MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is related to another one of my questions on DM stacks. In Brian Conrad's article 'The Keel-Mori Theorem via Stacks', a sufficient condition on for an Artin stack to have coarse moduli space is that it has finite inertia stack. This does not include DM stacks without finite inertia. My question is that, does every DM stack of finite type over a field have a coarse moduli space? And what's the reference? Thanks.

share|cite|improve this question
I don't know the answer yet, but Martin Olsson made this comment about weakening the finiteness condition in his class: "Quasi-finite is not enough, but maybe you can relax this slightly to say that the Aut_x are finite." – David Zureick-Brown Jan 21 '10 at 19:24
up vote 10 down vote accepted

No, not every DM-stack has a coarse moduli space. The following is a counter-example (see my paper on geometric quotients):

Let X be two copies of the affine plane glued outside the y-axis (a non-separated scheme). Let G=Z2 act on X by y → –y and by switching the two copies. Then G acts non-freely on the locally closed subset {y=0, x ≠ 0}. The quotient [X/G] is a DM-stack with non-finite inertia and it can be shown that there is no coarse moduli space (neither categorical nor topological) in the category of algebraic spaces.

share|cite|improve this answer
Precise reference: this is Example 6.14 of – Anton Geraschenko Jan 13 '11 at 21:38

Your Answer


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.