The Stacks project

Lemma 76.15.14. Let $S$ be a scheme. Let

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

be a commutative diagram of algebraic spaces over $S$ where $i$ and $j$ are formally unramified. 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 76.15.8 and the second from Lemma 76.15.13.

Proof. Since the maps have been defined, checking the sequence is exact reduces to the case of schemes by ├ętale localization, see Lemma 76.15.11 and Lemma 76.7.3. In this case the result is More on Morphisms, Lemma 37.7.11. $\square$

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