The $j$-Multiplicity of Monomial Ideals

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by  Jack Jeffries

THE j-MULTIPLICITY OF MONOMIAL IDEALS
JACKJEFFRIESANDJONATHANMONTAN˜O
3 Abstract. We prove a characterization of the j-multiplicity of a monomial
1 ideal as the normalized volume of a polytopal complex. Our result is an ex-
0 tensionofTessier’svolume-theoreticinterpretationoftheHilbert-Samuelmul-
2 tiplicity for m-primary monomial ideals. We also give a description of the
n ε-multiplicityofamonomialidealintermsofthevolumeofaregion.
a
J

7
2 1. Introduction

] The j-multiplicity was defined in 1993 by Achilles and Manaresi in [1] as a
C
generalizationoftheHilbert-SamuelmultiplicityforarbitraryidealsinaNoetherian
A
localring. SeveralresultsontheHilbert-Samuelmultiplicityhavebeensuccessfully
h. extended to more general classes of ideals using the j-multiplicity, for example [8],
t [14],and[4]. Themainresultofthispapermaybeviewedasoneoftheseextensions.
a
Let(R,m,k)beaNoetherianlocalringofdimensiond,andI ⊂Ranideal. The
m
j-multiplicity of I is defined as the limit
[
v2 j(I)=nl→im∞(dn−d−11)!λR(cid:0)H0m(In/In+1)(cid:1).
9 There have been previous approaches for computing the j-multiplicity. For ex-
1 ample, in [1] and [21] it is proven that if k is infinite, then for general elements
4 a ,...,a in I, and α=(a ,...,a ), we have
1 d 1 d−1
1
. j(I)=λ (cid:0)R/((α: I∞)+a R)(cid:1).
2 R R d
1 This formula is applied to compute specific examples in [12].
2
Let R denote now the polynomial ring k[x ,...,x ] over the field k, m the ho-
1 1 d
mogeneous maximal ideal (x ,...,x ), and I a monomial ideal of R. The Newton
: 1 d
v polyhedronofI istheconvexhullofthepointsinRd thatcorrespondtomonomials
i
X in I, which we will denote by conv(I). In this paper we generalize the classical
result that describes the Hilbert-Samuel multiplicity of an m-primary ideal as the
r
a normalized volume of the complement of its Newton polyhedron in Rd , see [17].
(cid:62)0
If I is not m-primary, the complement of conv(I) is infinite, but we can define the
analogue of this region in the general case by considering the truncated cone from
the origin to the union of the bounded faces of conv(I). This truncated cone will
be denoted by pyr(I). With this notation, we can state our main result:
Theorem 3.2. Let I ⊂R be a monomial ideal. Then j(I)=d!vol(pyr(I)).
EarlierunpublishedworkofJ.Validashtiobtainsthisformulaindimensiontwo.
The rest of the paper is organized as follows: In the second section we set up
the notation and also present some results that will be used in the proof of the
main theorem. The third section will include the proof of Theorem 3.2. In the
1
2 JACKJEFFRIESANDJONATHANMONTAN˜O
fourth section we provide an extension of this result to pointed normal affine toric
varieties. In the fifth section we will apply our characterization of the saturation
of a monomial ideal in R in Lemma 2.2 to give a geometric description of the
ε-multiplicity. The paper ends with some examples in a sixth section.
2. Preliminaries
LetR=k[x ,...,x ]beapolynomialringoverafieldk andm=(x ,...,x )its
1 d 1 d
homogeneousmaximalideal. LetI beamonomialidealofRminimallygeneratedby
xv1,xv2,...,xvn wherevi =(vi,1,...,vi,d)andxvi =xv1i,1···xvdi,d. Foramonomial
ideal L in R we denote by Γ(L) the set of lattice points in Rd corresponding to the
exponents. Additionally, if L ⊇ L are monomial ideals, we will write Γ(L /L )
1 2 1 2
for Γ(L )\Γ(L ).
