tailieunhanh - Báo cáo "A NEW NUMERICAL INVARIANT OF ARTINIAN MODULES OVER NOETHERIAN LOCAL RINGS "

Let (R, m) be a commutative Noetherian local ring the maximal ideal and A an Artinian R-module with Ndim A = d. For each system of parameters x (x1 , ., xd ) of A, we denote by e(x, A) the multipility of A with respect to x. Let n (n1 , n2 , ., nd ) be a d-tuple of positive integers. The paper concerns to the function d-variables I(x(n); A) := where | VNU. JOURNAL OF SCIENCE Mathematics - Physics. No2 - 2005 A NEW NUMERICAL INVARIANT OF ARTINIAN MODULES OVER NOETHERIAN LOCAL RINGS Nguyen Duc Minh Department of Mathematics Quy Nhon University Abstract. Let R m be a commutative Noetherian local ring the maximal ideal m and A an Artinian R-module with Ndim A d. For each system of parameters x xi . Xd of A we denote by e x A the multipility of A with respect to x. Let n ni n2 . nd be a d-tuple of positive integers. The paper concerns to the function of d-variables I x n A r 0 a x 1 . xnd R - e xn1 . xd d A where r - is the length of function. We show in this paper that this function may be not a polynomial in the general case but the least degree of all upper-bound polynomials for the function is a numerical invariant of A. A characterization for co Cohen-Macaulay modules in term of this new invariant is also given. Keywords Artinian module multiplicity 1. Introduction Throughout let R m denote a commutative Noetherian local ring with the maximal ideal m and A an Artinian R-module with Ndim A d 0. For each system of parameters x xi . xd for A we denote by e x A the multiplicity of A with respect to x in the sense of 3 . It has been shown by Kirby in 8 that there exist q n E Q x and no E N such that -Ểr 0 r xi . xd nA q n Vn A no. It is very important that the degree of q n equals d and if ad is the lead coefficient of q n then ad d agrees with e x A . Let n ni . nd E Nd and consider I x n A r 0 a x . xnd - ni nd e x A as a function on ni . nd. As shown in Example this function may be not a polynomial on ni . nd even when n large enough . The aim of this paper is to show that the above function is still interesting to investigate. First the least degree of all polynomials bounding this function from above is a numerical invariant of A. Moreover this invariant carries informations on structure of A. The existence of our invariant is proved in the third section. But before doing this in the second section we