Một số ứng dụng của d-dãy distinguish = Some application of distinguished d-sequences

PhamH6ngNam Tap chf KHOA HOC & CONG NGHE 132(02): 147-153

SOME APPLICATIONS OF DISTINGUISHED d-SEQUENCES

PHAM HONG NAM

College of Sciences, Thai Nguyen University

Thai Nguyen, Vietnam

e-mail: [email protected]

Abstract

Let (R, m) be a Noetherian local ring and M a finitely generated i?-module of dimension

d. Let I = (a:i,..., xj) be a system of parameters of M. For each d-tuple of positive integers

'n = («!,...,Tirf), set /M(ZI,S) -= iRiM/{xi\... ,x'^'')M) -ui-..nae{x\M), where e{x;M) is

the multiplicity of M with respect to x In this paper, we give some applications of dd-sequences

in the study of the function /M{Z*I^) and certain cohomology modules. Recall that the notion

of dd-sequence was introduced by N. T. Cuong and D. T. Cuong [6], which is a distinguished

type of d-sequence defined by C. Huneke [10].

1 Introduction

Throughout this paper, let {R, m) be a commutative local Noetherian ring and M a finitely gen￾erated /^-module of dimension d. Let x = (xi,... , x^) be a system of parameters of M. For each

ti-tuple of positive integers n = {rti,..., na), we set

lM{n,x) : - ERiM/ix'l',... ,x'^'')M)-m . ..nde{x;M),

where e{x\M) is the multiplicity of M with respect to x. Consider IM{I1,^) as a function in n. It

well known that this function in general is not a polynomial, but it is bounded above by polynomials

and the least degree of all polynomials in n bounding above this function is independent of the

choice of X, of. [3, Theorem 2.3]. Following N. T. Cuong [3, Definition 2.4], this least degree is called

the polynomial type of M and denoted by p{M). If we stipulate the degree of the zero polynomial to

be -1 , then M is Cohen-Macaulay if and only if p(M) = —1 and M is generalized Coben-Macaulay

if and only if p(M) < 0. In general we have

p{M) < maxdim{R/AnaRW^{M)) (1.1)

i<d

and the equality holds when R is universally catenary and all formal fibers of R are Cohen-Macaulay,

cf. [3, Theorem 3.1(ii)], [11, Theorem 1.1]. This fact shows that the greater p{M), the more different

M is from the Cohen-Macaulayness. Therefore, it is very useful to study the function /M(M>£) and

the invariant p{M).

Let x = (xi,. ..,Xd) be a system of parameters of M. Following C. Huneke [10], (xi,..., xj) is

called a d-sequence of M if for all integers i,j satisfying l^i^j^dwe have

{{xi,...,Xi-i)M -.MXj) = {{xi,... ,Xt-i)M :M XiXj).

Then, by N. T. Cuong and D. T. Cuong [6, Remark 3.2 (iii)], (xi, ...,Xd) is called a dd-sequence

of M iff for any i e {l,...,d } and any rf-tuple of positive integers (ni,...,nd), the sequence

Keywords' d-sequence, dd-sequcnce, polynomials type.

147