Abstract
We show that singular equivalences of Morita type with level between finite-dimensional Gorenstein algebras over a field preserve the (Fg) condition.
Highlights
Throughout the paper, we let k be a fixed field
We show that singular equivalences of Morita type with level between finite-dimensional Gorenstein algebras over a field preserve the (Fg) condition
We show that if the singular equivalence f is of Morita type with level, and induced by a tensor functor, the equivalence g is induced by the same tensor functor
Summary
Throughout the paper, we let k be a fixed field. Support varieties for modules over a group algebra kG were introduced by J.F. The main result of this paper, Theorem 7.4, answers this question affirmatively: A singular equivalence of Morita type with level between finite-dimensional Gorenstein algebras over a field preserves the (Fg) condition. In the terminology of [16], this implies that a tensor functor inducing a singular equivalence of Morita type with level between Gorenstein algebras is an eventually homological isomorphism The proof of this result builds on the result about stable categories of Cohen–Macaulay modules from Section 3. The main ingredients in the proof of this result are the isomorphism (1.1) of extension groups from Section 5 and the isomorphism (1.2) of Hochschild cohomology groups from Section 6
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