In this paper we introduce the total specialization of an finitely generated module over local ring. This total specialization preserves the Cohen-Macaulayness, the Gorensteiness and Buchsbaumness of a module. The length and multiplicity of a module are studied. 1. Introduction Given an object defined for a family of parameters u = (u1, . . . , um ) we can often substitute u by a family α = (α1, . . . , αm) of elements of an infinite field K to obtain a similar object which is called a specialization. The new object usually behaves like the given object.
