### Abstract

Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ^{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 λ^{W}, including an explicit way to calculate λ^{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.

Original language | English |
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Pages (from-to) | 405-435 |

Number of pages | 31 |

Journal | Pacific Journal of Mathematics |

Volume | 296 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2018 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 296, no. 2, pp. 405-435. https://doi.org/10.2140/pjm.2018.296.405

}

TY - JOUR

T1 - Localization functors and cosupport in derived categories of commutative Noetherian rings

AU - Nakamura, Tsutomu

AU - Yoshino, Yuji

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Colocalizing subcategory

KW - Cosupport

KW - Local homology

UR - http://www.scopus.com/inward/record.url?scp=85050363788&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050363788&partnerID=8YFLogxK

U2 - 10.2140/pjm.2018.296.405

DO - 10.2140/pjm.2018.296.405

M3 - Article

AN - SCOPUS:85050363788

VL - 296

SP - 405

EP - 435

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -