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Koszul homology annihilators with respect to distinguished D sequences

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Let (R, m) be a Noetherian local ring and M a finitely generated R-module of dimension d. Let x = (x1, . . . , xd) be a system of parameters of M. In this paper, we give some applications of dd-sequences in the study of certain Koszul homology modules. Recall that the notion of dd-sequence was introduced by N. T. Cuong and D. T. Cuong [4], which is a distinguished type of d-sequence defined by C. Huneke. | Phạm Hồng Nam Tạp chí KHOA HỌC & CÔNG NGHỆ 135(05): 103 - 108 KOSZUL HOMOLOGY ANNIHILATORS WITH RESPECT TO DISTINGUISHED d-SEQUENCES PHAM HONG NAM College of Sciences, Thai Nguyen University Thai Nguyen, Vietnam e-mail: phamhongnam2106@gmail.com Abstract Let (R, m) be a Noetherian local ring and M a finitely generated R-module of dimension d. Let x = (x1 , . . . , xd ) be a system of parameters of M . In this paper, we give some applications of dd-sequences in the study of certain Koszul homology modules. Recall that the notion of dd-sequence was introduced by N. T. Cuong and D. T. Cuong [4], which is a distinguished type of d-sequence defined by C. Huneke [8]. 1 Introduction Throughout this paper, let (R, m) be a commutative local Noetherian ring and M a finitely generated R-module of dimension d. Let x = (x1 , . . . , xd ) be a system of parameters of M . Following C. Huneke [8], (x1 , . . . , xd ) is called a d-sequence of M if for all integers i, j satisfying 1 6 i 6 j 6 d we have (x1 , . . . , xi−1 )M :M xj = (x1 , . . . , xi−1 )M :M xi xj . Then, by N. T. Cuong and D. T. Cuong [4, Remark 3.2 (iii)], (x1 , . . . , xd ) is called a dd-sequence of M iff for any i ∈ {1, . . . , d} and any d-tuple of positive integers (n1 , . . . , nd ), the sequence ni+1 xn1 1 , . . . , xni i is a d-sequence of M/(xi+1 , . . . , xnd d )M. It should be mentioned that every dd-sequence is a d-sequence, but the converse statement is not true, cf. [4, Example 3.10]. Moreover, if R is universally catenary and all formal fibers of R are Cohen-Macaulay then dd-sequences of M exist, cf. [5]. The purpose of this paper is to use dd-sequence to study certain Koszul homology modules with respect to dd-sequences of M . Denote by Hi (x; M ) the i-th Koszul homology module of M with respect to x. If x is a strong d-sequence of M , i.e. (xn1 1 , . . . , xnd d ) is a d-sequence for all n1 , . . . , nd , then (xk+1 , . . . , xd )Hj (x1 , . . . , xk ; M ) = 0 for all k = 1, . . . , d .