### Abstract

Original language | Undefined/Unknown |
---|---|

Journal | Pacific J. Math. |

DOIs | |

Publication status | Published - 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.

Research output: Contribution to journal › Article

}

TY - JOUR

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

AU - Nakamura, Tsutomu

AU - Yoshino, Yuji

N1 - 26 pages, to appear in Pacific J. Math

PY - 2017/10/24

Y1 - 2017/10/24

N2 - 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.

AB - 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.

KW - math.AC

KW - math.AG

KW - 13D09, 13D45, 55P60

U2 - 10.2140/pjm.2018.296.405

DO - 10.2140/pjm.2018.296.405

M3 - Article

JO - Pacific J. Math.

JF - Pacific J. Math.

ER -