Absolutely Clean, Level, and Gorenstein AC-Injective Complexes

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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 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.