English

Classifying finite localizations of quasi-coherent sheaves

Algebraic Geometry 2007-08-14 v1 K-Theory and Homology

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 YXY\subseteq X of the form Y=iΩYiY=\bigcup_{i\in\Omega}Y_i, with XYiX\setminus Y_i quasi-compact and open for all iΩi\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)),OQcoh(X))(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)\perf(X) and the tensor localizing subcategories of finite type in Qcoh(X) is established.

Keywords

Cite

@article{arxiv.0708.1622,
  title  = {Classifying finite localizations of quasi-coherent sheaves},
  author = {Grigory Garkusha},
  journal= {arXiv preprint arXiv:0708.1622},
  year   = {2007}
}