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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. == 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 Suppose the quotient exists as, say, 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,〔The ''T''-point is obtained by completing the diagram .〕 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. 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）』 ■ウィキペディアで「Quotient stack」の詳細全文を読む スポンサード リンク
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