Rational group actions on affine PI-algebras

TL;DR

This paper investigates the structure of affine PI-algebras under rational actions of algebraic groups, establishing finiteness conditions for G-prime ideals and extending classical results to the PI setting.

Contribution

It extends known results on algebraic group actions to affine PI-algebras, including finiteness of G-primes and catenarity theorems, under new hypotheses.

Findings

R has finitely many G-prime ideals iff the G-action on its center is multiplicity free

Extension of classical results on affine algebraic G-varieties to PI-algebras

A PI-version of Schelter's catenarity theorem for G-primes

Abstract

Let R be an affine PI-algebra over an algebraically closed field k and let G be an affine algebraic k-group that acts rationally by algebra automorphisms on R. For R prime and G a torus, we show that R has only finitely many G-prime ideals if and only if the action of G on the center of R is multiplicity free. This extends a standard result on affine algebraic G-varieties. Under suitable hypotheses on R and G, we also prove a PI-version of a well-known result on spherical varieties and a version of Schelter's catenarity theorem for G-primes.