On the multiplicity of multi-graded fiber cones

Chứng minh. By Theorem 4.1 [HHRT], f(n1, . . . , ns) is a polynomial for all sufficiently large n1, . . . , ns. Moreover, all monomials of highest degree in this polynomial have nonnegative coefficients. Denote this polynomial by P(n1, . . . , ns). We will prove that deg P(n1, . . . , ns) = `− 1. Set Q(n) = P(n, . . . , n). Since all monomials of highest degree in P(n1, . . . , ns) have non-negative coefficients, deg P(n1, . . . , ns) = deg Q(n). We have

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Journal of Science of Hanoi National University of Education Natural sciences, Volume 52, Number 4, 2007, pp. 23- 31 ON THE MULTIPLICITY OF MULTI-GRADED FIBER CONES Nguyen Tien Manh Math. Depart., Hung Vuong University, Phu Tho Abstract: Let (A,m) denote a Noetherian local ring with maximal ideal m, J an m-primary ideal, I1, . . . , Is ideals of A; M a finitely generated A-module. In this paper, we express the multiplicity of the multi-graded fiber cone FM (J, I1, . . . , Is) = ⊕ n1,...,ns>0 In1 1 · · · Inss M JIn1 1 · · · Inss M in terms of mixed multiplicities. 1 INTRODUCTION Throughout this paper, (A,m) denotes a Noetherian local ring with maximal ideal m, infinite residue field k = A/m; M a finitely generated A-module with Krull dimension dimM = d > 0. Let J be m-primary and I1, . . . , Is ideals of A. Set I = I1 · · · Is. Define FM (J, I1, . . . , Is) = ⊕ n1,...,ns>0 In11 · · · Inss M JIn11 · · · Inss M ; ` = dim (⊕ n>0 InM mInM ) to be the multi-graded fiber cone of M with respect to J, I1, . . . , Is and the analytic spread of I with respect to M , respectively. The multiplicity of blow-up algebras was concerned by many authors in the past years. Several of authors expressed the multiplicity of some Rees algebras in terms of mixed multiplicities, e.g. Verma in [Ve1, Ve2] for Rees algebras and multi-graded Rees algebras ; Katz and Verma in [KV] for extended Rees algebras; D'Cruz in [CD] for multi-graded extended Rees algebras. Herrmann et al. in [HHRT] for standard multi-graded algebras over an Artinian local ring... Set f(n1, . . . , ns) = lA ( In11 · · · Inss M JIn11 · · · Inss M ) . Then f(n1, . . . , ns) is a polynomial of degree `−1 for all large n1, . . . , ns (see Proposition 3.1, Section 3). The terms of total degree `− 1 in this polynomial have the form ∑ d1 + ···+ ds = `−1 EJ(I [d1] 1 , . . . , I [ds] s ;M) nd11 · · ·ndss d1! · · · ds! , then EJ(I [d1] 1 , . . . , I [ds] s ;M) is a non-negative integer and is called the mixed multiplicity of the multi-graded fiber cone FM (J, I1, . . . , Is). The purpose of this paper is to express 23 NGUYEN TIEN MANH the multiplicity of FM (J, I1, . . . , Is) in terms of mixed multiplicities in the case where ht ( I +AnnM AnnM ) > 0. This paper is divided into three sections. In Section 2, we give some results on weak- (FC)-sequences of modules and the analytic spread of ideals. Section 3 investigates the Krull dimension and the multiplicity of multi-graded fiber cones. The main result of this section is Theorem 3.3. 