Lemma 10.125.6. 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 an open neighbourhood $V$ of $\mathfrak q$ in $\mathop{\mathrm{Spec}}(S)$ such that $\dim _{\mathfrak q'}(S/R) \leq n$ for all $\mathfrak q' \in V$.

**Proof.**
By Lemma 10.125.2 we see that we may assume that $S$ is quasi-finite over a polynomial algebra $R[t_1, \ldots , t_ n]$. Considering the fibres, we reduce to Lemma 10.125.5.
$\square$

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