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

JOURNAL OF ALGEBRA

卷:281

Article

Cortiñas, G

关键词: commutator filtration;    poisson algebra;    d-smooth algebra;   

DOI  :  10.1016/j.jalgebra.2004.08.002

来源: Elsevier

【 摘 要 】

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). (C) 2004 Elsevier Inc. All rights reserved.

【 授权许可】

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