Hilbert polynomials of multigraded filtrations of ideals

Hilbert polynomials of multigraded filtrations of ideals

Journal of Algebra 444 (2015) 527–566 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Hilbert polynom...

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Journal of Algebra 444 (2015) 527–566

Contents lists available at ScienceDirect

Journal of Algebra www.elsevier.com/locate/jalgebra

Hilbert polynomials of multigraded filtrations of ideals Shreedevi K. Masuti a,1 , Parangama Sarkar b,2 , J.K. Verma b a

Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai, 600113, India b Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

a r t i c l e

i n f o

Article history: Received 5 December 2014 Available online 31 August 2015 Communicated by Kazuhiko Kurano Keywords: Hilbert polynomial Analytically unramified local ring Joint reductions Local cohomology of Rees algebra Joint reduction number

a b s t r a c t Hilbert functions and Hilbert polynomials of Zs -graded  ad missible filtrations of ideals {F (n)}n∈Zs such that λ FR (n) is finite for all n ∈ Zs are studied. Conditions are provided for the Hilbert function HF (n) := λ(R/F (n)) and the corresponding Hilbert polynomial PF (n) to be equal for all n ∈ Ns . A formula for the difference HF (n) − PF (n) in terms of local cohomology of the extended Rees algebra of F is proved which is used to obtain sufficient linear relations analogous to the ones given by Huneke and Ooishi among coefficients of PF (n) so that HF (n) = PF (n) for all n ∈ Ns . A theorem of Rees about joint reductions of the filtration {I r J s }r,s∈Z is generalised for admissible filtrations of ideals in two-dimensional Cohen–Macaulay local rings. Necessary and sufficient conditions are provided for the multi-Rees algebra of an admissible Z2 -graded filtration F to be Cohen–Macaulay. © 2015 Elsevier Inc. All rights reserved.

E-mail addresses: [email protected] (S.K. Masuti), [email protected] (P. Sarkar), [email protected] (J.K. Verma). 1 The first author is supported by Department of Atomic Energy, Government of India. 2 The second author is supported by CSIR Fellowship of Government of India. http://dx.doi.org/10.1016/j.jalgebra.2015.07.032 0021-8693/© 2015 Elsevier Inc. All rights reserved.

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1. Introduction The objective of this paper is to understand links among joint reductions of multigraded filtrations of ideals, the coefficients of their Hilbert polynomials, local cohomology modules of Rees algebra and depths of various associated multigraded rings of filtrations. In order to explain the principal results proved in this paper, we recall a few definitions, results and set up notation. Throughout this paper let (R, m) be a Noetherian local ring of dimension d with infinite residue field and I1 , . . . , Is be m-primary ideals of R. For s ≥ 1, we put e = (1, . . . , 1), 0 = (0, . . . , 0) ∈ Zs and for all i = 1, . . . , s, ei = (0, . . . , 1, . . . , 0) ∈ Zs where 1 occurs at ith position. For n = (n1 , . . . , ns ) ∈ Zs , we write I n = I1n1 · · · Isns and + n+ = (n+ 1 , . . . , ns ) where  n+ i

=

ni 0

if ni > 0 if ni ≤ 0.

For s ≥ 2 and α = (α1 , . . . , αs ) ∈ Ns , put |α| = α1 +· · ·+αs . Define m = (m1 , . . . , ms ) ≥ n = (n1 , . . . , ns ) if mi ≥ ni for all i = 1, . . . , s. By “for all large n” we mean n ∈ Ns and ni  0 for all i = 1, . . . , s. Definition 1.1. A set of ideals F = {F(n)}n∈Zs is called a Zs -graded I = (I1 , . . . , Is )-filtration if for all m, n ∈ Zs , (i) I n ⊆ F(n), (ii) F(n)F(m) ⊆ F(n + m) and (iii) if m ≥ n, F(m) ⊆ F(n). Two kinds of Rees rings encode information about filtrations of ideals. To define these, let t1 , t2 , . . . , ts be indeterminates and tn = t1 n1 · · · ts ns . The Ns -graded Rees ring of F is   R(F) = F(n)tn . The Zs -graded extended Rees ring of F is R (F) = F(n)tn . n∈Ns

For F = {I n }n∈Zs , we set R(F) = R(I) and R (F) = R (I).

n∈Zs

Definition 1.2. A Zs -graded I = (I1 , . . . , Is )-filtration F = {F(n)}n∈Zs of ideals in R is called an I = (I1 , . . . , Is )-admissible filtration if F(n) = F(n+ ) for all n ∈ Zs and R (F) is finite R (I)-module. For an R-module M of finite length, we write λR (M ) for length of M as an R-module. If the context is clear we simply write λ(M ). Let I be an m-primary ideal of R. In [31],   P. Samuel showed that for n  0, the Hilbert function HI (n) = λ IRn coincides with a polynomial  PI (n) = e0 (I)

 n+d−1 n+d−2 − e1 (I) + · · · + (−1)d ed (I) d d−1

of degree d, called the Hilbert polynomial of I. The coefficients ei (I) for i = 0, 1, . . . , d are integers, called the Hilbert coefficients of I. The multiplicity of I, namely e0 (I) is

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denoted by e(I). For m-primary ideals  I1 , . . . , Is , B. Teissier [33] proved that for all large n the Hilbert function HI (n) = λ IRn coincides with a polynomial PI (n) =



 ns + αs − 1 n1 + α1 − 1 ··· α1 αs

 (−1)d−|α| eα (I)

α=(α1 ,...,αs )∈Ns |α|≤d

of degree d, called the Hilbert polynomial of I = (I1 , . . . , Is ). Here we assume that s ≥ 2 in order to write PI (n) in the above form. This was proved by P.B. Bhattacharya for s = 2 in [1]. Here eα (I) are integers called the Hilbert coefficients of I. D. Rees [29] showed that eα (I) > 0 for |α| = d. These are called the mixed multiplicities of I. Rees [29] also showed that edei (I) = e0 (Ii ) for all i = 1, . . . , s. We now describe the main results proved in this paper. In Section 2, we prove preliminary results about superficial elements, the Hilbert polynomial of a filtration and some results about local cohomology modules of the Rees algebra of F. For an I-admissible filtration F = {F(n)}n∈Zs of ideals in a local ring (R, m) of dimension d, Rees showed existence of a polynomial  

ns + αs − 1 n1 + α1 − 1 d−|α| ··· PF (n) = (−1) eα (F) α1 αs s α=(α1 ,...,αs )∈N |α|≤d

  R of degree d which coincides with the Hilbert function HF (n) = λ F(n) for all large n [29]. The polynomial PF (n) plays an important role in the theory of singularities especially in the study of Milnor numbers [33]. Recall that x ∈ R is called integral over an ideal I of R if it satisfies the equation xn + a1 xn−1 + a2 xn−2 + · · · + an = 0 where ai ∈ I i for all i = 1, 2, . . . , n. The integral closure of I, denoted by I, is the set of all x ∈ R which are integral over I. For m-primary ideals I and J, using knowledge of the coefficients of the Hilbert polynomial of the bigraded (I, J)-admissible filtration {I r J s }r,s∈Z in two-dimensional Cohen–Macaulay analytically unramified local rings, Rees studied two-dimensional pseudo-rational local rings. One of the main purposes of this paper is to treat the filtrations {I n }n∈Zs and {I n }n∈Zs in a unified manner using the notion of admissibility. For an ideal I in a Noetherian ring R, L.J. Ratliff and D. Rush [26] introduced the ideal (I k+1 : I k ), I˜ = k≥1

called the Ratliff–Rush closure of I. The ideal I˜ has some nice properties such as for ˜ n = I n , I˜n = I n etc. If I is an m-primary ideal in a Noetherian local all large n, (I)

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ring (R, m) then I˜ is the largest ideal with respect to inclusion having the same Hilbert polynomial as that of I. C. Blancafort [3] introduced Ratliff–Rush closure filtration of an N-graded good filtration. In [14], A.V. Jayanthan and J.K. Verma studied properties of Ratliff–Rush closure of product of ideals. In Section 3, we introduce the Ratliff–Rush closure filtration of a Zs -graded filtration which unifies the above notions. Definition 1.3. The Ratliff–Rush closure filtration of F = {F(n)}n∈Zs is the filtration of ˘ ideals F˘ = {F(n)} n∈Zs given by ˘ (1) F(n) = k≥1 (F(n + ke) : F(e)k ) for all n ∈ Ns , ˘ ˘ + ) for all n ∈ Zs . (2) F(n) = F(n Let G(F) =



F(n) n∈Ns F(n+e)



be the associated multigraded ring of F with respect

F(n) to F(e) and Gi (F) = n∈Ns F(n+ei ) the associated multigraded ring of F with respect to F(ei ) for all i = 1, . . . , s. For an Ns (or a Zs )-graded ring T , the ideal generated by elements of degree at least e will be denoted by T++ . We discuss some properties of Ratliff–Rush closure filtration and compute the local cohomology modules 0 HG (Gi (F)) for all i = 1, . . . , s. i (F )++

Theorem 1.4. Let (R, m) be a Noetherian local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals in R such that grade(I1 · · · Is ) ≥ 1. Let F = {F(n)}n∈Zs be an I-admissible filtration. Then for all n ∈ Ns and i = 1, . . . , s, 0 [HG (Gi (F))]n = i (F )++

˘ + ei ) ∩ F(n) F(n . F(n + ei )

As a consequence we show that for the integral closure filtration F = {I n }n∈Zs in an analytically unramified local ring of dimension d ≥ 1, grade(Gi (F)++ ) ≥ 1 for all i = 1, . . . , s. In Section 4, we prove an analogue of Grothendieck–Serre formula (Theorem 4.3) for PF (n) − HF (n) in terms of local cohomology modules of extended Rees ring of F. This generalises results of Johnston–Verma [16], Blancafort [2], Jayanthan–Verma [14]. This will play a crucial role in our investigations of the Hilbert polynomial of Zs -graded admissible filtration of ideals. For a collection of m-primary ideals I1 , . . . , Is in a Noetherian local ring (R, m), Rees [29] introduced the concept of joint reduction of ideals. D. Kirby and Rees extended this concept for modules in [17]. Definition 1.5. Let (R, m) be a Noetherian local ring of dimension d, I1 , . . . , Is be m-primary ideals of R and F = {F(n)}n∈Zs be a Zs -graded I-admissible filtra-

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tion. A joint reduction of type q = (q1 , . . . , qs ) ∈ Ns is a collection of qi elements s

xi1 , . . . , xiqi ∈ Ii for all i = 1, . . . , s such that qj = d and j=1 qi s



xij F(n − ei ) = F(n) for all large n.

i=1 j=1

Definition 1.6. We say the joint reduction number of F with respect to a joint reduction {xij ∈ Ii : j = 1, . . . , qi ; i = 1, . . . , s} of type q is zero if qi s



xij F(n − ei ) = F(n) for all n ≥

i=1 j=1



ei , where A = {i|qi = 0}.

i∈A

Definition 1.7. We say the joint reduction number of F of type q is zero if joint reduction number of F with respect to any joint reduction of type q is zero. Definition 1.8. A joint reduction {xij ∈ Ii : j = 1, . . . , qi ; i = 1, . . . , s} of F of type q satisfies superficial conditions if for all j = 1, . . . , qi and i = 1, . . . , s, F(n) ∩ (xij ) = xij F(n − ei ) for all n ∈ Ns such that ni  0. In Section 5, we prove few results about Hilbert coefficients of PF (n). These results generalise results of D. Northcott, C. Huneke, A. Ooishi and T. Marley. We discuss vanishing of e0 (F) in two-dimensional Cohen–Macaulay local ring. Let F (i) denote the filtration {F(nei )}n∈Z for i = 1, 2, . . . , s and F Δ denote the filtration {F(ne)}n∈Z . If F is I-admissible then the filtrations F Δ and F (i) are I1 · · · Is -admissible and Ii -admissible respectively. For the Hilbert polynomial of a Z-graded I-admissible filtration G where I is an m-primary ideal in a d-dimensional local ring (R, m) we write, 

 n+d−1 n+d−2 PG (n) = e0 (G) − e1 (G) + · · · + (−1)d ed (G). d d−1 Theorem 1.9. Let (R, m) be a Cohen–Macaulay local ring of dimension one and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R and (xj ) a minimal reduction of Ij for j = 1, . . . , s. Then for fixed i ∈ {1, . . . , s},  

R e(Ii ) − e0 (F) = λ F(e if and only if xj F(n + ei ) = F(n + ei + ej ) for all n ≥ 0 and i) j = 1, . . . s. In this case, PF (n) = HF (n) for all n ≥ ei .

