Galois bimodules and integrality of PI comodule algebras over invariants

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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.