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

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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.

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##### Keywords
colocalizing subcategory, cosupport, local homology
##### Mathematical Subject Classification 2010
Primary: 13D09, 13D45, 55P60
##### Milestones
Received: 25 October 2017
Revised: 23 February 2018
Accepted: 26 February 2018
Published: 16 July 2018
##### Authors
 Tsutomu Nakamura Graduate School of Natural Science and Technology Okayama University Okayama Japan Yuji Yoshino Graduate School of Natural Science and Technology Okayama University Tsushima-Naka Okayama Japan