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

Abstract

Given an artin algebra Λ \Lambda with an idempotent element a a we compare the algebras Λ \Lambda and a Λ a a\Lambda a with respect to Gorensteinness, singularity categories and the finite generation condition Fg \mathrm {\textsf {Fg}} for the Hochschild cohomology. In particular, we identify assumptions on the idempotent element a a which ensure that Λ \Lambda is Gorenstein if and only if a Λ a a\Lambda a is Gorenstein, that the singularity categories of Λ \Lambda and a Λ a a\Lambda a are equivalent and that Fg \mathrm {\textsf {Fg}} holds for Λ \Lambda if and only if Fg \mathrm {\textsf {Fg}} holds for a Λ a 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.