The Specializations in a Scheme

Abstract

In this paper we will obtain some further properties for specializations in a scheme. Using these results, we will take a picture for a scheme and a picture for a morphism of schemes. In particular, we will prove that every morphism of schemes is specialization-preserving and of norm not greater than one (under some condition); a necessary and sufficient condition will be given for an injective morphism of irreducible schemes.