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Let $S$ be a Noetherian scheme and $X$ a projective $S$-scheme.

Reading Laumon-Moret--Bailly's "Champs Algebriques", Theorem 4.6.2.1:

$ \mathscr Coh_{X|S} $ and $ \mathscr Fib_{X|S,r} $ are algebraic stacks of finite type.

The proof of "finite type" part seems to be done by proving that the $\mathfrak Isom$ sheaf above $U\in \operatorname{Aff}/S$ is a separated $U$-scheme of finite type.

Is there any reference that this property of the $\mathfrak Isom$ sheaf implies that the stack is of finite type?

I know that geometric properties of the algebraic stacks are "transferred" to stacks by a presentation/atlas, but I am not aware of such a thing for the $\mathfrak Isom$ sheaf.

Thank you for any reference on the subject.

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    $\begingroup$ It is recommended to always use a top-level tag in your questions, see here: meta.mathoverflow.net/questions/1075/… for example the [algebraic-geometry] tag. $\endgroup$
    – user347489
    Jun 12, 2020 at 21:28
  • $\begingroup$ It is not true that every stack with finite type diagonal is of finite type. If that were true, then every algebraic space over $S$ would be of finite type. The proof of "finite type" for those two stacks uses "limit arguments" and the equivalence of "limit preserving" with "finitely presented". $\endgroup$ Jun 13, 2020 at 4:39

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