# Quotient stack

A quotient stack is an algebraic stack of the form \([X/G]\), the stack-theoretic quotient of an algebraic space \(X\) by the action of a group scheme \(G\). It records not only the orbits of the action but also the stabilizer (automorphism) group of each point, and it is the standard device for turning a group action into a geometric object when the naive orbit set fails to be a scheme.<sup>[1](https://stacks.math.columbia.edu/tag/04UI)</sup>

| Key fact | Detail |
|---|---|
| General form | For a groupoid \((U, R, s, t, c)\) in algebraic spaces over \(S\), the quotient stack \([U/R]\) is a stack in groupoids over \((\mathit{Sch}/S)_{fppf}\) whose Isom-sheaves are representable by algebraic spaces.<sup>[1](https://stacks.math.columbia.edu/tag/04UI)</sup> |
| Group action case | \([X/G]\) is the quotient stack \([X/G \times_B X]\) of the groupoid \((X, G \times_B X, s, t, c)\) attached to the action.<sup>[1](https://stacks.math.columbia.edu/tag/04UI)</sup> |
| Points over \(T\) | Objects of \([U/G]\) over an \(S\)-scheme \(T\) are principal \(G\)-bundles \(P \to T\) together with a \(G\)-equivariant morphism \(P \to U\).<sup>[2](https://web.ma.utexas.edu/users/ikmartin/notes/quotient-prestack-and-stack/)</sup> |
| Common convention | In much of the literature, "quotient stack" means \([X/G]\) with \(X\) an algebraic space and \(G\) a subgroup scheme of \(\mathrm{GL}_n\), rather than an arbitrary flat group scheme.<sup>[3](https://stacks.math.columbia.edu/tag/04UZ)</sup> |
| Moduli example | The moduli stacks \(\mathcal{M}_g\) and \(\overline{\mathcal{M}}_g\) are quotient stacks for the action of \(\mathrm{PGL}(r+1)\) on suitable subschemes of a Hilbert scheme.<sup>[4](https://arcadia.sba.uniroma3.it/bitstream/2307/426/1/CompactifiedPicardStacks.pdf)</sup> |
| Coarse moduli | GIT-geometric quotients by the action of an algebraic group on a scheme are coarse moduli spaces for the associated quotient stack.<sup>[4](https://arcadia.sba.uniroma3.it/bitstream/2307/426/1/CompactifiedPicardStacks.pdf)</sup> |
| Local structure | Every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is étale-locally a quotient stack near a point with linearly reductive stabilizer (Alper–Hall–Rydh, 2020).<sup>[5](https://annals.math.princeton.edu/2020/191-3/p01)</sup> |

## Why the naive quotient functor fails

The set-valued functor sending an \(S\)-scheme \(T\) to the orbit set \(X(T)/G(T)\) is not a sheaf, even when the action is free, so it cannot serve as the quotient.<sup>[6](https://sites.math.washington.edu/~jarod/math582C-winter21/moduli-6-24-21.pdf)</sup> The fix is to remember the data that gluing loses. A first attempt, the quotient prestack, takes objects over \(T\) to be maps \(T \to U\) up to the relation \(f^*u = g \cdot u'\) for some \(g \in G(T')\); its fiber over \(T\) is the quotient groupoid \([U(T)/G(T)]\).<sup>[2](https://web.ma.utexas.edu/users/ikmartin/notes/quotient-prestack-and-stack/)</sup> This prestack still fails the descent condition, so one passes to its stackification.<sup>[6](https://sites.math.washington.edu/~jarod/math582C-winter21/moduli-6-24-21.pdf)</sup>

The resulting stack has a torsor-based description: a \(T\)-point of \([U/G]\) is a principal \(G\)-bundle \(P \to T\) with a \(G\)-equivariant map \(P \to U\).<sup>[2](https://web.ma.utexas.edu/users/ikmartin/notes/quotient-prestack-and-stack/)</sup> Equivalently, \([X/G]\) is the quotient stack of the groupoid \((X, G \times_B X, s, t, c)\) in which the relation space is \(G \times_B X\), with source and target given by projection and the action.<sup>[1](https://stacks.math.columbia.edu/tag/04UI)</sup> More generally, any groupoid object \(R \rightrightarrows U\) with smooth source and target yields an algebraic stack \([U/R]\), and the group-action quotient is the special case \(R = G \times_S U\).<sup>[7](https://math-jh.com/en/math/stacks/algebraic_stacks)</sup>