Theorem 1.10. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals  in

R R and for all i = 1, . . . , s, grade(Gi (F)++ ) ≥ 1. Suppose e(Ii ) − eei (F) = λ F(e for i) all i = 1, . . . , s. Then PF (n) = HF (n) for all n ≥ 0 and for i, j = 1, . . . , s, there exist

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joint reductions of F of type ei + ej with respect to which the joint reduction number of type ei + ej is zero. Theorem 1.11. Let (R, m) be a Cohen–Macaulay local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration. Then (1) eα (F) ≥ 0 where α = (α1 , . . . , αs ) ∈ Ns , |α| ≥ d − 1. (2) eα (F) ≥ 0 where α = (α1 , . . . , αs ) ∈ Ns , |α| = d − 2 and d ≥ 2. In Section 6, we study links between joint reduction numbers and Hilbert polynomials. For this purpose, we obtain the following criterion for Zs -graded admissible filtrations to have joint reduction number zero. For s = 2, we give necessary and sufficient conditions for admissible filtrations to have joint reduction number zero. Theorem 1.12. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Let i = j be fixed. (1) Suppose that for all n ≥ ei + ej  λ

R F(n)



 =λ

R F(n − ni ei )



 + ni nj eei +ej (I) + λ

R F(n − nj ej )

.

Then the joint reduction number of F of type ei + ej is zero. (2) For s = 2, suppose joint reduction number of F with respect to a joint reduction of type e is zero. Then  λ

R F(n)



 =λ

R F(n − n1 e1 )



 + n1 n2 ee (I) + λ

R F(n − n2 e2 )

for all n ≥ e.

We find a cohomological criterion for a Z2 -graded admissible filtration of ideals in a two-dimensional Cohen–Macaulay local ring to have joint reduction number zero. We obtain a theorem of Rees for the bigraded (I, J)-admissible filtration {I r J s }r,s∈Z where I, J are m-primary to have joint reduction number zero. Theorem 1.13. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I, J be m-primary ideals in R. Let F = {F(n)}n∈Z2 be an (I, J)-admissible filtration of ideals in R. Suppose e(1,0) (F) = e1 (F (1) ) and e(0,1) (F) = e1 (F (2) ). Then for any joint reduction (a, b) of F of type e satisfying superficial conditions, 2 λR [H(at (R (F))](0,0) = e2 (F (1) ) + e2 (F (2) ) − e2 (F Δ ). 1 ,bt2 )

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Using Theorems 1.12 and 1.13, we prove a generalisation of a theorem by Rees [28, Theorem 2.5]. Theorem 1.14. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I, J be m-primary ideals of R. Let F = {F(n)}n∈Z2 be an (I, J)-admissible filtration of ideals in R. Suppose e(1,0) (F) = e1 (F (1) ), e(0,1) = e1 (F (2) ) and depth G(F (1) ), depth G(F (2) ) ≥ 1. Then the following statements are equivalent: 2 (1) For every joint reduction (a, b) of F of type e, [H(at (R (F))](0,0) = 0, 1 ,bt2 )  (1 ) there exists a joint reduction (a, b) of F of type e satisfying superficial conditions 2 such that [H(at (R (F))](0,0) = 0, 1 ,bt2 ) Δ (1) (2) e2 (F ) = e2 (F ) + e2 (F (2) ), (3) the joint reduction number of F of type e is zero.

In Section 7, we study the Cohen–Macaulay property of the Rees algebra R(F) of a Z2 -graded (I, J)-admissible filtration F = {F(n)}n∈Z2 where I, J are m-primary ideals in a two-dimensional Cohen–Macaulay local ring. In [8], the authors proved that if R is a Cohen–Macaulay local ring and joint reduction number of the filtration {I r J s }r,s∈Z of type e is zero then R(I, J) is Cohen–Macaulay if and only if R(I) and R(J) are Cohen–Macaulay. We obtain a generalisation of this theorem for admissible bigraded filtrations of ideals. Finally we prove our main result in dimension two which characterises the Cohen– Macaulayness of bigraded Rees algebra R(F) in terms of Hilbert coefficients, reduction numbers and joint reduction numbers. For an m-primary ideal I, a reduction of an I-admissible filtration I = {In } is an ideal J ⊆ I1 such that JIn = In+1 for all large n. A minimal reduction of I is a reduction of I minimal with respect to inclusion. For a minimal reduction J of I, we set rJ (I) = min{m : JIn = In+1 for n ≥ m} and r(I) = min{rJ (I) : J is a minimal reduction of I}. The next result gives necessary and sufficient conditions for R(F) to be Cohen– Macaulay. For I, J m-primary ideals, it was proved for the filtration {I r J s }r,s∈Z in [14] and the equivalence of (2) and (4) was done for the filtration {I r J s }r,s∈Z in [19]. The following theorem unifies all these results. Theorem 1.15. Let (R, m) be a Cohen–Macaulay local ring of dimension two, I, J be m-primary ideals of R and F = {F(n)}n∈Z2 be an (I, J)-admissible filtration of ideals in R. Then the following statements are equivalent. (1) (2) (3) (4)

e(I) − e(1,0) (F) = λ(R/F(1, 0)) and e(J) − e(0,1) (F) = λ(R/F(0, 1)), r(F (1) ), r(F (2) ) ≤ 1 and the joint reduction number of F of type e is zero, PF (r, s) = HF (r, s) for all r, s ≥ 0, R(F) is Cohen–Macaulay.

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2. Preliminary results In this section, we discuss existence of superficial elements and Hilbert polynomial for admissible multigraded filtrations. We also prove certain properties of local cohomology modules of extended Rees algebras of such filtrations needed later. Definition 2.1. An element xi ∈ Ii is said to satisfy superficial condition with respect to F = {F(n)}n∈Zs if there exists an integer ri such that for all n ≥ ri ei , (xi ) ∩ F(n) = xi F(n − ei ). Rees [29, Lemma 1.2] proved existence of these elements for the filtration {I n }n∈Zs . The same proof works for any I-admissible filtration of ideals F = {F(n)}n∈Zs . Lemma 2.2 (Rees’ Lemma). Let (R, m) be a Noetherian local ring of dimension d and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R and S be a finite set of prime ideals of R not containing I1 · · · Is . Then for each i = 1, . . . , s, there exists an element xi ∈ Ii not contained in any of the prime ideals of S and an integer ri such that for all n ≥ ri ei , F(n) ∩ (xi ) = xi F(n − ei ). The following theorem proves the existence of elements y1 , . . . , yd ∈ I1 · · · Is such that (y1 , . . . , yd )F(n) = F(n + e) for all large n. Rees [29] proved it for {I n }n∈Zs . The same proof works for any I-admissible filtration of ideals F = {F(n)}n∈Zs . Theorem 2.3. Let (R, m) be a Noetherian local ring of dimension d and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Then there exist elements {xij ∈ Ii : j = 1, . . . , d; i = 1, . . . , s} such that for each i = 1, . . . , s, xi1 satisfies superficial condition with respect to F and (y1 , . . . , yd )F(n) = F(n+ e) for all large n where yj = x1j · · · xsj ∈ I1 · · · Is for all j = 1, . . . , d. Existence of joint reduction of F of type q = (q1 , . . . , qs ) ∈ Ns follows from Theorem 2.3. For all large n, we have F(n + e) = (y1 , . . . , yd )F(n) ⊆

qi s



xij F(n + e − ei ).

i=1 j=1

The following two results follow from admissibility of F. Proposition 2.4. (See [29].) Let (R, m) be a Noetherian local ring, I1 , . . . , Is be m-primary ideals of R and F = {F(n)}n∈Zs be an I-admissible filtration. Then for each i = 1, . . . , s,

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there exist integers ri , ri such that (1) for all n ∈ Zs , where ni ≥ ri , F(n + ei ) = Ii F(n), (2) for all n ∈ Zs , where ni ≤ ri , F(n − ei ) = F(n). Proposition 2.5. Let (R, m) be a Noetherian local ring of dimension d and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Then R(F) is a finitely generated R(I)-module. Proof. By Proposition 2.4, for each i = 1, . . . , s, there exists ri ∈ N such that F(n + ei ) = Ii F(n) for all n ∈ Ns , where ni ≥ ri . For all 0 ≤ n ≤ r = (r1 , . . . , rs ), let F(n) = (x(n)1 , . . . , x(n)l )R. We prove that S = {x(n)i tn : 1 ≤ i ≤ l and 0 ≤ n ≤ r} is a generating set of R(F) as R(I)-module. Let n ∈ Ns . If n ≤ r then for any a ∈ F(n), l

atn = bj x(n)j tn for some bj ∈ R for all j = 1, . . . , l. j=1

Suppose n  r. Let I = {k ∈ {1, . . . , s} : nk = rk + pk for some pk ≥ 1}. Then  F(n) =



 Ikpk

F(n −

k∈I



pk ek ).

k∈I

q l

Denote pk ek by m. Hence for any a ∈ F(n), a = ag ( bgj x(n − m)j ) for some g=1 j=1 k∈Ipk ag ∈ Ik for all g = 1, . . . , q and for some bgj ∈ R for all j = 1, . . . , l, g = 1, . . . , q. k∈I

Thus n

at =

 q l



j=1

 ag bgj t

m

x(n − m)j tn−m .

2

g=1

A polynomial P ∈ Q[X1 , . . . , Xs ] is called a numerical polynomial if P (n) ∈ Z for all n ∈ Zs . For a numerical polynomial P ∈ Q[X1 , . . . , Xs ] of total degree m, it is well known that for all n ∈ Zs , we can write P (n) =

γ=(γ1 ,...,γs )∈Ns |γ|≤m

 cγ

 n1 + γ1 − 1 ns + γs − 1 ··· , γ1 γs

where cγ ∈ Z. For j = 1, . . . , s, we define the operators Δj as Δ1j (f (n)) = f (n) − f (n − ej ) and for all k ≥ 2, (Δ1j (f (n))) for all n ∈ Zs . Δkj (f (n)) = Δk−1 j

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Lemma 2.6. Let P ∈ Q[X1 , . . . , Xs ] be a numerical polynomial of total degree m such that for all large n,  

n1 + γ1 − 1 ns + γs − 1 m−|γ| ··· P (n) = (−1) cγ γ1 γs s γ∈N |γ|≤m



=

m−|ν|

(−1)

ν∈Ns |ν|≤m

  n1 + ν1 − 1 ns + νs − 1 uν ··· ν1 νs

where γ = (γ1 , . . . , γs ) and ν = (ν1 , . . . , νs ). Then cα = uα for α = (α1 , . . . , αs ) ∈ Ns and |α| ≤ m. Proof. Fix α ∈ Ns with |α| ≤ m. Let γ = (γ1 , . . . , γs ) ∈ Ns and |γ| ≤ m. Note that if γi < αi for some i, then   n1 + γ1 − 1 ns + γs − 1 α1 αs Δs · · · Δ1 ··· = 0. γ1 γs Hence α1 s Δα s · · · Δ1 (P (n)) ⎛

=

s Δα s

1 · · · Δα 1

⎜ ⎜ ⎜ ⎜ ⎝

⎞ 



m−|γ|

(−1)

γ∈Ns |γ|≤m γi ≥αi for all i



 ⎟ n1 + γ1 − 1 ns + γs − 1 ⎟ ⎟ ··· ⎟ γ1 γs ⎠

= A + B, where  α1 s A = Δα s · · · Δ1

 (−1)m−|α| cα

⎛ ⎜ ⎜ α1 ⎜ s B = Δα · · · Δ s 1 ⎜ ⎝



 n1 + α1 − 1 ns + αs − 1 ··· = (−1)m−|α| cα and α1 αs ⎞  (−1)m−|γ| cγ

γ∈Ns |γ|≤m γi ≥αi for all i and γ=α

 ⎟ n1 + γ1 − 1 ns + γs − 1 ⎟ ⎟. ··· ⎟ γ1 γs ⎠

Let γi ≥ αi for all i = 1, . . . , s and γ = α. Then there exists j such that γj > αj . Then    n1 + γ1 − 1 ns + γs − 1 cγ ··· γ1 γs   n1 + γ1 − α1 − 1 ns + γs − αs − 1 = cγ ··· (γ1 − α1 ) (γs − αs ) s Δα s

1 · · · Δα 1

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is a nonconstant polynomial in n1 , . . . , ns . Hence we get the constant term of Q(n) = α1 m−|α| s cα . A similar argument shows that the constant term of Δα s · · · Δ1 (P (n)) is (−1) α1 αs Q(n) = Δs · · · Δ1 (P (n)) is (−1)m−|α| uα . Hence cα = uα . 2 Rees proved existence of the Hilbert polynomial of any I-admissible filtration F = {F(n)}n∈Zs [29, Theorem 2.4]. We give an elementary proof here. For F = {I n }n∈Zs , we set Gi (F) = Gi (I) for i = 1, . . . , s. Theorem 2.7. Let (R, m) be a Noetherian local ring of dimension d and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Then there exists a numerical polynomial with rational coefficients  

ns + αs − 1 n1 + α1 − 1 ··· PF (n) = (−1)d−|α| eα (F) α1 αs s α=(α1 ,...,αs )∈N |α|≤d

of total degree d satisfying PF (n) = HF (n) for all large n. The coefficients eα (F) ∈ Z and all monomials of highest degree in PF (n) have nonnegative coefficients. Proof. Since for each i = 1, . . . , s, Gi (F) is finitely generated Gi (I)-module, by [8, Theorem 4.1], for all i= 1, . . . ,s, there exist numerical polynomials Qi ∈ Q[X1 , . . . , Xs ]

F(n) such that Qi (n) = λ F(n+e for all large n and all monomials of highest degree in i) these polynomials have nonnegative coefficients. Let for all n = (n1 , . . . , ns ) ≥ m =   F(n) (m1 , . . . , ms ) and i = 1, . . . , s, Qi (n) = λ F(n+ei ) . Then for all n ≥ m + e,

 λ

R F(n)

=

n

1 −1

Q1 (k1 , n2 , . . . , ns ) +

k1 =m1

+ ··· +

n

2 −1

Q2 (m1 , k2 , n3 , . . . , ns )

k2 =m2 n

s −1

 Qs (m1 , m2 , . . . , ms−1 , ks ) + λ

ks =ms

R F(m)

.

