Lemma 29.34.17. Let $i : Z \to X$ be an immersion of schemes over $S$. Assume that $Z$ is smooth over $S$. Then the canonical exact sequence

$0 \to \mathcal{C}_{Z/X} \to i^*\Omega _{X/S} \to \Omega _{Z/S} \to 0$

of Lemma 29.32.15 is short exact.

Proof. The algebraic version of this lemma is the following: Given ring maps $A \to B \to C$ with $A \to C$ smooth and $B \to C$ surjective with kernel $J$, then the sequence

$0 \to J/J^2 \to C \otimes _ B \Omega _{B/A} \to \Omega _{C/A} \to 0$

of Algebra, Lemma 10.131.9 is exact. This is Algebra, Lemma 10.139.2. $\square$

Comment #7217 by 羽山籍真 on

It is also split according to 06A8.

Comment #7218 by 羽山籍真 on

I mean, locally (when i is a closed immersion it splits)

Comment #7330 by on

Going to leave this as is for now. But yes what you say is true.

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