Absolutely Clean, Level, and Gorenstein AC-Injective Complexes

TL;DR

This paper extends the concepts of absolutely clean, level, and Gorenstein AC-injective modules to chain complexes, establishing model structures where Gorenstein AC-injective complexes are fibrant objects.

Contribution

It introduces and characterizes Gorenstein AC-injective and projective complexes, and constructs a cofibrantly generated model structure on chain complexes with these as fibrant objects.

Findings

Gorenstein AC-injective complexes characterized in chain complexes

A model structure on chain complexes with Gorenstein AC-injective complexes as fibrant objects

Extension of Gorenstein homological algebra to chain complexes

Abstract

Absolutely clean and level $R$-modules were introduced in [BGH13] and used to show how Gorenstein homological algebra can be extended to an arbitrary ring $R$. This led to the notion of Gorenstein AC-injective and Gorenstein AC-projective $R$-modules. Here we study these concepts in the category of chain complexes of $R$-modules. We define, characterize and deduce properties of absolutely clean, level, Gorenstein AC-injective, and Gorenstein AC-projective chain complexes. We show that the category $\text{Ch}(R)$ of chain complexes has a cofibrantly generated model structure where every object is cofibrant and the fibrant objects are exactly the Gorenstein AC-injective chain complexes.