Lemma 6

Let $\rho : B \to C$ be a ring homomorphism and S a multiplicative subset of B. Suppose that C is a B-algebra of finite type, and that $S^{-1}C$ is a finite $S^{-1}B$-algebra. Then there exists $f \in S$ such that $C_f$ is a finite $B_f$-algebra.

Let X be a finite generating set of the B-algebra C. For every $x \in X$, the image of x in $S^{-1}C$ is integral over $S^{-1}B$, and consequently there exist an integer $n(x) \geq 0$, elements $b_1(x), ..., b_{n(x)}(x) \in B$ and an element $f(x) \in S$ such that

Let $f = \prod_{x \in X} f(x)$; the image of every element x of X in $C_f$ is integral over $B_f$, hence $C_f$ is a finite $B_f$-algebra (V, § 1, No. 1, Prop. 4).