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The gorensteiness of modules over a noetherian local ring by specializations

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In this paper, we prove that the Gorensteiness, the injective resolution, the injective envelope and Gorenstein dimension of modules are preserved by specializations. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2013 Vol. 58 No. 7 pp. 66-71 This paper is available online at http stdb.hnue.edu.vn THE GORENSTEINESS OF MODULES OVER A NOETHERIAN LOCAL RING BY SPECIALIZATIONS Dam Van Nhi1 and Luu Ba Thang2 1 School for Gifted Student Hanoi National University of Education 2 Faculty of Mathematics Hanoi National University of Education Abstract. In this paper we prove that the Gorensteiness the injective resolution the injective envelope and Gorenstein dimension of modules are preserved by specializations. Keywords. Specialization injective dimension Gorensteiness. 1. Introduction Specialization is a technique which considers algebraic equations with generic coefficients and substitute the generic coefficients by elements of the base field. This technique is usually used to show the existence of algebraic structures with a given property. The theory of specialization of ideals was introduced by W. Krull 3 where it was shown that the property of being a prime ideal is preserved by almost all specializations. Using specializations of finitely generated free modules and homomorphisms between them D.V. Nhi and N.V. Trung defined in 5 the specializations of finitely generated modules over a local ring. They showed that the basic properties of modules are preserved by specializations. Developing the ideas of 5 we show that the Gorensteiness the injective resolution of a module are also preserved by specializations. Notice that in 4 Minh and Nhi proved that the Gorensteiness is preserved by total specializations. 2. Specializations of RP -modules Throughout this paper we alway assume that k is an arbitrary perfect infinite field and K is an extension field of k. We denote the polynomial rings in n variables x1 . . . xn over k u and k α by R k u x and by Rα k α x respectively where u u1 . . . um is a family of parameters and α α1 . . . αm K m is a family of elements of an infinite field K. Received October 12 2013. Accepted .