  Let $H$ be a semisimple Hopf algebra, and let $R$ be a noetherian left
$H$-module algebra. If $R/R^H$ is a right $H^*$-dense Galois extension, then
the invariant subalgebra $R^H$ will inherit the AS-Cohen-Macaulay property from
$R$ under some mild conditions, and $R$, when viewed as a right $R^H$-module,
is a Cohen-Macaulay module. In particular, we show that if $R$ is a noetherian
complete semilocal algebra which is AS-regular of global dimension 2 and
$H=\operatorname{\bf k} G$ for some finite subgroup $G\subseteq Aut(R)$, then
all the indecomposable Cohen-Macaulay module of $R^H$ is a direct summand of
$R_{R^H}$, and hence $R^H$ is Cohen-Macaulay-finite, which generalizes a
classical result for commutative rings. The main tool used in the paper is the
extension groups of objects in the corresponding quotient categories.
