Pham Hong Nam Tap chi KHOA HQC & CONG NGHE 135(05): 103 - 108
K O S Z U L H O M O L O G Y A N N I H I L A T O R S W I T H R E S P E C T T O D I S T I N G U I S H E D d - S E Q U E N C E S
P H A M H O N G N A M College of Sciences, T h a i Nguyen University
T h a i Nguyen, V i e t n a m e-mail: p h a m h o n g n a m 2 1 0 6 @ g m a i l . c o m
A b s t r a c t
Let {R, m) be a Noetherian local ring and M a finitely generated iS-nioduIe of dimension d.
Let X = {xi,. ., Xd) be a system of parameters of M . In this paper, we give j,onie 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-sequenoe defined by C Huneke [8].
1 Introduction
T h r o u g h o u t t h i s p a p e r , let {R, m) b e a c o m m u t a t i v e local N o e t h e r i a n r i n g a n d M a finitely gener- ated /^-module of d i m e n s i o n d. Let x = ( x i , . . . , x^) b e a system of p a r a m e t e r s of M. Following C.
Huneke [8], ( x i , . . ,Xrf) is called a d-sequence of M if for all integers i.j satisfying 1 ^ i ^ j $ d we have
{{xi,...,Xi-i)M iM Xj) = {{xi,...,Xi-i)M :M ^i^j)-
T h e n , by N. T . C u o n g a n d D, T , C u o n g [4, R e m a r k 3.2 (iii)], {xi,. .,Xd) is called a dd-sequence of M iff for a n y i G { l , . . . , d } a n d any d-tuple of positive integers (TII, . . . ,71^). t h e sequence x " ' , . - . , x " ' is a d-sequence of M/{x^^^, • • •, 3:j'*)M. It should b e m e n t i o n e d t h a t every dd-sequence is a d-sequence, b u t t h e converse s t a t e m e n t is not t r u e , cf. [4, E x a m p l e 3.10]. Moreover, if R is universally c a t e n a r y a n d all foniial fibers of R are C o h e n - M a c a u l a y t h e n dd-sequences of M exist, cf. [5].
T h e p u r p o s e of t h i s p a p e r is to use dd-sequence t o s t u d y c e r t a i n Koszul homology modules w i t h respect t o d d - s e q u e n c e s of M.
D e n o t e by Hi{x-, M) t h e i-th Koszul homology m o d u l e of M w i t h r e s p e c t t o x. If x is a s t r o n g d-sequence of M , i.e. {x"^,... ,x'^'') is a d-sequence for all n^,... ,na, t h e n
( X f c + i , , . , , X d ) i / j ( x i , , . ,Xk;M) = Q
for all fc = 1 , . . . , d a n d all j < fc, cf. [2, L e m m a 2.9). N o t e t h a t each dd-sequence is a s t r o n g d-sequence. For dd-sequences, we have t h e following t h e o r e m , which is t h e second m a i n result of this p a p e r .
T h e o r e m 1 . 1 . Let x — ( x i , . . . ,Xd) be a system of papameters of M which is a dd-sequence. Let
/k\ fc!
fc < d 6e a positive integer. For every integer i such that 1 < i < k, set ik := [ , = _ . ' , •
\i/ i.(lc-i).
Then for all integers n i , . . . , n ^ > 0 we have
(xfc_i+i,...,xrf)'*//.(<S---,4*;^) = o
Keywords' d-sequence, dd-sequence, Koszul homology module.
Pham Hong Nam Tap chi KHOA HOC & CONG NGHE 135(05): 103 - 108
anden{Hi{x'l^ 4'-^M))=j^(^'.^_~^)in{0:^^^^^._^^'-,^^x,^,).
In t h e next section, we give some preliminaries on dd-sequences t h a t will b e used in t h e sequel.
T h e proofs of t h e m a i n r e s u l t s , T h e o r e m 1.1, are presented in t h e last section.
2 Preliminaries on dd-sequences
Firstly, we recall t h e notion of d-sequence i n t r o d u c e d by C. H u n e k e [8]. For a s u b m o d u l e N of M and an ideal / of R, we set [N :M I) = {m € M \ Im C N}. I t is clear t h a t {N :M I) is a s u b m o d u l e of M containing N.
