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quotient stack : ウィキペディア英語版
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.
== Definition ==
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''.
Suppose the quotient X/G exists as, say, an algebraic space (for example, by the Keel–Mori theorem). The canonical map
:() \to X/G,
that sends a bundle ''P'' over ''T'' to a corresponding ''T''-point,〔The ''T''-point is obtained by completing the diagram T \leftarrow P \to X \to X/G.〕 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, () 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".〔http://www.math.uwo.ca/~jardine/papers/preprints/book.pdf〕

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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