Mixed Hilbert Coefficients of Homogeneousd-Sequences and Quadratic Sequences

195, 211]232 Ž1997. JA966938

JOURNAL OF ALGEBRA ARTICLE NO.

Mixed Hilbert Coefficients of Homogeneous d-Sequences and Quadratic Sequences K. N. Raghavan* School of Mathematics, SPIC Science Foundation, 92 G.N. Chetty Road, T. Nagar, Madras 600 017, India

and J. K. Verma† Department of Mathematics, Indian Institute of Technology, Powai, Bombay 400 076, India Communicated by Craig Huneke Received July 24, 1996

0. INTRODUCTION This paper is the outcome of our attempt to answer the following question. Let R s k w X x be the polynomial ring obtained by adjoining to a field k the entries of an m = n matrix X of indeterminates Žwe assume m F n., let M denote the maximal ideal of R generated by the entries of X, and let I be the ideal of R generated by the m = m minors of X. It is well known Žsee the next paragraph for hint of a proof. that the dimension M rq1 I s as a vector space over RrM M s k is, for large r and s, of M r I srM given by a polynomial in r and s of total degree one less than the dimension of R:

dim k

M rI s M

rq1 s

I

s

Ý

r , sc0 iqjFdim Ry1

ai j Ž M ¬ I .

rqi i

ž /ž

sqj . j

/

Ž 1.

* E-mail: [email protected]. † E-mail: [email protected]. 211 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

212

RAGHAVAN AND VERMA

The coefficients a i j Ž M ¬ I . of top degree, that is, with i q j s dim R y 1, are called the mixed multiplicities of the ordered pair Ž M ¬ I .. The problem of obtaining formulas for the mixed multiplicities of Ž M ¬ I . in terms of m and n was what started us out on this paper. Our approach to mixed multiplicities is through the following slightly more general problem. The associated graded ring grŽ M , I t . Rw It x of the Rees algebra Rw It x with respect to the maximal ideal Ž M , It . has a natural bigrading on it, and the graded piece of degree Ž r, s . in this bigrading is M rq1 I s. Thus the problem of computing this bigraded Hilbert series M r I srM in terms of m and n would, if solved, lead to formulas for all the coefficients a i j Ž M ¬ I ., not just for those of top degree. We solve this problem. The expression we obtain for the bigraded Hilbert series H Ž grŽ M , I t . R w It x ; l1 , l 2 . s

dim k

Ý r , sG0

M rI s M rq1 I s

l1r l2s

is Žsee Corollary 3.2.1 and Example 3.2.3. H Ž grŽ M , I t . R w It x ; l1 , l 2 . s H Ž R; l1 . q l2

Ý

vgV

H

ž

R RP v

; l1 H Ž Fv ; l2 . ,

/

Ž 2. where H Ž R; l1 . is the Hilbert series Ý iG 0 Ždim k R i . l1i of the graded k-algebra R; P is the poset of all minors of all sizes of the matrix X; V is the ideal of P consisting of the m = m minors; P v s p g P ¬ p h v 4 , the ideal of P ‘‘cogenerated by v ’’; RP v is the ideal of R generated by the elements of P v ; Fv is the face ring with coefficients in the field k of the poset P v s p g P ¬ p F v 4 . The face rings Fv are determined entirely by the combinatorics of the underlying poset V; recall that the face ring of a poset P, with coefficients in a field k, is the quotient of the polynomial ring k w x p ¬ p g P x, where the x p are indeterminates, by the ideal Ž x p x q ¬ Ž p, q . g D ., where D is the set of all incomparable pairs of elements in P. The rings RrRP v are rather well studied quotients of R}formulas for the Hilbert series of these rings have been obtained by Abhyankar wAx and by several authors after him. Equation Ž2. therefore gives an effective method for calculating the bigraded Hilbert series in terms of m and n.

213

MIXED HILBERT COEFFICIENTS

Using the fact that all maximal chains in P have the same length, we deduce from eq. Ž2. the following expression for the mixed multiplicities of Ž M ¬ I . Žsee Corollary 3.2.2.: ai j Ž M ¬ I . s

e

Ý

vgV rk Ž v .sjq1

R

e v.

ž /Ž RP v

for i q j s dim R y 1, Ž 3 .

where eŽ RrRP v . is the multiplicity as a graded k-algebra of RrRP v ; rkŽ v . is the length of any maximal chain v 1 z ??? z v p s v in V; eŽ v . is the number of distinct maximal chains of the type v 1 z ??? z v p s v. The referee has urged that the authors provide for the reader’s convenience explicit expressions for the above mixed multiplicities in terms of m and n. Unfortunately, however, we do not know a closed-form expression for the right-hand side of eq. Ž3.. For more information and for more admissions of ignorance about closed-form expressions, the reader is referred to Examples 2.6 and 2.8 of wRSx. Equation Ž2. holds more generally Žsee Corollary 3.2.1. for an ideal I generated by elements of a straightening-closed ideal V of a graded algebra R with straightening law over a field k on a finite poset P, provided that degree v F degree v X as elements of R whenever v F v X g V. ŽHere M denotes the graded maximal ideal of R.. In fact, the setup we work with is even more general and includes within its ambit ideals generated by d-sequences and Huckaba]Huneke ideals of analytic deviation 1 and 2 Žsee Sect. 3.. Having discussed above the results of the paper, we now give a more formal introduction to its content and technique. We restrict our attention in this introduction to the Rees algebra alone, but statements analogous to the ones to be made here hold for the associated graded ring and the extended Rees algebra. We begin with a basic definition. Let R be a Noetherian ring, M a maximal ideal of R, and I an ideal of R. There exists a natural bigrading on the associated graded ring grŽ M , I t . Rw It x of the Rees algebra Rw It x with respect to the maximal ideal Ž M , It ., and the graded piece of degree Ž r, s . M rq1 I s. The dimension of M r I srM M rq1 I s as a vector space over is M s I srM M is therefore a polynomial in r and s for large values of r and s. We RrM write dim R r M

M rI s M

rq1 s

I

s

Ý ai j Ž r , sc0

M¬I.

