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arXiv · 2109.12981

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

Sebastian Nitsche
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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.

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Sebastian Nitsche. 2021-09-27. On stable equivalences, perfect exact sequences and Gorenstein-projective modules. https://arxiv.org/abs/2109.12981

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