Hochschild cohomology of deformation quantizations over $\mathbb{Z}/p^n\mathbb{Z}$

TL;DR

This paper establishes a deep connection between Hochschild cohomology of deformation quantizations over p-adic integers and de Rham-Witt complexes for affine smooth symplectic varieties over finite fields, providing explicit computations of centers.

Contribution

It proves that Hochschild cohomology of deformation quantizations over p-adic integers aligns with de Rham-Witt complexes and computes centers of certain affine Poisson variety deformations.

Findings

Hochschild cohomology is isomorphic to de Rham-Witt complex for all n.

Centers of deformations of specific affine Poisson varieties are explicitly computed.

The result links deformation quantization with p-adic cohomological structures.

Abstract

Let X be a an affine smooth symplectic variety over $\mathbb{Z}/p\mathbb{Z},$ and A be its deformation quantization over the p-adic integers. We prove that for all $n\geq 1,$ the Hochschild cohomogy of $A/p^nA$ is isomorphic to the de Rham-Witt complex of X over $\mathbb{Z}/p^n\mathbb{Z}$. We also compute centers of deformations of certain affine Poisson varieties over $\mathbb{Z}/p\mathbb{Z}.$