Ideals in deformation quantizations over $\mathbb{Z}/p^n\mathbb{Z}$

TL;DR

This paper investigates deformation quantizations of Azumaya algebras over smooth affine symplectic varieties in characteristic p, showing that flat two-sided ideals are generated by central elements, revealing structural properties of such quantizations.

Contribution

It establishes that in deformation quantizations over rac{p^n}{p^n}ractions, all flat two-sided ideals are centrally generated, a new structural insight.

Findings

Flat two-sided ideals are generated by central elements.

Deformation quantizations preserve certain ideal structures.

Results apply to smooth affine symplectic varieties over rac{p}{p}ractions.

Abstract

Let $X$ be a smooth affine symplectic variety over $\mathbb{Z}/p\mathbb{Z},$ and let $A_1$ be an Azumaya algebra over $X.$ In this note we show that if $A$ is a deformation quantization of $A_1$ over $\mathbb{Z}/p^n\mathbb{Z}$, then any two-sided $\mathbb{Z}/p^n\mathbb{Z}$-flat ideal in $A$ is generated by central elements.