The Stacks project

Remark 29.32.6. The lemma above gives a second way of constructing the module of differentials. Namely, let $f : X \to S$ be a morphism of schemes. Consider the collection of all affine opens $U \subset X$ which map into an affine open of $S$. These form a basis for the topology on $X$. Thus it suffices to define $\Gamma (U, \Omega _{X/S})$ for such $U$. We simply set $\Gamma (U, \Omega _{X/S}) = \Omega _{A/R}$ if $A$, $R$ are as in Lemma 29.32.5 above. This works, but it takes somewhat more algebraic preliminaries to construct the restriction mappings and to verify the sheaf condition with this ansatz.


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