Degreewise n-projective and n-flat model structures on chain complexes

TL;DR

This paper develops new abelian model structures on chain complexes by generalizing existing methods to include degreewise n-projective and n-flat complexes, expanding the framework for homological algebra over noetherian rings.

Contribution

It introduces a generalized construction of abelian model structures on chain complexes for each n>1, based on projective and flat dimension constraints, extending prior work.

Findings

Constructed model structures with cofibrant objects as dw-n-projective complexes.

Established model structures with cofibrant objects as dw-n-flat complexes.

Generalized Kaplansky's theorem for broader classes of chain complexes.

Abstract

In the paper "Cotorsion Pairs in C(R-Mod)", the authors construct an abelian model structure on the category of chain complexes Ch(R), where the class of cofibrant objects is given by the class of degreewise projective chain complexes. Using a generalization of a well known theorem by I. Kaplansky, we generalize the method used in that paper in order to obtain, for each integer n>1, a new abelian model structure on Ch(R), where the class of cofibrant objects is the class of chain complexes whose terms have projective dimension smaller or equal than n (dw-n-projective complexes), provided the ring R is noetherian. We also present another method, based on the paper "Covers and Envelopes in Grothendieck Categories: Flat Covers of Complexes with Applications", to construct this model structure. This method also works to construct an abelian model structure whose cofibrant objects are the dw-n-flat complexes.