In this section we discuss an induction principle for algebraic spaces analogous to what is Cohomology of Schemes, Lemma 30.4.1 for schemes. To formulate it we introduce the notion of an elementary distinguished square; this terminology is borrowed from [MV]. The principle as formulated here is implicit in the paper [GruRay] by Raynaud and Gruson. A related principle for algebraic stacks is [Theorem D, rydh_etale_devissage] by David Rydh.
Definition 75.9.1. Let $S$ be a scheme. A commutative diagram of algebraic spaces over $S$ is called an elementary distinguished square if
$U$ is an open subspace of $W$ and $j$ is the inclusion morphism,
$f$ is étale, and
setting $T = W \setminus U$ (with reduced induced subspace structure) the morphism $f^{-1}(T) \to T$ is an isomorphism.
\[ \xymatrix{ U \times _ W V \ar[r] \ar[d] & V \ar[d]^ f \\ U \ar[r]^ j & W } \]
We will indicate this by saying: “Let $(U \subset W, f : V \to W)$ be an elementary distinguished square.”
Note that if $(U \subset W, f : V \to W)$ is an elementary distinguished square, then we have $W = U \cup f(V)$. Thus $\{ U \to W, V \to W\} $ is an étale covering of $W$. It turns out that these étale coverings have nice properties and that in some sense there are “enough” of them.
Lemma 75.9.2. Let $S$ be a scheme. Let $(U \subset W, f : V \to W)$ be an elementary distinguished square of algebraic spaces over $S$.
If $V' \subset V$ and $U \subset U' \subset W$ are open subspaces and $W' = U' \cup f(V')$ then $(U' \subset W', f|_{V'} : V' \to W')$ is an elementary distinguished square.
If $p : W' \to W$ is a morphism of algebraic spaces, then $(p^{-1}(U) \subset W', V \times _ W W' \to W')$ is an elementary distinguished square.
If $S' \to S$ is a morphism of schemes, then $(S' \times _ S U \subset S' \times _ S W, S' \times _ S V \to S' \times _ S W)$ is an elementary distinguished square.
Proof. Omitted. $\square$
Lemma 75.9.3. 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
$P$ holds for every affine object of $X_{spaces, {\acute{e}tale}}$,
for every elementary distinguished square $(U \subset W, f : V \to W)$ such that
$W$ is a quasi-compact and quasi-separated object of $X_{spaces, {\acute{e}tale}}$,
$U$ is quasi-compact,
$V$ is affine, and
$P$ 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 étale 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
and the morphisms $f_ p : V_ p \to U_ p$ of Decent Spaces, Lemma 68.8.6. 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
for $i = 1, \ldots , m$. These are quasi-compact open subspaces of $X$. Then we have
and the pairs
are elementary distinguished squares by Lemma 75.9.2. 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 $W_{p, 1}, \ldots , W_{p, m} = U_ p$. $\square$
Lemma 75.9.4. 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
every $W \in \mathcal{B}$ is quasi-compact and quasi-separated,
if $W \in \mathcal{B}$ and $U \subset W$ is quasi-compact open, then $U \in \mathcal{B}$,
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 $V$,
for every elementary distinguished square $(U \subset W, f : V \to W)$ such that
$W \in \mathcal{B}$,
$U$ is quasi-compact,
$V$ is affine, and
$P$ 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 75.9.3. (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).) $\square$
Remark 75.9.5. How to choose the collection $\mathcal{B}$ in Lemma 75.9.4? Here are some examples:
If $X$ is quasi-compact and separated, then we can choose $\mathcal{B}$ to be the set of quasi-compact and separated objects of $X_{spaces, {\acute{e}tale}}$. Then $X \in \mathcal{B}$ and $\mathcal{B}$ satisfies (1), (2), and (3)(a). With this choice of $\mathcal{B}$ Lemma 75.9.4 reproduces Lemma 75.9.3.
If $X$ is quasi-compact with affine diagonal over $\mathbf{Z}$ (as in Properties of Spaces, Definition 66.3.1), then we can choose $\mathcal{B}$ to be the set of objects of $X_{spaces, {\acute{e}tale}}$ which are quasi-compact and have affine diagonal over $\mathbf{Z}$. Again $X \in \mathcal{B}$ and $\mathcal{B}$ satisfies (1), (2), and (3)(a).
If $X$ is quasi-compact and quasi-separated, then the smallest subset $\mathcal{B}$ which contains $X$ and satisfies (1), (2), and (3)(a) is given by the rule $W \in \mathcal{B}$ if and only if either $W$ is a quasi-compact open subspace of $X$, or $W$ is a quasi-compact open of an affine object of $X_{spaces, {\acute{e}tale}}$.
Here is a variant where we extend the truth from an open to larger opens.
Lemma 75.9.6. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W \subset X$ be a quasi-compact open subspace. Let $P$ be a property of quasi-compact open subspaces of $X$. Assume that
$P$ holds for $W$, and
for every elementary distinguished square $(W_1 \subset W_2, f : V \to W_2)$ where such that
$W_1$, $W_2$ are quasi-compact open subspaces of $X$,
$W \subset W_1$,
$V$ is affine, and
$P$ holds for $W_1$,
then $P$ holds for $W_2$.
Then $P$ holds for $X$.
Proof. We can deduce this from Lemma 75.9.4, but instead we will give a direct argument by explicitly redoing the proof of Lemma 75.9.3. We will use the filtration
and the morphisms $f_ p : V_ p \to U_ p$ of Decent Spaces, Lemma 68.8.6. We will prove that $P$ holds for $W_ p = W \cup U_ p$ by descending induction on $p$. This will finish the proof as $W_1 = X$. Note that $P$ holds for $W_{n + 1} = W \cap U_{n + 1} = W$ by (1). Assume $P$ holds for $W_{p + 1}$. Observe that $W_ p \setminus W_{p + 1}$ (with reduced induced subspace structure) is a closed subspace of $U_ p \setminus U_{p + 1}$. Since $(U_{p + 1} \subset U_ p, f_ p : V_ p \to U_ p)$ is an elementary distinguished square, the same is true for $(W_{p + 1} \subset W_ p, f_ p : V_ p \to W_ p)$. However (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 $W_{p, 0} = W_{p + 1}$ and
for $i = 1, \ldots , m$. These are quasi-compact open subspaces of $X$ containing $W$. Then we have
and the pairs
are elementary distinguished squares by Lemma 75.9.2. Now (2) applies to each of these and we inductively conclude $P$ holds for $W_{p, 1}, \ldots , W_{p, m} = W_ p$. $\square$
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