1 2
We denote by conv(I) the Newton polyhedron of I, that is:
conv(I):=conv(v ,...,v )+Rd ,
1 n (cid:62)0
where + denotes the Minkowski sum. It is worth noting that the collection of
boundedfacetsoftheNewtonpolyhedronisnotconvex,andthusisnotapolytope,
but rather has the structure of a polytopal complex. Notice also that conv(I) =
conv(Γ(I)). Since every polyhedron is defined by the intersection of finitely many
closed half spaces, we can define H ={x∈Rn |(cid:104)x,b (cid:105)=c }, with b ∈Qd, c ∈Q
i i i i i
for i=1,...,w to be the supporting hyperplanes of conv(I) such that
conv(I)=H+∩H+∩···∩H+,
1 2 w
where H+ = {x ∈ Rn | (cid:104)x,b (cid:105) (cid:62) c }. Let F = H ∩conv(I) for i = 1,...,h
i i i i i
be the facets of conv(I). We will assume that H ,...,H , are the hyperplanes
1 u
corresponding to unbounded facets.
It can be shown that all the vector b have nonnegative components, and that
i
b ∈ Rd if and only if F is a bounded facet, as in [15, Lemma 1.1]. This forces
i >0 i
the c to be nonnegative, and in fact positive in the case of a bounded facet.
i
Wedenotebyvert(I)thesetofverticesofconv(I),andsetbd(I)=(cid:83)w F for
i=u+1 i
theunionoftheboundedfacetsofconv(I). IfP isapolytope,wewillwritepyr(P)
for conv(P,0), the truncated cone, or pyramid, over P. By abuse of notation, we
willwritepyr(I)for(cid:83)w pyr(F ). Notethatthemonomialscorrespondingtothe
i=u+1 i
points in vert(I) are part of the set of minimal generators of I, so we will assume
vert(I)={v ,...,v } for some 1(cid:54)s(cid:54)n. We will also find it convenient to define
1 s
the nth cone section of a polytope P as
(cid:0) (cid:1) (cid:0) (cid:1)
cone (P):= (n+1)pyr(P)\(n+1)P \ npyr(P)\nP ,
n
(cid:83)
which we may alternatively write as sP. We again abuse notation by
n(cid:54)s<n+1
writing cone (I) for (cid:83)w cone (F ).
n i=u+1 n i
For the monomial ideal I = (xy5,x2y3,x3y2), in Figure 1, on the left we mark
vert(I) with black dots, bd(I) with dark red lines, and the unbounded facets with
pink lines. In this example, the green region with its boundary forms pyr(I), and
the yellow region with its boundary forms conv(I). In the graph on the right, we
shade cone (I) in purple, where the top dotted segments of the boundary are not
2
included.
The following description of the faces of the Newton polyhedron will be useful.
THE j-MULTIPLICITY OF MONOMIAL IDEALS 3
Figure 1. Various regions for the monomial ideal I =(xy5,x2y3,x3y2)
Lemma2.1. [15,Lemma3.1]LetF beafaceofconv(I)withsupportinghyperplane
H ={x∈Rn |(cid:104)x,b(cid:105)=c}. Then F ∩vert(I)={v ,...,v } is non-empty, and
i1 ir
(cid:88)
F =conv(vi1,...,vir)+ R(cid:62)0ej
j:bj=0
where e is the unit vector with nonzero jth component.
j
Recall that for any submodule N of an R-module M, the saturation of N, de-
noted (N : m∞), is the set of elements a in M for which there exists n∈N such
M
thatamn ∈N. Thezerothlocalcohomologymodule ofM isdefinedtobe(0: m∞)
M
and is denoted by H0(M). Notice that in general, H0(M/N)=(N : m∞)/N. If
m m M
I is a monomial ideal, then (I : m∞) is also monomial.
R
The integral closure of an arbitrary ideal J is the set of elements x in R that
satisfy an integral relation xn + a xn−1 + ··· + a x + a where a ∈ Ji for
1 n−1 n i
i=1,...,n. ItisdenotedbyJ anditisanideal. Formonomialideals,itispossible
togiveageometricdescriptionoftheintegralclosure, namelyΓ(I)=Zd∩conv(I),
i.e., conv(I)=conv(I); see [19, Proposition 7.25].
Proposition 2.2. Γ(I : m∞)=H+∩...∩H+∩Zd .
