The Stacks project

Lemma 73.9.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$. Let $P$ be a property of the elements of $\mathcal{B}$. Assume that

  1. every $W \in \mathcal{B}$ is quasi-compact and quasi-separated,

  2. if $W \in \mathcal{B}$ and $U \subset W$ is quasi-compact open, then $U \in \mathcal{B}$,

  3. if $V \in \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$ is affine, then (a) $V \in \mathcal{B}$ and (b) $P$ holds for $V$,

  4. for every elementary distinguished square $(U \subset W, f : V \to W)$ such that

    1. $W \in \mathcal{B}$,

    2. $U$ is quasi-compact,

    3. $V$ is affine, and

    4. $P$ holds for $U$, $V$, and $U \times _ W V$,

    then $P$ holds for $W$.

Then $P$ holds for every $W \in \mathcal{B}$.

Proof. This is proved in exactly the same manner as the proof of Lemma 73.9.3. (We remark that (4)(d) makes sense as $U \times _ W V$ is a quasi-compact open of $V$ hence an element of $\mathcal{B}$ by conditions (2) and (3).) $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 73.9: Induction principle

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Your email address will not be published.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 08GQ. Beware of the difference between the letter 'O' and the digit '0'.