Localization functors and cosupport in derived categories of commutative Noetherian rings

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.