Finite generation of the invariants of finite dimensional Hopf algebras (Q1327056)

Let \(H\) be a finite-dimensional commutative Hopf algebra over a field \(k\), \(A\) a commutative \(H\)-comodule algebra. The author shows that \(A\) is then an integral extension of \(A^ H\), the algebra of \(H\)-invariants of \(A\). Moreover, if \(A\) is also finitely-generated as a \(k\)-algebra, then \(A^ H\) is also finitely-generated as a \(k\)-algebra. The proof is Hopf theoretic. The final result also appears as a result on quotient schemes in the book ``Groupes Algébriques'' by \textit{M. Demazure} and \textit{P. Gabriel} [Masson, Paris (1970; Zbl 0203.234)].