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.
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