## Algebraicity

The Stacks project constructs \([U/R]\) as a stack in groupoids for any groupoid in algebraic spaces, and shows its Isom-sheaves are algebraic spaces; these quotient stacks are described as of fundamental importance to the theory of algebraic stacks.<sup>[1](https://stacks.math.columbia.edu/tag/04UI)</sup> When \(G \to B\) is flat and locally of finite presentation, the torsor interpretation agrees with the principal-bundle interpretation in the fppf topology.<sup>[8](https://stacks.math.columbia.edu/tag/04UV)</sup>

Not every quotient stack is algebraic. The easiest examples are \([S/G]\) where \(S\) is a scheme and \(G\) is a group scheme over \(S\) acting trivially.<sup>[9](https://stacks.math.columbia.edu/tag/06PI)</sup> A concrete case is the classifying stack \([S/\mu_p]\) over \(\mathrm{Spec}(\mathbf{F}_p)\), which classifies fppf \(\mu_p\)-torsors but is not algebraic; more generally, the classifying stack in the étale topology for étale locally trivial torsors of a group scheme \(G\) over \(B\) is algebraic if and only if \(G\) is smooth over \(B\).<sup>[10](https://stacks.math.columbia.edu/tag/077C)</sup> A related example is \([\mathrm{Spec}(k)/G]\) for \(G = (\mu_{2,k})^\infty\), which satisfies two of the three defining properties of an algebraic stack but not the third.<sup>[11](https://stacks.math.columbia.edu/tag/04AG)</sup>

## Examples and moduli

Quotient stacks provide concrete models for general algebraic stacks, much as toric varieties do for schemes, and they serve as a testing ground for conjectures; in addition, many algebraic stacks are quotient stacks, or at least locally quotient stacks, so local properties of quotient stacks transfer to many algebraic stacks.<sup>[6](https://sites.math.washington.edu/~jarod/math582C-winter21/moduli-6-24-21.pdf)</sup>

The central example in moduli theory is the stack of curves. One first parametrizes curves together with extra data, such as an embedding \(C \hookrightarrow \mathbb{CP}^n\), inside a [Hilbert scheme](https://www.edgepedia.org/hilbert-scheme), and then quotients by \(\mathrm{PGL}(r+1)\), which acts transitively on the set of choices; \(\mathcal{M}_g\) and \(\overline{\mathcal{M}}_g\) arise this way as quotient stacks.<sup>[4](https://arcadia.sba.uniroma3.it/bitstream/2307/426/1/CompactifiedPicardStacks.pdf)</sup><sup> • </sup><sup>[12](https://doi.org/10.1090/noti3280)</sup> The GIT quotient of the parameter scheme is the coarse moduli space of the quotient stack, and the map from the stack to that space is bijective on geometric points.<sup>[4](https://arcadia.sba.uniroma3.it/bitstream/2307/426/1/CompactifiedPicardStacks.pdf)</sup><sup> • </sup><sup>[12](https://doi.org/10.1090/noti3280)</sup> For a finite group \(G\) acting on an affine variety \(\mathrm{Spec}(A)\), the quotient \(\mathrm{Spec}(A^G)\) is the coarse space of \([\mathrm{Spec}(A)/G]\), and the map from stack to variety is a homeomorphism on topological spaces.<sup>[12](https://doi.org/10.1090/noti3280)</sup>

