Subsection 3.1. (0NZH) · ScalingStacks

3.1. Definition of perfect stacks

Our main objects of study are $\infty$ -categories of quasi-coherent sheaves on derived stacks. For the foundations of derived stacks and quasi-coherent sheaves on them, we refer the reader to [ToVe1, ToVe2, To2]. (Note that a theory of descent for sheaves on higher stacks was developed previously by Hirschowitz and Simpson [HS].)

Given a derived stack $X$ , we have the stable symmetric monoidal $\infty$ -category $\qc(X)$ of quasi-coherent sheaves on $X$ . To recall its construction, consider first a derived commutative $k$ -algebra $A$ , and the representable affine derived scheme $X=\Spec A$ . In this case, one defines $\qc(X)$ to be the $\infty$ -category of $A$ -modules $\Mod_{A}$ (i.e., module objects over $A$ in $k$ -modules). Its homotopy category is the unbounded derived category of $A$ .

In general, any derived stack $X$ can be written as a colimit of a diagram of affine derived schemes $X\simeq\colim_{U\in\it{Aff}_{/X}}U$ . Then one defines $\qc(X)$ to be the limit (in the $\infty$ -category of $\infty$ -categories) of the corresponding diagram of $\infty$ -categories

\qc(X):=\lim_{U\in\it{Aff}{/X}}\qc(U).

One can think of an object $F\in\qc(X)$ as collections of quasi-coherent sheaves $F|_{U}$ on the terms $U$ together with compatible identifications between their pullbacks under the diagram maps.

When $X$ is quasi-compact and has affine diagonal, by choosing an affine cover $U\to X$ with induced C̆ech simplicial affine derived scheme $U_{*}\to X$ , we can realize $\qc(X)$ by a smaller limit, the totalization of the cosimplicial diagram $\qc(U_{*})$ .

An important feature of the $\infty$ -category $\qc(X)$ is that it is cocomplete, that is, closed under all small colimits (or equivalently, since $\qc(X)$ is stable, all small coproducts). Nevertheless, it can be difficult to control $\qc(X)$ algebraically via reasonable generators. In general, it is convenient (and sometimes indispensable) to work with $\infty$ -categories that are “generated by finite objects” in a suitable sense. Let us summarize well known approaches to this idea. In a moment, we will provide a more detailed discussion.

There are two common notions of when a small $\infty$ -subcategory $\mathcal{C}^{\circ}$ generates an $\infty$ -category $\mathcal{C}$ . On the one hand, we could ask that $\mathcal{C}$ be the inductive limit $\operatorname{Ind}\mathcal{C}^{\circ}$ . On the other hand, we could ask that in $\mathcal{C}$ the right orthogonal of $\mathcal{C}^{\circ}$ vanishes.

There are three common notions of when an object should be considered finite: perfect objects, dualizable objects, and compact objects, which refer respectively to the geometry, monoidal structure, and categorical structure of $\qc(X)$ .

We now introduce the class of perfect stacks. We will check below that for perfect stacks, the above notions of generators and finite objects all coincide.

0NWB

Definition 3.1. Let $A$ be a derived commutative ring. An $A$ -module $M$ is perfect if lies in the smallest $\infty$ -subcategory of $\Mod_{A}$ containing $A$ and closed under finite colimits and retracts. For a derived stack $X$ , $\infty$ -category $\operatorname{Perf}(X)$ is the full $\infty$ -subcategory of $\qc(X)$ consisting of those sheaves $M$ whose restriction $f^{*}M$ to any affine $f:U\rightarrow X$ over $X$ is a perfect module.

0NWC

Definition 3.2. A derived stack $X$ is said to be perfect if it has affine diagonal and the $\infty$ -category $\qc(X)$ is the inductive limit

\qc(X)\simeq\operatorname{Ind}\operatorname{Perf}(X)

of the full $\infty$ -subcategory $\operatorname{Perf}(X)$ of perfect complexes.

A morphism $X\to Y$ is said to be perfect if its fibers $X\times_{Y}U$ over affines $U\to Y$ are perfect.

See [L2, 5.3.5] for the construction of Ind-categories of $\infty$ -categories, and [L3, 8] where it is shown that Ind-categories of stable $\infty$ -categories are stable. Let us mention that in the Ind-category $\operatorname{Ind}\mathcal{C}$ of an $\infty$ -category $\mathcal{C}$ , morphisms between Ind-objects can be calculated via the expected formula

\Hom_{\operatorname{Ind}\mathcal{C}}(\varinjlim[X_{i}],(\varinjlim[Y_{j}])\simeq\lim\colim\Hom_{\mathcal{C}}(X_{i},Y_{j}).

