Quillen equivalences for stable categories

TL;DR

This paper explores conditions under which stable categories derived from an abelian category are triangulated equivalent, using model categories and Quillen equivalences, and extends these results to related categories like chain complexes and quiver representations.

Contribution

It provides new criteria for establishing Quillen equivalences between stable categories of an abelian category and related categories, enhancing understanding of their homotopy-theoretic relationships.

Findings

Identifies conditions for Quillen equivalences between stable categories

Shows transfer of Quillen equivalences to categories of chain complexes

Extends results to categories of representations of quivers

Abstract

For an abelian category $\mathcal{A}$ we investigate when the stable categories $\underline{\mathrm{GPro}}\mathrm{j}(\mathcal{A})$ and $\underline{\mathrm{GIn}}\mathrm{j}(\mathcal{A})$ are triangulated equivalent. To this end, we realize these stable categories as homotopy categories of certain (non-trivial) model categories and give conditions on $\mathcal{A}$ that ensure the existence of a Quillen equivalence between the model categories in question. We also study when such a Quillen equivalence transfers from $\mathcal{A}$ to categories naturally associated to $\mathcal{A}$, such as $\mathrm{Ch}(\mathcal{A})$, the category of chain complexes in $\mathcal{A}$, or $\mathrm{Rep}(Q,\mathcal{A})$, the category of $\mathcal{A}$-valued representations of a quiver $Q$.