On singular equivalences of Morita type with level and Gorenstein algebras

Abstract

Rickard proved that for certain self-injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Broué. In this paper we study singular equivalences of finite dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex of bimodules, induces a singular equivalence of Morita type with level, in the sense of Wang. This recovers Rickard's theorem in the self-injective case.

Singular equivalences and Auslander-Reiten conjecture

“…See [8,Proposition 4.8] for recent progress in this direction. In the general case, it was proved recently by Dalezios [12,Theorem 3.6] that F induces a singular equivalence if…”

“…Then we have done by Theorem 3.6. Now we focus on singular equivalences from change of rings, which was studied in [27,12]. Let f : A → B be a morphism of algebras with proj.…”

“…), then B B⊗ L A − and A B⊗ L B − induce mutual equivalences between the singularity categories; see [12,Corollary 3.7].…”

Reduction techniques of singular equivalences

“…In [11], Chen-Liu-Wang gave a sufficient condition on when a tensor functor with a bimodule defines a singular equivalence Morita type with level, and in [16], Dalezios proved that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a complex of bimodules always induces a singular equivalence of Morita type with level. Our first theorem is a complex version of Chen-Liu-Wang's work, and it generalizes the result of Dalezios to arbitrary algebra (not limited to Gorenstein algebra).…”

Singular equivalences induced by bimodules and quadratic monomial algebras

“…By a singular equivalence between A and B, we mean a triangle equivalence between D sg (A) and D sg (B). As in [24,23,9], the question when the tensor functor by an A-B-bimodule M yields a singular equivalence is of interest. We assume that M is projective on each side.…”