Comodule Algebras and Galois Extensions Relative to Polynomial Algebras, Free Algebras, and Enveloping Algebras

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Abstract (in LaTex)

In this paper we study the question of when an $H$-comodule
algebra is a faithfully flat Galois extension of its
subalgebra of coinvariants for certain Hopf algebras $H$.
We note that if $H$ is connected, a faithfully flat Galois
extension must actually be cleft and hence a crossed product,
and we show that with a different hypothesis, a faithfully
flat Galois extension must be a smash product.  We also
describe faithfully flat Galois extensions when $H$ is
pointed cococummutative.
We give an explicit description of $H$-comodule algebras when
$H$ is a polynomial algebra, a divided power Hopf algebra, a
free algebra, or a shuffle algebra.  We give necessary and
sufficient conditions for an $H$-extension to be faithfully
flat Galois in these cases and in the case where $H$ is the
enveloping algebra of a Lie algebra; a key ingredient in our
analysis is the existence and description of a total integral.
In the case where $H=k[x]$, we give a simple example of a
flat Galois extension that is not faithfully flat.

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