The Stacks project

Lemma 28.10.4. Let $X$ be a scheme. Let $x \in X$. Then $x$ is a generic point of an irreducible component of $X$ if and only if $\dim (\mathcal{O}_{X, x}) = 0$.

Proof. This follows from Lemma 28.10.3 for example. $\square$

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