Quotient stack

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In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack.

The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a strarification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks or toric stacks.


A quotient stack is defined as follows. Let G be an affine smooth group scheme over a scheme S and X a S-scheme on which G acts. Let [X/G] be the category over the category of S-schemes: an object over T is a principal G-bundle PT together with equivariant map PX; an arrow from PT to P'T' is a bundle map (i.e., forms a cartesian diagram) that is compatible with the equivariant maps PX and P'X.

Suppose the quotient X/G exists as, say, an algebraic space (for example, by the Keel–Mori theorem). The canonical map

[X/G] \to X/G,

that sends a bundle P over T to a T-point,[1] need not be an isomorphism of stacks; that is, the space "X/G" is usually coarser. The canonical map is an isomorphism if and only if the stabilizers are trivial (in which case X/G usually exists.)

In general, [X/G] is an Artin stack (also called algebraic stack). If the stabilizers of the geometric points are finite and reduced, then it is a Deligne–Mumford stack.

Remark: It is possible to approach the construction from the point of view of simplicial sheaves; cf. 9.2. of Jardine's "local homotopy theory".[2]


If X = S with trivial action of G (often S is a point), then [S/G] is called the classifying stack of G (in analogy with the classifying space of G) and is usually denoted by BG. Borel's theorem describes the cohomology ring of the classifying stack.

Example:[3] Let L be the Lazard ring; i.e., L = \pi_* \operatorname{MU}. Then the quotient stack [\operatorname{Spec}L/G] by G,

G(R) = \{g \in R[\![t]\!] | g(t) = b_0 t + b_1t^2 \dots, b_0 \in R^\times \},

is called the moduli stack of formal group laws, denoted by \mathcal{M}_\text{FG}.

See also[edit]


Some other references are