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Russian Mathematical Surveys, 2018, Volume 73, Issue 5, Pages 865–918
DOI: https://doi.org/10.1070/RM9844
(Mi rm9844)
 

This article is cited in 5 scientific papers (total in 5 papers)

Derived noncommutative schemes, geometric realizations, and finite dimensional algebras

D. O. Orlov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
References:
Abstract: 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.
Keywords: differential graded categories, triangulated categories, derived noncommutative schemes, finite-dimensional algebras, geometric realizations.
Funding agency Grant number
Russian Science Foundation 14-50-00005
This work was supported by the Russian Science Foundation under grant no. 14-50-00005.
Received: 20.07.2018
Bibliographic databases:
Document Type: Article
UDC: 512.7
Language: English
Original paper language: Russian
Citation: D. O. Orlov, “Derived noncommutative schemes, geometric realizations, and finite dimensional algebras”, Russian Math. Surveys, 73:5 (2018), 865–918
Citation in format AMSBIB
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\by D.~O.~Orlov
\paper Derived noncommutative schemes, geometric realizations, and finite dimensional algebras
\jour Russian Math. Surveys
\yr 2018
\vol 73
\issue 5
\pages 865--918
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Linking options:
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  • https://doi.org/10.1070/RM9844
  • https://www.mathnet.ru/eng/rm/v73/i5/p123
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
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    Abstract page:700
    Russian version PDF:150
    English version PDF:24
    References:56
    First page:33
     
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