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Abstract

Let $H$ be a Hopf algebra over a field $k$ such that every finite-dimensional (left) $H$-module is semisimple. We give a counterpart of the first fundamental theorem of the classical invariant theory for locally finite, finitely generated (commutative) $H$-module algebras, and for local, complete $H$-module algebras. Also, we prove that if $H$ acts on the $k$-algebra $A=k[[X_{1},\dots,X_{n}]]$ in such a way that the unique maximal ideal in $A$ is invariant, then the algebra of invariants $A^{H}$ is a noetherian Cohen–Macaulay ring.