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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