VNU. JOURNAL OF SCIENCE, Mathematics - Physics. T.XXI, N
0
2 - 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 =
(x
1
, ,x
d
) of A, we denote by e(x,A) the multipility of A with respect to x. Let n =
(n
1
,n
2
, ,n
d
) be a d-tuple of positive integers. The paper concerns to the function of
d-variables
I(x(n); A):=f
R
(0 :
A
(x
n
1
1

to x
in the sense of [3]. It has been shown by Kirby in [8] that there exist q(n) ∈ Q[x]
and n
0
∈ N such that f
R
(0 :
R
(x
1
, ,x
d
)
n
A)=q(n), ∀n  n
0
. It is very important that
thedegreeofq(n)equalsd and if a
d
is the lead coefficient of q(n)thena
d
· d!agreeswith
e(x
; A).
Let n
=(n
1
, ,n
d
) ∈ N

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 recall basic terminologies
and resuls which are needed later. Some relations between the new invariant with local
homology modules are presented in the last section.
Typeset by A
M
S-T
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34
A new numerical invariant of Artinian Modules over 35
2. Preliminaries
In this section, K isanonzeroArtinianR-module.
2.1. The residuum, residual lengthandwidthofArtinianmodules
We devote this subsection to recall some basic terminologies and results from [11]
and [13].
Let K =
h
3
i=1
C
i
be a minimal secondary representation of K. Set
p
i
=
0
0:

K
(a
1
, ,a
n
)R =0anda
i
is
0:
K
(a
1
, ,a
i−1
)R-coregular element for every i =1, ,n. We denote by Width(K)
the suprem um of lengths of all K-cosequences in m. It should be mentioned that a ∈ R is
K-coregular if a nd only if a ∈

p∈Att (K)
p.
An element a ∈ m is called pseudo-K-coregular if a ∈

p∈Att(K)−{m}
p. We define the
stability index s = s(K)ofK to be the least integer i  0 such that m
i
K = m
i+1
K.
Note that m

R
(K
n+1
/K
n
) <t
for all n>n
0
.
Asystemx
=(x
1
, ,x
t
)ofelementsinm is called a multiplicity system of K if
f
R
(0 :
K
(x
1
, ,x
t
)R) < ∞. Assume that N − dim
R
K = d, then a multiplicity system of
K is called a system of parameter (s.o.p for short) for K if t = d.
Let x
=(x
1

1
, ,n
d
) ∈ N
d
. Then
I(x
(n); A) a n
1
···n
d
I(x
1
, ,x
d
; A).
Proof: By [6], Lemma 2
f
R
(0 :
A
y
m
) a mf
R
(0 :
A
y), ∀y ∈ A, ∀m ∈ N.
Using an induction on d, we get
f

e(x
1
, ,x
d
; A). (2)
The proposition then com es from (1) and (2).
The proposition 3.1 leads to an immediate consequence as follows.
3.2. Corollary. If I(x
(n); L) is a polynomial, then it is linear in each n
i
,i=1, ,d.
The main result of this section is the following.
3.3. Theorem. Let x
=(x
1
, ,x
d
) be a s.o.p of A. Then, the least degree of all
polynomials in n
1
, ,n
d
bounding the function I(x
n
1
1
, ,x
n
d
d

forms a s.o.p of

R-module A. Furthermore,
(0 :
A
(/x
1
, ,/x
d
)

R)=(0:
A
(x
1
, ,x
d
)R)
and therefore,
I(x
n
1
1
, ,x
n
d
d
; A)=I(/x
1
, ,/x

is a Noetherian over R and we have
m
l
R ⊆ Ann
R
(0 :
A
(x
1
, ,x
d
)R)
∨
=Ann
R
(A
∨
/(x
1
, ,x
d
)A
∨
)
⊆
0
Ann
R
(A
∨

1
, ,x
d
)R +Ann
R
(A))
Q
t
⊆ ((x
1
, ,x
d
)R +Ann
R
(A)).
To finish our claim one just set k = tl. q
3.5. Lemma. Let x
1
,x
2
, ,x
d
and y
1
,y
2
, ,y
d
be two s.o.p’s of R with x
1

n
d
d
)R =(z
n
1
1
,x
n
2
2
, ,x
n
d
d
)R (3)
and
(y
n
1
1
,y
n
2
2
, ,y
n
d−1
d−1
,y

n
d−1
d−1
,x
n
d
d
.y
n
d
d
)R +Ann
R
(A)(5)
for some k ∈ N. Let A =
h
3
i=1
S
i
be a minimal secondary representation of A. Then
0
Ann
R
(A)=
>


