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

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.

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