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A scheme is separated if the diagonal inclusion $X \to X \times X$ is a closed immersion. I what to know if there is a good generalization of `separated' for algebraic stacks?

My usual stack reference, Anton Gerashchenko's stack notes, doesn't seem to provide an answer.

In a previous MO question several related notions came up. The most similar is quasi-separated where you require the diagonal to be quasi-compact. You can check wikipedia for some relevant algebraic geometry terminology. How does this compare to separatedness?

The main obstacle that I can see in defining separated for stacks is that the property of a map of schemes $X \to Y$ being separated does not appear to be local in the target. Since maps between affines are separated, it seems that every map of schemes is locally separated. This means that we shouldn't expect the usual trick of replacing an algebraic stack by a scheme which covers it to work very well.

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2 Answers 2

up vote 7 down vote accepted

Look at Def. 4.7 of Deligne--Mumford for the definition when $X$ is DM: they define $X$ to be separated if $X \to X \times X$ is proper (or equivalently, finite).

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Does this reproduce the usual notion of separated when X is a scheme? It seems stronger. If it doesn't give back the usual notion, then it is not a good generalization. Also there are many many stacks which are not DM. I'd like to see a notion which works for general algebraic stacks. –  Chris Schommer-Pries Jan 29 '10 at 18:08
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For a scheme, the diagonal is always an immersion, and so is proper if and only if it is finite if and only if its a closed immersion. I don't know the non-DM case well enough to be sure, but it doesn't seem unreasonable. –  Emerton Jan 29 '10 at 18:26

One can first define a 'proper algebraic space' $X,$ using its 'underlying space' $|X|,$ and then define a morphism of algebraic spaces $f: X \to Y$ to be proper if for any affine (or just quasi-compact) scheme mapping into $Y,$ the fiber product gives a proper algebraic space. Finally define an Artin stack $X$ to be separated if the diagonal (which is representable) is proper, i.e. for any algebraic space mapping into $X \times X,$ the fiber product....

One can also define a 'proper Artin stack' similarly. See Laumon and Moret-Bailly.

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Precise reference: Laumon, Moret-Bailly. Champs Algebriques Lemme 7.7. –  S. Carnahan Nov 8 '10 at 14:46

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