Lemma 33.18.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ with image $s \in S$. Let $V \subset S$ be an affine open neighbourhood of $s$. If $f$ is locally of finite type and $\dim _ x(X_ s) = d$, then there exists an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ and a factorization

$U \xrightarrow {\pi } \mathbf{A}^ d_ V \to V$

of $f|_ U : U \to V$ such that $\pi$ is quasi-finite.

Proof. This follows from Algebra, Lemma 10.125.2. $\square$

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