R 1 u (cid:62)0
Proof. Let v ∈ H1+∩···∩Hu+; then (cid:104)v,bi(cid:105) (cid:62) ci for i = 1,...,u. For t ∈ R(cid:62)0, one
has
(cid:104)v+te ,b (cid:105)(cid:62)c +tb (cid:62)c
j i i i,j i
forj =1,...,dandi=1,...,u. Also, ifu+1(cid:54)i(cid:54)w thenalltheentriesofb are
i
positive, so we also have that (cid:104)v+te ,b (cid:105)(cid:62)c for t(cid:29)0. We conclude xtxv ∈I for
j i i j
j =1,...,d and t(cid:29)0, that is v ∈Γ(I : m∞).
R
Conversely, if v ∈ Γ(I : m∞), then v +te ∈ conv(I) ⊂ H+ ∩...∩H+ for
R j 1 u
j =1,...,dandt(cid:29)0. Now, supposev (cid:54)∈H+ forsome1(cid:54)i(cid:54)u, then(cid:104)v,b (cid:105)<c .
i i i
By Lemma 2.1, since F is an unbounded facet, we can pick j such that b = 0,
i i,j
and hence (cid:104)v+te ,b (cid:105)=(cid:104)v,b (cid:105)<c for every t∈R, which is a contradiction. (cid:3)
j i i i
4 JACKJEFFRIESANDJONATHANMONTAN˜O
Recallthattheanalytic spreadofanidealI,denotedbyl(I),isdefinedtobethe
dimension of its special fiber ring F(I) = gr (R)⊗ R/m = (cid:76)∞ In/mIn, where
I R n=0
gr (R)istheassociatedgradedalgebraofI, i.e., gr (R)=(cid:76)∞ In/In+1. Wewill
I I n=0
say that I has maximal analytic spread if l(I)=dim(R).
Lemma 2.3. Γ(I : m∞)⊂conv(I)∪pyr(I).
R
Proof. The result follows immediately by Proposition 2.2 if conv(I) does not have
bounded facets. We will assume that u<w.
Let v be a nonzero vector in Zd \ (pyr(I) ∪ conv(I)). We will proceed by
(cid:62)0
contradiction. Supposev ∈Γ(I : m∞),thenv ∈H+∩···∩H+byProposition2.2.
R 1 u
Notethatsinceb haspositiveentriesforeachi=u+1,...,w,wehave(cid:104)v,b (cid:105)=(cid:54) 0.
i i
For each u+1 (cid:54) i (cid:54) w we can find a real number t such that t v ∈ H , each
i i i
of which is positive because c > 0. Suppose, without loss of generality, that
i
t =max{t ,...,t }. Sincev (cid:54)∈H+ forsomeu+1(cid:54)i(cid:54)w wehave(cid:104)v,b (cid:105)<c ,
w u+1 w i i i
then t >1, and so t >1.
i w
Now, (cid:104)t v,b (cid:105)(cid:62)(cid:104)t v,b (cid:105)=c for i=u+1,...,w, so
w i i i i
t v ∈H+∩H+∩...∩H+ ∩H =F .
w 1 2 w−1 w w
Then we have v ∈pyr(I) which is a contradiction. (cid:3)
Remark 2.4. By [15, Corollary 3.4], nconv(I) = conv(In), npyr(I) = pyr(In),
and nbd(I)=bd(In) for every n(cid:62)1. It follows from this and the previous lemma
that Γ(cid:0)(In+1 : m∞)/In+1(cid:1)⊆cone (I).
In n
Inthefollowinglemma,wewillusethenotionoftheHausdorff distance between
compact sets A and B in Rd, which is defined as
ρ(A,B):=inf{λ(cid:62)0|A⊆B+λU, B ⊆A+λU},
where U is the unit ball. We will use a related notion for polytopes: for a convex
polytopeP =conv(v ,...,v )inRd,wewillsaythatanotherconvexpolytopewith
1 t
t vertices P(cid:48) =conv(v(cid:48),...,v(cid:48)) is an ε-shaking of P if |v −v(cid:48)|<ε for all j.
1 t j j
Lemma2.5. FixP =conv(v1,...,vt)inRd. Let(P1(n),...,Ps(n))n∈N beasequence
of s-tuples of polytopes such that each P(n) is a (1/n)-shaking of P. Then,
j
(cid:32) s (cid:33)
lim vol P ∩ (cid:92)P(n) =vol(P).
i
n→∞
i=1
Proof. Let P(cid:48) =conv(v(cid:48),...,v(cid:48)) be an ε-shaking of P, and write
1 t
P(cid:48)(cid:113)P =conv(v(cid:48),...,v(cid:48),v ,...,v ).
