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The Norm of a Morphism of Schemes

By Weil's theory of specializations, we will define a distance (i.e., length) between two points in a scheme in an evident manner and then give a definition for the norm of a morphism of schemes reflecting its global behavior. To see such a norm is well-defined, we will prove that any morphism of schemes keeps the specializations. And it will be showed that any specialization in a scheme is contained in some affine open subset. Then we can take pictures for schemes and their morphisms: Schemes look like trees and morphisms exactly scale down or up the trees depending on their norms greater than one or less than one. The main theorems of the paper will be devoted to study the morphisms of norms not greater than one, especially the length-preserving morphisms.