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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)