The Stacks project

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

  1. 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

  2. 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 ├ętale. Thus we see in particular that (1) and (2) hold. $\square$

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