1 t 1 t
Note that P(cid:48)∪P ⊆P(cid:48)(cid:113)P. Also, ρ(P,P (cid:113)P(cid:48))<ε, since for q ∈P (cid:113)P(cid:48), we may
write q =(cid:80)λ v +(cid:80)λ(cid:48)v(cid:48) with (cid:80)λ +(cid:80)λ(cid:48) =1, and
j j j j i i
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
|q− λ v − λ(cid:48)v | = | λ(cid:48)v(cid:48) − λ(cid:48)v | (cid:54) λ(cid:48)|v(cid:48) −v | < ε,
j j j j j j j j j j j
where (cid:80)λ v +(cid:80)λ(cid:48)v ∈P. Similarly ρ(P,P(cid:48))<ε.
j j j j
THE j-MULTIPLICITY OF MONOMIAL IDEALS 5
We have
s s
0(cid:54)vol(P)−vol(P ∩ (cid:92)P(n))(cid:54)(cid:88)(cid:0)vol(P)−vol(P ∩P(n))(cid:1)
i i
i=1 i=1
s
=(cid:88)(cid:0)vol(P ∪P(n))−vol(P(n))(cid:1)
i i
i=1
s
(cid:54)(cid:88)(cid:0)vol(P (cid:113)P(n))−vol(P(n))(cid:1).
i i
i=1
Then, by continuity of volume with respect to Hausdorff distance, see [20, The-
orem 6.2.17], we have that vol(P(n)) and vol(P(cid:113)P(n)) both converge to vol(P) as
i i
n→∞. Thus, vol(P ∩(cid:84)s P(n))→vol(P) as n→∞, as required. (cid:3)
i=1 i
Recall that the Ehrhart function of a polytope P ⊂Rd is defined as
E (n):=#(Zd∩nP).
P
Ehrhart [7] showed that if the vertices of P have integer coordinates, E (n) is a
P
polynomial of degree dim(P) with leading coefficient equal to the relative volume
of P (cf., [11, Chapter 12]). Recall that a function f: N → Z is called a quasi-
polynomial if there is an m ∈ N and polynomials f ,...,f such that f(n) =
0 m−1
f (n) for all n ∈ N. If P has vertices in Qd, then the Ehrhart function E
(nmodm) P
is a quasi-polynomial. We will employ a stengthening of this fact.
Proposition 2.6. Let P ⊂ Rd be a (d−1)-dimensional polytope with vertices in
Qd. Suppose that the affine span of P contains a point in the integer lattice Zd.
Then #(Zd ∩cone (P)) as a function of n is given by a quasi-polynomial of the
n
form
p(n)=dvol(pyr(P))nd−1+O(nd−2).
Proof. Write
#(Zd∩cone (P))=#(cid:18)Zd∩(cid:16)(cid:0)(n+1)pyr(P)\(n+1)P(cid:1)\(cid:0)npyr(P)\nP(cid:1)(cid:17)(cid:19)
n
(cid:0) (cid:1) (cid:0) (cid:1)
= E (n+1)−E (n) − E (n+1)−E (n) .
pyr(P) pyr(P) P P
Byhypothesis,theaffinespanofeach(d−1)-dimensionalfaceofpyr(P)containsa
latticepoint,becausetheaffinespanofP hasalatticepointandeveryother(d−1)-
dimensional face contains 0. A conjecture of Ehrhart [7], proved by McMullen [10]
and Stanley [16] (see also [3] for a proof based on monomial ideal techniques),
confirms that in this situation E and E are quasi-polynomials of the form
pyr(P) P
E (n)=vol(pyr(P))nd+a nd−1+O(nd−2)
pyr(P) d−1
E (n)=b nd−1+O(nd−2),
P d−1
where a and b are constants; specifically, they do not depend on n. The
d−1 d−1
claim follows from the formula above. (cid:3)
Remark 2.7. Wecanapplytheresultofthepreviouspropositiontoeachbounded
facet F of conv(I). Using the definition cone (I) = (cid:83)h cone (F ), we can con-
i n u+1 n i
clude that #(Zd∩cone (I)) is given by a polynomial of the form
n
p(n)=dvol(pyr(I))nd−1+O(nd−2),
6 JACKJEFFRIESANDJONATHANMONTAN˜O
asintheproposition. Noticethatdouble-countingofthepointsinthefaceincom-
mon of two regions cone (F ) for distinct i does not change the leading coefficient
n i
of the polynomial, because these faces have lower dimension and the degree of the
Ehrhart polynomial of a polytope is equal to its dimension.
