The homotopy category of injectives

Abstract

Krause studied the homotopy category $K\left(Inj\mathcal{A}\right)$ of complexes of injectives in a locally noetherian Grothendieck abelian category $\mathcal{A}$. Because $\mathcal{A}$ is assumed locally noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category $K\left(Inj\mathcal{A}\right)$ has coproducts. It turns out that $K\left(Inj\mathcal{A}\right)$ is compactly generated, and Krause studies the relation between the compact objects in $K\left(Inj\mathcal{A}\right)$, the derived category $D\left(\mathcal{A}\right)$, and the category $K_{ac}\left(Inj\mathcal{A}\right)$ of acyclic objects in $K\left(Inj\mathcal{A}\right)$.

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 $\mathcal{A}$, the category $K\left(Inj\mathcal{A}\right)$ has coproducts and is $\mu$-compactly generated for some sufficiently large $\mu$.

The existence of coproducts follows easily from a result of Krause: the point is that the natural inclusion of $K\left(Inj\mathcal{A}\right)$ into $K\left(\mathcal{A}\right)$ 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 $\mu$-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 $D\left(\mathcal{A}\right)$ is not compactly generated. I believe this is the first known example of such a thing.