D e f i n i t i o n 2 . 1 . A sequence x = ( x i , . . , x^) of elements in R is called a d-sequence of M if
^t ^ (3:1,. •- , x , - i , x , + i , . . . , X s ) i ? a n d
{{xu•••,X^-l)M :MX;) = {{xi, . ,X^-l)M -.M XiXj)
for all integers i,j such t h a t 1 ^ i < j ^ d. We say t h a t x is a strong d-sequence of M if x{n) = ( x " ' , . . , , x "*) is a d-sequence of M for a n y s-tuple of positive i n t e r g e r s n= {ni,...,%).
It should b e m e n t i o n e d t h a t t h e notion of d-sequence is a very useful tool m m a n y different topics of C o m m u t a t i v e Algebra (for example, see [8], [4], [3]). A d i s t i n g u i s h e d kind of d-sequences, called dd-sequences, was introduced by N T , C u o n g a n d D . T . C u o n g [4, Definition 3.2] as follows.
D e f i n i t i o n 2 . 2 . A sequence x= {xi,.. .,Xs) of elements in m is called a dd-sequence of M ifx is a strong d-sequence and the following inductive conditions are satisfied:
(i) either s = I, or
(li) s > 1 and {xi,..., Xg^j) is a dd-sequence of M/x^M for all integers n > 0.
Next, we recall some basic properties of dd-sequences, t h a t will b e used for t h e proof of the main results of t h i s p a p e r . F r o m now on, for each mteger i > 1, we d e n o t e by ( x i , . . . ,Xi,.. .,Xs) t h e system ( x j , . . . . X i _ i , X i + i , . , , ,Xs).
P r o p o s i t i o n 2 . 3 . (See [4, P r o p o s i t i o n 3.4]). The following statements are true.
(i) If x_= {xi,... ,Xi) is a dd-sequence of M then ( x i , . . . , £ i , . . . ,Xs) is a dd-sequence of M/xjM for any integer z G { 1 , . , . , s } ,
(ii) If s > 3 then {xi x^) is a dd-sequence of M if and only if ( x i , . . . , x ^ , . . . , x^) is a dii- sequence of M/x^' M for all i e { 1 , . . . , s} and all positive integers n j .
Pi-om now on, let x — ( x i , , . . ,Xd) b e a s y s t e m of p a r a m e t e r s of M For d-tuple of positive integers n — ( n i , . . ,nd), set x ( n ) — ( x " \ . , , i X ^ ) .
L e m m a 2 . 4 . (See [9. L e m m a 2.5]) Let d > 2. Ifx — {xi,...,Xd) is a dd-sequence of M then for alii = 2 d we have £R[0 :A;/(X,+I, ,xa)M a^i) < 00 In particular,
e{x2,.•.,X^•,{0•.M/{x,+l, ,X^)M3;I))^0
Pham Hdng Nam Tap chi KHOA HOC & CONG NGHE 135(05): 103 - 108
3 Proofs of main results
Before giving the proof of main results, we need the following lemma.
Lemma 3.1. (See [7, Theorem 1.14]). ie( ( i i , . . . , i j ) be a strong d-sequence of M. Then JOT all k = 1,.. .,d and j < k, we have
{xt+i,...,Xi)Hjixi,.. ,xt;M)=0.
Proof of Theorem 1.1. For given integers i, k such that 1 < i < fc, we set ik-^ { 1 ^ '• .
\i) t\(k - iy.
We proceed by mduction on fc. If A; = 1 then Hi{x"^\M) = (0 :M X " ' ) . Since ^ is dd-sequence, Hl(xY\M) = (0 : „ xY) = (0 :M XI) C (0 : „ xix,) = (0 „ ij),
for all 1 < i < d. It follows that i,fl'i(x"';M) = 0 for all integer i = \,...,d. So, we get {xi i j ) H i ( x " ' ; M) = 0 and «n(Hi(iJ';M)) = £^(0 :« n ) , the result is true for fc = 1.
Assume that fc > 2 and the result is true for fc - 1, it means that ( i t _ j , . , . , x j ) " — / f , ( i j ' , . . . .x^^i'; Af) = 0, and
^ „ ( a ( x r . , . . . , x - - - ; M ) ) = ' f V ' : : * - M < „ ( ( 0 : ^ ^ ^ ^ ^ » , , „ x , « ) ) , for any integer i.