## Étale-local structure

A guiding question, posed by Jarod Alper of the University of Washington, is whether an Artin stack is locally near a point a quotient stack by the stabilizer at that point.<sup>[13](https://sites.math.washington.edu/~jarod/papers/quotient-structure.pdf)</sup> In 2020, Jarod Alper, Jack Hall ([University of Arizona](https://www.edgepedia.org/university-of-arizona) and [University of Melbourne](https://www.edgepedia.org/university-of-melbourne)), and David Rydh ([KTH Royal Institute of Technology](https://www.edgepedia.org/kth-royal-institute-of-technology)) established in the Annals of Mathematics that, given an algebraically closed field, any algebraic stack that is locally of finite type and has affine stabilizers becomes, in the étale topology, a quotient stack around any point whose stabilizer group is linearly reductive; their argument relies on an equivariant form of Artin's algebraization theorem.<sup>[5](https://annals.math.princeton.edu/2020/191-3/p01)</sup> Their Theorem 1.1 gives, at a smooth closed point \(x\) with linearly reductive stabilizer \(G_x\), a cartesian diagram relating \(\mathcal{X}\) to \([N_x/G_x]\), where \(N_x\) is the normal space at \(x\) viewed as a \(G_x\)-representation, so that \(\mathcal{X}\) and \([N_x/G_x]\) share an étale neighborhood of the form \([\mathrm{Spec}(A)/G_x]\).<sup>[14](https://arxiv.org/abs/1504.06467)</sup>

The linear reductivity hypothesis is necessary, and in characteristic \(p\) it is restrictive: an algebraic group over an algebraically closed field of characteristic \(p\) is linearly reductive if and only if its identity component is a torus and the order of its component group is coprime to \(p\).<sup>[15](https://sites.math.washington.edu/~jarod/papers/luna.pdf)</sup> The same circle of results recovers classical statements: a stack-theoretic proof of Luna's étale slice theorem shows that for \(\mathcal{X} = [\mathrm{Spec}(A)/G]\) with \(G\) linearly reductive, \(\mathcal{X}\) is étale-locally on the GIT quotient \(\mathrm{Spec}(A^G)\) a quotient stack by the stabilizer.<sup>[3](https://stacks.math.columbia.edu/tag/04UZ)</sup> Earlier results in the same direction include the Abramovich–Vistoli lemma that any separated [Deligne–Mumford stack](https://www.edgepedia.org/deligne-mumford-stack) is étale-locally on its coarse moduli space a quotient stack \([U/G]\) with \(U\) affine and \(G\) finite, and the theorem of Abramovich, Olsson, and Vistoli that a tame Artin stack is étale locally on the coarse moduli space a quotient of an affine by the stabilizer group.<sup>[3](https://stacks.math.columbia.edu/tag/04UZ)</sup>

## Which stacks are quotient stacks

Determining which algebraic stacks admit a global quotient presentation is an active problem with arithmetic consequences. Edidin and Graham related it to the old question of whether, for a scheme \(X\), the natural map from the [Brauer group](https://www.edgepedia.org/brauer-group) (equivalence classes of Azumaya algebras) to the cohomological Brauer group, the [torsion subgroup](https://www.edgepedia.org/torsion-subgroup) of \(H^2(X, \mathbf{G}_m)\), is surjective.<sup>[16](https://arxiv.org/abs/math/9905049)</sup> Kresch and Vistoli showed that in characteristic 0 and for fixed \(n\), the statement that every smooth Deligne–Mumford stack of dimension \(n\) is a quotient stack is equivalent to the Azumaya Brauer group coinciding with the cohomological Brauer group for smooth schemes of dimension \(n\).<sup>[3](https://stacks.math.columbia.edu/tag/04UZ)</sup> Kresch also showed that a reduced Artin stack of finite type over a field with affine stabilizer groups admits a stratification by quotient stacks.<sup>[3](https://stacks.math.columbia.edu/tag/04UZ)</sup>

