Azumaya algebras and the cohomological Brauer group

Our work is strongly influenced by Toën’s excellent paper [Reference ToënToë12] on derived Azumaya algebras and generators of twisted derived categories. In [Reference ToënToë12], Toën showed that compact generation of certain linear categories on derived schemes is an fppf-local question. The salient example is the derived category of twisted sheaves $\mathsf{D}(\mathsf{QCoh}^{\unicode[STIX]{x1D6FC}}(X))$ , where the twisting is given by a Brauer class $\unicode[STIX]{x1D6FC}$ of $H^{2}(X,\mathbb{G}_{m})$ . A compact generator of $\mathsf{D}(\mathsf{QCoh}^{\unicode[STIX]{x1D6FC}}(X))$ gives rise to a derived Azumaya algebra: the endomorphism algebra of the generator [Reference ToënToë12, Proposition 4.6]. More classically, a twisted vector bundle that is generating gives rise to an Azumaya algebra ${\mathcal{A}}$ .

The Brauer group $\operatorname{Br}(X)$ classifies Azumaya algebras ${\mathcal{A}}$ up to Morita equivalence, that is, up to equivalence of the category of modules $\mathsf{Mod}({\mathcal{A}})$ . Moreover, $\mathsf{Mod}({\mathcal{A}})\simeq \mathsf{QCoh}^{\unicode[STIX]{x1D6FC}}(X)$ for a unique element $\unicode[STIX]{x1D6FC}$ in the cohomological Brauer group $\operatorname{Br}^{\prime }(X):=H^{2}(X,\mathbb{G}_{m})_{\text{tors}}$ . Existence of twisted vector bundles thus answers the question of whether $\operatorname{Br}(X)\rightarrow \operatorname{Br}^{\prime }(X)$ is surjective.

Constructing twisted vector bundles, or equivalently Azumaya algebras, is difficult. Indeed, the question is not local as vector bundles rarely extend over open immersions. When $X$ is affine, or the separated union of two affines, Gabber proved in his thesis that $\operatorname{Br}(X)=\operatorname{Br}^{\prime }(X)$  [Reference GabberGab81] or, equivalently, that $\mathsf{D}(\mathsf{QCoh}^{\unicode[STIX]{x1D6FC}}(X))$ is compactly generated by a twisted vector bundle [Reference LieblichLie04, Theorem 2.2.3.3]. The state of the art is also due to Gabber: twisted vector bundles exist if $X$ is quasi-projective [Reference de JongdeJ03].

Compact objects of the derived category are typically easier to construct as we may extend them over open immersions using Thomason’s localization theorem (Corollary 3.13). With this technique, Lieblich proved that $\mathsf{D}(\mathsf{QCoh}^{\unicode[STIX]{x1D6FC}}(X))$ is compactly generated when $X$ is any quasi-compact and quasi-separated scheme [Reference LieblichLie04, Corollary 2.2.4.14]. Lieblich has also studied twisted vector bundles in great detail and obtained a number of arithmetic applications.

In § 9, we prove that compact generation is quasi-finite flat local for twisted derived categories. In particular, we prove that on a quasi-compact algebraic stack with quasi-finite and separated diagonal every twisted derived category has a compact generator (Example 9.3). We thus establish a derived analogue of $\operatorname{Br}(X)=\operatorname{Br}^{\prime }(X)$ for such stacks, extending the results of Toën [Reference ToënToë12] and Antieau and Gepner [Reference Antieau and GepnerAG14].

Sheaves of linear categories on derived Deligne–Mumford stacks

Although we work with non-derived schemes and stacks, our methods are strong enough to deduce similar results for derived (and spectral) Deligne–Mumford stacks (Example 9.4). Indeed, if $X$ is a derived Deligne–Mumford stack, then the small étale topos of $X$ is equivalent to the small étale topos of the non-derived $0$ -truncation $\unicode[STIX]{x1D70B}_{0}X$ . Thus, (local) compact generation of a presheaf of triangulated categories on $X$ can be studied on $\unicode[STIX]{x1D70B}_{0}X$ .

Sometimes results for stacks can be deduced from schemes using a similar approach: if $\unicode[STIX]{x1D70B}:X\rightarrow X_{\text{cms}}$ is a coarse moduli space, then a presheaf ${\mathcal{T}}$ of triangulated categories on $X$ induces a presheaf $\unicode[STIX]{x1D70B}_{\ast }{\mathcal{T}}$ of triangulated categories on $X_{\text{cms}}$ . If $\unicode[STIX]{x1D70B}_{\ast }{\mathcal{T}}$ is locally compactly generated, then it is enough to show that compact generation is local on $X_{\text{cms}}$ to deduce compact generation of ${\mathcal{T}}(X)$ . This is how Toën extended his result to Deligne–Mumford stacks admitting a coarse moduli scheme [Reference ToënToë12, Corollary 5.2].

Perfect and compact objects

As we already have mentioned, some care has to be taken since perfect objects are not necessarily compact. The perfect objects are the locally compact objects or, equivalently, the dualizable objects. If $X$ is a quasi-compact and quasi-separated algebraic stack, then the following conditions are equivalent (Remark 4.6):

1. – every perfect object of $\mathsf{D}_{\text{qc}}(X)$ is compact;

2. – the structure sheaf ${\mathcal{O}}_{X}$ is compact;

3. – there exists an integer $d_{0}$ such that for all quasi-coherent sheaves $M$ on $X$ , the cohomology groups $H^{d}(X,M)$ vanish for all $d>d_{0}$ ; and

4. – the derived global section functor $\mathsf{R}\unicode[STIX]{x1D6E4}:\mathsf{D}_{\text{qc}}(X)\rightarrow \mathsf{D}(\mathsf{Ab})$ commutes with small coproducts.

We say that a stack is concentrated when it satisfies the conditions above.

In [Reference Hall and RydhHR15, Theorem B], we give a complete list of the group schemes $G/k$ such that $BG$ is concentrated: every linear group and certain non-affine groups in characteristic zero but only the linearly reductive groups in positive characteristic. Note that [Reference Hall and RydhHR15, Theorem A] also gives many examples of classifying stacks that are not concentrated, yet compactly generated.

Drinfeld and Gaitsgory have proved that noetherian algebraic stacks with affine stabilizer groups in characteristic zero are concentrated [Reference Drinfeld and GaitsgoryDG13, Theorem 1.4.2]. This is generalized in [Reference Hall and RydhHR15, Theorem C] to positive characteristic. In particular, a stack with finite stabilizers is concentrated if and only if it is tame.

Perfect stacks

Ben-Zvi et al. introduced the notion of a perfect (derived) stack in [Reference Ben-Zvi, Francis and NadlerBFN10]. In our context, an algebraic stack $X$ is perfect if and only if it has affine diagonal, it is concentrated, and its derived category $\mathsf{D}_{\text{qc}}(X)$ is compactly generated [Reference Ben-Zvi, Francis and NadlerBFN10, Proposition 3.9]. A direct consequence of our main theorems and [Reference Hall and RydhHR15, Theorem C] is that the following classes of algebraic stacks are perfect:

1. (i) quasi-compact tame Deligne–Mumford stacks with affine diagonal; and

2. (ii) $\mathbb{Q}$ -stacks of s-global type with affine diagonal.

The affine diagonal assumption is needed only because it is required in the definition of a perfect stack. It is useful though: if $X$ is perfect, then $\mathsf{D}(\mathsf{QCoh}(X))=\mathsf{D}_{\text{qc}}(X)$ by [Reference Hall, Neeman and RydhHNR14].

In the terminology of Lurie [Reference LurieLur11a, Definition 8.14], an algebraic stack is perfect if it has quasi-affine diagonal, is concentrated, and $\mathsf{D}_{\text{qc}}(X)$ is compactly generated. Thus, in Lurie’s terminology, we have shown that:

1. (i) quasi-compact tame Deligne–Mumford stacks with quasi-compact and separated diagonals; and

2. (ii) $\mathbb{Q}$ -stacks of s-global type

are perfect.

Coherence

Compact generation is extremely useful and we will illustrate this with a simple application; also the origin of this paper. Let $A$ be a commutative ring and $\mathsf{Mod}(A)$ the category of $A$ -modules. A functor $F:\mathsf{Mod}(A)\rightarrow \mathsf{Ab}$ is coherent if there exists a homomorphism of $A$ -modules $M_{1}\rightarrow M_{2}$ together with isomorphisms

$$\begin{eqnarray}F(N)\cong \operatorname{coker}(\operatorname{Hom}_{A}(M_{2},N)\rightarrow \operatorname{Hom}_{A}(M_{1},N))\end{eqnarray}$$

natural in $N$ . This definition is due to Auslander [Reference AuslanderAus66], who initiated the study of coherent functors. Hartshorne [Reference HartshorneHar98] studied in detail coherent functors when $A$ is noetherian and $M_{1}$ and $M_{2}$ are finitely generated and obtained some very nice applications to classical algebraic geometry. For background material on coherent functors, we refer the reader to Hartshorne’s article. Recently, the first author has used coherent functors to prove ‘cohomology and base change’ for algebraic stacks [Reference HallHal14] and to give a new criterion for algebraicity of a stack [Reference HallHal17].

Using the compact generation results of this article, we can give a straightforward proof of the following theorem (combine Theorem A with Corollary 4.16).

Theorem D. Let $A$ be a noetherian ring and let $\unicode[STIX]{x1D70B}:X\rightarrow \operatorname{Spec}A$ be a proper morphism of algebraic stacks with finite diagonal. If ${\mathcal{F}}\in \mathsf{D}_{\text{qc}}(X)$ and ${\mathcal{G}}\in \mathsf{D}_{\mathsf{Coh}}^{b}(X)$ , then the functor

$$\begin{eqnarray}\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{F}},{\mathcal{G}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }(-)):\mathsf{Mod}(A)\rightarrow \mathsf{Mod}(A)\end{eqnarray}$$

is coherent.

