The nice m-system of parameters for Artinian modules

Nguyen Thi Khanh Hoa- Volume 2 - Issue 2-2020, p.158-165. 
The nice m-system of parameters for Artinian modules 
 by Nguyen Thi Khanh Hoa (Thu Dau Mot University) 
Article Info: Received 02 Jan. 2020, Accepted 29 Feb. 2020, Available online 15 June. 2020 
 Corresponding author: hoanguyenthikhanh@gmail.com 
 https://doi.org/10.37550/tdmu.EJS/2020.02.042 
 ABSTRACT 
 This paper restates the definition of the nice m-system of parameters for 
 Artinian modules. It also shows its effects on the differences between lengths and 
 multiplicities of certain systems of parameters for Artinian modules: 
 n1 n 2nndd n 1 n 2
 Ixn;A R 0: A x 1 ,x 2 ,...,x d R ex 1 ,x 2 ,...,x d ;A 
 In particular, if x is a nice m-system of parameters then the function 
 I x n ; A is a polynomial having very nice form. Moreover, we will prove 
 some properties of the nice m-system of parameters for Artinian modules. 
 Especially, its effect on the annihilation of local homology modules of Artinian 
 module A. 
 Keywords: annihilation, Artinian module, function of certain systems of 
 parameters, local homology, nice m-system of parameters 
1. Introduction 
Throughout this paper, (R, m) is a commutative Noetherian local ring with the maximal 
ideal m, and A is an Artinian R-module with N-dimA = d. 
 Rt
 + The R-module limTori R m ; A is called ith-local homology module of A 
 t
 m
with respect to m and denoted by HAi . 
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 Thu Dau Mot University Journal of Science - Volume 2 - Issue 2-2020 
 + Let n n1 ,n 2 ,...,n d be a d-tuple of positive integers. For each system of 
parameters (s.o.p) x x1 ,x 2 ,...,x d of A, we consider 
 n1 n 2nndd n 1 n 2
 Ixn;A R 0: A x 1 ,x 2 ,...,x d R ex 1 ,x 2 ,...,x d ;A 
as a function d-variables on n1 ,n 2 ,...,n d . 
Let I A sup I x; A where x runs over all s.o.p of A. 
 x 
The value of function I x; A and the annihilation of local homology modules of A 
help us classify many different types of modules. Moreover, they also give us lots of 
information about different types of modules (see [3]). Such as: 
 + I A 0 : A is a co-Cohen-Macaulay module. 
 + IA : A is a Generalized co-Cohen-Macaulay module. 
 + I x; A is a constant for all s.o.p of A: A is a co-Buchsbaum module. 
 + If A is a Generalized co-Cohen-Macaulay module, there exists an m-primary 
 m
ideal q such that qHi A 0 for all i = 1, , d – 1. 
 + If A is a co-Buchsbaum module, mHm A 0 for all i = 1, , d – 1. 
 i 
However, I x n ; A may be not a polynomial on n ,n ,...,n even when 
 1 2 d 
large enough (see [1]), but [2] has shown that if is a nice m-systems of parameters, 
 is a polynomial with simple form. In addition, a nice s.o.p of A also 
annihilates local homology modules of A. Thus, in this paper we will restate the 
definition of the nice m-s.o.p for Artinian modules, the effect of the nice m-s.o.p on the 
calculation formula of function and continue studying some its properties. 
Especially its effect on the annihilation of local homology modules of A. 
2. Preliminaries 
 ˆ
Lemma 2.1([1]). Assume AnnR A R AnnRˆ A and x x1 ,x 2 ,...,x d is a s.o.p of 
Artinian R-module A. Then, there exits j 1,2,...,d such that x j is a pseudo-A-
coregular element. 
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Nguyen Thi Khanh Hoa- Volume 2 - Issue 2-2020, p.158-165. 
Lemma 2.2 ([3]). Let xR be a pseudo-A-coregular element. Then R A xA . 
Lemma 2.3 ([4]). Let M be an R-module, I be an ideal of R. Then for all i ≥ 0, 
 sI
 IHMi 0. 
 s 0
Lemma 2.4 ([1]). Let s a positive integer such that mts A m A,  t s. Then 
 ms
 H0 A A m A. 
Lemma 2.5. ([3]). For every s.o.p x of A, we have 
 d 1
 d 1 m
 R 0: AxR e x ; A  R H i A . 
 i 0 i
Moreover, if HAm for all i < d, then there exists an m-primary ideal q such 
 Ri 
that the equality holds for every s.o.p contained in q. 
Definition 2.6 ([2]). 
