Lemma 37.7.11. Let
be a commutative diagram of schemes where $i$ and $j$ are formally unramified. Then there is a canonical exact sequence
where the first arrow comes from Lemma 37.7.5 and the second from Lemma 37.7.10.
Proof. Denote $Z \to Z'$ the universal first order thickening of $Z$ over $X$. Denote $Z \to Z''$ the universal first order thickening of $Z$ over $Y$. By Lemma 37.7.10 here is a canonical morphism $Z' \to Z''$ so that we have a commutative diagram
Apply Morphisms, Lemma 29.32.18 to the left triangle to get an exact sequence
As $Z''$ is formally unramified over $Y$ (see Lemma 37.7.4) we have $\Omega _{Z'/Z''} = \Omega _{Z/Y}$ (by combining Lemma 37.6.7 and Morphisms, Lemma 29.32.9). Then we have $(i')^*\Omega _{Z'/Y} = i^*\Omega _{X/Y}$ by Lemma 37.7.9. $\square$
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