For the reader unaccustomed to Ind-categories, we will momentarily give an alternative formulation of perfect stack in the more familiar language of compactly generated categories.

We next proceed with a review of the various notions of generators and finite objects. In Section 3.2, we show that perfect morphisms satisfy base change and the projection formula. In Section 3.3, we show that the class of perfect stacks includes many common examples of interest, and check that any morphism between perfect stacks is itself perfect.

3.1.1. Finite objects

We review here the three common notions of finite objects and their interrelations (See [L3, 17] and [L4, 4.7] for more details, as well as [BV, HPS, Ke, LMS] among many other sources). We remind the reader that we are working in the context of $\infty$ -categories, so constructions such as colimits correspond to homotopy colimits in the context of model categories.

0NWD

(1)

Definition 3.3. An object $M$ of a stable $\infty$ -category $\mathcal{C}$ is said to be compact if $\Hom_{\mathcal{C}}(M,-)$ commutes with all coproducts (equivalently, with all colimits).

(2)

An object $M$ of a stable symmetric monoidal $\infty$ -category $\mathcal{C}$ is said to be (strongly) dualizable if there is an object $M^{\vee}$ and unit and trace maps

such that the composite map

is the identity.

Suppose $\mathcal{C}$ is a stable presentable $\infty$ -category (such as $\qc(X)$ ). Then an object $M\in\mathcal{C}$ is compact if and only if maps from $M$ to any small coproduct factor through a finite coproduct. Furthermore, a functor $F:\mathcal{C}\to\mathcal{D}$ between stable presentable $\infty$ -categories that preserves finite colimits preserves small colimits if and only if it preserves small coproducts. (See [L3, Proposition 17.1].)

In a closed symmetric monoidal $\infty$ -category $\mathcal{C}$ (such as $\qc(X)$ ), an object $M\in\mathcal{C}$ is dualizable if and only if there exists a coevaluation map

1\to M\otimes{\mathcal{H}om}(M,1)

satisfying the appropriate conditions (since one already has an evaluation map). If an object $M\in\mathcal{C}$ is dualizable, then we can turn internal Hom from $M$ into tensor product with $M^{\vee}$ in the sense that there is a canonical equivalence

{\mathcal{H}om}(M,-)\simeq M^{\vee}\otimes(-).

In particular, this implies that ${\mathcal{H}om}(M,-)$ preserves colimits and $M\otimes-$ preserves limits:

M\otimes\lim N_{\alpha}\simeq\Hom_{\mathcal{C}}(1_{\mathcal{C}},M\otimes\lim N_{\alpha})\simeq\Hom_{\mathcal{C}}(M^{\vee},\lim N_{\alpha})\simeq\lim\Hom_{\mathcal{C}}(M^{\vee},N_{\alpha})\simeq\lim M\otimes N_{\alpha}.

It is enlightening to note the following characterization of dualizable objects, which parallels the definition of compact objects (but will not be used in this paper).

0NWE

Lemma 3.4. Let $\mathcal{C}$ be a symmetric monoidal presentable stable $\infty$ -category, whose monoidal structure distributes over colimits. An object $M$ of $\mathcal{C}$ is then dualizable if and only if tensoring with $M$ preserves all limits.

0NWF

Proof. The necessity of $M\otimes-$ preserving limits is noted above ( $\mathcal{C}$ is closed by virtue of being presentable with monoidal structure distributing over colimits, see [L4, Proposition 2.1.12]). To demonstrate sufficiency, assume that $M\otimes-$ preserves limits, and then consider the endofunctor of $\mathcal{C}$ defined by tensoring with $M$ . By assumption on $M$ and $\mathcal{C}$ , this functor preserves all limits and colimits. We may now apply the adjoint functor theorem of [L2] to deduce the existence of a left adjoint $F$ to $M\otimes-$ . Denote by $M^{\vee}$ the value $F(1_{\mathcal{C}})$ of $F$ applied to the unit of $\mathcal{C}$ . The existence of unit and trace maps $1_{\mathcal{C}}\rightarrow M\otimes M^{\vee}\rightarrow 1_{\mathcal{C}}$ is now a particular instance of the unit and counit maps for this adjunction, which implies that $M$ and $M^{\vee}$ are in duality. Hence $M$ is dualizable.

