The Stacks project

Lemma 31.16.5. Let $X$ be a separated locally Noetherian scheme. Let $U \subset X$ be an affine open. For every generic point $\xi $ of an irreducible component of $X \setminus U$ the local ring $\mathcal{O}_{X, \xi }$ has dimension $\leq 1$. If $U$ is dense or if $\xi $ is in the closure of $U$, then $\dim (\mathcal{O}_{X, \xi }) = 1$.

Proof. This follows from Lemma 31.16.4 because the morphism $U \to X$ is affine by Morphisms, Lemma 29.11.11. $\square$

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