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Transactions of the American Mathematical Society Series B

ISSN 2330-0000

   
 
 

 

Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements


Authors: Chrysostomos Psaroudakis, Øystein Skartsæterhagen and Øyvind Solberg
Journal: Trans. Amer. Math. Soc. Ser. B 1 (2014), 45-95
MSC (2010): Primary 18Exx, 18E30, 16E30, 16E40, 16E65; Secondary 16E10, 16Gxx, 16G50
DOI: https://doi.org/10.1090/S2330-0000-2014-00004-6
Published electronically: November 6, 2014
MathSciNet review: 3274657
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Abstract: Given an artin algebra $ \Lambda $ with an idempotent element $ a$ we compare the algebras $ \Lambda $ and $ a\Lambda a$ with respect to Gorensteinness, singularity categories and the finite generation condition $ \textup {\textsf {Fg}}$ for the Hochschild cohomology. In particular, we identify assumptions on the idempotent element $ a$ which ensure that $ \Lambda $ is Gorenstein if and only if $ a\Lambda a$ is Gorenstein, that the singularity categories of $ \Lambda $ and $ a\Lambda a$ are equivalent and that $ \textup {\textsf {Fg}}$ holds for $ \Lambda $ if and only if $ \textup {\textsf {Fg}}$ holds for $ a\Lambda a$. We approach the problem by using recollements of abelian categories and we prove the results concerning Gorensteinness and singularity categories in this general setting. The results are applied to stable categories of Cohen-Macaulay modules and classes of triangular matrix algebras and quotients of path algebras.


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Additional Information

Chrysostomos Psaroudakis
Affiliation: Institutt for Matematiske fag, NTNU, N-7491 Trondheim, Norway
Address at time of publication: Universität Stuttgart, Institut für Algebra und Zahlentheorie, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
Email: Chrysostomos.Psaroudakis@mathematik.uni-stuttgart.de

Øystein Skartsæterhagen
Affiliation: Institutt for Matematiske fag, NTNU, N-7491 Trondheim, Norway
Email: Oystein.Skartsaterhagen@math.ntnu.no

Øyvind Solberg
Affiliation: Institutt for Matematiske fag, NTNU, N-7491 Trondheim, Norway
Email: Oyvind.Solberg@math.ntnu.no

DOI: https://doi.org/10.1090/S2330-0000-2014-00004-6
Keywords: Recollements of abelian categories, finite generation condition, Hochschild cohomology, Gorenstein categories, Gorenstein artin algebras, singularity categories, Cohen--Macaulay modules, triangular matrix algebras, path algebras
Received by editor(s): March 7, 2014
Received by editor(s) in revised form: September 1, 2014
Published electronically: November 6, 2014
Article copyright: © Copyright 2014 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)

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