The Stacks project

Lemma 65.13.1. 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 65.10.4 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 65.11.4 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 65.4.1 and 65.4.3.) Since $j$ is an equivalence relation it is a monomorphism. Hence it is separated by Schemes, Lemma 26.23.3. 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 29.15.8 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 29.20.7. $\square$

Comments (1)

Comment #3836 by slogan_bot on

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

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02X4. Beware of the difference between the letter 'O' and the digit '0'.