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.

Characterization of eventually periodic modules in the singularity categories

“…It follows from Definition 3.1 that singular equivalences of Morita type with level 0 are noting but stable equivalences of Morita type in the sense of Broué [7] (cf. [13,Remark 4.3]). The next observation tells us that any two self-injective algebras are in fact stably equivalent of Morita type whenever they are singularly equivalent of Morita type with level (compare [31,Proposition 3.7]).…”

“…(1) Let Λ be a connected Nakayama algebra whose admissible sequence is given by (13,13,12,12,12). Note that it can be found in [26, Since w(Λ ′ ) = 3 = 1, Corollary 3.12 shows that Λ ′ is eventually periodic.…”

“…The notion of singular equivalences of Morita type with level was introduced by Wang [35], and it is known that such equivalences induce triangle equivalences between singularity categories (see Remark 3.2). Recently, singular equivalences of Morita type with level have been intensively studied (see [11,13,24,31,35]). In particular, Skartsaeterhagen [31], Qin [24] and Wang [35,37] have discovered invariants under singular equivalence of Morita type with level.…”