3. The j-multiplicity of monomial ideals
In order to be consistent with the definition of j-multiplicity, which is defined
for ideals in a local ring, in this chapter we will consider R = k[x ,...,x ] and
1 d m
I an ideal generated by monomials. All the results of the second section still hold
in this setting, because all the ideals involved are monomial ideals. Moreover, the
analytic spread does not change.
For an R-module M, we can define the j-multiplicity of I with respect to M as
j(I,M)= lim (d−1)!λ (cid:0)H0(InM/In+1M)(cid:1).
n→∞ nd−1 R m
In the case M =R, j(I,R) will be denoted j(I) as in the introduction.
The following proposition shows that we can compute j(I) using the filtration
{In}n∈N. The proof is similar to [8, Proposition 2.10].
Proposition 3.1. Let I be a monomial ideal, then
j(I)= lim (d−1)!λ (cid:0)H0(In/In+1)(cid:1).
n→∞ nd−1 R m
Proof. By [19, Theorem 7.29], In+1 = IIn for n (cid:62) d. From [12, Theorem 3.11],
and the following exact sequence of R-modules
0→Id →R→R/Id →0,
weobtainj(I,R)=j(I,Id)+j(I,R/Id). ButId(R/Id)=0,soj(I,R/Id)=0and
then j(I)=j(I,R)=j(I,Id).
Now,
j(I)=j(I,Id)= lim (d−1)!λ (cid:0)H0(ItId/It+1Id)(cid:1)
t→∞ td−1 R m
= lim (d−1)!λ (cid:0)H0(It+d/It+d+1)(cid:1)
t→∞ td−1 R m
= lim nd−1 (d−1)!λ (cid:0)H0(In/In+1)(cid:1),
n→∞(n−d)d−1 nd−1 R m
nd−1
and the result follows from the equality lim =1. (cid:3)
n→∞(n−d)d−1
Theorem 3.2. Let I ⊂R be a monomial ideal. Then j(I)=d!vol(pyr(I)).
Proof. By Lemma 3.1, we compute
j(I)= lim (d−1)!λ (cid:0)H0(In/In+1)(cid:1).
n→∞ nd−1 R m
THE j-MULTIPLICITY OF MONOMIAL IDEALS 7
Step 1: The proof that j(I)(cid:54)d!vol(pyr(I)):
Recall that H0(In/In+1)=(cid:0)(In+1 :m∞)∩In(cid:1)/In+1, so that
m
λ (cid:0)H0(In/In+1)(cid:1)=#(cid:16)Zd∩(cid:0)Γ(In+1 :m∞)∩conv(In)\conv(In+1)(cid:1)(cid:17)
R m
(cid:54)#(cid:16)Zd∩(cid:0)conv(In+1)∪pyr(In+1)(cid:1)∩conv(In)\conv(In+1)(cid:17)
where the last inequality holds by Lemma 2.3. By [15, Corollary 3.4], we have
nconv(I) = conv(In), npyr(I) = pyr(In), and nbd(I) = bd(In) for every n (cid:62) 1.
