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 $\lambda^W$ with cosupports in arbitrary subsets $W$ of $\text{Spec}\, R$; 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 $\lambda^W$ by the notion of Cech 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 languageUndefined/Unknown
JournalPacific J. Math.
DOIs
Publication statusPublished - Oct 24 2017

Keywords

  • math.AC
  • math.AG
  • 13D09, 13D45, 55P60

Cite this

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

In: Pacific J. Math., 24.10.2017.

Research output: Contribution to journalArticle

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