Lemma 31.22.11. Let

be a commutative diagram of morphisms of schemes. Assume that $Y \to S$ is smooth, $X \to S$ smooth, and $i$ an immersion. Then $i$ is a regular immersion.

Lemma 31.22.11. Let

\[ \xymatrix{ Y \ar[rd] \ar[rr]_ i & & X \ar[ld] \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume that $Y \to S$ is smooth, $X \to S$ smooth, and $i$ an immersion. Then $i$ is a regular immersion.

**Proof.**
This is a special case of Lemma 31.22.10 because a smooth morphism is syntomic, see Morphisms, Lemma 29.34.7.
$\square$

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