

  1. The functor MDerX(T,M) preserves direct limits.

  2. If, in addition, X is limit preserving, then the functor MExalX(T,M) preserves direct limits.

Proof.

Throughout we fix a directed system of quasi-coherent 𝒪T-modules {Mj}jJ with direct limit M. In particular, the natural map

T[M]limjT[Mj]

is an isomorphism of X-schemes. For (1), by Lemma 2.3, there are natural isomorphisms

DerX(T,M)RetX(T/T[M])limjRetX(T/T[Mj])limjDerX(T,Mj).

For (2), we first show that the map

(3.1)limjExalX(T,Mj)ExalX(T,M)

is injective. Lemma 2.3 shows that an X-extension (TT′′) of T by a quasi-coherent 𝒪T-module N represents 0 in ExalX(T,N) if and only if RetX(T/T′′). So, consider a compatible collection of X-extensions (TTj) of T by Mi with limit (TT). Since

RetX(T/T)=limjRetX(T/Tj),

it follows that the map (3.1) is injective.

We now show that the natural map (3.1) is surjective. First, we prove the result in the case where X=S and S and T are affine. Since T is affine and of finite presentation over S, there exist an integer n and a closed immersion k:T𝔸Sn. By [17, 0IV.20.2.3], there is a functorial surjection Hom𝒪T(k*Ω𝔸Sn/S,K)ExalS(T,K) for every quasi-coherent 𝒪T-module K. Since the 𝒪T-module k*Ω𝔸Sn/S is finite free, it follows that the functor KHom𝒪T(k*Ω𝔸Sn/S,K) preserves direct limits. Direct limits are exact, so the map

(3.2)limjExalS(T,Mj)ExalS(T,M)

is surjective.

If S and T are no longer assumed to be affine, then a straightforward Zariski descent argument combined with the affine case already considered shows that the map (3.2) is bijective. For the general case, let (TT)ExalX(T,M). By the above considerations, there exist a j0 and an S-extension of T by Mj0, (TTj0), such that its pushforward along Mj0M is isomorphic to (TT) as an S-extension. If jj0, then denote the pushforward of (TTj0) along the morphism Mj0Mj by (TTj). There is a natural morphism of S-schemes TjTj0 and the resulting inverse system {Tj}jj0 has limit T. Since X is a limit preserving S-groupoid, there exist j1j0 and an X-scheme structure on Tj1 such that the resulting inverse system of X-schemes {Tj}jj1 has limit T. The result follows. ∎

4 Formal smoothness and formal versality

In this section we prove the main result of the paper.

Definition 4.1

Fix a scheme S, an S-groupoid X, and an X-scheme V. Consider the following lifting problem in the category of X-schemes: given a pair of morphisms of X-schemes (V𝑝T,V𝑗V), where j is a locally nilpotent closed immersion, complete the following diagram so that it commutes:

(4.1)

(4.1)

We say that the X-scheme T is

  1. formally smooth if the lifting problem above can always be solved Zariski-locally on V;

  2. formally versal at t|T| if the lifting problem can be solved whenever V is local artinian with closed point v such that p(v)=t, the induced map κ(t)κ(v) is an isomorphism, and j is an X-extension of V by κ(v).

We certainly have the following implication:

formally smoothformally versal at every t|T|.

In general, there is no reverse implication. We will see, however, that this subtlety vanishes once the S-groupoid is 𝐀𝐟𝐟-homogeneous. The following lemma is hopefully clarifying. Note that we cannot immediately apply [25, Proposition 4.15 (ii)], because there G. Laumon and L. Moret-Bailly assume that solutions to the lifting problem exist étale-locally on V, whereas we only assume that they exist Zariski-locally.

Lemma 4.2

Fix an S-groupoid X and an X-scheme T. If the 1-morphism TX is representable by algebraic spaces that are locally of finite presentation, then the X-scheme T is formally smooth if and only if the 1-morphism TX is representable by smooth morphisms of algebraic spaces.

Proof.

Suppose that T is a formally smooth X-scheme. To prove that TX is representable by smooth morphisms, it is sufficient to prove that if W is an X-scheme, then the induced morphism of algebraic spaces TWW, obtained by pulling back TX along W, is smooth. Since TW is an algebraic space, there exists an étale and surjective morphism T~WTW, where T~W is a scheme. It remains to prove that the morphism of schemes T~WW is smooth. Since the morphism in question is locally of finite presentation, it remains to show that it satisfies the infinitesimal lifting criterion for smooth morphisms. We will use [17, IV.17.14.1], thus we must show that we can complete every 2-commutative diagram

where AA0 is a surjection of local rings with square 0 kernel. Since TX is formally smooth and A is a local ring, there exists a morphism SpecAT that makes the diagram 2-commute. The universal property of the 2-fiber product further implies that there is an induced morphism SpecATW that makes the diagram commute. But T~WTW is étale, surjective, and representable by schemes. It now follows from étale descent and again from [17, IV.17.14.1] that there is a unique morphism SpecAT~W completing the diagram. The result follows. The other direction is similar, thus is omitted. ∎

There is a tight connection between formal smoothness (resp. formal versality) and X-extensions in the affine setting. The next result has arguments similar to those of [15, Satz 3.2], but the definitions are slightly different.

Lemma 4.3

Fix a scheme S, an S-groupoid X, and an affineX-scheme T.

  1. If X is 𝐀𝐟𝐟-homogeneous and the abelian group ExalX(T,I) is trivial for every quasi-coherent 𝒪T-module I, then the X-scheme T is formally smooth.

  2. If X is 𝐫𝐂𝐥-homogeneous and ExalX(T,κ(t))=0 at a closed point t|T|, then the X-scheme T is formally versal at t.

  3. If X is 𝐂𝐥-homogeneous and T is noetherian and formally versal at a closed point t|T|, then ExalX(T,κ(t))=0.

Proof.

For (1), fix a locally nilpotent closed immersion of X-schemes j:VV. It suffices to construct an X-morphism VT Zariski-locally on V that makes the diagram (4.1) commute. Thus we may assume V and V are affine and the locally nilpotent closed immersion j:VV is defined by a quasi-coherent 𝒪V-ideal J such that Jn=0 for some integer n. By induction on the integer n, we may further reduce to the situation where J2=0. In particular, j is a square zero extension of V by J and (V𝑝T,V𝑗V) is an 𝐀𝐟𝐟-nil pair over X. Since X is 𝐀𝐟𝐟-homogeneous, Lemma 1.5(3) implies that there is a cocartesian 𝐀𝐟𝐟-nil square over X as in (1.1) that is geometric. In particular, the resulting X-morphism i:TT defines an X-extension of T by p*J. By hypothesis, ExalX(T,p*J)=0. Lemma 2.3 now provides an X-retraction TT. The composition

VpTT

gives the required lifting.

The claim (2) follows from an identical argument just given for (1).

For (3), given an X-extension TT~ of T by κ(t), write T=SpecR, T~=SpecR~, 𝔪=t|T|, and I=ker(R~R)R/𝔪. Let the ideal 𝔪~R~ denote the (unique) maximal ideal induced by 𝔪. For n0 define Rn=R/𝔪n+1, R~n=R~/𝔪~n+1, and In=ker(R~nRn). There is an induced surjective morphism ln:IIn and since I is an R-module of length 1, there is an n00 such that ln0 is an isomorphism. Let V=SpecRn0 and V=SpecR~n0 and let j:VV be the resulting X-extension of V by κ(t).

Formal versality at t|T| gives an X-morphism VT as in (4.1). If p:VT is the induced closed immersion, then (V𝑝T,V𝑗V) is a 𝐂𝐥-nil pair. By Lemma 1.5(3), there exists a cocartesian 𝐂𝐥-nil square over X as in (1.1) which is geometric. In particular, the resulting X-morphism i:TT defines an X-extension of T by κ(t). The compatible X-morphism VT and the cocartesian 𝐂𝐥-nil square (1.1) prove that the X-extension i:TT admits a retraction r:TT, thus defines a trivial extension of T by κ(t) over X (Lemma 2.3). The cocartesian 𝐂𝐥-nil square (1.1) also produces a morphism of X-extensions of T by κ(t) from TT~ to TT, which is automatically an isomorphism. The result follows. ∎

Fix an affine scheme T and an additive functor F:𝐐𝐂𝐨𝐡(T)𝐀𝐛. The functor F is finitely generated if there exist a quasi-coherent 𝒪T-module I and an object ηF(I) such that for all M𝐐𝐂𝐨𝐡(T) the induced morphism of abelian groups Hom𝒪T(I,M)F(M) given by ff*η is surjective. The notion of finite generation of a functor is due to M. Auslander [9].

The functor F is half-exact if for every short exact sequence 0MMM′′0 in 𝐐𝐂𝐨𝐡(T), the sequence F(M)F(M)F(M′′) is exact.

If, in addition, T is noetherian and sends coherent 𝒪T-modules to coherent 𝒪T-modules, then A. Ogus and G. Bergman have shown [33, Theorem 2.1] that if for all closed points t|T| we have F(κ(t))=0, then F is the zero functor. If F is finitely generated, then this result can be refined. Indeed, it is shown in [20, Corollary 6.7] that if F(κ(t))=0, then there exists an affine open subscheme p:UT such that the composition Fp*(-):𝐐𝐂𝐨𝐡(U)𝐀𝐛 is identically zero. We now use this to prove the main technical result of the paper.