Theorem D generalizes a result of the first author for algebraic spaces [Reference HallHal14, Theorem E], which was proved using a completely different argument. The first author has also proved a non-noetherian and infinite-stabilizer variant of Theorem D at the expense of assuming that ${\mathcal{G}}$ has flat cohomology sheaves over $S$ [Reference HallHal14, Theorem C].

Related results

The first proof that $\mathsf{D}_{\text{qc}}(X)$ is compactly generated when $X$ is a quasi-compact separated scheme appears to be due to Neeman [Reference NeemanNee96, Proposition 2.5] although he attributed the ideas to Thomason [Reference Thomason and TrobaughTT90]. Bondal and Van den Bergh [Reference Bondal and Van den BerghBVdB03, Theorem 3.1.1] adapted the proof to deal with quasi-separated schemes and noted that there is a single compact generator. Lipman and Neeman further refined the result by giving an effective bound on the existence of maps from the compact generator [Reference Lipman and NeemanLN07, Theorem 4.2]. As noted by Ben-Zvi et al., the proof of Bondal and Van den Bergh readily extends to derived schemes [Reference Ben-Zvi, Francis and NadlerBFN10, Proposition 3.19].

In [Reference LurieLur11a, Theorem 6.1] and [Reference LurieLur11b, Theorem 1.5.10], Lurie proved that compact generation is étale local on $\mathbb{E}_{\infty }$ -algebras and on spectral algebraic spaces for quasi-coherent stacks (sheaves of linear $\infty$ -categories). Lurie used scallop decompositions, which are a special type of étale neighborhoods (or Nisnevich squares). Unfortunately, scallop decompositions do not exist for algebraic stacks. This is what necessitates our stronger inductive assumption, $\unicode[STIX]{x1D6FD}$ -crispness, for our local-global principle, Theorem C. Indeed, to apply Thomason’s localization theorem it is necessary to establish the existence of compact objects with prescribed support. On affine schemes (which appear in the scallop decompositions) this is done using Koszul complexes, cf. Bökstedt and Neeman [Reference Bökstedt and NeemanBN93, Proposition 6.1]. This is the basis for our induction and also used in all previous proofs, e.g., [Reference ToënToë12, Lemma 4.10] and [Reference Antieau and GepnerAG14, Proposition 6.9].

Drinfeld and Gaitsgory [Reference Drinfeld and GaitsgoryDG13, Theorem 8.1.1] proved that on an algebraic stack of finite type over a field of characteristic zero with affine stabilizers, the derived category of $D$ -modules is compactly generated. They remarked that compact generation of $\mathsf{D}_{\text{qc}}(X)$ is much subtler and open in general [Reference Drinfeld and GaitsgoryDG13, 0.3.3].

Antieau [Reference AntieauAnt14] has considered local-global results for the telescope conjecture. Some of these are generalized in [Reference Hall and RydhHR17].

Krishna [Reference KrishnaKri09] has considered the K-theory and G-theory for tame Deligne–Mumford stacks with the resolution property admitting projective coarse moduli schemes (i.e., projective stacks).

Future extensions

In [Reference Lipman and NeemanLN07, Theorem 4.1], Lipman and Neeman proved that pseudo-coherent complexes can be approximated arbitrarily well by perfect complexes on a quasi-compact and quasi-separated scheme. Local approximability by perfect complexes is essentially the definition of pseudo-coherence, so this is a local-global result in the style of Theorem C. This result has been extended to algebraic spaces in [Sta, 08HH] and we expect that it can be extended to stacks with quasi-finite diagonal using the methods of this paper amplified with $t$ -structures. Similarly, we expect that there is an effective bound on the compact generator in Theorem A as in [Reference Lipman and NeemanLN07, Theorem 4.2].

Contents of this paper

In §§1–2, we recall and develop some generalities on unbounded derived categories of quasi-coherent sheaves on stacks and concentrated morphisms; working in the unbounded derived category is absolutely essential for this range of mathematics. Unfortunately, some important foundational notions, such as concentrated morphisms, had not been considered in the literature before.

In § 3, we recall the concept of compact objects and Thomason’s localization theorem for triangulated categories.

In § 4, we address fundamental results on perfect and compact objects in the derived categories of quasi-coherent sheaves on algebraic stacks. Using this, we establish a general projection formula for stacks, tor-independent base change, and finite flat duality. We also prove Theorem D assuming Theorem A.

In §§5–6, we introduce presheaves of triangulated categories and Mayer–Vietoris triangles. We also prove our main result on descent of compact generation (Theorem 6.9).

In § 7, we introduce the $\unicode[STIX]{x1D6FD}$ -resolution property, which gives a convenient method to keep track of the number of vector bundles needed for generating the derived category of a stack with the resolution property.

In § 8, we introduce compact generation with supports and $\unicode[STIX]{x1D6FD}$ -crispness and relate these to Koszul complexes.

In § 9, we prove the main theorems.

Notations and assumptions

For an abelian category ${\mathcal{A}}$ , denote by $\mathsf{D}({\mathcal{A}})$ its unbounded derived category. For a complex $M\in \mathsf{D}({\mathcal{A}})$ , denote its $i$ th cohomology group by ${\mathcal{H}}^{i}(M)$ . For a sheaf of rings $A$ on a topos $E$ , denote by $\mathsf{Mod}(A)$ (respectively $\mathsf{QCoh}(A)$ ) the category of $A$ -modules (respectively the category of quasi-coherent $A$ -modules). If the sheaf of rings $A$ on the topos $E$ is implicit, it will be convenient to denote $\mathsf{Mod}(A)$ as $\mathsf{Mod}(E)$ and $\mathsf{D}(\mathsf{Mod}(A))$ as $\mathsf{D}(E)$ .

For algebraic stacks, we adopt the conventions of the Stacks Project [Sta]. This means that algebraic stacks are stacks over the big fppf site of some scheme, admitting a smooth, representable, surjective morphism from a scheme (note that there are no separation hypotheses here). A morphism of algebraic stacks is quasi-separated if its diagonal and double diagonal are represented by quasi-compact morphisms of algebraic spaces.

For a scheme $X$ , denote its underlying topological space by $|X|$ . For a scheme $X$ and a point $x\in |X|$ , denote by $\unicode[STIX]{x1D705}(x)$ the residue field at $x$ .

Let $f:X\rightarrow Y$ be a $1$ -morphism of algebraic stacks. Then, for any other $1$ -morphism of algebraic stacks $g:Z\rightarrow Y$ , we denote by $f_{Z}:X_{Z}\rightarrow Z$ the pullback of $f$ by $g$ .

1.1 Hypercoverings and simplicial sites

We now recall the relationship between the unbounded derived categories of quasi-coherent sheaves on an algebraic stack and those on a smooth hypercovering (i.e., cohomological descent). Our approach follows [Reference OlssonOls07] and [Reference Laszlo and OlssonLO08]. Let $X$ be an algebraic stack and let $p_{\bullet }:U_{\bullet }\rightarrow X$ be a smooth hypercovering by algebraic spaces. Typically, we will take $U_{\bullet }$ to be the $0$ -coskeleton associated to a smooth covering $p_{0}:U_{0}\rightarrow X$ , where $U_{0}$ is an algebraic space. In plainer language, $U_{n}$ is the $(n+1)$ th fiber product of $U_{0}$ over $X$ and the simplicial structure (i.e., face and degeneracy maps) comes from the various projections and diagonals between the $U_{n}$ as $n$ varies.

The simplicial algebraic space $U_{\bullet }$ gives rise to two semi-simplicial topoi: $U_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+}$ and $U_{\bullet ,\acute{\text{e}}\text{t}}^{+}$ . The semi-simplicial topos $U_{\bullet ,\acute{\text{e}}\text{t}}^{+}$ is formed as follows: for each integer $n\geqslant 0$ there is the étale topos $U_{n,\acute{\text{e}}\text{t}}$ and for each injective map $\unicode[STIX]{x1D6FF}:[n]=\{0,\ldots ,n\}\rightarrow [m]=\{0,\ldots ,m\}$ there is a morphism of topoi $\unicode[STIX]{x1D6FF}:U_{m,\acute{\text{e}}\text{t}}\rightarrow U_{n,\acute{\text{e}}\text{t}}$ . A sheaf $F_{\bullet }$ on $U_{\bullet ,\acute{\text{e}}\text{t}}^{+}$ is a sheaf $F_{n}$ on each $U_{n,\acute{\text{e}}\text{t}}$ together with transition maps $\unicode[STIX]{x1D6FF}^{-1}F_{n}\rightarrow F_{m}$ for each injective map $\unicode[STIX]{x1D6FF}:[n]\rightarrow [m]$ that are compatible with composition; the sheaf $F_{\bullet }$ is cartesian if the transition maps are always isomorphisms.

The topos $U_{\bullet ,\acute{\text{e}}\text{t}}^{+}$ is naturally ringed by the flat sheaf ${\mathcal{O}}_{U_{\bullet ,\acute{\text{e}}\text{t}}}^{+}$ . Here flat means that the transition maps $\unicode[STIX]{x1D6FF}^{-1}{\mathcal{O}}_{U_{n},\acute{\text{e}}\text{t}}\rightarrow {\mathcal{O}}_{U_{m},\acute{\text{e}}\text{t}}$ are flat. Let $\mathsf{Mod}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ denote the associated category of modules and $\mathsf{Mod}_{\text{cart}}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ the subcategory of cartesian sheaves. Here cartesian means that the transition maps $\unicode[STIX]{x1D6FF}^{\ast }F_{n}=\unicode[STIX]{x1D6FF}^{-1}F_{n}\otimes _{\unicode[STIX]{x1D6FF}^{-1}{\mathcal{O}}_{U_{n,\acute{\text{e}}\text{t}}}}{\mathcal{O}}_{U_{m,\acute{\text{e}}\text{t}}}\rightarrow F_{m}$ are isomorphisms.

