The Stacks project

Lemma 17.28.14. Let $f : X \to Y$, $g : Y \to S$ be morphisms of ringed spaces Then there is a canonical exact sequence

\[ f^*\Omega _{Y/S} \to \Omega _{X/S} \to \Omega _{X/Y} \to 0 \]

where the maps come from applications of Lemma 17.28.12.

Proof. By taking induced maps in stalks at $x \in X$ and using Lemma 17.28.7 we obtain a sequence

\[ \mathcal{O}_{X, x} \otimes _{\mathcal{O}_{Y, f(x)}} \Omega _{\mathcal{O}_{Y, f(x)}/\mathcal{O}_{S, g(f(x))}} \to \Omega _{\mathcal{O}_{X, x}/\mathcal{O}_{S, g(f(x))}} \to \Omega _{\mathcal{O}_{X, x}/\mathcal{O}_{Y, f(x)}} \to 0 \]

It suffices to see that the maps of the sequence are the same as the ones in Algebra, Lemma 10.131.7. This is because the characterization of the maps in Lemma 17.28.12 and because via the isomorphism of Sheaves, Lemma 6.26.4 we have $(f^*s)_ x = s_ x \otimes 1$, for any local section $s$ of a sheaf of $\mathcal{O}_ Y$-modules. $\square$

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