The Stacks project

Lemma 73.9.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P$ be a property of the quasi-compact and quasi-separated objects of $X_{spaces, {\acute{e}tale}}$. Assume that

  1. $P$ holds for every affine object of $X_{spaces, {\acute{e}tale}}$,

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

    1. $W$ is a quasi-compact and quasi-separated object of $X_{spaces, {\acute{e}tale}}$,

    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 quasi-compact and quasi-separated object of $X_{spaces, {\acute{e}tale}}$ and in particular for $X$.

Proof. We first claim that $P$ holds for every representable quasi-compact and quasi-separated object of $X_{spaces, {\acute{e}tale}}$. Namely, suppose that $U \to X$ is ├ętale and $U$ is a quasi-compact and quasi-separated scheme. By assumption (1) property $P$ holds for every affine open of $U$. Moreover, if $W, V \subset U$ are quasi-compact open with $V$ affine and $P$ holds for $W$, $V$, and $W \cap V$, then $P$ holds for $W \cup V$ by (2) (as the pair $(W \subset W \cup V, V \to W \cup V)$ is an elementary distinguished square). Thus $P$ holds for $U$ by the induction principle for schemes, see Cohomology of Schemes, Lemma 30.4.1.

To finish the proof it suffices to prove $P$ holds for $X$ (because we can simply replace $X$ by any quasi-compact and quasi-separated object of $X_{spaces, {\acute{e}tale}}$ we want to prove the result for). We will use the filtration

\[ \emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X \]

and the morphisms $f_ p : V_ p \to U_ p$ of Decent Spaces, Lemma 66.8.6. We will prove that $P$ holds for $U_ p$ by descending induction on $p$. Note that $P$ holds for $U_{n + 1}$ by (1) as an empty algebraic space is affine. Assume $P$ holds for $U_{p + 1}$. Note that $(U_{p + 1} \subset U_ p, f_ p : V_ p \to U_ p)$ is an elementary distinguished square, but (2) may not apply as $V_ p$ may not be affine. However, as $V_ p$ is a quasi-compact scheme we may choose a finite affine open covering $V_ p = V_{p, 1} \cup \ldots \cup V_{p, m}$. Set $W_{p, 0} = U_{p + 1}$ and

\[ W_{p, i} = U_{p + 1} \cup f_ p(V_{p, 1} \cup \ldots \cup V_{p, i}) \]

for $i = 1, \ldots , m$. These are quasi-compact open subspaces of $X$. Then we have

\[ U_{p + 1} = W_{p, 0} \subset W_{p, 1} \subset \ldots \subset W_{p, m} = U_ p \]

and the pairs

\[ (W_{p, 0} \subset W_{p, 1}, f_ p|_{V_{p, 1}}), (W_{p, 1} \subset W_{p, 2}, f_ p|_{V_{p, 2}}),\ldots , (W_{p, m - 1} \subset W_{p, m}, f_ p|_{V_{p, m}}) \]

are elementary distinguished squares by Lemma 73.9.2. Note that $P$ holds for each $V_{p, 1}$ (as affine schemes) and for $W_{p, i} \times _{W_{p, i + 1}} V_{p, i + 1}$ as this is a quasi-compact open of $V_{p, i + 1}$ and hence $P$ holds for it by the first paragraph of this proof. Thus (2) applies to each of these and we inductively conclude $P$ holds for $W_{p, 1}, \ldots , W_{p, m} = U_ p$. $\square$


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