* The sequence x1t ,...,x m is called an m-sequence for A if: 
 (i) xks  x R for all k = 1, , t, 
 sk 
 (ii) x 0: x,...,x R xx 0: x,...,x R for all 1 i k t x 0 . 
 kA1 i1 kiA1 i1 0 
 n1 nt
* The sequence x1t ,...,x m is called a strong m-sequence for A if x1t ,...,x is m-
 t
sequence for all n1t ,...,n . 
* A strong m-sequence x1t ,...,x m is called a nice m-sequence for A if: 
 (i) t = 1; or 
 (ii) t > 1 and x ,...,x is a strong m-sequence of 0 : xnnit ,...,x R for 
 1 i 1 A i t 
all 2 it and for all nit ,...,n . 
* A s.o.p for A is called a nice m-s.o.p if it is a nice m-sequence. 
Lemma 2.7 ([2]). Let x ,...,x be an m-sequence for A. Then: 
 1t 
 n
 (i) xi 0: A x,...,x 1 i 1 R x i 0: A x,...,x 1 i 1 R for all 1 it and n; 
 (ii) for every (i, k) with 1 i k t we have 
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 Thu Dau Mot University Journal of Science - Volume 2 - Issue 2-2020 
 xk 0: A x,...,x 1 i 1 R  x i 0: A x,...,x 1 i 1 R; 
 (iii) x2t ,...,x is an m-sequence for 0 :A1 x . 
The following theorem shows that I x n ; A will be a polynomial when 
x x1 ,x 2 ,...,x d is a nice m-s.o.p for A. Furthermore, in this case it has a nice form. 
Theorem 2.8 ([2]). Let be a s.o.p for A. Then the following three 
conditions are equivalent: 
 (i) x is a nice m-s.o.p for A; 
 (ii) there exist non-negative intergers 01 x, A ,...,d x , A such that 
 d 1
 I x n ; A 01 x , A  n ... nii . x , A 
 i 1
for all nn1,...,d 1; 
 (iii) 
 d 1 
 0:A x22 ,..., x d R 0: A x i ,..., x d R
I x n ; A R n11 ... n i . e x ,..., x i ; 
 x0: x ,..., x R  x 0: x ,..., x R 
 1 A 2 d i 1 i 1 A i 2 d 
for all nn1,...,d 1. 
3. Main results 
In this section, we give some corollaries of Theorem 2.8. 
Corollary 3.1. Let x1 ,x 2 ,...,x d be a nice m-s.o.p for A with N-dimA = d 2. Then 
 i) For all nn,..., we have 
 1 d 
 n1 n 2nd n 1 n 2
 I x1, x 2 ,..., xdd ; A I x 1 , x 2 ,..., x ; A . 
 ii) For all nn,..., we have 
 2 d 
 nn22nndd
 I x2,..., xd ;0: A x 1 I x 1 , x 2 ,..., x d ; A . 
Proof. 
 nn12 nd
 i) From (iii) of Theorem 2.8, we find that I x12, x ,..., xd ; A doesn’t depend on 
nd. So we have 
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Nguyen Thi Khanh Hoa- Volume 2 - Issue 2-2020, p.158-165. 
 n1 n 2nd n 1 n 2
 I x1, x 2 ,..., xdd ; A I x 1 , x 2 ,..., x ; A . 
 d 1
 ii) For any nn2 ,..., d , we have 
 n2nd n 2 n d n 2 n d
 Ix 2,..., xd ;0: A x 1 Ixx 1 , 2 ,..., xAex d ; 2 ,..., xAxA d ; 1 . 
 n2
Because x1 ,x 2 ,...,x d is an m-sequence, by Lemma 2.7 we have x2 A x 2 A x 1 A. 
 n2 n2 nd
Hence, x21 A x A 0. So that e x21,..., xd ; A x A 0. 
 nn22nndd
This deduce I x2,..., xd ;0: A x 1 I x 1 , x 2 ,..., x d ; A . 
Next, we give some example for nice m-s.o.p. 
Remark 3.2. 
 i) Let A be an co-Cohen-Macaulay R-module. Then every s.o.p of A is a nice m-
s.o.p. 
 ii) Let A be an co-Buchsbaum R-module. Then every s.o.p of A is a nice m-s.o.p. 
 iii) Let A be an generalized co-Cohen-Macaulay R-module. Then there exists an 
m-primary ideal q such that every s.o.p contain in q is a nice m-s.o.p. 
Proof. 
 i) As A is an co-Cohen-Macaulay module, I A supx I x ; A 0 with x run 
over all s.o.p of A. From Theorem 2.8, we get is a nice m-s.o.p. 
 ii) As A is an co-Buchsbaum module, I x; A is a constant (not depending on 
s.o.p of A). From Theorem 2.8, we get is a nice m-s.o.p. 
 iii) As A is an generalized co-Cohen-Macaulay R-module, HAm for 
 Ri 
all i < d. Thus, from Lemma 2.5 and Theorem 2.8, there exists an m-primary ideal q 
such that every s.o.p contain in q is a nice m-s.o.p. 
