### 111.5.4 Quotient stacks

Quotient stacks^{1} form a very important subclass of Artin stacks which include almost all moduli stacks studied by algebraic geometers. The geometry of a quotient stack $[X/G]$ is the $G$-equivariant geometry of $X$. It is often easier to show properties are true for quotient stacks and some results are only known to be true for quotient stacks. The following papers address: When is an algebraic stack a global quotient stack? Is an algebraic stack “locally” a quotient stack?

Laumon, Moret-Bailly: [Chapter 6, LM-B]

Chapter 6 contains several facts about the local and global structure of algebraic stacks. It is proved that an algebraic stack $\mathcal{X}$ over $S$ is a quotient stack $[Y/G]$ with $Y$ an algebraic space (resp. scheme, resp. affine scheme) and $G$ a finite group if and only if there exists an algebraic space (resp. scheme, resp. affine scheme) $Y'$ and an finite étale morphism $Y' \to \mathcal{X}$. It is shown that any Deligne-Mumford stack over $S$ and $x : \mathop{\mathrm{Spec}}(K) \to \mathcal{X}$ admits an representable, étale and separated morphism $\phi : [X/G] \to \mathcal{X}$ where $G$ is a finite group acting on an affine scheme over $S$ such that $\mathop{\mathrm{Spec}}(K) = [X/G] \times _\mathcal {X} \mathop{\mathrm{Spec}}(K)$. The existence of presentations with geometrically connected fibers is also discussed in detail.

Edidin, Hassett, Kresch, Vistoli:

*Brauer Groups and Quotient stacks*[ehkv]First, they establish some fundamental (although not very difficult) facts concerning when a given algebraic stack (always assumed finite type over a Noetherian scheme in this paper) is a quotient stack. For an algebraic stack $\mathcal{X}$ : $\mathcal{X}$ is a quotient stack if and only if there exists a vector bundle $V \to \mathcal{X}$ such that for every geometric point, the stabilizer acts faithfully on the fiber if and only if there exists a vector bundle $V \to \mathcal{X}$ and a locally closed substack $V^0 \subset V$ such that $V^0$ is representable and surjects onto $\mathcal{X}$. They establish that an algebraic stack is a quotient stack if there exists finite flat cover by an algebraic space. Any smooth Deligne-Mumford stack with generically trivial stabilizer is a quotient stack. They show that a $\mathbf{G}_ m$-gerbe over a Noetherian scheme $X$ corresponding to $\beta \in H^2(X, \mathbf{G}_ m)$ is a quotient stack if and only if $\beta $ is in the image of the Brauer map $\text{Br}(X) \to \text{Br}'(X)$. They use this to produce a non-separated Deligne-Mumford stack that is not a quotient stack.

Totaro:

*The resolution property for schemes and stacks*[totaro_resolution]A stack has the resolution property if every coherent sheaf is the quotient of a vector bundle. The first main theorem is that if $\mathcal{X}$ is a normal Noetherian algebraic stack with affine stabilizer groups at closed points, then the following are equivalent: (1) $\mathcal{X}$ has the resolution property and (2) $\mathcal{X} = [Y/\text{GL}_ n]$ with $Y$ quasi-affine. In the case $\mathcal{X}$ is finite type over a field, then (1) and (2) are equivalent to: (3) $\mathcal{X} = [\mathop{\mathrm{Spec}}(A)/G]$ with $G$ an affine group scheme finite type over $k$. The implication that quotient stacks have the resolution property was proven by Thomason. The second main theorem is that if $\mathcal{X}$ is a smooth Deligne-Mumford stack over a field which has a finite and generically trivial stabilizer group $I_\mathcal {X} \to \mathcal{X}$ and whose coarse moduli space is a scheme with affine diagonal, then $\mathcal{X}$ has the resolution property. Another cool result states that if $\mathcal{X}$ is a Noetherian algebraic stack satisfying the resolution property, then $\mathcal{X}$ has affine diagonal if and only if the closed points have affine stabilizer.

Kresch:

*On the Geometry of Deligne-Mumford Stacks*[kresch_geometry]This article summarizes general structure results of Deligne-Mumford stacks (of finite type over a field) and contains some interesting results concerning quotient stacks. It is shown that any smooth, separated, generically tame Deligne-Mumford stack with quasi-projective coarse moduli space is a quotient stack $[Y/G]$ with $Y$ quasi-projective and $G$ an algebraic group. If $\mathcal{X}$ is a Deligne-Mumford stack whose coarse moduli space is a scheme, then $\mathcal{X}$ is Zariski-locally a quotient stack if and only if it admits a Zariski-open covering by stack quotients of schemes by finite groups. If $\mathcal{X}$ is a Deligne-Mumford stack proper over a field of characteristic 0 with coarse moduli space $Y$, then: $Y$ is projective and $\mathcal{X}$ is a quotient stack if and only if $Y$ is projective and $\mathcal{X}$ possesses a generating sheaf if and only if $\mathcal{X}$ admits a closed embedding into a smooth proper DM stack with projective coarse moduli space. This motivates a definition that a Deligne-Mumford stack is

*projective*if there exists a closed embedding into a smooth, proper Deligne-Mumford stack with projective coarse moduli space.Kresch, Vistoli

*On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map*[kresch-vistoli]It is shown that in characteristic 0 and for a fixed $n$, the following two statements are equivalent: (1) every smooth Deligne-Mumford stack of dimension $n$ is a quotient stack and (2) the Azumaya Brauer group coincides with the cohomological Brauer group for smooth schemes of dimension $n$.

Kresch:

*Cycle Groups for Artin Stacks*[kresch_cycle]It is shown that a reduced Artin stack finite type over a field with affine stabilizer groups admits a stratification by quotient stacks.

Abramovich-Vistoli:

*Compactifying the space of stable maps*[abramovich-vistoli]Lemma 2.2.3 establishes that for any separated Deligne-Mumford stack is étale-locally on the coarse moduli space a quotient stack $[U/G]$ where $U$ affine and $G$ a finite group. [Theorem 2.12, olsson_homstacks] shows in this argument $G$ is even the stabilizer group.

Abramovich, Olsson, Vistoli:

*Tame stacks in positive characteristic*[tame]This paper shows that a tame Artin stack is étale locally on the coarse moduli space a quotient stack of an affine by the stabilizer group.

Alper:

*On the local quotient structure of Artin stacks*[alper_quotient]It is conjectured that for an Artin stack $\mathcal{X}$ and a closed point $x \in \mathcal{X}$ with linearly reductive stabilizer, then there is an étale morphism $[V/G_ x] \to \mathcal{X}$ with $V$ an algebraic space. Some evidence for this conjecture is given. A simple deformation theory argument (based on ideas in [tame]) shows that it is true formally locally. A stack-theoretic proof of Luna's étale slice theorem is presented proving that for stacks $\mathcal{X} = [\mathop{\mathrm{Spec}}(A)/G]$ with $G$ linearly reductive, then étale locally on the GIT quotient $\mathop{\mathrm{Spec}}(A^ G)$, $\mathcal{X}$ is a quotient stack by the stabilizer.

*quotient stack*often means a stack of the form $[X/G]$ with $X$ an algebraic space and $G$ a subgroup scheme of $\text{GL}_ n$ rather than an arbitrary flat group scheme.

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