Hence for all large n, HF (n) coincides with a numerical polynomial PF ∈ Q[X1 , . . . , Xs ] and all monomials of highest degree in this polynomial have nonnegative coefficients. Since F is an I = (I1 , . . . , Is )-admissible filtration, there existsan integer k∈ N such   n−ke R R that for all n ≥ ke, F(n) ⊆ I . Hence for all n ≥ ke, we get λ I n−ke ≤ λ F(n) ≤   λ IRn . Thus for all large n, we have PI (n − ke) ≤ PF (n) ≤ PI (n). Since PI (n) is a polynomial of total degree d, we get the required result. 2 The next result follows from [29, Theorem 2.4] Proposition 2.8. Let (R, m) be a Noetherian local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Then eα (F) = eα (I) for all α ∈ Ns such that |α| = d.

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Proposition 2.9. Let (R, m) be a Noetherian local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals of R such that grade(I1 · · · Is ) ≥ 1. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Then for any α ∈ Ns such that |α| = d, the set of joint reductions of F of type α is same as the set of joint reductions of {I n }n∈Zs of type α. In particular, eα (F) = eα (I) = e(x11 , . . . , x1α1 , . . . , xs1 , . . . , xsαs ) for any joint reduction {xij ∈ Ii : j = 1, . . . , αi ; i = 1, . . . , s} of F of type α = (α1 , . . . , αs ) ∈ Ns where |α| = d. Proof. Fix α = (α1 , . . . , αs ) ∈ Ns such that |α| = d. Let S = {k : αk ≥ 1}. By Proposition 2.4, for each i ∈ {1, . . . , s}, there exists ri ∈ N such that for all n ∈ Ns where ni ≥ ri , F(n + ei ) = Ii F(n). Let {xij ∈ Ii : j = 1, . . . , αi ; i = 1, . . . , s} be a joint reduction of F of type α. Then for all large n, we have αi s



xij F(n − ei ) = F(n).

(∗)

i=1 j=1

Let ti ∈ N for all i = 1, . . . , s such that (∗) holds for all n ≥ (t1 . . . , ts ). Therefore for all n ∈ Ns such that ni ≥ ti + ri + 1 for all i = 1, . . . , s, we get (



Ik )F(n −

k∈S



ek ) = F(n)

k∈S αi s



=

xij F(n − ei )

i=1 j=1

⎛ ⎞ αi s





⎜ ⎟ =⎝ xij ( Ik )⎠ F(n − ek ). i=1 j=1

Hence by [29, Lemma 1.5],

αi s

i=1 j=1

xij (



k∈S k=i

Ik ) is reduction of

k∈S k=i

k∈S



Ik . Thus {xij ∈ Ii : j =

k∈S

1, . . . , αi ; i = 1, . . . , s} is a joint reduction of {I n }n∈Zs of type α. Let {xij ∈ Ii : j = 1, . . . , αi ; i = 1, . . . , s} be a joint reduction of {I n }n∈Zs of type α. Then for all large n, we have αi s

i=1 j=1

xij I n−ei = I n .

(∗∗)

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Let ti ∈ N for all i = 1, . . . , s such that (∗∗) holds for all n ≥ (t1 . . . , ts ). Therefore for all n ∈ Ns such that ni ≥ ti + ri + 1 for all i = 1, . . . , s, we get F(n) = I t F(n − t) ⎛ ⎞ αi s

=⎝ xij I t−ei ⎠ F(n − t) i=1 j=1



αi s



xij F(n − ei ) ⊆ F(n).

i=1 j=1

The last part follows from [29], [32, Theorem 17.4.9]. 2 Lemma 2.10. (See [22, Lemma 3.3].) Let (R, m) be a Noetherian local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals in R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Let grade(Gi (F)++ ) ≥ 1 for all i = 1, . . . , s. For fixed j, suppose xj ∈ Ij \ mIj is a nonzerodivisor satisfying superficial condition. Then F(n) ∩ (xj ) = xj F(n − ej ) for all n ≥ ej . Lemma 2.11. Let I1 , . . . , Is be m-primary ideals in a Noetherian local ring (R, m) of dimension d ≥ 1 such that grade(I1 · · · Is ) ≥ 1. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Denote R(I)++ as R++ . Then d d λR [HR (R (F))]n ≤ λR [HR (R (F))]n−ei ++ ++

for all n ∈ Zs and i = 1, . . . , s. Proof. By Lemma 2.2 and Theorem 2.3, there exists an ideal J = (y1 , . . . , yd ) ⊆ I1 · · · Is such that y1 = x11 · · · xs1 is a nonzerodivisor, xi1 ∈ Ii for all i = 1, . . . , s and JF(n) =   F(n + e) for all large n. Hence R(I)++ = (y1 t, . . . , yd t). Consider the short exact sequence of R(I)-modules, i1 i 0 −→ R (F)(−ei ) −→ R (F) −→

x t

R (F) −→ 0. xi1 ti R (F)

This gives a long exact sequence of n-graded components of local cohomology modules,   d  d  d · · · −→ [HR(I) (R (F ))] −→ [H (R (F ))] −→ H n−e n R(I)++ R(I)++ i ++

R (F ) xi1 ti R (F )

 −→ 0. n

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   R (F ) R(I) Let T = xi1R(I) . Now is a T -module and = (y2 t, · · · , yd t)T . ti R(I) xi1 ti R (F ) xi1 ti R(I) ++   R (F ) d Hence HR(I)++ xi1 ti R (F ) = 0 which implies the required result. 2 We recall following theorem about change of grading principle of local cohomology modules. Theorem 2.12. (See [11].) Let T be Zs -graded ring defined over a local ring and U be a homogeneous ideal of T . Given a homomorphism φ : Zs → Zq , set Mφ =

 m∈Zq

⎛ ⎝



⎞ Mn ⎠

φ(n)=m

for any Zs -graded T -module M . Then for all i ≥ 0 (HUi (M ))φ = HUi φ (M φ ). Let T be a standard N2 -graded ring defined over a local ring. We define the a-invariants [11] of an N2 -graded T -module M as: dim M a1 (M ) = sup{k ∈ Z | [HM (M )](k,q) = 0 for some q ∈ Z} dim M (M )](p,k) = 0 for some p ∈ Z} a2 (M ) = sup{k ∈ Z | [HM

where M is the maximal homogeneous ideal of T . We need the following lemmas due to Hyry to prove our next result. For an N-graded  ring S, set S+ = n>0 Sn . Lemma 2.13. (See [11, Lemma 2.3].) Let S be a graded ring defined over a local ring (R, m). Let M be the homogeneous maximal ideal of S. Let a ⊂ m be an ideal. Let M be i a finitely generated graded S-module and n0 ∈ Z. Then [HM (M )]n = 0 for all n ≥ n0 i and i ≥ 0 if and only if [H(a,S+ ) (M )]n = 0 for all n ≥ n0 and i ≥ 0. Lemma 2.14. (See [11, Lemma 2.4]. Let T be a bigraded ring defined over a local ring (R, m). Let M be the homogeneous maximal ideal of T . Set M+ = (m, T++ ) ⊂ T . Let i M be a finitely generated bigraded T -module. Then [HM + (M )](p,q) = 0 for all p, q ≥ 0 i and i ≥ 0 if and only if [HT++ (M )](p,q) = 0 for all p, q ≥ 0 and i ≥ 0. Theorem 2.15. Let T be a standard bigraded ring defined over a local ring and let  2 M = r,s∈N M(r,s) be a finitely generated N -graded Cohen–Macaulay T -module. Let 1 2 a (M ), a (M ) < 0. Then [HTi ++ (M )](p,q) = 0 for all p, q ≥ 0 and i ≥ 0.

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 Proof. We follow the argument given in [11, Theorem 2.5]. Set M+ = (m p>0 T(p,0) )T , 1   + M+ T++ )T . By Lemma 2.14, it suffices to prove that 2 = (m q>0 T(0,q) )T , M = (m i [HM + (M )](p,q) = 0 for all p, q ≥ 0 and i ≥ 0. + + + + We have M+ 1 + M2 = M and M1 ∩ M2 = M . Consider the Mayer–Vietoris sequence of local cohomology modules i i · · · → HM (M ) → HM + (M ) 1



i i HM + (M ) → HM+ (M ) → · · · .

(2.15.1)

2

 Consider φ : Z2 → Z defined as φ(p, q) = q. Let T1 = p≥0 T(p,0) and n the maximal homogeneous ideal of T1 . Let N = M φ ⊗T1 (T1 )n and S = T φ ⊗T1 (T1 )n . Then S is an N-graded algebra defined over a local ring (T1 )n and N is an N-graded S-module. i φ i Note that [HN (M φ ⊗T1 (T1 )n )]q = 0 for any φ (M )]q = 0 if and only if [HN φ ⊗ T (T1 )n 1

i homogeneous ideal N in T . Since a2 (M ) < 0 and M is Cohen–Macaulay, [HM (M )](p,q) = i φ 0 for all q ≥ 0 and i ≥ 0. Hence by Theorem 2.12, [HMφ (M )]q = 0 for all q ≥ 0 and  i ≥ 0. Let P = nS0 q>0 Sq be the maximal homogeneous ideal of S. Then [HPi (N )]q = 0  i (N )]q = 0 for all q ≥ 0 and i ≥ 0. Let Q = mS0 q>0 Sq . Then by Lemma 2.13, [HQ i φ for all q ≥ 0 and i ≥ 0. Hence [H + φ (M )]q = 0 for all q ≥ 0 and i ≥ 0. Therefore by M2

i Theorem 2.12, [HM + (M )](p,q) = 0 for all q ≥ 0 and i ≥ 0. 2

Considering φ : Z2 → Z defined as φ(p, q) = p, a similar argument shows that i [HM+ (M )](p,q) = 0 for all p ≥ 0 and i ≥ 0. Hence, using the exact sequence (2.15.1), we 1

i get [HM 2 + (M )](p,q) = 0 for all p, q ≥ 0 and i ≥ 0.

3. Ratliff–Rush closure filtration and computation of local cohomology modules In this section we define Ratliff–Rush closure filtration of a Zs -graded filtration and discuss its few basic properties. We derive formulas for local cohomology modules 0 1 HG (Gi (F)) for all i = 1, . . . , s and HR(F) (R(F)) in terms of Ratliff–Rush cloi (F )++ ++ 0 sure filtration of F. In [3], Blancafort computed local cohomology modules HG(I) (G(I)) + 1 and HR(I)+ (R(I)) in terms of Ratliff–Rush closure filtration of an I-admissible filtration I. In [14], Jayanthan and Verma computed the local cohomology modules of R(I, J) in terms of Ratliff–Rush closure of product of ideals. We unify the above results in the setting of Zs -graded admissible filtrations of ideals. Proposition 3.1. Let (R, m) be a Noetherian local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals in R such that grade(I1 · · · Is ) ≥ 1. Let F = {F(n)}n∈Zs be an I-admissible filtration. Then (1) (2) (3)

˘ F(n) ⊆ F(n) for all n ∈ Zs and equality holds for all large n. F˘ is an I-filtration. ˘ F(n) ⊆ F(n) for all n ≥ e.

