### 111.5.2 Coarse moduli spaces

Papers discussing coarse moduli spaces.

• Keel, Mori: Quotients in Groupoids [K-M]

It had apparently long been “folklore” that separated Deligne-Mumford stacks admitted coarse moduli spaces. A rigorous (although terse) proof of the following theorem is presented here: if $\mathcal{X}$ is an Artin stack locally of finite type over a Noetherian base scheme such that the inertia stack $I_\mathcal {X} \to \mathcal{X}$ is finite, then there exists a coarse moduli space $\phi : \mathcal{X} \to Y$ with $\phi$ separated and $Y$ an algebraic space locally of finite type over $S$. The hypothesis that the inertia is finite is precisely the right condition: there exists a coarse moduli space $\phi : \mathcal{X} \to Y$ with $\phi$ separated if and only if the inertia is finite.
• Conrad: The Keel-Mori Theorem via Stacks [conrad]

Keel and Mori's paper [K-M] is written in the groupoid language and some find it challenging to grasp. Brian Conrad presents a stack-theoretic version of the proof which is quite transparent although it uses the sophisticated language of stacks. Conrad also removes the Noetherian hypothesis.
• Rydh: Existence of quotients by finite groups and coarse moduli spaces

Rydh removes the hypothesis from [K-M] and [conrad] that $\mathcal{X}$ be finitely presented over some base.
• Abramovich, Olsson, Vistoli: Tame stacks in positive characteristic [tame]

They define a tame Artin stack as an Artin stack with finite inertia such that if $\phi : \mathcal{X} \to Y$ is the coarse moduli space, $\phi _*$ is exact on quasi-coherent sheaves. They prove that for an Artin stack with finite inertia, the following are equivalent: $\mathcal{X}$ is tame if and only if the stabilizers of $\mathcal{X}$ are linearly reductive if and only if $\mathcal{X}$ is étale locally on the coarse moduli space a quotient of an affine scheme by a linearly reductive group scheme. For a tame Artin stack, the coarse moduli space is particularly nice. For instance, the coarse moduli space commutes with arbitrary base change while a general coarse moduli space for an Artin stack with finite inertia will only commute with flat base change.
• Alper: Good moduli spaces for Artin stacks

For general Artin stacks with infinite affine stabilizer groups (which are necessarily non-separated), coarse moduli spaces often do not exist. The simplest example is $[\mathbf{A}^1 / \mathbf{G}_ m]$. It is defined here that a quasi-compact morphism $\phi : \mathcal{X} \to Y$ is a good moduli space if $\mathcal{O}_ Y \to \phi _* \mathcal{O}_\mathcal {X}$ is an isomorphism and $\phi _*$ is exact on quasi-coherent sheaves. This notion generalizes a tame Artin stack in [tame] as well as encapsulates Mumford's geometric invariant theory: if $G$ is a reductive group acting linearly on $X \subset \mathbf{P}^ n$, then the morphism from the quotient stack of the semi-stable locus to the GIT quotient $[X^{ss}/G] \to X//G$ is a good moduli space. The notion of a good moduli space has many nice geometric properties: (1) $\phi$ is surjective, universally closed, and universally submersive, (2) $\phi$ identifies points in $Y$ with points in $\mathcal{X}$ up to closure equivalence, (3) $\phi$ is universal for maps to algebraic spaces, (4) good moduli spaces are stable under arbitrary base change, and (5) a vector bundle on an Artin stack descends to the good moduli space if and only if the representations are trivial at closed points.

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