Theorem 4.4

Fix a locally noetherian scheme S, an Aff-homogeneous and limit preserving S-groupoid X, and an affine X-scheme T, locally of finite type over S. If the functor MExalX(T,M) is finitely generated and T is formally versal at a closed point t|T|, then it is formally smooth in an open neighborhood of t.

Proof.

By Lemma 4.3(3), we have ExalX(T,κ(t))=0. By Corollary 2.5, the functor MExalX(T,M) is half-exact, and by Proposition 3.4 it commutes with direct limits. As ExalX(T,-) is finitely generated, Corollary 6.7 of [20] now applies. Thus, there exists an affine open neighborhood p:UT of t such that the functor ExalX(T,p*(-)):𝐐𝐂𝐨𝐡(U)𝐀𝐛 is the zero functor. By Corollary 2.6, ExalX(U,-) is also the zero functor. By Lemma 4.3(1), we conclude that U is a formally smooth X-scheme. ∎

5 Coherent functors

Fix a ring A. An additive functor F:𝐌𝐨𝐝(A)𝐀𝐛 is coherent, if there exist an A-module homomorphism f:IJ and an element ηF(I), inducing an exact sequence for every A-module M,

HomA(J,M)HomA(I,M)F(M)0.

We refer to the data (f:IJ,η) as a presentation for F. For a comprehensive account of coherent functors, we refer the interested reader to [9]. Some stronger results that are available in the noetherian situation are developed in [23]. Here we record some simple consequences of [9, p. 200].

Lemma 5.1

Fix a ring A. For each i=1,,5, let Hi:Mod(A)Ab be an additive functor fitting into an exact sequence

H1H2H3H4H5.

  1. If H2 and H4 are finitely generated and H5 is coherent, then H3 is finitely generated.

  2. If H1 is finitely generated and H2, H4, and H5 are coherent, then H3 is coherent.

We now have the following important example of a coherent functor, which is a special case of [9, p. 200].

Example 5.2

Let A be a ring. If Q is a bounded above complex of A-modules, then the functor MExtAi(Q,M) is coherent for every integer i. Indeed, there is a quasi-isomorphism PQ, where P is a complex of A-modules that is term-by-term projective. By definition, for every A-module M and integer i there is a natural isomorphism:

ExtAi(Q,M)=ker(HomA(Pi,M)HomA(Pi-1,M))im(HomA(Pi+1,M)HomA(Pi,M)).

Since the functor MHomA(Pj,M) is coherent for every integer j, Lemma 5.1(2) implies that the functor MExtAi(Q,M) is coherent for every integer i. Using Spaltenstein resolutions [43], this example extends to where Q is unbounded.

Example 5.3

Let R be a noetherian ring. If Q is a bounded above complex of R-modules with coherent cohomology, then the functor MToriR(Q,M) is coherent for every integer i. Indeed, there is a quasi-isomorphism FQ, where F is a bounded above complex of finitely generated free R-modules. Arguing as in Example 5.2, it remains to show that if F is a finitely generated and free R-module, then the functor MFRM is coherent. But this is clear: there is a natural isomorphism FRM=HomR(F,M) for every R-module M. Thus the functor in question is isomorphic to HomR(F,-), which is coherent.

The following lemma is crucial for the proof of Theorem A.

Lemma 5.4

Fix a scheme S and an algebraic S-stack X. If T is an affine X-scheme, then the functors MDerX(T,M) and MExalX(T,M) are coherent.

Proof.

By [36, Theorem 1.1], there is a bounded above complex of 𝒪T-modules LT/X, with quasi-coherent cohomology sheaves, as well as functorial isomorphisms

DerX(T,M)Ext𝒪T0(LT/X,M)andExalX(T,M)Ext𝒪T1(LT/X,M)

for all quasi-coherent 𝒪T-modules M. By Example 5.2, the result follows. ∎

The next example is [20, Theorem C] and is crucial for the applications in Sections 89.

Example 5.5

Fix an affine scheme S and a morphism of algebraic stacks f:XS that is locally of finite presentation. If 𝖣qc(X) and 𝒩𝐐𝐂𝐨𝐡(X), where 𝒩 is of finite presentation, flat over S, with support proper over S, then the functor

Hom𝒪X(,𝒩𝒪Xf*(-)):𝐐𝐂𝐨𝐡(S)𝐀𝐛

is coherent. Stated in this generality, the coherence of the above functor is non-trivial. If S is noetherian, f is projective, and 𝐂𝐨𝐡(X), then a direct proof of the coherence of the above functor can be found in [23, Example 2.7]. If S is noetherian and admits a dualizing complex (e.g., when S is of finite type over a field or ; see [22, V.2]), f is a proper morphism of algebraic stacks, and 𝖣Coh-(X), then the coherence is simpler (see [20, Proposition 2.1], which extends Flenner’s arguments in the analytic case [16, Satz 2.1] to algebraic stacks).

If S is noetherian, 𝖣Coh-(X), and f is a proper morphism of schemes or algebraic spaces, then the coherence is also a consequence of some (now) standard facts. Indeed, by [28, Theorem 4.1] (if X is a scheme) or [44, Tag 08HP] (if X is an algebraic space), there exist a perfect complex 𝒫 on X and a morphism p:𝒫 that induces a quasi-isomorphism τ0𝒫τ0. There is now a natural sequence of isomorphisms for every 𝐐𝐂𝐨𝐡(S):

Hom𝒪X(,𝒩𝒪Xf*)Hom𝒪X(𝒫,𝒩𝒪Xf*)
0(𝖱Γ(X,𝒫𝒪X𝖫[𝒩𝒪Xf*]))(𝒫is perfect)
0(𝖱Γ(X,𝒫𝒪X𝖫𝒩𝒪X𝖫𝖫f*))(𝒩 is flat over S)
0(𝖱Γ(X,𝒫𝒪X𝖫𝒩)𝒪S𝖫).

The last isomorphism is the projection formula, see [32, Proposition 5.3] (if X is a scheme) or [44, Tag 08IN] (if X is an algebraic space). Since f is proper, 𝖱Γ(X,𝒫𝒪X𝖫𝒩) is a bounded above complex of R=Γ(S,𝒪S)-modules with coherent cohomology, see [10, III.2.2.1] (if X is a scheme) or [44, Tag 08GK] (if X is an algebraic space). The result now follows from Example 5.3.

6 Automorphisms, deformations, obstructions, and composition

A hypothesis in Theorem 4.4 is that the functor MExalX(T,M) is finitely generated. We have found the direct verification of this hypothesis to be difficult. In this section, we provide some exact sequences to remedy this situation. We also take the opportunity to formalize and relativize obstruction theories.

Fix a scheme S and a 1-morphism of S-groupoids Φ:YZ. We write 𝐃𝐞𝐟Φ for the category with objects the pairs (i:TT,r:TT), where i is a Y-extension and r is a Z-retraction of i. A morphism (i1:T1T1,r1:T1T1)(i2:T2T2,r2:T2T2) in 𝐃𝐞𝐟Φ is a morphism of Y-extensions i1i2 such that the resulting diagram of Z-schemes commutes,

By Lemma 2.3, 𝐃𝐞𝐟Φ can be viewed as the category of completions of the following diagram:

where I is a quasi-coherent 𝒪T-module and T[I] is the trivial Z-extension of T by I. There is a natural functor 𝐃𝐞𝐟Φ𝐄𝐱𝐚𝐥Y, which sends (i:TT,r:TT) to (i:TT). If T is a Y-scheme, then we denote the fiber of this functor over 𝐄𝐱𝐚𝐥Y(T) by 𝐃𝐞𝐟Φ(T). It follows that there is an induced functor 𝐃𝐞𝐟Φ(T)𝐐𝐂𝐨𝐡(T). We denote the fiber of this functor over a quasi-coherent 𝒪T-module I as 𝐃𝐞𝐟Φ(T,I). This category is naturally pointed by the trivial Y-extension of T by I. The following example is related to Example 2.1.

Example 6.1

Let S be a scheme and let Φ:YZ be a 1-morphism of algebraic stacks. If T is a Y-scheme, which we regard as being given by a 1-morphism t:TY, and I is a quasi-coherent 𝒪T-module, then the category 𝐃𝐞𝐟Φ(T,I) is naturally equivalent to the Picard category represented by the complex τ0𝖱Hom𝒪T(τ0𝖫t*τ0LY/Z,I), where τ0LY/Z is the truncated cotangent complex defined in [25, Chapter 17]. In particular, many of the results of this section, when Φ is a 1-morphism of algebraic stacks, can be viewed as mild generalizations or simple consequences of the results appearing in the latter parts of [25, Chapter 17].

The following lemma, which is related to [25, Lemme 17.15.1], will be important and explains why the groupoids 𝐃𝐞𝐟Φ(T,I) are more amenable to calculation than 𝐄𝐱𝐚𝐥Y(T,I) and 𝐄𝐱𝐚𝐥Z(T,I). Indeed, the groupoid 𝐃𝐞𝐟Φ has base change properties, while 𝐄𝐱𝐚𝐥Y(T,I) typically does not. This will be revisited in Lemma 6.9 and Corollary 6.14.

