Homological dimensions and Abelian model structures on chain complexes

Abstract

We construct Abelian model structures on the category of chain complexes over a ring~$R$, from the notion of 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 over $R$, 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 \cite {Rada}, 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 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 flat dimension.