
Lemma 57.13.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism. Then

1. $\Delta$ is locally of finite type,

2. $\Delta$ is a monomorphism,

3. $\Delta$ is separated, and

4. $\Delta$ is locally quasi-finite.

Proof. Let $F = U/R$ be a presentation of $F$. As in the proof of Lemma 57.10.4 the diagram

$\xymatrix{ R \ar[r] \ar[d]_ j & F \ar[d]^\Delta \\ U \times _ S U \ar[r] & F \times F }$

is cartesian. Hence according to Lemma 57.11.4 it suffices to show that $j$ has the properties listed in the lemma. (Note that each of the properties (1) – (4) occur in the lists of Remarks 57.4.1 and 57.4.3.) Since $j$ is an equivalence relation it is a monomorphism. Hence it is separated by Schemes, Lemma 25.23.3. As $R$ is an étale equivalence relation we see that $s, t : R \to U$ are étale. Hence $s, t$ are locally of finite type. Then it follows from Morphisms, Lemma 28.14.8 that $j$ is locally of finite type. Finally, as it is a monomorphism its fibres are finite. Thus we conclude that it is locally quasi-finite by Morphisms, Lemma 28.19.7. $\square$

Comment #3836 by slogan_bot on

Suggested slogan: "The diagonal of any algebraic space is a separated, locally quasi-finite monomorphism"

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).