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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 \[ (X,\mathcal {O}_X)\overset {\sim }{\longrightarrow } (\sf {spec}(\operatorname {Qcoh}(X)), \mathcal {O}_{\operatorname {Qcoh}(X)}) \] 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
MR Author ID: 660286
ORCID: 0000-0001-9836-0714

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