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valuative criterion of separatedness

The valuative criterion of separatedness is EGA II, 7.2.3. (numdam) which translated into English says

Proposition. Let XX be a scheme (resp. a locally noetherian scheme), f:XYf: X\to Y a morphism of schemes (resp. a morphism locally of finite type). The following conditions are equivalent

a) ff is separated.

b) The diagonal morphism XX× YXX\to X\times_Y X is quasicompact, and for every affine scheme Y=SpecAY' = Spec A in which AA is a valuation ring (resp. a discrete valuation ring), any two morphisms from YXY'\to X which coincide at the generic point of YY' are equal.

c) The diagonal morphism XX× YXX\to X\times_Y X is quasicompact, and for every affine scheme of the form Y=SpecAY' = Spec A in which AA is a valuation ring (resp. a discrete valuation ring), any two sections of X=X(Y)X' = X(Y') which coincide at the generic point of YY' are equal.

Compare the valuative criterion of properness, EGA II, 7.3.8.

Revised on March 6, 2013 18:57:00 by Zoran Škoda (161.53.130.104)