Krause studied the homotopy category
of complexes of injectives in a locally noetherian Grothendieck abelian category
. Because
is assumed locally
noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category
has coproducts.
It turns out that
is compactly generated, and Krause studies the relation between the compact objects in
, the derived
category
, and
the category
of
acyclic objects in
.
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 , the category
has coproducts and is
-compactly generated for
some sufficiently large
.
The existence of coproducts follows easily from a result of Krause: the point is that the natural
inclusion of
into
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
-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
is not
compactly generated. I believe this is the first known example of such a
thing.
Keywords
locally presentable categories, well generated categories,
homotopy categories of injectives
Centre for Mathematics and its
Applications
Mathematical Sciences Institute
Australian National University
John Dedman Building 27
Canberra ACT 0200
Australia
