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On the local structure of quantizations in characteristic p

Let A be a central quantization of an affine Poisson variety X over a field of characteristic p>0 . We show that the completion of A with repect to a closed point y∈X is isomorphic to the tensor product of the Weyl algebra with a local Poisson algebra. This result can be thought of as a positive characteristic analogue of results of Losev and Kaledin about slice algebras of quantizations in characteristic 0.