Finally, we continue studying the effect of a nice m-s.o.p on the annihilation of local 
homology modules of A. 
 ˆ
Proposition 3.3. Assume AnnR A R AnnRˆ A and x x1 ,x 2 ,...,x d is a s.o.p and a 
strong m-sequence of Artinian R-module A. Then 
 m
 xji H A 0 for all 0. i j d 
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 Thu Dau Mot University Journal of Science - Volume 2 - Issue 2-2020 
Proof. We proceed by introduction on d = N-dimA. 
 For d = 1 and let x1 be a s.o.p of A. Because of A is an Artinian R-module, the 
system mAt  is stationary, i.e there exists a positive interger s such that mts A m A, 
for all t ≥ s. 
 ms
It follows from Lemma 2.4 that x1 H 0 A x 1 A m A. 
 s s
Since is m-sequence for A and x11 A x A , we have x1 A m A. This implies 
 m
x10 H A 0. 
 Assume that d > 1 and our assertion is true for all Artinian R-module of N-dim 
smaller than d. 
 m
First, we shall prove xj H0 A 0 for all 1 ≤ j ≤ d. Similar proof in case d = 1, from 
 ss
Lemma 2.7, we get xj A x11 A x A  m A. 
 m
Next, we shall prove xji H A 0 for all 1 ≤ i < j ≤ d. 
According to Lemma 2.1 and Lemma 2.2, there exists kd 1,...,  such that xk is a 
pseudo-A-coregular element and Rk A x A . Since x1 ,x 2 ,...,x d is a s.o.p and a 
strong m-sequence of A, we have xk A x1 A. Thus R A x1 A R A x k A . This 
 m
deduces that N-dim A x1 A ≤ 0. So Hi A x1 A 0 for all i > 0. 
The exact sequence 00 x11 A A A x A generates the long exact sequence 
 m m m m
 Hi 1 AxA 1 HxA i 1 HA i HAxA i 1 
 mm mm
Since Hii 1 A x 1 A H A x 1 A 0 we have Hii x1 A  H A for all i > 0. 
 n
Moreover, because x1 ,x 2 ,...,x d is an m-sequence of A, we get x11 A x A for all n > 0. 
This deduces 
 m m m n
 Hi A  H i x11 A H i x A for all i, n > 0. 
 nn
Combining this result and the exact sequence 0 0:A x11 A x A 0 we have 
the long exact sequence: 
 nn
 mxx11 m i m n m m
 Hi A  H i A  H i 1 0: A x 1 H i 1 A  H i 1 A 
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Nguyen Thi Khanh Hoa- Volume 2 - Issue 2-2020, p.158-165. 
 mm
 n n m HAHAii 
Since Ker ii Im x11 x H A ,∀ n > 0 we have Im i  nm . 
 Ker ii x1 H A 
As x1 ,x 2 ,...,x d is a strong m-sequence of A then x2d ,...,x is a strong m-sequence of 
 n mn
0:A x1 . Applying the inductive hypothesis for to have xj H i 11 0: A x 0 for all 
1 ≤ i < j ≤ d. Therefore x jiIm 0 . Combining this result and Lemma 2.3 we get 
 m n m n m m n m
xHAj i  xHAmHAn1 i  i ,  0 xHA j i  mHA i 0. 
 n 0
Our proof is complete. 
 ˆ
Corollary 3.4. Assume AnnR A R AnnRˆ A. 
 i) Let be a s.o.p and strong m -sequence of A. Then 
 m n1 nk
 xj H i 0: A x1 ,..., x k 0 for all 0 ≤ i, k < j ≤ d. 
 ii) Let be a nice m-s.o.p of A. Then 
 m nnkd
 xj H i 0: A x k ,..., x d 0 for all 0 ≤ i < j < k ≤ d. 
Proof. 
 i) Because is a s.o.p and a strong m-sequence of A, 
xn1 ,..., xnk , x n k 1 ,..., x n d is also a s.o.p and an m-sequence of A for all n, n ,..., n .
 11k k d 12 d 
 n1 nk
Therefore xk 1 ,...,x d is a s.o.p and a strong m-sequence of 0:Ak xx1 ,..., . 
By Proposition 3.3, we have x Hm 0: xn1 ,..., xnk 0 for all 0 ≤ i, k < j ≤ d. 
 j i A 1 k 
 ii) Because is a nice s.o.p, x1 ,...,x k 1 is also a s.o.p and a strong m-
 nnkd
sequence of 0:A xx k ,..., d for all nnkd,..., . 
 m nnkd
By Proposition 3.3, we have xj H i 0: A x k ,..., x d 0 . 
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 Thu Dau Mot University Journal of Science - Volume 2 - Issue 2-2020 
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