Classifying finite localizations of quasi-coherent sheaves

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.