Geometricity for derived categories of algebraic stacks

Abstract

We prove that the dg category of perfect complexes on a smooth, proper Deligne–Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.

Appendix A: Bounded derived category of coherent modules

Our aim is to show that the bounded derived category of coherent modules on a regular, quasi-compact, separated algebraic stack with finite stabilizers is equivalent to the derived category of perfect complexes (see Remark A.3).

The category of coherent ${\mathcal {O}}_X$-modules on a locally noetherian algebraic stack X is denoted by ${{\text {Coh}}}(X)$. We use the usual decorations for full subcategories of derived categories. For example, the symbol ${{\text {D}}}^-_{{\text {Coh}}}({{\text {Qcoh}}}(X))$ denotes the full subcategory of the derived category ${{\text {D}}}({{\text {Qcoh}}}(X))$ of quasi-coherent modules whose objects have bounded above coherent cohomology modules.

The following proposition generalizes a well-known result for noetherian schemes [46, Exposé II, Proposition 2.2.2], [22, Proposition 3.5] to noetherian algebraic stacks.

Proposition A.1

Let X be an noetherian algebraic stack. Then the obvious functor defines an equivalence

$$\begin{aligned} {{\text {D}}}^-({{\text {Coh}}}(X)) \xrightarrow {\sim }{{\text {D}}}^-_{{\text {Coh}}}({{\text {Qcoh}}}(X)). \end{aligned}$$

Proof

It is certainly enough to show that each bounded above complex of quasi-coherent modules with coherent cohomology modules has a quasi-isomorphic subcomplex of coherent modules. This is an easy consequence of the proof of [22, Proposition 3.5] as soon as we know the following fact: given any epimorphism $G \rightarrow F$ from a quasi-coherent module G to a coherent module F, there is a coherent submodule $G'$ of G such that the composition $G' \subset G \rightarrow F$ is still an epimorphism. This latter statement follows from the fact that every quasi-coherent module is the filtered colimit of its coherent submodules [29, Proposition 15.4] and [46, Exposé II, Lemma 2.1.1.a)]. $\square$

Proposition A.2

Let X be a regular and quasi-compact algebraic stack. Then we have an equality ${{{\text {D}}}_{{{\text {pf}}}}}(X) = {{\text {D}}}^{\mathrm {b}}_{{\text {Coh}}}(X)$.

Proof

Since X is quasi-compact we have ${{{\text {D}}}_{{{\text {pf}}}}}(X) \subset {{\text {D}}}^{\mathrm {b}}_{{\text {Coh}}}(X)$. In order to show equality it is enough to prove that any coherent module is perfect. Let ${\text {Spec}}A \rightarrow X$ be any smooth morphism where A is a ring. Then A is regular. It is enough to prove that any finitely generated A-module M has a finite resolution by finitely generated projective A-modules. Let $P \rightarrow M$ be a resolution by finitely generated projective A-modules. Let ${\mathfrak {p}}\in {\text {Spec}}A$. Since $A_{\mathfrak {p}}$ is regular, it has finite global dimension by the Auslander–Buchsbaum–Serre theorem. Therefore, there is a natural number $n=n({\mathfrak {p}})$ such that the kernel of the differential $d^{-n} :(P^{-n})_{\mathfrak {p}}\rightarrow (P^{-n+1})_{\mathfrak {p}}$ is a finitely generated projective $A_{\mathfrak {p}}$-module. Since A is noetherian, there is some open neighborhood ${\text {Spec}}A_f$ of ${\mathfrak {p}}$ in ${\text {Spec}}A$ such that the kernel of $d^{-n} :(P^{-n})_f \rightarrow (P^{-n+1})_f$ is a finitely generated projective $A_f$-module. Then also all kernels $d^{-i} :(P^{-i})_f \rightarrow (P^{-i+1})_f$, for $i \ge n$, are finitely generated projective $A_f$-modules. Since ${\text {Spec}}A$ is quasi-compact there is a natural number N such that the kernel of $d^{-N} :P^{-N} \rightarrow P^{-N+1}$ is a finitely generated projective A-module. $\square$

Remark A.3

If X is a noetherian, separated algebraic stack with finite stabilizers, we have equivalences

$$\begin{aligned} {{\text {D}}}^-({{\text {Coh}}}(X)) \xrightarrow {\sim }{{\text {D}}}^-_{{\text {Coh}}}({{\text {Qcoh}}}(X)) \xrightarrow {\sim }{{\text {D}}}^-_{{\text {Coh}}}(X). \end{aligned}$$

This follows immediately from Proposition A.1 and the equivalence ${{\text {D}}}({{\text {Qcoh}}}(X)) \xrightarrow {\sim }{{{\text {D}}}_{{{\text {qc}}}}}(X)$ from (2.2). If we assume in addition that X is regular, then Proposition A.2 together with the above equivalences shows that

$$\begin{aligned} {{\text {D}}}^{\mathrm {b}}({{\text {Coh}}}(X)) \xrightarrow {\sim }{{\text {D}}}^{\mathrm {b}}_{{\text {Coh}}}(X) = {{{\text {D}}}_{{{\text {pf}}}}}(X) \end{aligned}$$

is an equivalence.