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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