Lemma 29.32.9. Let $f : X \to Y$, $g : Y \to S$ be morphisms of schemes. 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 29.32.8.

Proof. This is the sheafified version of Algebra, Lemma 10.131.7. Alternatively, there is a general version for morphisms of ringed spaces, see Modules, Lemma 17.28.14. $\square$

Comment #8578 by on

Just curious: is there some specific reason why this isn't stated in Modules, Section 17.28? Here's the proof for arbitrary ringed spaces $X,Y,S$: The sequence is the same in the ringed spaces case, this time the maps come from Modules, Lemma 17.28.12. Call $g:X\to S$ and $h:Y\to S$ to the structure morphisms. By taking induced maps in stalks at $x\in X$ and using Modules, Lemma 17.28.7, we obtain a sequence 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 (i) the “characterizing property” at the end of Modules, Lemma 17.28.12 and (ii) by means of the isomorphism from Sheaves, Lemma 6.26.4, we can identify $(f^*s)_x=s_x\otimes 1$, for a local section $s$ of a sheaf of $\mathcal{O}_Y$-modules.

Comment #9157 by on

To save time. OK, I added this as you suggested. Thanks! See here. Of course, it would be better to prove this not using stalks and then add it to the chapter on modules on sites....

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