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

TL;DR

This paper explores the relationship between stable module categories, perfect exact sequences, and Gorenstein-projective modules, establishing conditions under which stable equivalences imply Morita-type equivalences and preserve Gorenstein-projective modules.

Contribution

It introduces new conditions linking stable equivalences and Morita equivalences via perfect exact sequences, advancing understanding of module category structures.

Findings

Stable equivalences induce Morita-type equivalences under separability.

Stable equivalences can preserve Gorenstein-projective modules.

Conditions are provided for stable equivalences to preserve perfect exact sequences.

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.