Lemma 6.2

Fix a scheme S and a 2-cartesian diagram of S-groupoids:

Let T be a YW-scheme and let IQCoh(T). Then the natural functor

𝐃𝐞𝐟ΦW(T,I)𝐃𝐞𝐟Φ(T,I)

induces an equivalence of categories.

Proof.

We prove that the functor in question induces an equivalence of categories by constructing a quasi-inverse. If (i:TT,r:TT) belongs to 𝐃𝐞𝐟Φ(T,I), then the retraction r endows T with a structure of a W-scheme, which as a Z-scheme is isomorphic to its other Z-scheme structure obtained from its Y-scheme structure. The universal property of the 2-fiber product implies that T becomes a YW-scheme, the Y-morphism i is a YW-morphism, and the Z-morphism r is a W-morphism. It follows that we have functorially defined an object of 𝐃𝐞𝐟ΦW(T,I), thus there is an induced functor 𝐃𝐞𝐟Φ(T,I)𝐃𝐞𝐟ΦW(T,I). That this functor is quasi-inverse to 𝐃𝐞𝐟ΦW(T,I)𝐃𝐞𝐟Φ(T,I) is clear. ∎

We record for future reference the following trivial observations.

Lemma 6.3

Fix a scheme S, 1-morphisms of S-groupoids XΨYΦZ, an X-scheme T, and a quasi-coherent OT-module I. If the 1-morphism Ψ:XY is formally étale, then the natural functor

𝐃𝐞𝐟ΦΨ(T,I)𝐃𝐞𝐟Φ(T,I)

is an equivalence of categories.

Lemma 6.4

Fix a scheme S, a class of morphisms PAff, a 1-morphism of P-homogeneous S-groupoids Φ:YZ, a morphism of Y-schemes p:VT where pP, and KQCoh(V). Then the natural functor

𝐃𝐞𝐟Φ(T,p*K)𝐃𝐞𝐟Φ(V,K)

is an equivalence of categories.

The proof of the next result is similar to Proposition 2.4, thus is omitted.

Proposition 6.5

Fix a scheme S, a 1-morphism of Nil-homogeneous S-groupoids Φ:YZ, a Y-scheme T, and a quasi-coherent OT-module I. Then the category DefΦ(T,I) admits a natural structure as a Picard category.

Denote the set of isomorphism classes of the Picard category 𝐃𝐞𝐟Φ(T,I) by DefΦ(T,I). Thus, by Proposition 6.5, we obtain functors

DefΦ(T,-):𝐐𝐂𝐨𝐡(T)𝐀𝐛,IDefΦ(T,I)

and

AutΦ(T,-):𝐐𝐂𝐨𝐡(T)𝐀𝐛,IAut𝐃𝐞𝐟Φ(T,I)(iT,I).

We include the following corollary for its intended reference in [21]. Its proof is almost identical to that of Corollary 2.5 and [48, Proposition 2.2 (iv)], thus is omitted.

Corollary 6.6

Fix a scheme S, a 1-morphism of rNil-homogeneous S-groupoids Φ:YZ, and a Y-scheme T. Then for every short exact sequence in QCoh(T),

0K𝑘M𝑐C0.

there is a natural exact sequence of abelian groups

We now have an exact sequence that greatly aids computations.

Proposition 6.7

Fix a scheme S, a 1-morphism of Nil-homogeneous S-groupoids Φ:YZ, a Y-scheme T, and a quasi-coherent OT-module I. Then there is a natural exact sequence of abelian groups

Proof.

By Lemma B.1, it is sufficient to show that the following sequence of Picard categories is left-exact:

(iT,I,rT,I)𝐃𝐞𝐟Φ(T,I)𝐄𝐱𝐚𝐥Y(T,I)𝐄𝐱𝐚𝐥Z(T,I).

By Lemma 2.3 and the explicit description of the 2-fiber product of Picard categories given in Appendix B, this is clear, and the result follows. ∎

We now introduce multi-step relative obstruction theories. For single-step obstruction theories, this definition is similar to [7, (2.6)] and [34, Definition A.10].

Definition 6.8

Fix a scheme S, a 1-morphism of 𝐍𝐢𝐥-homogeneous S-groupoids Φ:YZ, and an integer n1. For a Y-scheme T, an n-step relative obstruction theory for Φ at T is a sequence of additive functors (the obstruction spaces)

Oi(T,-):𝐐𝐂𝐨𝐡(T)𝐀𝐛,IOi(T,I)for i=1,,n

as well as natural transformations of functors (the obstruction maps)

o1(T,-):ExalZ(T,-)O1(T,-),
oi(T,-):keroi-1(T,-)Oi(T,-)for i=2,,n,

such that the natural transformation of functors

ExalY(T,-)ExalZ(T,-)

has image keron(T,-). For an affineY-scheme T, an n-step relative obstruction theory at T is coherent if the functors {Oi(T,-)}i=1n are all coherent.

We feel that it is important to point out that simply taking the cokernel of the last morphism in the exact sequence of Proposition 6.7 produces a 1-step relative obstruction theory, which we denote as (obsΦ,ObsΦ) and call the minimal relative obstruction theory. This obstruction theory generalizes to the relative setting the minimal obstruction theory described in [15]. In practice, the minimal obstruction theory is a difficult object to explicitly describe. The following base change result is useful, however.

Lemma 6.9

Fix a scheme S and a 2-cartesian diagram of Nil-homogeneous S-groupoids:

If T is a YW-scheme and IQCoh(T), then

ObsΦW(T,I)ObsΦ(T,I).

Proof.

By Proposition 6.7, there is a commutative diagram with exact rows:

It follows that there is a naturally induced morphism ObsΦW(T,I)ObsΦ(T,I), which we will now prove to be injective. Fix a W-extension (TT) of T by I. If this W-extension lifts, as a Z-extension, to a Y-extension, then the universal property of the 2-fiber product implies that it lifts to a YW-extension. A standard diagram chase now shows that this proves the injectivity of the map in question. ∎

Example 6.10

Note that the injection of Lemma 6.9 is rarely a bijection. Indeed, if Φ:YZ admits a section s, then for any Z-scheme T and quasi-coherent 𝒪T-module I it follows that ExalY(T,I)ExalZ(T,I) also admits a section and is thus surjective. In particular, ObsΦ(T,I)=0. Note that this implies that ObsΦY(T,I)=0 for every Y-scheme T and quasi-coherent 𝒪T-module I. To obtain an explicit counterexample, it suffices to find a Φ, a T, and an I such that ObsΦ(T,I)0. For this, let Φ:YZ be the 1-morphism of S-groupoids given by a non-smooth morphism of affine schemes. Let T=Y, which we view as a Y-scheme in the obvious way. Then ExalY(T,I)=0 for every quasi-coherent 𝒪T-module I. Since TZ is not smooth, Lemmas 4.2 and 4.3 imply that ExalZ(T,I0)0 for some quasi-coherent 𝒪T-module I0. In particular, ObsΦ(T,I0)=ExalZ(T,I0)0.

Combining Lemmas 6.3 and 2.2, we obtain the following.

Lemma 6.11

Fix a scheme S, 1-morphisms of Nil-homogeneous S-groupoids

XΨYΦZ,

an X-scheme T, and a quasi-coherent OT-module I. If Ψ is formally étale, then every n-step relative obstruction theory for Φ at T lifts to an n-step relative obstruction theory for ΦΨ with the same obstruction spaces.

What follows is an immediate consequence of Proposition 6.7 and Lemma 5.1.

Corollary 6.12

Fix a scheme S, a 1-morphism of Nil-homogeneous S-groupoids Φ:YZ, an affine Y-scheme T, and an integer n1. Suppose there exists a coherent n-step relative obstruction theory at T.

  1. If the functor MExalZ(T,M) is coherent, then the minimal obstruction theory (obsΦ,ObsΦ) is coherent at T.

  2. If the functors MDefΦ(T,M), ExalZ(T,M) are finitely generated, then the functor MExalY(T,M) is finitely generated.

Proof.

For (1), we note that for every quasi-coherent 𝒪T-module M and i=2,,n there are natural exact sequences

0kero1(T,M)ExalZ(T,M)o1(T,M)O1(T,M),
0keroi(T,M)keroi-1(T,M)oi(T,M)Oi(T,M),
0keron(T,M)ExalZ(T,M)ObsΦ(T,M)0.

Combining the first exact sequence with Lemma 5.1(2), we see that the functor kero1(T,-) is coherent. Working by induction on i, the second exact sequence combined with Lemma 5.1(2) proves that the functor keron(T,-) is coherent. The third exact sequence and Lemma 5.1(2) now prove that ObsΦ(T,-) is coherent.

The claim (2) is an immediate consequence of the exact sequence of Proposition 6.7 and Lemma 5.1(1). ∎

The result that follows shows the stability of the conditions of Theorem A under composition, in the sense of J. Starr [45]. The following result also extends – by four terms to the right – the exact sequence [34, §A.15].

Proposition 6.13

Fix a scheme S and 1-morphisms of Nil-homogeneous S-groupoids

XΨYΦZ,

an X-scheme T, and a quasi-coherent OT-module I. Then there is a natural 9-term exact sequence of abelian groups

In particular, there are natural isomorphisms

AutΨ(T,I)DefΔΨ(T,I)𝑎𝑛𝑑DefΨ(T,I)ObsΔΨ(T,I).

