On stable equivalences, perfect exact sequences and Gorenstein-projective modules

Abstract

We consider the equivalence from the stable module category to a subcategory ${\mathcal{L}}_A$ of the homotopy category constructed by Kato. This equivalence induces a correspondence between distinguished triangles in the homotopy category and perfect exact sequences in the module category. We show that an exact equivalence between categories ${\mathcal{L}}_A$ and ${\mathcal{L}}_B$ induces a stable equivalence of Morita type between two finite dimensional algebra A and B under a separability assumption. Moreover, we provide further sufficient conditions for a stable equivalence induced by an exact functor to be of Morita type. This is shown using perfect exact sequences. In particular, we study when a stable equivalence preserves perfect exact sequences up to projective direct summands. As an application, we show that a stable equivalence preserves the category of stable Gorenstein-projective modules if it preserves perfect exact sequences.

26 pages

References in corpus (2)

• Gorenstein Homological Algebra of Artin Algebras
• A triangulated hull and a Nakayama closure of the stable module category inside the homotopy category

Cited by in corpus (1)

• A triangulated hull and a Nakayama closure of the stable module category inside the homotopy category