Localization functors and cosupport in derived categories of commutative Noetherian rings

TL;DR

This paper introduces generalized localization functors with cosupports in derived categories of Noetherian rings, providing explicit calculations and applications to projective dimension and pure-injective replacements.

Contribution

It develops a unified framework for localization functors with cosupports, extending classical notions and offering new computational and theoretical tools.

Findings

Explicit calculation method for localization functors using Cech complexes

Simplified proof of the bound on projective dimension of flat modules

Functorial replacement of complexes with pure-injective modules

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.