Proof.

The latter claims follow by combining the result with the triple

XΔΨX×YXX

and Lemma 6.2.

By Lemma B.1, we will obtain the first seven terms of the exact sequence if the following sequence of Picard categories is left-exact:

(iT,I,rT,I)𝐃𝐞𝐟Ψ(T,I)𝐃𝐞𝐟ΦΨ(T,I)𝐃𝐞𝐟Φ(T,I).

By Lemma 2.3 and the explicit description of the 2-fiber product of Picard categories given in Appendix B, this is clear. For the remaining four terms of the exact sequence: we first apply the Snake Lemma to the commutative diagram with exact rows

which produces an exact sequence

(6.1)KΦΨ(T,I)KΦ(T,I)ObsΨ(T,I)ObsΦΨ(T,I)ObsΦ(T,I)0,

where

KΦΨ(T,I)=ker(ExalX(T,I)ExalZ(T,I)),
KΨ(T,I)=ker(ExalY(T,I)ExalZ(T,I)).

By Proposition 6.7, we obtain a commutative diagram with exact rows

By the Snake Lemma, we thus obtain an isomorphism

coker(DefΦΨ(T,I)DefΦ(T,I))coker(KΦΨ(T,I)KΦ(T,I)).

Combining this isomorphism with the exact sequence (6.1), we deduce that the sequence

DefΦΨ(T,I)DefΦ(T,I)ObsΨ(T,I)ObsΦΨ(T,I)ObsΦ(T,I)0

is exact. Splicing the 7-term exact sequence which we earlier obtained from the left-exact sequence of Picard categories to the 6-term exact sequence above gives the result. ∎

We now arrive at the final result of this section, which is instrumental to the bootstrapping argument employed to prove Theorem A.

Corollary 6.14

Let Ψ:XY be a Nil-homogeneous 1-morphism of S-groupoids. Let W be an (X×Ψ,Y,ΨX)-scheme and let (ΔΨ)W:DΨ,WW be the pullback of ΔΨ along W. If T is a DΨ,W-scheme and M is a quasi-coherent OT-module, then

Aut(ΔΨ)W(T,M)=0,
Def(ΔΨ)W(T,M)AutΨ(T,M),
Obs(ΔΨ)W(T,M)DefΨ(T,M).

Proof.

The third diagonal of Ψ is an isomorphism, so ObsΔΔΔΨ(T,M)=0. By Lemmas 6.2 and 6.9 and Proposition 6.13, there are natural isomorphisms:

Aut(ΔΨ)W(T,M)AutΔΨ(T,M)DefΔΔΨ(T,M)ObsΔΔΔΨ(T,M)0,
Def(ΔΨ)W(T,M)DefΔΨ(T,M)AutΨ(T,M),
Obs(ΔΨ)W(T,M)ObsΔΨ(T,M)DefΨ(T,M).

7 Proof of Theorem A

In this section we prove Theorem A. Before we do this, however, we will prove the following theorem.

Theorem 7.1

Fix an excellent scheme S. An S-groupoid X is an algebraic S-stack that is locally of finite presentation over S if and only if the following conditions are satisfied.

  1. X is a stack over the site (𝐒𝐜𝐡/S)E´t.

  2. X is limit preserving.

  3. X is 𝐀𝐟𝐟-homogeneous.

  4. The diagonal ΔX/S:XX×SX is representable by algebraic spaces.

  5. For any 𝔪-adically complete local noetherian ring (B,𝔪) with an S-scheme structure SpecBS such that the induced morphism Spec(B/𝔪)S is locally of finite type, the natural functor

    X(SpecB)limnX(Spec(B/𝔪n))

    induces an equivalence of categories.

  6. For any affine X-scheme T that is locally of finite type over S, the functor

    MExalX(T,M)

    is finitely generated.

Proof.

Fix a morphism x:Spec𝕜S, where 𝕜 is a field. Denote by 𝒜S(x) the category whose objects are pairs (A,a), where A is a local artinian ring with residue field 𝕜, and a:SpecAS is a morphism of schemes, such that the composition SpecAredSpecAS agrees with x. Morphisms (A,a)(B,b) in 𝒜S(x) are ring homomorphisms AB that preserve the data. For ξX(x), there is an induced category fibered in groupoids Xξ:𝒞ξ𝒜S(x). The 𝐀𝐟𝐟-homogeneity of the S-groupoid X implies the homogeneity (in the sense of [12, Exposé VI, Definition 2.5]) of the cofibered category Xξ:𝒞ξ𝒜S(x).

If the morphism x is locally of finite type, then, by (6) and [20, Lemma 6.6], the 𝕜-vector space ExalX(ξ,𝕜) is finite dimensional. By Lemma 5.4 and again by [20, Lemma 6.6], the 𝕜-vector space DerS(x,𝕜) is also finite dimensional. Thus, by Proposition 6.7, the 𝕜-vector space DefX/S(ξ,𝕜) is finite dimensional. By definition, DefX/S(ξ,𝕜) is the set of isomorphism classes of the category Xξ(ξ[ϵ]).

Thus, by (5), Theorem 1.5 of [11] applies, and so for every such ξ, there is a pointed and affine X-scheme (Qξ,q), locally of finite type over S, such that the X-scheme Specκ(q) is isomorphic to ξ, and Qξ is formally versal at q, and q is a closed point of Qξ. We now apply Theorem 4.4 to conclude that we may (by passing to an open subscheme) assume that Qξ is a formally smooth X-scheme containing q. Condition (4) and Lemma 4.2 now imply that the X-scheme Qξ is representable by smooth morphisms.

Define K to be the set of all morphisms x:Spec𝕜S that are locally of finite type, where 𝕜 is a field. Set Q=κK,ξX(κ)Qξ. Then, we have seen that the X-scheme Q is representable by smooth morphisms, and it remains to show that it is representable by surjective morphisms. Since the stack X is limit preserving, it suffices to show that if V is an affine X-scheme that is locally of finite type over S, then the morphism of algebraic S-spaces Q×XVV is surjective. But Q×XVV smooth and its image contains all the points v|V| such that the morphism Specκ(v)S is locally of finite type. Since V is locally of finite type over S, the result follows. ∎

To deduce Theorem A from Theorem 7.1 we will use a bootstrapping process. This process begins with the following corollary.

Corollary 7.2

Fix an excellent scheme S and an S-groupoid X. If X satisfies the conditions of Theorem A and ΔX/S:XX×SX is representable, then X is an algebraic stack that is locally of finite presentation over S.

Proof.

Note that conditions (1) and (2), combined with Lemma 3.2(1), imply that the S-groupoid X is limit preserving. Also, conditions (1) and (3) combined with Lemma 1.5(4) imply that X is 𝐀𝐟𝐟-homogeneous. Conditions (5) and (6), together with Corollary 6.12, imply that for every affine X-scheme V that is locally of finite type over S, the functor MExalX(V,M) is finitely generated. Thus, Theorem 7.1 implies that X is an algebraic stack that is locally of finite presentation over S. ∎

We now come to the proof of Theorem A.

Proof of Theorem A.

By Corollary 7.2, it remains to prove that ΔX/S is representable. To show this, it remains to prove that for any (X×SX)-scheme T, the T-groupoid DX/S,T, which is obtained by pulling back ΔX/S along T, is an algebraic stack.

Arguing as in the proof of Corollary 7.2, X is limit preserving and 𝐀𝐟𝐟-homogeneous. By Lemmas 1.5(8) and 3.2(6), the diagonal 1-morphism ΔX/S:XX×SX is 𝐀𝐟𝐟-homogeneous and limit preserving. By Lemmas 1.5(7), (5) and 3.2(5), (3), the S-groupoid X×SX is 𝐀𝐟𝐟-homogeneous and limit preserving. Thus, by Lemmas 1.5(7) and 3.2(5) the T-groupoid DX/S,T is limit preserving and 𝐀𝐟𝐟-homogeneous. Representability of DX/S,T is local on T for the Zariski topology, thus we may assume that T is an affine scheme. Since X×SX is limit preserving, every affine (X×SX)-scheme T factors through an affine (X×SX)-scheme T0 that is locally of finite type over S. Thus, we may assume henceforth that T is locally of finite type over S, and is consequently excellent. By Corollary 6.14, the T-groupoid DX/S,T satisfies all the conditions of Theorem A. Thus, by Corollary 7.2, it remains to prove that ΔDX/S,T/T is representable. Replacing XS by DX/S,TT and repeating the above analysis we are further reduced to proving that the diagonal of DDX/S,T,VV is representable for every affine (DX/S,T×TDX/S,T)-scheme V that is locally of finite type over T. Since ΔΔX/S is a monomorphism, it follows that the diagonal of DX/S,T is a monomorphism. In particular, DDX/S,T,VV is a monomorphism. Thus, ΔDDX/S,T/T,V/V is an isomorphism, which is representable, and the result follows. ∎

8 Application I: The stack of quasi-coherent sheaves

Fix a scheme S. For an algebraic S-stack Y and a property P of quasi-coherent 𝒪Y-modules, denote by 𝐐𝐂𝐨𝐡P(Y) the full subcategory of 𝐐𝐂𝐨𝐡(Y) consisting of objects which are P. We will be interested in the following properties P of quasi-coherent 𝒪Y-modules and their combinations:

  1. fp – finitely presented,

  2. flY-flat,

  3. flbS-flat,

  4. prbS-proper support.

