The Stacks project

Lemma 27.29.5. Let $X$ be a scheme. Assume either

  1. The scheme $X$ is quasi-affine.

  2. The scheme $X$ is isomorphic to a locally closed subscheme of an affine scheme.

  3. There exists an ample invertible sheaf on $X$.

  4. 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 27.18.1 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 27.26.13 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 10.56.6 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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