# $\mathscr Coh_{X|S}$ is algebraic and of finite type

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.

• 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. Jun 12, 2020 at 21:28
• 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". Jun 13, 2020 at 4:39