The structure of smooth algebras in Kapranov's framework for noncommutative geometry

In [M. Kapranov, Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998) 73-118] a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the noncommutative coordinate rings of affine open subsets of such of those varieties w...

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Detalles Bibliográficos
Publicado: 2004
Materias:
Commutator filtration
d-smooth algebra
Poisson algebra
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v281_n2_p679_Cortinas
http://hdl.handle.net/20.500.12110/paper_00218693_v281_n2_p679_Cortinas
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paper:paper_00218693_v281_n2_p679_Cortinas2023-06-08T14:42:21Z The structure of smooth algebras in Kapranov's framework for noncommutative geometry Commutator filtration d-smooth algebra Poisson algebra In [M. Kapranov, Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998) 73-118] a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the noncommutative coordinate rings of affine open subsets of such of those varieties which are smooth (Theorem 3.4). The theorem describes the local ring of a point as a truncation of a quantization of the enveloping Poisson algebra of a smooth commutative local algebra. An explicit description of this quantization is given in Theorem 2.5. A description of the A-module structure of the Poisson envelope of a smooth commutative algebra A was given in loc. cit., Theorem 4.1.3. However the proof given in loc. cit. has a gap. We fix this gap for A local (Theorem 1.4) and prove a weaker global result (Theorem 1.6). © 2004 Elsevier Inc. All rights reserved. 2004 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v281_n2_p679_Cortinas http://hdl.handle.net/20.500.12110/paper_00218693_v281_n2_p679_Cortinas
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Commutator filtration
d-smooth algebra
Poisson algebra
spellingShingle Commutator filtration
d-smooth algebra
Poisson algebra
The structure of smooth algebras in Kapranov's framework for noncommutative geometry
topic_facet Commutator filtration
d-smooth algebra
Poisson algebra
description In [M. Kapranov, Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998) 73-118] a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the noncommutative coordinate rings of affine open subsets of such of those varieties which are smooth (Theorem 3.4). The theorem describes the local ring of a point as a truncation of a quantization of the enveloping Poisson algebra of a smooth commutative local algebra. An explicit description of this quantization is given in Theorem 2.5. A description of the A-module structure of the Poisson envelope of a smooth commutative algebra A was given in loc. cit., Theorem 4.1.3. However the proof given in loc. cit. has a gap. We fix this gap for A local (Theorem 1.4) and prove a weaker global result (Theorem 1.6). © 2004 Elsevier Inc. All rights reserved.
title The structure of smooth algebras in Kapranov's framework for noncommutative geometry
title_short The structure of smooth algebras in Kapranov's framework for noncommutative geometry
title_full The structure of smooth algebras in Kapranov's framework for noncommutative geometry
title_fullStr The structure of smooth algebras in Kapranov's framework for noncommutative geometry
title_full_unstemmed The structure of smooth algebras in Kapranov's framework for noncommutative geometry
title_sort structure of smooth algebras in kapranov's framework for noncommutative geometry
publishDate 2004
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v281_n2_p679_Cortinas
http://hdl.handle.net/20.500.12110/paper_00218693_v281_n2_p679_Cortinas
_version_ 1768542924443746304

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