Invariants and constructions of separable equivalences

Abstract

In this paper, we first establish relationships between Gorenstein projective modules linked by the separable equivalence of rings, and prove that Gorenstein, CM-finite and CM-free algebras are invariant under separable equivalences. Secondly, we provide a new method to produce separable equivalences. As applications, the following results are obtained. Let Λ and Γ be Artin algebras such that Λ is separably equivalent to Γ. (1) For representation-finite algebras Λ and Γ, their Auslander algebras are separably equivalent; (2) For CM-finite algebras Λ and Γ, the endomorphism algebras of their representative generators are separably equivalent. Finally, we discuss when tilted algebras are invariant under separable equivalences, and give an example to illustrate it.