Note that bd(I)=conv(I)∩pyr(I); then,
(pyr(In+1)∩conv(In))\conv(In+1)
(cid:0) (cid:1)
= (n+1)pyr(I)∩nconv(I) \(n+1)conv(I)
(cid:0) (cid:1)
= (n+1)pyr(I)∩nconv(I) \(n+1)bd(I)
(cid:0) (cid:1) (cid:0) (cid:1)
= (n+1)pyr(I)\(n+1)bd(I) \ npyr(I)\nbd(I) =cone (I).
n
It follows that λ (cid:0)H0(In/In+1)(cid:1)(cid:54)#(Zd∩cone (I)). Therefore, by Remark 2.7 ,
R m n
j(I)= lim (d−1)!λ (cid:0)H0(In/In+1)(cid:1)(cid:54)d!vol(pyr(I)).
n→∞ nd−1 R m
Step 2: The proof that j(I)(cid:62)d!vol(pyr(I)):
Step 2a: Reduction to the case of an ideal corresponding to a single facet:
Firstweclaimthatitsufficestoverifytheinequalityforamonomialidealwhose
Newton polyhedron has a single bounded facet. Indeed, if the inequality holds
for such ideals, write J ,...,J ⊂ I for the monomial ideals corresponding to the
1 t
bounded facets of I and P ,...,P for the corresponding facets, so that we have
1 t
(cid:83)
bd(I)= P and bd(J )=P . Then since we have
i i i i
t
(cid:91)Γ(cid:0)(Jn+1 : m∞)/Jn+1(cid:1) ⊆ Γ(cid:0)(In+1 : m∞)/In+1(cid:1),
i Jn i In
i
i=1
and
t
(cid:91)
cone (P )=cone (I),
n i n
i=1
we have a containment
t t
cone (I)\Γ(cid:0)(In+1 : m∞)/In+1(cid:1)⊆ (cid:91)cone (P )\ (cid:91)Γ(cid:0)(Jn+1 : m∞)/Jn+1(cid:1)
n In n i i Jn i
i
i=1 i=1
t
⊆ (cid:91)(pyrn(P )\Γ(cid:0)(Jn+1 : m∞)/Jn+1)(cid:1).
i i Jn i
i
i=1
Notice #Γ(cid:0)(Jn+1 : m∞)/Jn+1(cid:1)=λ (cid:0)H0(Jn/Jn+1)(cid:1) for any ideal J, so
Jn R m
#(cone (I)∩Zd)−λ (cid:0)H0(In/In+1)(cid:1)
n R m
t
(cid:54)(cid:88)(cid:16)#(cone (P )∩Zd)−λ (cid:0)H0(Jn/Jn+1)(cid:1)(cid:17).
n i R m i i
i=1
8 JACKJEFFRIESANDJONATHANMONTAN˜O
Thus, if the claimed inequality holds for each J , we have that
i
lim (d−1)!(cid:16)#(cid:0)cone (I)∩Zd(cid:1)−λ (cid:0)H0(In/In+1)(cid:1)(cid:17)
n→∞ nd−1 n R m
(cid:18) (cid:19)
(cid:54)(cid:88) lim (d−1)!#(cid:0)cone (P )∩Zd(cid:1)− lim (d−1)!λ (cid:0)H0(Jn/Jn+1)(cid:1)
n→∞ nd−1 n i n→∞ nd−1 R m i i
i
=(cid:88)(cid:0)d!vol(pyr(J ))−j(J )(cid:1)(cid:54)0.
i i
i
It follows that
j(I)= lim (d−1)!λ (cid:0)H0(In/In+1)(cid:1)
n→∞ nd−1 R m
(cid:62) lim (d−1)!#(cid:0)cone (I)∩Zd(cid:1)=d!vol(pyr(I)),
n→∞ nd−1 n
where the last equality follows from Remark 2.7.
We subsequently assume that the Newton polyhedron of I has a single bounded
facet P.
Step 2b: Descriptionofarationalpolytopecontainingpointscontributingtoj(I):
LetP bebd(I)andletH betheaffine(d−1)-planespannedbyP. Let(cid:104)x,b(cid:105)=c
be a defining equation for H. Recall that each entry of b is positive for a bounded
face, so that after rescaling b, we may assume that c=1, with each b >0.
j
We now describe a region R ⊂ cone (P) such that for any α ∈ Zd ∩ R ,
n n n
xα ∈(In+1 : m∞). Note that xα ∈(In+1 : x∞) if and only if α is in the image
In R i
of π , the projection in the e direction onto (cid:104)α,b(cid:105)H. That is,
i i
Γ(In+1 : x∞)∩(cid:104)α,b(cid:105)H =Zd∩π (cid:0)(n+1)P(cid:1).
