Perfect complexes on algebraic stacks

TL;DR

This paper develops a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks, proving they are compactly generated by perfect complexes under certain conditions, and extends key results to derived Deligne--Mumford stacks.

Contribution

It establishes the local nature of compact generation of derived categories on algebraic stacks and extends foundational results to derived stacks, broadening the scope of derived algebraic geometry.

Findings

Derived categories are compactly generated by perfect complexes for certain stacks.

Compact generation is local for the quasi-finite flat topology.

Extended results on derived Azumaya algebras to derived Deligne--Mumford stacks.

Abstract

We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend To\"en and Antieau--Gepner's results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne--Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.