The Stacks project

[EGA IV, Corollaire 21.12.7, EGA4]

Lemma 31.16.4. Let $X$ be a locally Noetherian scheme. Let $U \subset X$ be an open subscheme such that the inclusion morphism $U \to X$ is affine. 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. Since $\xi $ is a generic point of $X \setminus U$, we see that

\[ U_\xi = U \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi }) \subset \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi }) \]

is the punctured spectrum of $\mathcal{O}_{X, \xi }$ (hint: use Schemes, Lemma 26.13.2). As $U \to X$ is affine, we see that $U_\xi \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi })$ is affine (Morphisms, Lemma 29.11.8) and we conclude that $U_\xi $ is affine. Hence $\dim (\mathcal{O}_{X, \xi }) \leq 1$ by Lemma 31.16.1. If $\xi \in \overline{U}$, then there is a specialization $\eta \to \xi $ where $\eta \in U$ (just take $\eta $ a generic point of an irreducible component of $\overline{U}$ which contains $\xi $; since $\overline{U}$ is locally Noetherian, hence locally has finitely many irreducible components, we see that $\eta \in U$). Then $\eta \in \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi })$ and we see that the dimension cannot be $0$. $\square$

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