This is related to another one of my questions on DM stacks. In Brian Conrad's article 'The KeelMori 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.

$\begingroup$ I don't know the answer yet, but Martin Olsson made this comment about weakening the finiteness condition in his class: "Quasifinite is not enough, but maybe you can relax this slightly to say that the Aut_x are finite." $\endgroup$– David ZureickBrownJan 21, 2010 at 19:24
1 Answer
No, not every DMstack has a coarse moduli space. The following is a counterexample (see my paper on geometric quotients):
Let X be two copies of the affine plane glued outside the yaxis (a nonseparated scheme). Let G=Z_{2} act on X by y → –y and by switching the two copies. Then G acts nonfreely on the locally closed subset {y=0, x ≠ 0}. The quotient [X/G] is a DMstack with nonfinite inertia and it can be shown that there is no coarse moduli space (neither categorical nor topological) in the category of algebraic spaces.

$\begingroup$ Precise reference: this is Example 6.14 of arxiv.org/abs/0708.3333 $\endgroup$ Jan 13, 2011 at 21:38