Lemma 28.29.5. Let X be a scheme. Assume either
The scheme X is quasi-affine.
The scheme X is isomorphic to a locally closed subscheme of an affine scheme.
There exists an ample invertible sheaf on X.
The scheme X is isomorphic to a locally closed subscheme of \text{Proj}(S) for some graded ring S.
Then for any finite subset E \subset X there exists an affine open U \subset X with E \subset U.
Proof.
By Properties, Definition 28.18.1 a quasi-affine scheme is a quasi-compact open subscheme of an affine scheme. Any affine scheme \mathop{\mathrm{Spec}}(R) is isomorphic to \text{Proj}(R[X]) where R[X] is graded by setting \deg (X) = 1. By Proposition 28.26.13 if X has an ample invertible sheaf then X is isomorphic to an open subscheme of \text{Proj}(S) for some graded ring S. Hence, it suffices to prove the lemma in case (4). (We urge the reader to prove case (2) directly for themselves.)
Thus assume X \subset \text{Proj}(S) is a locally closed subscheme where S is some graded ring. Let T = \overline{X} \setminus X. Recall that the standard opens D_{+}(f) form a basis of the topology on \text{Proj}(S). Since E is finite we may choose finitely many homogeneous elements f_ i \in S_{+} such that
E \subset D_{+}(f_1) \cup \ldots \cup D_{+}(f_ n) \subset \text{Proj}(S) \setminus T
Suppose that E = \{ \mathfrak p_1, \ldots , \mathfrak p_ m\} as a subset of \text{Proj}(S). Consider the ideal I = (f_1, \ldots , f_ n) \subset S. Since I \not\subset \mathfrak p_ j for all j = 1, \ldots , m we see from Algebra, Lemma 10.57.6 that there exists a homogeneous element f \in I, f \not\in \mathfrak p_ j for all j = 1, \ldots , m. Then E \subset D_{+}(f) \subset D_{+}(f_1) \cup \ldots \cup D_{+}(f_ n). Since D_{+}(f) does not meet T we see that X \cap D_{+}(f) is a closed subscheme of the affine scheme D_{+}(f), hence is an affine open of X as desired.
\square
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