Absolutely Clean, Level, and Gorenstein AC-Injective Complexes

Abstract

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 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.

Gorenstein AC-projective complexes

“…On the other hand, the dual Gorenstein AC-injectives are the fibrant objects of another model structure on R-Mod. These concepts were extended to the category Ch(R), of chain complexes of R-modules, in [BG16]. In particular, the Gorenstein AC-injective and Gorenstein AC-projective chain complexes were studied, and, the Gorenstein AC-injective complexes were shown to be the fibrant objects of a (cofibrantly generated) abelian model structure on Ch(R).…”

“…In particular, the Gorenstein AC-injective and Gorenstein AC-projective chain complexes were studied, and, the Gorenstein AC-injective complexes were shown to be the fibrant objects of a (cofibrantly generated) abelian model structure on Ch(R). However, as noted in the introduction to [BG16], the Gorenstein ACprojective model structure was not constructed there; it is much more technical to construct. It is the purpose of this paper to give this construction to complete the work in [BG16].…”

“…However, as noted in the introduction to [BG16], the Gorenstein ACprojective model structure was not constructed there; it is much more technical to construct. It is the purpose of this paper to give this construction to complete the work in [BG16].…”

“…It was shown in [BG16,Theorem 4.13] that X is Gorenstein AC-projective if and only if each X n is a Gorenstein AC-projective R-module and any chain map f : X → L is null homotopic whenever L is a level complex. For most rings commonly occurring in practice, the Gorenstein AC-projective complexes coincide with the usual Gorenstein projective complexes of [GR99], or at least with the Ding projective complexes of [YLL13,Section 3].…”

“…The result proved, Theorem 6.4, says a few things about the model structure of Theorem 1.2. First, the identify functor from it to the Gorenstein ACinjective model structure of [BG16] is a Quillen equivalence in this case. Second, the model structure is finitely generated and so it follows from a result of Hovey [Hov99,Corollary 7.4.4] that the associated homotopy category is compactly generated.…”