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Perfect complexes on algebraic stacks

Part of: Algebraic geometry: Foundations Abelian categories (Co)homology theory Commutative algebra: Homological methods

Published online by Cambridge University Press:  17 August 2017

Jack Hall  and
David Rydh
Jack Hall
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, USA email jackhall@math.arizona.edu
David Rydh
Affiliation:
Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden email dary@math.kth.se
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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.

Keywords

derived categoriesalgebraic stackscompact generationperfect complexes

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions
Secondary: 13D09: Derived categories 14A20: Generalizations (algebraic spaces, stacks) 18E30: Derived categories, triangulated categories

Information

Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 11 , November 2017 , pp. 2318 - 2367
Copyright
© The Authors 2017 

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