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The Stacks project

Lemma 28.29.7. Let X be a quasi-affine scheme. Let \mathcal{L} be an invertible \mathcal{O}_ X-module. Let E \subset W \subset X with E finite and W open. Then there exists an s \in \Gamma (X, \mathcal{L}) such that X_ s is affine and E \subset X_ s \subset W.

Proof. The proof of this lemma has a lot in common with the proof of Algebra, Lemma 10.15.2. Say E = \{ x_1, \ldots , x_ n\} . If E = W = \emptyset , then s = 0 works. If W \not= \emptyset , then we may assume E \not= \emptyset by adding a point if necessary. Thus we may assume n \geq 1. We will prove the lemma by induction on n.

Base case: n = 1. After replacing W by an affine open neighbourhood of x_1 in W, we may assume W is affine. Combining Lemmas 28.27.1 and Proposition 28.26.13 we see that every quasi-coherent \mathcal{O}_ X-module is globally generated. Hence there exists a global section s of \mathcal{L} which does not vanish at x_1. On the other hand, let Z \subset X be the reduced induced closed subscheme on X \setminus W. Applying global generation to the quasi-coherent ideal sheaf \mathcal{I} of Z we find a global section f of \mathcal{I} which does not vanish at x_1. Then s' = fs is a global section of \mathcal{L} which does not vanish at x_1 such that X_{s'} \subset W. Then X_{s'} is affine by Lemma 28.26.4.

Induction step for n > 1. If there is a specialization x_ i \leadsto x_ j for i \not= j, then it suffices to prove the lemma for \{ x_1, \ldots , x_ n\} \setminus \{ x_ i\} and we are done by induction. Thus we may assume there are no specializations among the x_ i. By either Lemma 28.29.5 or Lemma 28.29.6 we may assume W is affine. By induction we can find a global section s of \mathcal{L} such that X_ s \subset W is affine and contains x_1, \ldots , x_{n - 1}. If x_ n \in X_ s then we are done. Assume s is zero at x_ n. By the case n = 1 we can find a global section s' of \mathcal{L} with \{ x_ n\} \subset X_{s'} \subset W \setminus \overline{\{ x_1, \ldots , x_{n - 1}\} }. Here we use that x_ n is not a specialization of x_1, \ldots , x_{n - 1}. Then s + s' is a global section of \mathcal{L} which is nonvanishing at x_1, \ldots , x_ n with X_{s + s'} \subset W and we conclude as before. \square


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