Classifying finite localizations of quasicoherent sheaves. (English. Russian original) Zbl 1211.14008
St. Petersbg. Math. J. 21, No. 3, 433-458 (2010); translation from Algebra Anal. 21, No. 3, 93-129 (2009).
In his famous thesis on abelian categories, P. Gabriel [cf. Bull. Soc. Math. France 90, 323–448 (1962; Zbl 0201.35602)] showed that a Noetherian scheme can be reconstructed up to an isomorphism from its abelian category of quasicoherent sheaves. This theorem has later been generalized by Rosenberg and others and has ever since been a source of inspiration in different areas of mathematics. In particular, it established a new perspective to think of all kinds of spaces in terms of their categories of quasicoherent modules over them. Since these notions also make sense in noncommutative settings, this led to interesting new conceptions for noncommutative spaces.
In the paper under review, the author reconsiders these types of classification of schemes by its category of quasicoherent sheaves and slightly generalizes some known results in the commutative algebraic setting. More precisely, in the main theorem of the paper, he shows the following correspondence for any quasicompact and quasiseparated scheme \(X\). There is a bijection between the set of all subsets of the form \(V=\bigcup_i V_i\) such that each \(V_i\) has a quasicompact open complement in \(X\) and the set of all tensor localizing subcategories of finite type in the category of quasicoherent sheaves over \(X\). As an application, the author obtains a reconstruction theorem that shows that there is an isomorphism of ringed spaces between \(X\) and a ringed space that is associated to the lattice of tensor localizing subcategories of finite type in the category of quasicoherent sheaves over \(X\).
In the paper under review, the author reconsiders these types of classification of schemes by its category of quasicoherent sheaves and slightly generalizes some known results in the commutative algebraic setting. More precisely, in the main theorem of the paper, he shows the following correspondence for any quasicompact and quasiseparated scheme \(X\). There is a bijection between the set of all subsets of the form \(V=\bigcup_i V_i\) such that each \(V_i\) has a quasicompact open complement in \(X\) and the set of all tensor localizing subcategories of finite type in the category of quasicoherent sheaves over \(X\). As an application, the author obtains a reconstruction theorem that shows that there is an isomorphism of ringed spaces between \(X\) and a ringed space that is associated to the lattice of tensor localizing subcategories of finite type in the category of quasicoherent sheaves over \(X\).
Reviewer: Gereon Quick (Cambridge)
MSC:
| 14A15 | Schemes and morphisms |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
Citations:
Zbl 0201.35602References:
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