The homotopy category of injectives

Abstract

Krause studied the homotopy category K.(Inj A) of complexes of injectives in a locally noetherian Grothendieck abelian category A. Because A is assumed locally noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category K.InjA/ has coproducts. It turns out that K. (Inj A) is compactly generated, and Krause studies the relation between the compact objects in K. (Inj A)/, the derived category D.A/, and the category Kac. (Inj A) of acyclic objects in K. (Inj A). We wish to understand what happens in the nonnoetherian case, and this paper begins the study. We prove that, for an arbitrary Grothendieck abelian category A, the category K. (Inj A) has coproducts and is μ-compactly generated for some sufficiently large μ. The existence of coproducts follows easily from a result of Krause: the point is that the natural inclusion of K. (Inj A) into K.A has a left adjoint and the existence of coproducts is a formal corollary. But in order to prove anything about these coproducts, for example the μ-compact generation, we need to have a handle on this adjoint. Also interesting is the counterexample at the end of the article: we produce a locally noetherian Grothendieck abelian category in which products of acyclic complexes need not be acyclic. It follows that D.A is not compactly generated. I believe this is the first known example of such a thing.