Lemma 33.28.6. Let $k$ be a field. Let $X$ be a variety over $k$. The group $\mathcal{O}(X)^*/k^*$ is a finitely generated abelian group provided at least one of the following conditions holds:

1. $k$ is integrally closed in $\Gamma (X, \mathcal{O}_ X)$,

2. $k$ is algebraically closed in $k(X)$,

3. $X$ is geometrically integral over $k$, or

4. $k$ is the “intersection” of the field extensions $\kappa (x)/k$ where $x$ runs over the closed points of $x$.

Proof. We see that (1) is enough by Proposition 33.28.5. We omit the verification that each of (2), (3), (4) implies (1). $\square$

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