### 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 |

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### Keywords

- Colocalizing subcategory
- Cosupport
- Local homology

### ASJC Scopus subject areas

- Mathematics(all)