rqi i

ž /ž

sqj j

/

214

RAGHAVAN AND VERMA

and christen the a i j the mixed Hilbert coefficients of the ordered pair Ž M ¬ I .. Let R be a quotient by a graded ideal of a polynomial ring in finitely many variables over a field k. Let M denote the irrelevant maximal ideal of R. Let x 1 , . . . , x n be a d-sequence of homogeneous elements in R with degŽ x 1 . F ??? F degŽ x n .. Let I denote the ideal Ž x 1 , . . . , x n .. The multiplicity of the Rees algebra Rw It x of I with respect to the maximal ideal Ž M , It . is computed in wHTUx by Herzog, Trung, and Ulrich. The answer is in terms of the multiplicities as graded algebras over k of R and the quotients of R by the related ideals I j s ŽŽ x 1 , . . . , x jy1 . : x j ., 1 F j F n, of the d-sequence. In wRSx the technique and results of wHTUx are extended to linearizations of quadratic sequences with nondecreasing degrees. Here we push the work in wHTUx and wRSx to its logical conclusion: we write down the Hilbert series and Hilbert polynomial of Rw It x with respect to Ž M , It . in terms of those of R and RrIj as graded algebras over k. In fact, we write down the bigraded Hilbert series of grŽ M , I t . Rw It x. From this we get not only the Žsingly graded. Hilbert series and Hilbert polynomial of Rw It x with respect to Ž M , It . but also formulas for the mixed Hilbert coefficients of the pair Ž M ¬ I .. Our starting point here is the main technical lemma of wHTUx and this we describe now. The authors of that paper are interested in the multiplicity of Rw It x with respect to Ž M , It ., that is, the multiplicity of grŽ M , I t . Rw It x as a graded algebra over the base field k. To calculate this multiplicity}or for that matter the Hilbert series of grŽ M , I t . Rw It x as a graded k-algebra} we are free to look instead at the associated graded ring grF Ž Rw It x. of Rw It x with respect to F , where F is any filtration on Rw It x finer than the Ž M , It .-adic filtration. Now they look at a particular such filtration. To describe this, write Rw It x as a homomorphic image of the polynomal ring A s Rw T1 , . . . , Tn x under the map Ti ¬ x i t, and let N denote the maximal ideal Ž M , T1 , . . . , Tn . of A. The degree-lexicographic filtration F on A is finer than the N-adic filtration. Since Ž M , It . is the image of N under the above map from A to Rw It x, the induced filtration F on Rw It x is finer than the Ž M , It .-adic filtration. This is the filtration they consider. The associated graded ring grF Ž Rw It x. is presented as a quotient of grF Ž A . ( A, and under the hypothesis that x 1 , . . . , x n is a d-sequence with nondecreasing degrees, the presentation ideal is shown to have the form of a monomial ideal. This is their main technical result. From this the formula for the multiplicity of Rw It x is deduced by first writing the presentation ideal as an intersection and then using the associativity formula for multiplicities. The argument in wRSx runs exactly parallel to the argument in wHTUx just described.

MIXED HILBERT COEFFICIENTS

215

Our work here is based on two observations. First, the bigrading on grŽ M , I t . Rw It x is induced from the one on A, and since F respects the bigrading on A, the bigraded Hilbert series of grF Ž Rw It x. is the same as that of grŽ M , I t . Rw It x. Second, the associativity formula for multiplicities is a false lead}it gives the formula for the multiplicity all right but it hides other information that can just as readily be extracted from the presentation ideal. In our main result, namely, Theorem 2.4 below, we deduce from the presentation ideal an expression for the bigraded Hilbert series of grŽ M , I t . Rw It x. The argument is not any more difficult than the argument in wHTUx and wRSx based on the associativity formula. The organization of this paper should be clear from the titles of its sections. We would like to draw the reader’s attention to our rather special notation for the Hilbert series and the Hilbert polynomial, to the remarks in Sect. 4, and to the question posed in connection with Example 3.2.3. To end this section, here are some bibliographical notes. Mixed multiplicities were introduced first by Teissier and Risler wTx in their work on Milnor numbers of hypersurface singularties. They calculated them in terms of sufficiently general elements. Later Rees wRex introduced joint reductions to replace sufficiently general elements. This made the task of calculating the mixed multiplicities easier. Paul Roberts wRox has used mixed multiplicities to define multiplicity of a homomorphism between free modules. The remarks in Sect. 4 below are in the spirit of the formula in wKVx for the multiplicity of the extended Rees algebra in terms of mixed multiplicities. For more information about mixed multiplicities, see wKVx and the references therein. There are similarities between the proof of Theorem 2.4 below and the proof in wJRx.

1. NOTATION FOR HILBERT SERIES AND HILBERT POLYNOMIALS Let k be a field and let k wU1 , . . . , Up x be the polynomial ring in p variables over k with the usual grading. Let R s [iG 0 R i be a graded quotient of k wU1 , . . . , Up x. It is well known Že.g., it follows immediately from Hilbert’s syzygy theorem . that the Hilbert series H Ž R; t . [ Ý m G 0 dim k Ž R m . t m is a rational function of the form QŽ t .rŽ1 y t . p , where QŽ t . is a polynomial with integer coefficients. Thus dim k Ž R m . is a polynomial P Ž R; m. for large m. This is the Hilbert polynomial of R and we write

P Ž R; m . s

Ý ci iG0

ž

mqi . i

/

216

RAGHAVAN AND VERMA

The coefficients c i are all integers and of course c i s 0 for large i since this is a polynomial. To calculate the Hilbert polynomial from the Hilbert series, write the Hilbert series as a Laurent polynomal in 1 y t s X. The following proposition shows that the Hilbert polynomial can be read off the principal part of this Laurent polynomial. PROPOSITION 1.1. If H Ž R; X . s Ýa i X i is the Hilbert series written as a Laurent polynomial in X s 1 y t, then the Hilbert polynomal is gi¨ en by i. P Ž R;m. s Ý iG 0 ayiy1Ž m q . i Proof. Note that 1rŽ1 y t . m s Ý jG 0 Ž m y j1 q j . t j for m G 1. For a bigraded quotient R of the bigraded polynomial ring k wU1 , . . . , Up , V1 , . . . , Vq x, where each Ui has degree Ž1, 0. and each Vj has degree Ž0, 1., the bigraded Hilbert series HŽ R; t 1 , t 2 . [ Ý r, s G 0 dim k Ž R r, s . t 1r t 2s is similarly a rational function of the form QŽ t 1 , t 1 .rŽ1 y t 1 . p Ž1 y t 2 . q, and the bigraded Hilbert polynomial PŽ R; m, n., whose value for large r and s equals dim k R r, s can similarly be read off the principal part of the Laurent polynomial form of writing the Hilbert series: if HŽ R; X, Y . s Ýa i j X i Y j, where X s 1 y t 1 and Y s 1 y t 2 , then

P Ž R; m, n . s

Ý i , jG0

ayiy1, yjy1

ž

mqi i

/ž

nqj . j

/

All Hilbert series appearing in the sequel are written as Laurent polynomials in the above fashion: H Ž R; X . s Ýa i Ž R . X i and HŽ R; X, Y . s Ýa i j Ž R . X i Y j. We define a i j Ž M ¬ I . [ a i j ŽgrŽ M , I t . Rw It x. and bi j Ž M ¬ I . [ a i j ŽgrŽ M r I, Ir I 2 .ŽgrI Ž R ..., where M and I are ideals in a ring R and the bigradings on the rings appearing on the right-hand side are to be specified later by context. Any bigraded ring can be viewed as a singly graded ring by considering elements of degree Ž i, j . to have degree i q j. Under this process, the singly graded Hilbert series is obtained from the bigraded Hilbert series by setting Y s X. Thus the coefficient a i j in the bigraded Hilbert polynomial contributes towards the coefficient a iqj in the singly graded Hilbert polynomial. But terms of the bigraded Hilbert series of the form X i Y j with i q j - 0 but i G 0 or j G 0, while they contribute to the singly graded Hilbert polynomial, are irrelevant to the bigraded Hilbert polynomial.

