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.
[Math] coarse moduli space of DM stacks
ag.algebraic-geometrystacks
Related Solutions
An example is Deligne's theorem on the existence of good notion of quotient $X/G$ of a separated algebraic space $X$ under the action of a finite group $G$, or relativizations or generalizations (with non-constant $G$) due to D. Rydh. See Theorem 3.1.13 of my paper with Lieblich and Olsson on Nagata compactification for algebraic spaces for the statement and proof of Deligne's result in a relative situation, and Theorem 5.4 of Rydh's paper "Existence of quotients..." on arxiv or his webpage for his generalization.
Note that in the above, there is no mention of DM stacks, but they come up in the proof! The mechanism to construct $X/G$ (say in the Deligne situation or its relative form) is to prove existence of a coarse space for the DM stack $[X/G]$ via Keel-Mori and show it has many good properties to make it a reasonable notion of quotient. Such quotients $X/G$ are very useful when $X$ is a scheme (but $X/G$ is "only" an algebraic space), such as for reducing some problems for normal noetherian algebraic spaces to the scheme case; cf. section 2.3 of the C-L-O paper. I'm sure there are numerous places where coarse spaces are convenient to do some other kinds of reduction steps in proofs of general theorems, such as reducing a problem for certain DM stacks to the case of algebraic spaces.
Also, Mazur used a deep study of the coarse moduli scheme associated to the DM stack $X_0(p)$ in his pioneering study of torsion in and rational isogenies between elliptic curves over $\mathbf{Q}$ (and these modular curves show up in numerous other places). But those specific coarse spaces are schemes and can be constructed and studied in more concrete terms without needing the fact that they are coarse spaces in the strong sense of the Keel-Mori theorem, so I think the example of Deligne's theorem above is a "better" example.
As noted in the comments, the question has a positive answer for tame Artin stacks (see e.g. [Ols12, Prop 6.1]) and also for non-tame Deligne–Mumford stacks (see [KV04, Lem. 2]). It is however also true in general. Throughout, let $\mathscr{X}$ be an algebraic stack with finite inertia and let $\pi\colon \mathscr{X}\to X$ denotes its coarse moduli space.
Proposition 1. The map $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$ is injective and if $\mathscr{X}$ is quasi-compact, then $\mathrm{coker}(\pi^*)$ has finite exponent, i.e., there exists a positive integer $n$ such that $\mathcal{L}^{n}\in \mathrm{Pic}(X)$ for every $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$.
For simplicity, assume that $\pi$ is of finite type (as is the case if $\mathscr{X}$ is of finite type over a noetherian base scheme) although this is not necessary for any of the results (the only property that is used is that $\mathscr{X}\to X$ is a universal homeomorphism and that invertible sheaves are trivial over semi-local rings etc).
Lemma 1. The functor $\pi^*\colon \mathbf{Pic}(X)\to \mathbf{Pic}(\mathscr{X})$ is fully faithful. In particular:
If $\mathcal{L}\in \mathrm{Pic}(X)$, then the adjunction map $\mathcal{L}\to\pi_*\pi^*\mathcal{L}$ is an isomorphism and the natural map $H^0(X,\mathcal{L})\to H^0(\mathscr{X},\pi^*\mathcal{L})$ is an isomorphism.
The natural map $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$ is injective.
Moreover, a line bundle on $\mathscr{X}$ that is locally trivial on $X$ comes from $X$, that is:
- Let $g\colon X'\to X$ be faithfully flat and locally of finite presentation and let $f\colon \mathscr{X}':=\mathscr{X}\times_X X'\to \mathscr{X}$ denote the pull-back. If $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$ is such that $f^*\mathcal{L}$ is in the image of $\pi'^*\colon\mathrm{Pic}(X')\to \mathrm{Pic}(\mathscr{X}')$, then $\mathcal{L}\in \mathrm{Pic}(X)$.
Proof. Statement 1 follows immediately from the isomorphism $\mathcal{O}_X\to \pi_*\mathcal{O}_{\mathscr{X}}$.
