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Keel–Mori theorem
In algebraic geometry, the Keel–Mori theorem gives conditions for the existence of the quotient of an algebraic space by a group. The theorem was proved by .
A consequence of the Keel–Mori theorem is the existence of a coarse moduli space of a separated algebraic stack, which is roughly a "best possible" approximation to the stack by a separated algebraic space.
==Statement==
All algebraic spaces are of finite type over a locally Noetherian base. Suppose that ''j'':''R''→''X''×''X'' is a flat groupoid whose stabilizer ''j''−1Δ is finite over ''X'' (where Δ is the diagonal of ''X''×''X''). The Keel–Mori theorem states that there is an algebraic space that is a geometric and uniform categorical quotient of ''X'' by ''j'', which is separated if ''j'' is finite.
A corollary is that for any flat group scheme ''G'' acting properly on an algebraic space ''X'' with finite stabilizers there is a uniform geometric and uniform categorical quotient ''X''/''G'' which is a separated algebraic space. proved a slightly weaker version of this and described several applications.
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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