Absolutely Clean, Level, and Gorenstein AC-Injective Complexes

摘要

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