# coarse moduli space of DM stacks

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.

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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

## 1 Answer

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.

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Precise reference: this is Example 6.14 of arxiv.org/abs/0708.3333 – Anton Geraschenko Jan 13 '11 at 21:38