Let R be a commutative Noetherian ring. We introduce the notion of localization functors λW with cosupports in arbitrary subsets W of SpecR; 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 λW, including an explicit way to calculate λW 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 R-module is at most the Krull dimension of R. As another application, it is possible to give a functorial way to replace complexes of flat R-modules or complexes of finitely generated R-modules by complexes of pure-injective R-modules.
|Number of pages||31|
|Journal||Pacific Journal of Mathematics|
|Publication status||Published - Jan 1 2018|
- Colocalizing subcategory
- Local homology
ASJC Scopus subject areas