Extended Rees algebras and mixed multiplicities

Math. Z. 202, 111-128 (1989) Mathematische zeitschrift

�9 Springer-Verlag 1989

Extended Rees Algebras and Mixed Multiplicities

Daniel Katz* and LK. Verma Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA

w 1. Introduction

The aim of this paper is to study mixed multiplicities of ideals and use them to calculate multiplicities of certain homogeneous ideals of extended Rees algebras and to give a complete characterization of those parameter ideals whose extended Rees algebras are Cohen-Macaulay with minimal multiplicity at their maximal homogeneous ideals.

Let (R, m) be a local ring of dimension d. Let I be an m-primary ideal. It is well known that the length of the artinian ring R/F, I(R/F), is a polynomial of degree d in r for all large r. The coefficient of nd/d ! in this polynomial, called the multiplicity of I, denoted by e(I), is a well-understood and useful invariant of/. Suppose J is another m-primary ideal. Then it is natural to consider l (R/FJ s) for positive integers r and s. It is proved in [B] that for large values of r and s, I(R/FJ ~) is given by a polynomial P(r, s) of total degree d in r and s. Moreover, the terms of total degree d in P(r, s) have the form

1 d ~T. {eo(IlJ) r -I-el (I[J) rd-l s-I- ... d-(~) ei(i]J) rd-i si--I- ... -}-ed(IlJ) sd 1.

D. Rees showed in [-R1] that eo(IlJ)=e(I) and ed(IlJ)=e(J). B. Teissier and J.J. Risler studied the middle coefficients e~(II J) in [T] and called them mixed multiplicities of ! and J. Their fundamental contribution is to provide an interpretation of mixed multiplicities. They proved [% p. 302] that if R/m is infinite, then e~(! l J) is the multiplicity of an ideal generated by i elements from J and d - i elements from I chosen "sufficiently generally". It follows, in particular, that eo(IlJ) . . . . . ed(IlJ) are all positive integers.

On the other hand, Rees has obtained a similar interpretation of mixed multiplicities in [R3]. This interpretation is given in terms of joint reductions of a set of ideals (I1, Iz . . . . . Id) in a local ring R. A set of elements (xl . . . . . xa)

* Partially supported by the general research fund at the University of Kansas

1!2 D. Katz and J.K. Verma

is called a joint reduction of (I~ . . . . . 14) if xisI~ for i= 1, 2, ..., d and there exists a positive integer n so that

xi I1 I2 ... [i ... Id (I1.. . Id)"-I = (i1. . . ia)." i=

Rees has proved in [R3] that for any joint reduction (x~, ...,xa) of a set of ideals consisting of i copies of J and d - i copies of I,

ei( I I J) = e((xl, x2 . . . . , Xd)).

Bhattacharya also considered the function I (FJ~/ l '+ lJ ~) where J is arbitrary and I is m-primary. Not much is known about this function. Suppose the height of J is positive. It is proved in [B] that for large r and s, I (FJ~ /F+I j ~) is a polynomial Q(r, s) in r and s of total degree d - 1 (cf. Theorem 7 of [B]). Thus we can write the terms of total degree d - 1 in Q (r, s) as

( d - l ) ! eo(IlJ) . . .+ e i ( I l J ) r ~ - l - i s i + . . . + e a _ l ( I l J ) s ~-1 .

Note that if J is m-primary then the coefficient o f ( d ~ . l ) re- l - is~ ~-)y in Q(r,s) \ t ]

is equal to the coefficient of \ / ] ~ in P(r, s). Hence we continue to call

the coefficients ei(IIJ) in Q(r,s) mixed multiplicities of I and J. At present, an interpretation of mixed multiplicities of I and J, when I is m-primary and J is arbitrary, which is analogous to Rees' joint reduction interpretation, is not known. Finally, P. Roberts has recently used the function l (FJ~/Ir+lJ s) in [Ro] to define the multiplicity of a homomorphism between free modules which extends the usual concept of mul t ip l id ty of an ideal.

We now briefly describe the contents of the various sections of the paper. In w 2, we study the function I(FJ~/F +1JS) and the associated polynomial Q(r, s).

0o

Suppose that the analytic spread of J =d im @ j n / j , m=a . We shall prove that n = 0

the only monomials of total degree d - 1 appearing in Q(r,s) are rd-1, rd-2 s 2, -.., rd-1 s,-1. In w 3 we calculate the multiplicity of the homogeneous ideal L = (t-1, I, J t) in the extended Rees algebra R [J t, t-1] where I is m-primary and J is an ideal of positive height. Our formula for e(L) involves mixed multiplicities of J + 12 and J. As a consequence we are able to recover several results already known.

In w we give a criterion for the extended Rees algebra R[ I t , t - ' ] to be Cohen-Macaulay and use it to deduce theorems of J.D. Sally [$2] and G. Valla [Va 2] as quick corollaries. Finally in w 5, as an application of the multiplicity formula for R [It, t - tIN where N = ( t - 1 m, It), we present a complete characterization of parameter ideals I of a Cohen-Macaulay local ring (R, m) for which the extended Rees algebra R [ I t , t -1] is Cohen-Macaulay with minimal multiplicity at N.

Extended Rees Algebras and Mixed Multiplicities 113

w 2. Vanishing of Mixed Multiplicities

We begin by fixing notation. Throughout this paper we deal with commutative Noetherian rings with identity. A Noetherian ring R with unique maximal ideal m is called a local ring and it is denoted by (R, m). By " c " (resp. " < ") between sets we mean inclusion (resp. proper inclusion). We use v for embedding dimension, dim for dimension, e for multiplicity, l for length and ht for height.

