Classifying finite localizations of quasicoherent sheaves

Abstract:

Given a quasicompact, quasiseparated scheme $X$, a bijection between the tensor localizing subcategories of finite type in $\operatorname {Qcoh}(X)$ and the set of all subsets $Y\subseteq X$ of the form $Y=\bigcup _{i\in \Omega }Y_i$, with $X\setminus Y_i$ quasicompact and open for all $i\in \Omega$, is established. As an application, an isomorphism of ringed spaces \[ (X,\mathcal {O}_X)\overset {\sim }{\longrightarrow } (\sf {spec}(\operatorname {Qcoh}(X)), \mathcal {O}_{\operatorname {Qcoh}(X)}) \] is constructed, where $(\sf {spec}(\operatorname {Qcoh}(X)), \mathcal {O}_{\operatorname {Qcoh}(X)})$ is a ringed space associated with the lattice of tensor localizing subcategories of finite type. Also, a bijective correspondence between the tensor thick subcategories of perfect complexes $\mathcal {D}_{\operatorname {per}}(X)$ and the tensor localizing subcategories of finite type in $\operatorname {Qcoh}(X)$ is established.