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Mixed multiplicities of arbitrary ideals in local ringsDuong Quôc Viêt aa Department of Mathematics , Hanoi University of Technology , Dai Co Viet, Hanoi,Vietnam E-mail:Published online: 27 Jun 2007.

To cite this article: Duong Quôc Viêt (2000) Mixed multiplicities of arbitrary ideals in local rings, Communications inAlgebra, 28:8, 3803-3821, DOI: 10.1080/00927870008827059

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COMMUNICATIONS IN ALGEBRA, 28(8), 3803-3821 (2000)

MIXED MULTIPLICITIES OF ARBITRARY IDEALS

IN LOCAL RINGS

DUONG QUBC VI@T

Department of Mathematics, Hanoi University of Technology

Dai Co Viet, Hanoi, Vietnam

E-mail: duongquocvietO bdvn.vnrnail.vnd.net

Throughout this paper we will be concerned with a local ring ( A , m, k , d) , where A is a Noetherian local ring

with maximal ideal m, infinite residue field k = A/m, and Krulf dimension dim A = d > 0 . Now suppose

tha t J is an m-primary ideal and ( I l , ..., I,) is a set of positive height ideals of A. Then there exists a positive

integer u such that the Bhattacharya function

is a polynomial of degree d - 1 for all values of n, n l , ..., n , > u

If we write the terms of total degree d - 1 in this polynomial in the form

B ( n , n l , ..., n , ) = e r ( ~ [ d o + l l , f i [ d l l , ..., l , ~ d , ~ ) ~ ~ ~ ~ ~ ~ ' " ' ~ ' ~ * do+d,+. +d.=d-1 d ~ ! d l ! ... d,! '

then e a ( ~ l d o + l ] , h [ d ' l , ,.., 1 ~ 1 ~ ~ 1 ) are non-negative integers not all zero and are called the mized multiplicity

of a set of ideals ( J , I , , ..., I , ) of the type (do + l , d l , ..., d , ) , (see 151).

A reason for interest in mixed multiplicities is their relation with multiplicities of graded rings, see e.g.

ll51, P61, 1171, [51, 1141.

Mixed multiplicities were first introduced by Teissier and Risler in 1121 for two m-primary ideals and in this

case they can be interpreted as the multiplicity of general elements. Next, Rees in [ l o ] showed the existence

of joint reductions of d m-primary ideals and the mixed multiplicity of those ideals is the multiplicity of the

ideal generated by a joint reduction.

In general, mixed multiplicities have been mentioned in the works of Verma, Katz, Swanson and other

authors, see e.g. 1151, [16] , [17], 151, [6 ] , [14] . However, how to find general formulas in the case, when J

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is m-primary and (I1, I z , ..., I,) is a set of arbitrary ideals, which are analogous to Rees'joint reduction or

Teissier-Risler'general elements interpretation, is an open question (see e.g.[6],[14]) and this problem has

attracted much attention.

In this paper, we will link mixed multiplicities of a set of arbitrary ideals with Zariski-Samuel multiplicities

of local rings. To connect the above multiplicities, we construct a sequence of elements what is called a

sequence satisfying the condttion (FC). The results of this paper will show that a sequence satisfying the

condition (FC) carries important informations on mixed multiplicities.

Let U = (11, ..., I,) be a set of ideals of A, I = 11.1 z...I,, N := U (0 : I n ) , A' = AIN, I,' = &A'; " t o i = 1,2, ..., s. We say that an element z E A satisfies the condition (FC) with respect to U if there exists an

ideal I, of U and integer ni such that

(i) z E I, \ m.1, and

for all n, > n:, where z' is the initial form of x in A*

(ii) x is a filter-regular element with respect to I = 11 ... I..

(iii) dim [A/ U (x : In) ] = dim A/N - 1. nzo

4 sequence XI, xz, ..., xt of A is said to satisfy the condition (FC) with respect to U if E,+: is an element sat-

isfying the condition (FC) with respect to = (11, ..., 1,) for each i = @, 1, ..., t - 1, where A = A j ( s l , ..., zi) ,

Zit] is an initial form of xi+l in A, 7, = I IA, ..., f, = 1,A. This notion was introduced in order to combine

the notions of filter-regular sequence in 1131 and a complete reduction in [I@], That is why we choose the

provisional name "(FC)".

Our approach is based on the idea? of Rees in [lo], Teissier and Risler in [12], Katz and Verma (the proof

of main theorem in [6]), Herrmann et al. [ 5 ] . We see that if x E I1 is an element satisfying the condition

(FC) with respect to (J,1:, ..., I,) then

where A = AIxA, = J A , 71 = IIA, ..., f, = I,A (see Proposition 3.3, Section 3). This is a key fact of our

paper.

By using a sequence of elements satisfying the condition (FC) to study mixed mutiplicities, we get inter-

esting results. Now, we summarize some important results of this paper as follows.

M a i n t h e o r e m ( T h e o r e m 3.5). Let (I , , ..., I.) be a set of arbitrary ideals of positive height. Let J be an

m-primary ideal of A. Let ka, kl , ..., k, be non-nagatiue integers with sum erpal to d - 1. Then

for any sequence x l , z z , ..., xt ( t = kl + ... + k,) satisfying the condition (FC) with respect to ( J , 11, ..., I,)

consisting of k, elements of 4, kz elements of Iz, ..., ks elements of I., where I = 11 ... I., A = A/ U "20

[(.I, zz, ..., ~ t ) : I"], J = J A .

