# Model structures, n-Gorenstein flat modules and PGF dimensions

## TL;DR

This paper constructs a new abelian model structure on modules over a ring using Gorenstein flat and PGF modules, explores the homotopy category, and relates PGF dimension to Gorenstein projective dimension, offering new insights into ring and module theory.

## Contribution

It introduces a novel abelian model structure based on Gorenstein flat and PGF modules, and establishes the equivalence of PGF and Gorenstein projective dimensions.

## Key findings

- Homotopy category is triangulated equivalent to a stable category.
- Model structure is compactly generated for rings with finite global Gorenstein AC-projective dimension.
- Global Gorenstein projective dimension equals global PGF dimension.

## Abstract

Given a non-negative integer $n$ and a ring $R$ with identity, we construct an abelian model structure on the category of left $R$-modules where the class of cofibrant objects coincides with $\mathcal{GF}_n(R)$ the class of left $R$-modules with Gorenstein flat dimension less than $n$, the class of fibrant objects coincides with $\mathcal{F}_n(R)^\perp$ the right ${\rm Ext}$-orthogonal class of left $R$-modules with flat dimension less than $n$, and the class of trivial objects coincides with $\mathcal{PGF}(R)^\perp$ the right ${\rm Ext}$-orthogonal class of PGF left $R$-modules recently introduced by \v{S}aroch and \v{S}\v{t}ov\'{\i}\v{c}ek. The homotopy category of this model structure is triangulated equivalent to the stable category $\underline{\mathcal{GF}(R)\cap\mathcal{C}(R)}$ modulo flat-cotorsion modules and it is compactly generated when $R$ has finite global Gorenstein AC-projective dimension. The second part of this paper deals with the PGF dimension of modules and rings. Our results suggest that this dimension could serve as an alternative definition of the Gorenstein projective dimension. We show, among other things, that ($n$-)perfect rings can be characterized in terms of Gorenstein homological dimensions, similar to the classical ones, and the global Gorenstein projective dimension coincides with the global PGF dimension.