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Báo cáo " The total specialization of modules over a local ring"

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. | VNU Journal of Science Mathematics - Physics 25 2009 39-45 The total specialization of modules over a local ring Dao Ngoc Minh Dam Van Nhi Department of Mathematics Hanoi National University of Education 136 Xuan Thuy Road Hanoi Vietnam Received 23 March 2009 Abstract. 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 a a1 . am 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 for almost all a that is for all a except perhaps those lying on a proper algebraic subvariety of Km. Though specialization is a classical method in Algebraic Geometry there is no systematic theory for what can be specialized . The first step toward an algebraic theory of specialization was the introduction of the specialization of an ideal by W. Krull in 1 . Given an ideal I in a polynomial ring R k u x where k is a subfield of K he defined the specialization of I as the ideal Ia f a x f u X G I n k u x of the polynomial ring Ra k a x . For almost all a G Km Ia inherits most of the basic properties of I. Let pu be a separable prime ideal of R. In 2 we introduced and studied the specializations of finitely generated modules over a local ring Rpu at an arbitrary associated prime ideal of pa For specialization of modules see 3 . Now we will introduce the notation about the total specializations of modules. We showed that the Cohen-Macaulayness the Gorensteiness and Buchsbaumness of a module are preserved by the total specializations. 2. Specializations of prime separable ideals Let pu be an arbitrary prime ideal of R. The first obstacle in defining the specialization of Rpu is