Localization in Categories of Complexes and Unbounded Resolutions

Abstract

AbstractIn this paper we show that for a Grothendieck category and a complex E in C() there is an associated localization endofunctor ℓ in D(). This means that ℓ is idempotent (in a natural way) and that the objects that go to 0 by ℓ are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of D() that contains E. As applications, we construct K-injective resolutions for complexes of objects of and derive Brown representability for D() from the known result for D(R-mod), where R is a ring with unit.