Gorenstein flat modules with respect to duality pairs

TL;DR

This paper introduces Gorenstein $( ext{X}, ext{Y})$-flat modules as a unifying framework for various Gorenstein flat modules, establishing their stability, functorial properties, and model structures.

Contribution

It generalizes several known Gorenstein flat modules by defining Gorenstein $( ext{X}, ext{Y})$-flat modules and studies their properties and categorical structures.

Findings

Gorenstein $( ext{X}, ext{Y})$-flat modules are strongly stable.

Functorial descriptions of Gorenstein $( ext{X}, ext{Y})$-flat dimension are provided.

A hereditary abelian model structure with these modules as cofibrant objects is constructed.

Abstract

Let $\mathcal{X}$ be a class of left $R$-modules, $\mathcal{Y}$ be a class of right $R$-modules. In this paper, we introduce and study Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat modules as a common generalization of some known modules such as Gorenstein flat modules \cite{EJT93}, Gorenstein $n$-flat modules \cite{SUU14}, Gorenstein $\mathcal{B}$-flat modules \cite{EIP17}, Gorenstein AC-flat modules \cite{BEI17}, $\Omega$-Gorenstein flat modules \cite{EJ00} and so on. We show that the class of all Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat modules have a strong stability. In particular, when $(\mathcal{X}, \mathcal{Y})$ is a perfect (symmetric) duality pair, we give some functorial descriptions of Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat dimension, and construct a hereditary abelian model structure on $R$-Mod whose cofibrant objects are exactly the Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat modules. These results unify the corresponding results of the aforementioned modules.