The Stacks project

Lemma 27.29.1. Let $X$ be a quasi-separated scheme. Let $Z_1, \ldots , Z_ n$ be pairwise distinct irreducible components of $X$, see Topology, Section 5.8. Let $\eta _ i \in Z_ i$ be their generic points, see Schemes, Lemma 25.11.1. There exist affine open neighbourhoods $\eta _ i \in U_ i$ such that $U_ i \cap U_ j = \emptyset $ for all $i \not= j$. In particular, $U = U_1 \cup \ldots \cup U_ n$ is an affine open containing all of the points $\eta _1, \ldots , \eta _ n$.

Proof. Let $V_ i$ be any affine open containing $\eta _ i$ and disjoint from the closed set $Z_1 \cup \ldots \hat Z_ i \ldots \cup Z_ n$. Since $X$ is quasi-separated for each $i$ the union $W_ i = \bigcup _{j, j \not= i} V_ i \cap V_ j$ is a quasi-compact open of $V_ i$ not containing $\eta _ i$. We can find open neighbourhoods $U_ i \subset V_ i$ containing $\eta _ i$ and disjoint from $W_ i$ by Algebra, Lemma 10.25.4. Finally, $U$ is affine since it is the spectrum of the ring $R_1 \times \ldots \times R_ n$ where $R_ i = \mathcal{O}_ X(U_ i)$, see Schemes, Lemma 25.6.8. $\square$


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