Lemma 29.34.17 (06AA)—The Stacks project

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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. $\square$

Comments (3)

Comment #7217 by 羽山籍真 on April 29, 2022 at 10:50

It is also split according to 06A8.

Comment #7218 by 羽山籍真 on April 29, 2022 at 10:52

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

Comment #7330 by Johan on May 04, 2022 at 17:12

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

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