The Stacks project

Lemma 33.28.2. Let $k$ be an algebraically closed field. Let $X$ be an integral scheme locally of finite type over $k$. Then $\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group.

Proof. As $X$ is integral the restriction mapping $\mathcal{O}(X) \to \mathcal{O}(U)$ is injective for any nonempty open subscheme $U \subset X$. Hence we may assume that $X$ is affine. Choose a closed immersion $X \to \mathbf{A}^ n_ k$ and denote $\overline{X}$ the closure of $X$ in $\mathbf{P}^ n_ k$ via the usual immersion $\mathbf{A}^ n_ k \to \mathbf{P}^ n_ k$. Thus we may assume that $X$ is an affine open of a projective variety $\overline{X}$.

Let $\nu : \overline{X}^\nu \to \overline{X}$ be the normalization morphism, see Morphisms, Definition 29.54.1. We know that $\nu $ is finite, dominant, and that $\overline{X}^\nu $ is a normal irreducible scheme, see Morphisms, Lemmas 29.54.5, 29.54.9, and 29.18.2. It follows that $\overline{X}^\nu $ is a proper variety, because $\overline{X} \to \mathop{\mathrm{Spec}}(k)$ is proper as a composition of a finite and a proper morphism (see results in Morphisms, Sections 29.41 and 29.44). It also follows that $\nu $ is a surjective morphism, because the image of $\nu $ is closed and contains the generic point of $\overline{X}$. Hence setting $X^\nu = \nu ^{-1}(X)$ we see that it suffices to prove the result for $X^\nu $. In other words, we may assume that $X$ is a nonempty open of a normal proper variety $\overline{X}$. This case is handled by Lemma 33.28.1. $\square$

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