groupoid scheme について

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groupoid scheme ：ウィキペディア英語版

groupoid scheme
In algebraic geometry, a groupoid scheme is a pair of schemes

R, U

together with five morphisms

s, t: R \to U, e: U \to R, m: R \times_ R \to R, i: R \to R

satisfying

s \circ e, t \circ e

are the identity morphisms,

s \circ m = s \circ p_1, t \circ m = t \circ p_2

and other obvious conditions that generalize the axioms of group action; e.g., associativity. In practice, it is usually written as

R \rightrightarrows U

(cf. coequalizer.)
Example: Suppose an algebraic group ''G'' acts from the right on a scheme ''U''. Then take

R = U \times G

, ''s'' the projection, ''t'' the given action.
The main use of the notion is that it provides an atlas for a stack. More specifically, let

(\rightrightarrows U)

be the category of

(R \rightrightarrows U)

-torsors. Then it is a category fibered in groupoids; in fact, a Deligne–Mumford stack. Conversely, any DM stack is of this form.
== Notes ==

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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