The Stacks project

Lemma 31.10.3. Let $f : X \to S$ be a morphism of schemes. Let $d \geq 0$ be an integer. Assume

  1. $f$ is flat,

  2. $f$ is locally of finite presentation, and

  3. every nonempty fibre of $f$ is equidimensional of dimension $d$.

Let $Z \subset X$ be the closed subscheme cut out by the $d$th fitting ideal of $\Omega _{X/S}$. Then $Z$ is exactly the set of points where $f$ is not smooth.

Proof. By Lemma 31.9.6 the complement of $Z$ is exactly the locus where $\Omega _{X/S}$ can be generated by at most $d$ elements. Hence the lemma follows from Morphisms, Lemma 29.34.14. $\square$

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