Lemma 31.10.2. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. The closed subscheme $Z \subset X$ cut out by the $0$th fitting ideal of $\Omega _{X/S}$ is exactly the set of points where $f$ is not unramified.

Proof. By Lemma 31.9.3 the complement of $Z$ is exactly the locus where $\Omega _{X/S}$ is zero. This is exactly the set of points where $f$ is unramified by Morphisms, Lemma 29.35.2. $\square$

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