Proposition 33.42.7. Let X be a separated scheme such that every quasi-compact open has a finite number of irreducible components. Let x_1, \ldots , x_ r \in X be points such that \mathcal{O}_{X, x_ i} is Noetherian of dimension \leq 1. Then there exists an affine open subscheme of X containing all of x_1, \ldots , x_ r.
Proof. We can replace X by a quasi-compact open containing x_1, \ldots , x_ r hence we may assume that X has finitely many irreducible components. By Lemma 33.42.6 we reduce to the case where X is integral. This case is Lemma 33.42.4. \square
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