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 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 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 exists as an algebraic space (for example, by the Keel–Mori theorem). The canonical map


that sends a bundle P over T to a corresponding 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 usually exists.)

In general, 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.

(Totaro & 04) has shown: let X be a normal Noetherian algebraic stack whose stabilizer groups at closed points are affine. Then X is a quotient stack if and only if it has the resolution property; i.e., every coherent sheaf is a quotient of a vector bundle. Earlier, Thomason proved that a quotient stack has the resolution property.

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 with trivial action of G (often S is a point), then 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., . Then the quotient stack by ,


is called the moduli stack of formal group laws, denoted by .

See also[edit]


  1. ^ The T-point is obtained by completing the diagram .
  2. ^ http://www.math.uwo.ca/~jardine/papers/preprints/book.pdf
  3. ^ Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf

Some other references are