## Equivariant geometry on the stack

The practical advantage of a quotient presentation is that the geometry of \([X/G]\) is the \(G\)-equivariant geometry of \(X\).<sup>[17](https://mat.uc.cl/~urzua/est7.pdf)</sup> This identification underlies several structure theorems. For \(X\) a separated algebraic space of finite type over an algebraically closed field \(k\) with an action of a \(k\)-algebraic group \(G\), and \(\Lambda\) a noetherian ring annihilated by an integer invertible in \(k\), the equivariant étale cohomology algebra \(H^*([X/G], \Lambda)\) is finitely generated over \(\Lambda\), with structure theorems generalizing Quillen's 1971 theory of the spectrum of an equivariant cohomology ring.<sup>[18](https://arxiv.org/abs/1305.0365)</sup> On the \(K\)-theory side, Vezzosi and Vistoli defined the geometric part of the rational equivariant \(K\)-theory \(K(X,G)\) and conjectured it isomorphic to the rational \(K\)-theory of \(X/G\); a 2025 refinement constructs the rational \(K\)-theory of \([X/G]\) over an arbitrary excellent base as part of an intrinsic decomposition of the \(K\)-theory of the stack.<sup>[19](https://arxiv.org/abs/2505.12357)</sup>

## References

1. [The Stacks Project, Section 95.13 (Tag 04UI): Quotient stacks](https://stacks.math.columbia.edu/tag/04UI)
2. [Isaac Martin, Quotient Prestack and Stack, University of Texas notes](https://web.ma.utexas.edu/users/ikmartin/notes/quotient-prestack-and-stack/)
3. [The Stacks Project, Section 112.5.4 (Tag 04UZ): Quotient stacks, papers in the literature](https://stacks.math.columbia.edu/tag/04UZ)
4. [Compactified Picard Stacks over the Moduli Space of Stable Curves (thesis, Università Roma Tre)](https://arcadia.sba.uniroma3.it/bitstream/2307/426/1/CompactifiedPicardStacks.pdf)
5. [Jarod Alper, Jack Hall, David Rydh (2020). A Luna étale slice theorem for algebraic stacks. Annals of Mathematics 191(3), 675–738.](https://annals.math.princeton.edu/2020/191-3/p01)
6. [Jarod Alper, Introduction to Stacks and Moduli (lecture notes, University of Washington)](https://sites.math.washington.edu/~jarod/math582C-winter21/moduli-6-24-21.pdf)
7. [Algebraic Stacks (reference notes)](https://math-jh.com/en/math/stacks/algebraic_stacks)
8. [The Stacks Project, Section 95.15 (Tag 04UV): Quotients by group actions](https://stacks.math.columbia.edu/tag/04UV)
9. [The Stacks Project, Section 97.18 (Tag 06PI): When is a quotient stack algebraic?](https://stacks.math.columbia.edu/tag/06PI)
10. [The Stacks Project, Section 110.55 (Tag 077C): A non-algebraic classifying stack](https://stacks.math.columbia.edu/tag/077C)
11. [The Stacks Project, Section 110.53 (Tag 04AG): Stack with quasi-compact flat covering which is not algebraic](https://stacks.math.columbia.edu/tag/04AG)
12. [Notices of the American Mathematical Society (2024), on the local quotient structure and moduli spaces of algebraic stacks](https://doi.org/10.1090/noti3280)
13. [Jarod Alper, On the local quotient structure of Artin stacks](https://sites.math.washington.edu/~jarod/papers/quotient-structure.pdf)
14. [Jarod Alper, Jack Hall, David Rydh (2015). A Luna étale slice theorem for algebraic stacks. arXiv:1504.06467.](https://arxiv.org/abs/1504.06467)
15. [Jarod Alper, Jack Hall, David Rydh, A Luna étale slice theorem for algebraic stacks (full text, KTH)](https://sites.math.washington.edu/~jarod/papers/luna.pdf)
16. [Dan Edidin, William Graham (1999). Brauer groups and quotient stacks. arXiv:math/9905049.](https://arxiv.org/abs/math/9905049)
17. [Quotient stacks (lecture notes, Universidad de Chile)](https://mat.uc.cl/~urzua/est7.pdf)
18. [Quotient stacks and equivariant étale cohomology algebras: Quillen's theory revisited. arXiv:1305.0365.](https://arxiv.org/abs/1305.0365)
19. [The geometric K-theory of quotient stacks. arXiv:2505.12357.](https://arxiv.org/abs/2505.12357)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Algebraic geometry › Schemes, stacks, and morphisms › Algebraic stacks: definitions and properties*

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