  Action of finite-dimensional Hopf algebra $H$ on commutative $k-$algebra $A$
is considered. As a generalization of the well-known fact for finite groups S.
Montgomery raised a problem in 1993 whether $A$ is integral over subalgebra of
invariants $A^H$. Recently some new results were obtained. Using the properties
of coradical filtration of pointed Hopf algebras we verified the truth of the
hypothesis in tree different cases: 1) Hopf algebra $H$ is commutative; 2) char
$k = p > 0$; 3) $A$ is integral domain. In spite of numerous partial positive
results it turned out that hypothesis of S. Montgomery isn't true in general.
The counteraxamples were built for series of pointed Hopf algebras $A_N, N \ge
2$.
