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.
Documento: | Artículo |
Título: | The structure of smooth algebras in Kapranov's framework for noncommutative geometry |
Autor: | Cortiñas, G. |
Filiación: | Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, Pabellón 1, 1428 Buenos Aires, Argentina Depto. de Algebra/Geometria/Topol., Facultad de Ciencias, Universidad de Valladolid, Prado de la Magdalena s/n, 4705 Valladolid, Spain |
Palabras clave: | Commutator filtration; d-smooth algebra; Poisson algebra |
Año: | 2004 |
Volumen: | 281 |
Número: | 2 |
Página de inicio: | 679 |
Página de fin: | 694 |
DOI: | http://dx.doi.org/10.1016/j.jalgebra.2004.08.002 |
Título revista: | Journal of Algebra |
Título revista abreviado: | J. Algebra |
ISSN: | 00218693 |
CODEN: | JALGA |
Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v281_n2_p679_Cortinas |