An ${\mathcal{O}}_{U_{\bullet ,\acute{\text{e}}\text{t}}}^{+}$ -module is quasi-coherent if it is cartesian and its restriction to each $U_{n,\acute{\text{e}}\text{t}}$ is quasi-coherent. Let $\mathsf{QCoh}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ denote the category of quasi-coherent sheaves. Let $\mathsf{D}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ be the unbounded derived category of $\mathsf{Mod}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ , let $\mathsf{D}_{\text{cart}}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ denote the subcategory whose objects have cartesian cohomology sheaves, and let $\mathsf{D}_{\text{qc}}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ denote the subcategory with quasi-coherent cohomology sheaves. The semi-simplicial topos $U_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+}$ and its various module categories are defined similarly.

Thus, there are natural morphisms of ringed topoi:

(1.1) $$\begin{eqnarray}X_{\text{lis}\text{-}\acute{\text{e}}\text{t}}\xleftarrow[{}]{p_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+}}U_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+}\xrightarrow[{}]{\operatorname{res}_{U_{\bullet }}}U_{\bullet ,\acute{\text{e}}\text{t}}^{+}.\end{eqnarray}$$

One way of phrasing smooth descent of quasi-coherent sheaves is that these morphisms of topoi induce equivalences of abelian categories:

$$\begin{eqnarray}\mathsf{QCoh}(X)\xleftarrow[{}]{(p_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})_{\ast }}\mathsf{QCoh}(U_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})\xrightarrow[{}]{(\operatorname{res}_{U_{\bullet }})_{\ast }}\mathsf{QCoh}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+}).\end{eqnarray}$$

In [Reference Laszlo and OlssonLO08, Example 2.2.5], it is shown that this can be improved to unbounded cohomological descent, that is, these morphisms of topoi induce equivalences of triangulated categories:

(1.2) $$\begin{eqnarray}\mathsf{D}_{\text{qc}}(X)\xleftarrow[{}]{\mathsf{R}(p_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})_{\ast }}\mathsf{D}_{\text{qc}}(U_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})\xrightarrow[{}]{\mathsf{R}(\operatorname{res}_{U_{\bullet }})_{\ast }}\mathsf{D}_{\text{qc}}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+}).\end{eqnarray}$$

The morphisms $p_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+}$ and $\operatorname{res}_{U_{\bullet }}$ have left-exact inverse image functors.

1.2 Operations in unbounded categories of modules

We now record for future reference some useful formulas [Reference Kashiwara and SchapiraKS06, §§18.4 and 18.6]. If ${\mathcal{M}}$ and ${\mathcal{N}}\in \mathsf{D}(X)$ , then there are

$$\begin{eqnarray}\displaystyle {\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}{\mathcal{N}} & \in & \displaystyle \mathsf{D}(X)\quad \text{(the derived tensor product),}\nonumber\\ \displaystyle \mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{M}},{\mathcal{N}}) & \in & \displaystyle \mathsf{D}(X)\quad \text{(the derived sheaf Hom functor),}\nonumber\\ \displaystyle \mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{M}},{\mathcal{N}}) & \in & \displaystyle \mathsf{D}(\mathsf{Ab})\quad \text{(the derived global Hom functor).}\nonumber\end{eqnarray}$$

If in addition ${\mathcal{P}}\in \mathsf{D}(X)$ , then we have a functorial isomorphism

(1.3) $$\begin{eqnarray}\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}{\mathcal{N}},{\mathcal{P}})\cong \operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{M}},\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{N}},{\mathcal{P}}))\end{eqnarray}$$

as well as a functorial quasi-isomorphism

(1.4) $$\begin{eqnarray}\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}{\mathcal{N}},{\mathcal{P}})\simeq \mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{M}},\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{N}},{\mathcal{P}})).\end{eqnarray}$$

Letting $\mathsf{R}\unicode[STIX]{x1D6E4}(X,-)=\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{O}}_{X},-)$ , there is also a natural quasi-isomorphism

(1.5) $$\begin{eqnarray}\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{M}},{\mathcal{N}})\simeq \mathsf{R}\unicode[STIX]{x1D6E4}\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{M}},{\mathcal{N}}).\end{eqnarray}$$

If ${\mathcal{M}}$ and ${\mathcal{N}}$ belong to $\mathsf{D}_{\text{qc}}(X)$ , then ${\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}{\mathcal{N}}\in \mathsf{D}_{\text{qc}}(X)$ . Indeed, using homotopy colimits and triangles we reduce to the case where ${\mathcal{M}}$ and ${\mathcal{N}}$ are quasi-coherent sheaves. Then it follows from the definition of quasi-coherence [Reference Laumon and Moret-BaillyLM00, Définition 13.2.2]. Since the category $\mathsf{D}_{\text{qc}}(X)$ is well generated [Reference Hall, Neeman and RydhHNR14, Theorem B.1] and the functor $-\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}{\mathcal{M}}:\mathsf{D}_{\text{qc}}(X)\rightarrow \mathsf{D}_{\text{qc}}(X)$ preserves small coproducts, it admits a right adjoint [Reference NeemanNee01b, Theorem 8.4.4]

$$\begin{eqnarray}\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}^{\text{qc}}({\mathcal{M}},-):\mathsf{D}_{\text{qc}}(X)\rightarrow \mathsf{D}_{\text{qc}}(X).\end{eqnarray}$$

In fact, if

$$\begin{eqnarray}{\mathcal{Q}}_{X}:\mathsf{D}(X)\rightarrow \mathsf{D}_{\text{qc}}(X)\end{eqnarray}$$

is the right adjoint to the inclusion $\mathsf{D}_{\text{qc}}(X)\subseteq \mathsf{D}(X)$ , which exists for the same reasons as above, then

$$\begin{eqnarray}\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}^{\text{qc}}({\mathcal{M}},-)\simeq {\mathcal{Q}}_{X}(\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{M}},-)).\end{eqnarray}$$

Note that while the formation of $\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{M}},-)$ is smooth local on $X$ , this is not true in general for $\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}^{\text{qc}}({\mathcal{M}},-)$ . It is true, however, if ${\mathcal{M}}$ is perfect (Lemma 4.3 (ii)).

1.3 Direct and inverse images

For a morphism of algebraic stacks $f:X\rightarrow Y$ , the induced morphism of ringed topoi $f_{\text{lis}\text{-}\acute{\text{e}}\text{t}}:X_{\text{lis}\text{-}\acute{\text{e}}\text{t}}\rightarrow Y_{\text{lis}\text{-}\acute{\text{e}}\text{t}}$ does not necessarily have a left-exact inverse image functor [Reference BehrendBeh03, 5.3.12]. Thus, the construction of the correct derived functors of $f^{\ast }:\mathsf{QCoh}(Y)\rightarrow \mathsf{QCoh}(X)$ is somewhat subtle. There are currently two approaches to constructing these functors. The first, due to Olsson [Reference OlssonOls07] and Laszlo and Olsson [Reference Laszlo and OlssonLO08], uses cohomological descent. The other approach appears in the Stacks Project [Sta, Tag 07BD]. In this article, we will employ the approach of Olsson and Laszlo–Olsson, which we now briefly recall.

Let $f:X\rightarrow Y$ be a morphism of algebraic stacks. Let $q:V\rightarrow Y$ be a smooth surjection from an algebraic space. Let $U\rightarrow X\times _{Y}V$ be another smooth surjection from an algebraic space. Let $\tilde{f}:U\rightarrow V$ be the resulting morphism of algebraic spaces and let $p:U\rightarrow X$ be the resulting smooth covering. By (1.1), there is an induced $2$ -commutative diagram of ringed topoi as follows.

(1.6)

The $2$ -commutativity of the diagram above induces natural transformations:

(1.7) $$\begin{eqnarray}\displaystyle \mathsf{R}(f_{\text{lis}\text{-}\acute{\text{e}}\text{t}})_{\ast } & \Rightarrow & \displaystyle \mathsf{R}(q_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})_{\ast }\mathsf{R}(\tilde{f}_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})_{\ast }\mathsf{L}(p_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})^{\ast }\end{eqnarray}$$

(1.8) $$\begin{eqnarray}\displaystyle \mathsf{R}(\tilde{f}_{\bullet ,\acute{\text{e}}\text{t}}^{+})_{\ast } & \Rightarrow & \displaystyle \mathsf{R}(\operatorname{res}_{V_{\bullet }})_{\ast }\mathsf{R}(\tilde{f}_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})_{\ast }\mathsf{L}(\operatorname{res}_{U_{\bullet }})^{\ast },\end{eqnarray}$$

which are natural isomorphisms for those complexes with quasi-coherent cohomology that are sent to complexes with quasi-coherent cohomology by $\mathsf{R}(\tilde{f}_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})_{\ast }$ or $\mathsf{R}(\tilde{f}_{\bullet ,\acute{\text{e}}\text{t}}^{+})_{\ast }$ .

