The Stacks project

Lemma 68.5.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the following conditions on $X$:

  • $(\alpha )$ For every $x \in |X|$, the equivalent conditions of Lemma 68.4.2 hold.

  • $(\beta )$ For every $x \in |X|$, the equivalent conditions of Lemma 68.4.3 hold.

  • $(\gamma )$ For every $x \in |X|$, the equivalent conditions of Lemma 68.4.5 hold.

  • $(\delta )$ The equivalent conditions of Lemma 68.4.6 hold.

  • $(\epsilon )$ The equivalent conditions of Lemma 68.4.7 hold.

  • $(\zeta )$ The space $X$ is Zariski locally quasi-separated.

  • $(\eta )$ The space $X$ is quasi-separated

  • $(\theta )$ The space $X$ is representable, i.e., $X$ is a scheme.

  • $(\iota )$ The space $X$ is a quasi-separated scheme.

We have

\[ \xymatrix{ & (\theta ) \ar@{=>}[rd] & & & & \\ (\iota ) \ar@{=>}[ru] \ar@{=>}[rd] & & (\zeta ) \ar@{=>}[r] & (\epsilon ) \ar@{=>}[r] & (\delta ) \ar@{=>}[r] & (\gamma ) \ar@{<=>}[r] & (\alpha ) + (\beta ) \\ & (\eta ) \ar@{=>}[ru] & & & & } \]

Proof. The implication $(\gamma ) \Leftrightarrow (\alpha ) + (\beta )$ is immediate. The implications in the diamond on the left are clear from the definitions.

Assume $(\zeta )$, i.e., that $X$ is Zariski locally quasi-separated. Then $(\epsilon )$ holds by Properties of Spaces, Lemma 66.6.6.

Assume $(\epsilon )$. By Lemma 68.4.7 there exists a Zariski open covering $X = \bigcup X_ i$ such that for each $i$ there exists a scheme $U_ i$ and a quasi-compact surjective ├ętale morphism $U_ i \to X_ i$. Choose an $i$ and an affine open subscheme $W \subset U_ i$. It suffices to show that $W \to X$ has universally bounded fibres, since then the family of all these morphisms $W \to X$ covers $X$. To do this we consider the diagram

\[ \xymatrix{ W \times _ X U_ i \ar[r]_-p \ar[d]_ q & U_ i \ar[d] \\ W \ar[r] & X } \]

Since $W \to X$ factors through $X_ i$ we see that $W \times _ X U_ i = W \times _{X_ i} U_ i$, and hence $q$ is quasi-compact. Since $W$ is affine this implies that the scheme $W \times _ X U_ i$ is quasi-compact. Thus we may apply Morphisms, Lemma 29.57.9 and we conclude that $p$ has universally bounded fibres. From Lemma 68.3.4 we conclude that $W \to X$ has universally bounded fibres as well.

Assume $(\delta )$. Let $U$ be an affine scheme, and let $U \to X$ be an ├ętale morphism. By assumption the fibres of the morphism $U \to X$ are universally bounded. Thus also the fibres of both projections $R = U \times _ X U \to U$ are universally bounded, see Lemma 68.3.3. And by Lemma 68.3.2 also the fibres of $R \to X$ are universally bounded. Hence for any $x \in X$ the fibres of $|U| \to |X|$ and $|R| \to |X|$ over $x$ are finite, see Lemma 68.3.6. In other words, the equivalent conditions of Lemma 68.4.5 hold. This proves that $(\delta ) \Rightarrow (\gamma )$. $\square$

Comments (2)

Comment #696 by Simon Pepin Lehalleur on

There are two numberings in the lemma, one 1-9 and one

Comment #697 by on

That is actually a bug of the Stacks website code, see also I will take care of this soon.

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