2 ON WEAK-(FC)-SEQUENCES OF MODULES This section presents some results on weak-(FC)-sequences and the analytic spread of ideals which will be used in the paper. Set a : b∞ = ⋃ n>0(a : b n). The notion of weak-(FC)-sequences in [Vi1] is extended to modules as follows. Definition 2.1 (see [MV, Definition 2.1]). Let U = (I1, . . . , Is) be a set of ideals of A such that I = I1 · · · Is is not contained in √ AnnM . Set M∗ = M 0M : I∞ . We say that an element x ∈ A is a weak-(FC)-element of M with respect to U if there exists an ideal Ii of U and a positive integer n′i such that (FC1) : x ∈ Ii \mIi and In11 · · · Inss M∗ ⋂ xM∗ = xIn11 · · · Ini−1i−1 Ini−1i Ini+1i+1 · · · Inss M∗ for all ni > n ′ i and all non-negative integers n1, . . . , ni−1, ni+1, . . . , ns. (FC2) : 0M : x ⊆ 0M : I∞. Let x1, . . . , xt be a sequence in ⋃s i=1 Ii. For each i = 0, 1, . . . , t−1, set A¯ = A (x1, . . . , xi) , I¯1 = I1A¯, . . . , I¯s = IsA¯,M = M (x1, . . . , xi)M . Let x¯i+1 denote the image of xi+1 in A¯. Then x1, . . . , xt is called a weak-(FC)-sequence of M with respect to U if x¯i+1 is a weak-(FC)- element of M with respect to (I¯1, . . . , I¯s) for i = 0, 1, . . . , t− 1. Remark 2.2. If I is contained in √ AnnM , then the conditions (FC1) and (FC2) are usually true for all x ∈ ⋃si=1 Ii. This only obstructs and does not carry useful. That is why in Definition 2.1, one has to exclude the case that I is contained in √ AnnM . In [MV], the authors showed the existence of weak-(FC)-sequences of modules by the following proposition. Proposition 2.3 (see [MV, Proposition 2.3]). Let (I1, . . . , Is) be a set of ideals such that I = I1 · · · Is is not contained in √ AnnM . Then for any 1 6 i 6 s, there exists a weak-(FC)-element xi ∈ Ii of M with respect to (I1, . . . , Is). So the existence of weak-(FC)-sequences is universal. 24 ON THE MULTIPLICITY OF MULTI-GRADED FIBER CONES Set RA(I) = ⊕ n>0 I ntn, RM (I) = ⊕ n>0 I nMtn. RA(I) and RM (I) are called the Rees algebra and the Rees module of I, respectively. Denote by LU (I1, . . . , Is;M) the set of lengths of maximal weak-(FC)-sequences in ⋃s i=1 Ii of M with respect to U . The parts (i), (ii) and (iv) of Theorem 3.4 in [Vi3] are stated in terms of modules as follows. Lemma 2.4 (see [Vi3, Theorem 3.4]). Let J1, . . . , Jt be m-primary ideals and I1, . . . , Is ideals such that I = I1 · · · Is is not contained in √ AnnM . Set U = (J1, . . . , Jt, I1, . . . , Is), ` = dim (⊕ n>0 InM mInM ) , `j = dim (⊕ n>0 InjM mInjM ) , Iˆi = I1 · · · Ii−1Ii+1 · · · Is if s > 1 and Iˆi = A if s = 1. Then the following statements hold. (i) For any 1 6 i 6 s, the length of maximal weak-(FC)-sequences in Ii of M with respect to U is an invariant and this invariant does not depend on t and