The Stacks project

Lemma 21.31.3. Let $X$ be a Hausdorff and locally quasi-compact space, in other words, an object of $\textit{LC}$.

  1. If $X' \to X$ is an isomorphism in $\textit{LC}$ then $\{ X' \to X\} $ is a qc covering.

  2. If $\{ f_ i : X_ i \to X\} _{i\in I}$ is a qc covering and for each $i$ we have a qc covering $\{ g_{ij} : X_{ij} \to X_ i\} _{j\in J_ i}$, then $\{ X_{ij} \to X\} _{i \in I, j\in J_ i}$ is a qc covering.

  3. If $\{ X_ i \to X\} _{i\in I}$ is a qc covering and $X' \to X$ is a morphism of $\textit{LC}$ then $\{ X' \times _ X X_ i \to X'\} _{i\in I}$ is a qc covering.

Proof. Part (1) holds by the remark above that open coverings are qc coverings.

Proof of (2). Let $x \in X$. Choose $i_1, \ldots , i_ n \in I$ and $E_ a \subset X_{i_ a}$ quasi-compact such that $\bigcup f_{i_ a}(E_ a)$ is a neighbourhood of $x$. For every $e \in E_ a$ we can find a finite subset $J_ e \subset J_{i_ a}$ and quasi-compact $F_{e, j} \subset X_{ij}$, $j \in J_ e$ such that $\bigcup g_{ij}(F_{e, j})$ is a neighbourhood of $e$. Since $E_ a$ is quasi-compact we find a finite collection $e_1, \ldots , e_{m_ a}$ such that

\[ E_ a \subset \bigcup \nolimits _{k = 1, \ldots , m_ a} \bigcup \nolimits _{j \in J_{e_ k}} g_{ij}(F_{e_ k, j}) \]

Then we find that

\[ \bigcup \nolimits _{a = 1, \ldots , n} \bigcup \nolimits _{k = 1, \ldots , m_ a} \bigcup \nolimits _{j \in J_{e_ k}} f_ i(g_{ij}(F_{e_ k, j})) \]

is a neighbourhood of $x$.

Proof of (3). Let $x' \in X'$ be a point. Let $x \in X$ be its image. Choose $i_1, \ldots , i_ n \in I$ and quasi-compact subsets $E_ j \subset X_{i_ j}$ such that $\bigcup f_{i_ j}(E_ j)$ is a neighbourhood of $x$. Choose a quasi-compact neighbourhood $F \subset X'$ of $x'$ which maps into the quasi-compact neighbourhood $\bigcup f_{i_ j}(E_ j)$ of $x$. Then $F \times _ X E_ j \subset X' \times _ X X_{i_ j}$ is a quasi-compact subset and $F$ is the image of the map $\coprod F \times _ X E_ j \to F$. Hence the base change is a qc covering and the proof is finished. $\square$

Comments (0)

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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09X1. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 21.31.3 as pdf