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

a) $f$ is separated.

b) The diagonal morphism $X\to X{\times }_{Y}X$ is quasicompact, and for every affine scheme $Y\prime =\mathrm{Spec}A$ in which $A$ is a valuation ring (resp. a discrete valuation ring), any two morphisms from $Y\prime \to X$ which coincide at the generic point of $Y\prime$ are equal.

c) The diagonal morphism $X\to X{\times }_{Y}X$ is quasicompact, and for every affine scheme of the form $Y\prime =\mathrm{Spec}A$ in which $A$ is a valuation ring (resp. a discrete valuation ring), any two sections of $X\prime =X(Y\prime )$ which coincide at the generic point of $Y\prime$ are equal.

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

• MathOverflow can-the-valuative-criteria-for-separatedness-properness-be-checked-formally

category: algebraic geometry