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

 Recent Issues Vol. 307: 1  2 Vol. 306: 1  2 Vol. 305: 1  2 Vol. 304: 1  2 Vol. 303: 1  2 Vol. 302: 1  2 Vol. 301: 1  2 Vol. 300: 1  2 Online Archive Volume: Issue:
 The Journal Editorial Board Subscriptions Officers Special Issues Submission Guidelines Submission Form Contacts ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Author Index To Appear Other MSP Journals
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.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/pjm

We have not been able to recognize your IP address 18.208.202.194 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.