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˘˘ (4) F(n) = F˘ (n) for all n ∈ Zs . (5) Let yi ∈ I1 · · · Is for all i = 1, . . . , d such that (y1 , . . . , yd )F(n) = F(n + e) for all large n. Then for all n ∈ Ns , ˘ F(n) =



(F(n + ke) : (y1k , . . . , ydk )).

k≥1

(6) The Hilbert polynomial of the filtration F is same as the Hilbert polynomial of the ˘ filtration F. Proof. (1) For any n ∈ Ns and k ≥ 1, F(n)F(e)k ⊆ F(n)F(ke) ⊆ F(n + ke). Hence ˘ ˘ ˘ + ) and F(n) = F(n+ ) for all n ∈ Zs \ Ns , F(n) ⊆ F(n) for all n ∈ Ns . Since F(n) = F(n s ˘ we get F(n) ⊆ F(n) for all n ∈ Z . Since grade(I1 · · · Is ) ≥ 1, by Lemma 2.2, there exist a nonzerodivisor xi ∈ Ii and an integer ri ≥ 1 for all i = 1, . . . , s, such that F(n) ∩ (xi ) = xi F(n − ei ) for all n ≥ ri ei . Let y = x1 · · · xs . Then (F(n + e) : y) = F(n) for n ≥ r = (r1 , . . . , rs ). Let n ≥ r. Then F(n) ⊆ (F(n + e) : F(e)) ⊆ (F(n + e) : y) = F(n). Hence (F(n + e) : F(e)) = F(n). Therefore for all k ≥ 1, we have (F(n + ke) : F(e)k ) = ((F(n + ke) : F(e)) : F(e)k−1 ) = (F(n + (k − 1)e) : F(e)k−1 ) .. . = (F(n + e) : F(e)) = F(n). ˘ Hence F(n) = F(n) for all n ≥ r. ˘ ˘ (2) For n ∈ Zs , we have I n ⊆ F(n) ⊆ F(n). Let n, m ∈ Ns , m ≥ n and x ∈ F(m). k Then there exists an integer k ≥ 1 such that xF(e) ⊆ F(m + ke) ⊆ F(n + ke). Hence ˘ ˘ ˘ x ∈ F(n). Thus F(m) ⊆ F(n). Suppose n ∈ Zs \Ns , m ∈ Zs and m ≥ n. Then m+ ≥ n+ . Hence ˘ ˘ + ) ⊆ F(n ˘ + ) = F(n). ˘ F(m) = F(m ˘ ˘ Let n, m ∈ Ns , x ∈ F(n) and y ∈ F(m). Then xF(e)k ⊆ F(n + ke) and yF(e)k ⊆ ˘ F(m+ke) for some integer k ≥ 1. Thus xyF(e)2k ⊆ F(n+m+2ke). Hence xy ∈ F(n+m).

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˘ F(m) ˘ ˘ + m). For all n, m ∈ Zs , F(n) ˘ ˘ +) Therefore for all n, m ∈ Ns , F(n) ⊆ F(n = F(n ˘ + ). Since n+ ≥ n and m+ ≥ m, n+ + m+ ≥ n + m. Hence ˘ and F(m) = F(m ˘ F(m) ˘ ˘ + )F(m ˘ + ) ⊆ F(n ˘ + + m+ ) ⊆ F(n ˘ + m). F(n) = F(n ˘ F(m) ˘ ˘ + m) for all n, m ∈ Zs . Thus F(n) ⊆ F(n ˘ (3) Fix r = (r1 , . . . , rs ) ≥ e. It suffices to show that F(r) ⊆ F(r) is a reduction of ˘ F (r). Consider the following Z-graded filtrations G = {F(r)n }n∈Z ,

˘ n }n∈Z H = {F(r)

and

˘ I = {F(nr)} n∈Z

where nr = (nr1 , . . . , nrs ). Enough to show that R(H) is a finite R(G)-module. First we prove that {F(nr)}n∈Z is an F(r)-admissible filtration, i.e. there exists an integer k ≥ 1, n−k such that for all n ≥ k, F(nr) ⊆ F(r) . Since F is an I-admissible filtration, there exist homogeneous generators m1 , . . . , mg of R (F) as an R (I)-module where deg mt = (t) (t) (j) αt = (α1 , . . . , αs ) for all t = 1, . . . , g. Let k = max{|αi | : i = 1, . . . , s, j = 1, . . . , g}. Then for all n ≥ k, F(nr) = I (nr−α1 ) F(α1 ) + · · · + I (nr−αg ) F(αg ) ⊆ I (nr−ke) . n−k

Using induction on n, we prove that for all n ≥ k, I (nr−ke) ⊆ F(r) I (kr−ke) ⊆ R. Let n ≥ k and the result is true for n. Then I ((n+1)r−ke) = I1nr1 −k · · · Isnrs −k I1r1 · · · Isrs ⊆ F(r)

n−k

. For n = k,

(n+1)−k

F(r) ⊆ F(r)

.

n−k n−k ˘ Hence for all n ≥ k, F(nr) ⊆ F(r) . Therefore for n  0, F(nr) = F(nr) ⊆ F(r) . Hence R(I) is a finite R(G)-module. Since R(H) ⊆ R(I) is a R(G)-submodule, R(H) is ˘ a finite R(G)-module. Hence F(r) is a reduction of F(r). ˘˘ ˘˘ ˘ ˘ (4) Suffices to show that F(n) = F(n) for n ∈ Ns . By part (1), F(n) ⊆ F(n). Let ˘ k ˘ ˘ ˘ ˘ k be large enough such that F(n) = (F(n + ke) : F(e) ), F(n + ke) = F(n + ke) and F˘ (n) = (F(n + ke) : F(e)k ). Hence by part (1),

˘˘ ˘ + ke) : F(e) ˘ k ) = (F(n + ke) : F(e) ˘ k ) ⊆ (F(n + ke) : F(e)k ) = F(n). ˘ F(n) = (F(n ˘ (5) Since yik ∈ F(e)k for all i = 1, . . . , s and k ≥ 1, F(n) ⊆ k≥1 (F(n + ke) : (y1k , . . . , ydk )) for all n ∈ Ns . Let x ∈ k≥1 (F(n + ke) : (y1k , . . . , ydk )). Then xyik ∈ F(n + ke) for some k ≥ 1 and all i = 1, . . . , d. Consider a large integer l such that (y1 , . . . , yd )F(n) = F(n + e) for all n ≥ le. Then xF(e)l+dk ⊆ xF((l + dk)e) = x(y1 , . . . , yd )dk F(le)

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=

xy1i1 · · · ydid F(le)

i1 +···+id =dk

⊆ F(n + dke)F(le) ⊆ F(n + (l + dk)e). ˘ ˘ Hence x ∈ (F(n + (l + dk)e) : F(e)l+dk ) ⊆ F(n). Thus F(n) = k≥1 (F(n + ke) : (y1k , . . . , ydk )). ˘ (6) By part (1), for all large n, F(n) = F(n). Hence by Lemma 2.6 result follows. 2 Now we prove that in any analytically unramified local ring of dimension d ≥ 1 with infinite residue field, for the filtration F = {I n }n∈Zs , we have grade(Gi (F)++ ) ≥ 1 for all i = 1, . . . , s. Proposition 3.2. Let (R, m) be a Noetherian local ring of dimension d ≥ 1 and I1 , . . . , Is ˘ be m-primary ideals in R such that grade(I1 · · · Is ) ≥ 1. Let F = {I n }n∈Zs . Then F = F. ˘ ˘ Proof. It is enough to prove that F(n) ⊆ F(n) for n ∈ Ns . Let n ∈ Ns and x ∈ F(n). k

Then for some k  0, x ∈ (I n+ke : I e ). Let P be any minimal prime ideal of R, S = R/P and x be the image of x in S. Then by [32, Proposition 6.8.1], for any discrete valuation ring V lying between S and its field of fractions, we get k

k

x V I ke V = x V (I e V )k = x V (I e V )k = x V I e V = x I e V ⊆ I n+ke V = I n+ke V = I n V I ke V. Since V is discrete valuation ring, there exists some α ∈ I ke V such that I ke V = αV . Thus x α = αv for some v ∈ I n V . Thus x ∈ I n V . Since by [32, Proposition 6.8.4], I nS =

 (I n V ∩ S) V

where V varies over those discrete valuation rings of rank one between S and its field of fractions for which maximal ideal of V contracts to a maximal ideal of S, we get x ∈ I n S. Therefore by [32, Proposition 1.1.5], x ∈ I n . 2 Theorem 3.3. Let (R, m) be a Noetherian local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals in R such that grade(I1 · · · Is ) ≥ 1. Let F = {F(n)}n∈Zs be an I-admissible filtration. Then for all n ∈ Ns and i = 1, . . . , s, 0 [HG (Gi (F))]n = i (F )++

˘ + ei ) ∩ F(n) F(n . F(n + ei )

0 Proof. Let x ∈ F(n) and x(i) = x + F(n + ei ) ∈ [HG (Gi (F))]n . Then i (F)++ k (i) k ˘ x Gi (F) = 0 for some k ≥ 1. Therefore xF(e) ⊆ F(n+ke +ei ). Hence x ∈ F(n+e i ). ++

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˘ + ei ) ∩ F(n). Then xF(e)k ⊆ F(n + ke + ei ) for all Conversely, suppose x ∈ F(n large k. Since F is an I-admissible filtration, there exists a large integer t such that for all n ≥ te, F(n + e) = F(e)F(n). Let r = (r1 , . . . , rs ) ≥ e and r = min{ri : i = 1, . . . , s}. Without loss of generality assume r = r1 . Then for l  0, F(r)l ⊆ F(lr) = F(e)r1 l−t F(t, r2 l − r1 l + t, . . . , rs l − r1 l + t). Hence for all l  0, xF(r)l ⊆ xF(e)r1 l−t F(t, r2 l − r1 l + t, . . . , rs l − r1 l + t) ⊆ F(n + (r1 l − t)e + ei )F(t, r2 l − r1 l + t, . . . , rs l − r1 l + t) ⊆ F(n + lr + ei ). Since Gi (F)++ is finitely generated, (x + F(n + ei ))Gi (F)m ++ = 0 for some m ≥ 1. Hence 0 (x + F(n + ei )) ∈ [HG (G (F))] . 2 i n i (F )++ Corollary 3.4. Let (R, m) be an analytically unramified local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals in R. Consider the filtration F = {I n }n∈Zs . Then grade(Gi (F)++ ) ≥ 1 for all i = 1, . . . , s. Proof. Since R is an analytically unramified local ring, by [27], F is an I-admissible 0 filtration. Then by Proposition 3.2 and Theorem 3.3, [HG (Gi (F))]n = 0 for all i (F)++ n ≥ 0. Hence grade(Gi (F)++ ) ≥ 1 for all i = 1, . . . , s. 2 Proposition 3.5. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I1 , . . . , Is be m-primary ideals in R. Let F = {F(n)}n∈Zs be an I-admissible filtration. Then for all n ∈ Ns , ˘ F(n) 1 . [HR(F) (R(F))]n ∼ = ++ F(n) Proof. By Lemma 2.2 and Theorem 2.3, there exists a regular sequence {y1 , y2 } such that (y1 , y2 )F(n) = F(n + e) for all large n. For all k ≥ 1, consider the following complex of R(F)-modules α

βk

k F k. : 0 −→ R(F) −→ R(F)(ke)2 −→ R(F)(2ke) −→ 0,

where αk (1) = (y1 k tke , y2 k tke ) and βk (u, v) = y2 k tke u − y1 k tke v. Since radical of the ideal (y1 t, y2 t)R(F) is same as radical of the ideal R(F)++ , by [4, Theorem 5.2.9], ker (βk )n 1 [HR(F) (R(F))]n ∼ . = lim ++ −→ (im αk )n k

(3.5.1)

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Suppose (u, v) ∈ (ker βk )n for any n ∈ Ns . Then y2 k u − y1 k v = 0. Since {y1 , y2 } is a regular sequence, u = y1 k p for some p ∈ R. Thus v = y2 k p. Hence (u, v) = (y1 k p, y2 k p). This implies for all n ∈ Ns , (u, v) ∈ (ker βk )n if and only if (u, v) = (y1 k p, y2 k p) for some p ∈ (F(n+ke) : (y1 k , y2 k )). For k  0, by Proposition 3.1, F˘ (n) = (F(n+ke) : (y1 k , y2 k )) ˘ for all n ∈ Ns . Hence for all n ∈ Ns and k  0, (ker βk )n ∼ Also for all n ∈ Ns , = F(n). (im αk )n = {(y1 k ptke , y2 k ptke ) : p ∈ R(F)n } ∼ = F(n). 1 ∼ Hence [HR(F )++ (R(F))]n =

˘ (n) F F(n)

for all n ∈ Ns .

2

4. The difference formula In this section we express the difference of the Hilbert polynomial and the Hilbert function of an I-admissible filtrations of ideals in terms of local cohomology modules of extended Rees algebra. This generalises results of Johnston–Verma [16], Blancafort [2], Jayanthan–Verma [14]. To prove the result we follow the lines of the proof [21, Theorem 1] which is done for bigraded integral closure filtration. Proposition 4.1. Let T be a standard Noetherian Ns -graded ring, T0 be local ring and M a finitely generated Zs -graded T -module. Then (1) For all i ≥ 0 and n ∈ Zs , [HTi ++ (M )]n is finitely generated T0 -module. (2) For all large n and i ≥ 0, [HTi ++ (M )]n = 0. (3) Suppose T0 is Artinian. Let HM (n) = λ(Mn ) and PM (X1 , . . . , Xs ) ∈ Q[X1 , . . . , Xs ] be a polynomial such that for all large n, HM (n) = PM (n). Then for all n ∈ Zs , HM (n) − PM (n) =

(−1)j λT0 [HTj ++ (M )]n . j≥0

Proof. The proof follows from [6, Proposition 2.4.2, Proposition 2.4.3]. Proposition 4.2. Let S  be a Zs -graded ring and S = HSi ++ (S) for all i > 1 and the sequence 0 −→ HS0++ (S) −→ HS0++ (S  ) −→

 n∈Ns

2

Sn . Then HSi ++ (S  ) ∼ =

S −→ HS1++ (S) −→ HS1++ (S  ) −→ 0 S

is exact. Proof. Consider the short exact sequence of S-modules 0 −→ S −→ S  −→

S −→ 0. S

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This gives the long exact sequence of S-modules · · · −→ HSi ++ (S) −→ HSi ++ (S  ) −→ HSi ++ 

Since SS is S++ -torsion, HS0++ result follows. 2



S S

 =

S S

and HSi ++





S S

S S

−→ · · · .

