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 $Y' = Spec A$ in which $A$ is a valuation ring (resp. a discrete valuation ring), any two morphisms from $Y'\to X$ which coincide at the generic point of $Y'$ are equal.

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

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

category: algebraic geometry

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