217

MIXED HILBERT COEFFICIENTS

2. THE THEOREM In this section we prove the main result ŽTheorem 2.4., which unfortunately is rather technical. The starting point of the proof is the expression proved in wRSx for the ideal of leading forms, with respect to the degreelexicographic filtration F , of the ideal defining the Rees algebra as a quotient of the polynomial ring over the base ring. This ideal of leading forms has the form of a monomial ideal which enables us to express it as an intersection of simpler ideals and deduce from there the formula for the bigraded Hilbert series. We begin by recalling from wR2x the definition of a quadratic sequence. A subset L of a finite poset Ž V, F. is an ideal if

l g L,

v g V , and

vFl

«

v g L.

If L is an ideal of V and v g V _ L is such that l g L for every l z v , then Ž L, v . is a pair of V. Given a set  xv ¬ v g V 4 of elements of a ring R and L : V, denote by XL the ideal Ž xl ¬ l g L . of R Ž XL s 0 if L is empty. and by I the ideal X V s Ž xv ¬ v g V .. Definition 2.1 w R2, Definition 3.3x. A set  xv ¬ v g V 4 : R is a quadratic sequence if for every pair Ž L, v . of V there exists an ideal Q of V such that 1. Ž XL : xv . l I s XQ ; 2.

xv XQ : XL I.

Such an ideal Q is said to be associated with the pair Ž L, v .. This association need not be unique}the set  xv ¬ v g V 4 of generators of I may not be unshortenable }but XQ is unique by 1. Definition 2.2. A linerization of a poset V of cardinality n is a bijective map a: V ª w1, n x [  1, . . . , n4 such that v F v X « aŽ v . F aŽ v X .. Let  xv ¬ v g V 4 be a quadratic sequence and, let a: V ª w1, n x be a fixed linearization. Identify V with w1, n x via a. Then Žw1, j y 1x, j . is a pair of V for every j g w1, n x. Let Qj s an ideal of V associated with Ž w 1, j y 1 x , j . , I j s Ž Ž x 1 , . . . , x jy1 . : x j . , D s  Ž j, k . ¬ 1 F j F k F n, x j ? x k g Ž x 1 , . . . , x jy1 . 4 .

218

RAGHAVAN AND VERMA

For k g w1, n x, set Ck s w 1, k y 1 x

D

Qj

A k s X Ck .

and

jFk Ž j, k .fD

Note that Ck is an ideal of V and that A k is independent of the choices of Qj . Definition 2.3. A linearization a: V ª w1, n x of the indexing poset V of a quadratic sequence is stable if Ik s Ž A k : x k . for every k, 1 F k F n. We can now state our main theorem. THEOREM 2.4. Let R be a standard graded algebra o¨ er a field k, that is, R s [iG 0 R i s R 0 w R1 x with R 0 s k. Let  xv ¬ v g V 4 : R be a quadratic sequence consisting of homogeneous elements of R, let a: V ª w1, n x be a stable linearization, and suppose that deg Ž x 1 . F ??? F deg Ž x n . . Let I denote the ideal Ž x 1 , . . . , x n . and M s [i) 0 R i the irrele¨ ant maximal ideal. Then the bigraded Hilbert series of grŽ M , I t .Ž Rw It x. is gi¨ en by H Ž grŽ M , I t . R w It x ; X , Y . N

s H Ž R; X . q Ž 1 y Y .

H

Ý ls1 Ž l , l .gD

R

ž / Il

n

q

Ý ls1 Ž l , l .fD

H

;X

R

H Ž Fl ; Y . 2yY

ž / Il

; X H Ž Fl ; Y . ,

where D is defined as abo¨ e and Fl s

k Ti , Tl ¬ i - l, Ž i , l . f D

Ž Tj Tk ¬ 1 F j F k F l, Ž j, k . g D , Ž j, l . f D , Ž k, l . f D . q Ž Tl2 ¬ only if Ž l, l . g D .

.

The formula for the bigraded Hilbert series of grŽ M r I, Ir I 2 .ŽgrI Ž R .. is the same as the one abo¨ e except that on the right-hand side we replace R by RrI and

219

MIXED HILBERT COEFFICIENTS

RrIl by RrŽ I q Il .: H Ž grŽ M r I , Ir I 2 . Ž grI Ž R . . ; X , Y . sH

ž

R I

n

; X q Ž1 y Y .

/

H

Ý ls1 Ž l , l .gD

ž

R I q Il

n

q

Ý ls1 Ž l , l .fD

H

ž

;X

R I q Il

/

H Ž Fl ; Y . 2yY

/

; X H Ž Fl ; Y . .

Proof. We prove only the formula for the Rees algebra. If in this proof we replace R by RrI and RrIj by RrŽ I q I j ., we get the proof of the formula for the associated graded ring. Map Rw T1 , . . . , Tn x ª Rw It x by sending Ti to x i t. Let F be the filtration on Rw T1 , . . . , Tn x defined in Sect. 1 of wRSx. The associated graded ring grF Ž Rw It x. of Rw It x with respect to the induced filtration F has, by Theorem 1.4 of wRSx, the following presentation: grF Ž R w It x . (

R w T1 , . . . , Tn x

Ž I1T1 , . . . , InTn , Tj Tk ¬ Ž j, k . g D .

.

Ž 4.

Since grŽ M , I t .Ž Rw It x. and grF Ž Rw It x. have the same bigraded Hilbert series, we may work with the latter ring. The ideal Ž I1T1 , . . . , InTn , Tj Tk ¬ Ž j, k . g D . is a ‘‘block monomial ideal,’’ i.e., it behaves like a monomial ideal if we treat I1 , . . . , In as blocks. To calculate the Hilbert series, we treat it as such. The first step then is to write it as an intersection of simpler ideals. In order to do this, we first observe that if j F k, Ž j, k . f D, and a g I j , then by the stability assumption ax k g I j l I s XQ j : Ik , so a g Ik . Thus, for j F k, either Ž j, k . g D or I j : Ik . We claim that n

Ž I1T1 , . . . , InTn , Tj Tk ¬ Ž j, k . g D . s

F Jl , ls0

where J 0 [ ŽT1 , . . . , Tn . and Jl [ Ž Il , Tlq1 , . . . , Tn , Ti , Tj Tk ¬ i - l, Ž i , l . g D ; 1 F j F k F l, Ž k, l . g D . .

