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User:TakuyaMurata/Quotient stack

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In algebraic geometry, a quotient stack is a stack that generalizes the quotient of a scheme or a variety by a group. It is defined as follows. Let G be an affine flat group scheme over a scheme S and X a S-scheme on which G acts. Let be the category over S: an object over T is a principal G-bundle ET (in etale topology) together with equivariant map EX; an arrow from ET to E'T' is a bundle map (i.e., forms a cartesian diagram) that is compatible with the equivariant maps EX and E'X. It is a theorem of Deligne–Mumford that is an algebraic stack. If with trivial action of G, then is called the classifying stack of G (in analogy with the classifying space of G) and is usually denoted by BG.

References

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  • Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS, 36 (36): 75–109, doi:10.1007/BF02684599, MR 0262240

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