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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