Galois bimodules and integrality of PI comodule algebras over invariants

Abstract

Let $A$ be a comodule algebra for a finite dimensional Hopf algebra $K$ over an algebraically closed field $k$, and let $A^K$ be the subalgebra of invariants. Let $Z$ be a central subalgebra in $A$, which is a domain with quotient field $Q$. Assume that $Q{\otimes }_{Z}A$ is a central simple algebra over $Q$, and either $A$ is a finitely generated torsion-free $Z$-module and $Z$ is integrally closed in $Q$, or $A$ is a finite projective $Z$-module. Then we show that $A$ and $Z$ are integral over the subring of central invariants $Z\cap A^K$. More generally, we show that this statement is valid under the same assumptions if $Z$ is a reduced algebra with quotient ring $Q$,and $Q{\otimes }_{Z}A$ is a semisimple algebra with center $Q$. In particular, the statement holds for a coaction of $K$ on a prime PI algebra $A$ whose center $Z$ is an integrally closed finitely generated domain over $k$. For the proof, we develop a theory of Galois bimodules over semisimple algebras finite over the center.