# 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 language Undefined/Unknown Pacific J. Math. https://doi.org/10.2140/pjm.2018.296.405 Published - Oct 24 2017

### Keywords

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

### Cite this

In: Pacific J. Math., 24.10.2017.

Research output: Contribution to journalArticle

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