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In algebraic geometry, a geometric quotient of an algebraic variety ''X'' with the action of an algebraic group ''G'' is a morphism of varieties such that :(i) For each ''y'' in ''Y'', the fiber is an orbit of ''G''. :(ii) The topology of ''Y'' is the quotient topology: a subset is open if and only if is open. :(iii) For any open subset , is an isomorphism. (Here, ''k'' is the base field.) The notion appears in geometric invariant theory. (i), (ii) say that ''Y'' is an orbit space of ''X'' in topology. (iii) may also be phrased as an isomorphism of sheaves . In particular, if ''X'' is irreducible, then so is ''Y'' and : rational functions on ''Y'' may be viewed as invariant rational functions on ''X'' (i.e., rational-invariants of ''X''). For example, if ''H'' is a closed subgroup of ''G'', then is a geometric quotient. A GIT quotient may or may not be a geometric quotient: but both are categorical quotients, which is unique; in other words, one cannot have both types of quotients (without them being the same). == Relation to other quotients == A geometric quotient is a categorical quotient. This is proved in Mumford's geometric invariant theory. A geometric quotient is precisely a good quotient whose fibers are orbits of the group. 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「geometric quotient」の詳細全文を読む スポンサード リンク
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