Lemma 53.12.3. Let $X \to \mathop{\mathrm{Spec}}(k)$ be smooth of relative dimension $1$ at a closed point $x \in X$. If $\kappa (x)$ is separable over $k$, then for any uniformizer $s$ in the discrete valuation ring $\mathcal{O}_{X, x}$ the element $\text{d}s$ freely generates $\Omega _{X/k, x}$ over $\mathcal{O}_{X, x}$.

Proof. The ring $\mathcal{O}_{X, x}$ is a discrete valuation ring by Algebra, Lemma 10.140.3. Since $x$ is closed $\kappa (x)$ is finite over $k$. Hence if $\kappa (x)/k$ is separable, then any uniformizer $s$ maps to a nonzero element of $\Omega _{X/k, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x)$ by Algebra, Lemma 10.140.4. Since $\Omega _{X/k, x}$ is free of rank $1$ over $\mathcal{O}_{X, x}$ the result follows. $\square$

## Comments (0)

There are also:

• 2 comment(s) on Section 53.12: Riemann-Hurwitz

## 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Your email address will not be published.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 0C1E. Beware of the difference between the letter 'O' and the digit '0'.