Gorenstein Homological Dimensions and Abelian Model Structures

TL;DR

This paper develops new Abelian model structures on modules and chain complexes over Gorenstein rings by constructing complete cotorsion pairs based on Gorenstein homological dimensions, with applications to differential graded complexes.

Contribution

It introduces novel cotorsion pairs related to Gorenstein dimensions and establishes a correspondence between differential graded complexes and modules over specific quotient rings.

Findings

Complete cotorsion pairs for Gorenstein-projective and Gorenstein-flat dimensions.

New Abelian model structures on module and chain complex categories.

Bijective correspondence between differential graded complexes and modules over R[x]/(x^2).

Abstract

We construct new complete cotorsion pairs in the categories of modules and chain complexes over a Gorenstein ring $R$, from the notions of Gorenstein homological dimensions, in order to obtain new Abelian model structures on both categories. If $r$ is a positive integer, we show that the class of modules with Gorenstein-projective (or Gorenstein-flat) dimension $\leq r$ forms the left half of a complete cotorsion pair. Analogous results also hold for chain complexes over $R$. In any Gorenstein category, we prove that the class of objects with Gorenstein-injective dimension $\leq r$ is the right half of a complete cotorsion pair. The method we use in each case consists in constructing a cogenerating set for each pair. Later on, we give some applications of these results. First, as an extension of some results by M. Hovey and J. Gillespie, we establish a bijective correspondence between the class of differential graded $r$-projective complexes and the class of modules over $R[x] / (x^2)$ with Gorenstein-projective dimension $\leq r$, provided $R$ is left and right Noetherian with finite global dimension. The same correspondence is also valid for the (Gorenstein-)injective and (Gorenstein-)flat dimensions.