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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.

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