R i i
Let P =conv(v ,...,v ). Then
1 t
(cid:0) (cid:1) (n+1)−(cid:104)α,b(cid:105)
π (n+1)v =(n+1)v − e ,
i j j (cid:104)e ,b(cid:105) i
i
so that
(cid:0) (cid:1) (cid:16) (n+1)−(cid:104)α,b(cid:105) (n+1)−(cid:104)α,b(cid:105) (cid:17)
π (n+1)P =conv (n+1)v − e ,...,(n+1)v − e
i 1 (cid:104)e ,b(cid:105) i t (cid:104)e ,b(cid:105) i
i i
Since each b >0, this is well-defined. Now,
j
d
(cid:92)
(In+1 : m∞)=In∩ (In+1 : x∞)
In R i
i=1
We define a region
d
Rn :=conen(P)∩ (cid:92)(cid:0)(n+1)P +R(cid:54)0ei(cid:1)
i=1
sothat,bytheabove,R ∩ZdconsistsofallthepointsαinΓ(cid:0)(In+1 : m∞)/In+1(cid:1).
n In
We remark that for n(cid:54)s<(n+1),
s s
R ∩sH ⊇ sH ∩(R ∩nH +Rd ) ⊇ (R ∩nH) = (R ∩nP)
n n (cid:62)0 n n n n
THE j-MULTIPLICITY OF MONOMIAL IDEALS 9
sothatifwesetτ = 1(R ∩nP),thencone (τ )⊂R . Notethatτ hasvertices
n n n n n n n
in Qd.
We also claim that τ ⊂τ . We have α∈τ if and only if
n n+1 n
n 1 1
α+ e ∈P
n+1 n+1(cid:104)e ,b(cid:105) i
i
for all i. By convexity of P since α ∈ P, if λα+(1−λ) 1 e ∈ P then also
(cid:104)ei,b(cid:105) i
λ(cid:48)α+(1−λ(cid:48)) 1 e ∈P for 0(cid:54)λ(cid:54)λ(cid:48) (cid:54)1. In particular,
(cid:104)ei,b(cid:105) i
n+1 1 1
α+ e ∈P
n+2 n+2(cid:104)e ,b(cid:105) i
i
for all i, so that α∈τ . It follows by induction that τ ⊂τ for n(cid:54)n(cid:48).
n+1 n n(cid:48)
Step 2c: Using pyr(τ ) to give a lower bound:
n
Consider the distance between a vertex of nP and the corresponding vertex of
(cid:0) (cid:1)
π (n+1)P . We compute this distance as
i
1 1
|nv −((n+1)v − e )|=|v − e |,
j j (cid:104)e ,b(cid:105) i j (cid:104)e ,b(cid:105) i
i i
which is bounded above uniformly in n by L := max {|v − 1 e |}. Then
i,j j (cid:104)ei,b(cid:105) i
nτ =R ∩nP is the intersection of (d+1) many polytopes, each of which is the
n n
convexhulloftpointsv(cid:48),...,v(cid:48) suchthat|v −v(cid:48)|<Lforallj. Thatis,eachsuch
1 t j j
polytope is an L-shaking of nP in the sense of Lemma 2.5. Dividing through by n
we see that τ is the intersection of (d+1) many polytopes that are all L-shakings
n n
of the polytope P.
Given 0 < c < 1, we may now apply Lemma 2.5 in the affine subspace H. We
obtain, for a sufficiently large M, an τ such that vol(τ ) (cid:62) cvol(P) in H, and
M M
hence vol(pyr(τ ))(cid:62)cvol(pyr(P)).
M
For n>M, we have from the previous step that τ ⊇τ , so
n M
cone (τ )⊆cone (τ )⊆R .
n M n n n
Thus, if α∈Zd∩cone (τ ), then
n M
xα ∈(In+1 : m∞)/In+1 =H0(In/In+1).
In m
Thus,
j(I)= lim (d−1)!λ (cid:0)H0(In/In+1)(cid:1)
n→∞ nd−1 R m
(cid:62) lim (d−1)!#(cid:0)Zd∩cone (τ )(cid:1)=d!vol(pyr(τ )),
n→∞ nd−1 n M M
where the last equality follows from Proposition 2.6. Therefore, for all c < 1, we
have the inequality j(I)(cid:62)c(d!vol(pyr(P))), so
j(I)(cid:62)d!vol(pyr(P))=d!vol(pyr(I)),
as required. (cid:3)
Remark 3.3. If I is an m-primary monomial ideal, the j-multiplicity is equal
to the Hilbert-Samuel multiplicity, and pyr(I) is the complement of the Newton
polyhedron in Rd . In this way, Theorem 3.2 agrees with Tessier’s result for m-
(cid:62)0
primary monomial ideals.
