Lemma 29.32.18 (067L)—The Stacks project

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Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.32: Sheaf of differentials of a morphism
Lemma 29.32.18 (cite)

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Lemma 29.32.18. Let

$\xymatrix{ Z \ar[r]_ i \ar[rd]_ j & X \ar[d] \\ & Y }$

be a commutative diagram of schemes where $i$ and $j$ are immersions. Then there is a canonical exact sequence

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

where the first arrow comes from Lemma 29.31.3 and the second from Lemma 29.32.15.

Proof. The algebraic version of this is Algebra, Lemma 10.134.7. $\square$

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