Lemma 53.8.4. Let $X$ be a smooth proper curve over a field $k$ with $H^0(X, \mathcal{O}_ X) = k$. Then
\[ \dim _ k H^0(X, \Omega _{X/k}) = g \quad \text{and}\quad \deg (\Omega _{X/k}) = 2g - 2 \]
where $g$ is the genus of $X$.
Proof. By Lemma 53.4.1 we have $\Omega _{X/k} = \omega _ X$. Hence the formulas hold by (53.8.1.1) and Lemma 53.8.3. $\square$
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