Nội dung

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 ) = in terms of mixed multiplicities. L n1 ,...,ns >0 I1n1 · · · Isns M JI1n1 · · · Isns M 1 INTRODUCTION (A, m) denotes a Noetherian local ring with maximal ideal m, k = A/m; M a finitely generated A-module with Krull dimension Throughout this paper, infinite residue field dim M = d > 0. Let J be m-primary and I1 , . . . , Is M FM (J, I1 , . . . , Is ) = n1 ,...,ns >0 to be the multi-graded fiber cone of of I with respect to M, M ideals of A. Set I = I1 · · · Is . Define M I n M  I1n1 · · · Isns M ; ` = dim JI1n1 · · · Isns M mI n M n>0 with respect to J, I1 , . . . , Is and the analytic spread 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 X d1 + ··· + ds = `−1 then [d ]  I1n1 · · · Isns M . JI1n1 · · · Isns M Then ` − 1 for all large n1 , . . . , ns (see Proposition 3.1, ` − 1 in this polynomial have the form is a polynomial of degree terms of total degree  [d ] EJ (I1 1 , . . . , Is s ; M ) f (n1 , . . . , ns ) Section 3). The nd11 · · · nds s [d1 ] [ds ] EJ (I1 , . . . , Is ; M ) , d1 ! · · · ds ! 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  FM (J, I1 , . . . , Is )  I + AnnM > 0. ht AnnM the multiplicity of in terms of mixed multiplicities in the case where 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∞ = S n>0 (a : bn ). The notion of weak-(FC)-sequences in [Vi1] is extended to modules as follows. Definition 2.1 (see [MV, Definition 2.1]). Let such that element of U I = I1 · · · Is x∈A is not contained in is a weak-(FC)-element of n0i such that √ U = (I1 , . . . , Is ) be a set of ideals of M AnnM . Set M ∗ = . We say 0M : I ∞ M with respect to U if there exists an A that an ideal Ii and a positive integer (FC1 ) : x ∈ Ii \ mIi and I1n1 · · · Isns M ∗ for all ni > n0i \ n n i+1 i−1 ni −1 · · · Isns M ∗ xM ∗ = xI1n1 · · · Ii−1 Ii Ii+1 and all non-negative integers n1 , . . . , ni−1 , ni+1 , . . . , ns . (FC2 ) : 0M : x ⊆ 0M : I ∞ . Let x1 , . . . , xt be a sequence in Ss i=1 Ii . For each i = 0, 1, . . . , t−1, set Ā = A , (x1 , . . . , xi ) M . Let x̄i+1 denote the image of xi+1 in Ā. Then (x1 , . . . , xi )M x1 , . . . , xt is called a weak-(FC)-sequence of M with respect to U if x̄i+1 is a weak-(FC)element of M with respect to (I¯1 , . . . , I¯s ) for i = 0, 1, . . . , t − 1. I¯1 = I1 Ā, . . . , I¯s = Is Ā, M = √ I isScontained in AnnM , then the conditions (FC1 ) and (FC2 ) are s usually true for all x ∈ i=1 Ii . This only obstructs and does not carry √ useful. That is why AnnM . in Definition 2.1, one has to exclude the case that I is contained in Remark 2.2. If 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 that I = I1 · · · Is is not contained in a weak-(FC)-element xi ∈ Ii of M with (I1 , . . . , Is ) be a set of AnnM . Then for any 1 6 i 6 s, respect to (I1 , . . . , Is ). So the existence of weak-(FC)-sequences is universal. 24 ideals such there exists ON THE MULTIPLICITY OF MULTI-GRADED FIBER CONES L RM (I) = n>0 I n M tn . 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 Ss of lengths of maximal weak-(FC)-sequences in i=1 Ii of M with respect to U . The parts Set RA (I) = L n>0 I n tn , (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 is not contained in AnnM . Set L L Ijn M  I nM  (J1 , . . . , Jt , I1 , . . . , Is ), ` = dim , ` = dim , j n>0 n>0 mI n M mIjn M ideals such that U= I = I1 · · · Is Iˆi = I1 · · · Ii−1 Ii+1 · · · Is (i) For any if s>1 Iˆi = A and 1 6 i 6 s, the length U is an invariant and respect to (ii) If p if s = 1. Then the following statements hold. t this invariant does not depend on is the length of maximal weak-(FC)-sequences in  p = dim S Ii of M with and J1 , . . . , Jt . of maximal weak-(FC)-sequences in Ij of M with respect to RA (Ij ) ˆ k k≥0 [m(mIj ) RM (Ij ) : (mIˆj )k RM (Ij )]  U, then 6 `j . (iii) maxLU (I1 , . . . , Is ; M ) = `. Next, we need the following lemma. Lemma 2.5. Let   =1 =2 + AnnM > 0. I, =1 , =2 be ideals such that ht   AnnM   RM (I) =2 RM (I) = dim . dim =1 =2 RM (I) =1 RM (I) Then Chùng minh. We have     =2 RM (I) RA (I) dim = dim =1 =2 RM (I) =1 =2 RM (I) : =2 RM (I)   RA (I) = dim =1 RA (I) + AnnRA (I) (=2 RM (I))   RA (I) q . = dim =1 RA (I) + AnnRA (I) (=2 RM (I)) On the other hand, Since Thus q L Tp AnnRA (I) (=2 RM (I)) = n>0 (I n AnnA (=2 M ))tn .     =1 =2 + AnnM =2 + AnnM ht > 0, it follows that ht > 0. This AnnM AnnM p √ AnnA (=2 M ) = AnnM . q AnnRA (I) (=2 RM (I)) = L n n>0 (I T√ AnnM )tn = q implies that AnnRA (I) (RM (I)). 25 NGUYEN TIEN MANH From these facts, we get  =2 RM (I) dim =1 =2 RM (I)   RA (I) q =1 RA (I) + AnnRA (I) (RM (I))   RA (I) = dim (I) (RM (I))  =1 RA (I) + AnnRA   RA (I) RM (I) = dim = dim . =1 RM (I) : RM (I) =1 RM (I)  = dim The following proposition is a sharpening of Lemma 2.4(ii). Proposition 2.6. Let Set `j = dim L Chùng minh. Set be an Ijn M  m -primary ideal and  I1 , . . . , Is I1 · · · Is + AnnM > 0. ht AnnM ideals such that (1 6 j 6 s). Suppose that p is the length mIjn M Ij of M with respect to (J, I1 , . . . , Is ). Then p = `j . n>0 (FC)-sequences in J Iˆj = I1 · · · Ij−1 Ij+1 · · · Is if s>1 and Iˆj = A if of maximal weak- s = 1. Using the same argument as in the proof of [Vi3, Theorem 3.4(ii)], there exists a positive integer that Since    (mIˆj )v RM (Ij ) p = dim = dim . m(mIˆj )v RM (Ij ) : (mIˆj )v RM (Ij ) m(mIˆj )v RM (Ij )     I1 · · · Is + AnnM m(mIˆj )v + AnnM > 0, we have ht > 0. By Lemma ht AnnM AnnM v such  RA (Ij ) 2.5,     M I n M  (mIˆj )v RM (Ij ) RM (Ij ) j dim = dim = dim n M = `j . v ˆ mR (I ) mI m(mIj ) RM (Ij ) M j j n>0 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 `1 = dim Then `1 6 `12 . L n>0 I1n M  , `12 mI1n M   I1 I2 + AnnM > 0. Set AnnM L (I1 I2 )n M  = dim . n>0 m(I1 I2 )n M ht U = (m, I1 , I2 ). Denote by p the length of maximal weak-(FC)-sequences respect to U . By Proposition 2.6, Chùng minh. Set in I1 of M with p = `1 . 