The Stacks project

Lemma 28.29.8. Let $X$ be a scheme and $x \in X$ a point. There exists an affine open neighbourhood $U \subset X$ of $x$ such that the canonical map $\mathcal{O}_ X(U) \to \mathcal{O}_{X, x}$ is injective in each of the following cases:

  1. $X$ is integral,

  2. $X$ is locally Noetherian,

  3. $X$ is reduced and has a finite number of irreducible components.

Proof. After translation into algebra, this follows from Algebra, Lemma 10.31.9. $\square$

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