On K-absolutely pure complexes

Abstract

In this paper, we examine the class of K-absolutely pure complexes. These are the complexes which are right orthogonal in the homotopy category K(R) to the acyclic complexes of pure-projective modules. The class K-abspure of these complexes is preenveloping in K(R); in fact, a Bousfield localization exists for the embedding K-abspure⊆K(R) and the quotient K(R)/K-abspure is equivalent to the homotopy category of acyclic complexes of pure-projective modules. We examine the role of K-absolutely pure complexes in the pure derived category Dpure(R) and show that K-abspure is the isomorphic closure of the class of K-injective complexes therein. We explore the relevance of strongly fp-injective modules in the study of K-absolutely pure complexes and characterize the K-absolutely pure complexes of strongly fp-injective modules. Finally, we show that a K-absolutely pure complex of strongly fp-injective modules admits a K-injective complex of injective modules as a K(PInj)-preenvelope, in the case where the ring is left coherent. The notion of K-absolute purity is dual to the notion of K-flatness in the homotopy category, in a way analogous to the duality between (strongly) fp-injectivity and flatness in the module category.