Perfect complexes on algebraic 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 Toën 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.

reordering of the Introduction and sections 3 and 4; additional material on perfect complexes; Appendix A on Mayer--Vietoris squares incorporated into arXiv:1606.08517; final version, to appear in Compositio Math

References in corpus (5)

• Algebraic groups and compact generation of their derived categories of representations
• Geometricity for derived categories of algebraic stacks
• The telescope conjecture for algebraic stacks
• The Balmer spectrum of a tame stack
• On the derived category of quasi-coherent sheaves on an Adams geometric stack

Cited by in corpus (17)

• Coherent Tannaka duality and algebraicity of Hom-stacks
• Conservative descent for semi-orthogonal decompositions
• Equivariant Hodge theory and noncommutative geometry
• Virtual cycles of stable (quasi)-maps with fields
• An improvement on the base-change theorem and the functor $f^{!}$
• K-theory and G-theory of derived algebraic stacks
• Algebraic K-theory of quasi-smooth blow-ups and cdh descent
• Categorical Milnor squares and K-theory of algebraic stacks
• Mayer-Vietoris squares in algebraic geometry
• Triangulated categories with a single compact generator and two Brown representability theorems
• Brane actions, Categorification of Gromov-Witten theory and Quantum K-theory
• Addendum: Étale dévissage, descent and pushouts of stacks
• Universal functors on symmetric quotient stacks of Abelian varieties
• Separable commutative algebras and Galois theory in stable homotopy theories
• The Moduli of Sections Has a Canonical Obstruction Theory
• The Atiyah class on algebraic stacks
• The derived Brauer map via twisted sheaves