A PI degree theorem for quantum deformations

TL;DR

This paper establishes a PI degree theorem for quantum deformations, showing that such deformations are commutative in characteristic zero and have PI degree a power of p in positive characteristic, extending to filtered deformations.

Contribution

It proves a PI degree theorem for quantum formal deformations of commutative domains, linking PI degree to the characteristic of the base field, and extends results to filtered deformations.

Findings

Deformations are commutative in characteristic zero.

PI degree is a power of p in positive characteristic.

Results apply to filtered deformations.

Abstract

Let $F$ be an algebraically closed field. We show that if a quantum formal deformation $A$ of a commutative domain $A_0$ over $F$ is a PI algebra, then $A$ is commutative if ${\rm char}(F)=0$, and has PI degree a power of $p$ if ${\rm char}(F)=p>0$. This implies the same result for filtered deformations (i.e., filtered algebras $A$ such that ${\rm gr}(A)=A_0$).