Let R be a commutative Noetherian ring. We introduce the notion of localization functors W 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 W, including an explicit way to calculate W by the notion of Cech 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.
MSC Codes 13D09, 13D45, 55P60
|Publication status||Published - Oct 24 2017|
- colocalizing subcategory
- local homology.
ASJC Scopus subject areas