Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements
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- by Chrysostomos Psaroudakis, Øystein Skartsæterhagen and Øyvind Solberg HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 1 (2014), 45-95
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 $\mathrm {\textsf {Fg}}$ 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 $\mathrm {\textsf {Fg}}$ holds for $\Lambda$ if and only if $\mathrm {\textsf {Fg}}$ 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.References
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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
- MR Author ID: 1041820
- Email: Chrysostomos.Psaroudakis@mathematik.uni-stuttgart.de
- Øystein Skartsæterhagen
- Affiliation: Institutt for Matematiske fag, NTNU, N-7491 Trondheim, Norway
- MR Author ID: 1084194
- Email: Oystein.Skartsaterhagen@math.ntnu.no
- Øyvind Solberg
- Affiliation: Institutt for Matematiske fag, NTNU, N-7491 Trondheim, Norway
- MR Author ID: 255073
- Email: Oyvind.Solberg@math.ntnu.no
- Received by editor(s): March 7, 2014
- Received by editor(s) in revised form: September 1, 2014
- Published electronically: November 6, 2014
- © Copyright 2014 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- 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
- MathSciNet review: 3274657