Another crucial observation here is that the morphism of topoi $\tilde{f}_{\bullet ,\acute{\text{e}}\text{t}}^{+}$ has a left-exact inverse image functor. The general theory now gives rise to an unbounded derived functor $\mathsf{L}(\tilde{f}_{\bullet ,\acute{\text{e}}\text{t}}^{+})^{\ast }:\mathsf{D}(V_{\bullet ,\acute{\text{e}}\text{t}}^{+})\rightarrow \mathsf{D}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ , which is left adjoint to $\mathsf{R}(\tilde{f}_{\bullet ,\acute{\text{e}}\text{t}}^{+})_{\ast }:\mathsf{D}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})\rightarrow \mathsf{D}(V_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ . The functor $\mathsf{L}(\tilde{f}_{\bullet ,\acute{\text{e}}\text{t}}^{+})^{\ast }$ is easily verified to preserve small coproducts and complexes with quasi-coherent cohomology. Using the equivalences of (1.2), we may now define a functor $\mathsf{L}f_{\text{qc}}^{\ast }:\mathsf{D}_{\text{qc}}(Y)\rightarrow \mathsf{D}_{\text{qc}}(X)$ such that ${\mathcal{H}}^{0}(\mathsf{L}f_{\text{qc}}^{\ast }{\mathcal{M}}[0])\cong f^{\ast }{\mathcal{M}}$ whenever ${\mathcal{M}}\in \mathsf{QCoh}(Y)$ . It is readily verified that the functor $\mathsf{L}f_{\text{qc}}^{\ast }$ is unique up to canonical natural isomorphism, that is, it does not depend on the choice of charts $U$ and $V$ . If $f:X\rightarrow Y$ is flat, then for all integers $q$ and all ${\mathcal{M}}\in \mathsf{D}_{\text{qc}}(Y)$ there is a natural isomorphism

(1.9) $$\begin{eqnarray}f^{\ast }{\mathcal{H}}^{q}({\mathcal{M}})\cong {\mathcal{H}}^{q}(\mathsf{L}f_{\text{qc}}^{\ast }{\mathcal{M}}).\end{eqnarray}$$

Since the category $\mathsf{D}_{\text{qc}}(Y)$ is well generated [Reference Hall, Neeman and RydhHNR14, Theorem B.1] and the functor $\mathsf{L}f_{\text{qc}}^{\ast }$ preserves small coproducts, it admits a right adjoint [Reference NeemanNee01b, Theorem 8.4.4]

$$\begin{eqnarray}\mathsf{R}(f_{\text{qc}})_{\ast }:\mathsf{D}_{\text{qc}}(X)\rightarrow \mathsf{D}_{\text{qc}}(Y).\end{eqnarray}$$

The functor above is closely related to the functors we have already seen. Indeed, since $\mathsf{L}f_{\text{qc}}^{\ast }{\mathcal{O}}_{Y}[0]\simeq {\mathcal{O}}_{X}[0]$ , it follows that if ${\mathcal{M}}\in \mathsf{D}_{\text{qc}}(X)$ , then

(1.10) $$\begin{eqnarray}\mathsf{R}\unicode[STIX]{x1D6E4}(Y,\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{M}})\simeq \mathsf{R}\unicode[STIX]{x1D6E4}(X,{\mathcal{M}}).\end{eqnarray}$$

We now describe $\mathsf{R}(f_{\text{qc}})_{\ast }$ locally. Let

$$\begin{eqnarray}{\mathcal{Q}}_{V_{\bullet ,\acute{\text{e}}\text{t}}^{+}}:\mathsf{D}(V_{\bullet ,\acute{\text{e}}\text{t}}^{+})\rightarrow \mathsf{D}_{\text{qc}}(V_{\bullet ,\acute{\text{e}}\text{t}}^{+})\end{eqnarray}$$

be a right adjoint to the natural inclusion functor $\mathsf{D}_{\text{qc}}(V_{\bullet ,\acute{\text{e}}\text{t}}^{+})\rightarrow \mathsf{D}(V_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ , which exists by [Reference NeemanNee01b, Theorem 8.4.4]. A straightforward calculation, utilizing the equivalences (1.2), induces a natural isomorphism of functors:

(1.11) $$\begin{eqnarray}\mathsf{R}(f_{\text{qc}})_{\ast }\simeq \mathsf{R}(q_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})_{\ast }\mathsf{L}(\operatorname{res}_{V_{\bullet }})^{\ast }{\mathcal{Q}}_{V_{\bullet ,\acute{\text{e}}\text{t}}^{+}}\mathsf{R}(\tilde{f}_{\bullet ,\acute{\text{e}}\text{t}}^{+})_{\ast }\mathsf{R}(\operatorname{res}_{U_{\bullet }})_{\ast }\mathsf{L}(p_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})^{\ast }.\end{eqnarray}$$

The following lemma clarifies the situation somewhat.

Proof. Claim (i) is [Reference OlssonOls07, Lemma 6.20]. Claim (ii) follows from cohomological descent (1.2), claim (i) and equations (1.7), (1.8), and (1.11).

For (iii), by (ii), we may replace $\mathsf{R}(f_{\text{qc}})_{\ast }$ by $\mathsf{R}(f_{\text{lis}\text{-}\acute{\text{e}}\text{t}})_{\ast }$ . The hypercohomology spectral sequence

(1.12) $$\begin{eqnarray}\mathsf{R}^{r}(f_{\text{lis}\text{-}\acute{\text{e}}\text{t}})_{\ast }{\mathcal{H}}^{s}({\mathcal{M}})\Rightarrow \mathsf{R}^{r+s}(f_{\text{lis}\text{-}\acute{\text{e}}\text{t}})_{\ast }{\mathcal{M}}\end{eqnarray}$$

now applies and it is thus sufficient to prove that the higher pushforwards

$$\begin{eqnarray}\mathsf{R}^{r}(f_{\text{lis}\text{-}\acute{\text{e}}\text{t}})_{\ast }:\mathsf{QCoh}(X)\rightarrow \mathsf{QCoh}(Y)\end{eqnarray}$$

preserve direct limits for every integer $r\geqslant 0$ . This is local on $Y$ for the smooth topology, so we may assume that $Y$ is an affine scheme. Thus, it suffices to prove that the cohomology functors $H^{r}(X_{\text{lis}\text{-}\acute{\text{e}}\text{t}},-):\mathsf{QCoh}(X)\rightarrow \mathsf{Ab}$ preserve direct limits for every integer $r\geqslant 0$ . Since $X$ is quasi-compact and quasi-separated, this is well known (e.g., [Sta, 0739]).

The base-change transformation of (iv) exists by functoriality of the adjoints. Applying (ii), we may replace $(f_{\text{qc}})_{\ast }$ and $(f_{\text{qc}}^{\prime })_{\ast }$ by $(f_{\text{lis}\text{-}\acute{\text{e}}\text{t}})_{\ast }$ and $(f_{\text{lis}\text{-}\acute{\text{e}}\text{t}}^{\prime })_{\ast }$ , respectively. The statement is now local on $Y$ and $Y^{\prime }$ for the smooth topology, so we may assume that both $Y$ and $Y^{\prime }$ are affine schemes. Small modifications to the argument of [Sta, 073K] complete the proof.◻

In § 2 we describe a class of morphisms for which the conclusions of Lemma 1.2 remain valid in the unbounded derived category.

1.4 Comparison with definitions in derived algebraic geometry

We will now compare the derived category of quasi-coherent sheaves $\mathsf{D}_{\text{qc}}(X)$ that we are using in this paper with derived categories of quasi-coherent sheaves as defined in derived algebraic geometry. This is not used in the remainder of the paper (but see Example 9.4).

First recall that if ${\mathcal{A}}$ is an abelian category, then the derived category $\mathsf{D}({\mathcal{A}})$ is the homotopy category of a natural stable $\infty$ -category ${\mathcal{D}}({\mathcal{A}})$  [Reference LurieLur16a, Definition 1.3.5.8]. If in addition ${\mathcal{C}}\subseteq {\mathcal{A}}$ is a weak Serre subcategory, then we can consider the full $\infty$ -subcategory ${\mathcal{D}}_{{\mathcal{C}}}({\mathcal{A}})$ of objects with cohomology in ${\mathcal{C}}$ and this has homotopy category $\mathsf{D}_{{\mathcal{C}}}({\mathcal{A}})$ . We thus define ${\mathcal{D}}_{\text{qc}}(X):={\mathcal{D}}_{\mathsf{QCoh}(X)}(\mathsf{Mod}(X))$ , which has homotopy category $\mathsf{D}_{\text{qc}}(X)$ . If $X$ is Deligne–Mumford, we also define ${\mathcal{D}}_{\text{qc}}(X_{\acute{\text{e}}\text{t}}):={\mathcal{D}}_{\mathsf{QCoh}(X_{\acute{\text{e}}\text{t}})}(\mathsf{Mod}(X_{\acute{\text{e}}\text{t}}))$ , which is equivalent to ${\mathcal{D}}_{\text{qc}}(X)$  [Reference Laumon and Moret-BaillyLM00, Proposition 12.10.1].

Proposition 1.3 (cf. [Reference Drinfeld and GaitsgoryDG13, Remark 1.2.3]).

Let $X$ be an algebraic stack. Then

$$\begin{eqnarray}{\mathcal{D}}_{\text{qc}}(X)=\underset{\operatorname{Spec}A\rightarrow X}{\varprojlim }{\mathcal{D}}(\mathsf{Mod}(A)),\end{eqnarray}$$

where the limit is taken in the $\infty$ -category of $\infty$ -categories and is over (i) the category of all affine schemes over $X$ , (ii) the full subcategory of those smooth over $X$ , or (iii) the subcategory of those smooth over $X$ with only smooth $X$ -morphisms $\operatorname{Spec}A\rightarrow \operatorname{Spec}B$ .

We denote the limit of the right-hand side going over all morphisms by ${\mathcal{Q}}{\mathcal{C}}\mathsf{oh}(X)$ . This is a stable $\infty$ -category which is left- and right-complete, and has a t-structure with heart the abelian category $\mathsf{QCoh}(X)$  [Reference LurieLur16b, Corollary 9.1.3.2 and Remark 9.1.3.3]. The $\infty$ -category ${\mathcal{Q}}{\mathcal{C}}\mathsf{oh}(X)$ also makes sense for any contravariant functor $X$ from affine schemes to groupoids and can be adapted to variants in derived algebraic geometry. This is how derived categories of quasi-coherent sheaves usually are defined in derived algebraic geometry; cf. [Reference Ben-Zvi, Francis and NadlerBFN10, §3.1], [Reference Drinfeld and GaitsgoryDG13, §1.2], [Reference LurieLur16b, §6.2.2], and [Reference Gaitsgory and RozenblyumGR17, I.3, §1.1].

The category $\mathsf{D}_{\text{qc}}(X)$ is left-complete [Reference Hall, Neeman and RydhHNR14, Theorem B.1]. We do not use this fact in the proof and so we obtain an independent proof of the left-completeness of both $\mathsf{D}_{\text{qc}}(X)$ and ${\mathcal{D}}_{\text{qc}}(X)$ .

Proof of Proposition 1.3.