We shall use the concept of reduction of an ideal introduced by Northcott and Rees in [NR]. An ideal J contained in an ideal I is called a reduction of I if JIn=I n+l for some n. The ideal J is called a minimal reduction of I if J is minimal with respect to inclusion among all reductions of I. If R/m is infinite then any minimal reduction of I is minimally generated by aft) ele-

ments, where a ( I )=d im (~ln/Pm is called the analytic spread of I. It is proved n = O

in JR2] that htI<a(1)<dimR. Let (R, m) be a local ring of positive dimension d. Let I be an m-primary

ideal and J be an ideal of positive height. Let B(r,s) denote the number I(FJ~/F+IJ~). We call B(r, s) the Bhattacharya function of I and J. Let Q(r, s) be the polynomial of total degree d - 1 in r and s so that B(r,s)=Q(r,s) for large r and s. We call Q(r, s) the Bhattacharya polynomial of I and J. Write the terms of total degree d - 1 in Q(r, s) as

l {eo(IlJ)rd-l+...+(d~l)ei(IlJ)ra-l-isi+...+ea_l(I[J)sd-11. (a-1)~

By combining Theorem 6 and Eq. (4.2) of [B] it would seem that ei(I]d)> 0 for i=0 ,1 ,2 , . . . , d - 1 . It turns out that ei(IIJ ) can be zero. The main result of this section shows that eo(/lJ) . . . . . e,_l(Ild), where a=a(Y), are nonzero and e, (/[ J) . . . . . ed- 1 (I [ J) = O. We begin with a remark and an example which illustrates this result. This is followed by several preparatory lemmas needed in the proof of Theorem (2.7).

(2.1) Remark and Example. In [B], Bhattacharya studies the characteristic function associated to a bigraded ideal d in the bigraded polynomial ring A IX0, ..., Xm, Yo . . . . , Y,] over an Artinian local ring A. (See also [VdW].) In Theorem 2 it is shown that such functions have the form

where z(r, s) is the length of the (r, s)-component of A [X; Y]/d for large r and s, where l is the dimension of d and aij are integers. Those a u for which i + j = l - 2 are called the degrees of d and are written eij. In this context l-B, Theorem 6] asserts that if d is (projectively) irrelevant, then each eu>O. Clearly this is not so as simple examples show. (In fact if d = 0 , then z(r,s)

=(r + m] (s + n] so e,,, # 0 and are all other eij are zero.) A close inspection \ m / \ n /

114 D. Katz and J.K. Verrna

of the proof reveals that the degrees are all non-negative and at least one eii> 0. If I and d are two ideals in a local ring (R, m) with I m-primary and J of positive height then for R/I = A, A i-X; Y] maps onto @ FJS/F +1J~ with kernel

S,r~0

d and in this case the characteristic function of d is the Bhattacharya function of I and J described above while the degrees of d are the mixed multiplicities of I and J. Thus Theorem 2.7 below shows that for such d there is a sort of rigidity of non-vanishing of the degrees until vanishing occurs: writing eij =ei(IIJ) for i = 0 . . . . , d - l , then e i j~0 for i = 0 , . . . , a - 1 and e i j=0 for i = a . . . . . d - 1, where a is the analytic spread of J and d = dim R (so i + j = d - 1). The following example illustrates the result as well as aspects of the proof.

Example. Let k be a field and S=k~X , Y,, Z, W~ be the formal power series ring over k in four indeterminates X, Y, Z, W.. Set M = ( X , Y,,Z, W), P=(Y,, W), Q = ( X , Z ) and F = ( X Y - Z W ) . Write R=S/(F), m = M R and q=QR. Then R is a 3-dimensional local ring and q is a height one analytic spread two (prime) ideal. We will calculate the mixed multiplicities ei(m[q) of m and q. We do this by considering the functions

C (r, s) = dimk M r Qr/Mr + 1 Qs and

B (r, s) = dimk m r q~/m r + 1/@

Now, MrQ~/Mr+IQ ~ is easily seen to be isomorphic to" the k-vector space of forms of degree r + s in X, Y, Z, W such that the Y-degree + W-degree ~ r. Thus C(r, s) may be obtained by counting the forms of degree r+s in the ideals piQr+S-i for i=0 , ..., r. So

/,3 /.2 S C(r, s) = ~ (r + s - i + 1) (i + 1) = ~- + ~ - + lower degree terms.

i=0

We claim (F) n M r Q~ = FM' - ~ Q~- 1 for r, s > 1. Clearly F M r- 1QS - 1 ~_ (F) c~ M r Q~. For the reverse containment it suffices to show that if G is a form of degree / . + s - 2 and GF6M'Q' , then GeMr-~Q ~-x. Let g x " y b z c w d be a term of G. Then GF6MrQ ~ implies b+d<__r-1. Set i=b+d . Then c t x~ybzcwa ePiQr+~-2-~=PiQ'- l - iQ ~-I ~_Mr-lQ s-1. Thus the claim holds. It follows that multiplication by F induces an exact sequence of S-modules

O~S/MrQ~-I F , S / M r + I Q ~ S / ( F , Mr+IQ~)~O

which in turn induces an exact sequence of k-vector spaces

0 ~ M r-i Q~-i/MrQS-1 ~ MrQ~/M ~+~ Q~ ~ M~Q~+(F)/M r+l Q~+(F)--* O.

Thus B(r, s) = C(r, s ) - C(/ . - 1, s - 1). Hence

B(/., s)--/.2 + r s + lower degree terms

1 2! { 2 r 2 + 2 r s } + ' " '


SO

eo(mlq)=2, el(mlq)=l and e2(m[q)=O.

In what follows we retain the notation from the third paragraph of this section.

(2.2) Lemma. eo(I[J ) = e(I). Proof Suppose that Q(r,s)=B(r,s) for all r>r o and S>So. Viewing js as an R-module, we calculate e(I, JS) in two ways where e(I, js) is the multiplicity of I on the R-module JS. Fix an s > So. Then

e (I, J~) = Lim I(F J~/F + ~ J~) ,-~oo ra-1/(d-1)!

