Relative cohomology of complexes II: Vanishing of relative cohomology

Abstract

Let R be a ring such that (GP,GP⊥) forms a cotorsion pair cogenerated by a set, where GP denotes the category of all Gorenstein projective R-modules. Recently, the first author defined for any complex X the relative cohomology functors ExtGP⁎(X,−) as H−⁎(Hom(G,−)) in which G is a special Gorenstein projective precover of X. In the present paper, we introduce the dimension of X related to Gorenstein projective precovers, and show that such a dimension of X is equal to the least integer n for which ExtGPi(X,Q)=0 for all i>n and all R-modules Q∈GP⊥. This result gives a “Gorenstein” version of the relationship between the projective dimension of complexes introduced by Avramov and Foxby and the absolutely cohomology functors ExtR⁎(−,−).