Lemma 33.18.5. 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
r = \dim _{\kappa (x)} \Omega _{X/S, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x) = \dim _{\kappa (x)} \Omega _{X_ s/s, x} \otimes _{\mathcal{O}_{X_ s, x}} \kappa (x) = \dim _{\kappa (x)} T_{X/S, x}
then there exist
an affine open U \subset X with x \in U and f(U) \subset V and a factorization
U \xrightarrow {j} \mathbf{A}^{r + 1}_ V \to V
of f|_ U such that j is an immersion, or
an affine open U \subset X with x \in U and f(U) \subset V and a factorization
U \xrightarrow {j} D \to V
of f|_ U such that j is a closed immersion and D \to V is smooth of relative dimension r.
Proof.
Pick any affine open U \subset X with x \in U and f(U) \subset V. Apply Lemma 33.18.4 to U \to V to get U \to \mathbf{A}^ r_ V \to V as in the statement of that lemma. By Lemma 33.18.3 we get a factorization
U \xrightarrow {j} D \xrightarrow {j'} \mathbf{A}^{r + 1}_ V \xrightarrow {p} \mathbf{A}^ r_ V \to V
where j and j' are immersions, p is the projection, and p \circ j' is standard étale. Thus we see in particular that (1) and (2) hold.
\square
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