Fori = 1, since xi.s a dd-sequence of M, we get by using Lemma 2,4 the following exact sequence Q'^Hi{x'{\...,xl''_\';M)lxl'Hi{x'l' xlt\';M) ^ Hi{x1\... ,xl'-,M) ^
^ ( 0 >;/(.- .7-:)M-T)^^-
By Lemma 3,1, we get x]^''i?i ( x " \ . , .,x^''_^\M) — 0. So, the above exact sequence becomes 0 ^ i^Ti(xf,..., x « - / ; M) ^ i?,(x« ij>; M) ^ (0 -^^i^.. ^ . . _ . , „ xJ') ^ 0.
Since x is a dd-sequence of M, it follows by Proposition 2.3(ii) that (x^,..., Xd) is a dd-sequence of M/{x^\ ..., x^^"j')A/ for all integers n i , . . . , Jifc_i > 1. Therefore
for all ( = fc,... ,d. Hence (xfc,,.. ,Xd)(0 '•^^,,^'•1 x''''~')M ^fc*^ ~ *^" -^^ follows that {Xk-u •••, Xd)''-Hxk,..., xd){0 :„^(^., ^^*^ i j ^ x^*) - 0.
By induction, we have {xk-i>- •• ,Xd)''~^Hi{x"'. . . , x ^ t j ' ; M ) ^ 0. It follows that
{xk-\,...,x,)'-'Txk,.--,Xd)Hi{x1\...,xlt\';M)^0.
Pham H6ng Nam Tap chi KHOA HQC & CONG NGHE 135(05): 103 - 10!
So we get
( x t , . . . , x j ' / f , ( x ; " , . . . , x ^ * l M ) C ( x t _ , x j ) ' - i { x i , . . . , x j ) i f , ( x r , . . . , x J ' ; M ) = 0 . Moreover, we have
«„(ff,(x?',..,,Xj"';M))=<„(if,(x?' x ; i j ' ; M ) ) + f « ( ( 0 : „ / , ^ . . ,;;._.)„ x^))
the result is true for 1 = 1.
Let i > 1 Since x is a dd-sequence, it follows by Lemma 2.4 the exact sequence 0 -> i ? , ( i j ' , . . . ,Xji-,'; M)/x;»/fi(xJ',... ,x;i-j'; Af) - t H,{x';',... ,xj»; M) -»
^(»^H, ,(.r x;i,')«^r)^''-
Because xU*ffi(x"',... ,x|^*j'; JW") ^ 0 by Lemma 3.1 and
(° ^ft-,(.r',. ,.:>T';«) ^J*) = - f f - i W . • • ••4-1-.M], we have an exact sequence
u ^ f t ( i J ' , . . . , X t l V ; - M ) - » « i W x J » ; M ) - » t f , _ i ( x J ' , . . . , i ; i - i ' ; M ) - t O Therefore, we get by induction that
( H _ . , . . . , I J ) * ' - ' i J , ( x f ' , . . . , i ^ ^ j ' ; M) = 0, and (xt_,+,,..., x j ) ( - ' ) ' - ffi_i ( x f . . ., xlt\'; M) = 0.
It follows that
(xt_i+i,... ,xj)<'-i>»->(xt_„... , x j ) " — i J i ( x J ' , . . . , x l f i ; Af) = 0,and (xt-i x j ) - » - ' ( x t _ . + i , . . . , x j ) ( ' - ' ' ' - i f f , - . i ( x J ' , . . . , i ^ l - ' ; M ) = 0 . Therefore
(Xk_,+i Xd)*>S.(xJ',...,xJ';Al-)
e ((xt_, Xd)"->(xt_,+i, . , , x j ) t ' - ' ) ' - ' i f , ( x J ' , . . . , i J ' ; Af) = 0.
Moreover,
<„(ff,(xJ',...,i;«;Af))=<H(ffj_i(x;' x ; i - ' ; M ) ) - l - « „ ( i f i ( x f ' , . . . , x ^ i - l ' i ' ^ ) )
J . 0
• fc-J-1
3=0
the result is proved completely.
§ ( ' 7 - 2 ' ) M ( 0 ^ „ / K . . ,.;.,„x,«))
Pham H6ng Nam Tap chf KHOA HOC & CONG NGHE 135(05): 1 0 3 - 108
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