Fix a morphism of algebraic stacks f:XS. For any S-scheme T, consider a property P of quasi-coherent 𝒪XT-modules. Define

QCoh¯X/SP

to be the category with objects a pair (T,), where T is an S-scheme and 𝐐𝐂𝐨𝐡P(XT). A morphism (a,α):(V,𝒩)(T,) in the category QCoh¯X/SP consists of an S-scheme morphism a:VT together with an 𝒪XV-isomorphism α:aXT*𝒩. Set

Coh¯X/S=QCoh¯X/Sflb,fp,prb.

In this section, we will prove the following theorem.

Theorem 8.1

Fix a scheme S and a morphism of algebraic stacks f:XS. If f is separated and locally of finite presentation, then Coh¯X/S is an algebraic stack that is locally of finite presentation over S with affine diagonal over S.

A proof of Theorem 8.1, without the statement about the diagonal, appeared in [26, Theorem 2.1], though was light on details. In particular, no explicit obstruction theory was given and, as we will see, the obstruction theory is subtle when f is not flat (and is not a standard fact). There was also a minor error in the statement – that the morphism f is separated is essential [29]. The statement about the diagonal of Coh¯X/S was addressed by M. Roth and J. Starr [40, §2]. Their approach, however, is completely different, and relies on [26, Proposition 2.3]. In the setting of analytic spaces, the properties of the diagonal were addressed by H. Flenner [16, Korollar 3.2].

Just as in [26, Proposition 2.7], an immediate consequence of Theorems 8.1 and [20, Theorem D] is the existence of Quot spaces. Recall that for a quasi-coherent 𝒪X-module , the presheaf Quot¯X/S():(𝐒𝐜𝐡/S)𝖲𝖾𝗍𝗌 is defined as follows:

Quot¯X/S()[T𝜏S]={τX*𝒬:𝒬𝐐𝐂𝐨𝐡flb,fp,prb(XT)}/.
Corollary 8.2

Fix a scheme S, a morphism of algebraic stacks f:XS, and FQCoh(X). If f is separated and locally of finite presentation over S, then Quot¯X/S(F) is an algebraic space that is separated over S. If, in addition, F is of finite presentation, then Quot¯X/S(F) is locally of finite presentation over S.

When is of finite presentation, Corollary 8.2 was proved by M. Olsson and J. Starr [38, Theorem 1.1] and M. Olsson [35, Theorem 1.5]. When is quasi-coherent and XS is locally projective, Corollary 8.2 was recently addressed by G. Di Brino [13, Theorem 0.0.1] using different methods.

To prove Theorem 8.1 we use Theorem A. Note that there are inclusions

QCoh¯X/Sflb,fp,prbQCoh¯X/Sflb,fpQCoh¯X/Sflb.

The first inclusion is trivially formally étale. By Lemma A.5(a) the second inclusion is also formally étale. Thus, by Lemma 1.5(9), if QCoh¯X/Sflb is 𝐀𝐟𝐟-homogeneous, then Coh¯X/S is 𝐀𝐟𝐟-homogeneous. Also, by Lemmas 6.3 and 6.11, it is sufficient to determine the automorphisms, deformations, and obstructions for QCoh¯X/Sflb.

Throughout, we fix a clivage for QCoh¯X/S. This gives an equivalence of S-groupoids QCoh¯X/S𝐒𝐜𝐡/QCoh¯X/S, which we will use without further comment.

Lemma 8.3

Fix a scheme S and a morphism of algebraic stacks f:XS. Then the S-groupoid QCoh¯X/Sflb is Aff-homogeneous.

Proof.

First we check (H1𝐀𝐟𝐟). Fix a commutative diagram of QCoh¯X/Sflb-schemes:

(8.1)

(8.1)

where p is affine and i is a locally nilpotent closed immersion. Set

(g,γ)=(i,ϕ)(p,π):(T0,0)(T3,3).

Lemma 1.5(1) implies that if the diagram (8.1) is cocartesian in the category of QCoh¯X/Sflb-schemes, then it remains cocartesian in the category of S-schemes. Conversely, suppose that the diagram (8.1) is cocartesian in the category of S-schemes. By Lemma 1.5(1) (applied to Y=Z=S), i is a locally nilpotent closed immersion and p is affine. Let (W,𝒩) be a QCoh¯X/Sflb-scheme and for k3 fix QCoh¯X/Sflb-scheme maps (yk,ψk):(Tk,k)(W,𝒩). Since the diagram (8.1) is cocartesian in the category of S-schemes, there exists a unique S-morphism y3:T3W that is compatible with this data. By adjunction, we obtain unique maps of 𝒪XW-modules

𝒩(y1)*1×(y0)*0(y2)*2{(y3)*p*1}×{(y3)*g*0}{(y3)*i*2}.

The functor (y3)* is left-exact, so there is a functorial isomorphism of 𝒪XW-modules

{(y3)*p*1}×{(y3)*g*0}{(y3)*i*2}(y3)*{p*1×g*0i*2}.

The commutativity of the diagram (8.1) posits a uniquely induced morphism

δ:3p*1×g*0i*2p*p*3×g*g*3i*i*3.

It suffices to prove that δ is an isomorphism, which is local for the smooth topology. So, we immediately reduce to the affine case, where S=SpecA, X=SpecD, and f:XS is given by a ring homomorphism AD. For each l we may take Tl=SpecAl and we set Dl=DAA3. Also, 3M~3, where M3 is a D3-module which is A3-flat. Now, there is an exact sequence of A3-modules

0A3A1×A2A00.

Applying the exact functor M3A3- to this sequence produces an exact sequence

0M3(M3A3A1)×(M3A3A2)M3A3A00.

Since M3A3AlM3D3Dl, we obtain the required isomorphism δ:

M3(M3A3A1)×(M3A3A0)(M3A3A2)
(M3D3D1)×(M3D3D0)(M3D3D2).

Next we check condition (H2𝐀𝐟𝐟). Fix a diagram of QCoh¯X/Sflb-schemes,

[(T1,1)(i,ϕ)(T0,0)(p,π)(T2,2)],

where i is a locally nilpotent closed immersion and p is affine. Given a cocartesian square of S-schemes:

(8.2)

(8.2)

write g=ip and set

3=ker((pXT3)*1×(iXT3)*2𝑑g*0)𝐐𝐂𝐨𝐡(XT3),

where d is the map (m1,m2)(g*ϕ)(m1)-(g*π)(m2). It remains to show that 3 is T3-flat, that the induced morphisms of quasi-coherent 𝒪X2-modules ϕ:iXT3*32 and π:pXT3*31 are isomorphisms, and that the following diagram commutes:

Indeed, this shows that the pairs (i,ϕ) and (p,π) define QCoh¯X/Sflb-morphisms, and that the resulting completion of the diagram (8.2) commutes.

Now, these claims may all be verified locally for the smooth topology. Thus, we reduce to the affine situation as before, with the modification that for k3 we have kM~k, where Mk is a Dk-module which is flat over Ak, and 3M~3M~1×M~0M~2. The result now follows from [14, Théorème 2.2]. ∎

We now determine the automorphisms, deformations, and obstructions. Let (T,) be a QCoh¯X/Sflb-scheme, and fix a quasi-coherent 𝒪T-module I. For an S-extension i:TT of T by I, form the 2-cartesian diagram

Set J=j*ker(𝒪XTj*𝒪XT). Fixing a QCoh¯X/S-extension (i,ϕ):(T,)(T,), we obtain a commutative diagram

By the local criterion for flatness, is T-flat if and only if the diagonal arrow above is an isomorphism. Thus, if a QCoh¯X/Sflb-extension (i,ϕ):(T,)(T,) exists, the top map must be an isomorphism. This is how we will describe our first obstruction.

Example 8.4

This obstruction can be non-trivial when f is not flat and i is not split. Indeed, let S=Spec[x,y] and take 0=(x,y)|S| to be the origin. Set

X=Bl0S=Proj¯S𝒪S[U,V]/(xV-yU),

f:XS the induced map, and let E=f-1(0) be the exceptional divisor. Now take =𝒪E and consider the S-extension T=Specκ(0)T=Spec[x,y]/(x2,y). A straightforward calculation shows that 𝒪XTJ is the skyscraper sheaf supported at the point of E corresponding to the y=0 line in S. Also, fT*I=𝒪XT and so 𝒪XTfT*I𝒪E. The resulting map 𝒪XTfT*I𝒪XTJ is not injective.

Observe that there is a short exact sequence of 𝒪T-modules

0i*I𝒪Ti*𝒪T0.

By Theorem C.1 we obtain an exact sequence of quasi-coherent 𝒪XT-modules

𝒯𝑜𝑟1S,τ,f(i*𝒪T,𝒪X)fT*i*Ij*J0.

Since there is a functorial isomorphism fT*i*Ij*fT*I, Corollary C.3 provides a natural exact sequence of quasi-coherent 𝒪XT-modules

𝒯𝑜𝑟1S,τ,f(𝒪T,𝒪X)fT*IJ0.

