The Stacks project

The Jacobian criterion for hypersurfaces, done right.

Lemma 15.48.2. Let $R$ be a regular ring. Let $f \in R$. Assume there exists a derivation $D : R \to R$ such that $D(f)$ is a unit of $R/(f)$. Then $R/(f)$ is regular.

Proof. It suffices to prove this when $R$ is a local ring with maximal ideal $\mathfrak m$ and residue field $\kappa $. In this case it suffices to prove that $f \not\in \mathfrak m^2$, see Algebra, Lemma 10.106.3. However, if $f \in \mathfrak m^2$ then $D(f) \in \mathfrak m$ by the Leibniz rule, a contradiction. $\square$

Comments (1)

Comment #1123 by Simon Pepin Lehalleur on

Suggested slogan: The Jacobian criterion for hypersurfaces, done right.

There are also:

  • 10 comment(s) on Section 15.48: Regularity and derivations

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07PF. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 15.48.2 as pdf