I have seen the use of the word "proper Deligne-Mumford stack". Now, it is clear to me what it means for a morphism f of stacks to be proper: as usual it should be representable, and every morphism between schemes obtained from f by base change should be proper.

Now, first I guess that "proper" here actually means "complete". A scheme over a field is complete when the structural morphism to a point is proper. But it does not make sense for a stack to ask that the morphism to the point is proper. Indeed it would be in particular representable, and since a point is a scheme this would imply that the stack itself is a scheme.

Another possibility is that the sentence means "a stack with a proper atlas", so that one cannot speak of proper stacks, but only of proper Deligne-Mumford stacks.

So I am asking here what the standard terminology is.

  • $\begingroup$ Why do you say that a proper morphism `as usual it should be representable'? There are proper morphisms that are not representable. For instance, the morphism from the classifying stack of a finite group to a point is proper. This is an example of a complete Deligne-Mumford stack (I also prefer to use 'complete' for spaces and 'proper' for morphisms). $\endgroup$ – t3suji Jan 26 '10 at 18:58
  • $\begingroup$ This is exactly what I'm asking. Can you argue a bit more on the standard terminology in an answer? The definition I have seen says that a morphism of stacks has property P iff it is representable and and every morphism between schemes obtained from it by base change has property P. $\endgroup$ – Andrea Ferretti Jan 26 '10 at 19:58
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    $\begingroup$ If you look in Deligne--Mumford, you will find the definition, as well as a statement of the valuative criterion. $\endgroup$ – Emerton Jan 26 '10 at 20:03

As requested, an answer on terminology My favorite reference on basics for DM stacks is Edidin's paper, which I find much easier to read than Laumon & Moret-Bailly (who of course deal with Artin stacks).

Short summary. Suppose $P$ is a property of morphisms $f:X\to Y$ in the category of schemes:

  • If $P$ is local on both $X$ and $Y$ (`local' in appropriate topology, e.g., etale for DM stacks), it makes sense for morphisms of stacks (pass to compatible presentation of both stacks).

  • If $P$ is local on $Y$ only, it is easy to define for representable morphisms $F:{\mathcal X}\to{\mathcal Y}$ by changing base to a presentation of ${\mathcal Y}$.

  • If $P$ is local on $Y$, but you want to make sense of it for all morphisms, you have to make a special definition --- there is no general approach that works for all properties. This is what happens with definitions of separated/proper morphism of stacks.

So proper morphism of stacks need not be representable.

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