 = 0 for all i > 0. Hence the

We set R(I)++ = R++ . Theorem 4.3. Let (R, m) be a Noetherian local ring of dimension d and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Then i (1) λR [HR (R (F))]n < ∞ for all i ≥ 0 and n ∈ Zs . ++

i (2) PF (n) − HF (n) = (−1)i λR [HR (R (F))]n for all n ∈ Zs . ++ i≥0

Proof. (1) Denote

R (F ) R (F )(ei )

j by Gi (F). By the change of ring principle, HG (Gi (F)) ∼ = i (I) ++

j (Gi (F)) for all i = 1, . . . , s and j ≥ 0. For a fixed i, consider the short exact seHR ++ quence of R(I)-modules

0 −→ R (F)(ei ) −→ R (F) −→ Gi (F) −→ 0.

(4.3.1)

This induces the long exact sequence of R-modules 0 0 0 1 0 −→ [HR (R (F))]n+ei −→ [HR (R (F))]n −→ [HR (Gi (F))]n −→ [HR (R (F))]n+ei −→ · · · . ++ ++ ++ ++

j By Propositions 4.1 and 4.2, [HR (R (F))]n = 0 for all large n and j ≥ 0. Since ++    Gi (F ) j = 0 for all n ∈ Ns or ni < 0, by Propositions 4.1 and 4.2, [HR (Gi (F))]n is Gi (F ) ++ n

finitely generated (Gi (I))0 -module for all n ∈ Ns or ni < 0 and j ≥ 0. Since (Gi (I))0 is j Artinian, [HR (Gi (F))]n has finite length for all n ∈ Ns or ni < 0 and j ≥ 0. Hence ++

j using decreasing induction on n, we get that λR [HR (R (F))]n < ∞ for all j ≥ 0 and ++ s n∈Z .

i (2) Let χM (n) = (−1)i λR [HR (M )]n where M is an R(I)-module. Then from ++ i≥0

the short exact sequence (4.3.1) and Propositions 4.1 and 4.2, for each i = 1, . . . , s and n ∈ Ns or ni < 0, χR (F ) (n + ei ) − χR (F ) (n) = −χGi (F ) (n) = −χGi (F ) (n) = PGi (F ) (n) − HGi (F) (n) = (PF (n + ei ) − PF (n)) − (HF (n + ei ) − HF (n)).

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Let h(n) = χR (F ) (n) − (PF (n) − HF (n)). Then h(n + ei ) = h(n) for all n ∈ Ns or ni < 0 and i = 1, . . . , s. Since h(n) = 0 for all large n, h(n) = 0 for all n ∈ Zs . 2 5. Analogues of Huneke–Ooishi Theorem Let (R, m) be a Cohen–Macaulay local ring of dimension d ≥ 1 and I be an m-primary ideal. Huneke [10] and Ooishi [25] independently proved that if e0 (I) − e1 (I) = λ(R/I) then ei (I) = 0 for i = 2, 3, . . . , d and HI (n) = PI (n) for all n ≥ 0. This was used effectively by Huneke [10] to obtain ideal theoretic conditions for the symbolic Rees algebra of dimension one prime ideals to be Noetherian. It also inspired other researchers to look for linear relations among Hilbert coefficients which give information about Hilbert functions. Our investigations in this section lead to several avatars of Huneke–Ooishi Theorem for Hilbert polynomials of multigraded filtrations. The following theorem gives linear conditions on Hilbert coefficients which ensure positive grade for the associated graded rings in dimension one Cohen–Macaulay local ring and then we generalise Huneke–Ooishi Theorem for multigraded filtrations in Cohen–Macaulay local ring of dimension one. Theorem 5.1. Let (R, m) be a Cohen–Macaulay local ring of dimension one and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an  I-admissible filtration of ideals in R and for fixed i ∈ {1, . . . , s}, e(Ii ) − e0 (F) = λ grade(G(F)++ ) ≥ 1.

R F(ei )

. Then grade(Gi (F)++ ) ≥ 1 and

1 Proof. By Theorem 4.3, λR (HR (R (F))ei ) = 0. Consider the short exact sequence of ++ R(I)-modules,

0 −→ R (F)(ei ) −→ R (F) −→ Gi (F) −→ 0. This induces a long exact sequence of local cohomology modules, 0 1 0 −→ [HR (Gi (F))]n −→ [HR (R (F))]n+ei −→ · · · . ++ ++

By Lemma 2.11, for any m ≥ ei , 1 1 λR [HR (R (F))]m ≤ λR [HR (R (F))]ei = 0. ++ ++ 0 Thus for all n ∈ Ns , [HR (Gi (F))]n = 0. Therefore by Proposition 4.2, ++ 0 [HR (Gi (F))]n = 0. Hence grade(Gi (F)++ ) ≥ 1. A similar argument shows that ++ grade(G(F)++ ) ≥ 1. 2

Theorem 5.2. Let (R, m) be a Cohen–Macaulay local ring of dimension one and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R and (xj ) a minimal reduction of Ij for j = 1, . . . , s. Then for fixed i ∈ {1, . . . , s},

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  R e(Ii ) − e0 (F) = λ F(e if and only if xj F(n + ei ) = F(n + ei + ej ) for all n ≥ 0 and i) j = 1, . . . s. In this case, PF (n) = HF (n) for all n ≥ ei .   R 1 Proof. Let e(Ii ) − e0 (F) = λ F(e (R (F))]ei = 0. . Then by Theorem 4.3, λR [HR ) ++ i 1 Hence by Lemma 2.11, λR [HR (R (F))]m = 0 for all m ≥ ei . Again by Theorem 4.3, ++ PF (n) = HF (n) for all n ≥ ei . Then for all n ≥ 0 and j = 1, . . . , s, we get

0 = [PF (n + ej + ei ) − HF (n + ej + ei )] − [PF (n + ei ) − HF (n + ei )]  s  

R = nk e(Ik ) + e(Ij ) + e(Ii ) − e0 (F) − λ F(n + ej + ei ) k=1  s  

R − nk e(Ik ) + e(Ii ) − e0 (F) − λ F(n + ei ) k=1   R R +λ = e(Ij ) − λ F(n + ej + ei ) F(n + ei )    R (xj ) R −λ +λ =λ (xj ) F(n + ej + ei ) xj F(n + ei )   R R −λ =λ xj F(n + ei ) F(n + ej + ei )  F(n + ej + ei ) . =λ xj F(n + ei ) Hence xj F(n + ei ) = F(n + ei + ej ) for all n ≥ 0 and j = 1, . . . s. Conversely, suppose xj F(n + ei ) = F(n + ei + ej ) for all n ≥ 0 and j = 1, . . . s. Since (xj ) is a minimal reduction of Ij , (xj ) is a joint reduction of F of type ej . Let f (n) = PF (n) − HF (n). Then for all n ≥ 0, we get   s

R R +λ [f (n + e + ei ) − f (n + ei )] = e(Ik ) − λ F(n + e + ei ) F(n + ei ) k=1    s

R R R −λ +λ λ = (xk ) F(n + e + ei ) F(n + ei ) k=1   R R −λ =λ (x1 · · · xs ) F(n + e + ei )  (x1 · · · xs ) +λ x1 · · · xs F(n + ei )   R R =λ −λ x1 · · · xs F(n + ei ) F(n + e + ei )  F(n + e + ei ) =λ x1 · · · xs F(n + ei ) = 0.

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Since f (m) = 0 for all large m, we get f (n + ei ) = 0 for all n ≥ 0. Thus taking n = 0 in f (n + ei ), we get the required result. 2 In following theorem we prove an analogue of Huneke–Ooishi Theorem for Cohen– Macaulay local ring of dimension two under some depth conditions. Theorem 5.3. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals  in

R R and for all i = 1, . . . , s, grade(Gi (F)++ ) ≥ 1. Suppose e(Ii ) − eei (F) = λ F(e for i) all i = 1, . . . , s. Then PF (n) = HF (n) for all n ≥ 0 and for i, j = 1, . . . , s, there exist joint reductions of F of type ei + ej with respect to which the joint reduction number of type ei + ej is zero.

Proof. Fix i. By Lemma 2.2, there exist a nonzerodivisor xi ∈ Ii \ mIi and an integer ri ≥ 1 such that F(n) ∩ (xi ) = xi F(n − ei ) for all n ≥ ri ei . Then by Lemma 2.10, F(n) ∩ (xi ) = xi F(n − ei ) for all n ≥ ei . For all n ≥ 0, consider the following exact sequence 0 −→

R .xi R (F(n + ei ) : (xi )) R −→ −→ −→ 0. −→ F(n) F(n) F(n + ei ) (xi , F(n + ei ))

Since for all n ≥ 0, (F(n + ei ) : (xi )) = F(n), we get  λ

R (xi , F(n + ei ))



 =λ

R F(n + ei )



 −λ

R F(n)

.

Let R = R/(xi ) and  denote the image of an ideal in R . Since HF  (n + ei ) = HF (n + e(Ii ) = e(Ii ) ei ) − HF (n), PF  (n + ei ) = PF (n + ei ) − PF (n). By Lemma 2.6, we  have  

and eei (F) = e0 (F  ). Since xi ∈ Ii , we get e(Ii ) − e0 (F  ) = λ F(eRi )R . Hence by Theorem 5.2, we have PF  (n + ei ) = HF  (n + ei ) for all n ≥ 0. This implies PF (n + ei ) − HF (n + ei ) = PF (n) − HF (n) for all n ≥ 0. Since this is true for all i = 1, . . . , s, we get PF (n) = HF (n) for all n ≥ 0. By Theorem 2.3 and Lemma 2.10, there exist elements {xik ∈ Ii : k = 1, 2; i = 1, . . . , s} such that for all n ≥ ei , xi1 ∩ F(n) = xi1 F(n − ei ) for all i = 1, . . . , s and (xi1 , xj2 ) is a joint reduction of F of type ei + ej . Let R = (xRi1 ) . Then xj2 R is    joint reduction of FR of type ej and e(Ii R ) − e0 (FR ) = λ F(eRi )R . Hence xj2 R is

a reduction of Ij R , by [29, Lemma 1.5]. Therefore by Theorem 5.2, xj2 F(n + ei )R = F(n + ei + ej )R for all n ≥ 0. Thus for all n ≥ 0,

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F(n + ei + ej ) ⊆ (F(n + ei + ej ) ∩ (xi1 )) + xj2 F(n + ei ) = xi1 F(n + ej ) + xj2 F(n + ei ).

2

Corollary 5.4. Let (R, m) be an analytically unramified Cohen–Macaulay local ring of n dimension two  and  I1 , . . . , Is be m-primary ideals of R. Let F = {I }n∈Zs . If e(Ii ) −

eei (F) = λ IR for all i = 1, . . . , s, then PF (n) = HF (n) for all n ≥ 0 and for i i, j = 1, . . . , s, there exist joint reductions of F of type ei + ej with respect to which the joint reduction number of type ei + ej is zero. Proof. The proof follows from Corollary 3.4 and Theorem 5.3.

