Let $X$ be an algebraic prevariety over an algebraically closed field $k$ (I.e. an integral scheme of finite type over $k$). In the valuative criteria for separatedness and properness of $X$ over $k$, is it enough to consider only DVR's of transcendence degree $1$ over $k$? If so, is it also enough for determining the separatedness and properness of a morphism of prevarieties over $k$?
For both, please point me in the direction of a proof/counterexample.