Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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). –  t3suji Jan 26 '10 at 18:58
    
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. –  Andrea Ferretti Jan 26 '10 at 19:58
3  
If you look in Deligne--Mumford, you will find the definition, as well as a statement of the valuative criterion. –  Emerton Jan 26 '10 at 20:03
add comment

1 Answer

up vote 5 down vote accepted

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.

share|improve this answer
    
Thank you, I will have a look in Edidin's paper. –  Andrea Ferretti Jan 27 '10 at 9:03
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.