J1, . . . , Jt. (ii) If p is the length of maximal weak-(FC)-sequences in Ij of M with respect to U , then p = dim ( RA(Ij)⋃ k≥0[m(mIˆj) kRM (Ij) : (mIˆj)kRM (Ij)] ) 6 `j. (iii) maxLU (I1, . . . , Is;M) = `. Next, we need the following lemma. Lemma 2.5. Let I,=1,=2 be ideals such that ht (=1=2 +AnnM AnnM ) > 0. Then dim ( =2RM (I) =1=2RM (I) ) = dim ( RM (I) =1RM (I) ) . Chùng minh. We have dim ( =2RM (I) =1=2RM (I) ) = dim ( RA(I) =1=2RM (I) : =2RM (I) ) = dim ( RA(I) =1RA(I) + AnnRA(I)(=2RM (I)) ) = dim ( RA(I) =1RA(I) + √ AnnRA(I)(=2RM (I)) ) . On the other hand,√ AnnRA(I)(=2RM (I)) = ⊕ n>0(I n ⋂√ AnnA(=2M))tn. Since ht (=1=2 +AnnM AnnM ) > 0, it follows that ht (=2 +AnnM AnnM ) > 0. This implies that √ AnnA(=2M) = √ AnnM. Thus √ AnnRA(I)(=2RM (I)) = ⊕ n>0(I n ⋂√ AnnM)tn = √ AnnRA(I)(RM (I)). 25 NGUYEN TIEN MANH From these facts, we get dim ( =2RM (I) =1=2RM (I) ) = dim ( RA(I) =1RA(I) + √ AnnRA(I)(RM (I)) ) = dim ( RA(I) =1RA(I) + AnnRA(I)(RM (I)) ) = dim ( RA(I) =1RM (I) : RM (I) ) = dim ( RM (I) =1RM (I) ) . The following proposition is a sharpening of Lemma 2.4(ii). Proposition 2.6. Let J be an m-primary ideal and I1, . . . , Is ideals such that ht ( I1 · · · Is +AnnM AnnM ) > 0. Set `j = dim (⊕ n>0 InjM mInjM ) (1 6 j 6 s). Suppose that p is the length of maximal weak- (FC)-sequences in Ij of M with respect to (J, I1, . . . , Is). Then p = `j. Chùng minh. Set Iˆj = I1 · · · Ij−1Ij+1 · · · Is if s > 1 and Iˆj = A if s = 1. Using the same argument as in the proof of [Vi3, Theorem 3.4(ii)], there exists a positive integer v such that p = dim ( RA(Ij) m(mIˆj)vRM (Ij) : (mIˆj)vRM (Ij) ) = dim ( (mIˆj) vRM (Ij) m(mIˆj)vRM (Ij) ) . Since ht ( I1 · · · Is +AnnM AnnM ) > 0, we have ht ( m(mIˆj) v +AnnM AnnM ) > 0. By Lemma 2.5, dim ( (mIˆj) vRM (Ij) m(mIˆj)vRM (Ij) ) = dim ( RM (Ij) mRM (Ij) ) = dim (⊕ n>0 InjM mInjM ) = `j . Hence p = `j . Proposition 2.6 gives an interesting consequence on the analytic spread of ideals as follows. Corollary 2.7. Let I1, I2 be ideals such that ht ( I1I2 +AnnM AnnM ) > 0. Set `1 = dim (⊕ n>0 In1M mIn1M ) , `12 = dim (⊕ n>0 (I1I2) nM m(I1I2)nM ) . Then `1 6 `12. Chùng minh. Set U = (m, I1, I2). Denote by p the length of maximal weak-(FC)-sequences in I1 of M with respect to U . By Proposition 2.6, p = `1. (*) 26 ON THE MULTIPLICITY OF MULTI-GRADED FIBER CONES Let LU(I1, I2;M) denote the set of lengths of maximal weak-(FC)-sequences in I1 ⋃ I2 of M with respect to U . By Lemma 2.4(iii), maxLU (I1, I2;M) = `12. (**) It is easy to see that p 6 maxLU (I1, I2;M). (***) By (∗), (∗∗) and (∗ ∗ ∗), we get `1 6 `12. 