:
0:

d−1
d−1
,x
n
d
d
.y
n
d
d
)R +
<
p∈Att(A)
p.
This implies x
1
R +(x
n
2
2
, ,x
n
d−1
d−1
,x
n
d
d
.y
n

We have now z
1
is a pseudo-A-coregular. Furthermore, for each n
1
∈ N, one can find
c
n
1
∈ (x
n
2
2
, ,x
n
d−1
d−1
,x
n
d
d
.y
n
d
d
)A such that z
n
1
1
= x
n

d−1
and
every n
=(n
1
, ,n
d
) ∈ N
d
, it holds
I(x
(n); A) a tI(y(n); A).
Proof: We proceed induction on d. For d =1, by [6] (Lemma 2),
I(x
(n); A)=f
R
(A/x
n
1
1
A)=f
R
(A/(x
n
1
1
A +Ann
R
A)A)
a f

1
L) < ∞ and
e(x
n
2
2
, ,x
n
d−1
d−1
,x
n
d
d
; L/x
n
1
1
L)=0; e(y
n
2
2
, ,y
n
d−1
d−1
,y
n
d
d

,x
n
d
d
;0:
A
x
n
1
1
)
and
e(y
n
1
1
,y
n
2
2
, ,y
n
d−1
d−1
,y
n
d
d
; A)=e(y
n

d
;0:
A
x
n
1
1
)(7)
and
I(y
(n); A)=I(y
n
2
2
, ,y
n
d−1
d−1
,y
n
d
d
;0:
A
y
n
1
1
)=I(y
n

A ⊆ (x
n
2
2
, ,x
n
d−1
d−1
,x
n
d
d
)A +Ann
R
(0 :
A
x
n
1
1
),
we can apply the inductive hypothesis for 0 :
A
x
n
1
1
to obtai n
I(x
n

). (9)
The proposition now follows from (7), (8) and (9). q
We now already to prove our main theorem.
Let y
=(y
1
, ,y
d
) be arbitrary s.o.p of A. Then we can connect x and y by a
sequence of not more than (2d + 1) s.o.p’s of A with the property that t wo consecutive
ones differ by at most one element. By repeated applications of Lemma 3.6, one can find
natural numbers t
1
,t
2
suc h that, ∀n ∈ N
d
,
I(x
(n); A) a t
1
I(y(n); A)andI(y(n); A) a t
2
I(x(n); A).
A new numerical invariant of Artinian Modules over 39
The p roof is then complete.
The above theorem means that the least degree of all polynomials bounding from
above I(x
(n); A) is a numerical invariant of A. From no w on, we denote this invariant
by ld

Y
3
,Y
2
Y
3
), where k is a field and We denote
by x
1
,x
2
the natural images of Y
1
+ Y
3
,Y
2
+ Y
3
in B, then x =(x
1
,x
2
)formsasystemof
parameters for the Noetherian module B (as B-module). It can be ve rified that
f
B
(B/(x
n
1

is an Artinian B-module and x is also a
system of parameters for B
∨
. It goes from basic facts of Matlist dual that
f
B
(B/(x
n
1
1
,x
n
2
2
)B)=f
B
((B/(x
n
1
1
,x
n
2
2
)B)
∨
)=f
B
(0 :
B

,x
2
; B)+min{n
1
,n
2
}.
Moreover, because
f
B
(B/(x
1
,x
2
)
t
B)=f
B
((B/(x
1
,x
2
)
t
B)
∨
)=f
B
(0 :
B

∨
)=1.
4. Connect to local homology modules
We devote this section to sho w some ralations between the invariant ld and local
homology modules. But let us first recall the definition of local homology which is first
introduced in [5].
4.1. Definition.Let I be an ideal in R and let i is a non-negative integer. Then the
R-module lim
←−
t
Tor
R
i
(R/I
t
; A)iscalledith- local homology module of A with respect to I
and denoted by H
I
i
(A).
Denote by

R be the m-completion of R. As A is Artinian over R, for all i  0and
t>0, on can check that Tor
R
i
(R/I
t
; A)isanArtineR-module. Thus Tor
R

4.2. Lemma. Let s = s(A) be the stability index of A. The n H
m
0
(A)=A/m
s
A.
Proof: H
m
0
(A)=lim
←−
t
D
Tor
R
0
(R/m
t
; A)
i
=lim
←−
t
(R/m
t
⊗
R
A)=lim
←−
t

m
i
(A))
for all system of parameters x
R contained in m
k2
d
Proof: It suffices to prove our lemma in the case R is complete. We make induction on
d. When d =1, and m
k
H
m
0
(A)=0andletx = x
1
is a s.o.p of A with x
1
R ⊆ m
2k
. As
0=m
k
H
m
0
(A)=m
k
(A/m
s
A), we have m