220

RAGHAVAN AND VERMA

That F Jl contains the left-hand side follows from the observation made above. For the converse, suppose aTi1 ??? Ti p does not belong to the left-hand side, where a g R and i1 F ??? F i p . Then no pair Ž i r , i s . with r - s belongs to D. Thus aTi1 ??? Ti p does not belong to Ji p and the claim is proved. Now let I0 [ J 0 and for 1 F l F n, Il [ Ily1 l Jl . Letting A denote the polynomial ring Rw T1 , . . . , Tn x we have exact sequences 0ª

A Il

A

ª

Ily1

A

[

Jl

ª

A Ily1 q Jl

ª0

for 1 F l F n. These yield H

A

A

A

ž / ž / ž / ž Il

sH

Ily1

qH

Jl

yH

A

/

,

Ily1 q Jl

/

Ily1 q Jl

where H stands for bigraded Hilbert series. Adding these n equations, we get H

A

A

ž / ž / In

sH

I0

n

q

ÝH ls1

A

ž / ž Jl

yH

A

.

It is proved easily by induction that Ily1 s Ž I1T1 , . . . , Ily1Tly1 , Tl , . . . , Tn , Tj Tk ¬ 1 F j F k F l y 1, Ž j, k . g D . . The expression for Ily1 q Jl is now immediate: Ily1 q Jl s Ž Il , Tl , . . . , Tn , Ti , Tj Tk ¬ i - l, Ž i , l . g D ; 1 F j F k - l, Ž j, k . g D . . In . is the desired Hilbert series, that ArIl ( RrIl mk Observe that HŽ ArI Fl , where Fl is as in the statement of the theorem, and that ArŽ Ily1 q Jl . ( RrIl mk Fl , where Fl [ FlrŽTl .. Either Ž l, l . g D, in which case Fl s Fl mk k w Tl xrŽTl2 ., or Ž l, l . f D, in which case Fl s Fl w Tl x. In the former case, the Hilbert series of Fl and Fl are related by H Ž Fl ; t 2 . s H Ž Fl ; t 2 .Ž1 q t 2 . Žwhich, if we set 1 y t 2 s Y, means H Ž Fl ; Y . s H Ž Fl ; Y .Ž2 y Y .. and in the latter case, by H Ž Fl ; t 2 . s H Ž Fl ; Y .rŽ1 y t 2 . Žwhich, if we set 1 y t 2 s Y, means H Ž Fl ; Y . s H Ž Fl ; Y .rY .. We thus get the desired formula.

221

MIXED HILBERT COEFFICIENTS

3. EXAMPLES 3.1. d-Sequences An ordered sequence x 1 , . . . , x n of elements of a ring is called a d-sequence if

Ž Ž x 1 , . . . , x jy1 . : x j x k . s Ž Ž x 1 , . . . , x jy1 . : x k .

; 1 F j F k F n. Ž 5 .

It is easy to see that Žsee, for example, wR1x. that the above condition is equivalent to

Ž Ž x 1 , . . . , x jy1 . : x j . l I s Ž x 1 , . . . , x jy1 .

; 1 F j F n,

Ž 6.

where I is the ideal Ž x 1 , . . . , x n .. It is clear from this last equation that d-sequences are quadratic sequences: indeed L s w1, j y 1x is itself associated with the pair Ž L, v s j .. It is also clear, since Ck s w1, k y 1x for every k g w1, n x, that the Žonly possible. linearization is stable. Thus Theorem 2.4 is applicable to homogeneous d-sequences whose elements have nondecreasing degrees. For Ž j, k . g D, eq. Ž5. shows that Ik is the unit ideal, so the term Tj Tk is redundant in the denominator of the right-hand side of eq. Ž4.. We may therefore assume, in the conclusion of Theorem 2.4, that D is empty, hence that Fl s k w T1 , . . . , Tl x, and hence that H Ž Fl ; Y . s Yyl . Thus Theorem 2.4 reduces in this case to COROLLARY 3.1.1. Let R be a standard graded algebra o¨ er a field, let x 1 , . . . , x n be a d-sequence consisting of homogeneous elements of R, and suppose that deg Ž x 1 . F ??? F deg Ž x n . . Let I denote the ideal Ž x 1 , . . . , x n . and M the irrele¨ ant maximal ideal. Then the bigraded Hilbert series of grŽ M , I t .Ž Rw It x. is gi¨ en by H Ž grŽ M , I t . R w It x ; X , Y . s H Ž R; X . q Ž 1 y Y .

n

ÝH ls1

R

; X Yyl .

ž / Il

The formula for the bigraded Hilbert series of grŽ M r I, Ir I 2 .ŽgrI Ž R .. is the same as the one abo¨ e except that on the right-hand side we replace R by RrI and RrIl by RrŽ I q Il .: H Ž grŽ M r I , Ir I 2 . Ž grI Ž R . ; X , Y . . sH

ž

R I

n

; X q Ž1 y Y .

/

ÝH ls1

ž

R I q Il

; X Yyl .

/

222

RAGHAVAN AND VERMA

In particular, n

a p Ž grŽ M , I t . Ž R w It x . . s a p Ž R . q

Ý a pyl ls1

ap q Ž M ¬ I . s ap a p Ž grŽ M r T , Ir I 2 . grI Ž R . . s a p

R

ž / Il

R

y a pylq1

ž / ž / ž / Ý ž / ž / ž / ž / y ap

Iqq 1

q

I

R

I q Iqq 1

Il

,

I q Il

ls1

I q Il

ž /

R

a pyl

R

R

,

Iqq2

n

R

y a py lq1 bp q Ž M ¬ I . s a p

R

,

y ap

R

I q Iqq2

.

For a detailed discussion of examples to which the above corollary can be applied, we refer the reader to wHTU, Sect. 3x. Here we stay content with the simplest application: EXAMPLE 3.1.2 Žregular sequences .. Let R be a standard graded algebra over a field and let x 1 , . . . , x n be a regular sequence of homogeneous elements of respective degrees d1 F ??? F d n . Let I denote the ideal Ž x 1 , . . . , x n . and M the irrelevant maximal ideal. Corollary 3.1.1 is applicable in this situation since a regular sequence is, evidently, a d-sequence. Since Il s Ž x 1 , . . . , x ly1 . for 1 F l F n, we have H Ž RrIl ; X . s H Ž R; X . f 1 ??? f l , where f i s Ž1 y Ž1 y X . d i ., so that H Ž grŽ M , I t . R w It x ; X , Y . n

s H Ž R; X . q Ž 1 y Y .

Ý H Ž R; X . f 1

??? f ly1Yyl

ls1

s

H Ž R; X . Y

n

ny1

f 1 ??? f ny1 q

Predictably, the formula for grŽ M r I, Ir I 2 . gri Ž R . reduces to

Ý Y ny l Ž 1 y f l . f 1

??? f ly1 .

ls1

the

bigraded

H Ž grŽ M r I , Ir I 2 . grI Ž R . ; X , Y . s

H Ž R; X . Yn

Hilbert

f 1 ??? f n .

series

of

223

MIXED HILBERT COEFFICIENTS

3.2. Straightening-Closed Ideals in Graded ASLs We borrow the notation, terminology, definitions, and references of the subsection of wRS, Sect. 2x that has the same title as the present one. Throughout our discussion of straightening-closed ideals, R denotes a graded ASL on a finite poset P over a field R 0 s k. For L : P, the ideal Ž l ¬ l g L . of R is denoted RL. Let V be a straightening-closed ideal of P. By Proposition 2.4 of wRSx,  v ¬ v g V 4 is a quadratic sequence in R. By Proposition 2.5 of wRSx, any linearization a: V ª w1, n x of V is stable. Assuming that degŽ x 1 . F ??? F degŽ x n . for a given linearization, we may therefore apply Theorem 2.4. Proposition 2.5 of wRSx gives D s  Ž j , n . ¬ j , n incomparable elements of V , and a Ž j . F a Ž n . 4 , Il s RP v ,

where v s ay1 Ž l . and P v s  p g P ¬ p h v 4 .

Thus Fl is the face ring of the simplicial complex with vertex set  s g V ¬ s F v [ ay1 Ž l .4 and faces the square-free standard monomials on this poset. We obtain the following COROLLARY 3.2.1. Let R be a graded ASL on a finite poset P o¨ er a field k and let M denote the graded maximal ideal of R. If V is a straighteningclosed ideal of P which admits a linearization a: V ª w1, n x satisfying deg Ž x 1 . F ??? F deg Ž x n . then the bigraded Hilbert series of grŽ M , Ž R V .t .Ž RwŽ RV . t x. is gi¨ en by H Ž grŽ M , Ž R V .t . R Ž RV . t ; X , Y . s H Ž R; Y . q Ž 1 y X .

Ý

H

vgV

ž

R RP v

/

; X H Ž Fv ; Y . ,

where Fv is the face ring of the poset  s g V ¬ s F v 4 . The formula for the bigraded Hilbert series of grŽ M r R V , R V rŽ R V . 2 .ŽgrR V Ž R .. is the same as the one abo¨ e except that on the right-hand side we replace R by RrRV and RrRP v by RrRŽ V j P v .: H Ž grŽ M r R V , R V rŽ R V . 2 . Ž grR V Ž R . . . sH

ž

R RV

; X q Ž1 y Y .

/

Ý

vgV

H

ž

R RŽ V j P v .

/

; X H Ž Fv ; Y . .

224

RAGHAVAN AND VERMA

In particular, a p Ž grŽ M , Ž R V .t . R Ž RV . t s ap Ž R. q

.

Ý Ý aj

vgV

a p q Ž M ¬ RV . s

j

Ý

vgV

ap

R

ž / ž / RP v R

RP v

a pyjy1 Ž Fv . y a pyj Ž Fv . , a q Ž Fv . y a qq1 Ž Fv . ,

a p Ž grŽ M r R V , R V rŽ R V . 2 . grR V Ž R . . s ap

R

ž / RV

q

Ý Ý aj

vgV

bp q Ž M ¬ RV . s

Ý

vgV

j

ap

ž

ž

R RŽ V j P v . R

RŽ V j P v .

a pyjy1 Ž Fv . y a pyj Ž Fv . ,

/

a q Ž Fv . y a qq1 Ž Fv . .

/

Note the formal analogy between the special case of the above formulas when V is linearly ordered and the formulas for a d-sequence of Corollary 3.1.1: if V s  1, . . . , n4 is linearly ordered, then Fl is the polynomial ring k w T1 , . . . , Tl x, so a q Ž Fl . is 1 if q s l y 1 and 0 otherwise. We now recover, from the above corollary, expressions for the multiplicities of grŽ M , Ž R V .t . RwŽ RV . t x and grŽ M r R V , R V rŽ R V . 2 . grR V Ž R .. The following statement contains within it Theorem 2.2 of wRSx. COROLLARY 3.2.2.

Suppose further that V is nonempty and that

all maximal chains in P ha¨ e the same length. Then the multiplicity of grŽ M , Ž R V .t . RwŽ RV . t x is gi¨ en by R e Ž grŽ M , Ž R V .t . R Ž RV . t . s Ý e eŽ v . , RP v vgV

Ž ).

ž /

Ž 7.

where eŽ RrRP v . is the multiplicity of the graded algebra RrRP v and eŽ v . is the number of maximal chains in V of the type v 1 z ??? z v p . The mixed multiplicities of Ž M ¬ RV . are gi¨ en by R a p q Ž M ¬ RV . s e eŽ v . , Ý RP v v gV , rk Ž v . ,sqq1

ž /

where p q q s dim R y 1.

Ž 8.

The multiplicity of grŽ M r R V , R V rŽ R V . 2 . grR V Ž R . is gi¨ en by e Ž grŽ M r R V , R V rŽ R V . 2 . grR V Ž R . . s

Ý

vg ­ V

e

ž

R RŽ V j P v .

/

eŽ v . ,

Ž 9.

where ­ V s  v g V j  y`4 ¬ v has an upper neighbor in P j  `4 not in V 4 , where y` and ` are the smallest and largest elements added to P.

225

MIXED HILBERT COEFFICIENTS

Proof. Since ASLs are reduced and Ž). implies that all minimal primes of R have maximal dimension, it is clear that dim grŽ M , R V . RwŽ RV . t x s d q 1, where d s dim R. Thus eŽgrŽ M , R V . RwŽ RV . t x. s a d ŽgrŽ M , R V . =RwŽ RV . t x. and a d Ž R . s 0. Since RrRP v is an ASL on P _ P v , it follows that dimension of RrRP v is rkŽ P _ P v ., so a j Ž RrRP v . s 0 for j G rkŽ P _ P v . and a j Ž RrRP v . s eŽ RrRP v . for j s rkŽ P _ P v . y 1. The dimension of Fv is rkŽ v . and its multiplicity is eŽ v ., so a k Ž Fv . s 0 for k G rkŽ v . and a k Ž Fv . s eŽ v . for k s rkŽ v . y 1. It follows from Ž). that rkŽ P _ P v . q rkŽ v . s rkŽ P . q 1 s d q 1 for every v . Using the formula for a p ŽgrŽ M , Ž R V .t . RwŽ RV . t x. in Corollary 3.2.1 to calculate a d , we get eq. Ž7.. From the above considerations we also get eq. Ž8.. The proof of eq. Ž9. is similar: note that rkŽ P _Ž V j P v .. q rkŽ v . F rkŽ P . q 1 and that equality holds if and only if v g ­ V. EXAMPLE 3.2.3 Žgeneric maximal minors.. Let R s k w X x be the polynomial ring obtained by adjoining to a field k the entries of an m = n matrix X of indeterminates Žwe assume m F n.. It is well known that R is a graded ASL over k on the poset P of all q-minors of X, 1 F q F m. A q-minor is denoted w r 1 ??? r q ¬ c1 ??? c q x, where r 1 , . . . , r q and c1 , . . . , c q indicate row and column indices. The partial order on P is given by the following rule: r 1 , . . . , r q ¬ c1 ??? c q F r 1X , . . . , r qX X ¬ cX1 , . . . , cXqX if and only if q G qX and ri F riX , c i F cXi for 1 F i F qX . The ideal V of all m-minors is a straightening-closed ideal ŽwH2, item 1.19x, wBST, Example 2.1.3x.. Since all elements of V have the same degree, Theorem 2.4 applies. The Hilbert series of the rings RrRP v have been calculated by Abhyankar wAx and by several authors after him, e.g., wGx, wCHx, and wGhx. An element v of V is characterised by its column indices c1 , . . . , c m . From eq. Ž2. of wCHx, we see that

H

ž

R RP

det v ; X s

/

ž

Ý k

ž

myi k

/ž

n y ci q i y j k Ž1 y X . kqiyj

/

X

d

/

i , js1, . . . , m

,

where d s mŽ2 n q m q 1.r2 y Ýc i is the dimension of RrRP v. Substituting the above equation and information about the face rings Fv into the formulas of Theorem 2.4, we can calculate the bigraded Hilbert series of grŽ M , R V . RwŽ RV . t x and grŽ M r R V , R V rŽ R V . 2 . grR V Ž R .. We will now describe a way of looking at this example which leads naturally to a question. Our reference for this paragraph is wSx. Let Gm, mqn be the Grassmannian of m-planes in Ž m q n.-space. The opposite

226

RAGHAVAN AND VERMA

big cell of Gm, mqn is an affine space of dimension mn. The Schubert varieties in Gm, mqn are indexed by the poset P of the above example. If we intersect the Schubert variety corresponding to w1, . . . , m y 1 ¬ 1, . . . , m y 1x with the opposite big cell, the resulting affine variety has ideal RV Žsee Proposition 6.10 of Chapter I of wSx.. This leads us to ask the following question: let V be some other Schubert variety and I be the ideal of the intersection of V with the opposite big cell; then what is the Hilbert series of grŽ M , I t . Rw It x? Note that the Grassmannian Gr, n is SL nrP for a maximal parabolic subgroup of SL n . The question can also be asked for Schubert varieties in the quotient spaces of SO Ž n. and SP Ž2 n. by maximal parabolic subgroups. Except for the situation of the above example and for some other rather special situations which can be reduced easily to the situation of the above example, we do not know the answer. 3.3. Defining Ideals of Projecti¨ e Monomial Space Cur¨ es Lying on the Quadric Surface xy y zw s 0 Let R s k w x, y, z, w x be the polynomial ring in four variables over a field k, and let I denote the homogeneous ideal of the projective monomial curve

Ž x : y : z : w . s Ž u bq c : u b ¨ c : u c ¨ b : ¨ bq c . , where b ) c and bcdŽ b, c . s 1. It is proved in Propositions 2.1 and 2.2 of wMSx that the ideal I is generated by a particular set of b y c q 2 elements which form a quadratic sequence in a particular linear ordering. Let us use this ordered set of generators of I to present Rw It x and grI Ž R . as quotients of the polynomial ring S [ Rw T1 , . . . , Tbycq2 x. Let J and K be the kernels, respectively, of the maps S ¸ Rw It x and S ¸ grI Ž R .. Let F be the degree-lexicographic filtration on S defined as in wRS, Sect. 1x. If F denotes also the induced filtration of Rw It x, the associated graded ring grF Ž Rw It x. has the same bigraded Hilbert series as Rw It x. Since grF Ž S . ( S, we have grF Ž Rw It x. ( grF Ž SrJ . ( SrJ#, where J# is the ideal generated by initial forms of elements of J with respect of F. Similar statements are true for grI Ž R .. Theorems of wRS, Sect. 1x, apply in this situation Žsee wRSx for details. and we get J# s Ž T2 , . . . , Tbycq2 . l Ž xw y yz , T3 , . . . , Tbycq2 . 2

l Ž x, y, Ž T3 , . . . , Tbycq1 . . , 2

K# s Ž I, T3 , . . . , Tbycq2 . l Ž x, y, z b , Ž T3 , . . . , Tbycq1 . . .

227

MIXED HILBERT COEFFICIENTS

Using this we can calculate, as in the proof of Theorem 2.4, the bigraded Hilbert series of grŽ M , I t . Rw It x and grŽ M r I, Ir I 2 .ŽgrI Ž R ... Among the ideals u occurring in the right-hand side of the above display, the only one for which the computation of Hilbert series of Sru may perhaps take some time is Ž I, T3 , . . . , Tbycq2 .. So we compute the Hilbert series of RrI below. Since y is a regular element of degree 1 in RrI, we have H Ž RrI; X . s Ž1rX . H Ž RrŽ I, y .; X .. Now Ž I, y . s Ž y, xw, x by c z c, . . . , xz by 1 , z b . is a monomial ideal. For k - b, the graded piece of degree k of RrŽ I, y . is spanned by the Ž2 k q 1. monomials x k , x ky1 z, . . . , xz ky1 , z k and w k , w ky 1 z, . . . , wz ky 1 ; for k G b, it is spanned by the b q c monomials x k , x ky 1 z, . . . , x ky cq1 z cy 1 and w k , w ky 1 z, . . . , w ky bq1 z by 1. Thus

H

ž

R I

;X s

/

bqc X

2

y

by1

1 X

žÝ

Ž b q c y 2 k y 1. Ž 1 y X .

k

ks0

/

.

Calculating the bigraded Hilbert series of grŽ M , I t . R w It x and grŽ M r I, Ir I 2 .ŽgrI Ž R .. as in the proof of Theorem 2.4, we obtain H Ž grŽ M , I t . R w It x ; X , Y . s

1 4

X Y q

H Ž grŽ M r I 2 , Ir I 2 . Ž grI Ž R . . . s

X Y 2

X Y

X Y q

3

2

q

y Ž b y c q 1.

bqc 2

2

q

2

q

XY

byc X 2Y 3

by1

1 2

2

žÝ

byc X 2Y 3 q

y2 3

X Y

q

1 X 2Y

,

Ž 1 y Y . Ž1 y Ž 1 y X .

b

.

j Ž2 j q 1 y b y c. Ž1 y X . .

js0

/

3.4. Huckaba]Huneke Ideals of Analytic De¨ iation 1 and 2 As shown in wR2, Sect. 4x, the ideals of analytic deviation 1 and 2 studied by Huckaba and Huneke in wHH1x and wHH2x are generated by quadratic sequences. While the ambient ring in wHH1x and wHH2x is always local, the structure theorems established in those papers and in wR2x for these ideals hold good for certain homogeneous ideals; a particular example is given below. So it seems worthwhile to derive formulas for grŽ M , I t . Rw It x and grŽ M r I, Ir I 2 .ŽgrI Ž R .. for homogeneous ideals having this structure. We proceed to do this. Let R be a standard graded algebra over a field k and let M be the irrelevant maximal ideal of R. Let x 1 , . . . , x m , ymq1 , . . . , yn be homoge-

228

RAGHAVAN AND VERMA

neous elements of R which are indexed by the poset V specified as follows. We denote the elements of V also by x 1 , . . . , x m , ymq1 , . . . , yn . The relations in V are x i F x j for 1 F i F j F m and x i F y j for every pair Ž i, j . with 1 F i F m and m q 1 F j F n. Let I be the ideal Ž x 1 , . . . , x m , ymq 1 , . . . , yn . and J the ideal Ž x 1 , . . . , x m .. Suppose that x 1 , . . . , x m form a d-sequence with respect to I, that is,

Ž Ž x 1 , . . . , x jy1 . : x j . l I s Ž x 1 , . . . , x jy1 .

;1FjFm

Ž 10 .

and that JI s I 2 . Further suppose that I is not generated by any proper subset of  x 1 , . . . , x m , ymq1 , . . . , yn4 . Then x 1 , . . . , x m , ymq1 , . . . , yn from a quadratic sequence on V wR2, Proposition 4.7x. For any linearization a, we have D s Ž i, j . ¬ m q 1 F i, j F n4 and thus a is stable. Assuming that V admits a linearization with nondecreasing degrees, we may apply Theorem 2.4 to this situation. For 1 F l F m, we have Fl s k w T1 , . . . , Tl x. For m q 1 F l F n, we have Fl s k w T1 , . . . , Tm , Tl xrŽTl2 .. Thus H Ž Fl ; Y . is 1rY l for 1 F l F m and Ž2 y Y .rY m for m q 1 F l F n. Substituting these into the formulas of Theorem 2.4, we obtain In the situation just described,

COROLLARY 3.4.1.

H Ž grŽ M , I t . R w It x ; X , Y . s H Ž R; X . q

1yY Y

m

m

ÝH ls1

R

n

ž / Il

; X Y my l q

H

Ý lsmq1

R

ž / Il

;X

H Ž grŽ M r I , Ir I 2 . Ž grI Ž R . . ; X , Y . sH

ž

q

R I

;X

1yY Y

m

/ m

Ý ls1

H

ž

R I q Il

n

/

; X Y my l q

Ý lsmq1

H

ž

R I q Il

;X

/

.

EXAMPLE 3.4.2. This example is discussed in wHH1x Žsee wHH1, Example 4.7x.. Let R s k w a, b, c, d, e x be the polynomial ring in five variables over a field k, char k / 2, and let M denote the irrelevant maximal ideal of R. Let I be the homogeneous ideal of R generated by the eight elements f 1 , . . . , f 8 displayed below Žthis ideal is a prime of codimension 2 and defines the base locus of a section of the Horrocks]Mumford bundle.: f 1 s 5abcde y a5 y b 5 y c 5 y d 5 y e 5 , f 2 s ab 3 c q bc 3d q a3 be q cd 3 e q ade 3 , f 3 s a2 bc 2 q b 2 cd 2 q a2 d 2 e q ab 2 e 2 q c 2 de 2 ,

229

MIXED HILBERT COEFFICIENTS

f 4 s abc 5 y b 4 c 2 d y 2 a2 b 2 cde q ac 3 d 2 e y a4 de 2 q bcd 2 e 3 q abe 5 , f5 s ab 2 c 4 y b 5 cd y a2 b 3 de q 2 abc 2 d 2 e q ad 4 e 2 y a2 bce 3 y cde 5 , f6 s a3 b 2 cd y bc 2 d 4 q ab 2 c 3 e y b 5de y d 6 e q 3abcd 2 e 2 y a2 be 4 y de 6 , f 7 s a4 b 2 c y abc 2 d 3 y ab 5 e y b 3 c 2 de y ad 5 e q 2 a2 bcde 2 q cd 2 e 4 , f 8 s b 6 c q bc 6 q a2 b 4 e y 3ab 2 c 2 de q c 4 d 2 e y a3 cde 2 y abd 3 e 2 q bce 5 . As can be verified on a computer Žsee also wHH1x., this example satisfies the hypothesis of Corollary 3.4.1}the integer m in this case is 3. Fixing the linearization f 1 , . . . , f 8 , we can calculate using a computer the Hilbert series H Ž RrIl ; X .. Plugging these series into the formula for HŽgrŽ M , I t .Ž Rw It x; X, Y ., we get H Ž grŽ M , I t . R w It x ; X , Y . s

1 X Y q q

y

5

q

5

10 3

X Y 5 XY

4

X Y q

q

68 X 3 Y2

2

q

25 2

2

X Y

54 X Y3 q

10 3

X Y q

y

20 X 4 Y3

y

3

5 XY 1 Y

q

3

q

5 X Y

y 54 X Y2

20 X 4 Y2

4

q

10 2

X Y q

q

20 X Y

q

99 X 2 Y3 2X5 Y3

3

q

2

10 XY q

y

2

15 X 2Y 3

y

1 Y3

99 X 2 Y2 2X5 Y2

q

68 X 3 Y3

.

4. REMARKS ABOUT THE HILBERT SERIES OF grŽ M , I t, ty1 . Rw It, ty1 x AND grŽ M r I, Ir I 2 . grI Ž R . In Theorem 2.4 and in the examples of Sect. 3, an expression for the bigraded Hilbert series of grŽ M , I t, ty1 . Rw It, ty1 x can also be derived similarly. But we have not bothered to do this for two reasons. First, as can be seen from the proposition below, the Hilbert series of grŽ M , I t, ty1 . Rw It, ty1 x can in most cases be recovered from those of grŽ M , I t . Rw It x and grŽ M r I, Ir I 2 . grI Ž R .. Second, it is not clear what the bigraded pieces of grŽ M , I t, ty1 . Rw It, ty1 x are and whether they are of interest.