3 THEMULTIPLICITYOFMULTI-GRADED FIBER CONES This section will give the multiplicity formula and the Krull dimension of multi-graded fiber cones. We first have the following proposition. Proposition 3.1. Let J be an m-primary ideal and I1, . . . , Is ideals of A. Set I = I1 · · · Is, ` = dim (⊕ n>0 InM mInM ) , f(n1, . . . , ns) = lA ( In11 · · · Inss M JIn11 · · · Inss M ) . Then f(n1, . . . , ns) is a polynomial of degree `− 1 for all large n1, . . . , ns. Chùng minh. By Theorem 4.1 [HHRT], f(n1, . . . , ns) is a polynomial for all sufficiently large n1, . . . , ns. Moreover, all monomials of highest degree in this polynomial have non- negative coefficients. Denote this polynomial by P (n1, . . . , ns). We will prove that degP (n1, . . . , ns) = `−1. Set Q(n) = P (n, . . . , n). Since all monomials of highest degree in P (n1, . . . , ns) have non-negative coefficients, degP (n1, . . . , ns) = degQ(n). We have Q(n) = P (n, . . . , n) = lA ( In1 · · · InsM JIn1 · · · InsM ) = lA ( InM JInM ) for all sufficiently large n. Hence degQ(n) = dim (⊕ n>0 InM JInM ) −1 = dim (⊕ n>0 InM mInM ) −1 = `− 1. Thus degP (n1, . . . , ns) = `− 1. Recall that a polynomialF (t1, . . . , ts) ∈ Q[t1, . . . , ts] is called a numerical polynomial if F (n1, . . . , ns) ∈ Z for all n1, . . . , ns ∈ Z. Using the same argument as in [HHRT, Lemma 4.2], we get the following. Lemma 3.2 (see [HHRT, Lemma 4.2]). Let F (n1, . . . , ns) be a numerical polynomial of degree p in n1, . . . , ns. Let u1, . . . , us be non-negative integers. Then the function G(n) = ∑ n1 + ···+ ns = n, n1>u1,...,ns>us F (n1, . . . , ns) is a numerical polynomial of degree 6 p + s − 1 in n for large n and the coefficient of np+s−1 in this polynomial is 1 (p + s− 1)! ∑ k1 + ···+ ks = p e(k1, . . . , ks), where e(k1, . . . , ks) k1! · · · ks! is the coefficient of nk11 · · ·nkss in F (n1, . . . , ns). 27 NGUYEN TIEN MANH Let J be an m-primary ideal and I1, . . . , Is ideals. Set FA(J, I1, . . . , Is) = ⊕ n1,...,ns≥0 In11 · · · Inss JIn11 · · · Inss , F+A = ⊕ n1 + ···+ ns > 0 In11 · · · Inss JIn11 · · · Inss . Denote by e(FM ) the multiplicity of FM (J, I1, . . . , Is). We get the following result on the relationship between the multiplicity and mixed multiplicities of multi-graded fiber cones. Theorem 3.3. Let J be an m-primary ideal and I1, . . . , Is ideals such that ht ( I +AnnM AnnM ) > 0, where I = I1 · · · Is. Set ` = dim (⊕ n>0 InM mInM ) . Then dimFM (J, I1, . . . , Is) = ` + s − 1 and e(FM ) = ∑ d1 + ···+ ds = `−1 EJ(I [d1] 1 , . . . , I [ds] s ;M). Chùng minh. By Proposition 3.1, f(n1, . . . , ns) = lA ( In11 · · · Inss M JIn11 · · · Inss M ) is a polynomial of degree `− 1 for all large n1, . . . , ns. Set F (n) = lA ( (F+A ) nFM (J, I1, . . . , Is) (F+A ) n+1FM (J, I1, . . . , Is) ) . It can be verified that F (n) = ∑ n1 + ···+ ns=n f(n1, . . . , ns). Denote by df the Krull dimension of FM (J, I1, . . . , Is). Then F (n) is a polynomial of degree df − 1 for large n and e(FM ) = limn→∞ (df − 1)!F (n) ndf−1 . Assume that u is a positive integer such that f(n1, . . . , ns) is a polynomial for all n1, . . . , ns > u. Set T = {1, . . . , s}. For each r = 1, . . . , s− 1, denote by Tr the set{ (i1, . . . , is)|1 6 i1 < · · · < ir 6 