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MIXED MULTIPLICITIES OF ARBITRARY IDEALS

(ii) If t = kl + ... + k, < htI then

e(Jlk0+'l, ..., = e(J1; A')

for any sequence ~ 1 ~ x 2 , ... ,zt satisfying the condition (FC) with respect to ( J , 11, ..., I.) consisting of kl

elements of 11, kz elements of l a ,..., k, elements of I,, where A' = A / ( x l , xz, ..., s t ) and J' = JA'.

if and only i f there exiats a sequence of kl + ... + k, elements satisfying the condition (FC) with respect to

( J , 11, ..., I.) consisting of kl elements of 11, kz elements of I2 ,..., k, elements of I,.

Of course, this result holds for arbitrary ideals (see Theorem 3.4, Section 3). Moreover, in the case ideals

are m-primary, we get some results which is sharper than the results of Teissier and Risler in [I21 and of

Rees in (101 (see Theorem 3.6 and Corollary 3.7, Section 3).

In the case for two-ideal, we have the following result.

Theorem (Theorem 4.1). Let J be m-primary. Let I be a non-nilpotent ideal of height h. Suppose

x1,x2, ..., x, is a maximal sequence of I satisfying the condition (FC) with respect to U = ( J , I ) . Then

(i) e(JId-'1, Ili]) = e(.& Af(s1, ..., xi ) ) for all i < h - 1 , where J, = J IA / ( x l , x z , ..., xi)]

for all i = 0, 1, ..., q.

(ii) e(Jld-'1, 11')) = e (J i ;A / u [ ( x l r x 2 , ..., xi ) : In ] ) , "20

where Ji = =[A/ Wu ( ( X I ,xz , ..., xi) : I n ] ) for h 5 i 5 q.

(iii) e(Jld-'1, 11'1) # 0 if and only if i < q.

Ftom Theorem 4.1 we immediately follow that "the length of any man'rnal sequence of I satisfying the

condition (FC) with respect to U = ( J , I ) is an invariant".

Note that if I is m-primary then Rees in (81 showed that e(.Jld], Il0l) = e ( J ) and e (~[Ol , 1 1 ~ 1 ) = e ( I ) . When

I is an ided of positive height, 0. I(& and J.K. Verme groved that e(.r[d],~lo]) = e(J) f i ! .

By combining Theorem 4.1 and Theorem 3.4 o f (151 we get multiplicity formulas for Rees algebras as

follows.

T h e o r e m (Theorem 4.2). Let R = A[It] be Rees algebra of ideal I of positive height ht I = h. Let J be a

m-primary ideal. Then

for any maximal sequence 2 1 , ..., x, of I satisfying the condition (FC) with respect to ( J , I ) .

Our main purpose is to describe mixed multiplicities of a set of ideals of positive height, but the problems

arising here are mixed multiplicities o f a set of ideals of zero-height, which causes some difficulties. W e will

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investigate the problem in general for a set of arbitrary ideals (see Proposition 3.1 and Theorem 3.4, Section

3).

The paper is organized as follows. The rest of this paper is divided into 3 sections.

In Section 2 we collect several facts about filter-regular sequence with respect to an ideal and Rees'lemma ... which will be used in the sequel.

In Section 3 we introduce the notion of a sequence satisfying the condition (FC) with respect to a set of

ideals and some properties of this sequence. Then we describe the results concerning mixed multiplicities of a

set of arbitrary ideals (Proposition 3.1 and Theorem 3.4). Consequently, we get the results concerning mixed

multiplicites of a set of ideals of positive height (Theorem 3.5) or m-primary (Theorem 3.6 and Corollary

3.7).

In Section 4 we will establish multiplicity formulas of Rees rings with respect to arbitrary ideals (Theorem

4.2) and in particular, we get interesting results concerning multiplicities of Rees rings of m-primary ideals

(Theorem 4.3, Theorem 4.4).

In this section we will select some results and notions which will be needed in this paper.

Let (A, m, k) denote a Noetherian local ring of dimA = d > 0 with maximal ideal m.

Let (Il ,Iz, ..., I,) be a set of ideals of A. Let a be an element of I1 such that dim (AIaA) = dim A - 1

and there exists a positive integer c for large nl and all nz,n,, ..., n,,

Then a is called superficial for 4 , Iz, ..., I, [12]

Risler and Teissier proved that if (11,12, ..., Is) i8 a set of m-primary ideals then there exists a sequence

a l ,a2 , ..., ad consisting of dl elements of 11, dz elements of Iz ,..., d, elements of I, (dl + d2 + ... + d, = d),

such that a1 is superficial for I l , Iz , ..., I,, a2 is superficial for the images of 1 2 , 13, ..., I. in AIalA, etc. In

this case, we have (see [12])

e(b[d'], ..., l,ld41) = e(a1, ..., ad; A).

Recall that an ideal (al , ..., ad) is called a joint reduction of m-primary ideals 11, ..., Id if a, E Ii for

i = I , ..., d and the ideal all2 ... ld+azl1.13 ... I d + ...+ adIl...Id-l is areduction of ideal Il...Id [lo]. Rees showed

(see [lo]) that if (al , ..., ad) is a joint reduction for kl copies of 11 ,..., k, copies of I,, then e(1~['~1, .,,, lT['*1) =

e(a1, ..., ad;A).

A sequence XI, x2, ..., x, of A is called filter-regular with respect to I if z, is not contained in the associated

prime ideals of ideal U [(zl,xz, ..., x,-I) : In], for i = 1, ..., n [13], 1141. Originally, the notion of a filter- n>o

regular sequence is introduced with respect to maximal ideal of a local ring [3]. It can be seen that a sequence

z l ,zz , ..., zn is filter-regular with respect to I if and only if

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for all i = O , l , ..., n - 1

A filter-regular sequence with respect to maximal ideal rn of A is called a filter-regular A-sequence (see

PI).

