Lemma 10.125.2. Let $R \to S$ be a finite type ring map. Let $\mathfrak q \subset S$ be a prime. Suppose that $\dim _{\mathfrak q}(S/R) = n$. There exists a $g \in S$, $g \not\in \mathfrak q$ such that $S_ g$ is quasi-finite over a polynomial algebra $R[t_1, \ldots , t_ n]$.

Proof. The ring $\overline{S} = S \otimes _ R \kappa (\mathfrak p)$ is of finite type over $\kappa (\mathfrak p)$. Let $\overline{\mathfrak q}$ be the prime of $\overline{S}$ corresponding to $\mathfrak q$. By definition of the dimension of a topological space at a point there exists an open $U \subset \mathop{\mathrm{Spec}}(\overline{S})$ with $\overline{q} \in U$ and $\dim (U) = n$. Since the topology on $\mathop{\mathrm{Spec}}(\overline{S})$ is induced from the topology on $\mathop{\mathrm{Spec}}(S)$ (see Remark 10.17.8), we can find a $g \in S$, $g \not\in \mathfrak q$ with image $\overline{g} \in \overline{S}$ such that $D(\overline{g}) \subset U$. Thus after replacing $S$ by $S_ g$ we see that $\dim (\overline{S}) = n$.

Next, choose generators $x_1, \ldots , x_ N$ for $S$ as an $R$-algebra. By Lemma 10.115.4 there exist elements $y_1, \ldots , y_ n$ in the $\mathbf{Z}$-subalgebra of $S$ generated by $x_1, \ldots , x_ N$ such that the map $R[t_1, \ldots , t_ n] \to S$, $t_ i \mapsto y_ i$ has the property that $\kappa (\mathfrak p)[t_1\ldots , t_ n] \to \overline{S}$ is finite. In particular, $S$ is quasi-finite over $R[t_1, \ldots , t_ n]$ at $\mathfrak q$. Hence, by Lemma 10.123.13 we may replace $S$ by $S_ g$ for some $g\in S$, $g \not\in \mathfrak q$ such that $R[t_1, \ldots , t_ n] \to S$ is quasi-finite. $\square$

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