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ONE POSITIVE AND TWO NEGATIVE RESULTS FOR DERIVED CATEGORIES OF ALGEBRAIC STACKS

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

Published online by Cambridge University Press:  22 January 2018

Jack Hall ,
Amnon Neeman  and
David Rydh
Jack Hall
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, USA (jackhall@math.arizona.edu)
Amnon Neeman
Affiliation:
Mathematical Sciences Institute, The Australian National University, Acton, ACT 2601, Australia (Amnon.Neeman@anu.edu.au)
David Rydh
Affiliation:
KTH Royal Institute of Technology, Department of Mathematics, SE-100 44 Stockholm, Sweden (dary@math.kth.se)
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Abstract

Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) $\mathsf{D}_{\text{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$ is noetherian or has affine diagonal, then the functor $\unicode[STIX]{x1D6F9}_{X}:\mathsf{D}(\mathsf{QCoh}(X))\rightarrow \mathsf{D}_{\text{qc}}(X)$ is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.

Keywords

derived categoriesalgebraic stacks

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions
Secondary: 13D09: Derived categories 14A20: Generalizations (algebraic spaces, stacks) 18G10: Resolutions; derived functors
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 5 , September 2019 , pp. 1087 - 1111
Copyright
© Cambridge University Press 2018 

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Footnotes

The first and second author are supported by the Australian Research Council (ARC), grant numbers FL100100137, DE150101799, and DP150102313. The third author is supported by the Swedish Research Council (VR), grant numbers 2011-5599 and 2015-05554.

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