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On the partial Euler-Poincare characteristics of certain systems of parameters in local rings

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Math. Z. 222, 383-390 (1996)
Mathematlsche Zeltschri t
© Springer-Verlag 1996
On the partial Euler-Poincare characteristics of certain systems of parameters in local rings
Nguyen Tu Cuong, Vu The Khoi
Institute of Mathematics, P.O. Box 631 Bo Ho, 10.000 Hanoi, Vietnam e-mail: ntcuong@thevinh.ac.vn Received 20 June 1994; in final form 1 December 1994
I Introduction
Let (A, m) be a local ring with the maximal ideal m and M a finitely generated A-module with dimM = d. Let x = {xl,...,Xd} be a system of parameters (abbr. s.o.p) of M. Denote by
Hi(x;M)
the i-th Koszul homology module of M with respect to the s.o.p x. Following Serre [9], Appendice' II, the partial Euler-Poincare characteristic of M with respect to x is defined by
Zk(x;M) = ~ (-1)i-kl(Hi(x;M))
i>k
for k = 0 ..... d, where
l(N)
will be denoted for the length of the A-module N. Let n = (nl .....
ha)
be a d-tuple of positive integers. We denote by
x (n)
the
s.o.p {x 7. .....
xa d }
of M. Then we can consider Zk(x(');M) as a function in n. This function in general is not a polynomial (cf. [6]). But, in [5] Garcia Roig has shown that for n] .....
nd= t
the function
Zk(x(t);M)
of variable t is bounded above by polynomials in t and that the least degree of all polynomials bounding above this function is independent of the choice of the s.o.p x. The first author has shown in [3] that the above statement is still true for the case of d variables
nl
.... nd
when k = 1 (ZI(x(');M) is denoted in [3] by
IM(n;x)).
And he proved in [2] that the least degree of all polynomials bounding above Zl(x(n);M) is exactly equal to
dimA/(ao(M)..,
ad_~(M)) ifA admits a dualizing complex, where a/(M) is the annihilator of the i-th local cohomology module
H~m(M)
of M with respect to the maximal ideal m. The purpose of this paper is to generalize the above results. Namely, we will show in Sect. 3 that the least degree of all polynomials in n bounding
The first author is partially supported by the National Basic Research Program, Vietnam, and, during the completion of this paper, by the Mathematics Section of the international Centre for Theoretical Physics, Italy
384 N.T. Cuong, V.T. Khoi above Xk(x(n);M) is independent of the choice ofx for k = 0,...,d. Let us denote this invariant by
pk(M).
Then our main result is the following theorem. Theorem
Let M be a finitely 9enerated ,4-module of dimension d. Suppose that ,4 admits a dualizin9 complex. Then pk(M) = dim,4/( ao(M) . . - ad-k(M) ) for k = O,...,d.
This theorem will be proved in Sect. 4. In Sect. 2 we will show a gener- alization of the formula for multiplicity of Auslander-Buchsbaum [1] which plays an important role in the proof of the theorem.
2 A formula for multiplicity
Throughout this paper we always denote by (A,m) a local commutative Noetherian ring with the maximal ideal m and by M a finitely generated ,4-module of dimension d. Let
x = (xt ..... xt)
be a multiplicity system of M. We denote by
e(x;M)
the Serre-multiplicity of M with respect to x. First of all, we have the following lemma.
Lemma
2.1
Let x = (xl .... ,xt) be a multiplicity system of M. Let j,k be two positive integers such that t > j > k > O. Then
J e(xj+l,... ,xt;Hk(Xl ..... xj;M)) = ~ e(xi+l .... ,xt; (0 • Xi)Hk_l(xl,...,xi_l;M)) i=k J + Z e(xi+l ..... xt; (0 : Xi)14k(xt,...,xi_l;M)), i=k + 1
where we set e(xi+l ..... Xt;(O : xi)h,k~xl..,xi_l;M)) = l((O : Xi)nk~xj,..,xt_l;M~) if i=t. Proof
We prove by induction on j. It is easy to see for the case j = 1. Suppose that j > 1. From the exact sequence of Koszul homology modules
0 ~ Hk(xl,... ,xj_~; M)/xjHk(xl ..... xj_ l; M)'--+ Hk(xl,...
