Homological dimension based on a class of Gorenstein flat modules

TL;DR

This paper explores a new homological dimension based on Gorenstein flat modules, revealing properties similar to Gorenstein projective dimension and establishing connections with classical invariants and triangulated categories.

Contribution

It introduces the PGF-dimension, studies its properties, and links it to classical homological invariants, generalizing known results to broader ring contexts.

Findings

PGF-dimension shares properties with Gorenstein projective dimension

Hereditary Hovey triple exists in modules of finite PGF-dimension

Connections established between PGF-global dimension and classical invariants

Abstract

In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying the finiteness of the PGF global dimension reveals a connection between classical homological invariants of left and right modules over the ring, that leads to generalizations of certain results by Jensen, Gedrich and Gruenberg that were originally proved in the realm of commutative Noetherian rings.