The Stacks project

Lemma 53.8.2. Let $k'/k$ be a field extension. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. Then $X_{k'}$ is a proper scheme over $k'$ having dimension $1$ and $H^0(X_{k'}, \mathcal{O}_{X_{k'}}) = k'$. Moreover the genus of $X_{k'}$ is equal to the genus of $X$.

Proof. The dimension of $X_{k'}$ is $1$ for example by Morphisms, Lemma 29.28.3. The morphism $X_{k'} \to \mathop{\mathrm{Spec}}(k')$ is proper by Morphisms, Lemma 29.41.5. The equality $H^0(X_{k'}, \mathcal{O}_{X_{k'}}) = k'$ follows from Cohomology of Schemes, Lemma 30.5.2. The equality of the genus follows from the same lemma. $\square$

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