Theorem 2

Let $A$ be a finitely generated $K$-algebra and $G$ a locally finite group of operators on $A$. Then $A$ is a finitely generated $A$-module; if further $K$ is Noetherian, $A^G$ is affinitely generated $K$-algebra.

Let $(a_j)_{1 \leq j \leq m}$ be a system of generators of the $K$-algebra $A$; as a fortiori $A = A^G[a_1, \ldots, a_l]$ and the $a_j$ are integral over $A^G$ by Proposition 22, the first assertion follows from no. 1, Proposition 4. The second is a consequence of the following lemma: