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

Published online by Cambridge University Press:  22 January 2018

Jack Hall
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, USA ([email protected])
Amnon Neeman
Affiliation:
Mathematical Sciences Institute, The Australian National University, Acton, ACT 2601, Australia ([email protected])
David Rydh
Affiliation:
KTH Royal Institute of Technology, Department of Mathematics, SE-100 44 Stockholm, Sweden ([email protected])

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.

Type
Research Article
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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