-- Lira B(r, s) (d - 1)!/r a- I r - ~ o o

= eo (I I d).

On the other hand since ht J > 0, the ideal (0:J ~) = {re R It J~ = (0)} is nilpotent. Hence dim J~ = dim R/(O :.IS) = d and dim R/J s < d. By applying the multiplicity symbol e(I, - ) on the exact sequence

O~ JS ~ R ~ R/J~ ~O

we get e(I, J~)=e(I, R)-e(I , R/JS). Since dimR/J~<d, e(I, R/J~)--O. Thus e(I) =e(I, J~)=eo(I[J) for all large s.

(2.3) Lemma. Let ~ denote the Rees algebra R[Jt]. For each r>=O, let M, be the ~-module I" ~/I" + ~ ~. Then dim M, = a (J).

Proof Set N,=Ir~ so M,=N~/IN,. Since I is m-primary, I ~ and m ~ have the same radical. Hence dim ~ / I ~ = dim ~ / m ~ = a(J). Since ht(I'~) > 0, ann~ N, is nilpotent. Consequently dim N, = dim ~. If f e a n n s M, then fN~ c IN,. Since annsN, is nilpotent, f is integral over I ~ (cf. Lemma 1.5 of [R3]). It follows that dim Mr = dim ~/I ~ = a(J). (2.4) Lemma. The terms of total degree d - 2 in the polynomial Q(r, s ) - Q(r, s - 1) are given by

l {re-2el(ilj)+ /d-2\a-2 . . .+sd-aea_,(llj)}. (d-2)! ""+~ i )r -'s'e,+i(IIJ)+

Proof

Q( r , s ) -Q( r , s -1 ) : (d l - l , la~. l (d~l )e i ( l lJ ' ra- ' - i [s f - (s -1) f]+. . . i=0

_ 1 ai~(d:l)e,(iiJ)ra_~_~is,_l+.." ( d - 1 ) ! i=i

_ 1 aii (d-:)ei(llJ)ra_i_~si_l +.." ( d - 2 ) ! i=i i - 1

- l a~2(d_21e,+l(llJ)rd-2-,si + .... ( d - 2 ) ! i=o\ t /

116 D. Katz and J.K. Verma

(2.5) Lemma. Let A denote the stable value of the sequence of ideals 0: J c 0: j2 ~ .... and " - "denote homomorphic images in R/A = R. Then for large r and s

Proof l(p y~/i -r+ l j~)= l(F J~/F + l J~).

l(I" J~/P+ l fS)=l[(I~ jS + A)/(Ir+ l J~ + A)]

=I[F J~/(I" J~ n(I '+ ~ J~ + A))]

=l[IrJ~/Ir+ a J~ +(F J~ na))] .

By the multi-graded version of the Artin-Rees lemma [M; Exercise 8.8] there exist integers m and n so that F J ~ n A = I * - m J ~ - " ( A n I ~ J ") for all r>m and s>n. Since AJ~=O for all large s, we get F J ~ n A = O for all large r and s. This completes the proof of the lemma.

(2.6) Rees' lemma (JR3, Lemma 1.2]). Suppose (R, m) is a local ring with infinite residue field. Let (11,12 . . . . . Ig) be a set of ideals of R and let ~ be a finite collection of prime ideals of R not containing 11, 12, ..., I , . Then for each i=1 ,2 , . . . ,g, there exists an element x~eI,, x, not contained in any prime ideal in ~ and an integer s~ such that for r~>s~ and all nonnegative integers rl, r2 , . . . , ~, . . . , rg;

xi R ~ I~* F2a ... F,* = xi I~ ... I'?- 1... I~'.

(2.7) Theorem. Let (R, m) be a local ring of dimension d > 2. Let I be an m-primary ideal. Let J be an ideal with analytic spread a and positive height. Let Q(r, s) denote the Bhattacharya polynomial corresponding to the Bhattacharya function B(r, s) = l(I" J~/F +1 Y). Write the terms of total degree d - 1 in Q(r, s) as

l { e o ( l l J ) r a - 1 . . + ( d T 1 ) e f l l j ) r a - ~ - i s i + +ea_l(IlJ)sa-1}. ( d - 1)! + . . . .

Then ( i ) e , ( I lJ)=ea+1(I lJ)=. . .=ea-l ( I lJ)=O.

(ii) e i ( I l J )>e( l+J) for i=0 , 1 . . . . , a - 1 .

Proof First, we prove (i). Let r o and So be positive integers so that B(r, s)= Q(r, s) for all r>ro and S>So. Fix an r~ro and put M r = F N / F + I N where N is the Rees algebra R[Jt] . The direct sum decompositions of F ~ and F + ~ are given by

I ' ~ = F O F J tG. . . (~FJ ~ t~G...

I'+ l~=I '+l (~Ir+l Jt@...@)I'+l JStS@ ....

Therefore M, = F/F + 1 G F J/I" + 1 j @... q) F JS/F + 1 y G ... . Hence M r is a finitely generated graded N/IN-module. Let (Mr)~ denote the component of degree s of Mr. Therefore

I•/x (M,)~ = IR (F J~/F + 1 j , ) = B (r, s)


is a po lynomia l in s hav ing degree d im M , - 1. By (2.3), d im M~ = a(J)= a. Hence for any fixed r>=ro, the s-degree of B(r, s) is a - 1 . I t follows tha t

ea(IIa)=e,+1(IIg) . . . . . ea-~(IIJ)=O.

W e prove (ii) by using induct ion on d. Before s tar t ing the induct ion we m a k e several observat ions . I f a = 1 then el (I I J) . . . . . ca- l (I ] J) = 0 and eo (I l J) =e(I)>e(I+J) by (i) and (2.2). I f a=d then ea_l(t]J)=~O since the s-degree of Q(r, s) is d - 1 by (i). We m a y assume, wi thout loss of generality, tha t Rim is infinite. By L e m m a (2.5) we m a y assume tha t 0 : J " = 0 for all large n. Hence J contains a nonzerodivisor . Wi th ~ = A s s R in Rees ' Iemma, there exists a nonzerod iv i sor b~J and a posi t ive integer s so tha t for all r, u > 0 and s~so,

b R~m~ F J~=bm~F J ~-1.

In the following calculat ions let L- - - I or m and t = r or u. F o r t > 0 and s __> so,

l[(Lt j~ + b R)/(L'+ a JS + b R)]

= l [ e JS/((L' +a js + bR) c~ IJ J~)]

= l[L t J~/(IJ + ' JS + (b R c~ I2 J~))]

=l[IJ J~/(Lt+~ j~+b L'JS-~)]

=l(Lt J~/Lt+ a j~)_l[(Lt+ ~ J~ + b Lt J~-~)/Lt+ ~ J']

=l(Lt J~/Lt+~ JS)-l[b Lt J~-l/(Lt+~ j ~ b Ltj~-l)]

=l(Lt J~/Lt+~ j~)_l(b Lt j~-~/b Lt+~ J ̀ -t)

=l(U J'/Lt+ a J ')- l(Lt J~- ~ /Lt+~ Js-1). (,)

By taking L = I and t = r in the above calculat ions we ob ta in

l(F J~+bR/F+~ J~+bR)=B(r,s)--B(r,s--1). (**)

Suppose d = 2 and a = 2. Then for large r and s, using (2.4), we get

B(r, s) - -B(r , s - 1)= Q(r, s)-Q(r, s - 1)

= e~ (I1J).

Since R/bR is a one-d imens iona l local ring, I(FJ~+bR/F+IJ~+bR) is a cons tan t for large r and s. By (2.2),

e~(I[ J) = eo(I + b R/b RlJ + b R/b e)

= e(I + b R/b R).

t t is easy to see tha t

e(I + b R/b R) > e(l + b R) > e(I + J).

Thus (ii) has been p roved for d = 2. Assume that d > 3 and (ii) is valid for all local rings of d imens ion d - 1 . Let " p r i m e " denote h o m o m o r p h i c images


in R' = R / b R and let B'(r, s)= l(l'rJ'S/I 'r+l j,s). By taking r and s large it follows from (**) and (2.4) that

ei(l ' lJ ' )=ei+l(IlJ) for i=O, 1o . . . , d - 2 .

We have already proved (ii) for a = 1 and any d >2. So assume that a__>2. By taking L = m and t = 0 and large s in (,) we get

l(J ~ + b Rim JS + b R) = l(J~/m J~) - l (J~- l /m J~- 1).

Hence a ( J ' ) = a ( J ) - l . By the induction hypothesis ei(I'lJ')>=e(I'+J') for i=0, 1, ..., a - 2 . Since ei(I'lJ')=ei+1(IlJ), we get ei(IlJ)>=e(I'+J')>_>e(I+J) for i = 1, 2, ..., a - 1. By (2.2), e o (I I J) = e (I) > e (I + J) and the proof is complete.

(2.8) Corollary. Let M denote the maximal homogeneous ideal (m, J t) of the Rees algebra R [J t]. Then

e (R [J tiM) >= a (J) e (R).

Proof. By the main result of [V5],

e (R [J t]u) = eo (m I J) +. . . + ed- 1 (m ] J).

Since e i (m lJ )>e (m+J)=e(R) for i=O, . . . , a ( J ) - - l , and ei(mlJ)=O for i= a(J) . . . . . d - 1, the inequality follows.

w 3. A Multiplicity Formula

Let (R, m) be a local ring of positive dimension d. For an ideal J of positive height, let T denote the extended Rees algebra R[J t , t -~] and N denote its maximal homogeneous ideal (t- 1, m, J t). By [Va 1], dim T= dim T N = d + 1. If I is an m-primary ideal of R, then the homogeneous ideal L = (t-1, I, Jr) is N- primary. The main result of this section is a formula for the multiplicity of the ideal L in terms of the mixed multiplicities of J + I 2 and J. In particular we obtain a formula for the multiplicity of the local ring TN. We shall use this multiplicity formula in w 5 to study extended Rees algebras of parameter ideals. Because the element t -1 brings powers of 12 into the calculation, our task is made easier by calculating e(L2).

(3.1) Lemma. Put K = J + I 2, L = ( t -1, I, Jr) and H = I + J. For any integer n> 1, the direct sum decomposition of the homogeneous ideal L 2~ is given by

n n - 1

Lzn= ( ~ R t - 2,-i ( ~ K i t - 2n+ 2' ( ~ H K i t -2,+2i+' i = 0 i = 0 i = 0

oo n - - 1 n - - 1

( ~ ( J t) 2n+i ( ~ Ki(J t) 2~- 2i ( ~ H K i ( j t) 2~- 2i-1. i = 0 i = 0 i = 0

Extended Rees Algebras and Mixed Multiplicities

Table 1

119

t -6 t -5 t -4 t -3 t -2 t-1 t o t 1 t 2 t 3 t 4 t 5 t 6

L R H J L 2 R H K J H j2

L a R H K K H J K j2 H ja L 4 R H K K H K 2 J K H j2 K j3 H j 4 L s R H K K H K 2 K 2 H J K 2 j 2 K H J 3 K j 4 H L 6 R H K K H K 2 K 2 H K 3 J K 2 H J2K2 J a K H J a K

j 5

j5 H j6

Proof We display the direct sum decomposition of the first six powers of L, in Table 1.

The undisplayed direct summands are easy to see. The lemma can be proved by using induction on n. We leave this routine verification for the reader.

(3.2) Lemma. For q = O, 1 . . . . . d - 1,

~ ( ~ ) ( q r ) ( d + l ) ( - 1 ) r - 1 . r=0 (d--q+r+l)

Proof This follows from (2.7) of [Vl].

(3.3) Lemma. Let n and d be positive integers. Then

nd+ l la + 2 a +... + n a = + �89 n a + lower degree terms.

d + l

Proof See [V1].

(3.4) Theorem. Let eq denote the mixed multiplicity e q ( J - } - 1 2 [ j ) . Then

e((t-l,I, Jt) TN)= e(J+I2)+ ~" 2"e, . q = O - )

Proof We calculate e (L 2) retaining the notation of (3.1). The direct sum decomposition of L 2" obtained in (3.1) enables us to calculate l(T/LZ"). The lengths of various modules appearing in the calculation for I(T/L 2n) are given by polynomials when the exponents are large. We may assume that these lengths are given by polynomials for all the values of the exponents involved, since this does not affect the leading coefficient of the polynomial for I(T/L zn) for large n. By using the direct sum decompositions of T and L z" we obtain

n - 1 n - 1 [ j 2 n - 2 i \ n - 1 / j 2 n - 2 1 - 1 \ I(T/L2n)= I(R/K~)+ ~, I(R/HK~)+ ~ 1 ~ - ~ 2 , - ] + ~, l/-. ~ l |

i=l i=o i=o r ~ a - / ~ = o \ n r ~ d - - /

= I(R/Ki) + 2 l(R/K~) + 2 l(K~/HKi) + 2 l K ~ 2 i i = 1 i = 0 i = 0 i = 0 a

. - l i - 1 / Kaj2.-2i-1 \ .-1 [ j2n-ai- lKi \

+i~=o,,~ol%Ka'-a+lj2-~-~-l)+ ~ = o l ~ - H ~ l ) "


n - 1

Since I(Ki/HK i) is a polynomial in i having degree d - 1 , ~ l(Ki/HK i) is a i = 0

polynomial in n of degree d by (3.3). Since dim T=d+ 1, I(T/L 2") is a polynomial n--1

of degree d+ 1 in n for large n. Hence we can ignore y, I(Ki/HK i) for the i = 0

purpose of calculating e(LZ). For similar reasons we may ignore the sum involv- ing l(Kij2"-zi-1/HKij2"-2i-1). Write

and l(R/K i) = e (K) ia/d! +...

l[ gajb ~ - S ~d-l [d - l~ eqaa-l-q~bq 4-

in the expression for I(T/L 2") to obtain

/(T/C a") =e(K ia/d! +... + ~,, ia/d! +... v . i=0 i = 0

n - l i ~ d ~ ( d - - 1 ) eqaa-l-q(2n--2i)q "-~ i=02 a = 0 q=O\ q (d - l ) ! t-...

n--1 i~, d~l (d-1~ eqaa-l-q(2rt--2i - 1) q

+,=oZ a=Oq=O\ q ] (d--l)!

= 2 e ( K ) ~ + Y , E E i = 0 a = 0 q = 0

�9 ad-l-qeq {(2n--2i-- 1)q +(2n--2i) q} + ... (d-1)~

n d+J` "-j`'~j`e~j`(d_l]aa-l-qeq. = 2 e ( K ) ~ +Z/=o a=Oq=O\ q } (d - l ) ]

'{,~o(qr)(2n)q-~(-2i)'}+ ...

t-...

=o'q ,d l,, ,d--q, nd+l

~=o\ q (d-1)!(d--q) ( d - q + r + l ) +''"

nd+l nd+l d-*(~o(d-1) ({ )d(d+l ) ( -1 /e~2q+*] = 2 e ( K ) ~ - ~ (d+l)l q~o q r ( d T - - q ) ~ - J ~+' '"

na+l na+, a - t ~ , [ ~ { q ~ (d+l) (__l) r \ = 2 e ( K ) ~ q - (d+ 1)! q~=o lr \q]\r] (d -q+r + 1)J eq2q+l + ....


By (3.2) the expression within the curly brackets is equal to 1. Therefore

d - I e(L2)=2e(K)+ ~ eq2 q+l.

q=O

Since dim TN=d+ 1, e(L2)=2 d+* e(L). Hence

e (L) = e (K) + ~, 2 q eq . q=0 3

(3.5) Corollary ([V2, Cor. 4.2]). Let (R, m) be a two-dimensional regular local ring. Let J c m 2 be an m-primary ideal. Let N denote the maximal homogeneous ideal of the extended Rees algebra T= R [J t, t - * I. Put o (J) = max {nl J c m"}. Then

e(TN)= 2 + o(J).

Proof Since Y c m 2, K = J + m 2 = m z. Hence

e (TN) = �88 {e (m 2) + eo (m z IS) + 2 e 1 (m 2 J J)}

=2 +�89

Since J is m-primary, it is easily seen that for any m-primary ideal I, ea (I I J) is equal to the coefficient of rs in the polynomial I (R/FY) for large r and s. By the proof of Theorem 4.2 of [V1], ea(mZIJ)=2ea(mlJ)=2o(d). Thus e(TN) = 2 + o(J). The reader may compare this with [HS, Thin. 3.2].

(3.6) Corollary ([V3, Thm. 2.2]). Let (R, m) be a local ring of positive dimension d. For r>_ 1, let N denote the maximal homogeneous ideal (t -a, m, tort) of the extended Rees algebra R [m" t, t-a] . Then

e(R[m't,t-a]N) re(R) for r = l = ( e ( R ) [ 2 + r + . . . + r a-a] for r>=2.

Proof Suppose that r = 1. Then

1 ( d-a e(R[m%t-1]N)=~e(m)+q~o2qeq(mlm) }

_e(R) {1 d-1 2d + q~o2~}

=e(R).


Now suppose that r > 2. It is easy to see that eq (m21m r) = 2 a -q r q e(R). Thus

e(R[mrt, t-1]N)= ~--~{e(m2 ) d-1 } + Z 2qeq(m2[ m')

q = 0

1 C d-1 2 d-qr qe(R); = ~7 ~2a e (n) + ~ 2 9 2 ( ~=o J

=e(R) [ 2 + r + ... + r e - l ] .

(3.7) Corollary. Let (R, m) be a local ring of positive dimension d. Let J c m 2 be an ideal of positive height. Put T = R [ J t , t - l] , N = ( t -1, m, Jt). Then

e(TN) > e(R) [a(J) + 1].

Proof. It is easy to see that e,(mZIJ)= 2 d-q eq(mlJ). Therefore

e(TN)= e(m2)+ ~ eq(m2[j)2 q = 0

1 ( d-1 =q-~2ae(R) + 2 2aeq(mlJ)}

2 ( q=o

d - 1

=e(R)+ ~, eq(mlJ ) q = 0

>=e(R) [a(J )+ 1] by (2.7).

w 4. Cohen-Macaulay Extended Rees Algebras

Let (R, m) be a Cohen-Macaulay local ring, I an m-primary ideal. Let T denote the extended Rees algebra R[I t , t -1] and N denote the maximal homogeneous ideal (t-1, m, I t). Several researchers have investigated necessary and sufficient conditions on I so that T is Cohen-Macaulay. The Cohen-Macaulayness of T is equivalent to that of the associated graded ring gr~(R)=R/IGI/ I2+. . . by virtue of the isomorphism

R [I t, t - 1]/(t- 1) ~ grr (R).

Hochster and Ratliff proved in [HR] that if I is a parameter ideal of R then R [I t, t - 1] is Cohen-Macaulay. J. Sally proved in IS 1] that if v ( R ) - dim R + l=e(R) , i.e., R has minimal multiplicity, then grin(R) is Cohen-Macaulay. G. Valla generalized it in [Va2] as follows: Suppose Rim is infinite. If J is a minimal reduction of I with J I = I 2 then gr1(R) is Cohen-Macaulay. This was generalized further in [$2] for I=m: Suppose R/m is infinite and J is a minimal reduction of m with Jm 2 =m 3, then gr,,(R) is Cohen-Macaulay.


By [MR] or [HR], T is Cohen-Macaulay if and only if TN is Cohen-Macau- lay and a local ring A is Cohen-Macaulay if and only if for some parameter ideal I, l(A/I)=e(I). We use this characterization to obtain a necessary and sufficient condition on I for T to be Cohen-Macaulay. As consequences, we are able to recover the above mentioned results of J. Sally, G. Valla and Hoch- ster-Ratliff. We wish to remark that Theorem (4.1) has been observed by K. Shah independently.

(4.1) Theorem. Let (R, m) be a d-dimensional Cohen-Macaulay local ring with infinite residue field. Let J be any minimal reduction of an m-primary ideal I. Then the extended Rees algebra T = R [ I t , t -1] is Cohen-Macaulay if and only

if JI" c~ I ~ + 2 = j i ~ + 1 for all n >= O. (*)

Proof. Set N = ( t -1, m, It), A = ( t -1, It) and B=( t -1, Jr). It is easily seen that for n > l ,

CJO n n Q O

A n = O R t - i O I k t - n + 2 k - l O I k t - n + 2 k @ Iiti" i = n k = l k = l i = n + l

Thus I(A~/A"+I)= I(R/P+I). Since I is m-primary, A is N-primary. Consequently

l(A" TN/A ~ + 1 TN ) = I(A,/A ~ + 1) = l (R/P + 1).

Thus e(ATN)=e(I). Since J is a minimal reduction of I, there is an r with JF = F +1. Hence for n>r,

j i~ n i~+ 2 = I,+ l n i,+ 2= i .+ 2=dI .+1.

Note that BTs is a parameter ideal of TN. Hence the Cohen-Macaulayness of T is equivalent to having I(TN/BTN)= e(BTN)= e(ATN)= e(I). We now compute I(Ts/BTN) = l(T/B). For this, find a direct sum decomposition of B.

B =(t -1, Jt) T = t - 1 T + J t T

= @ In+ltn+ @ din-it" n = - - ~ n = - - o o

= @ R t - n @ ( J I ~ - l + I " + l ) t ~ p t ~. n = l n = 0 n = r + l

Thus

I(T/B)= ~ l[In/(JI "-1 + p + l ) ] n = O

= I(R/F + 1)_ ~ 1 [(dI k + I k + 2)/Ik + 2] k = O

= l(R/J) + l(J/dF) = ~ 1 [(JI ~ + I k + 2)/Ik + 2]. k = 0

124 D. K a t z an d J .K. V e r m a

For k = 0, 1, ..., r - 1 , consider the natural maps

J'k: Jig~ JIg + 1 _~ ( j i k + i k + 2)/ik + 2.

Each fk is a surjective R-module homomorphism. Set Lk=Kernel fk = ( j1 k ~ i k + 2 ) / j p + 1. Then

Hence

l [(JI k + I k + 2)/Ik + 2] = i ( j i k / j I k + 1)_ i(Lk).

r - 1

l(T/B) = l(R/J) + l (J /dF) - ~ { l (JIk /JI k + 1)_ l(Lk)} k=O

r - 1

= l(R/J) + ~ l(Lk) k=O

r - 1

= e ( I ) + ~' l(Lk). k=O

r - 1

Hence TN is Cohen-Macaulay if and only if l ( T / B ) = e ( I ) + ~ l (Lk)=e(B ) k = 0

=e(I) if and only if Lk=O for k=0 , 1, ..., r - 1 if and only if J Ikc~Ik+Z=JI k+l for k = 0, 1, ..., r-- 1. Finally, TN is Cohen-Macaulay if and only if T is so, by [MR] or [HR].

(4.2) Corollary (J. Sally). Let (R, m) be a Cohen-Macaulay local ring with infinite residue field. I f there exists a minimal reduction J o f m with j m 2 = m 3, then the associated graded ring grin R is Cohen-Macaulay.

Proo f Since t -1 is a regular element in T = R [ m t , t -1] and g r m R ~ - T / t - l T , it is enough to show that T is Cohen-Macaulay. The equation J m 2 = m a implies (in context of (*)) that

Jmnc3mn+2-.~.mn+l(qmn+2mmn+2 for n>2 .

For n=0 , J n m 2 = J m by Lemma 3 of Sect. 2 in [NR]. For n = l , J m c 3 m 3 = J m n j m 2 = j m 2. Hence (.) is satisfied for all n__>0. Hence gr,,R is Cohen- Macaulay.

(4.3) Corollary (G. Valla). Let (R, m) be a Cohen-Macaulay local ring. Suppose J is a parameter ideal with J c I and J I = I z. Then gr I R is Cohen-Macaulay.

Proo f We may pass to R ( x ) = R [X],~Rtx], without loss of generality, to create an infinite residue field. Then J c~ 12= J n J I = JI. Hence (.) is satisfied for n = 0. Since J I = 12, for n __> 1, JI" ~ I" + 2 = j i n + 1 n I" + 2 = i n + 2 = JI" + t. Hence (.) is satisfied for all n->__0. Thus R U t , t -1] is Cohen-Macaulay. Equivalently, grIR is Cohen-Macaulay.


w 5. The Extended Rees Algebra of a Parameter Ideal

Suppose that (R, m) is a Cohen-Macaulay local ring of positive dimension d, multiplicity e and embedding dimension v. Abhyankar showed in [-A] that v - d + 1 < e. If v - d + 1 = e then R is said to have minimal multiplicity. J. Sally showed in [S 1] that R has minimal multiplicity if and only if for any minimal reduction J of m, J m = m 2 provided Rim is infinite. In I-V4] it is proved that for a parameter ideal I, the Rees algebra R [I t] is Cohen-Macaulay with minimal multiplicity at its maximal homogeneous ideal (m, I t) if and only if R is regular and l(I + mZ/m 2) => d - 1 . As an application of the multiplicity formula in (3A), we present an analogous theorem for extended Rees algebras.

(5.1) Lemma. Let (R, m) be a Cohen-Macaulay local ring of positive dimension d, multiplicity e and embedding dimension v. I f a parameter ideal K of R contains m 2 then either ( i ) R has minimal multiplicity and K m = m 2 or (ii) R is regular and e(R)=2.

Proof By passing to R [x]mRtxl we may assume, without loss of generality, that Rim is infinite. We apply induction on d. Suppose d = 1 and K = x R for some x eR. If m 2--- x m then R has minimal multiplicity. If x m < m z then there exists b ~ m Z \ x m , hence b = x r for some unit r. Thus x = b r - l ~ m 2, so m g = x R . There- fore l (mZ"/mg"+l)=l(x"R/mx"R) = 1. Since l(mZ"/m2"+l)=e for large n, we conclude that e = 1 and hence R is regular. Since K = x R =m 2, e (K)= 2.

Suppose that d>2 . We claim that m 2<K . Suppose to the contrary that m2= K. Then l(mZ"/m2"+l)= l(K"/mK") for all r. Since elements in any minimal basis of K are analytically independent [M, Theorem 14.5], the "fibre ring"

o0

( ~ K " / m K " is a polynomial ring in d variables over R/m. Thus l(K"/mK") n = O =(n+d-1)

\ d - 1 for all n. On the other hand l(m2"/mZ"+l)--e(2n)d-1/(d--1)! + . . . ,

hence e 2 a- a _- 1 which gives the contradiction d = 1. Therefore m z < K and hence we can select a b ~ K \ m 2 so that b is superficial for K. Let " - " denote homomorphic images in R/bR. Then d im/~= d - 1 and rh2= K. By induction hypothesis, /~ is either Cohen-Macaulay with minimal multiplicity and K rh=rfi 2 or /~ is regular and e(/s I f / ~ is regular then R is regular since b ~ K \ m z. Also e(K)=e(ff~)=2 since b is superficial for K. Suppose t h a t /~ has minimal multiplicity a n d / s rfi = rh z. Then m2= K m + b R. Hence any r~m 2 can be written as r = c + b s for some c e K m and sER. If sq~m then s is a unit and therefore bern 2 which is a contradiction. Thus sere which implies that m 2 = Kin. By Sally's theorem [S 1] it follows that R has minimal multiplicity.

(5.2) Theorem. Let (R, m) be a Cohen-Macaulay local ring of positive dimension d. For a parameter ideal I, put T = R [ I t , t -~] and N = ( t - ~ , m , It). Then T N is Cohen-Macaulay with minimal multiplicity if and only if either ( i ) R has minimal multiplicity and I m = m 2 or (ii) R is regular and e(I + m2)=2.

Proof Put K = I + m 2. We prove that if TN is Cohen-Macaulay with minimal multiplicity then K is a parameter ideal. It is easily checked that v(TN)--dim TN


+ 1 = I(R/K). We may assume without loss of generality that R / m is infinite. By the Risler-Teissier interpretation of mixed multiplicities [T, w 2] or by Rees' theorem [R3] for each q=0 , 1, . . . , d - l , there exist a parameter ideal J q c K so that eq(KII) = e(Jq). Hence for each q, e(Jq) = l(R/Jq) > l(R/K). By the multiplicity formula for TN we get

Thus

Write

Therefore

e(TN) = e (K) + eq (K [I) 2 = I(R/K). q = 0

d - n

e ( K ) + ~ eq(KII) 2 q = U l ( R / K ) . q=O

d - 1

2 a l (R/K) = I (R/K) + ~, 2 q l (R/K) . q=O

, ' /-1

e (K) - - l (R /K) + ~ [eq (K I I ) - l (R/K)] 2 q = O. q=O

Since eq(K[I)-I(R/K)>>_O for each q, we conclude that e ( K ) = l ( R / K ) . There exists a minimal reduction J of K and e ( J ) = l ( R / J ) = e ( K ) = l ( R / K ) , since R is Cohen-Macaulay. Thus J = K and consequently K is a parameter ideal and r n 2 c K . By (5.1) either (i) R has minimal multiplicity and K m = m 2 or (ii) R is regular and e(K)=2. In the (i) case, K m = ( I + m 2 ) m = m 2 gives I m = m 2 by Nakayama's lemma.

Conversely suppose that (i) holds. Then TN is Cohen-Macaulay by [HR]. The ideal (t- 1, I t) TN is a parameter ideal in TN and

(t -n, I t) (t -n, m, I t ) = ( t -2, m t -1, I + I m , m I t, 12 t 2)

= ( t - 2, m t - 1, I + m 2, m I t, 12 t 2)

= N 2.

Therefore T N is Cohen-Macaulay with minimal multiplicity by [S I]. Suppose that (ii) holds. Since R is regular, there exists a minimal reduction

L of K so that e(K) = I(R/L) = 2. Hence

l(R/m) + l (m/K) + I(K/L) = 2.

This implies that either K = m or K = L . Since e(K)=2, K < m . Thus K = L and consequently l ( I + m 2 / m Z ) = d - 1 . It follows that we can choose x l , . . . , x a ~ m and an integer e > 2 so that m = (xn, x2, . . . , Xd) and 1 = (xn . . . . . xa- 1, x~). Consider the ideal

J = ( t - 1 + x~ t, x l t, x2 t . . . . , xa- n t, xa).


Then J c N = (t- 1, m, I t) and J N = N 2. Indeed,

N 2 = ( t - 2, m t - 1, ( x l , x 2 . . . . . xa - 1, x~) , m I t, 12 t2).

Since e > 2, x ~ a J N . Consequently the equation

t -~( t -1 + x ~ t ) = t - 2 + x~

shows tha t t - 2 ~ d N , Since m c J , m t - X c j N . F o r i = 1 , 2 , . . . , d - l , we h a v e xi

= ( x l t ) ( t - 1 ) ~ J N . T o see t h a t m l t c J N , n o t e t ha t m c J a n d I t c N . I t is c lear t ha t x i x j t 2 ~ d N for 1 -< i,j<= d - 1 a n d xixed t 2 =(X i t)(x~ t ) e J N for i = 1, 2 . . . . , d - 1.

The e q u a t i o n

( t - 1 + X~l t) (x~ t) = x~ + x~ e t 2

shows tha t x 2~ t2~dN. There fo re 12 t 2 c J N which impl ies t ha t , ] N = N 2. H e n c e

TN is Cohen-Macaulay with minimal multiplicity.

Acknowledgement. We would like to thank Professors Jeffrey Lang and Michinori Sakaguchi for many stimulating discussions.

References

[A]

[B]

[MR]

[I~S3

[M] [MR]

[NR]

[R1]

[R2]

[R3]

[Ro] [s l ]

is2] IT]

[Vl]

[V2]

[V3] ~V4]

Abhyankar, S.S.: Local rings of high embedding dimension. Am. J. Math. 89, 1073-1077 (1967) Bhattacharya, P.B.: The Hilbert function of two ideals. Proc. Cambridge Philos. Soc. 53, 568-575 (1957) Hochster, M., Ratliff, L.J., Jr.: Five theorems on Macaulay rings. Pac. J. Math. 44, 147-172 (1973) Huneke, C., Sally, J.D.: Birational extensions in dimension two and integrally closed ideals. J. Algebra 115, 481-500 (1988) Matsumura, H.: Commutative ring theory. Cambridge: Cambridge University Press 1987 Matijevic, J., Roberts, P.: A conjecture of Nagata on graded Cohen-Macaulay rings. J. Math. Kyoto Univ. 14, 125-128 (1974) Northcott, D.G., Rees, D.: Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50, 145-158 (1954) Rees, D.: A-transforms of local rings and a theorem on multiplicities of ideals. Proc. Cam- bridge Philos. Soc. 57, 8-17 (1961) Rees, D.: Rings associated witla ideals and analytic spread. Proc. Cambridge Philos. Soc. 89, 423-432 (1981) Rees, D.: Generalizations of reductions and mixed multiplicities. J. Lond. Math. Soc. 29, 397-414 (1984) Roberts, P.: Local Chern classes, multiplicities and perfect complexes (1988) (preprint) Sally, J.D.: On the associated graded ring of a local Cohen-Macaulay ring. J. Math. Kyoto Univ. 17, 19-21 (1977) Sally, J.D.: Tangent cones at Gorenstein singularities. Compos. Math. 40, 167-175 (1980) Teissier, B.: Cycles 6vanescents, sections planes, et conditions de Whitney, Singularities

Carg~se, 1972. Ast~risque 7-8, 285-362 (1973) Verma, J.K.: Rees algebras and mixed multiplicities. Proc. Am. Math. Soc. 104, 1036-1044 (1988) Verma, J.K.: Rees algebras of contracted ideals in two dimensional regular local rings (1988) (preprint) Verma, J.K.: Rees algebras with minimal multiplicity. Commun. Algebra (to appear) Verma, J.K.: Rees algebras of parameter ideals. J. Pure Appl. Algebra (to appear)


[v5] [Val]

[Va2] [VdW]

Verma, J.K.: Multigraded Rees algebras and mixed multiplicities (1988) (preprint) Valla, G.: Certain graded algebras are always Cohen-Macaulay. J. Algebra 42, 537-548 (1976) Valla, G.: On form rings which are Cohen-Macaulay. J. Algebra 58, 247-250 (1979) Van der Waerden, B.L.: On Hilbert's function, series of composition of ideals and a general- ization of the theorem of Bezout. Proc. K. Akad. Wet. Amst. 31, 749-770 (1928)

Received October 24, 1988

Extended Rees algebras and mixed multiplicities

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