Lemma 115.16.1. Let S be a scheme. Let X be a reasonable algebraic space over S. Then |X| is Kolmogorov (see Topology, Definition 5.8.6).
115.16 Very reasonable algebraic spaces
Material that is somewhat obsolete.
Proof. Follows from the definitions and Decent Spaces, Lemma 68.12.3. \square
In the rest of this section we make some remarks about very reasonable algebraic spaces. If there exists a scheme U and a surjective, étale, quasi-compact morphism U \to X, then X is very reasonable, see Decent Spaces, Lemma 68.4.7.
Lemma 115.16.2. A scheme is very reasonable.
Proof. This is true because the identity map is a quasi-compact, surjective étale morphism. \square
Lemma 115.16.3. Let S be a scheme. Let X be an algebraic space over S. If there exists a Zariski open covering X = \bigcup X_ i such that each X_ i is very reasonable, then X is very reasonable.
Proof. This is case (\epsilon ) of Decent Spaces, Lemma 68.5.2. \square
Lemma 115.16.4. An algebraic space which is Zariski locally quasi-separated is very reasonable. In particular any quasi-separated algebraic space is very reasonable.
Proof. This is one of the implications of Decent Spaces, Lemma 68.5.1. \square
Lemma 115.16.5. Let S be a scheme. Let X, Y be algebraic spaces over S. Let Y \to X be a representable morphism. If X is very reasonable, so is Y.
Proof. This is case (\epsilon ) of Decent Spaces, Lemma 68.5.3. \square
Remark 115.16.6. Very reasonable algebraic spaces form a strictly larger collection than Zariski locally quasi-separated algebraic spaces. Consider an algebraic space of the form X = [U/G] (see Spaces, Definition 65.14.4) where G is a finite group acting without fixed points on a non-quasi-separated scheme U. Namely, in this case U \times _ X U = U \times G and clearly both projections to U are quasi-compact, hence X is very reasonable. On the other hand, the diagonal U \times _ X U \to U \times U is not quasi-compact, hence this algebraic space is not quasi-separated. Now, take U the infinite affine space over a field k of characteristic \not= 2 with zero doubled, see Schemes, Example 26.21.4. Let 0_1, 0_2 be the two zeros of U. Let G = \{ +1, -1\} , and let -1 act by -1 on all coordinates, and by switching 0_1 and 0_2. Then [U/G] is very reasonable but not Zariski locally quasi-separated (details omitted).
Warning: The following lemma should be used with caution, as the schemes U_ i in it are not necessarily separated or even quasi-separated.
Lemma 115.16.7. Let S be a scheme. Let X be a very reasonable algebraic space over S. There exists a set of schemes U_ i and morphisms U_ i \to X such that
each U_ i is a quasi-compact scheme,
each U_ i \to X is étale,
both projections U_ i \times _ X U_ i \to U_ i are quasi-compact, and
the morphism \coprod U_ i \to X is surjective (and étale).
Proof. Decent Spaces, Definition 68.6.1 says that there exist U_ i \to X such that (2), (3) and (4) hold. Fix i, and set R_ i = U_ i \times _ X U_ i, and denote s, t : R_ i \to U_ i the projections. For any affine open W \subset U_ i the open W' = t(s^{-1}(W)) \subset U_ i is a quasi-compact R_ i-invariant open (see Groupoids, Lemma 39.19.2). Hence W' is a quasi-compact scheme, W' \to X is étale, and W' \times _ X W' = s^{-1}(W') = t^{-1}(W') so both projections W' \times _ X W' \to W' are quasi-compact. This means the family of W' \to X, where W \subset U_ i runs through the members of affine open coverings of the U_ i gives what we want. \square
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