The limit restricted to smooth morphisms or to smooth morphisms with smooth maps between them is also equivalent to ${\mathcal{Q}}{\mathcal{C}}\mathsf{oh}(X)$  [Reference Gaitsgory and RozenblyumGR17, I.3, §1.4.2]. Moreover, if $U\rightarrow X$ is a smooth presentation and $U_{\bullet }^{+}$ is the corresponding semi-simplicial algebraic space, then restricting the limit to the diagram $U_{\bullet }^{+}$ gives the same limit (use that $X$ is a stack and that $\unicode[STIX]{x1D6E5}^{+}\subseteq \unicode[STIX]{x1D6E5}$ is right cofinal; cf. [Reference LurieLur16b, Proposition 6.2.3.1 and proof of Proposition 9.1.3.1] or [Reference Gaitsgory and RozenblyumGR17, I.3, Corollary 1.3.11]). We may also instead take the limit over the site $U_{\bullet ,\acute{\text{e}}\text{t}}^{+}$ since $U_{\bullet }^{+}:\unicode[STIX]{x1D6E5}^{+}\rightarrow U_{\bullet ,\acute{\text{e}}\text{t}}^{+}$ is right cofinal.

We have a sequence of maps between $\infty$ -categories

$$\begin{eqnarray}\displaystyle {\mathcal{D}}_{\text{qc}}(X)\xrightarrow[{}]{\;\unicode[STIX]{x1D6FC}\;}{\mathcal{D}}_{\text{qc}}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+}) & \xrightarrow[{}]{\;\unicode[STIX]{x1D6FD}\;} & \displaystyle \underset{V\in U_{\bullet ,\acute{\text{e}}\text{t}}^{+}}{\varprojlim }{\mathcal{D}}_{\text{qc}}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+}/V)\nonumber\\ \displaystyle & \xrightarrow[{}]{\;\unicode[STIX]{x1D6FE}\;} & \displaystyle \underset{V\in U_{\bullet ,\acute{\text{e}}\text{t}}^{+}}{\varprojlim }{\mathcal{D}}_{\text{qc}}(V)\xleftarrow[{}]{\;\unicode[STIX]{x1D6FF}\;}{\mathcal{Q}}{\mathcal{C}}\mathsf{oh}(X).\nonumber\end{eqnarray}$$

That $\unicode[STIX]{x1D6FC}$ is an equivalence follows by unbounded cohomological descent (1.2). That $\unicode[STIX]{x1D6FD}$ is an equivalence follows from Lemma 1.6 applied to the semi-simplicial étale site $U_{\bullet ,\acute{\text{e}}\text{t}}^{+}$ since having quasi-coherent cohomology can be verified locally. The map $\unicode[STIX]{x1D6FE}$ comes from the morphism of topoi $\unicode[STIX]{x1D716}:U_{\bullet ,\acute{\text{e}}\text{t}}^{+}/V\rightarrow V_{\acute{\text{e}}\text{t}}$ . Since $\unicode[STIX]{x1D716}_{\ast }$ (restriction) is exact and $\unicode[STIX]{x1D716}^{\ast }$ is exact and fully faithful with essential image the cartesian modules, we obtain equivalences between cartesian modules and between derived categories of modules with cartesian cohomology sheaves; cf. [Reference Laumon and Moret-BaillyLM00, Proposition 12.10.1]. This shows that $\unicode[STIX]{x1D6FE}$ is an equivalence. We saw that $\unicode[STIX]{x1D6FF}$ was an equivalence above.◻

Lemma 1.6. Let $({\mathcal{T}},{\mathcal{O}})$ be a ringed topos and let $({\mathcal{T}}/U,{\mathcal{O}}|_{U})$ denote the localized topos for any $U\in {\mathcal{T}}$ . Then the assignment $U\mapsto {\mathcal{D}}(\mathsf{Mod}({\mathcal{O}}|_{U}))$ is a sheaf, that is, there is a limit-preserving functor

$$\begin{eqnarray}\displaystyle {\mathcal{T}}^{\circ } & \longrightarrow & \displaystyle \mathsf{Cat}_{\infty }\nonumber\\ \displaystyle U & \longmapsto & \displaystyle {\mathcal{D}}(\mathsf{Mod}({\mathcal{O}}|_{U})).\nonumber\end{eqnarray}$$

4.1 Perfect complexes

We recall some notions from [Reference Berthelot, Grothendieck and IllusieSGA6, Exposé II] (also see [Sta, Tag 08FK]). If $A$ is a ring, then a complex of $A$ -modules $P$ is strictly perfect if it is a bounded complex of finitely generated and projective $A$ -modules. More generally, if $A$ is a sheaf of rings on a site $E$ , then a complex $P=(P^{k})$ of $A$ -modules is strictly perfect if it is a bounded complex and each term $P^{k}$ is a direct summand of a finite free $A$ -module. A complex $P\in \mathsf{D}(A)$ is perfect if it is locally strictly perfect.

Let $X$ be an algebraic stack. Then a complex ${\mathcal{P}}$ on $X$ is perfect if it is a perfect object of $\mathsf{D}(X)$ . Note that all perfect complexes on $X$ belong to $\mathsf{D}_{\text{qc}}(X)$ . The following lemma provides a useful criterion for perfection.

We recall the following well-known definition (e.g., [Reference BrandenburgBra14, Definition 4.7.1] or [Reference Ben-Zvi, Francis and NadlerBFN10, Definition 3.3]). Let $X$ be an algebraic stack. An object ${\mathcal{P}}\in \mathsf{D}_{\text{qc}}(X)$ is dualizable if there exists ${\mathcal{P}}^{\vee }\in \mathsf{D}_{\text{qc}}(X)$ together with morphisms $e:{\mathcal{P}}^{\vee }\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}{\mathcal{P}}\rightarrow {\mathcal{O}}_{X}$ , $c:{\mathcal{O}}_{X}\rightarrow {\mathcal{P}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}{\mathcal{P}}^{\vee }$ such that the two induced maps

are isomorphisms. It is standard that ${\mathcal{P}}$ dualizable implies that ${\mathcal{P}}^{\vee }\simeq \mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}^{\text{qc}}({\mathcal{P}},{\mathcal{O}}_{X})$ and the $e$ and $c$ maps are simply those arising from the adjunction between $-\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}{\mathcal{P}}$ and $\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}^{\text{qc}}({\mathcal{P}},-)$ .

The following lemma is straightforward but crucial.

4.2 Compact complexes

We now move on to a description of the compact objects of $\mathsf{D}_{\text{qc}}(X)$ and their relationship to perfect complexes.

Proof. We first prove (i). Consider a smooth morphism $p:\operatorname{Spec}A\rightarrow X$ . It follows from the quasi-separatedness of $X$ that $p$ is quasi-compact, quasi-separated, and representable. In particular, $p$ is concentrated (Lemma 2.5 (iii)) and so $\mathsf{L}p_{\text{qc}}^{\ast }{\mathcal{Q}}\in \mathsf{D}_{\text{qc}}(\operatorname{Spec}A)^{c}$ by Example 3.9. The claim now follows from Example 3.7 and Lemma 4.1.

To prove (ii), we note that if ${\mathcal{M}}\in \mathsf{D}(X)$ , then (1.3) implies that

$$\begin{eqnarray}\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}{\mathcal{P}},{\mathcal{M}})\simeq \operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{P}},{\mathcal{M}})).\end{eqnarray}$$

By Lemma 4.3 (ii), the restriction of $\mathsf{R}{\mathcal{H}}om({\mathcal{P}},-)$ to $\mathsf{D}_{\text{qc}}(X)$ preserves small coproducts. Since ${\mathcal{Q}}\in \mathsf{D}_{\text{qc}}(X)^{c}$ , it follows that the restriction of $\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{Q}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}{\mathcal{P}},-)$ to $\mathsf{D}_{\text{qc}}(X)$ preserves small coproducts. Hence, ${\mathcal{Q}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}{\mathcal{P}}\in \mathsf{D}_{\text{qc}}(X)^{c}$ .

For (iii), we note that if ${\mathcal{M}}\in \mathsf{D}(X)$ , then by definition $\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{O}}_{X},{\mathcal{M}})=\mathsf{R}\unicode[STIX]{x1D6E4}(X_{\text{lis}\text{-}\acute{\text{e}}\text{t}},{\mathcal{M}})$ . Since $X$ is concentrated, Theorem 2.6 (iii) implies that the restriction of $\mathsf{R}\unicode[STIX]{x1D6E4}(X_{\text{lis}\text{-}\acute{\text{e}}\text{t}},-)$ to $\mathsf{D}_{\text{qc}}(X)$ preserves small coproducts. Thus, ${\mathcal{O}}_{X}\in \mathsf{D}_{\text{qc}}(X)^{c}$ . The latter claim follows from the former and (ii).◻

In the following lemma we provide criteria for a perfect complex on an algebraic stack to be compact. We have not seen this characterization in the literature before.

Proof. Assume that (ii) does not hold. Then there are an infinite sequence of quasi-coherent ${\mathcal{O}}_{X}$ -modules ${\mathcal{M}}_{1},{\mathcal{M}}_{2},\ldots$ and a strictly increasing sequence of integers $d_{1}<d_{2}<\cdots \,$ such that $\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{P}},{\mathcal{M}}_{i}[d_{i}])\neq 0$ for every $i$ . Since $\mathsf{D}_{\text{qc}}(X)$ is left-complete [Reference Hall, Neeman and RydhHNR14, Theorem B.1], there is a quasi-isomorphism in $\mathsf{D}_{\text{qc}}(X)$ :

$$\begin{eqnarray}\bigoplus _{i=1}^{\infty }{\mathcal{M}}_{i}[d_{i}]\simeq \mathop{\prod }_{i=1}^{\infty }{\mathcal{M}}_{i}[d_{i}].\end{eqnarray}$$

This implies that the natural morphism

$$\begin{eqnarray}\bigoplus _{i=1}^{\infty }\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{P}},{\mathcal{M}}_{i}[d_{i}])\rightarrow \operatorname{Hom}_{{\mathcal{O}}_{X}}\biggl({\mathcal{P}},\bigoplus _{i=1}^{\infty }{\mathcal{M}}_{i}[d_{i}]\biggr)\simeq \mathop{\prod }_{i=1}^{\infty }\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{P}},{\mathcal{M}}_{i}[d_{i}])\end{eqnarray}$$

is not an isomorphism. In particular, ${\mathcal{P}}$ is not compact. Thus, by the contrapositive, we have proved the implication (i) $\Rightarrow$ (ii).

