Galois bimodules and integrality of PI comodule algebras over invariants

TL;DR

This paper proves that under certain conditions, a comodule algebra over a finite dimensional Hopf algebra is integral over its invariants, extending previous results to noncommutative PI algebras using Galois bimodule theory.

Contribution

It generalizes integrality results for comodule algebras to noncommutative PI algebras by developing Galois bimodule theory over semisimple algebras.

Findings

A comodule algebra is integral over invariants under specified conditions.

Extension of Skryabin's results to noncommutative PI algebras.

Development of Galois bimodule theory over semisimple algebras.

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. This generalizes the results of S. Skryabin in the case when A is commutative. For the proof, we develop a theory of Galois bimodules over semisimple algebras finite over the center.