Lemma 31.10.3 (0C3K)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.10: The singular locus of a morphism
Lemma 31.10.3 (cite)

previous lemma

numberstags

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

$f$ is flat,
$f$ is locally of finite presentation, and
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. $\square$

Comments (0)

There are also:

5 comment(s) on Section 31.10: The singular locus of a morphism

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment