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.
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30
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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