For (ii) $\Rightarrow$ (iii), first choose a diagram as in (1.6) with $V=Y=\operatorname{Spec}\mathbb{Z}$ . Let ${\mathcal{P}}_{\bullet ,\acute{\text{e}}\text{t}}^{+}=\mathsf{R}(\operatorname{res}_{U_{\bullet }})_{\ast }\mathsf{L}(p_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})^{\ast }{\mathcal{P}}$ and ${\mathcal{M}}_{\bullet ,\acute{\text{e}}\text{t}}^{+}=\mathsf{R}(\operatorname{res}_{U_{\bullet }})_{\ast }\mathsf{L}(p_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})^{\ast }{\mathcal{M}}$ . Note that

$$\begin{eqnarray}\mathsf{R}(\operatorname{res}_{U_{\bullet }})_{\ast }\mathsf{L}(p_{\bullet ,\text{lis}\text{-}\acute{\text{e}}\text{t}}^{+})^{\ast }\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{P}},{\mathcal{M}})\simeq \mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{U_{\bullet ,\acute{\text{e}}\text{t}}^{+}}}({\mathcal{P}}_{\bullet ,\acute{\text{e}}\text{t}}^{+},{\mathcal{M}}_{\bullet ,\acute{\text{e}}\text{t}}^{+})\end{eqnarray}$$

has quasi-coherent cohomology (Lemma 4.3) and that $\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{U_{\bullet ,\acute{\text{e}}\text{t}}^{+}}}({\mathcal{P}}_{\bullet ,\acute{\text{e}}\text{t}}^{+},{\mathcal{N}}[i])=0$ for all $i>r$ and ${\mathcal{N}}\in \mathsf{QCoh}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ . It is enough to prove that

$$\begin{eqnarray}\unicode[STIX]{x1D70F}^{{\geqslant}j}\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{U^{+}}}({\mathcal{P}}_{\bullet ,\acute{\text{e}}\text{t}}^{+},{\mathcal{M}})\rightarrow \unicode[STIX]{x1D70F}^{{\geqslant}j}\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{U^{+}}}({\mathcal{P}}_{\bullet ,\acute{\text{e}}\text{t}}^{+},\unicode[STIX]{x1D70F}^{{\geqslant}j-r}{\mathcal{M}})\end{eqnarray}$$

for every integer $j$ and ${\mathcal{M}}\in \mathsf{D}_{\text{qc}}(U_{\bullet ,\acute{\text{e}}\text{t}}^{+})$ . This follows as in the proof of [Reference Laszlo and OlssonLO08, Lemma 2.1.10] with $\unicode[STIX]{x1D716}_{\ast }$ replaced by $\operatorname{Hom}_{{\mathcal{O}}_{U_{\bullet ,\acute{\text{e}}\text{t}}^{+}}}({\mathcal{P}}_{\bullet ,\acute{\text{e}}\text{t}}^{+},-)$ .

Finally, for (iii) $\Rightarrow$ (i), we note that because ${\mathcal{P}}$ is perfect, it is dualizable. Thus, for all ${\mathcal{M}}\in \mathsf{D}_{\text{qc}}(X)$ and integers $j$ , there are natural quasi-isomorphisms:

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}^{{\geqslant}j}\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{P}},{\mathcal{M}}) & \simeq & \displaystyle \unicode[STIX]{x1D70F}^{{\geqslant}j}\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{P}},\unicode[STIX]{x1D70F}^{{\geqslant}j-r}{\mathcal{M}})\nonumber\\ \displaystyle & \simeq & \displaystyle \unicode[STIX]{x1D70F}^{{\geqslant}j}\mathsf{R}\unicode[STIX]{x1D6E4}(X_{\text{lis}\text{-}\acute{\text{e}}\text{t}},{\mathcal{P}}^{\vee }\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}\unicode[STIX]{x1D70F}^{{\geqslant}j-r}{\mathcal{M}}).\nonumber\end{eqnarray}$$

Also, since ${\mathcal{P}}$ is perfect, the restriction of the functor ${\mathcal{P}}^{\vee }\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}(-)$ to $\mathsf{D}_{\text{qc}}^{{\geqslant}j-r}(X)$ factors through $\mathsf{D}_{\text{qc}}^{{\geqslant}j-r^{\prime }}(X)$ for some fixed integer $r^{\prime }$ . The result now follows from Lemma 1.2 (iii).◻

For future reference, we summarize the situation in the following remark.

4.3 Supports and generation

Let $X$ be a scheme and let $M\in \mathsf{QCoh}(X)$ . The support of $M$ is the subset

$$\begin{eqnarray}\operatorname{supp}(M)=\{x\in |X|:M_{x}\neq 0\}.\end{eqnarray}$$

More generally, if $X$ is an algebraic stack and $M\in \mathsf{QCoh}(X)$ , then we define the support of $M$ as follows: let $p:U\rightarrow X$ be a smooth surjection from a scheme $U$ ; then $\operatorname{supp}(M)=p(\operatorname{supp}(p^{\ast }M))$ . It is easily verified that this is well defined (i.e., independent of the cover $p$ ).

Definition 4.7. Let $X$ be an algebraic stack and let $E\in \mathsf{D}_{\text{qc}}(X)$ . Define the cohomological support of $E$ to be the subset

$$\begin{eqnarray}\operatorname{supph}(E)=\mathop{\bigcup }_{n\in \mathbb{Z}}\operatorname{supp}({\mathcal{H}}^{n}(E))\subseteq |X|.\end{eqnarray}$$

Recall that if $X$ is a quasi-separated algebraic stack, then, for every $x\in |X|$ , there is a quasi-affine monomorphism $i_{x}:{\mathcal{G}}_{x}{\hookrightarrow}X$ with image $x$ , where ${\mathcal{G}}_{x}$ is a gerbe over a field $\unicode[STIX]{x1D705}(x)$ , the residue field at $x$ [Reference RydhRyd11, Theorem B.2]. We refer to this data as the residual gerbe at $x$ . We now have the following lemma.

If $X$ is an algebraic stack and $Z\subseteq |X|$ , then we define

$$\begin{eqnarray}\mathsf{D}_{\text{qc},|Z|}(X)=\{{\mathcal{M}}\in \mathsf{D}_{\text{qc}}(X):\operatorname{supph}({\mathcal{M}})\subseteq Z\}.\end{eqnarray}$$

If $j:U{\hookrightarrow}X$ is a flat monomorphism, e.g., an open immersion, then

$$\begin{eqnarray}\mathsf{D}_{\text{qc},|X\setminus U|}(X)=\{{\mathcal{M}}\in \mathsf{D}_{\text{qc}}(X):j^{\ast }{\mathcal{M}}\simeq 0\}.\end{eqnarray}$$

4.4 Projection formula

A typical application of Corollary 3.14 is given by the following proposition. The given argument is a variant of [Reference NeemanNee96, Proposition 5.3], though we have not seen this proposition in the literature before.

Proposition 4.11 (Strong projection formula).

Let $A$ be a ring and let $\unicode[STIX]{x1D70B}:X\rightarrow \operatorname{Spec}A$ be a morphism of algebraic stacks. Let ${\mathcal{Q}}$ be a compact object of $\mathsf{D}_{\text{qc}}(X)$ and let ${\mathcal{G}}\in \mathsf{D}_{\text{qc}}(X)$ . Then, for every $I\in \mathsf{D}(A)$ , there is a natural quasi-isomorphism

$$\begin{eqnarray}\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\otimes _{A}^{\mathsf{L}}I\simeq \mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }I).\end{eqnarray}$$

Proof. First we describe the morphism: by adjunction, there is a natural morphism

$$\begin{eqnarray}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }\mathsf{R}(\unicode[STIX]{x1D70B}_{\text{qc}})_{\ast }\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\rightarrow \mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\end{eqnarray}$$

and so by (1.3) there is a natural morphism

$$\begin{eqnarray}(\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }\mathsf{R}(\unicode[STIX]{x1D70B}_{\text{qc}})_{\ast }\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}}))\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}{\mathcal{Q}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }I\rightarrow {\mathcal{G}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }I.\end{eqnarray}$$

By (1.3) again, there is a natural morphism

$$\begin{eqnarray}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }[\mathsf{R}(\unicode[STIX]{x1D70B}_{\text{qc}})_{\ast }\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\otimes _{A}^{\mathsf{L}}I]\rightarrow \mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }I).\end{eqnarray}$$

By adjunction and (1.5) and (1.10), we deduce the existence of the required natural morphism

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{I}:\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\otimes _{A}^{\mathsf{L}}I\rightarrow \mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }I).\end{eqnarray}$$

Let ${\mathcal{K}}\subseteq \mathsf{D}(A)$ be the full subcategory with objects those $I$ such that $\unicode[STIX]{x1D719}_{I}$ is a quasi-isomorphism. It remains to show that ${\mathcal{K}}=\mathsf{D}(A)$ . Clearly, ${\mathcal{K}}$ is a triangulated subcategory that contains $A[k]$ for every integer $k$ . Moreover, since ${\mathcal{Q}}$ is a compact object of $\mathsf{D}_{\text{qc}}(X)$ , ${\mathcal{K}}$ is closed under small coproducts. The result now follows from Corollary 3.14.◻

A straightforward implication is the usual projection formula for concentrated morphisms of algebraic stacks.

Corollary 4.12 (Projection formula).

Let $f:X\rightarrow Y$ be a concentrated $1$ -morphism of algebraic stacks. The natural map

$$\begin{eqnarray}(\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{M}})\otimes _{{\mathcal{O}}_{Y}}^{\mathsf{ L}}{\mathcal{N}}\rightarrow \mathsf{R}(f_{\text{qc}})_{\ast }({\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}f_{\text{qc}}^{\ast }{\mathcal{N}})\end{eqnarray}$$

is a quasi-isomorphism for every ${\mathcal{M}}\in \mathsf{D}_{\text{qc}}(X)$ and ${\mathcal{N}}\in \mathsf{D}_{\text{qc}}(Y)$ .

