Lemma 33.42.4. Let X be an integral separated scheme. Let x_1, \ldots , x_ r \in X be a finite set of 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. Let K be the field of rational functions of X. Set A_ i = \mathcal{O}_{X, x_ i}. Then A_ i \subset K and K is the fraction field of A_ i. Since X is separated, and x_ i \not= x_ j there cannot be a valuation ring \mathcal{O} \subset K dominating both A_ i and A_ j. Namely, considering the diagram
\xymatrix{ \mathop{\mathrm{Spec}}(\mathcal{O}) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A_1) \ar[d] \\ \mathop{\mathrm{Spec}}(A_2) \ar[r] & X }
and applying the valuative criterion of separatedness (Schemes, Lemma 26.22.1) we would get x_ i = x_ j. Thus we see by Lemma 33.37.3 that A_ i \otimes A_ j \to K is surjective for all i \not= j. By Lemma 33.37.7 we see that A = A_1 \cap \ldots \cap A_ r is a Noetherian semi-local ring with exactly r maximal ideals \mathfrak m_1, \ldots , \mathfrak m_ r such that A_ i = A_{\mathfrak m_ i}. Moreover,
\mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{Spec}}(A_1) \cup \ldots \cup \mathop{\mathrm{Spec}}(A_ r)
is an open covering and the intersection of any two pieces of this covering is \mathop{\mathrm{Spec}}(K). Thus the given morphisms \mathop{\mathrm{Spec}}(A_ i) \to X glue to a morphism of schemes
\mathop{\mathrm{Spec}}(A) \longrightarrow X
mapping \mathfrak m_ i to x_ i and inducing isomorphisms of local rings. Thus the result follows from Lemma 33.42.3. \square
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