,xy; M) (0 : Xj)/~k_~xl,..,xj_l;M)~ 0 and the additive property of the multiplicity, we get
e(xj+l ..... xt; Hk(Xl ..... xj;M) ) = e(xj+l .....
xt;(0 : Xy)~/k_l~x ...,xj--I M))
q- e(Xj+ l ..... xt; Hk(Xl ..... xj_
1;
M )/xjHk(xl ....
, xj-1; M ) )
= e(xj+l ..... xt;
(0 Xj)Hk_l(Xl,...,Xj_l;M) )
+ e(xj+l
.... ,Xt ~ (0 :Xj)Hk(Xl,...,Xj_l;M))
+ e(xj+b... ,xt; H,(xt .... ,xj-l;M)).
Thus the lemma easily follows from the inductive hypothesis.
Certain systems of parameters in local rings 385 As an immediate consequence of Lemma 2.1 we have the following corollary which will be often used in next sections. Corollary 2.2
Let x--(xl,...,xt) be a multiplicity system of M. Then
t
Zk(x;M) : ~
e(xi+l,... ,xt; (0 : Xi)Hk_l(Xl,...,Xi_l;M)). i=k
Corollary 2.2 has some interesting consequences. First, since the multiplicity is a non-negative integer, we obtain again the following well-known result of Serre (see [9], Appendice II). Corollary 2.3
Let x = (xl ..... xt) be a multiplicity system of M. Then
~(k(x;M) >
O for all k > O.
Note that
Zo(x;M)= e(x;M)
(see [1] or [9]) it follows that
Zl(x;M) = l(Ho(x;M)) -
X0(x;M)=
I(M/(x)M)- e(x;M) .
Therefore Corollary 2.2 leads to the following formula for the multiplicity (see [1], Corollary 4.3). Corollary 2.4
Let x = (xl ..... xt) be a multiplicity system of M. Then
t
e(x; M) = I(M/(x)M) - Y~ e(xi+ 1
... ,Xt;
(XI,...
,Xi--1
)M : xi/(Xl,...
,X i-
1 M) .
i=1
3 The invariant
pk(M)
Keep all notations in Sects. 1 and 2. We begin with the following lemmas. Lemma 3.1 (see [5], Lemma 2).
Let a E A and let t be a positive integer. Then
(i) l(0 :at)M <
tl(O
:a)M; (ii) l(0 :
at)M <- l(0
: att)M
for all t t ~ t. Proof
Straightforward.
Lemma
3.2 Let x = {xl ..... :ca}
be a s.o.p of M and
n = (nl .....
nd) a d-tuple of positive integers. Then, for k = 0 ..... d, it holds
(i)
Xk(x~' ..... X~dd;M) ~ nh...,ndZk(x;M);
(ii)
nl
nd tl t d .... = =
Zk~x 1 ..... x d
;M) ~
Zktx I ..... Xd ;M ) Jor all tl > nl,...,td > rid. Proof
(i) Since Z0(x;M) =
e(x;M) the
lemma is clear for k = 0. Suppose that k > 1. Note that the partial Euler-Poincare characteristic is indepen- dent of the order of the sequence
xl ..... :ca
therefore it suffices to show that
386 N.T. Cuong, V.T. Khoi
nd -~-
Zk(x~
.....
Xd_l,X,t
;M) <
ndZk(x;M).
Thus by Corollary 2.2 we only have to verify that
nd
1(0 : x d )Hk_l(Xb...,xa_l;M ) <= ndl(O xd)n~_l(xl,...,xa_l;M) •
But, this is clear by Lemma 3. We also get (ii) by the same method• An immediate consequence of Lemma 3.2 is the following corollary.
Corollary
3.3
Keep x and n as in Lemma 3.2. Then, for k = 0 .... , d, it holds
n 1 nd
(i)
l(Hk(x I ..... x a
;M)) < nl .....
ndl(Hk(x;M));
(ii)
l(Hk(Xl I nd .... x a
;M)) =<
l(Hk(Xtl',...,xda;M)) for all tl => n l, ..., td => nd.
,• nd
Lemma-3.2 shows that if we consider
Zk(x~ t ..,x d
;M) as a function in d variables nl .....
nd
then this function is bounded above by the polynomial n~ .....
ndZk(X; M).