Proof. By adjunction, there is a natural morphism

$$\begin{eqnarray}\mathsf{L}f_{\text{qc}}^{\ast }\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}f_{\text{qc}}^{\ast }{\mathcal{N}}\rightarrow {\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}f_{\text{qc}}^{\ast }{\mathcal{N}}.\end{eqnarray}$$

By adjunction again, we deduce the existence of a natural morphism

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{{\mathcal{N}}}:(\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{M}})\otimes _{{\mathcal{O}}_{Y}}^{\mathsf{ L}}{\mathcal{N}}\rightarrow \mathsf{R}(f_{\text{qc}})_{\ast }({\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}f_{\text{qc}}^{\ast }{\mathcal{N}}).\end{eqnarray}$$

It remains to show that $\unicode[STIX]{x1D713}_{{\mathcal{N}}}$ is a quasi-isomorphism for every ${\mathcal{N}}\in \mathsf{D}_{\text{qc}}(Y)$ . Note that the verification of this is smooth local on $Y$ , so, by Theorem 2.6 (iv), we may reduce to the situation where $Y$ is an affine scheme. By Lemma 4.4 (iii), ${\mathcal{O}}_{X}$ is compact and the result now follows from Proposition 4.11.◻

4.5 Tor-independent base change

Let $f:X\rightarrow Y$ and $g:Y^{\prime }\rightarrow Y$ be morphisms of algebraic stacks. We say that $f$ and $g$ are tor-independent if for every smooth morphism $\operatorname{Spec}A\rightarrow Y$ and every pair of smooth morphisms $\operatorname{Spec}B\rightarrow X\times _{Y}\operatorname{Spec}A$ and $\operatorname{Spec}A^{\prime }\rightarrow Y^{\prime }\times _{Y}\operatorname{Spec}A$ , we have that $\operatorname{Tor}_{i}^{A}(B,A^{\prime })=0$ for all $i>0$ . Equivalently, ${\mathcal{T}}or_{i}^{Y,f,g}({\mathcal{O}}_{X},{\mathcal{O}}_{Y^{\prime }})=0$ for every integer $i>0$ (see [Reference HallHal17, Appendix C] for details). Note that if $g$ is flat, then it is tor-independent of every $f$ . The projection formula of Corollary 4.12 is powerful enough to prove a very general tor-independent base-change result which extends Theorem 2.6 (iv).

Corollary 4.13. Fix the following $2$ -cartesian square of algebraic stacks.

If $f$ and $g$ are tor-independent and $f$ is concentrated, then there is a natural quasi-isomorphism for every ${\mathcal{M}}\in \mathsf{D}_{\text{qc}}(X)$ :

$$\begin{eqnarray}\mathsf{L}g_{\text{qc}}^{\ast }\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{M}}\simeq \mathsf{R}(f_{\text{qc}}^{\prime })_{\ast }\mathsf{L}{g^{\prime }}_{\text{qc}}^{\ast }{\mathcal{M}}.\end{eqnarray}$$

Proof. By Theorem 2.6 (iv), the result can be verified smooth-locally on $Y$ and $Y^{\prime }$ . Thus, we may assume that $Y=\operatorname{Spec}A$ and $Y^{\prime }=\operatorname{Spec}A^{\prime }$ . In particular, $g$ and $g^{\prime }$ are affine. Since $g$ is affine, $\mathsf{R}(g_{\text{qc}})_{\ast }$ is conservative (Corollary 2.8). Hence, it is sufficient to verify that the morphism in question is a quasi-isomorphism after application of the functor $\mathsf{R}(g_{\text{qc}})_{\ast }$ . By the projection formula applied to $g$ and then $f$ , there are natural quasi-isomorphisms

$$\begin{eqnarray}\displaystyle \mathsf{R}(g_{\text{qc}})_{\ast }\mathsf{L}g_{\text{qc}}^{\ast }\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{M}} & \simeq & \displaystyle \mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{M}}\otimes _{{\mathcal{O}}_{Y}}\mathsf{R}(g_{\text{qc}})_{\ast }{\mathcal{O}}_{Y^{\prime }}\nonumber\\ \displaystyle & \simeq & \displaystyle \mathsf{R}(f_{\text{qc}})_{\ast }({\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}\mathsf{L}f_{\text{qc}}^{\ast }\mathsf{R}(g_{\text{qc}})_{\ast }{\mathcal{O}}_{Y^{\prime }}).\nonumber\end{eqnarray}$$

Note, however, that because $f$ and $g$ are tor-independent and $g$ is affine, the natural map

$$\begin{eqnarray}\mathsf{L}f_{\text{qc}}^{\ast }\mathsf{R}(g_{\text{qc}})_{\ast }{\mathcal{O}}_{Y^{\prime }}\rightarrow \mathsf{R}(g_{\text{qc}}^{\prime })_{\ast }{\mathcal{O}}_{X^{\prime }}\end{eqnarray}$$

is a quasi-isomorphism. Indeed, this may be verified smooth-locally on $X$ , so we may assume that $X=\operatorname{Spec}C$ . The morphism in question corresponds to the map $C\otimes _{A}^{\mathsf{L}}A^{\prime }\rightarrow (C\otimes _{A}A^{\prime })[0]$ in $\mathsf{D}(C)$ , which is a quasi-isomorphism because $f$ and $g$ are tor-independent. With the projection formula and functoriality, we now obtain the following natural sequence of quasi-isomorphisms:

$$\begin{eqnarray}\displaystyle \mathsf{R}(f_{\text{qc}})_{\ast }({\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}\mathsf{L}f_{\text{qc}}^{\ast }\mathsf{R}(g_{\text{qc}})_{\ast }{\mathcal{O}}_{Y^{\prime }}) & \simeq & \displaystyle \mathsf{R}(f_{\text{qc}})_{\ast }({\mathcal{M}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}\mathsf{R}(g_{\text{qc}}^{\prime })_{\ast }{\mathcal{O}}_{X^{\prime }})\nonumber\\ \displaystyle & \simeq & \displaystyle \mathsf{R}(f_{\text{qc}})_{\ast }\mathsf{R}(g_{\text{qc}}^{\prime })_{\ast }\mathsf{L}{g^{\prime }}_{\text{qc}}^{\ast }{\mathcal{M}}\nonumber\\ \displaystyle & \simeq & \displaystyle \mathsf{R}(g_{\text{qc}})_{\ast }\mathsf{R}(f_{\text{qc}}^{\prime })_{\ast }\mathsf{L}{g^{\prime }}_{\text{qc}}^{\ast }{\mathcal{M}}.\nonumber\end{eqnarray}$$

The result follows. ◻

Note that in the setting of derived algebraic geometry, tor-independence is not necessary to obtain a base-change result [Reference Ben-Zvi, Francis and NadlerBFN10, Proposition 3.10].

4.6 Finite duality

Using the projection formula, we can also establish finite duality.

Proof. By Theorem 2.6 (iii), the functor $\mathsf{R}(f_{\text{qc}})_{\ast }$ preserves small coproducts. Since $\mathsf{D}_{\text{qc}}(X)$ is well generated [Reference Hall, Neeman and RydhHNR14, Theorem B.1], the existence of $f^{\times }$ follows from [Reference NeemanNee01b, Proposition 1.20]. Now fix ${\mathcal{N}}\in \mathsf{D}_{\text{qc}}(Y)$ ; then there are natural isomorphisms

$$\begin{eqnarray}\displaystyle \operatorname{Hom}_{{\mathcal{O}}_{Y}}({\mathcal{N}},\mathsf{R}(f_{\text{qc}})_{\ast }f^{\times }({\mathcal{M}})) & \cong & \displaystyle \operatorname{Hom}_{{\mathcal{O}}_{X}}(\mathsf{L}f_{\text{qc}}^{\ast }{\mathcal{N}},f^{\times }({\mathcal{M}}))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{{\mathcal{O}}_{Y}}(\mathsf{R}(f_{\text{qc}})_{\ast }\mathsf{L}f_{\text{qc}}^{\ast }{\mathcal{N}},{\mathcal{M}})\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{{\mathcal{O}}_{Y}}((\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{O}}_{X})\otimes _{{\mathcal{O}}_{Y}}^{\mathsf{L}}{\mathcal{N}},{\mathcal{M}})\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{{\mathcal{O}}_{Y}}({\mathcal{N}},\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}^{\text{qc}}(\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{O}}_{X},{\mathcal{M}})).\nonumber\end{eqnarray}$$

The penultimate isomorphism follows from the projection formula (Corollary 4.12). By the Yoneda lemma, this proves (ii).

We now address (iii). Let $\tilde{f}^{\times }:\mathsf{D}_{\text{qc}}(Y)\rightarrow \mathsf{D}_{\text{qc}}(X)$ be the functor

$$\begin{eqnarray}\tilde{f}^{\times }({\mathcal{M}})=\overline{f}^{\ast }\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}^{\text{qc}}(f_{\ast }{\mathcal{O}}_{X},{\mathcal{M}}),\end{eqnarray}$$

where $\overline{f}^{\ast }$ comes from the equivalence of Corollary 2.7. We claim that there is a natural transformation of functors $\tilde{f}^{\times }\Rightarrow f^{\times }$ . To see this, let ${\mathcal{N}}\in \mathsf{D}_{\text{qc}}(X)$ and ${\mathcal{M}}\in \mathsf{D}_{\text{qc}}(Y)$ ; then there are natural morphisms

$$\begin{eqnarray}\displaystyle \operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{N}},\tilde{f}^{\times }({\mathcal{M}})) & \rightarrow & \displaystyle \operatorname{Hom}_{{\mathcal{O}}_{Y}}(\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{N}},\mathsf{R}(f_{\text{qc}})_{\ast }\tilde{f}^{\times }({\mathcal{M}}))\nonumber\\ \displaystyle & = & \displaystyle \operatorname{Hom}_{{\mathcal{O}}_{Y}}(\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{N}},\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}^{\text{qc}}(f_{\ast }{\mathcal{O}}_{X},{\mathcal{M}}))\nonumber\\ \displaystyle & \rightarrow & \displaystyle \operatorname{Hom}_{{\mathcal{O}}_{Y}}(\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{N}},\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}^{\text{qc}}({\mathcal{O}}_{Y},{\mathcal{M}}))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{{\mathcal{O}}_{Y}}(\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{N}},{\mathcal{M}})\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{N}},f^{\times }({\mathcal{M}})).\nonumber\end{eqnarray}$$

By the Yoneda lemma, we have the claim. Since $f$ is affine, to prove that the natural transformation $\tilde{f}^{\times }\Rightarrow f^{\times }$ is an isomorphism, it is sufficient to prove that it is after application of $\mathsf{R}(f_{\text{qc}})_{\ast }$ (Lemma 2.8). This follows from the definition of $\tilde{f}^{\times }$ and (ii).

