The Stacks project

Lemma 27.29.4. Let $X$ be a quasi-separated scheme. Let $Z_1, \ldots , Z_ n$ be pairwise distinct irreducible components of $X$. Let $\eta _ i \in Z_ i$ be their generic points. Let $x \in X$ be arbitrary. There exists an affine open $U \subset X$ containing $x$ and all the $\eta _ i$.

Proof. Suppose that $x \in Z_1 \cap \ldots \cap Z_ r$ and $x \not\in Z_{r + 1}, \ldots , Z_ n$. Then we may choose an affine open $W \subset X$ such that $x \in W$ and $W \cap Z_ i = \emptyset $ for $i = r + 1, \ldots , n$. Note that clearly $\eta _ i \in W$ for $i = 1, \ldots , r$. By Lemma 27.29.1 we may choose affine opens $U_ i \subset X$ which are pairwise disjoint such that $\eta _ i \in U_ i$ for $i = r + 1, \ldots , n$. Since $X$ is quasi-separated the opens $W \cap U_ i$ are quasi-compact and do not contain $\eta _ i$ for $i = r + 1, \ldots , n$. Hence by Algebra, Lemma 10.25.4 we may shrink $U_ i$ such that $W \cap U_ i = \emptyset $ for $i = r + 1, \ldots , n$. Then the union $U = W \cup \bigcup _{i = r + 1, \ldots , n} U_ i$ is disjoint and hence (by Schemes, Lemma 25.6.8) a suitable affine open. $\square$


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