s, 1 6 ir+1 < · · · < is 6 s, {i1, . . . , is} = T } . For each (i1, . . . , is) ∈ Tr (r = 1, . . . , s− 1) and ar+1, . . . , as < u, denote by Sar+1,...,asi1,...,ir the set {(n1, . . . , ns)| ∑s i=1 ni = n, ni1 > u, . . . , nir > u, nir+1 = ar+1, . . . , nis = as}. First, to prove dimFM (J, I1, . . . , Is) = df = ` + s − 1, we need to show that F (n) is a polynomial of degree `+ s− 2 for large n. The proof is by induction on s. For s = 1, F (n) = ∑ n1=n f(n1) = f(n) = lA ( In1M JIn1M ) is a polynomial of degree ` − 1 for large n by Proposition 3.1. The result is true in this case. For s > 1, assume that the result has been true for 1, 2, . . . , s− 1. We will prove that it is also true for s. For n > su, F (n) = ∑ n1+···+ns=n; n1,...,ns>u f(n1, . . . , ns) + s−1∑ r=1 ( ∑ 16i1<···<ir6s ( ∑ ar+1,...,as<u ( ∑ (n1,...,ns)∈S ar+1,...,as i1,...,ir f(n1, . . . , ns) ))) . 28 ON THE MULTIPLICITY OF MULTI-GRADED FIBER CONES By Lemma 3.2, G(n) = ∑ n1+···+ns=n,n1,...,ns>u f(n1, . . . , ns) is a polynomial of degree `+ s− 2 for large n and the coefficient of n`+s−2 in this polynomial is 1 (`+ s− 2)! ∑ d1 + ···+ ds = `−1 EJ(I [d1] 1 , . . . , I [ds] s ;M). (1) Set F ar+1,...,as i1,...,ir (n) = ∑ (n1,...,ns)∈S ar+1,...,as i1,...,ir f(n1, . . . , ns). Then F (n) = G(n) + s−1∑ r=1 ( ∑ 16i1<···<ir6s ( ∑ ar+1,...,as<u F ar+1,...,as i1,...,ir (n) )) . (2) Set F ′A = ⊕ ni1 ,...,nir≥0 I ni1 i1 · · · Inirir JI ni1 i1 · · · Inirir , F ′A + = ⊕ ni1+···+nir>0 I ni1 i1 · · · Inirir JI ni1 i1 · · · Inirir ; F ′M = ⊕ ni1 ,...,nir≥0 I ni1 i1 · · · Inirir M JI ni1 i1 · · · Inirir M , F ′′M = ⊕ ni1 ,...,nir≥0 I ni1 i1 · · · Inirir Iui1 · · · IuirI ar+1 ir+1 · · · Iasis M JI ni1 i1 · · · Inirir Iui1 · · · IuirI ar+1 ir+1 · · · Iasis M . It is clear that F ′M and F ′′ M are multi-graded F ′ A-modules and dimF ′′ M 6 dimF ′ M . Set `′ = dim (⊕ n≥0 (Ii1 · · · Iir)nM m(Ii1 · · · Iir)nM ) . By Lemma 2.7, `′ 6 `. By the inductive assumption applied to r < s, dimF ′M = ` ′ + r − 1 6 `+ r − 1. From the above facts and note that r < s, dimF ′′M 6 dimF ′ M 6 `+ r − 1 < `+ s− 1. (3) Set v = ar+1 + · · ·+ as and f ar+1,...,as i1,...,ir (ni1 , . . . , nir) = lA ( I ni1 i1 · · · Inirir Iui1 · · · IuirI ar+1 ir+1 · · · Iasis M JI ni1 i1 · · · Inirir Iui1 · · · IuirI ar+1 ir+1 · · · Iasis M ) . For any (n1, . . . , ns) ∈ Sar+1,...,asi1,...,ir , we have f(n1, . . . , ns) = lA ( I ni1−u i1 · · · Inir−uir Iui1 · · · IuirI ar+1 ir+1 · · · Iasis M JI ni1−u i1 · · · Inir−uir Iui1 · · · IuirI ar+1 ir+1 · · · Iasis M ) = f ar+1,...,as i1,...,ir (ni1 − u, . . . , nir − u). 