Quotient stack
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 stratification 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.
An orbifold is an example of a quotient stack.
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 be the category over the category of S-schemes: an object over T is a principal G-bundle P →T together with equivariant map P →X; an arrow from P →T to P' →T' is a bundle map (i.e., forms a cartesian diagram) that is compatible with the equivariant maps P →X and P' →X.
![[X/G]](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c10708eac95ed8fcce0dd132f18c6d844536ebf)
Suppose the quotient exists as, say, an algebraic space (for example, by the Keel–Mori theorem). The canonical map
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