Model structures and relative Gorenstein flat modules and chain complexes

TL;DR

This paper extends the theory of Gorenstein flat modules to a relative setting using a class of modules, establishing new model structures on modules and complexes, and analyzing their homotopy categories.

Contribution

It generalizes existing model structures to Gorenstein -flat modules relative to a class , and explores their properties in module and complex categories.

Findings

Established conditions for closure under extensions.

Constructed new model structures on modules and complexes.

Compared homotopy categories of these models.

Abstract

A recent result by J. \v{S}aroch and J. \v{S}\v{t}ov\'{\i}\v{c}ek asserts that there is a unique abelian model structure on the category of left $R$-modules, for any associative ring $R$ with identity, whose (trivially) cofibrant and (trivially) fibrant objects are given by the classes of Gorenstein flat (resp., flat) and cotorsion (resp., Gorenstein cotorsion) modules. In this paper, we generalise this result to a certain relativisation of Gorenstein flat modules, which we call Gorenstein $\mathcal{B}$-flat modules, where $\mathcal{B}$ is a class of right $R$-modules. Using some of the techniques considered by \v{S}aroch and \v{S}\v{t}ov\'{\i}\v{c}ek, plus some other arguments coming from model theory, we determine some conditions for $\mathcal{B}$ so that the class of Gorenstein $\mathcal{B}$-modules is closed under extensions. This will allow us to show approximation properties concerning these modules, and also to obtain a relative version of the model structure described before. Moreover, we also present and prove our results in the category of complexes of left $R$-modules, study other model structures on complexes constructed from relative Gorenstein flat modules, and compare these models via computing their homotopy categories.