∎

In a general stable presentable symmetric monoidal $\infty$ -category $\mathcal{C}$ , the classes of compact and dualizable objects do not coincide. In particular, the monoidal unit $1\in\mathcal{C}$ is always dualizable but not necessarily compact.

In the case of a derived stack $X$ , the unit $\mathcal{O}_{X}\in\qc(X)$ is compact if and only if the global sections functor $\Gamma(X,-)$ preserves colimits. (This fails if the global sections $\Gamma(X,\mathcal{O}_{X})$ are too big such as in the following examples: ind-schemes such as the formal disk $\operatorname{Spf}k[[t]]$ ; the classifying space of a topological group such as $BS^{1}$ ; the classifying space of finite groups in modular characteristics.) However, if the unit $\mathcal{O}_{X}\in\qc(X)$ is itself compact then all dualizable objects are compact, since Hom from a dualizable object $M$ is the composition of the colimit preserving functors internal Hom ${\mathcal{H}om}(M,-)$ and global sections $\Gamma(X,-)$ .

0NWG

Lemma 3.5 ([BoN], 6.4, [EKMM] III.7.9, [L4] 4.7.2). For the $\infty$ -category $\Mod_{k}=\qc(\Spec k)$ of modules over a commutative derived ring (that is, quasi-coherent sheaves on an affine derived scheme), all three notions of finiteness coincide: $M$ compact $\iff$ $M$ dualizable $\iff$ $M$ perfect.

0NWH

Proof. First, note that the free module $k$ , which is the monoidal unit, is clearly compact. Hence all dualizable objects are compact. Moreover, we can write any object as a colimit of free modules. For $M$ compact, the identity map ${\rm id}_{M}\in\Hom(M,M)$ has to factor through a finite colimit, showing that $M$ is perfect. Finally, perfect modules are dualizable since we can explicitly exhibit their dual as a finite limit of free modules. ∎

It is useful to note that the notion of dualizable is local. On the one hand, pullback for any map of stacks (for example, restriction to an affine) preserves dualizable objects. On the other hand, a dual object with its unit and trace maps is functorially characterized, thus if it exists locally, it will glue together to a global object. This observation leads to the identification of perfect and dualizable objects in $\qc(X)$ for any $X$ :

0NWI

Proposition 3.6. For a derived stack $X$ , an object of $\qc(X)$ is dualizable if and only if it is perfect.

0NWJ

Proof. Let $M\in\qc(X)$ be dualizable with dual $M^{\vee}$ . Then for any map $\eta:\Spec A\rightarrow X$ , the pullback $\eta^{*}M$ is dualizable with dual $\eta^{*}M^{\vee}$ . Dualizable objects of $\Mod_{A}$ are perfect, hence $\eta^{*}M$ is perfect and so by definition, $M$ is perfect.

Now suppose $M\in\qc(X)$ is perfect. Recall that by definition, we have

\qc(X)\simeq\lim_{\Spec A\in{\it Aff}/X}\Mod_{A}.

Since $M$ is perfect, for any map $\eta:\Spec A\to X$ , the pullback $\eta^{*}M$ is perfect, hence dualizable. We take the value of the dual $M^{\vee}$ along a map $\eta:\Spec A\to X$ to be the dual of the pullback $(\eta^{*}M)^{\vee}$ . Note that $M^{\vee}$ is well-defined, since for any composite $\eta\circ\nu:\Spec B\rightarrow X$ , there is a natural equivalence $(\nu^{*}\eta^{*}M)^{\vee}\simeq\nu^{*}((\eta^{*}M)^{\vee})$ .

To exhibit $M$ and $M^{\vee}$ as dual to one another, we must construct the requisite unit and counit maps $u:\mathcal{O}_{X}\rightarrow M\otimes M^{\vee}$ and $c:M^{\vee}\otimes M\rightarrow\mathcal{O}_{X}$ . Again using the definition of $\qc(X)$ as a limit, to produce one of these maps, it suffices to define analogous maps for the pullbacks under each $\eta:\Spec A\rightarrow X$ which themselves are compatible under pullbacks. But the existence of such maps are an immediate consequence of the definition $\eta^{*}M^{\vee}=(\eta^{*}M)^{\vee}$ . Finally, to verify that the usual composititions $M\rightarrow M\otimes M^{\vee}\otimes M\rightarrow M$ and $M^{\vee}\rightarrow M^{\vee}\otimes M\otimes M^{\vee}\rightarrow M^{\vee}$ are equivalences, it suffices to check under pullbacks to affines. But this is a direct consequence of our definition of $M^{\vee}$ and the fact that pullbacks preserve tensor products. ∎

3.1.2. Generators

Now we review notions of what it means for compact objects to generate a stable $\infty$ -category. (See [L3, 17] for more details, and [L2, 5.5.7] for the general setting of presentable $\infty$ -categories.)

