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

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).