Applying the functor 𝒪XT- to this sequence produces another exact sequence

𝒪XT𝒯𝑜𝑟1S,τ,f(𝒪T,𝒪X)o1((T,),I)(i)𝒪XTfT*I𝒪XTJ0.

Thus, we have defined a natural class

o1((T,),I)(i)Hom𝒪XT(𝒪XT𝒯𝑜𝑟1S,τ,f(𝒪T,𝒪X),𝒪XTfT*I),

whose vanishing is necessary and sufficient for the map 𝒪XTfT*I𝒪XTJ to be an isomorphism. By functoriality of the class o1((T,),I)(i), we obtain a natural transformation of functors

o1((T,),-):ExalS(T,-)Hom𝒪XT(𝒪XT𝒯𝑜𝑟1S,τ,f(𝒪T,𝒪X),𝒪XTfT*(-)).

Suppose that the S-extension i:TT now has the property that the map

𝒪XTfT*I𝒪XTJ

is an isomorphism. Let γ,I denote the inverse to this map. Then [24, IV.3.1.12] gives a naturally defined obstruction

o2((T,),I)(i)Extj*𝒪XT2(j*,j*(𝒪XTfT*I))Ext𝒪XT2(,𝒪XTfT*I)

whose vanishing is a necessary and sufficient condition for there to exist a lift of over T. Thus, there is a natural transformation

o2((T,),-):kero1((T,),-)Ext𝒪XT2(,𝒪XTfT*(-))

such that the pair {o1((T,),-),o2((T,),-)} defines a 2-step obstruction theory for the S-groupoid QCoh¯X/Sflb at (T,).

In the case where i=iT,I:TT[I], the trivial X-extension of T by I, then the map 𝒪XTfT*I𝒪XTJ is an isomorphism. By [24, IV.3.1.12], we obtain natural isomorphisms of abelian groups:

AutQCoh¯X/Sflb/S((T,),I)Homj*𝒪XT(j*,j*(𝒪XTfT*I)),
Hom𝒪XT(,𝒪XTfT*I),
DefQCoh¯X/Sflb/S((T,),I)Extj*𝒪XT1(j*,j*(𝒪XTfT*I)),
Ext𝒪XT1(,𝒪XTfT*I).

Proof of Theorem 8.1.

Using standard reductions [41, Appendix B], we are free to assume that f is, in addition, finitely presented, and the scheme S is affine and of finite type over Spec (in particular, it is noetherian and excellent). We now verify the conditions of Theorem A. Certainly, the S-groupoid Coh¯X/S is a limit preserving étale stack over S. By Lemma 8.3, it is also 𝐀𝐟𝐟-homogeneous. Also, for a noetherian local ring (B,𝔪) that is 𝔪-adically complete and a map SpecBS, the canonical functor

𝐐𝐂𝐨𝐡flb,fp,prb(XSpecB)limn𝐐𝐂𝐨𝐡flb,fp,prb(XSpec(B/𝔪n))

is an equivalence of categories [35, Theorem 1.4].

If (T,) is a Coh¯X/S-scheme, then we have proved that

AutCoh¯X/S/S((T,),-)=Hom𝒪XT(,𝒪XTfT*(-)),
DefCoh¯X/S/S((T,),-)=Ext𝒪XT1(,𝒪XTfT*(-)),
O1((T,),-)=Hom𝒪XT(𝒪XT𝒯𝑜𝑟1S,τ,f(𝒪T,𝒪X),𝒪XTfT*(-)),
O2((T,),-)=Ext𝒪XT2(,𝒪XTfT*(-)),

where {O1((T,),-),O2((T,),-)} are the obstruction spaces for a 2-step obstruction theory. If T is assumed to be locally noetherian, then by Theorem C.1, the 𝒪XT-module 𝒯𝑜𝑟1S,τ,f(𝒪T,𝒪X) is coherent. In addition, if T is affine, Example 5.5 implies that the functors listed above are coherent. Having met the conditions of Theorem A, we see that the S-groupoid QCoh¯X/Sflb,fp,prb is algebraic and locally of finite presentation over S.

It remains to show that the diagonal of Coh¯X/S is affine. If (T,) and (T,𝒩) are Coh¯X/S-schemes, then the commutative diagram in the category of T-presheaves

where the morphism along the base is (μ,ν)(νμ,μν), is cartesian. By [20, Theorem D], we deduce the result. ∎

We conclude this section with the following observations. Let XS be a morphism of algebraic stacks and let be a quasi-coherent 𝒪X-module. Let

(T𝜏S,φ:τX*𝒢)=(T,φ)

be a Quot¯X/S()-scheme. Then minor variations of the arguments given in the determination of the deformation and obstruction theory for Coh¯X/S show that there is a 2-step obstruction theory for Quot¯X/S():

o1((T,φ),-):ExalS(T,-)Hom𝒪XT(𝒯𝑜𝑟1S,τ,f(𝒪T,),𝒢𝒪XTfT*(-)),
o2((T,φ),-):kero1((T,φ),-)Ext𝒪XT1(kerφ,𝒢𝒪XTfT*(-)).

Moreover, we also have a functorial isomorphism

DefQuot¯X/S()((T,φ),-)Hom𝒪XT(kerφ,𝒢𝒪XTfT*(-)).

9 Application II: The Hilbert stack and spaces of morphisms

Fix a scheme S and a 1-morphism of algebraic stacks f:XS. For an S-scheme T, consider a property P of a morphism ZXT. Such properties P could be (but not limited to):

  1. qf – quasi-finite,

  2. lfpb – the composition ZXTT is locally of finite presentation,

  3. prb – the composition ZXTT is proper,

  4. flb – the composition ZXTT is flat.

Define Mor¯X/SP to be the category with objects pairs (T,Z𝑔XT), where T is an S-scheme and g:ZXT is a representable morphism of algebraic S-stacks that is P. Morphisms

(p,π):(V,WXV)(T,Z𝑔XT)

in the category Mor¯X/SP are 2-cartesian diagrams

If the property P is reasonably well-behaved, the natural functor Mor¯X/SP𝐒𝐜𝐡/S defines an S-groupoid. We define the Hilbert stack, HS¯X/S, to be the S-groupoid Mor¯X/Sflb,lfpb,prb,qf. This Hilbert stack contains Vistoli’s Hilbert stack [47] as well as the stack of branchvarieties [3]. We will prove the following theorem.

Theorem 9.1

Fix a scheme S and a morphism of algebraic stacks f:XS that is separated and locally of finite presentation. Then HS¯X/S is an algebraic stack, which is locally of finite presentation over S with affine diagonal over S.

Theorem 9.1 was the result alluded to in the title of M. Lieblich’s paper [26], though a precise statement was not given. Theorem 9.1 was established in [26] using an auxiliary representability result [26, Proposition 2.3] combined with [26, Theorem 2.1] (Theorem 8.1 above). In the non-flat case, the obstruction theory used in the proof of [26, Proposition 2.3] is incorrect (a variant of Example 8.4 can be made into a counterexample in this setting also). The stated obstruction theory can be made into the second step of a 2-step obstruction theory, however. The properties of the diagonal of HS¯X/S have not been addressed previously.

Corollary 9.2

Fix a scheme S and morphisms of algebraic stacks f:XS and g:YS. Assume that f is locally of finite presentation, proper, and flat and that g is locally of finite presentation with finite diagonal. Then Hom¯S(X,Y) is an algebraic stack, which is locally of finite presentation over S with affine diagonal over S.

Corollary 9.2 can be used in the construction of the stack of twisted stable maps [1, Proposition 4.2]. The original construction of the stack of twisted stable maps utilized an incorrect obstruction theory in the non-flat case [2, Lemma 5.3.3]. The original proof of Corollary 9.2, due to M. Aoki [4, §3.5], also has an incorrect obstruction theory in the case of a non-flat target. The stated obstruction theories, as before, can be realized as the second step of a 2-step obstruction theory. A variant of Example 8.4 can be made into counterexamples in these settings too.

To prove Theorem 9.1, we will apply Theorem A directly (though as mentioned previously, this could be done as in [26] using Theorem 8.1). With Theorem 9.1 proven it is easy to deduce Corollary 9.2 via the standard method of associating to a morphism its graph, thus the proof is omitted. Now, just as in Section 8, there are inclusions

Mor¯X/Sflb,lfpb,prb,qfMor¯X/Sflb,lfpb,prbMor¯X/Sflb,lfpbMor¯X/Sflb.

The first two inclusions are trivially formally étale. By Lemma A.6, the third inclusion is formally étale. By Lemma 1.5(9), they will all be 𝐀𝐟𝐟-homogeneous if Mor¯X/Sflb is 𝐀𝐟𝐟-homogeneous. Also, by Lemmas 6.3 and 6.11, descriptions of the automorphisms, deformations, and obstructions for Mor¯X/Sflb descend to the subcategories listed above.

Lemma 9.3

Fix a scheme S and a morphism of algebraic stacks f:XS. Then the S-groupoid Mor¯X/Sflb is Aff-homogeneous.

Proof.