2

One fundamental result about the Hilbert coefficients of an m-primary ideal I in a   Cohen–Macaulay local ring of dimension d is the Northcott’s inequality λ R I ≥ e0 (I) − e1 (I) with equality if and only if r(I) ≤ 1 [24,10]. S. Huckaba and Marley [9] generalised this result for Z-graded I-admissible filtrations. Jayanthan proved the above result for the filtration {I r J s }r,s∈Z for m-primary ideals I, J [15]. We generalise this result for I-admissible multigraded filtration of ideals. Theorem 5.5. Let (R, m) be a Cohen–Macaulay local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Then for all i = 1, . . . , s, (1) e(d−1)ei (F) ≥ e1 (F (i) ), (2) e(Ii ) − e(d−1)ei (F) ≤ λ(R/F(ei )), (3) e(Ii ) − e(d−1)ei (F) = λ(R/F(ei )) if and only if r(F (i) ) ≤ 1 and e(d−1)ei (F) = e1 (F (i) ). Proof. (1) We apply induction on d. Let d = 1. Then by Theorem 4.3, 1 PF (rei ) − λ(R/F(rei )) = −λR [HR (R (F))](rei ) for all r ≥ 0. ++

Since F (i) is Ii -admissible, we have e(F (i) ) = e(Ii ). Hence using PF (i) (r) = e(Ii )r − e1 (F (i) ), we get 1 PF (i) (r) − λ(R/F(rei )) + [e1 (F (i) ) − e0 (F)] = −λR [HR (R (F))](rei ) ≤ 0. ++

Taking r  0, we get e0 (F) ≥ e1 (F (i) ). Let d ≥ 2. By Lemma 2.2, there exists a nonzerodivisor xi ∈ Ii such that (xi ) ∩ F(n) = xi F(n − ei ) for n ∈ Ns where ni  0. Let R = R/(xi ), F  = {F(n)R } and F (i) = {F(nei )R }. For all n ∈ Ns such that ni  0, consider the following exact sequence

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0 −→

R R (F(n) : (xi )) R .xi −→ −→ −→ 0. −→ F(n − ei ) F(n − ei ) F(n) (xi , F(n))

Since for all n ∈ Ns where ni  0, (F(n) : (xi )) = F(n − ei ), we get HF  (n) = HF (n) − HF (n − ei ) and hence Δ1i (PF (n)) = PF  (n). By Lemma 2.6, e(d−2)ei (F  ) = e(d−1)ei (F) and e1 (F (i) ) = e1 (F (i) ). Therefore by induction, the result follows. (2) Using part (1), for all i = 1, . . . , s, we have e(Ii ) − e(d−1)ei (F) ≤ e(Ii ) − e1 (F (i) ) ≤ λ(R/F(ei )) where the last inequality follows from [9, Corollary 4.9]. (3) Let e(Ii ) − e(d−1)ei (F) = λ(R/F(ei )). Then by part (1), λ(R/F(ei )) = e(Ii ) − e(d−1)ei (F) ≤ e(Ii ) − e1 (F (i) ) ≤ λ(R/F(ei )), where the last inequality follows by [9, Corollary 4.9]. Hence e(d−1)ei (F) = e1 (F (i) ) and e(Ii ) − e1 (F (i) ) = λ(R/F(ei )). Therefore, by [9, Corollary 4.9], r(F (i) ) ≤ 1. Conversely, suppose r(F (i) ) ≤ 1 and e(d−1)ei (F) = e1 (F (i) ). Again, by [9, Corollary 4.9], e(Ii ) − e1 (F (i) ) = λ(R/F(ei )). Hence e(Ii ) − e(d−1)ei (F) = λ(R/F(ei )). 2 Northcott [24] proved that e1 (I) ≥ 0 for an m-primary ideal I in a Cohen–Macaulay local ring. M. Narita [23] showed that e2 (I) is also nonnegative. Marley [18] proved that ei (I) ≥ 0 for any Z-graded I-admissible filtration I and i = 1, 2. In [20], the authors proved that eα (F) ≥ 0 for |α| ≥ d − 2 where F = {I r J s }r,s∈Z and I, J are m-primary ideals of R. We show that eα (F) ≥ 0 for any I-admissible multigraded filtration of ideals where |α| ≥ d − 2. We characterise the vanishing of e0 (F) in dimension two and obtain a result of S. Itoh [13] as a consequence. Theorem 5.6. Let (R, m) be a Cohen–Macaulay local ring of dimension d ≥ 1 and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration. Then (1) eα (F) ≥ 0 where α = (α1 , . . . , αs ) ∈ Ns , |α| ≥ d − 1. (2) eα (F) ≥ 0 where α = (α1 , . . . , αs ) ∈ Ns , |α| = d − 2 and d ≥ 2. Proof. (1) For |α| = d, the result follows from Theorem 2.7. Suppose |α| = d − 1. We use induction on d. Let d = 1. Then putting n = 0 in the difference formula (Theorem 4.3), 1 we get e0 (F) = λR [HR (R (F))]0 ≥ 0. Let d ≥ 2 and the result is true for rings of ++ dimension d − 1. Then there exists i such that αi ≥ 1. Without loss of generality assume α1 ≥ 1. By Lemma 2.2, there exists a nonzerodivisor x ∈ I1 such that (x) ∩ F(n) = R and F  = {F(n)R }n∈Zs . For xF(n − e1 ) for all n ∈ Ns such that n1  0. Let R = (x) all large n, consider the following short exact sequence 0 −→

R R (F(n) : (x)) R .x −→ −→ −→ 0. −→ F(n − e1 ) F(n − e1 ) F(n) (x, F(n))

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Since for all large n, (F(n) : (x)) = F(n − e1 ), we get PF  (n) = PF (n) − PF (n − e1 ). Hence by Lemma 2.6, (−1)d−1−|(α−e1 )| b(α−e1 ) (F  ) = (−1)d−|α| eα (F) where PF  (n) =

 ns + γs − 1 n1 + γ1 − 1 ··· . γ1 γs





(−1)d−1−|γ| bγ (F  )

γ=(γ1 ,...,γs )∈Ns |γ|≤d−1

Since |(α − e1 )| = d − 2 = (d − 1) − 1, by induction b(α−e1 ) (F  ) ≥ 0. Hence for |α| = d − 1, eα (F) ≥ 0. (2) We prove the result using induction on d. For d = 2 the result follows from the Difference Formula (Theorem 4.3) for n = 0 and Proposition 3.5. The rest is same as for the case |α| = d − 1. 2 Theorem 5.7. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Then ˘ (1) e0 (F) = 0 if and only if r(G) ≤ 1 whereG = { F(ne)} n∈Z . R (2) If e0 (F) = 0 then e(Ii ) − eei (F) = λ F(e all i = 1, . . . , s. Suppose F˘ is for ˘ i) I-admissible filtration, then the converse is also true. Proof. (1) Consider the filtration F Δ = {F(ne)}n∈Z . Then PF (ne) = PF Δ (n) = PG (n) where the last equality follows from Proposition 3.1. This implies that e0 (F) = 0 if and only if e2 (G) = 0. By Proposition 3.1 and Theorem 3.3, for all n ≥ 0, ˘ + 1) ∩ G(n) G(n = 0. G(n + 1)  

G(n) Hence by [9, Proposition 4.6], we get e2 (G) = (n − 1)λ JG(n−1) where J is any 0 [HG(G) (G(G))]n = +

n≥2

minimal reduction of G. Thus we get the required result. 1 (2) Let e0 (F) = 0. By Proposition 3.5, [HR (R (F))]0 = 0. Hence by Theorem 4.3, ++ 2  2 λR [HR++ (R (F))]0 = e0 (F) = 0. By Lemma 2.11, λR [HR (R (F))]ei = 0 for all ++ ˘  F (ei ) i = 1, . . . , s. Then using Theorem 4.3 and Proposition 3.5, PF (ei ) −HF (ei ) = −λ F(e i)   R for all i = 1, . . . , s. Hence e(Ii ) − eei (F) = λ F(e for all i = 1, . . . , s. ˘ i)   R Suppose F˘ is I-admissible filtration and e(Ii ) − eei (F) = λ F(e ˘ i ) for all i = 1, . . . , s. Then by Proposition 3.1 (4) and Theorem 3.3, for all n ≥ 0 and i = 1, . . . , s, 0 [HG

˘˘ ˘ F(n + ei ) ∩ F(n) ˘ = 0. (G ( F))] = i n ˘ i (F )++ ˘ F(n + ei )

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Since the Hilbert polynomial of F˘ is same as the Hilbert polynomial of F, by Theorem 5.3, PF˘ (n) = HF˘ (n) for all n ≥ 0.

(5.7.1)

˘ = 0. 2 Thus taking n = 0 in the equation (5.7.1), we get e0 (F) = e0 (F) As a consequence of the above theorem we get a result by Itoh [13, Corollary 5]. Corollary 5.8. (See [13, Corollary 5].) Let (R, m) be a Cohen–Macaulay local ring of dimension two and I be m-primary ideal of R. Let Q be any minimal reduction of I. Then the following are equivalent.   (1) e1 (I) − e0 (I) + λ R = 0. I˘ 2 ˘ ˘ (2) I = QI. ! ˘ (2 ) I 2 = QI. ! (3) I n+1 = Qn I˘ for all n ≥ 1. (4) e2 (I) = 0. Proof. We prove (4) ⇒ (3) ⇒ (2 ) ⇒ (2) ⇒ (1) ⇒ (4). (4) ⇒ (3): Taking F = {I n }n∈Z in Theorem 5.7, the result follows. (3) ⇒ (2 ): Put n = 1 in (3). (2 ) ⇒ (2) Consider the filtration F = {I n }n∈Z . Then by [2, Proposition 3.2.3], for ! ! ˘ Then all n ≥ 0, I n = ∪k≥1 (I nk+n : I nk ). It suffices to show that I˘2 ⊆ I 2 . Let x, y ∈ I. k k+1 k k+1 2k 2k+2 for some large k, xI ⊆ I and yI ⊆ I . Hence xyI ⊆ I . This implies that ! I˘2 ⊆ I 2 . (2) ⇒ (1): Follows from [10, Theorem 2.1]. (1) ⇒ (4): Let F = {I n }n∈Z . Since F˘ is an I-admissible filtration, the result follows by Theorem 5.7. 2 6. Joint reduction of admissible multigraded filtrations In this section, we explore relationship between coefficients of Hilbert polynomials of bigraded admissible filtrations, joint reductions and local cohomology. These explorations are inspired by a theorem of Rees [28] about the bigraded filtration {I r J s }r,s∈Z , where I and J are m-primary ideals, to have joint reduction number zero of type e in terms of e2 (I), e2 (J) and e2 (IJ) in two-dimensional Cohen–Macaulay analytically unramified local rings. We obtain a generalisation of Rees’ Theorem for bigraded filtrations in Cohen–Macaulay local rings of dimension two.

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Lemma 6.1. Let (a, b) be a regular sequence in a local ring (R, m). Then the sequence 0 −→

R φ R (a, b) ψ R ⊕ −→ 0, −→ −→ (I : a) ∩ (J : b) I J aJ + bI

where ψ(¯ r) = (ar, br) and φ(¯ x, y¯) = ay − bx, is exact. Proof. It suffices to prove that im ψ = ker φ. Clearly, im ψ ⊆ ker φ. Let φ(¯ x, y¯) = 0. Then ay − bx = ap + bq for some p ∈ J and q ∈ I. Thus a(y − p) = b(q + x). Hence y − p = br and q + x = ar for some r ∈ R. Thus (¯ x, y¯) = (ar, br) = ψ(¯ r). 2 In [34], Verma gave an equivalent criterion for joint reduction number zero in terms of Hilbert coefficients for the filtration {I r J s }r,s∈Z in dimension 2. In the following theorem we give a sufficient condition for joint reduction number zero for Zs -graded filtrations. Theorem 6.2. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I1 , . . . , Is be m-primary ideals of R. Let F = {F(n)}n∈Zs be an I-admissible filtration of ideals in R. Let i = j be fixed. (1) Suppose  λ

R F(n)



 =λ  +λ

R F(n − ni ei )



R F(n − nj ej )

+ ni nj eei +ej (I) for all n ≥ ei + ej .

Then the joint reduction number of F of type ei + ej is zero. (2) For s = 2, suppose joint reduction number of F with respect to a joint reduction of type e is zero. Then  λ

R F(n)



 =λ

R F(n − n1 e1 )



 + n1 n2 ee (I) + λ

R F(n − n2 e2 )

for all n ≥ e.

Proof. (1) Let (xi , xj ) be a joint reduction of F of type ei + ej . First we show that joint reduction number of F with respect to (xi , xj ) is zero if and only if for all n ≥ ei + ej , F(n) = xi ni F(n − ni ei ) + xj nj F(n − nj ej ).

(6.2.1)

Suppose joint reduction number of F with respect to (xi , xj ) is zero. We use induction on ni + nj . We have F(n) = xi F(n − ei ) + xj F(n − ej ) for all n ≥ ei + ej . If ni + nj = 2 then ni = nj = 1 and hence the assertion is true in this case. Let ni + nj > 2. Then ni ≥ 2 or nj ≥ 2. Without loss of generality we may assume that ni ≥ 2. If nj = 1 then using induction

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F(n) = xi F(n − ei ) + xj F(n − ej ) = xi (xi ni −1 F(n − ni ei ) + xj F(n − ei − ej )) + xj F(n − ej ) ⊆ xi ni F(n − ni ei ) + xj F(n − ej ). Thus the assertion follows. Hence we may assume that nj ≥ 2. Then F(n) = xi F(n − ei ) + xj F(n − ej ) = xi (xi ni −1 F(n − ni ei ) + xj nj F(n − ei − nj ej )) + xj (xi ni F(n − ni ei − ej ) + xj nj −1 F(n − nj ej )) ⊆ xi ni F(n − ni ei ) + xj nj F(n − nj ej ). Conversely, suppose the equation (6.2.1) holds true. Hence for all n ≥ ei + ej , F(n) = xi ni F(n − ni ei ) + xj nj F(n − nj ej ) ⊆ xi F(n − ei ) + xj F(n − ej ). For any joint reduction (wi , wj ) of F of type ei + ej and for all n ≥ ei + ej , we have  λ 

F(n) n i wi F(n − ni ei ) + wj nj F(n − nj ej )



 R +λ n (F(n − nj ej ) : wini ) ∩ (F(n − ni ei ) : wj j )   R R −λ =λ wi ni F(n − ni ei ) + wj nj F(n − nj ej ) F(n)   R +λ n (F(n − nj ej ) : wini ) ∩ (F(n − ni ei ) : wj j )    (wi ni , wj nj ) R R +λ −λ =λ (wi ni , wj nj ) wi ni F(n − ni ei ) + wj nj F(n − nj ej ) F(n)   R +λ n ni (F(n − nj ej ) : wi ) ∩ (F(n − ni ei ) : wj j )    R R R =λ + λ + λ (wi ni , wj nj ) F(n − ni ei ) F(n − nj ej )  R , by Lemma 6.1 −λ F(n)   R R = ni nj eei +ej (I) + λ +λ F(n − ni ei ) F(n − nj ej )  R , by [29, Theorem 2.4]. −λ F(n)