For the other statements, let $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$ and identify $\mathcal{L}$ with a class $c_\mathcal{L}\in H^1(\mathscr{X},\mathcal{O}_{\mathscr{X}}^*)$. If $\mathcal{L}$ is in the image of $\pi^*$ or locally in its image, then there exists an fppf covering $g\colon X'\to X$ such that $f^*\mathcal{L}$ is trivial. This means that we can represent $c_\mathcal{L}$ by a Čech $1$-cocycle for $f\colon \mathscr{X}'\to \mathscr{X}$. But since $H^0(\mathscr{X}\times_X U,\mathcal{O}_{\mathscr{X}}^*)=H^0(U,\mathcal{O}_X^*)$ for any flat $U\to X$, this means that $c_\mathcal{L}$ is given by a Čech $1$-cocycle for the covering $X'\to X$ giving a unique class in $H^1(X,\mathcal{O}_X^*)$. QED
As mentioned in the comments, when $\mathscr{X}$ is tame, then 3. can be replaced with: $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$ is in the image of $\pi^*$ if and only if the restriction to the residual gerbe $\mathcal{L}|_{\mathscr{G}_x}$ is trivial for every $x\in |\mathscr{X}|$ (see [Alp13, Thm 10.3] or [Ols12, Prop 6.1]).
Lemma 2. If there exists an algebraic space $Z$ and a finite morphism $p\colon Z\to \mathscr{X}$ such that $p_*\mathcal{O}_Z$ is locally free of rank $n$, then the cokernel of $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$ is $n$-torsion.
Proof. If $\mathcal{L}\in\mathrm{Pic}(\mathscr{X})$, then $\mathcal{L}^{n}=N_p(p^*\mathcal{L})$ (the norm is defined and behaves as expected since $p$ is flat). Since $Z\to X$ is finite, we can trivialize $p^* \mathcal{L}$ étale-locally on $X$. This implies that the norm is trivial étale-locally on $X$, i.e., $\mathcal{L}^{n}$ is trivial étale-locally on $X$. The result follows from 3. in the previous lemma. QED
Proof of Proposition 1. There exist an étale covering $\{X'_i\to X\}_{i=1}^r$ such that $\mathscr{X}'_i:=\mathscr{X}\times_X X'_i$ admits a finite flat covering $Z_i\to \mathscr{X}'_i$ of some constant rank $n_i$ for every $i$. By the two lemmas above, the integer $n=\mathrm{lcm}(n_i)$ then kills every element in the cokernel of $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$. QED
It is also simple to prove things like:
Proposition 2. Let $f\colon \mathscr{X}'\to \mathscr{X}$ be a representable morphism and let $\pi'\colon \mathscr{X}'\to X'$ denote the coarse moduli space and $g\colon X'\to X$ the induced morphism between coarse moduli spaces. If $\mathscr{X}$ is quasi-compact and $\mathcal{L}\in \mathrm{Pic}(\mathscr{X}')$ is $f$-ample, then there exists an $n$ such that $\mathcal{L}^{n}=\pi'^*\mathcal{M}$ and $\mathcal{M}$ is $g$-ample.
Proof. The question is local on $X$ so we may assume that $X$ is affine and $\mathscr{X}$ admits a finite flat morphism $p\colon Z\to \mathscr{X}$ of constant rank $n$ with $Z$ affine. Let $p'\colon Z'\to \mathscr{X}'$ be the pull-back. Then $p'^*\mathcal{L}$ is ample. We have seen that $\mathcal{L}^{n}=\pi'^*\mathcal{M}$ for $\mathcal{M}\in \mathrm{Pic}(X')$. It is enough to show that sections of $\mathcal{M}^m=\mathcal{L}^{mn}$ for various $m$ defines a basis for the topology of $X'$. Thus let $U'\subseteq X'$ be an open subset and pick any $x'\in U'$. Since $\mathcal{L}|_{Z'}$ is ample, we may find $s\in \Gamma(Z',\mathcal{L}^m)$ such that $D(s)=\{s\neq 0\}$ is an open neighborhood of the preimage of $x'$ (which is finite) contained in the preimage of $U'$. Let $t=N_p(s)$. Then $t\in H^0(\mathscr{X}',\mathcal{L}^{mn})=H^0(X',\mathcal{M}^m)$. But $\mathscr{X}\setminus D(t)=p(Z\setminus D(s))$ so $D(t)=X\setminus \pi(p(Z\setminus D(s)))$ is an open neighborhood of $x'$ contained in $U'$. QED
Acknowledgments I am grateful for comments from Jarod Alper and Daniel Bergh.
References
[Alp13] Alper, J. Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2349–2402.
[KV04] Kresch, A. and Vistoli, A. On coverings of Deligne–Mumford stacks and surjectivity of the Brauer map, Bull. London Math. Soc. 36 (2004), no. 2, 188–192.
[Ols12] Olsson, M. Integral models for moduli spaces of G-torsors, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 4, 1483–1549.
Best 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.