26 (*) ON THE MULTIPLICITY OF MULTI-GRADED FIBER CONES Let M LU (I1 , I2 ; M ) denote the set of lengths of maximal weak-(FC)-sequences in with respect to U. I1 By Lemma 2.4(iii), max LU (I1 , I2 ; M ) = `12 . S I2 of (**) It is easy to see that p 6 max LU (I1 , I2 ; M ). By (∗), (∗∗) and (∗ ∗ ∗), we get (***) `1 6 `12 . 3 THE MULTIPLICITY OF MULTI-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 m-primary ideal and I1 , . . . , Is ideals A. Set   nof L I1 1 · · · Isns M I nM  , f (n1 , . . . , ns ) = lA . I = I1 · · · Is , ` = dim n>0 mI n M JI1n1 · · · Isns M f (n1 , . . . , ns ) is a polynomial of degree ` − 1 for all large n1 , . . . , ns . Then J be an Chùng minh. By Theorem 4.1 [HHRT], large n1 , . . . , ns . f (n1 , . . . , ns ) is a polynomial for all sufficiently Moreover, all monomials of highest degree in this polynomial have non- 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    n  n I M I1 · · · Isn M = lA Q(n) = P (n, . . . , n) = lA JI1n · · · Isn M JI n M for all sufficiently large n. Hence L L I nM  I nM  −1 = dim −1 = ` − 1. deg Q(n) = dim n>0 n>0 JI n M mI n M Thus deg P (n1 , . . . , ns ) = ` − 1. negative coefficients. Denote this polynomial by Recall that a polynomial F (t1 , . . . , ts ) if F (n1 , . . . , ns ) ∈ Z for all n 1 , . . . , n s ∈ Z. ∈ Q[t1 , . . . , ts ] is called a numerical polynomial Using the same argument as in [HHRT, Lemma 4.2], we get the following. Lemma 3.2 (see [HHRT, Lemma 4.2]). Let degree p in n1 , . . . , ns . Let u1 , . . . , us G(n) = F (n1 , . . . , ns ) be a numerical polynomial of be non-negative integers. Then the function X F (n1 , . . . , ns ) n1 + ··· + ns = n, n1 >u1 ,...,ns >us 6 p + s − 1 in n for large n and the coefficient of P 1 e(k1 , . . . , ks ) k1 + ··· + ks = p e(k1 , . . . , ks ), where (p + s − 1)! k1 ! · · · ks ! k1 k s n1 · · · ns in F (n1 , . . . , ns ). is a numerical polynomial of degree np+s−1 in this polynomial is is the coefficient of 27 NGUYEN TIEN MANH Let J be an m-primary FA (J, I1 , . . . , Is ) = Denote by e(FM ) ideal and I1 , . . . , Is M I1n1 · · · Isns , JI1n1 · · · Isns n1 ,...,ns ≥0 the multiplicity of ideals. Set FA+ = M n1 + ··· + ns > 0 FM (J, I1 , . . . , Is ). I1n1 · · · Isns . JI1n1 · · · Isns We get the following result on the relationship between the multiplicity and mixed multiplicities of multi-graded fiber cones. Theorem 3.3. Let where and I = I1 · · · Is . J m-primary I1 , . . . , Is ideals such that  ideal and  I + AnnM > 0, ht AnnM L  n I M Set ` = dim . Then dim FM (J, I1 , . . . , Is ) = ` + s − 1 n>0 mI n M X [d ] e(FM ) = EJ (I1 1 , . . . , Is[ds ] ; M ). be an d1 + ··· + ds = `−1 Chùng minh. By Proposition 3.1, I1n1 · · · Isns M JI1n1 · · · Isns M  is a polynomial of n1 , . . . , ns . Set   (FA+ )n FM (J, I1 , . . . , Is ) . F (n) = lA (FA+ )n+1 FM (J, I1 , . . . , Is ) P 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 (df − 1)!F (n) n and e(FM ) = limn→∞ . Assume that u is a positive integer such that ndf −1 f (n1 , . . . , ns ) is a polynomial for all n1 , . . . , ns > u. Set T = {1, . . . , s}. For each r = 1, . . . , sn− 1, denote by Tr the set o (i1 , . . . , is )|1 6 i1 < · · · < ir 6 s, 1 6 ir+1 < · · · < is 6 s, {i1 , . . . , is } = T . degree `−1 f (n1 , . . . , ns ) = lA  For each set for all large (i1 , . . . , is ) ∈ Tr (r = 1, . . . , s − 1) and ar+1 , . . . , as < u, denote by a ,...,as Si1r+1 ,...,ir the P {(n1 , . . . , ns )| si=1 ni = n, ni1 > u, . . . , nir > u, nir+1 = ar+1 , . . . , nis = as }. First, to prove dim FM (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  on s. For s = 1,  induction P I1n M F (n) = n1 =n f (n1 ) = f (n) = lA JI1n M 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, X F (n) = f (n1 , . . . , ns ) n1 +···+ns =n; n1 ,...,ns >u + s−1  X r=1 28 X  X 16i1 <···u f (n1 , . . . , ns ) is a polynomial `+s−2 in this polynomial is large n and the coefficient of n By Lemma 3.2, `+s−2 for X 1 (` + s − 2)! [d ] EJ (I1 1 , . . . , Is[ds ] ; M ). a ,...,as (1) d1 + ··· + ds = `−1 Set Fi1r+1 ,...,ir of degree X (n) = f (n1 , . . . , ns ). a ,...,as (n1 ,...,ns )∈Si r+1 1 ,...,ir Then F (n) = G(n) + s−1  X r=1 X 16i1 <···0 JIi1 · · · Iir ; , a n a r+1 JIi1 1 · · · Iirir Iiu1 · · · Iiur Iir+1 · · · Iiass M . 00 6 dim F 0 . FA0 -modules and dim FM M   L (Ii1 · · · Iir )n M 0 ` = dim . n≥0 m(Ii1 · · · Iir )n M 0 By Lemma 2.7, ` 6 `. By the inductive assumption applied to r < s, 0 = `0 + r − 1 6 ` + r − 1. dim FM From the above facts and note that r < s, It is clear that and 00 FM n Ii1 1 · · · Iirir r+1 Ii1 1 · · · Iirir Iiu1 · · · Iiur Iir+1 · · · Iiass M M (2) n Ii1 1 · · · Iirir M ni1 ,...,nir ≥0 0 FM  a ,...,as Fi1r+1 (n) ,...,ir ar+1 ,...,as u X = (ni1 − u, . . . , nir − u) a ,...,as fi1r+1 (mi1 , . . . , mir ) ,...,ir = lA mi1 +···+mir =n−ru−v; mi1 ,...,mir >0  00  (FA0 + )n−ru−v FM . 00 (FA0 + )n−ru−v+1 FM a ,...,a s 1 6 i1 < · · · < ir 6 s and ar+1 , . . . , as < u (r = 1, . . . , s − 1), then Fi1r+1 (n) ,...,ir ar+1 ,...,as 00 (n) is a polynomial of degree dim FM − 1 for large n. By (3), for large n then Fi ,...,i r 1 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 So for all 1 (` + s − 2)! Thus X [d ] EJ (I1 1 , . . . , Is[ds ] ; M ). d1 + ··· + ds = `−1 dim FM (J, I1 , . . . , Is ) = ` + s − 1 and X e(FM ) = [d ] EJ (I1 1 , . . . , Is[ds ] ; M ). d1 + ··· + ds = d−1 The proof of Theorem 3.3 is complete. I1 , . . . , Is are m-primaryideals, it is  easily seen L I + AnnM I nM  = ht = d > 0, ` = dim n>0 mI n M AnnM In the case where where result. I = I1 · · · Is . that As an immediate consequence of Theorem 3.3, we have the following Corollary 3.4. Let J, I1 , . . . , Is and e(FM ) = be m-primary. X Then dim FM (J, I1 , . . . , Is ) = d + s − 1 [d ] EJ (I1 1 , . . . , Is[ds ] ; M ). d1 + ··· + ds = d−1 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 m-ngu¶n (A, m) l mët I1 , . . . , Is l sì, v nh àa ph÷ìng Noether vîi i¶an cüc ¤i nhúng i¶an cõa A; M l mët A-mæ m, J l mët i¶an 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 ) = theo c¡c sè bëi trën. L n1 ,...,ns >0 I1n1 · · · Isns M JI1n1 · · · Isns M 31

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.