10 JACKJEFFRIESANDJONATHANMONTAN˜O
4. Normal Affine Semigroup Rings
We next record that, with slight modifications, we can use the same proof to
establish a similar result for a wider class of rings. In this section, we state this
generalization, and describe necessary modifications to the argument.
By an affine semigroup ring, we mean a ring A = k[Q] that has as a k vector
space basis {xq|q ∈ Q}, where Q is a subsemigroup of Zd (with the operation
+), and multiplication given by xq1xq2 = xq1+q2. We denote by mA its maximal
homogeneous ideal. If A is a normal ring, then there is a cone σ ⊆Rd with finitely
many extremal rays, each of which contains a lattice point (σ is a rational cone)
and such that A ∼= k[Zd∩σ], see [11, Chapters 7 and 10]. We assume henceforth
that A is presented in this form. Additionally, suppose that σ is pointed, i.e., that
it contains no nontrivial linear subspace of Rd, and that dim(σ)=d.
Set r ,...,r to be ray generators for σ, i.e., minimal lattice points along the
1 s
extremal rays of σ. Let I be a monomial ideal of R minimally generated by
xv1,xv2,...,xvn. In this context, we define
conv(I):=conv(v ,...,v )+σ,
1 n
and, as in section two, H = {x ∈ Rn | (cid:104)x,b (cid:105) = c }, with b ∈ Qd, c ∈ Q for
i i i i i
i=1,...,w to be the supporting hyperplanes of conv(I) so that
conv(I)=H+∩H+∩···∩H+,
1 2 w
where H+ = {x ∈ Rn | (cid:104)x,b (cid:105) (cid:62) c }. We again assume that H ,...,H , are the
i i i 1 u
hyperplanes corresponding to unbounded facets. We retain the other definitions,
e.g., pyr and cone , from the preliminary section. Note that since we have chosen
n
an embedding of our semigroup in Zd ⊂ Rd, it makes sense to talk about volume.
Our main result from the previous section holds in this context:
Theorem 4.1. Let A=k[Zd∩σ] , where σ is a d-dimensional pointed rational
mA
cone. Let I ⊂A be a monomial ideal. Then j(I)=d!vol(pyr(I)).
Proof. The proof of Theorem 3.2 applies, after some slight changes:
It is easy to see that the inequalities (cid:104)r ,b (cid:105)(cid:62)0 hold for all i,j, and (cid:104)r ,b (cid:105)>0
j i j i
for u+1(cid:54)i(cid:54)w and all j, as in [15, Lemma 1.1]. Apply these inequalities in the
proof of Lemma 2.3 mutatis mutandis to obtain the same conclusion. Step 1 in the
proof of Theorem 3.2 follows.
To prove the other inequality in this setting, first note that for any monomial
ideal J, we have α ∈ Γ(J :A m∞A) if and only if there exist aj ∈ R(cid:62)0 such that
α+a r ∈Γ(J) for all j. Thus, for a monomial ideal I with a single bounded face
j j
P, we may define the region
m
Rn :=conen(P)∩ (cid:92) (cid:0)(n+1)P +R(cid:54)0rj(cid:1)
j=1
so that Γ(cid:0)(In+1 : m∞)/In+1(cid:1) = Zd ∩ R . This region has all of the salient
In n
properties from the case of the polynomial ring R (i.e., R is a rational polytope
n
such that R ∩sP ⊇ s(R ∩nP) for n (cid:54) s < (n+1), and we employ this to
n n n
complete the proof of Step 2 as before. (cid:3)
Remark4.2. Thej-multiplicityofanidealI isgreaterthanzeroifandonlyifI has
maximalanalyticspread([12, Lemma3.1]). ThenitfollowsfromTheorems3.2and
4.1 that a monomial ideal of a normal affine semigroup ring has maximal analytic