Let
be a commutative Noetherian ring. We introduce the notion of localization functors
with cosupports in
arbitrary subsets
of
; it
is a common generalization of localizations with respect to multiplicatively
closed subsets and left derived functors of ideal-adic completion
functors. We prove several results about the localization functors
, including an explicit
way to calculate
using the notion of Čech complexes. As an application, we can give a simpler proof of a
classical theorem by Gruson and Raynaud, which states that the projective dimension of a flat
-module is at most the
Krull dimension of
.
As another application, it is possible to give a functorial way to replace complexes of flat
-modules or complexes of finitely
generated
-modules by complexes
of pure-injective
-modules.
Keywords
colocalizing subcategory, cosupport, local homology
