Classifying finite localizations of quasicoherent sheaves
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G. Garkusha
Translated by: the author - St. Petersburg Math. J. 21 (2010), 433-458
- DOI: https://doi.org/10.1090/S1061-0022-10-01102-7
- Published electronically: February 26, 2010
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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.References
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Bibliographic 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
- Email: G.Garkusha@swansea.ac.uk
- Received by editor(s): July 20, 2008
- Published electronically: February 26, 2010
- © Copyright 2010 American Mathematical Society
- Journal: St. Petersburg Math. J. 21 (2010), 433-458
- MSC (2000): Primary 14A15, 18F20
- DOI: https://doi.org/10.1090/S1061-0022-10-01102-7
- MathSciNet review: 2588764
Dedicated: In memory of Vera Puninskaya