The Stacks project

Remark 76.7.5. Now that we know that $\Omega _{X/Y}$ is quasi-coherent we can attempt to construct it in another manner. For example we can use the result of Properties of Spaces, Section 66.32 to construct the sheaf of differentials by glueing. For example if $Y$ is a scheme and if $U \to X$ is a surjective ├ętale morphism from a scheme towards $X$, then we see that $\Omega _{U/Y}$ is a quasi-coherent $\mathcal{O}_ U$-module, and since $s, t : R \to U$ are ├ętale we get an isomorphism

\[ \alpha : s^*\Omega _{U/Y} \to \Omega _{R/Y} \to t^*\Omega _{U/Y} \]

by using Morphisms, Lemma 29.34.16. You check that this satisfies the cocycle condition and you're done. If $Y$ is not a scheme, then you define $\Omega _{U/Y}$ as the cokernel of the map $(U \to Y)^*\Omega _{Y/S} \to \Omega _{U/S}$, and proceed as before. This two step process is a little bit ugly. Another possibility is to glue the sheaves $\Omega _{U/V}$ for any diagram as in Lemma 76.7.3 but this is not very elegant either. Both approaches will work however, and will give a slightly more elementary construction of the sheaf of differentials.

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