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

Psaroudakis, Chrysostomos; Skartsæterhagen, Øystein; Solberg, Øyvind

doi:10.1090/S2330-0000-2014-00004-6

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.

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