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 Since λ have

R F (n)



 =λ

R F(n−ni ei )



 + ni nj eei +ej (I) + λ

R F(n−nj ej )



557

for all n ≥ ei + ej , we

F(n) = wi ni F(n − ni ei ) + wj nj F(n − nj ej ) for all n ≥ ei + ej for any joint reduction (wi , wj ) of F of type ei + ej . (2) For s = 2, (F(n − n2 e2 ) : w1n1 ) ∩ (F(n − n1 e1 ) : w2n2 ) = R for any joint reduction (w1 , w2 ) of F of type e. Hence for all n ≥ e,  λ

F(n) n 1 w1 F(n − n1 e1 ) + w2 n2 F(n − n2 e2 )



R = n1 n2 ee (I) + λ F(n − n1 e1 )   R R −λ . +λ F(n − n2 e2 ) F(n) 

Since joint reduction number of F with respect to a joint reduction of type e is zero, by equation (6.2.1), the result follows. 2 Remark 6.3. Note that if we take s = 2 in Theorem 5.3, under the assumptions that grade(Gi (F)++ ) ≥ 1, for all i = 1, 2, the linear relations of Hilbert coefficients e(Ii ) −   R eei (F) = λ F (e imply PF (n) = HF (n) for all n ≥ 0 and joint reduction number of i) F of type e is zero. The next theorem can be proved using the ideas in [19, Theorem 2.9] which is done for the filtration {I r J s }r,s∈Z . Theorem 6.4. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I, J be m-primary ideals in R and F = {F(n)}n∈Z2 be an (I, J)-admissible filtration. Then there exists a joint reduction (x1 , x2 ) of F of type e which satisfies superficial conditions. For a two-dimensional Noetherian local ring R and m-primary ideals I, J, we write 

 r+1 s+1 P(I,J) (r, s) = e(I) + e(I|J)rs + e(J) − e(1,0) (I, J)r 2 2 − e(0,1) (I, J)s + e2 (IJ). 2 We compute λR ([H(at (R (F))](0,0) ) where (a, b) is a joint reduction of F of type e 1 ,bt2 ) satisfying superficial conditions.

Theorem 6.5. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I, J be m-primary ideals of R. Let F = {F(n)}n∈Z2 be an (I, J)-admissible filtration of

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ideals in R. Suppose e(1,0) (F) = e1 (F (1) ) and e(0,1) (F) = e1 (F (2) ). Then for any joint reduction (a, b) of F of type e satisfying superficial conditions, 2 λR [H(at (R (F))](0,0) = e2 (F (1) ) + e2 (F (2) ) − e2 (F Δ ). 1 ,bt2 )

Proof. Let (a, b) be any joint reduction of F of type e. Consider the Koszul complex on ((at1 )k , (bt2 )k ): .

βk

α

k F k : 0 −→ R (F) −→ R (F)(ke1 ) ⊕ R (F)(ke2 ) −→ R (F)(ke) −→ 0,

where the maps are defined as, αk (1) = ((at1 )k , (bt2 )k )

and βk (u, v) = −(bt2 )k u + (at1 )k v.

We have the following commutative diagram of Koszul complexes, α

0 −−−−−→ R (F ) −−−−k−−−→

R (F )(ke1 ) ⊕ R (F )(ke2 )

⏐ ⏐ fk #

β

−−−−k−−−→

⏐ ⏐ gk #

0 −−−−−→ R (F )

αk+1

−−−−−→

R (F )(ke)

−−−−−→ 0

⏐ ⏐ hk #

R (F )((k + 1)e1 ) ⊕ R (F )((k + 1)e2 )

βk+1

−−−−−→

R (F )((k + 1)e) −−−−−→ 0

where fk (1) = 1, gk (1, 0) = (at1 , 0), gk (0, 1) = (0, bt2 ) and hk (1) = abt1 t2 . Then by [4, Theorem 5.2.9], 2 [H(at (R (F))](0,0) ∼ = lim 1 ,bt2 ) −→ k

F(k, k) F(k, k) . = lim k −→ (im βk )(0,0) a F(0, k) + bk F(k, 0) k

(6.5.1)

We claim that if (a, b) is a joint reduction of F of type e satisfying superficial conditions, then the map μk involved in the above direct limit μk :

F(k, k) F(k + 1, k + 1) ab −→ k+1 ak F(0, k) + bk F(k, 0) a F(0, k + 1) + bk+1 F(k + 1, 0)

is injective for k  0. Let  denote the image of an element in respective quotients. Suppose μk (x ) = 0 for some x ∈ F(k, k). Then xab = ak+1 p + bk+1 q for some p ∈ F(0, k + 1) and q ∈ F(k + 1, 0). Hence p ∈ (b) ∩ F(0, k + 1) = bF(0, k) for k  0. Let p = bp1 for some p1 ∈ F(0, k). Similarly, let q = aq1 for some q1 ∈ F(k, 0). Hence x = ak p1 + bk q1 ∈ ak F(0, k) + bk F(k, 0). Thus x = 0. Next we claim that for all k  0 and a joint reduction (a, b) of F of type e satisfying superficial conditions,  λ

F(k, k) ak F(0, k) + bk F(k, 0)

= e2 (F (1) ) + e2 (F (2) ) − e2 (F Δ ).

(6.5.2)

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Since e(1,0) (F) = e1 (F (1) ) and e(0,1) (F) = e1 (F (2) ), for k  0, we have  λ 

F(k, k) ak F(0, k) + bk F(k, 0)



 R R − λ ak F(0, k) + bk F(k, 0) F(k, k)    R (ak , bk ) R =λ + λ − λ (ak , bk ) ak F(0, k) + bk F(k, 0) F(k, k)     R R R R +λ +λ −λ = k2 λ (a, b) F(0, k) F(k, 0) F(k, k)   k+1 k+1 2 (2) (2) = k e(I|J) + e(J) − e1 (F )k + e2 (F ) + e(I) − e1 (F (1) )k 2 2   k+1 k+1 (1) 2 + e2 (F ) − [e(I) + e(I|J)k + e(J) 2 2 =λ

− (e1 (F (1) ) + e1 (F (2) ))k + e2 (F Δ )] = e2 (F (1) ) + e2 (F (2) ) − e2 (F Δ ).  Thus λ we get



F (k,k)

ak F (0,k)+bk F(k,0)

is independent of k for k  0. Since μk is injective for k  0,

2 [H(at (R (F))](0,0) ∼ = 1 ,bt2 )

F(k, k) for k  0. + bk F(k, 0)

ak F(0, k)

Hence 2 λR [H(at (R (F))](0,0) = e2 (F (1) ) + e2 (F (2) ) − e2 (F Δ ). 1 ,bt2 )

2

The conditions e(1,0) (F) = e1 (F (1) ) and e(0,1) (F) = e1 (F (2) ) are not true in general. Let R = k[|X, Y |] and m = (X, Y ). Let I = m2 and J = (X 2 , Y 2 ). Consider the filtration {I r J s }r,s∈Z . Then F (1) = {I r }r∈Z and F (2) = {J s }s∈Z . For n ≥ 1,  λ

R In



 =λ

R m2n



 =

2n + 1 2





n+1 =4 − n = PI (n). 2

Since J is parameter ideal and JI = I 2 ,  λ  λ

R (IJ)n



 =λ

R Jn

R m4n



 n+1 =4 = PJ (n), 2   4n + 1 n+1 = = 16 − 6n = PIJ (n) 2 2

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and  λ

R I 2n J n



 =λ

R m6n



 =

6n + 1 2

= 36

 n+1 − 15n = PI 2 J (n). 2

  R Hence e0 (I) = e0 (J) = 4. Now for large n, PIJ (n) = λ (IJ) = PI (ne) and PI 2 J (n) = n  R  λ I 2n J n = PI (2n, n). Comparing the coefficients on both sides we get 

 r+1 s+1 P(I,J) (r, s) = 4 +4 + 4rs − r − s. 2 2 Hence 0 = e1 (J) = e(0,1) (I, J) = 1. C. D’Cruz and S.K. Masuti [5, Theorem 5.2 and Theorem 5.3] gave necessary and sufficient conditions for the equalities e(1,0) (F) = e1 (F (1) ) and e(0,1) (F) = e1 (F (2) ) to hold true. Using Theorems 6.2 and 6.5, we prove: Theorem 6.6. Let (R, m) be a Cohen–Macaulay local ring of dimension two and I, J be m-primary ideals of R. Let F = {F(n)}n∈Z2 be an (I, J)-admissible filtration of ideals in R. Suppose e(1,0) (F) = e1 (F (1) ), e(0,1) (F) = e1 (F (2) ) and depth G(F (1) ), depth G(F (2) ) ≥ 1. Then following statements are equivalent. 2 (1) For every joint reduction (a, b) of F of type e, [H(at (R (F))](0,0) = 0, 1 ,bt2 ) (1 ) there exists a joint reduction (a, b) of F of type e satisfying superficial conditions 2 such that [H(at (R (F))](0,0) = 0, 1 ,bt2 ) (2) e2 (F Δ ) = e2 (F (1) ) + e2 (F (2) ), (3) the joint reduction number of F of type e is zero.

Proof. (1) ⇒ (1 ) and (1 ) ⇒ (2) follow from Theorem 6.4 and Theorem 6.5 respectively. (2) ⇒ (3): Let (a, b) be a joint reduction of F of type e satisfying superficial conditions. Then (a) ∩ F(k, 0) = aF(k − 1, 0) for k  0. Since depth G(F (1) ) ≥ 1, by [9, Lemma 2.1], (a) ∩ F(k, 0) = aF(k − 1, 0) for all k ≥ 1. Similarly, using depth G(F (2) ) ≥ 1, we get (b) ∩ F(0, k) = bF(0, k − 1) for all k ≥ 1. First we prove that μk :

F(k, k) F(k + 1, k + 1) ab −→ k+1 k + b F(k, 0) a F(0, k + 1) + bk+1 F(k + 1, 0)

ak F(0, k)

is injective for k ≥ 1. Let  denote the image of an element in respective quotients. Suppose μk (x ) = 0 for some x ∈ F(k, k). Then xab = ak+1 p + bk+1 q for some p ∈ F(0, k + 1) and q ∈ F(k + 1, 0). Hence p ∈ (b) ∩ F(0, k + 1) = bF(0, k). Let p = bp1 for some p1 ∈ F(0, k). Similarly, we get q = aq1 for some q1 ∈ F(k, 0). Hence x = ak p1 + bk q1 ∈ ak F(0, k) + bk F(k, 0). Thus x = 0.

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Then by [30, Lemma 5.30], for each k ≥ 1, φk :

F(k, k) F(k, k) → lim k k −→ + b F(k, 0) a F(0, k) + bk F(k, 0) k

ak F(0, k)

is injective. Since e2 (F Δ ) = e2 (F (1) ) + e2 (F (2) ), by Theorem 6.5 and equation (6.5.1), we have lim −→ k

F(k, k) = 0. ak F(0, k) + bk F(k, 0)

Therefore F(k, k) = ak F(0, k) + bk F(k, 0) for k ≥ 1. Now we prove that if F(r, s) = ar F(0, s) + bs F(r, 0) for some r ≥ 2 and s ≥ 1 then F(r − 1, s) = ar−1 F(0, s) + bs F(r − 1, 0). Let x ∈ F(r − 1, s). Then ax ∈ F(r, s) = ar F(0, s) + bs F(r, 0). Let ax = ar p + bs q for some p ∈ F(0, s) and q ∈ F(r, 0). Hence q ∈ (a) ∩ F(r, 0) = aF(r − 1, 0). Let q = aq  for some q  ∈ F(r − 1, 0). Hence x = ar−1 p + bs q  ∈ ar−1 F(0, s) + bs F(r − 1, 0). Similarly, if F(r, s) = ar F(0, s) + bs F(r, 0) for some r ≥ 1 and s ≥ 2 then F(r, s − 1) = ar F(0, s − 1) + bs−1 F(r, 0). Let r, s ≥ 1 and without loss of generality assume r = max{r, s}. Now F(r, r) = ar F(0, r) + br F(r, 0). Using the above procedure, we get F(r, r − 1) = ar F(0, r − 1) + br−1 F(r, 0), · · · , F(r, s) = ar F(0, s) + bs F(r, 0). Thus by equation (6.2.1), the joint reduction number of F with respect to the joint reduction (a, b) is zero. Hence by Theorem 6.2, joint reduction number of F of type e is zero. (3) ⇒ (1): Let (a, b) be a joint reduction of F of type e. Since joint reduction number of F of type e is zero, by equation (6.2.1), F(r, s) = ar F(0, s) + bs F(r, 0) F(k,k) for all r, s ≥ 1. Hence ak F(0,k)+b k F(k,0) = 0 for all k ≥ 1. Thus, by equation (6.5.1), 2  [H(at (R (F))] = 0. 2 (0,0) 1 ,bt2 ) As a consequence of this theorem we get the following result. The equivalence of (2) and (3) was proved by Rees for good joint reductions [28, Theorem 2.5]. Corollary 6.7. Let (R, m) be an analytically unramified Cohen–Macaulay local ring of dimension two and I, J be m-primary ideals of R. Then the following are equivalent: 2 (1) For every joint reduction (a, b) of {I r J s }r,s∈Z of type e, [H(at (R (I, J))](0,0) = 0, 1 ,bt2 )  (1 ) there exists a joint reduction (a, b) of {I r J s }r,s∈Z of type e satisfying superficial 2 conditions such that [H(at (R (I, J))](0,0) = 0, 1 ,bt2 ) (2) e2 (IJ) = e2 (I) + e2 (J),