29 NGUYEN TIEN MANH Direct computation shows that F ar+1,...,as i1,...,ir (n) = ∑ ni1+···+nir=n−v; ni1 ,...,nir>u f ar+1,...,as i1,...,ir (ni1 − u, . . . , nir − u) = ∑ mi1+···+mir=n−ru−v;mi1 ,...,mir>0 f ar+1,...,as i1,...,ir (mi1 , . . . ,mir) = lA ( (F ′A +)n−ru−vF ′′M (F ′A +)n−ru−v+1F ′′M ) . So for all 1 6 i1 < · · · < ir 6 s and ar+1, . . . , as < u (r = 1, . . . , s− 1), then F ar+1,...,asi1,...,ir (n) is a polynomial of degree dimF ′′M − 1 for large n. By (3), for large n then F ar+1,...,asi1,...,ir (n) is a polynomial of degree < ` + s − 2 for all 1 6 i1 < · · · < ir 6 s and ar+1, . . . , as < u (r = 1, . . . , s− 1). Hence by (1) and (2), F (n) is a polynomial of degree `+ s− 2 for large n and the coefficient of n`+s−2 in this polynomial is 1 (`+ s− 2)! ∑ d1 + ···+ ds = `−1 EJ(I [d1] 1 , . . . , I [ds] s ;M). Thus dimFM (J, I1, . . . , Is) = `+ s− 1 and e(FM ) = ∑ d1 + ···+ ds = d−1 EJ(I [d1] 1 , . . . , I [ds] s ;M). The proof of Theorem 3.3 is complete. In the case where I1, . . . , Is are m-primary ideals, it is easily seen that ` = dim (⊕ n>0 InM mInM ) = ht ( I +AnnM AnnM ) = d > 0, where I = I1 · · · Is. As an immediate consequence of Theorem 3.3, we have the following result. Corollary 3.4. Let J, I1, . . . , Is be m-primary. Then dimFM (J, I1, . . . , Is) = d + s − 1 and e(FM ) = ∑ d1 + ···+ ds = d−1 EJ(I [d1] 1 , . . . , I [ds] s ;M). References [Ba] P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge. Philos. Soc. 53(1957), 568-575. [CD] C. D'Cruz, A formula for the multiplicity of the multi-graded extended Rees algebras, Comm. Algebra. 31(6)(2003), 2573-2585. [HHRT] M. Herrmann, E. Hyry, J. Ribbe, Z. Tang, Reduction numbers and multiplicities of multigraded structures, J. Algebra 197(1997), 311-341. [KV] D. Katz, J. K. Verma, Extended Rees algebras and mixed multiplicities, Math. Z. 202(1989), 111-128. 30 ON THE MULTIPLICITY OF MULTI-GRADED FIBER CONES [MV] N. T. Manh, D. Q. Viet, Mixed multiplicities of modules over Noetherian local rings, Tokyo J. Math. Vol. 29, No. 2 (2006), 325-345. [NR] D. G. Northcott, D. Rees, Reduction of ideals in local rings, Proc. Cambridge Phil. Soc. 50(1954), 145-158. [Re] D. Rees, Generalizations of reductions and mixed multiplicities, J. London. Math. Soc. 29(1984), 397-414. [Ve1] J. K. Verma, Rees algebras and mixed multiplicities, Proc. Amer. Mat. Soc. 104(1988), 1036-1044. [Ve2] J. K. Verma, Multigraded Rees algebras and mixed multiplicities, J. Pure and Appl. Algebra 77(1992), 219-228. [Vi1] D. Q. Viet, Mixed multiplicities of arbitrary ideals in local rings, Comm. Algebra. 28(8) (2000), 3803-3821. [Vi2] D. Q. Viet, On some properties of (FC)-sequences of ideals in local rings, Proc. Amer. Math. Soc. 131(2003), 45-53. [Vi3] D. Q. Viet, Sequences determining mixed mutiplicities and reductions of ideals, Comm. Algebra. 31(10)(2003),5047-5069. Tâm t­t V· bëi cõa Fiber cone a ph¥n bªc Nguy¹n Ti¸n M¤nh ¤i håc Hòng V÷ìng-Phó Thå Cho (A,m) l  mët v nh àa ph÷ìng Noether vîi i¶an cüc ¤i m, J l  mët i¶an m-ngu¶n sì, I1, . . . , Is l  nhúng i¶an cõa A; M l  mët A-mæ un húu h¤n sinh. Trong b i b¡o n y, chóng tæi biºu di¹n bëi cõa fiber con a ph¥n bªc FM (J, I1, . . . , Is) = ⊕ n1,...,ns>0 In11 · · · Inss M JIn11 · · · Inss M theo c¡c sè bëi trën. 31