Homological dimensions and abelian model structures on chain complexes

TL;DR

This paper develops new Abelian model structures on chain complexes over a ring, based on homological dimensions of modules, extending previous models by incorporating modules with bounded projective or flat dimensions.

Contribution

It constructs a unique Abelian model structure on chain complexes where cofibrant objects have modules with bounded homological dimensions, generalizing prior models.

Findings

Established a model structure with modules of projective dimension ≤ n as cofibrant objects.

Proved the existence of a model structure with modules of flat dimension ≤ n as cofibrant objects.

Extended previous work by incorporating modules with bounded homological dimensions.

Abstract

We construct Abelian model structures on the category of chain complexes over a ring $R$, from the notion homological dimensions of modules. Given an integer $n > 0$, we prove that the left modules over a ringoid $\mathfrak{R}$ with projective dimension at most $n$ form the left half of a complete cotorsion pair. Using this result we prove that there is a unique Abelian model structure on the category of chain complexes where the exact complexes are the trivial objects and the complexes with projective dimension at most $n$ form the class of trivially cofibrant objects. In the paper "Cotorsion pairs in C($R$-Mod)", the authors construct an Abelian model structure on chain complexes, where the trivial objects are the exact complexes and the class of cofibrant objects is given by the complexes whose terms are all projective. We extend this result by finding a new Abelian model structure with the same trivial objects and where the cofibrant objects are given by the class of complexes whose terms are modules with projective dimension at most $n$. We also prove similar results concerning the flat dimension.