Hochschild Cohomology of Deformation Quantizations over ℤ/p n ℤ

Abstract

Let A 1 be an Azumaya algebra over a smooth affine symplectic variety X over Spec F p , where p is an odd prime. Let A be a deformation quantization of A 1 over the p-adic integers. In this note we show that for all n ≥ 1, the Hochschild cohomology of A/p n A is isomorphic to the de Rham-Witt complex \(W_{n}{\Omega }^{\ast }_{X}\) of X over \(\mathbb {Z}/p^{n}\mathbb {Z}\). We also compute the center of deformations of certain affine Poisson varieties over F p .