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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Autor principal: Cortiñas, G.
Formato: JOUR
Materias:
Commutator filtration
d-smooth algebra
Poisson algebra
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00218693_v281_n2_p679_Cortinas
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00218693_v281_n2_p679_Cortinas2023-10-03T14:21:21Z The structure of smooth algebras in Kapranov's framework for noncommutative geometry Cortiñas, G. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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
Cortiñas, G.
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.
format JOUR
author Cortiñas, G.
author_facet Cortiñas, G.
author_sort Cortiñas, G.
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
url http://hdl.handle.net/20.500.12110/paper_00218693_v281_n2_p679_Cortinas
work_keys_str_mv AT cortinasg thestructureofsmoothalgebrasinkapranovsframeworkfornoncommutativegeometry
AT cortinasg structureofsmoothalgebrasinkapranovsframeworkfornoncommutativegeometry
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