#### Vol. 296, No. 2, 2018

 Recent Issues Vol. 296: 1  2 Vol. 295: 1  2 Vol. 294: 1  2 Vol. 293: 1  2 Vol. 292: 1  2 Vol. 291: 1  2 Vol. 290: 1  2 Vol. 289: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear ISSN: 0030-8730
Localization functors and cosupport in derived categories of commutative Noetherian rings

### Tsutomu Nakamura and Yuji Yoshino

Vol. 296 (2018), No. 2, 405–435
##### Abstract

Let $R$ be a commutative Noetherian ring. We introduce the notion of localization functors ${\lambda }^{W}$ 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 ${\lambda }^{W}$, including an explicit way to calculate ${\lambda }^{W}$ 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.

##### Keywords
colocalizing subcategory, cosupport, local homology
##### Mathematical Subject Classification 2010
Primary: 13D09, 13D45, 55P60