0NWK

Definition 3.7. A stable category $\mathcal{C}$ is said to be compactly generated if there is a small $\infty$ -category $\mathcal{C}^{\circ}$ of compact objects $C_{i}\in\mathcal{C}$ whose right orthogonal vanishes: if $M\in\mathcal{C}$ satisfies $\operatorname{\Hom}_{\mathcal{C}}(C_{i},M)\simeq 0$ , for all $i$ , then $M\simeq 0$ .

As explained in [L3, Remark 17.3], whether a stable $\infty$ -category is compactly generated can be studied completely in the underlying homotopy category. In particular, the notion for stable $\infty$ -categories is compatible with that for triangulated categories.

0NWL

Example 3.8. For a commutative derived ring $k$ , the stable $\infty$ -category $\Mod_{k}$ of $k$ -modules is compactly generated. In fact, it is generated by the free module $k$ itself.

On the one hand, for a stable small $\infty$ -category $\mathcal{C}^{\circ}$ , the inductive limit $\mathcal{C}=\operatorname{Ind}(\mathcal{C}^{\circ})$ is a compactly generated stable presentable $\infty$ -category. Furthermore (see [L2, 5.3.4]), if $\mathcal{C}^{\circ}$ is closed under finite colimits and idempotent complete, then it can be recovered as the compact objects of $\mathcal{C}$ . In particular, we have that $\mathcal{C}$ is the Ind-category of its compact objects $\mathcal{C}^{\circ}$ .

On the other hand, given a stable $\infty$ -category $\mathcal{C}$ with a small full $\infty$ -subcategory $\mathcal{C}^{\circ}$ of compact generators, one can recover all compact objects of $\mathcal{C}$ by a result of Neeman [N1] (see also [L2, Proposition 5.3.4.17]): the compact objects are precisely direct summands of the objects of the smallest stable $\infty$ -subcategory $\mathcal{C}_{s}$ containing $\mathcal{C}^{\circ}$ (that is, they are direct summands of finite iterated extensions of objects of $\mathcal{C}^{\circ}$ ). In particular, if $\mathcal{C}^{\circ}$ is stable and idempotent complete, then it consists precisely of the compact objects of $\mathcal{C}$ .

If we further assume that $\mathcal{C}$ is a presentable stable $\infty$ -category $\mathcal{C}$ with a small full $\infty$ -subcategory $\mathcal{C}^{\circ}$ of compact generators, then a theorem of Schwede and Shipley [SSh] guarantees that we can recover $\mathcal{C}$ as the cocompletion of $\mathcal{C}_{s}$ (see [L4, 4.4] for the $\infty$ -categorical version, and [Ke] for the differential graded version). In other words, we recover $\mathcal{C}$ by passing to the category of colimit preserving $k$ -linear functors to $k$ -modules

\mathcal{C}\simeq\Fun(\mathcal{C}_{s}^{\rm op},\Mod_{k}).

In particular, we now can check that our notion of perfect stack is equivalent to more familiar assumptions on a symmetric monoidal $\infty$ -category.

0NWM

Proposition 3.9. For a derived stack $X$ with affine diagonal, the following are equivalent:

(1)

$X$ is perfect.
(2)

$\qc(X)$ is compactly generated, and its compact and dualizable objects coincide.

0NWN

Proof. If $X$ is perfect, so that $\qc(X)=\operatorname{Ind}\operatorname{Perf}(X)$ , we claim that compact and dualizable objects agree, and hence compact objects generate, so that (1) implies (2).

To see the claim, it suffices to show that the full $\infty$ -subcategory of dualizable objects of $\qc(X)$ is idempotent complete (since it is also stable). Since $\qc(X)$ is idempotent complete, this is equivalent to showing that dualizable objects are closed under retracts. However, for a retract $N$ of a dualizable object $M$ one can explicitly write down the unit and trace maps for $N\otimes{\mathcal{H}om}(N,\mathcal{O}_{X})$ and confirm the necessary conditions. We leave this to the reader.

Conversely, if $\qc(X)$ is compactly generated, then $\qc(X)$ is the Ind-category of its compact objects, and hence by assumption also the Ind-category of its dualizable objects, so that (2) implies (1). ∎

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5