Absolutely Clean, Level, and Gorenstein AC-Injective Complexes

Abstract

Absolutely clean and level R-modules were introduced in [2 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring. Submitted, arXiv:1405.5768 (under review). [Google Scholar]] 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 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.