R +Ann
R
(A), then
f
R
(0 :
A
xR) − e(x; A)=f
R
(A/x
1
A)=f
R
p
A/((x
1
R +Ann
R
(A))A)
Q
a f
R
(A/m
r
A) a f
R
(A/m
s
A)=f
R

x
1
R) −→ A
x
1
−→ x
1
A −→ 0. (13)
Because f
R
(A/x
1
A) < ∞ we get H
m
i
(A/x
1
A)=0, ∀i>0. The exact sequence (12) then
implies that
H
m
i
(A)
∼
=
H
m
i
(x
1

x
1
) −→ H
m
0
(A)
x
1
−→ H
m
0
(x
1
A) −→ 0. (15)
By our assumption, x
1
H
m
i
(A)=0, ∀i<d,there is an isomorphism
H
m
0
(0 :
A
x
1
)
∼
=

)=0, ∀j =0, ,d− 2 (16)
and moreover,
f
R
D
H
m
i−1
(0 :
A
x
1
)
i
= f
R
D
H
m
i
(A)
i
+ f
R
D
H
m
i−1
(A)
i

= m
2k.2
d−1
,
(16) and (17) enable us to apply the inductive hypothesis for the s.o.p (x
2
, ,x
d
)ofR-
module (0 :
A
x
1
) and then obtain
f
R
(0 :
(0:x
1
)
(x
2
, ,x
d
)R) − e(x
2
, ,x
d
;0:
A

m
i
(A)).
The inductive step completes by the observation that
f
R
(0 :
A
x) − e(x; A)=f
R
(0 :
(0:
A
x
1
)
(x
2
, ,x
d
)R) − e(x
2
, ,x
d
;0:
A
x
1
)
+ e(x

4.4. Lemma. Let x
be a s.o.p of A. Let m =(m
1
, ,m
d
),n =(n
1
, ,n
d
) ∈ N
d
with
m
i
a n
i
, ∀i =1, ,d. Then
I(x
(m); A) a I(x(n); A).
Proof: As usually, we can assume addition that R is complete. Moreover, because the
function I(x
(n); L) is not dependent on oder of x
1
, ,x
d
, it reduces our lemma to the case
m
1
= n
1

3
i=0
w
d − 1
i
W
f
R
(H
m
i
(A)).
Proof: If f
R
(H
m
i
(A)) = +∞ for some i ∈ {0, , d − 1}, then we have nothing to pro ve.
When f
R
(H
m
i
(A)) < ∞, ∀i<dwe can find k ∈ N such that m
k
(H
m
i
(A)) = 0, ∀i<d.
Taking n

d
d
; A)
=
d−1
3
i=0
w
d − 1
i
W
f
R
(H
m
i
(A))
by Lemma 4.4 and Lemma 4.3.
4.6. Theorem. ld(A)=−∞ ⇐⇒ H
m
i
(A)=0, ∀i<d.
Proof: If H
m
i
(A)=0, ∀i<dthen it follows from Corollary 4.5 that ld(A)=−∞.
We prove the inverse by induction on d. For d =1andletx
= x
1
be a s .o.p of A.

1
, ,x
d
)beas.o.pofA. As ld(A)=−∞, then f
R
(0 :
A
xR) − e(x; A)=0. By [3] (5.3), x
1
is A-coregular. The exact sequence 0 −→ (0 :
A
x
1
) −→
A
x
1
−→ A −→ 0 then generates the long e xact sequence
···−→ H
m
i
(0 :
A
x
1
) −→ H
m
i
(A)
x

1
is A-coregular,
0=f
R
(0 :
A
(xR)) − e(x; A)=f
R
(0 :
(0:x
1
)
(x
2
, ,x
d
)R) − e(x
2
, ,x
d
;0:
A
x
1
)
and thus ld(0 :
A
x
1
)=−∞. Now we c an apply t he inductive hypothesis for (0 :

m
i
(A) and consequently, ∀k ∈ N,
H
m
i
(A)=x
k
1
H
m
i
(A) ⊆ m
k
H
m
i
(A).
This deduces that
H
m
i
(A) ⊆
<
k0
m
k
H
m
i

(i) ⇐⇒ (iv)and(iii) ⇐⇒ (v) are essentially Theorem 5.3 in [[CN]].
In oder to prove (v) ⇐⇒ (vi)wefirst recall that Width
R
(A) a N − dim
R
(A)by
[17] (2.11). Observe that every A-cosequence is also a subset of a system of parameter of
A (see [17] (2.14)). This proves (v)=⇒ (vi). The inverse is clear by definition of Width
and the previous observation.
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