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

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

2000-02-22

arxiv.org/abs/math/0002177v1

J. of Algebra 281 (2004) 679-694

Mathematics

Rings and Algebras

13 pages, AmsTex, uses xypic for diagrams

Scientific paper

In Kapranov, M. {\it Noncommutative geometry based on commutator expansions,} J. reine angew. Math {\bf 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 descripition 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).

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