Applications of cotorsion triples

TL;DR

This paper explores the homotopy categories derived from cotorsion triples, establishing equivalences between stable categories, and applies these results to Gorenstein rings, providing new proofs and characterizations.

Contribution

It introduces new equivalences between homotopy categories of complexes from cotorsion triples and applies these to Gorenstein rings, including a novel proof of Gorenstein global dimension equality.

Findings

Equivalence of homotopy categories via Quillen equivalence.

Characterization of homological dimensions with respect to cotorsion triples.

New proof of Gorenstein global dimension equality.

Abstract

We study homotopy categories of model categories arising from a cotorsion triple, and the equivalences between corresponding stable categories. We characterize homological dimensions with respect to a cotorsion triple. Then, we lift cotorsion triple to complexes, and get the equivalence of homotopy categories of complexes via Quillen equivalence of model categories. Finally, we specify to Gorenstein cotorsion triple $(\mathcal{GP}, \mathcal{W}, \mathcal{GI})$. By Quillen equivalence, it is shown that for a left-Gorenstein ring, there is an equivalence $\mathrm{K}(\mathcal{GP})\simeq \mathrm{K}(\mathcal{GI})$, which restricts to an equivalence $\mathrm{K}(\mathcal{P})\simeq \mathrm{K}(\mathcal{I})$ (compare to [J. Algebra, 2010, 324:2718-2731]); a new proof for Bennis and Mahdou's equality of Gorenstein global dimension is also given.