Lemma 5

*Let $K$ be a Noetherian ring, $B$ a finitely generated $K$-algebra and $C$ a sub-K-algebra of B such that B is integral over C. Then C is a finitely generated A-algebra.

Let $(x_i)_{1 \leq i \leq n}$ be a finite system of generators of the K-algebra B. For all i, there exists by hypothesis a monic polynomial $P_i \in C[X]$ such that $P_i(x_i) = 0$. Let C' be the sub-K-algebra of C generated by the coefficients of the $P_i$ ($1 \leq i \leq n$); clearly the $x_i$ are integral over C' and $B = C'[x_1, \ldots, x_n]$; hence B is a finitely generated C'-module (no. 1, Proposition 4). On the other hand $C'$ is a Noetherian ring (Chapter III, § 2, no. 10, Corollary 3 to Theorem 2); hence C is a finitely generated $C'$-module, which proves that C is a finitely generated K-algebra.