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Derived noncommutative schemes, geometric realizations, and finite dimensional algebras
D. O. Orlov
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined as differential graded categories of special type. Different properties of both noncommutative schemes and morphisms between them are reviewed and discussed. In addition, the concept of a geometric realization for a derived noncommutative scheme is introduced and problems of existence and construction of such realizations are discussed. Also studied are the construction of gluings of noncommutative schemes via morphisms, along with certain new phenomena such as phantoms, quasi-phantoms, and Krull–Schmidt partners which arise in the world of noncommutative schemes and which enable us to find new noncommutative schemes. The last sections consider noncommutative schemes connected with basic finite-dimensional algebras. It is proved that such noncommutative schemes have special geometric realizations under which the algebra goes to a vector bundle on a smooth projective scheme. Such realizations are constructed in two steps, the first of which is the well-known construction of Auslander, while the second step is connected with the new concept of a well-formed quasi-hereditary algebra, for which there are very special geometric realizations sending standard modules to line bundles.
Bibliography: 50 titles.
differential graded categories, triangulated categories, derived noncommutative schemes, finite-dimensional algebras, geometric realizations.
|Russian Science Foundation
|This work was supported by the Russian Science Foundation under grant no. 14-50-00005.
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Russian Mathematical Surveys, 2018, 73:5, 865–918
MSC: 14A22, 14F05, 16E45, 18E30
D. O. Orlov, “Derived noncommutative schemes, geometric realizations, and finite dimensional algebras”, Uspekhi Mat. Nauk, 73:5(443) (2018), 123–182; Russian Math. Surveys, 73:5 (2018), 865–918
Citation in format AMSBIB
\paper Derived noncommutative schemes, geometric realizations, and finite dimensional algebras
\jour Uspekhi Mat. Nauk
\jour Russian Math. Surveys
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D. O. Orlov, “Finite-dimensional DG algebras and their properties”, Russian Math. Surveys, 74:4 (2019), 764–766
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