# Galois bimodules and integrality of PI comodule algebras over invariants

Research paper by **Pavel Etingof**

Indexed on: **17 Jun '13**Published on: **17 Jun '13**Published in: **Mathematics - Quantum Algebra**

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#### 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.