## 73.9 Induction principle

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 73.9.1. Let $S$ be a scheme. A commutative diagram

$\xymatrix{ U \times _ W V \ar[r] \ar[d] & V \ar[d]^ f \\ U \ar[r]^ j & W }$

of algebraic spaces over $S$ is called an elementary distinguished square if

1. $U$ is an open subspace of $W$ and $j$ is the inclusion morphism,

2. $f$ is étale, and

3. setting $T = W \setminus U$ (with reduced induced subspace structure) the morphism $f^{-1}(T) \to T$ is an isomorphism.

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 73.9.2. Let $S$ be a scheme. Let $(U \subset W, f : V \to W)$ be an elementary distinguished square of algebraic spaces over $S$.

1. 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.

2. 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.

3. 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 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 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

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

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 $W_{p, 1}, \ldots , W_{p, m} = U_ p$. $\square$

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).) $\square$

Remark 73.9.5. How to choose the collection $\mathcal{B}$ in Lemma 73.9.4? Here are some examples:

1. 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 73.9.4 reproduces Lemma 73.9.3.

2. If $X$ is quasi-compact with affine diagonal, 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. Again $X \in \mathcal{B}$ and $\mathcal{B}$ satisfies (1), (2), and (3)(a).

3. 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 73.9.6. 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

1. $P$ holds for $W$, and

2. for every elementary distinguished square $(W_1 \subset W_2, f : V \to W_2)$ where such that

1. $W_1$, $W_2$ are quasi-compact open subspaces of $X$,

2. $W \subset W_1$,

3. $V$ is affine, and

4. $P$ holds for $W_1$,

then $P$ holds for $W_2$.

Then $P$ holds for $X$.

Proof. We can deduce this from Lemma 73.9.4, but instead we will give a direct argument by explicitly redoing the proof of Lemma 73.9.3. 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 $W_ p = W \cup U_ p$ 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

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

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

$W_{p + 1} = W_{p, 0} \subset W_{p, 1} \subset \ldots \subset W_{p, m} = W_ p$

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. 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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