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. 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.