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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Classifying finite localizations of quasicoherent sheaves

Author: G. Garkusha
Translated by: the author
Original publication: Algebra i Analiz, tom 21 (2009), nomer 3.
Journal: St. Petersburg Math. J. 21 (2010), 433-458
MSC (2000): Primary 14A15, 18F20
Published electronically: February 26, 2010
MathSciNet review: 2588764
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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

$\displaystyle (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.

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Additional Information

G. Garkusha
Affiliation: Department of Mathematics, Swansea University, Singleton Park, SA2 8PP Swansea, United Kingdom

Keywords: Quasicompact, quasiseparated schemes, quasicoherent sheaves, localizing subcategories, thick subcategories
Received by editor(s): July 20, 2008
Published electronically: February 26, 2010
Dedicated: In memory of Vera Puninskaya
Article copyright: © Copyright 2010 American Mathematical Society