Lemma 29.32.12. Let $f : X \to S$ be a morphism of schemes. Assume $f$ is smooth. Then the module of differentials $\Omega _{X/S}$ of $X$ over $S$ is finite locally free and

for every $x \in X$.

Lemma 29.32.12. Let $f : X \to S$ be a morphism of schemes. Assume $f$ is smooth. Then the module of differentials $\Omega _{X/S}$ of $X$ over $S$ is finite locally free and

\[ \text{rank}_ x(\Omega _{X/S}) = \dim _ x(X_{f(x)}) \]

for every $x \in X$.

**Proof.**
The statement is local on $X$ and $S$. By Lemma 29.32.11 above we may assume that $f$ is a standard smooth morphism of affines. In this case the result follows from Algebra, Lemma 10.136.7 (and the definition of a relative global complete intersection, see Algebra, Definition 10.135.5).
$\square$

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