One of the most important examples of a stack is a quotient stack [X/G] arising from an action of a smooth algebraic groupGon a schemeX. The geometry of [X/G] couldn’t be simpler: it’s theG-equivariant geometry ofX.
Similar to how toric varieties provide concrete examples of schemes, quotient stacks provide both concrete examples useful to gain geometric intuition of general algebraic stacks and a fertile testing ground for conjectural results. On the other hand, it turns out that many algebraic stacks are quotient stacks (or at least locally quotient stacks) and therefore any (local) property that holds for quotient stacks also holds for many algebraic stacks.
0.8.1
Motivating the definition of the quotient stack
The quotient functor Sch→Sets defined byS7→X(S)/G(S) is not a sheaf even when the action is free (seeExample 0.4.31). We therefore first need to consider a better notion for a family of orbits.
For simplicity, let’s assume thatGand X are defined over C. Forx∈X(C),
there is aG-equivariant mapσx: G→Xdefined byg7→g·x. Note that two points
x, x0 are in the same G-orbit (sayx=hx0), if and only if there is aG-equivariant
morphismϕ:G→G(say byg7→gh) such thatσx=σx0◦ϕ. We can try the same thing for aT-pointT −→f X by considering
G×T f // p2 X, (g, t) //g·f(t) T
and noting thatf:G×T →X is aG-equivariant map. If we define a prestack consisting of such families, it fails to be a stack as objects don’t glue: given a Zariski-cover{Ti} of T, mapsT −→fi X and isomorphisms of the restrictions to
Tij, the trivial bundlesG×Ti →Ti will glue to aG-torsor P→T but it will not
necessarily be trivial (i.e. P ∼=G×T). It is clear then how to correct this using the language ofG-torsors (see Section C.3):
Definition 0.8.1 (Quotient stack). We define [X/G] as the category over Sch whose objects over a schemeS are diagrams
P f // X S
where P → S is a G-torsor and f: P → X is a G-equivariant morphism. A morphism (P0 →S0, P0 f
0
−→X)→(P →S, P −→f X) consists a maps g: S0 →S
andϕ:P0→P of schemes such that the diagram
P0 ϕ // f0 # # P f // X S0 g //S
commutes with the left square cartesian.
There is an object of [X/G] overX given by the diagram
G×X p2 σ // X X,
whereσ denotes the action map. This corresponds to a mapX →[X/G] via a 2-categorical version of Yoneda’s lemma.
The map X →[X/G] is aG-torsor even if the action of GonX is not free. We state that again: the map X →[X/G] is a G-torsor even if the action ofG on X is not free. Pause for a moment to appreciate how remarkable that is!
In particular, the map X → [X/G] is smooth and it follows that [X/G] is algebraic. At the expense of enlarging our category from schemes to algebraic stacks, we are able to (tautologically) construct the quotient [X/G] as a ‘geometric space’ with desirable geometric properties.
Example 0.8.2. Specializing to the case thatX = SpecCis a point, we define
theclassifying stack ofGas the categoryBG:= [SpecC/G] ofG-torsorsP →S.
The projection SpecC→BGis not only aG-torsor; it is the universalG-torsor.
Given any otherG-torsorP →S, there is a unique mapS→BGand a cartesian diagram P / /SpecC S //BG.
Exercise 0.8.3. What is the universal family over the quotient stack [X/G]?
0.8.2
Moduli as quotient stacks
Moduli stacks can often be described as quotient stacks, and these descriptions can be leveraged to establish properties of the moduli stack.
Example 0.8.4 (Moduli stack of smooth curves). In Example 0.7.8, the em- bedding of a smooth curve C via C |ω
⊗3
C |
,→ P5g−6 depends on a choice of basis
H0(C, ω⊗3
C ) ∼=C5g−5 and therefore is only unique up to a projective automor-
phism, i.e. an element of PGL5g−5 = Aut(P5g−6). The action of the algebraic
group PGL5g−5 on the schemeH0, parameterizingsmoothsubschemes such that
ωC∼=OC(3), yields an identificationMg= [∼ H0/PGL5g−6]. See??.
Example 0.8.5(Moduli stack of vector bundles). InExample 0.7.9, the presenta- tion of a vector bundleEas a quotientOC(−m)N Edepends on a choice of basis
H0(C, E(m))∼=CN. The algebraic group PGLN−1 acts on the schemeQ0m, pa-
rameterizing vector bundle quotients ofOC(−m)N such thatCN →∼ H0(C, E(m)),
yields an identificationMC,r,d∼=Sm0[Q0m/PGLN−1]. See??.
0.8.3
Geometry of
[X/G]
While the definition of the quotient stack [X/G] may appear abstract, its geometry is very familiar. The table below provides a dictionary between the geometry of a quotient stack [X/G] and theG-equivariant geometry ofX. The stack-theoretic concepts on the left-hand side will be introduced later. For simplicity we work overC.
Table 2: Dictionary
Geometry of [X/G] G-equivariant geometry ofX
C-pointx∈[X/G] orbitGx
automorphism group Aut(x) stabilizerGx
functionf ∈Γ([X/G],O[X/G]) G-equivariant function f ∈Γ(X,OX)G
map [X/G]→Y to a schemeY G-equivariant mapX →Y
line bundle G-equivariant line bundle (or lineariza- tion)
quasi-coherent sheaf G-equivariant quasi-coherent sheaf tangent spaceT[X/G],x normal spaceTX,x/TGx,x to the orbit
coarse moduli space [X/G]→Y geometric quotientX→Y
good moduli space [X/G]→Y good GIT quotientX →Y