First we check (H2𝐀𝐟𝐟). Fix a diagram of Mor¯X/Sflb-schemes

[(T1,Z1g1XT1)(i,ϕ)(T0,Z0g0XT0)(p,π)(T2,Z2g2XT2)],

where i is a locally nilpotent closed immersion and p is affine, and a cocartesian square of S-schemes

By Proposition A.2, there exists a 2-commutative diagram of algebraic S-stacks

where the left and rear faces of the cube are 2-cartesian, and the top and bottom faces are 2-cocartesian in the 2-category of algebraic S-stacks. Thus, the universal properties guarantee the existence of a unique T3-morphism Z3g3XT3. By Lemma A.4, the morphism Z3T3 is flat and all faces of the cube are 2-cartesian. In particular, the resulting Mor¯X/Sflb-scheme diagram

is cocartesian in the category of Mor¯X/S-schemes. Condition (H1𝐀𝐟𝐟) follows from a similar argument as that given in the proof Lemma 8.3. ∎

Fix a Mor¯X/Sflb-scheme (T,Z𝑔XT) and a quasi-coherent 𝒪T-module I. Unravelling the definitions and applying the results of [36] demonstrates that there are natural isomorphisms of abelian groups

AutMor¯X/Sflb/S((T,Z𝑔XT),I)Hom𝒪Z(LZ/XT,g*fT*I),
DefMor¯X/Sflb/S((T,Z𝑔XT),I)Ext𝒪Z1(LZ/XT,g*fT*I).

Using identical ideas to those developed in Section 8, together with [36], we obtain a 2-term obstruction theory for the S-groupoid Mor¯X/Sflb at (T,Z𝑔XT):

o1((T,Z𝑔XT),-):ExalS(T,-)Hom𝒪Z(g*𝒯𝑜𝑟1S,τ,f(𝒪T,𝒪X),g*fT*(-)),
o2((T,Z𝑔XT),-):kero1((T,Z𝑔XT),-)Ext𝒪Z2(LZ/XT,g*fT*(-)).

Proof of Theorem 9.1.

The proof that the S-groupoid HS¯X/S is algebraic and locally of finite presentation is essentially identical to the proof of Theorem 8.1, thus is omitted. It remains to show that the diagonal is affine. If

(T,Z1g1XT)and(T,Z2g2XT)

are HS¯X/S-schemes, then the inclusion of T-presheaves

Isom¯HS¯X/S((T,Z1g1XT),(T,Z2g2XT))
  Isom¯QCoh¯X/S((T,(g2)*𝒪Z2),(T,(g1)*𝒪Z1))

is representable by closed immersions. By Theorem 8.1, the result follows. ∎

A Homogeneity of stacks

The results of this section are routine bootstrapping arguments. They are included so that 𝐀𝐟𝐟-homogeneity can be proved for moduli problems involving stacks.

Definition A.1

Fix a 2-commutative diagram of algebraic stacks

where i and i are closed immersions and f and f are affine. If the induced map

𝒪X3i*𝒪X2×(if)*𝒪X0f*𝒪X1

is an isomorphism of sheaves, then we say that the diagram is a geometric pushout, and that X3 is a geometric pushout of the diagram

[X2𝑓X0𝑖X1].

The main result of this section is the following proposition.

Proposition A.2

Any diagram of algebraic stacks [X2𝑓X0𝑖X1], where i is a locally nilpotent closed immersion and f is affine, admits a geometric pushout X3. The resulting geometric pushout diagram is 2-cartesian and 2-cocartesian in the 2-category of algebraic stacks.

We now need to collect some results which aid with the bootstrapping process.

Lemma A.3

Fix a 2-commutative diagram of algebraic stacks

  1. If the diagram above is a geometric pushout diagram, then it is 2 -cartesian.

  2. If the diagram above is a geometric pushout diagram, then it remains so after flat base change on X3.

  3. If after fppf base change on X3, the above diagram is a geometric pushout diagram, then it was a geometric pushout prior to base change.

Proof.

The claim (1) is local on X3 for the smooth topology, thus we may assume that everything in sight is affine; whence the result follows from [14, Théorème 2.2]. Claims (2) and (3) are trivial applications of flat descent. ∎

Lemma A.4

Consider a 2-commutative diagram of algebraic stacks

where the back and left faces of the cube are 2-cartesian, the top and bottom faces are geometric pushout diagrams, and for j=0, 1, 2, the morphisms UjXj are flat. Then all faces of the cube are 2-cartesian and the morphism U3X3 is flat.

Proof.

By Lemma A.3(2), this is all smooth local on X3 and U3, thus we immediately reduce to the case where everything in sight is affine. Fix a diagram of rings [A2A0𝑝A1] where p:A1A0 is surjective. For j=0, 1, 2 fix flat Aj-algebras Bj, and A0-isomorphisms B2A2A0B0 and B1A1A0B0. Set A3=A2×A0A1 and B3=B2×B0B1, then we have to prove that the A3-algebra B3 is flat, the natural maps B3A3AjBj are isomorphisms, and that these isomorphisms are compatible with the given isomorphisms. This is an immediate consequence of [14, Théorème 2.2], since these are questions about modules. ∎

We omit the proof of the following easy result from commutative algebra.

Lemma A.5

Fix a surjection of rings AA0 and let I=ker(AA0). Suppose that there is a k such that Ik=0.

  1. Given a map of A-modules u:MN such that uAA0 is surjective, then u is surjective.

  2. For an A-module M, if MAA0 is finitely generated, then M is finitely generated.

  3. Given an A-algebra B and a B-module M, let M0=A0AM.

    1. If M is A-flat and M0 is B0-finitely presented, then M is B-finitely presented.

    2. If B0 is a finite type A0-algebra, then B is a finite type A-algebra.

    3. If B is a flat A-algebra and B0 is a finitely presented A0-algebra, then B is a finitely presented A-algebra.

Lemma A.6

Fix a flat morphism f:XY of algebraic stacks and a locally nilpotent closed immersion Y0Y. Then f is locally of finite presentation, respectively smooth, if and only if the same is true of X×YY0Y0.

Proof.

Observe that for flat morphisms which are locally of finite presentation, smoothness is a fibral condition, thus follows from the first claim. The first claim is smooth local on Y and X, thus follows from Lemma A.5(c). ∎

Lemma A.7

Let XX be a locally nilpotent closed immersion of algebraic stacks and let UX be a smooth morphism, where U is an affine scheme. Then there exists a smooth morphism UX which pulls back to UX.

Proof.

Since U is quasi-compact, it is sufficient to treat the case where the locally nilpotent closed immersion XX is square zero. By [36, Theorem 1.4], the only obstruction to the existence of a flat lift of UX over X lies in an abelian group of the form Ext𝒪U2(LU/X,M), where M is a quasi-coherent 𝒪U-module. The morphism UX is smooth, U is affine, and the 𝒪U-module 𝑜𝑚𝒪U(ΩU/X,M) is quasi-coherent, thus

Ext𝒪U2(LU/X,M)=H2(U,𝑜𝑚𝒪U(ΩU/X,M))=0.

Finally, by Lemma A.6, any such lift that is flat, is also smooth. ∎

The following result is a variation of [48, Proposition 2.1].

Lemma A.8

Fix a 2-commutative diagram of algebraic stacks

If the diagram is a geometric pushout diagram and i is a locally nilpotent closed immersion, then it is 2-cartesian and 2-cocartesian in the 2-category of algebraic stacks.

Proof.

That the diagram is 2-cartesian is Lemma A.3(1). It remains to show that we can uniquely complete all 2-commutative diagrams of algebraic stacks

By smooth descent, this is smooth-local on X3, so we may reduce to the situation where the Xj=SpecAj are all affine schemes. Since X3 is a geometric pushout of the diagram

[X2𝑓X0𝑖X2],

it follows that A3A2×A0A1.

Let q:SpecBW be a smooth morphism such that the pullback vj:UjXj of q along ψj is surjective for j{0,1,2}, which exists because the Xj are all quasi-compact. There are compatibly induced morphisms of algebraic spaces ψj,B:UjSpecB for j=1 and 2 and fB:U0U2 and iB:U0U1.

Let c2:SpecC2Uj be an étale morphism such that v2c2 is smooth and surjective. The morphism c2 pulls back along fB to give an étale morphism c0:SpecC0U0 such that v0c0 is smooth and surjective. Let f~:SpecC0SpecC2 and ψ~2:SpecC2SpecB be the resulting morphisms.

Since c0 is étale and i is a locally nilpotent closed immersion, there is an étale morphism c1:SpecC1X1 whose pullback along iB is isomorphic to c0; see [17, IV.18.1.2]. Let C3=C2×C0C1. Then there is a uniquely induced ring homomorphism A3C3. By Lemma A.4, the morphism c3:SpecC3SpecA3 is flat and surjective and by Lemma A.6 it follows that c3 is smooth and surjective. Hence, we may replace SpecAj by SpecCj and further assume that the ψj for j=0, 1, and 2 factor through some smooth morphism q:SpecBW. In particular, there is an induced morphism ψ3:SpecA3SpecBW. It remains to prove that the morphism ψ3 is unique up to a unique choice of 2-morphism. Let ψ3 and ψ3:SpecA3W be two compatible morphisms. That these morphisms are isomorphic can be checked smooth-locally on SpecA3. But smooth-locally, the morphisms ψ3 and ψ3 both factor through some SpecBW and the morphisms SpecAjSpecA3SpecB coincide for j=0, 1, and 2, thus ψ3 and ψ3 are isomorphic. To show that the isomorphism between ψ3 and ψ3 is unique, we just repeat the argument, and the result follows. ∎

We finally come to the proof of Proposition A.2.

