Lemma 31.15.3. Let $X$ be a locally Noetherian scheme.

Let $D \subset X$ be a locally principal closed subscheme. Let $\xi \in D$ be a generic point of an irreducible component of $D$. Then $\dim (\mathcal{O}_{X, \xi }) \leq 1$.

Let $D \subset X$ be an effective Cartier divisor. Let $\xi \in D$ be a generic point of an irreducible component of $D$. Then $\dim (\mathcal{O}_{X, \xi }) = 1$.

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