Moreover, we have Proposition 3.4
Let x = {xl ..... xd} be a s.o.p of M. Then the least degree of all polynomials in nl ..... nd bounding above the function Zk~x n~ ,... 'Xdnd ~vl '') is independent of the choice of x for all
k : 0 ....
d• We will denote this invariant by pk(M). Proof
Let t be a positive integer• By [5], Theorem 6, the least degree of all polynomials in t bounding above Xk(x] ..... x~; M) is independent of the choice of x. Denote by
p~(M)
this invariant and by
pk(x;M)
the least degree of all
• nd
polynomials in
nl ..... na
bounding above
Zk(x~ ~ ... x a
;M). Then is clear that
p~(M) ~ pk(x;M).
On the other hand, by Lemma 3.2 we get
-~- n I nd.
Zk(X~,...,Xtd;M) > Zk(x 1 ..... x d ,M)
for all t > max{nl .....
ha};
it follows that
p~(M) > pk(X;M).
Thus
pk(x;M) --- p~(M).
So the proposition is proved. Applying Corollary 3.3 we also have the following corollary by the same method used in the proof of the proposition 3.4.
Corollary
3.5
Let x = {xl ..... xu} be a s.o.p of M. Then the least degree of all polynomials in nb...,nd bounding above the function l(Hk(x~ ~, nd ... X a
;M))
is independent of the choice of x for k = 0 ..... d. We will denote this invariant by qk(M). Remark 3.6
(i) Note that X0(x;M) =
e(x;M)
and
Zd(X;M) = l(Ha(x;M)) =
1(0 : X)M. Hence
po(M)= d
and
pd(M) < O.
(ii) We will take -c~ for the degree of the zero-polynomial then d - 1 >
pk(M) > -c~
for k = 1 .... d - 1.
Proof pl(M)
is just the polynomial type of M defined in [3] therefore
d - 1 >__ pl(M)
by 2.5, (ii) of [3]. Moreover, since Zk(x;M) +
Zk+l(x;M) = l(H~(x;M))
and
qk(M) < d - k
by Corollary 7 of [5], we can easily obtain the above inequalities by induction on k.
Certain systems of parameters in local rings 387 (iii) Let A~ be the m-adic completion of M. Since every s.o.p of M is also a s.o.p of A;/ and
Hk(x;M) ~-
Hk(x;)tT/) it follows that
pk(M)
= Pk(~/) for k=0 ..... d.
4 The main result
In this section we always assume that the ring A admits a dualizing complex. The basic properties of dualizing complex used in this section are from [8]. First, we need some definitions and notions as follows. Let k be a positive integer. A sequence of elements
xl,... ,xj
of A is called
k-reducin9 sequence
of M if the following conditions hold: Either j < k or
xi ~P
for all PE Ass(Hk_l(xl,...,xi-1;M)) with
dimA/P
> d-i for alli=k, .... j. Now, we define
rk(M)=min{j:every
subset of a s.o.p, of M having
(d-j-
1)-elements is a k-reducing sequence of M}. For the case k = 0 we set
to(M) = d. Remark 4.1
(i) For the case k = 1 the notion 1-reducing sequence is just the reducing sequence defined in [2] which is a generalization of the reducing s.o.p of Auslander-Buchsbaum in [ 1 ]. (ii) It is not difficult to verify that
xi~P
for all P E Ass(Hk-l(xl,
.... xi-l;M))
with
dimA/P > d - i
if and only if
dim(O:xi)t-tk_l(xl,...,xi_~;M ) <
d - i. Thus, in the case j > k, a sequence Xl .....
xj,
which is a subset of a s.o.p x --- {xl,...,Xd} of M, is a k-reducing sequence if and only if
e(xi+l
....
Xd;
(0 Xi)Hk_l(Xl,._,Xi_l;M) )
:
0 for all i : k, .... j. (iii) A sequence
xl ..... xj
of A with j < k is always a k-reducing sequence. Therefore
d- k ~ rk(M).
For short we set bk(M) =
ao(M).., ak(M), k = 0 .... d.
Recall that
ai(M)
is the annihilator of the i-th local cohomology module
Hi(M)
of M. Now we are able to prove the main theorem announced in the introduction.
Theorem
4.2
Suppose that A admits a dualizing complex then pk(M) = rk(M) = dimA/bd_k(M) for k = 0 ..... d. Here we stipulate that the zero-module is of dimension -c~. Proof
For k = 0 then
po(M) = d
by 3.6, (i). Since ha(M) C_ AnnM by [8], Lemma 2.4.4, the theorem is true in this case. Assume now that d > k > 0. We will prove that
pk(M) > rk(M) >-- dimA/bd_k(M) > pk(M)
for
k=
1 .... d. Claim 1
pk(M) > rk(M).

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