Lemma 15.48.3. Let $(R, \mathfrak m, \kappa )$ be a regular local ring. Let $m \geq 1$. Let $f_1, \ldots , f_ m \in \mathfrak m$. Assume there exist derivations $D_1, \ldots , D_ m : R \to R$ such that $\det _{1 \leq i, j \leq m}(D_ i(f_ j))$ is a unit of $R$. Then $R/(f_1, \ldots , f_ m)$ is regular and $f_1, \ldots , f_ m$ is a regular sequence.

**Proof.**
It suffices to prove that $f_1, \ldots , f_ m$ are $\kappa $-linearly independent in $\mathfrak m/\mathfrak m^2$, see Algebra, Lemma 10.106.3. However, if there is a nontrivial linear relation the we get $\sum a_ i f_ i \in \mathfrak m^2$ for some $a_ i \in R$ but not all $a_ i \in \mathfrak m$. Observe that $D_ i(\mathfrak m^2) \subset \mathfrak m$ and $D_ i(a_ j f_ j) \equiv a_ j D_ i(f_ j) \bmod \mathfrak m$ by the Leibniz rule for derivations. Hence this would imply

which would contradict the assumption on the determinant. $\square$

## 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Your email address will not be published.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.

## Comments (0)