This is related to [another one of my questions on DM stacks](https://mathoverflow.net/questions/12472/is-the-inertia-stack-of-a-deligne-mumford-stack-always-finite). 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.



coarse moduli space of DM stacks
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.


