The Stacks project

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. $\square$

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