Localization functors and cosupport in derived categories of commutative Noetherian rings

Tsutomu Nakamura, Yuji Yoshino

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish
Pages (from-to)405-435
Number of pages31
JournalPacific Journal of Mathematics
Volume296
Issue number2
DOIs
Publication statusPublished - Jan 1 2018

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Derived Category
Noetherian Ring
Commutative Ring
Functor
Module
Projective Dimension
Krull Dimension
Subset
Injective
Finitely Generated
Completion
Calculate
Closed
Arbitrary
Theorem

Keywords

  • Colocalizing subcategory
  • Cosupport
  • Local homology

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Localization functors and cosupport in derived categories of commutative Noetherian rings. / Nakamura, Tsutomu; Yoshino, Yuji.

In: Pacific Journal of Mathematics, Vol. 296, No. 2, 01.01.2018, p. 405-435.

Research output: Contribution to journalArticle

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