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(3) for every joint reduction (a, b) of {I r J s }r,s∈Z of type e and for all r, s ≥ 1, I r J s = aI r−1 J s + bI r J s−1 . Proof. Consider F = {I r J s }r,s∈Z . By [27, Theorem 1.4], F is an (I, J)-admissible filtration. By [28, Theorem 1.2], e(1,0) (I, J) = e1 (I) and e(0,1) (I, J) = e1 (J). Consider the I-admissible filtration {I r }r∈Z and J-admissible filtration {J s }s∈Z . Let G(I) =   In Jn n≥0 I n+1 and G(J) = n≥0 J n+1 . Then by Corollary 3.4, depth G(I) ≥ 1, depth G(J) ≥ 1. Hence by Theorem 6.6, we get the required result. 2 7. Cohen–Macaulay property of the Rees algebra of bigraded filtrations In this section we relate the Cohen–Macaulayness of the bigraded Rees algebra R(F) with Hilbert coefficients, reduction numbers and joint reduction numbers. Let I, J be m-primary ideals in a Cohen–Macaulay local ring (R, m) of dimension two with infinite residue field and F be a Z2 -graded (I, J)-admissible filtration of ideals in R. M. Herrmann, E. Hyry, J. Ribbe and Z. Tang [8] proved that if R is a Cohen–Macaulay local ring and joint reduction number of {I r J s }r,s∈Z of type e is zero then R(I, J) is Cohen– Macaulay if and only if R(I) and R(J) are Cohen–Macaulay. We show that this is also valid for bigraded filtrations. Theorem 7.1. Let (R, m) be a Cohen–Macaulay local ring of dimension two, I, J be m-primary ideals and F = {F(n)}n∈Z2 be an (I, J)-admissible filtration of ideals in R. Let the joint reduction number of F with respect to some joint reduction of type e  be zero. Then R(F) is Cohen–Macaulay if and only if R(F (1) ) = n≥0 F(n, 0)tn and  R(F (2) ) = n≥0 F(0, n)tn are Cohen–Macaulay. Proof. By Theorem 6.4, there exists a joint reduction (a, b) of F of type e satisfying superficial conditions. By Theorem 6.2, joint reduction number of F with respect to (a, b) is zero. Therefore F(r, s) = aF(r − 1, s) + bF(r, s − 1) for all r, s ≥ 1. Let S := R(F)/(at1 , bt2 ) = R ⊕ S1+ ⊕ S2+ , where S1+ = that

 n≥1

F(n, 0)/aF(n − 1, 0) and S2+ =

 n≥1

F(0, n)/bF(0, n − 1). Note

S1 = R ⊕ S1+  R(F (1) )/(at) and S2 = R ⊕ S2+  R(F (2) )/(bt). The elements at and bt are nonzerodivisors in R(F (1) ) and R(F (2) ) respectively. Hence R(F (1) ) and R(F (2) ) are Cohen–Macaulay if and only if S1 and S2 are Cohen–Macaulay.

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By [8, Lemma 3.2], it is enough to prove that R(F) is Cohen–Macaulay if and only if S is Cohen–Macaulay. Suppose R(F) is Cohen–Macaulay. By [8, Lemma 3.2], dim S = 2. Let M be the maximal homogeneous ideal of R(F). Hence R(F)M is Cohen–Macaulay. Since dim R(F)M = 4 and dim SM = 2, it follows that ht((at1 , bt2 )M ) = 2. Let φ : R(F) −→ R(F)M be the natural ring homomorphism. Since R(F)M is Cohen– Macaulay, φ(at1 ), φ(bt2 ) is a regular sequence. Thus at1 , bt2 is a regular sequence in R(F). Hence S is Cohen–Macaulay. Conversely, suppose that S is Cohen–Macaulay. Since S = R(F)/(at1 , bt2 ), it suffices to show that (at1 , bt2 ) is a regular sequence on R(F). First we show that (a) ∩ F(r, s) = aF(r − 1, s) for all r > 0 and s ≥ 0.

(7.1.1)

By [8, Lemma 3.2], S1 and S2 are Cohen–Macaulay. Hence R(F (1) ) and R(F (2) ) are Cohen–Macaulay. Then by [35, Theorem 2.3], G(F (1) ) and G(F (2) ) are Cohen–Macaulay. Since a is superficial for F (1) , by [9, Lemma 2.1], (a) ∩F(r, 0) = aF(r − 1, 0) for all r ≥ 1. Thus the assertion (7.1.1) is true for s = 0. Suppose r, s ≥ 1 and ax ∈ F(r, s) for some x ∈ R. Since joint reduction number of F of type e is zero, for all r, s ≥ 1, we have F(r, s) = ar F(0, s) + bs F(r, 0). Hence ax = ar p + bs q for some p ∈ F(0, s) and q ∈ F(r, 0). Thus q ∈ (a) ∩ F(r, 0) = aF(r − 1, 0). Let q = aq  for some q  ∈ F(r − 1, 0). Then x = ar−1 p + bs q  ∈ F(r − 1, s). This proves the equation (7.1.1). In order to prove that (at1 , bt2 ) is R(F)-regular we prove that bt2 is R(F)/(at1 )-regular. Let zbt2 ∈ (at1 ). We may assume z = vtr1 ts2 for some v ∈ F(r, s) and r > 0. Then vb = aw for some w ∈ F(r − 1, s + 1). Thus v ∈ (a) ∩ F(r, s) = aF(r − 1, s). Hence z ∈ (at1 ). Therefore (at1 , bt2 ) is a regular sequence. This implies R(F) is Cohen– Macaulay. 2 In [7, Lemma 2.1], the authors proved that for a homogeneous ideal I of positive height in a multigraded ring B, the a-invariant, a(R(I)) = −1. Hyry proved that ai (R(I)) = −1 for all i = 1, . . . , s. In [22] authors generalised this result for an I-admissible filtrations. Proposition 7.2. (See [22, Lemma 3.7].) Let (R, m) be a Cohen–Macaulay local ring of dimension two, I, J be m-primary ideals of R and F = {F(n)}n∈Z2 be an (I, J)-admissible filtration of ideals in R. Then a1 (R(F)) = a2 (R(F)) = −1. Next we prove our main result in dimension two. Theorem 7.3. Let (R, m) be a Cohen–Macaulay local ring of dimension two, I, J be m-primary ideals of R and F = {F(n)}n∈Z2 be an (I, J)-admissible filtration of ideals in R. Then the following statements are equivalent.

564

(1) (2) (3) (4)

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e(I) − e(1,0) (F) = λ(R/F(1, 0)) and e(J) − e(0,1) (F) = λ(R/F(0, 1)), r(F (1) ), r(F (2) ) ≤ 1 and the joint reduction number of F of type e is zero, PF (r, s) = HF (r, s) for all r, s ≥ 0, R(F) is Cohen–Macaulay.

Proof. (1) ⇒ (2): From Theorem 5.5, e(1,0) (F) = e1 (F (1) ), e(0,1) (F) = e1 (F (2) ), r(F (1) ) ≤ 1 and r(F (2) ) ≤ 1. Thus, by [2, Theorem 4.3.6], G(F (1) ), G(F (2) ) are Cohen– Macaulay and e2 (F (1) ) = e2 (F (2) ) = 0. Therefore from Theorem 6.5, 2 λR [H(at (R (F))](0,0) = −e2 (F Δ ) 1 ,bt2 )

for every joint reduction (a, b) of F of type e satisfying superficial conditions. Hence e2 (F Δ ) ≤ 0. But e2 (F Δ ) ≥ 0 in any Cohen–Macaulay local ring of dimension d ≥ 2 [18, Proposition 3.23]. Hence e2 (F Δ ) = 0. Therefore, by Theorem 6.6, the joint reduction number of F of type e is zero. (2) ⇒ (3): Since joint reduction number of F of type e is zero, we have a joint reduction (a, b) of F of type e, such that F(r, s) = aF(r − 1, s) + bF(r, s − 1) for all r, s ≥ 1. Then by Theorem 6.2, for all r, s ≥ 1, λ(R/F(r, s)) = rse(I|J) + λ(R/F(r, 0)) + λ(R/F(0, s)).

(7.3.1)

If r = 0 or s = 0 the equation λ(R/F(r, s)) = rse(I|J) + λ(R/F(r, 0)) + λ(R/F(0, s)) is still true. Since r(F (1) ) ≤ 1, by [2, Theorem 4.3.6], G(F (1) ) is Cohen–Macaulay. Hence by [18, Corollary 3.8], PF (1) (r) = λ(R/F(r, 0)) for all r ≥ 0. Similarly PF (2) (s) = λ(R/F(0, s)) for all s ≥ 0. Therefore for all r, s ≥ 0 λ(R/F(r, s)) = rse(I|J) + PF (1) (r) + PF (2) (s). (3) ⇒ (1) ⇒ (4): Since PF (r, s) = λ(R/F(r, s)) for all r, s ≥ 0, it follows that e2 (F Δ ) = 0, e(I) − e(1,0) (F) = λ(R/F(1, 0)) and e(J) − e(0,1) (F) = λ(R/F(0, 1)). By (1) ⇒ (2), r(F (1) ), r(F (2) ) ≤ 1 and joint reduction number of F of type e is zero. Hence by [2, Theorem 4.3.6], G(F (1) ), G(F (2) ) are Cohen–Macaulay. Thus R(F (1) ) and R(F (2) ) are Cohen–Macaulay, by [35, Theorem 2.3]. Hence by Theorem 7.1, R(F) is Cohen–Macaulay. (4) ⇒ (1): From Proposition 7.2, a1 (R(F)), a2 (R(F)) < 0. Hence, by Theorem 2.15, we get i [HR (R(F))](r,s) = 0 for all r, s ≥ 0 and i = 0, 1, 2. ++

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i i Since by Proposition 4.2, [HR (R(F))](r,s)  [HR (R (F))](r,s) for all r, s ≥ 0, we ++ ++ get that i [HR (R (F))](r,s) = 0 for all r, s ≥ 0 and i = 0, 1, 2. ++

Hence using Theorem 4.3, we get that PF (r, s) = λ(R/F(r, s)) for all r, s ≥ 0.

(7.3.2)

Taking r = s = 0 in equation (7.3.2), we get e2 (F Δ ) = 0. Hence taking (r, s) = (1, 0) and (r, s) = (0, 1) in equation (7.3.2), we get the desired result. 2 As a consequence of this we generalise a very special case of a theorem by Hyry [12, Theorem 6.1]. Theorem 7.4. Let (R, m) be a Cohen–Macaulay local ring of dimension two, I, J be m-primary ideals of R and F = {F(n)}n∈Z2 be an (I, J)-admissible filtration of ideals in R. If R(F) is Cohen–Macaulay, then   α1

α2 

α1 α2 R 2−n1 −n2 (−1) eα (F) = λ n1 n2 F(n) n =0 n =0 1

2

where α = (α1 , α2 ), |α| = 2 and n = (n1 , n2 ). Proof. Proof follows from Theorem 7.3.

2

Remark 7.5. Jayanthan and Verma [14] showed that Theorem 7.3 is not true if we consider dimension of the ring greater than 2. They also gave an example showing that we cannot drop any conditions in (1) of Theorem 7.3 to conclude (2) and (3). References [1] P.B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge Philos. Soc. 53 (1957) 568–575. [2] C. Blancafort, Hilbert functions: combinatorial and homological aspects, Ph.D. Thesis, Universitat De Bercelona, 1997. [3] C. Blancafort, On Hilbert functions and cohomology, J. Algebra 192 (1997) 439–459. [4] M.P. Brodmann, R.Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge University Press, 1998. [5] C. D’Cruz, S.K. Masuti, Local cohomology of bigraded Rees algebras, Bhattacharya coefficients and joint reductions, arXiv:1405.1550. [6] G. Colomé-Nin, Multigraded structures and the depth of blow-up algebras, Ph.D. Thesis, Universitat de Barcelona, 2008. [7] M. Herrmann, E. Hyry, J. Ribbe, On the Cohen–Macaulay and Gorenstein properties of multigraded Rees algebras, Manuscripta Math. 79 (1993) 343–377. [8] M. Herrmann, E. Hyry, J. Ribbe, Z. Tang, Reduction numbers and multiplicities of multigraded structures, J. Algebra 197 (1997) 311–341.

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