Quiver representations over a quasi-Frobenius ring and Gorenstein-projective modules

TL;DR

This paper develops a model structure on quiver representations over a quasi-Frobenius ring, linking it to Gorenstein-projective modules, and generalizes existing characterizations to broader module classes using model category theory.

Contribution

It introduces a novel model structure on quiver representations over a quasi-Frobenius ring, connecting it to Gorenstein-projective modules and extending prior characterizations.

Findings

Model category structure on quiver representations over R

Equivalence between homotopy category and stable Gorenstein-projective modules

Generalization of Gorenstein-projective module characterization

Abstract

We consider a finite acyclic quiver $\mathcal{Q}$ and a quasi-Frobenius ring $R$. We endow the category of quiver representations over $R$ with a model structure, whose homotopy category is equivalent to the stable category of Gorenstein-projective modules over the path algebra $R\mathcal{Q}$. As an application, we then characterize Gorenstein-projective $R\mathcal{Q}$-modules in terms of the corresponding quiver $R$-representations; this generalizes a result obtained by Luo-Zhang to the case of not necessarily finitely generated $R\mathcal{Q}$-modules, and partially recover results due to Enochs-Estrada-Garc\'ia Rozas, and to Eshraghi-Hafezi-Salarian. Our approach to the problem is completely different since the proofs mainly rely on model category theory.