Proof of Proposition A.2.

By Lemma A.8, it suffices to prove the existence of geometric pushouts. Let 𝒞0 denote the category of affine schemes. For d=1, 2, 3, let 𝒞d denote the full 2-subcategory of the 2-category of algebraic stacks with affine dth diagonal. Note that 𝒞3 is the full 2-category of algebraic stacks. We will prove by induction on d0 that if

(A.1)[X2𝑓X0𝑖X1]

belongs to 𝒞d, then it admits a geometric pushout. For the base case, where d=0, take X3=Spec(𝒪X2(X2)×𝒪X0(X0)𝒪X1(X1)) and the result is clear. Now let d>0 and assume that (A.1) belongs to 𝒞d. Fix a smooth surjection lΛX2lX2, where X2l is an affine scheme for all lΛ. Set X0l=X2l×X2X0. Then as f is affine, the scheme X0l is also affine. By Lemma A.7, the resulting smooth morphism X0lX0 lifts to a smooth morphism X1lX1, with X1l affine, and X0lX1l×X1X0. For m=0, 1, and 2 and u, v, wΛ set

Xmuv=Xmu×XmXmvandXmuvw=Xmu×XmXmv×XmXmw.

Note that for m=0, 1, and 2 and all u, v, wΛ we have Xmuv, Xmuvw𝒞d-1. By the inductive hypothesis, for I=u, uv or uvw, a geometric pushout X3I of the diagram [X2IX0IX1I] exists. By Lemma A.8, there are uniquely induced morphisms XmuvXmu. For m3, these morphisms are clearly smooth, and by Lemmas A.4 and A.6 the morphisms X3uvX3u are smooth. It is easily verified that the universal properties give rise to a smooth groupoid [u,vΛX3uvwΛX3w]. The quotient X3 of this groupoid in the category of stacks is algebraic. By Lemma A.3(3), it is also a geometric pushout of (A.1). The result follows. ∎

B Fibre products of Picard categories

For background material and conventions on Picard categories, we refer the reader to [8, XVIII.1.4]. In this appendix, we describe a variant of the exact sequence appearing in [19, (2.5.2)].

Let f:PP and g:PQ be 1-morphisms of Picard categories. Define P×f,P,gQ to be the groupoid with objects (p,q,ξ), where pP and qQ and ξ:f(p)g(q), and morphisms (ϕ,χ):(p1,q1,ξ1)(p2,q2,ξ2), where ϕ:p1p2 and χ:q1q2 are morphisms such that the following diagram commutes:

It is easily shown that P×f,P,gQ admits a natural structure of a Picard category such that the induced projections f:P×f,P,gQQ and g:P×f,P,gQP are 1-morphisms of Picard categories. There is also a canonically induced 2-morphism α:fggf. In particular, there is a 2-commutative diagram of Picard categories,

It is easily shown that the 2-commutative diagram above is 2-cartesian in the 2-category of Picard categories.

Let denote the Picard category with one object, whose abelian group of automorphisms is 0. If P is a Picard category and 0P is a zero object of P, then there is an induced 1-morphism of Picard categories 0P:P. Also, there is a unique 1-morphism of Picard categories 0:P. Finally, let P¯ be the abelian group of isomorphism classes of P.

Let f1:P1P2 and f2:P2P3 be 1-morphisms of Picard categories and let 0P1 be a 0-object of P1. Let 0P3=f2f1(0P1). We say that the sequence of Picard categories

 0P1P1f1P2f2P3

is left-exact if there exists a 2-morphism δ:f2f10P30 that makes the 2-commutative diagram

2-cartesian in the 2-category of Picard categories.

The main result of this appendix is the following lemma.

Lemma B.1

Consider a left-exact sequence of Picard categories

 0P1P1f1P2f2P3.

Let 0P2=f1(0P1) and 0P3=f2(0P2). Then there is an exact sequence of abelian groups

Proof.

By the explicit description of the 2-fiber product of Picard categories, we may assume that P1 is expressed in the following way: it is the Picard category with objects pairs (p2,a), where p2P2 and a:f2(p2)0P3 is a morphism in P3, and morphisms ϕ:(p2,a)(p2,a), where ϕ:p2p2 is a morphism such that a(f2)*(ϕ)=Id0P3. The functor f1:P1P2 is the forgetful functor: (p2,a)p2. We also take 0P1 to be (0P2,Id0P3). Finally, the 2-morphism δ:f2f10P30 sends (f2f1)(p2,a)=f2(p2) to (0P30)(p2,a)=0P3 via a.

In this case it is trivial from the definitions that the sequence

0AutP1(0P1)(f1)*AutP2(0P2)(f2)*AutP3(0P3)

is exact. The morphism is described as follows: it sends an automorphism l:0P30P3 to the isomorphism class of the object (0P2,l)P1. Note that this shows, in particular, that (l)=0 if and only if there is an isomorphism ϕ:(0P2,l)(0P2,Id0P3). That is, (l)=0 if and only if there is an automorphism ϕ:0P20P2 such that l(f2)*(ϕ)=IdP3. It follows that (l)=0 if and only if lim(f2)*, thus the sequence

AutP2(0P2)(f2)*AutP3(0P3)P1¯

is exact. If (p2,a)P1, then f1¯(p2,a)=0 in P2¯ if and only if there is an isomorphism q:p20P2. In particular, it follows that q induces an isomorphism

(p2,a)(0P2,(f2)*(q)a-1)

in P1 and so (p2,a) belongs to im if and only if f1¯(p2,a). Finally, if p2P2, then f2¯(p2)=0 in P3¯ if and only if there is an isomorphism m:f2(p2)0P3. This is manifestly equivalent to p2 lying in the image of f1¯. We have thus shown that the sequence

AutP3(0P3)P1¯f1¯P2¯f2¯P3¯

is exact. The result follows. ∎

C Local Tor functors on algebraic stacks

The aim of the section is to state some easy generalizations of [17, III.6.5] to algebraic stacks.

Theorem C.1

Fix a scheme S and a 2-cartesian diagram of algebraic S-stacks:

Then for every integer i0 there exists a natural bifunctor

𝒯𝑜𝑟iX0,f1,f2(-,-):𝐐𝐂𝐨𝐡(X1)×𝐐𝐂𝐨𝐡(X2)𝐐𝐂𝐨𝐡(X3),

such that the family of bifunctors {T𝑜𝑟iX0,f1,f2(-,-)}i0 forms a -functor in each variable. Moreover, there is a natural isomorphism for all MQCoh(X1) and NQCoh(X2),

𝒯𝑜𝑟0X0,f1,f2(M,N)f2*M𝒪X3f1*N.

If M or N is X0-flat, then for all i>0 we have T𝑜𝑟iX0,f1,f2(M,N)=0. In addition, if the algebraic stacks X1 and X0 are locally noetherian and the morphism f2 is locally of finite type, then the bifunctor above restricts to a bifunctor:

𝒯𝑜𝑟iX0,f1,f2(-,-):𝐂𝐨𝐡(X1)×𝐂𝐨𝐡(X2)𝐂𝐨𝐡(X3).

Proof.

We will describe the quasi-coherent sheaves 𝒯𝑜𝑟iX0,f1,f2(M,N) smooth-locally on X3 and deduce their existence via descent. The other properties will be trivial consequences of this construction. We begin by observing that X3 admits a smooth cover by affine schemes of the form Spec(A1A0A2), where for each j=0, 1, and 2 there is a smooth morphism SpecAjX. For each integer i0 let

𝒯𝑜𝑟iX0,f1,f2(M,N)|Spec(A1A0A2)=ToriA0(Γ(SpecA1,M),Γ(SpecA2,N)).

Clearly, the above is an (A1A0A2)-module with the relevant properties. The result follows. ∎

Remark C.2

Unless X1 and X2 are tor-independent over X0, the quasi-coherent 𝒪X3-modules 𝒯𝑜𝑟iX0,f1,f2(M,N) are not isomorphic to -i([𝖫f1*N]𝒪X3𝖫[𝖫f2M]).

An immediate consequence of the proof of Theorem C.1 is the following corollary.

Corollary C.3

Fix a scheme S and a 2-cartesian diagram of algebraic S-stacks

where the morphism h is affine. Then, for any MQCoh(W), NQCoh(Y), and i0, there is a natural isomorphism of quasi-coherent OX×ZY-modules:

𝒯𝑜𝑟iZ,f,g(h*M,N)h*𝒯𝑜𝑟iZ,fh,g(M,N).

Acknowledgements

I would like to thank the Royal Institute of Technology, Sweden (KTH) for their support while this research was completed. I also would like to thank R. Ile, R. Skjelnes, and B. Williams for some interesting conversations. A special thanks goes to J. Wise for explaining to me the notion of homogeneity. A very special thanks is due to D. Rydh for his tremendous patience and enthusiasm. Finally, I would like to thank the anonymous referee for their many excellent suggestions.

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Received: 2013-8-11
Revised: 2014-3-21
Published Online: 2014-8-9
Published in Print: 2017-1-1

© 2017 by De Gruyter

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