A system of parameters al,az, ..., ad of A is called reducing (see [I]) if

with q = (a l ,az , ..., ad)A and q d - 1 = ( a l , a ~ , ..., ad-1)A.

Note that a filter-regular A-sequence is reducing (see [3]).

Our approach will be based on the following Lemma 1.2 in [lo] of Rees.

Rees'lemma. Suppose ( A , m ) is a local ring with infinite residue field. Let ( I l , Iz, ..., I,) be a set of ideals of

A and let P be a finite collection of prime ideals of A not containing I1,12, ..., Is. Then for each i = 1,2 , ..., s,

there exists an element x E l i , x not contained in any prime ideal in P and an integer ki such that for

ni 2 k; and all non-negattve integers 711,712, ..., n,,n,+l, ..., n,,

Artin-Rees lemma. Suppose A is a Noetherian local ring, A-module M is finitely generated, N is a

submodule of M and ( I 1 , 12, ..., I,) is a set of ideals of A. Then there exist integers d l , ..., d, such that

for all nl > dl,nz 2 dz, ..., n , > d,.

Let I,, I z , ..., I, be ideals of A. Then a multi-Rees algebra corresponding to a set I l , 12, ..., I, is

t l , t 2 , , t ; heing indeterminates.

In particular, Il = Iz = . .I, = I we denote the multi-Rees algebra

3. A SEQUENCE SATISFYING THE CONDITION (FC) AND MIXED MULTIPLICITY

First, we define a sequence satisfying the condition (FC) and give some remarks on this sequence.

Definition. Let U = (4 , ..., I,) be a set of ideals of A, I = IlIz ... I,, N := U (0 : In ) , A' = AIN, n>O

I,* = IiA*; i = 1,2 , ..., 8. We say that an element x E A satisfies the condition (FC) with respect to U if

there exists an ideal I, of U and an integer n: such that

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(i) z E Ii \ m.1, and

for all ni 2 n: and all non-negative integers n l , n z , ..., ni-l,ni+l, ..., n r , where z' is the initial form of z in

A*.

(ii) x is a filter-regular element with respect to I = I1...I,

(iii) dim [A/ U (x : In)] = dim AIN - 1 . " 2 0

A sequence X I , xz, ..., zt of A is called a sequence satisfying the condition (FC) with respect to U if f ,+1

is an element satisfying the condition (FC) with respect to

- U = (TI, ..., L ) for all i = Os 1, ..., t - 1,

where A = A/(xl, ..., x.), 71 = I IA, ..., 1, = IJ and is initial form of xi+l in A.

Remark 1. Let, ( I I , Iz, ..., I,) be a set of ideals of A such that I = 111 2...I, is non-nilpotent. Set + = AssA*.

Since I is non-nilpotent, it follows that + # 0. On the other hand + is finite collection of prime ideals of

A* not containing I;, I;, ..., I,'. Hence, by Rees'lemma, then for each i = 1,2, ..., s, there exists an element

x E I, \ m.I;, x' (the initial form of x in A*) not contained in any prime ideal in F and an integer ki such

that for all n, 2 k,;

I; "l...~,:n'...~,'n' n (x*) = I;~~...I,*~~-~...I;~~.x*.

And since x* 6 p for all p E +, it follows that x contained in no all the associated prime ideals of N.

Hence x is a filter-regular element with respect to I . Thus, if U = (11, Iz, ..., I,) is a set of ideals such that

I = I l . I z...I, is non-nilpotent, then there exists an element satisfying the conditions (i) and (ii) of (FC).

Remark 2. If x is an element in I; satisfying the condition (FC) with respect to U = ( I I , ..., I,, ..., I,) then

I;"' ... I;"' ... 1;"' n (x') = 1;"' ... I,*"'-'...I;~..X*,

for large n,. By the above, it follows that

(4n'...I;nx...Isn' + N ) n ( x + N ) = I ~ ~ ' . . . I ; ~ ' - ~ . . . I , ~ ' . x + N.

Note that

and I ~ ~ ~ . . . I ~ ~ . - ~ . . . I ~ ~ ~ . x + N & 4"l...I,"'...I,". n (x) + N

E I,"~...I~"....I,". n ( z + N ) + N.

From the above facts, we have

(I~"'...~"'-~...I,"'.X + N ) = (Gn'...Iin'...I,ns n (I) + N).

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By above equality,

(Ilnl...Ii"'-'...I,"'.x + N ) n N = ( I lnl...Iin....I.n, n (2) + N) n N.

By Artin-Rees lemma, 4"' ... Iin' ... I,"' n N = 0 for all large n l , n l , ..., n, (see proof of Proposition 3.1).

Hence ( I ~ ~ ~ . . . I , ~ ~ - ~ ... ISn*.x + N) n Ilnl , . ,I in~,. .18n~

= I~"~...I,"~-~...I,"~.X + Nnl,nL.. . l i"~. . .I ,"a

= I ~ ~ ~ . . . I ~ ~ ~ - ~ . . . I # ~ , . X

and ( ~ n l . . . ~ i n i . . . ~ . n s n (x) + N) ~ I ~ ~ ~ . . . I , " - . . . I . ~ ,

= TIn1 ... 1,". ... 1,". n (5) + N n I1nl...I,n~...I,n*

= nl , . . I ,~% ... lgnr n (I)

for all large n l , nz, ..., fa. . Consequently, we get

R e m a r k 3. Let U = (4, I=, ..., Id) be a set of m-primary ideals. By Remark 1, there exists an element x

satisfying the conditions (i) and (ii) of (FC) with respect to U. Since 1 1 . I ~ ... Id is m-primary, it follows that

dim AIxA = dim A - 1. Then by induction, there exists a sequence x l , x 2 , ..., xd satisfying the conditions

(i) and (ii) of (FC) with respect to U such that a, E I, for j = 1,2, ..., d. It can be verified that s f , ..., xd is

also a system of parameters of A, a reducsing sequence, a filter-regular A-sequence, and a joint reduction of

U. And by

dim A/ U [(XI, ..., xt) : I n ] = dim A/(xl, ..., s t ) = d - t "20

for all t < d, x l , ..., xd-1 is also a sequence satisfying the condition (FC) with respect to U. In particular, if

It = ... = Id = I then 21, ..., xd is a minimal rcductiori of I

R e m a r k 4. Let U = (11, I?, ..., I,) be a set of m-primary ideals and an element x E 11 satisfying the

conditions (i) and jiij of (FCj with respect io li. Set I = Il ... I,. By Xemark 2, for large nl and all nz, ng, ..., nd,

we have

I ~ ~ ~ I ~ " ~ . . . I , ~ . n (2) = I ~ " ~ - ~ I , ~ ~ . . . I , ~ . . X .

Since x is a filter-regular element with respect to I , it follows that 0 : x U 0 : In (see [14]). Thus, for "20

large c and n l > c, we get

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is a polynomial of degree q - 1 for all large n , n l , ..., n,.

Proof.

By Artin-Rees lemma, there exist integers do, d l , ..., d, such that

for all no 2 d0,nl 2 dl , ..., n, 2 d,. For all large t , we have N I L 0, it follows that

and

for all large no, ..., n,. Set 7 = ASSA. Since I is non-nilpotent, it follows that 3 # 0. On the other hand 7

is a collection of prime ideals of A not containing = IA. This fact follows that depht f > 0. Consequently,

is a polynon~ial of degree dim A - 1. Thus,

is a polynomial of degree dim A/N - 1 for all large no, ..., n , and

Lemma 3.2. Let ( I l , I z , ..., I,) be a set of ideals of A such that I = 11 ... I , is non-nilpotent and let J be

m-primary. Set N := U (0 : In) . Then "20

(i) e ( ~ [ ~ o + ~ ] , l P ~ , . . . , l p l ) # 0 and ~ ( J [ ' ~ + ~ I , I ~ ~ , . . . , I ~ ~ ] ) = e ( J ; A / N ) , where J = J ( A / N ) .

(ii) e ~ ( J [ ~ o + ' ] , I!], ..., = ea (J ; A ) if ht I > 0.

Proof. The proof ( i ): Set

i = 1 ,2 , ..., s. Since I is non-nilpotent, dim A = q > 0. Then there exists a positive interger u such that the

Bhattacharya function

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is a polynomial of degree q - 1 for all n , n l , ..., n, 2 u . We shall denote by P (n ,n t , n z , ..., n,) the above

polynomial. By taking n l = n z = ... = n , = u and fix the u . we get

(q - l ) ! P ( n , u, ..., u ) eA(J[ql, I ~ [ O I , ..., I . [ ~ ] ) = eA(1lq1, fp ' , ..., fp l ) = lirn

n-+m nP-1

Viewing iU as an A-module and e ( J ; f U ) is the multiplicity of j on the A-module fU. Since the A-module

iu has dimension q, it follows that P (n ,u ,u , ..., u ) = lA(-) is a polynomial of degree q - 1 for all large

n . Hence ( q - l)!M&) (q - l ) ! P ( n , u, ..., u)

e ( j ; f u ) = lirn n-+m nq-1

= lim n+m nr-l

The above facts show that e 4 ( ~ i q l , I l [ ~ I , ..., I,[") = e ( j ; i U ) and ea(Jlq1, I~[O], ..., 1,[01) # 0. Since I is non-

nilpotent, by Remark 1, it follows that fu contains an A-regular element. Now Assume that Q is the total

ring of fractions of A. Then f u 8 Q is a free Q-module of rank 1. Hence fU is a A-module o f rank 1. On

the other hand, since e ( J ; fU) = e (J ; A)rankfu, we get e ( J ; f U ) = e (J ; A) . The proof (ii): Since ht I > 0, it follows that N is nilpotent. Hence e ( J ; A) = e ( j ; A) . From this fact and ( i ) , we get

The proof of Lemma 3.2 is complete.

The main tool for our proofs will be the following proposition

Proposition 3.3. Let J be an m-primary ideal and let ( I l , I z , ..., I,, ..., I,) be a set of ideals of A such that

I = Il .I 2...I, is non-nilpotent. Suppose that an element x E I, satisfies the condition (FC) with respect to

U = (J,11,12 ,..., I ,,..., I,). Then

where k, is a positive integer and A = AIxA, 1 = JA , fi = I,A; i = 1,2, ..., s.

(ii) 1f ea(JIkot1I, I L [ ~ ' ] , . . . , z ' ~ ' ~ ) # 0 then for any j (1 5 j 5 s), there exists an element of I, satisfying

the rendition (PC! with respect to U.

Proof. The proof ( i ) : The following proof is based on an idea of Katz and Verma in [6, proof of Theorem

2.71. Set I = I112 ... I, and N := U ( 0 : In ) ; A' = AIN, J* = JA*, dim A* = q, I,' = I,A*; i = 1,2 ,..., s. n t o

B(n0, n l , ..., n,) denotes the Bhattacharya polynomial of function

Since dim A/N = q, by Proposition 3.1 we have deg B(no,nl , ..., n.) = q - 1. Now assume that an element

z E I, satisfies the conditions ( i ) and (ii) of (FC) with respect to ( J , 11, I z , ..., I.). Set P = AssA* and x* is

the initial form of x in A*, x* is not contained in any prime ideal belonging to P. Hence z* is a non-zero-

divisor in A'. From the property o f x (see ( i ) , Definition) it follows that there exists a positive integer u > u

(see u , Convention) such that for all no,nl, ..., n, > v,

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Set

B=A' /z 'A' ,b=J'B,b ,=I ,*B; i=1 ,2 ,..., s.

Then for all no,nl, ..., n, > v , we have

Since x' is non-zero-divisor in A', it follows that

Hence

Now if z satisfies the condition (iii) of (FC), that mean

dim [ A / U (x : In)] = dim A* - 1 = q - I nt0

(see Remark 5), then it can be verified that

is a polynomial of degree q - 2 for all large no, nl, ..., n, and the terms of total degree q - 2 in this polyno-

mial,i.e., the Bhattacharya polynomial of function

is equal to the terms of total degree q - 2 in the polynomial

B(no,nl, ..., nj, ..., n8) - B ( n ~ , n l , ..., n, - 1, ..., ns),

where B(no, nl , ..., n,) is the Bhattacharya polynomial of function

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The above facts show that if an element x E I, satisfies the condition (FC) with respect to U then

eB(b[ko+ll, b1lk1l ,..,, b j i k~ - ' ] ,..., b,[kal) = e A . ( ~ * [ k o + l l , ~ l * [ k ~ l ,,,., Is*[k']).

Set

N* := U 0 : ( b l ... b,)";T := U ( ( x ) : In) "20 "20

B = BIN' , b = bB, b, = b$;i = 1, ..., s ,

A' = A / T , J' = JA', I: = I A ' ; i = 1, ..., s

A = A / x A , J = J A , i i = I , A ; i = l , ..., s.

By B A' and from Proposition 3.1, it is easy to see that

and

ea.(J*Iko+'l, ~ ~ ' [ ~ ' l , ..., I,*Ik'l) = eA(Jiko+'1, 4[k11, ...,

Thus,

The proof (ii): I is non-nilpotent, by Remark 1, for any j (1 5 j 5 s), there exists an element x E I,

satisfying the conditions (i) and (ii) of (FC). From the proof of (i) we have

Since e a . ( ~ * [ ~ ' + ~ ~ , ..., # 0, it follows that

is a polynomial of degree q - 2. Thus,

dim [A/ U (x : In)] = q - 1 = dim A' - 1 "20

and x satisfies the condition (FC). The proof of Proposition 3.3 is complete.

The following theorem establishs mixed multiplicity formulas by means of Zariski-Samuel multiplicity.

Theorem 3.4. Let (11, I*, ..., I,) be a set of ideals o f A such that I = 1112 ... I , is non-nilpotent. Let be J a

m-primary ideal of A. Assume that dim A/ U (0 : In ) = q. Let b, k l , ..., k. be non-negative intergers with n>O

sum equal to q - 1. Then

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for any sequence xl ,x2 , ..., zt of t = kl + ... + k, elements satisfying the condition (FC) with respect to

( J , I l , ..., I,) consisting of kl elements of Il ,..., k, elements of I., where A = A/ U [ ( z l , x z , ..., x t ) : In] , n?O

f = JA .

if and only if there exists o sequence of kl + ... + k, elements satisfying the condition (FC) with respect to

( J , I l , ..., I,) consisting of kl elements of Il ,..., k, elements of I,.

Proof. The proof of (i): We shall begin with showing that if I = 11.12 ... 1, is non-nilpotent and x l r x 2 , ..., xt

is a sequence o f t = kl+ ... + k, elements satisfying the condition (FC) with respect t o ( J , 11, ..., I,) consisting

of kl elements of 11, ..., k, elements of I , then

where A' = A / ( x l , x z , ..., st) , J' = JA' , I'i = liA' ; i = 1 ,2 , ..., s. The proof is by induction on t = kl+ ...+ k,.

For t = 0, since I is non-nilpotent, by Lemma 3.2 we have

The result is true.

Suppose that the result has been proved for t - 1 we need show that the result is true for t. Now, assume

that x l , z z , ..., xt is a sequence o f t = kl + ... + k, elements satisfying the condition (FC) with respect to

( J , 11, ..., 1,) consisting of kl elements of 11, ..., k, elements of I,. Since t > 0, there exists j (1 5 j < s) such

that k j > 0 and X I E I,. By Proposition 3.3, we get

where A' = A / ( x l ) and J' = JA', I' = IA ' , I'i = I,A1; i = 1,2, ..., s. Since xl,x2, ..., xt is a sequence satis-

fying the condition (FC) with respect to ( J , I ] , ..., I.), it follows that x'z,xf3, ..., x't is a sequence satisfying

the condition (FC) with respect to ( J ' , 1'1, ..., I f , ) , where x'i the initial form of xi in A', i = 2, ..., t. Since

kl + ... + (k, - 1) + ... + k. = t - 1, then it follows by inductive assumption applied to (t - 1) that

e ~ , ( J ' [ k o c l l , I ~ [ k l ! , ,,,, I ; ' ~ J - " , .,,, I : ' ~ ' ] ) = e A u ( ~ " l k o + l l , I " ] [ ~ ] , ..,, I " , ' ~ ] ) # 0,

where A" = At / ( z ; , ..., si) = A / ( z l , x z , ..., xt ) , J" = JA", I1Ij = IjAU ; j = 1,2, ..., 9, I" = IA". Thus,

The induction is complete. We now turn to the proof of (i). Let I = 11.12 ... 1, be non-nilpotent. For any

sequence z l , x z , ..., xt o f t = kl + ... + k , elements satisfying the condition (FC) with respect to ( J , L , ..., 1.)

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consisting of kl elements of 11, ..., k, elements of I,, as shown above, we get

where A" = A / ( s l , z 2 , . , . ,x t ) , Jo = JAIr. Since ea , , (~" lko+l l , I " ~ ~ ~ ~ , . . . , I",[']) # 0, it follows that I" is

non-nilpotent. Now, from Lemma 3.2 we have

( J"'~"+", l;'[O1, ..., I$O]) = eA(J ; A),

where A = A/ U [(xl,x2, ..., z t ) : In] , J = J A . With above facts, we immediately see "20

e ( ~ [ " + ~ ] , 1 ~ [ ' ~ 1 , ..., ISIkb]) = e(J ;A/ U [ (x l ,zz , ..., s t ) : I"]) nzo

The proof of (i) is complete.

The proof of (ii): We prove first the necessity. The proof is by induction on (t = kl + ... + k,). If t = 0, in

this case the result is trivial. For suppose the result has been proved for t - 1. As the next step, we claim that

the result is true for t. On the one hand t = kl + ...+ k, > 0 on the other hand e ( ~ [ ~ o + ' l , I ~ [ ~ ~ ] , ..., ISlkal) # 0,

by Proposition 3.3, there exists j (1 5 j 5 s) such that k, > 0. and an element XI E Ij satisfying the

condition (FC) with respect to ( J , 11, ..., I.) and

where = A/(z l ) and J = JA, f, = I,A; i = 1,2, ..., s. From this equality we have

On the other hand kl + ... + (k, - 1) + ... + k, = t - 1, then it follows by our inductive assumption applied

to ( t - 1) that we can choose ( t - 1) elements 22, ..., xt consisting of kl elements of I l , kz elements of 12, ..., (k, - 1) elements of I,, ..., k, elements of I, such that Z2, Z3, ..., l t is a sequence satisfying the condition (FC)

with respect to ( f , i 1 , ..., I ,) , where = A/(xt) , j = JA and f, = I+$; i = 1,2, ..., s, 5; the initial form of

x, in .4, i = 2, ..., t . Since x1 satisfies the condition (FC) with respect to (J , I l , ..., I,) and Z2,53, ..., & is a

sequence satisfying the condition (FC) with respect to ( f , f l , ..., is) , it follows that 11, xz, ..., z t is a sequence

satisfying the condition (FC) with respect to ( J , I I , ..., I,).

We turn to the proof of sufficiency. Suppose that there exists a sequence x l ,xz , ..., z t o f t = kl + ... + k. elements satisfying the condition (FC) with respect to (J , Il, ..., I,) consisting of kl elements of 11, ..., k,

elements of I,. By (i),

e (~Iko+l l , ~ ~ [ ~ l l , ,,,, Is[k,l) = e ( ~ l k ~ + l l , fpl, ,,,, fpl) ,

where A = A/(xI ,..., zt), f = I A and J = J A . Since z l , zz , ..., zt is a sequence of t = kl + ... + k. elements satisfying the condition (PC) with respect to ( J , 11, ..., I,), it follows that dim A/ U [(xl, ..., s t ) :

"20 In] = ko + 1 > 0. This fact follows f non-nilpotent. By Lemma 3.2, e ( J [ ' ~ + ' l , f j O ~ , . . . , ~ ~ ~ ) # 0. Thus,

e ( ~ [ ' ~ + ~ ] , I~[~"] , ..., I,[';']) # 0. The proof of Theorem 3.4 is complete.

Our main purpose is to describe the mixed multiplicities of a set of ideals of positive height. If ( I l , Iz, ..., I,) is a set of ideals of positive height then there exists a positive integer u such that the Bhattacharya function

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is a polynomial of degree d - 1 for all n , n l , ..., n, 2 u. Consequently, by Theorem 3.4, we get the following

theorem.

Theorem 3.5. Let ( I 1 , ..., 1,) be a set of ideals of positive height. Let J be a m-primary ideal of A with

dim A = d. Let ko, k l , ..., ks be non-negat~ve intergers with sum equal to d - 1. Then

for any sequence z l , z z , ..., st (t = kl + ... + k.) satisfying the condition (FC) with respect to ( J , 11, ..., I,) consisting of kl elements of 11, k2 elements of 12, ..., k. elements of I,, where I = h... I,, A = A/ ,120 U

[ ( x ~ , z 2 , ..., xi) : PI, j = JA.

(ii) If t = kl + ... + k, < ht l then

for any sequence z l , x z , ,.,,z( satisfying the condition (FC) with respect to ( J , I l , ..., I.) consisting of kl

elements of h, kz elements of I2 ,..., ka elements of I., where A' = A / ( z l , x ~ , ..., z t ) and J' = JA'.

(iii)

e ( ~ [ ~ a + ' ] , l1 l k 1 1 , ..., # 0

~f and only if there ezists a seyuence of kl + ... + k, elements satisfying the condition (FC) with respect to

( J , I l , ..., I,) consisting of kl elements of 11, kz elements of Iz ,..., k. elements of I,.

Proof. By Theorem 3.4, we immediately get ( i ) and (iii). We now prove (ii). Since t = kl + k2 + ... + k, < ht I = h, it is easy to see that i f z1,x2,...,xt ( t = kl + ... + k,) is a sequence satisfying the condition

(FC) with respect to ( J , I l , ..., I,) consisting of kl elements of I l , ..., k, elements of I, then the ideal I' =

[I + (21, ..., z L ) ] / ( z l , ..., st) o f A' = A / ( z l , ..., zt) has a height ht (1') > 0. Set J' = JA'; I: = I jA t for

j = 1,2, ..., 3. In this c z e , by Lemma 3.2, -e get

~ h ~ ~ , e(~lka+ll, IIIkll, ..,,I,Ik*l) = e (J1 ;A ' ) . The proof is complete

In particular, i f 11, ..., I. are m-primary ideals. Then the Bhattacharya function

is a polynomial of degree d for large values of n l , ..., n,. Moreover, the terms of total degree d in this

polynomial have the form nlk l ... risk* C e A ( ~ ~ [ ~ l l , ..., I~~~+----

k1+ ...+ L,=d kl!k2! ... k,!'

then e A ( ~ l [ k l l , 121k21, ..., is the mixed multiplicity of I1,12, ..., I , of the type ( k l , ..., k.).

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Let (h, 1 2 , ..., I s ) be a set of m-primary ideals. Let k l , k2, ..., k , be non-negative integers with sum equal

to d. Suppose that x l , x ~ , ..., xd is a sequence satisfying the conditions (i) and (ii) of (FC) with respect to

U = ( I l , I z , ..., I.) consisting of kl elements of I I , ..., k, elements of I,. By Remark 3, X I , ..., xd-1 is a sequence

satisfying the condition (FC) , x l , x 2 , ..., xd is a filter-regular A-sequence, reducing, and also a joint reduction

of U of the type ( k l , ..., k.). Consequently, we get

where A = A / ( x l , ..., 2,) and %,+I, ..., Ed are initial forms of xi+1, ..., xd in A, respectively. From these facts,

we have a result similar to that of Risler and Teissier in [12] and Rees in [lo] but in term of sequence

satistifying the condition (FC) .

T h e o r e m 3.6. Let (4, ..., I.) be a set of m-primary ideals. Let e a ( ~ l [ ~ ' l , ..., I,Ika1) be the mixed multiplicity

of ( I l , ..., I,) of the type ( k l , ..., k,). Suppose that X I . x2, ..., xd is a sequence satisfying the conditions ( i ) and

(ii) of (FC) with respect to U = ( I l , I z , ..., I,) consisting of kl elements of h, ..., k, elements of I,. Then for

all 0 < i < d we have

e(11[~'1, ..., I ~ I ~ . ~ ) = e ( x l , x z r ..., xd; A ) = e ( ~ i + l , ..., z ~ ; A),

where A = A l ( x 1 , ... , x i ) and % + I , ..., Z,i are initial forms of xi+l, ..., xd i n A, respectively.

From Theorem 3.6 we immediately get the following consequence which has been mentioned in [12], [ lo] ,

[51.

Corollary 3.7. Let (I1,12, ..., I,) be a set of m-primary ideals of Noetherian local ring A with dimA = d.

Let k l , kz , ..., k. be non-negative intergers with k l + kz + ... + k t = d ( 1 < j < s ) . Then

and in particular,

e ( 1 1 ~ ~ ~ , ..., I j - ~ [ ~ l , ~ ~ [ ~ l , I , + ~ [ ~ ] ,..., = e ( I , ; A ) .

4. MULTIPLICITY OF REES RINGS

In this section, we will give some results on the multiplicity of Rees rings

Let J be an m-primary ideal and let I be a non-nilpotent ideal of height h = ht I .

Assume that x l , x z , ..., x , is a maximal seqnrnce in I satisfying the condition (FC) with respect to U =

( J , I ) . Then as a consequence of Theorem 3.5 we get the following theorem.

T h e o r e m 4.1. Let J be an m-primary ideal. Let I be a non-nilpotent ideal of hetght h. Suppose x l , x z , ..., x ,

is a maximal sequence i n I satisfying the condition (FC) with respect to U = ( J , I ) . Then

(i) e(JId-'1, 1 1 ~ 1 ) = e(&; A / ( x l , ..., 2,)) for all i < h - 1, where j , = J [ A / ( x l , x 2 , ..., x i ) ]

for all i = 0 , 1, ..., q.

(ii) e ( ~ [ ~ - ' l , 11'1) = e(&; A / U [ ( X I , x z , ..., x i ) : I n ] ) , "20

where .& = J [ A / U [ ( x l , x z , ..., z i ) : I n ] ) for h < i 5 q. n>o

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MIXED MULTIPLICITIES OF ARBITRARY IDEALS 3819

(iii) e ( ~ [ ~ - ' ] , I[ ']) # 0 if and only if i < q

Pmof. Ftom Theorem 3.5 we immediately get (i) and (ii). We now prove the part (iii). The "if" part.

Assuming e ( ~ [ ~ - ' 1 , 1 [ 4 ) # 0, we shall show that i < q. Now, Assume the contrary, that i > q. Since

x l , z z , ..., z , of I is a sequence satisfying the condition (FC) with respect to U = ( J , I ) , by Proposition 3.3,

it follows that e(jld-'1,17'-91) = e ( ~ [ ~ - ' l , I [ ' l ) # 0, where J = J [ A / ( x l , z l , ..., x ~ ) ] , = I [ A / ( z l , z z , ..., x,)].

Rom e(jld-'1, i I i - q l ) # 0, it follows, by Proposition 3.3, that there exists an element z in I such that Z

(the initial form of z in A / ( x l , ..., z q ) ) satisfying the condition (FC) with respect to ( J , i ) . The above fact

follows that xl, x2, ..., x,,z of I is a sequence satisfying the condition (FC) with respect to ( J , I ) . We thus

arrive at a contradiction. Thus, i < q. We turn to the proof of sufficiency. Since i < q , there exists a

sequence z l , z z , ..., xi of I satisfying the condition (FC) with respect to U = ( J , I ) , by Theorem 3.5, we get

e(.Tld-'], I [ ;] ) # 0. The proof is complete.

As one might expect, from Theorem 4.1 and the results in (151 and (181 we get some multiplicity formulas

for Rees rings.