230

RAGHAVAN AND VERMA

PROPOSITION 4.1. 1. Under the hypothesis of Theorem 2.4, H Ž grŽ M , I t , ty1 . R w It , ty1 x ; X , Y .

¡1 y X H Ž R; X . q H Žgr

s

Ž M , I t . R w It x ; X , Y . X if deg Ž x i . G 3 ; i , 1 F i F n,

~

1

¢

Y

H Ž grŽ M r I , Ir I 2 . grI Ž R . ; X , Y . if deg Ž x i . F 2 ;i , 1 F i F n.

2. In any Noetherian ring R, if I is an ideal and M is a maximal ideal such that I : M 2 , then H Ž grŽ M , I t , ty1 . R w It , ty1 x ; X . s

1yX X

H Ž gr M Ž R . ; X . q H Ž grŽ M , I t . R w It x ; X . .

Proof. 1. Fix notation as in wRS, Sect. 1x. If degŽ x i . F 2 for all i, then from Theorems 1.2 and 1.3 of wRSx it is clear that L# is the extension of K# to Rw T1 , . . . , Tn , U x. This proves the second half of the statement. Now suppose degŽ x i . G 3 for all i. Then by Theorems 1.1 and 1.3 of wRSx, we have L# s ŽUT1 , . . . , UTn , J#.. Thus L# s ŽU, J#. l ŽT1 , . . . , Tn .. Letting A s Rw T1 , . . . , Tn , U x, we have ArŽU, J#. ( Rw T1 , . . . , Tn xrJ#, Ar ŽT1 , . . . , Tn . ( RwU x, and ArŽU, J#, T1 , . . . , Tn . ( R. The first part of the statement follows. 2. As a routine calculation shows,

Ž M , It , ty1 .

n

y1 nq 1

Ž M , It , t

s

.

R M [

tyn [ Mn M nq1

ny1

s

žÝ

js0

M M2 [

tynq1 [ ??? M ny 1 I M nI

t [ ??? [

[ w gr M R x j [

/

In MI n

Ž M , It .

tn

n

nq1 Ž M , It .

.

231

MIXED HILBERT COEFFICIENTS

Thus dim

Ž M , It , ty1 . Ž M , It , ty1 .

n

nq1

n

s dim

žÝ

[ w gr M R x j

js0

y dim gr M Ž R .

n

/

q dim

Ž M , It .

n

nq1 Ž M , It .

.

Note that the first term on the right-hand side is just the dimension of the nth piece of a polynomial ring in one variable over gr M Ž R . which has Hilbert series H Žgr M Ž R .; X .rX. Remark 4.2. A routine calculation shows that the graded piece of degree Ž r, s . of grŽ M r I, Ir I 2 . grI Ž R . is Ž M r I s q I sq1 .rŽ M rq1 I s q I sq1 .. From this bigraded Hilbert series we can therefore read off the polynomial in r and s whose value for large r and s equals dimŽ M r I s q I sq1 .r Ž M rq1 I s q I sq1 ..

ACKNOWLEDGMENTS The authors thank Murali Srinivasan for combinatorial discussions. Part of this work was done when both authors were visiting Pondicherry University. Most of this work was done during two visits of the first named author to Indian Institute of Technology, Bombay. A grant of SPIC Science Foundation by the Department of Science and Technology of the Government of India enabled purchase of computing equipment on which calculations using Macaulay were performed. We thank all these institutions for their patronage.

REFERENCES wAx

S. S. Abhyankar, Combinatoire des tableaux de Young, varietes et ´ ´ determinantielles ´ calcul de functions de Hilbert, Rend. Sem. Mat. Uni¨ . Politec. Torino 42 Ž1984., 65]88. wBSTx W. Bruns, A. Simis, and N. V. Trung, Blow-up of straightening-closed ideals in ordinal Hodge algebras, Trans. Amer. Math. Soc. 326 Ž1991., 507]528. wCHx A. Conca and J. Herzog, On the Hilbert function of determinantal rings and their canonical module, preprint, 1992. wGx A. Galigo, Computations of some Hilbert functions related with Schubert calculus, in ‘‘Algebraic Geometry, Sitges ŽBarcelona.’’ ŽE. Casas]Alvero, G. E. Welters, and S. Xambo-Descamps, Eds.., Lecture Notes in Mathematics, Vol. 1124, pp. 79]97, ` Springer-Verlag, BerlinrNew York. wGhx S. R. Ghorpade, Young bitableaux, lattice paths and Hilbert functions, preprint, 1994. wH2x C. Huneke, Powers of ideals generated by weak d-sequences, J. Algebra 68 Ž1981., 471]509.

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RAGHAVAN AND VERMA

wHH1x S. Huckaba and C. Huneke, Powers of ideals having small analytic deviation, Amer. J. Math. 114 Ž1992., 367]403. wHH2x S. Huckaba and C. Huneke, Rees algebras of ideals having small analytic deviation, Trans. Amer. Math. Soc. 339 Ž1993., 373]402. wHTUx J. Herzog, N. V. Trung, and B. Ulrich, On the multiplicity of blow-up rings of ideals generated by d-sequences, J. Pure Appl. Algebra 80 Ž1992., 273]297. wJRx M. R. Johnson and K. N. Raghavan, Cohen]Macaulayness of blow-ups of homogeneous weak d-sequences, Proc. Amer. Math. Soc. 123Ž7. ŽJuly 1995., 1991]1994. wKVx D. Katz and J. K. Verma, Extended Rees algebras and mixed multiplicities, Math. Z. 202 Ž1989., 111]128. wMSx M. Morales ` and A. Simis, Symbolic powers of monomial curves in P 3 lying on xw y yz s 0, Comm. Algebra 20 Ž1992., 1109]1121. wR1x K. N. Raghavan, A simple proof that d-sequences generate ideals of linear type, Comm. Algebra 19Ž10. Ž1991., 2827]2831. wR2x K. N. Raghavan, Powers of ideals generated by quadratic sequences, Trans. Amer. Math. Soc. 343Ž2. Ž1994., 727]747. wRSx K. N. Raghavan and A. Simis, Multiplicities of blow-ups of homogeneous quadratic sequences, J. Algebra 175 Ž1995., 537]567. wRex D. Rees, Generalizations of reductions and mixed multiplicities, J. London Math. Soc. 29 Ž1984., 397]414. wRox P. Roberts, Multiplicities and Chern classes, Contemp. Math. 159, 333]350. wSx C. S. Seshadri, ‘‘Introduction to the Theory of Standard Monomials,’’ Brandeis Lecture Notes, Vol. 4, 1985. wTx B. Teissier, Cycles ´ evanescents, sections planes, et conditions de Whitney, ‘‘Singularities 7–8 Ž1973., 285]362. ´ `a Cargese, ´ 1972,’’ Asterisque ´