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

2021/09/27 by Sebastian Nitsche, Nitsche, Sebastian
Mathematics · #16E05 #16G10 #18G65 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2109.12981

openalex publication_date 2021/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the equivalence from the stable module category to a subcategory 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 L_A and 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.

Citations

• Gorenstein Homological Algebra of Artin Algebras

Cited by

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

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