We now treat (iv). By adjunction and the projection formula (Corollary 4.12) we obtain a natural morphism for every ${\mathcal{M}}\in \mathsf{D}_{\text{qc}}(Y)$ :

$$\begin{eqnarray}f^{\times }({\mathcal{O}}_{Y})\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}f_{\text{qc}}^{\ast }({\mathcal{M}})\rightarrow f^{\times }({\mathcal{M}}).\end{eqnarray}$$

Since $f$ is affine, it is sufficient to prove that this morphism is an isomorphism after application of $\mathsf{R}(f_{\text{qc}})_{\ast }$ (Lemma 2.8). By (ii) and the projection formula (Corollary 4.12), we see that it is sufficient to prove that the induced morphism

$$\begin{eqnarray}\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}^{\text{qc}}(f_{\ast }{\mathcal{O}}_{X},{\mathcal{O}}_{Y})\otimes _{{\mathcal{O}}_{Y}}^{\mathsf{ L}}{\mathcal{M}}\rightarrow \mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}^{\text{qc}}(f_{\ast }{\mathcal{O}}_{X},{\mathcal{M}})\end{eqnarray}$$

is a quasi-isomorphism for every ${\mathcal{M}}\in \mathsf{D}_{\text{qc}}(Y)$ . But $f_{\ast }{\mathcal{O}}_{X}$ is perfect, so Lemma 4.3 (ii) now gives the claim.

For the compatibility of $f^{\times }$ with base change, we consider the following tor-independent $2$ -cartesian diagram of algebraic stacks.

Adjointness and tor-independent base change (Corollary 4.13) provide a natural transformation $\mathsf{L}{g^{\prime }}_{\text{qc}}^{\ast }f^{\times }\rightarrow (f^{\prime })^{\times }\mathsf{L}g_{\text{qc}}^{\ast }$ of functors that we must show is an isomorphism. Tor-independent base change also implies that there is a quasi-isomorphism: $\mathsf{L}g_{\text{qc}}^{\ast }\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{O}}_{X}\simeq \mathsf{R}(f_{\text{qc}}^{\prime })_{\ast }\mathsf{L}{g^{\prime }}_{\text{qc}}^{\ast }{\mathcal{O}}_{X}$ . Since $\mathsf{R}(f_{\text{qc}})_{\ast }{\mathcal{O}}_{X}$ is perfect, it follows that $\mathsf{R}(f_{\text{qc}}^{\prime })_{\ast }{\mathcal{O}}_{X^{\prime }}$ is perfect. By the formula just determined for $f^{\times }$ , we thus see that it is sufficient to prove that $\mathsf{L}{g^{\prime }}_{\text{qc}}^{\ast }f^{\times }({\mathcal{O}}_{Y})\rightarrow (f^{\prime })^{\times }\mathsf{L}g_{\text{qc}}^{\ast }({\mathcal{O}}_{Y})$ is a quasi-isomorphism. Since $f$ (and so $f^{\prime }$ ) is affine, it is sufficient to verify this after application of $\mathsf{R}(f_{\text{qc}}^{\prime })_{\ast }$ . This observation, together with (ii) and tor-independent base change, shows that it is sufficient to prove that the morphism

$$\begin{eqnarray}\mathsf{L}g_{\text{qc}}^{\ast }\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}^{\text{qc}}(f_{\ast }{\mathcal{O}}_{X},{\mathcal{O}}_{Y})\rightarrow \mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y^{\prime }}}(\mathsf{L}g_{\text{qc}}^{\ast }(f_{\ast }{\mathcal{O}}_{X}),{\mathcal{O}}_{Y^{\prime }})\end{eqnarray}$$

is a quasi-isomorphism. But $f_{\ast }{\mathcal{O}}_{X}$ and $\mathsf{L}g_{\text{qc}}^{\ast }(f_{\ast }{\mathcal{O}}_{X})$ are both perfect and so dualizable (Lemma 4.3). In particular, the derived pullback of the dual of $f_{\ast }{\mathcal{O}}_{X}$ coincides with the dual of $\mathsf{L}g_{\text{qc}}^{\ast }(f_{\ast }{\mathcal{O}}_{X})$ . It follows that the asserted map is a quasi-isomorphism and the claim follows.

Finally, we address the conservativity. For this, it is sufficient to observe that if $f$ is surjective, then $\operatorname{supph}(f_{\ast }{\mathcal{O}}_{X})=|Y|$ . But $\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}(f_{\ast }{\mathcal{O}}_{X},{\mathcal{M}})\simeq 0$ if and only if ${\mathcal{M}}\simeq 0$ (Lemma 4.9).◻

Corollary 4.15. If $f:X\rightarrow Y$ is a finite and faithfully flat morphism of finite presentation between algebraic stacks, then the functor

$$\begin{eqnarray}f^{\times }({\mathcal{M}})=\overline{f}^{\ast }\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}(f_{\ast }{\mathcal{O}}_{X},{\mathcal{M}}),\quad \text{where }{\mathcal{M}}\in \mathsf{D}_{\text{qc}}(Y),\end{eqnarray}$$

is right adjoint to $\mathsf{R}(f_{\text{qc}})_{\ast }:\mathsf{D}_{\text{qc}}(X)\rightarrow \mathsf{D}_{\text{qc}}(Y)$ . Moreover, $f^{\times }$ is compatible with arbitrary base change on $Y$ , $f^{\times }({\mathcal{O}}_{Y})\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}f^{\ast }(-)\simeq f^{\times }(-)$ , preserves small coproducts, and is conservative.

4.7 Coherent functors

Combining the strong projection formula of Proposition 4.11 with the characterization of compact objects in Lemma 4.5, we can prove most of Theorem D.

Corollary 4.16. Let $A$ be a noetherian ring and let $\unicode[STIX]{x1D70B}:X\rightarrow \operatorname{Spec}A$ be a morphism of finite type between noetherian algebraic stacks. Suppose that:

1. (i) for every $i\geqslant 0$ and ${\mathcal{M}}\in \mathsf{Coh}(X)$ , the cohomology $H^{i}(X_{\text{lis}\text{-}\acute{\text{e}}\text{t}},{\mathcal{M}})$ is a finitely generated $A$ -module (e.g., $\unicode[STIX]{x1D70B}$ proper); and

2. (ii) $\mathsf{D}_{\text{qc}}(X)$ is compactly generated.

Then, for every ${\mathcal{F}}\in \mathsf{D}_{\text{qc}}(X)$ and ${\mathcal{G}}\in \mathsf{D}_{\mathsf{Coh}}^{b}(X)$ , the functor

$$\begin{eqnarray}\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{F}},{\mathcal{G}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }(-)):\mathsf{Mod}(A)\rightarrow \mathsf{Mod}(A)\end{eqnarray}$$

is coherent.

Proof. We begin by observing that the coherent functors $\mathsf{Mod}(A)\rightarrow \mathsf{Mod}(A)$ constitute a full abelian subcategory of the category of $A$ -linear functors, which is closed under products (where everything is computed ‘pointwise’) [Reference HallHal14, Example 4.9]. Let ${\mathcal{T}}\subseteq \mathsf{D}_{\text{qc}}(X)$ denote the full subcategory with objects those ${\mathcal{F}}\in \mathsf{D}_{\text{qc}}(X)$ where the functor $\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{F}},{\mathcal{G}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }(-))$ is coherent for every ${\mathcal{G}}\in \mathsf{D}_{\mathsf{Coh}}^{b}(X)$ . In particular, ${\mathcal{T}}$ is closed under small coproducts, shifts, and triangles. By Corollary 3.14, it is enough to prove that ${\mathcal{T}}$ contains the compact objects of $\mathsf{D}_{\text{qc}}(X)$ . If ${\mathcal{Q}}\in \mathsf{D}_{\text{qc}}(X)$ is compact, then the strong projection formula (Proposition 4.11) implies that there is a natural quasi-isomorphism

$$\begin{eqnarray}\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\otimes _{A}^{\mathsf{L}}I\simeq \mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{ L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }I).\end{eqnarray}$$

Since ${\mathcal{Q}}$ is compact, it is perfect (Lemma 4.4 (i)) and so $\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\in \mathsf{D}_{\mathsf{Coh}}^{b}(X)$ . The assumption on preservation of coherence implies that $\mathsf{R}(\unicode[STIX]{x1D70B}_{\text{qc}})_{\ast }$ sends $\mathsf{D}_{\mathsf{Coh}}^{+}(X)$ to $\mathsf{D}_{\mathsf{Coh}}^{+}(A)$ . In particular, $\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\simeq \mathsf{R}(\unicode[STIX]{x1D70B}_{\text{qc}})_{\ast }\mathsf{R}{\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\in \mathsf{D}_{\mathsf{Coh}}^{+}(A)$ . By Lemma 4.5, we also have $\mathsf{R}\!\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}})\in \mathsf{D}^{b}(A)$ . Thus, the functor $\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{Q}},{\mathcal{G}}\otimes _{{\mathcal{O}}_{X}}^{\mathsf{L}}\mathsf{L}\unicode[STIX]{x1D70B}_{\text{qc}}^{\ast }(-))$ is coherent [Reference HallHal14, Example 4.13] and we deduce the result.◻