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).) $\square$


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