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LOCAL DUALITY FOR THE SINGULARITY CATEGORY OF A FINITE DIMENSIONAL GORENSTEIN ALGEBRA

Part of: Representation theory of rings and algebras Homological methods

Published online by Cambridge University Press:  11 March 2020

DAVE BENSON Open the ORCID record for DAVE BENSON [Opens in a new window] ,
SRIKANTH B. IYENGAR Open the ORCID record for SRIKANTH B. IYENGAR [Opens in a new window] ,
HENNING KRAUSE Open the ORCID record for HENNING KRAUSE [Opens in a new window]  and
JULIA PEVTSOVA
DAVE BENSON
Affiliation:
Institute of Mathematics, University of Aberdeen, King’s College, AberdeenAB24 3UE, Scotland, UK email d.j.benson@abdn.ac.uk
SRIKANTH B. IYENGAR
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT84112, USA email iyengar@math.utah.edu
HENNING KRAUSE
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, 33501Bielefeld, Germany email hkrause@math.uni-bielefeld.de
JULIA PEVTSOVA
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA98195, USA email pevtsova@uw.edu
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Abstract

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the derived category, for each homogeneous prime ideal $\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology.

MSC classification

Primary: 16G10: Representations of Artinian rings
Secondary: 16G50: Cohen-Macaulay modules 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 16E35: Derived categories
Type
Article
Information
Nagoya Mathematical Journal , Volume 244 , December 2021 , pp. 1 - 24
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Copyright
© 2020 Foundation Nagoya Mathematical Journal

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Footnotes

The second author was partly supported by the National Science Foundation grants DMS-1503044 and DMS-1700985. The fourth author was partly supported by the National Science Foundation grants DMS-1501146 and DMS-1901854.

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