Theorem 4.2. Let R = A[It] be Rees algebra of an ideal I of positive height ht I = h. Let J be an m-primary

ideal of A. Then

h-1 V

e ( ( J , I t ) ; R ) = x e ( J ; A / ( x l , ..., x,)) + x e ( ~ ; A/ "yo [ ( z l , x z r ..., z i ) : In ] ) i=O i= h

for any maximal sequence z l , ..., xq in I satisfying the condition (FC) with respect to ( J , I ) .

Proof. Ftom Theorem 4.1, by the formula e ( ( J , I t ) ; R ) = e(~l~- ' ) ,1[*1) [15], the proof is straight-

forward.

In the case I is m-primary, we have h = ht I = d so from Theorem 4.2, we get the following result.

Theorem 4.3. Let R = A[It] be Rees algebra of an m-primary ideal I of A. Let J be an m-primary ideal.

Then there exists a sequence z l , ..., xd-1 in I satisfying the condition (FC) with respect to ( J , I ) and we haue

d- 1

e ( ( J , I t ) ; R ) = e ( j i ; A / ( z ~ , ..., x d), i=o

where j; = J IA / ( x l , ..., xi )] for i = 0,1, ..., d - 1. In particular, if J = m, we get

Proof. Since I is m-primary, by Remark 3, there exists a sequence z l , ...,xd-1 in I satisfying the condition

(FC) with respect to ( J , I ) . Hence, by Theorem 4.2, d-1

e ( ( J , I t ) ; R ) = e(Ji; A / ( ~ I , ...,zi)), i=O

where 1; = J [ A / ( z l , ..., x,)]. Since J = m, it follows that

Thus.

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3820 VIET

By an argument analogous to that used for the proof of Theorem 4.3 and Theorem 1.4 (161 we have the

following result for multi-Rees algebras.

T h e o r e m 4.4. Let R = RA(I[']) be multi-Rees algebra of an m-primary ideal I of ring A. Let J be an

m-primary ideal of A. Suppose that xl,...,xd-1 of I is a sequence satisfying the condition (FC) with respect

where J , = J [ A / ( z l , ..., x,)]. In particular, if J = m, we get

It is well-know that if x1,z2, ..., xn in m satisfying the conditions (i) and (ii) of (FC) with respect to m

then ( z l , z z , ..., xd) is a minimal reduction of m and a filter-regular A-sequence. By Remark 3, X I , ..., zd-1 is

a sequence (FC). Above facts follow e ( A / ( z l , 2 2 , ..., z,)) = e(A) , for all i < d and we get the following result.

Corollary 4.5. Let R = R ~ ( ~ [ S ] ) be multi-Rees algebra of maximal ideal m. Then

Finally, it should be noted that all results of this paper can be carried over modules with minor modifications.

Acknowledgement. The author is grateful to N.T. Cuong, L.T. Hoa, N.Q. Thang and N.V. Trung for

many tliscusstions on the results of this paper. He would like to express his sincere thanks to the editor for

his help and encouragement. Special thanks are due to the referee whose remarks substantially improved

the paper.

The author was informed that N.V. Trung has obtained related results.

REFERENCES

[l]. A. Ausiander and D. Buchsbaum. Codimension and multtplicity, Ann. Math. 68 (1958).

[Z]. P.B. Bhattacharya, The Hdbert function of two %deals., Proc. Cambridge Philoa. Soc. 53(1957), 568-575.

[3]. N. T. Cuong. P. Schenzei and N. V. Rung, Vemllgemeinsrte Cohen-Mocoulay-Modun, Math. Nachr. 85(1978), 57-73.

(41. J . Herzog, N.V. Dung and 8. Uirich, On the multiplicity of blow-up rings of rdeals genemted by d-sequences, J. Pure Appi. Algebra. 80(1992), 273-297.

[5]. M. Herrmann, E. Hyry, 3. Ribbe and 2. Tang, Reduction numbers and multiplicities of multigmded rtruelures, Preprint,l997.

[6]. D. Katz and J.K. Verma, Eztended Rees algebms and mized mult~plietties, Math. 2. 202(1989), 111-128.

[7]. D. G . Northcott and D. Rees, Reductson of ideals In local rings, Math. Proc. Cambridge Phil. Soc. 50 (1954), 145-158.

[a]. D. Rees, A-tmnsforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge. Philos. Soc. 57(1961), 8-17.

[9]. D. Rees, Hilbert functions and pseudo-mtional local rings of dimension two, J. London Math. Soc. 24(1981), 467-479.

[10].D. Rees, Genernlisations of nduetions ond mized multiplicities, J . London Math. Soc. 29(1984), 397-414.

[ll].I. Swanson, Tight Closure, Joint Reductions, ond mized multipl~cities, Thesis. Purdue University (1992).

[12].B. Teissier, Cycles 8(1973), 285-362.

sections planes, Whitney, Singularitien

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(13l.N.V. 'hung, Reduction ezponents and d q n e bound for thr defining equation of gmded rings, Proc. Amer. Soc. 101(1987), 229-234.

(14l.N.V. 'hung, Filter-regular sequences and multiplicity of blow-up rings of ideal* of the principal clma, J . Math. Kyoto. Univ. 33(1993), 665-683.

[15]J.K. Verma, Rees olgebms and mized multiplicities, Proc. Am. Math. Soc. 104(1988), 10361044.

[16]J. K. Verma, Rees algebmr with minimal multiplicity, Communications in Agebra 17(12), 2999-3024 (1988).

[17]5. K. Verma, Multi-gmded Reer olgebms and mized multiplicities, Proc. Amer. Math. Soc. 104(1988), 10361044.

[18].J. K. Verma, Multi-gmded Rees olgebms and mized multiplicities, J. Pure Appl. Algebra. 77(1992), 219-228.

Received: February 1999

Revised: January 2000

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