Valuative criterion
= Valuative criterion
{wiki}
Valuative criteria test properties of scheme morphisms by extending a lift from the generic point of the spectrum of a valuation ring to the whole spectrum. Under the usual finiteness hypotheses, uniqueness characterizes separatedness, existence characterizes universal closedness, and existence with uniqueness characterizes properness. The <valuative criterion for separatedness> and <valuative criterion for properness> are the respective specific statements.