Strongly flat modules via universal localization

TL;DR

This paper extends the concept of strongly flat modules to non-commutative rings using universal localization, analyzing their homotopy categories and establishing properties of associated subcategories.

Contribution

It introduces a non-commutative version of strongly flat modules via universal localization and studies their homotopy category and subcategory properties.

Findings

The class of σ-strongly flat modules forms a precovering class.

The quotient map from the homotopy category has a fully faithful right adjoint.

Thick subcategory of acyclic complexes is well-behaved in this setting.

Abstract

In this paper, we investigate a non-commutative version of strongly flat modules, which is based on the concept of universal localization introduced by Cohn. We consider a set $\sigma$ consisting of maps of finitely generated projective $R$-modules, where $R$ is not necessarily a commutative ring. Let $R_{\sigma}$ denote the universal localization of $R$ with respect to $\sigma$. The class of $\sigma$-strongly flat modules is defined as the left class in the cotorsion pair generated by $R_{\sigma}$. We examine the homotopy category of $\sigma$-strongly flat modules and demonstrate that the thick subcategory $\mathscr{S}_{\sigma}$, consisting of acyclic complexes, wherein all syzygies are $\sigma$-strongly flat, forms a precovering class within this homotopy category. This implies that the quotient map from $\mathbb{K}({\sigma\mbox{-}\mathcal{SF}})$ to $\mathbb{K}({\sigma\mbox{-}\mathcal{SF}})/\mathscr{S}_{\sigma}$ always has a fully faithful right adjoint.