Classifying finite localizations of quasi-coherent sheaves

TL;DR

This paper establishes a correspondence between certain subcategories of quasi-coherent sheaves on a scheme and subsets of the scheme, leading to a new understanding of the scheme's structure via ringed spaces.

Contribution

It introduces a bijection between tensor localizing subcategories of finite type in Qcoh(X) and specific subsets of X, and constructs an isomorphism of ringed spaces linking these concepts.

Findings

Bijection between tensor localizing subcategories and subsets of X.

Construction of an isomorphism of ringed spaces.

Correspondence between tensor thick subcategories and localizing subcategories.

Abstract

Given a quasi-compact, quasi-separated scheme X, a bijection between the tensor localizing subcategories of finite type in Qcoh(X) and the set of all subsets $Y\subseteq X$ of the form $Y=\bigcup_{i\in\Omega}Y_i$, with $X\setminus Y_i$ quasi-compact and open for all $i\in\Omega$, is established. As an application, there is constructed an isomorphism of ringed spaces (X,O_X)-->(Spec(Qcoh(X)),O_{Qcoh(X)}), where $(Spec(Qcoh(X)),O_{Qcoh(X)})$ is a ringed space associated to the lattice of tensor localizing subcategories of finite type. Also, a bijective correspondence between the tensor thick subcategories of perfect complexes $\perf(X)$ and the tensor localizing subcategories of finite type in Qcoh(X) is established.