arXiv:1302.0312v2 [] 20 Sep 2013

ON THE INTERSECTION ALGEBRA OF PRINCIPAL IDEALS
SARA MALEC
Abstract. We study the finite generation of the intersection algebra of two principal
ideals I and J in a unique factorization domain R. We provide an algorithm that produces
a list of generators of this algebra over R. In the special case that R is a polynomial
ring, this algorithm has been implemented in the commutative algebra software system
Macaulay2. A new class of algebras, called fan algebras, is introduced.

1. Introduction
In this paper, we study the intersection of powers of two ideals in a commutative Noetherian ring. This is achieved by looking at the structure called the intersection algebra, a
recent concept, which is associated to the two ideals.
The purpose of the paper is to study the finite generation of this algebra, and to show
that it holds in the particular important case of principal ideals in a unique factorization
domain (UFD). In the general case, not much is known about the intersection algebra, and
there are many questions that can be asked. Various aspects of the intersection algebra
have been studied by J. B. Fields in [3, 4]. There, he proved several interesting things, including the finite generation of the intersection algebra of two monomial ideals in the power
series ring over a field. He also studied the relationship between the finite generation of the
intersection algebra and the polynomial behavior of a certain function involving lengths of
Tors. It is interesting to note that this algebra is not always finitely generated, as shown
by Fields. The finite generation of the intersection algebra has also appeared in the work
of Ciupercă, Enescu, and Spiroff in [1] in the context of asymptotic growth powers of ideals.
We will start with the definition of the intersection algebra. Throughout this paper, R
will be a commutative Noetherian ring.
Definition 1.1. Let
LR be ar rings with two ideals I and J. Then the intersection algebra
of I and J is B =
r,s∈N I ∩ J . If we introduce two indexing variables u and v, then
P
r
BR (I, J) = r,s∈N I ∩ J s ur v s ⊆ R[u, v]. When R, I and J are clear from context, we will
simply denote this as B. We will often think of B as a subring of R[u, v], where there is a
natural N2 -grading on monomials b ∈ B given by deg(b) = (r, s) ∈ N2 . If this algebra is
finitely generated over R, we say that I and J have finite intersection algebra.
Example 1.2. If R = k[x, y], I = (x2 y) and J = (xy 3 ), then an example of an element in
B is 2 + 3x5 y 9 u2 v 3 + x10 y 15 u4 v , since 2 ∈ I 0 ∩ J 0 = k, x5 y 9 u2 v 3 ∈ I 2 ∩ J 3 u2 v 3 = (x4 y 2 ) ∩
(x3 y 9)u2 v 3 = (x4 y 9 )u2 v 3 , and x10 y 15u4 v ∈ I 4 ∩ Ju4 v = (x8 y 4) ∩ (xy 3 )u4 v = (x8 y 4)u4 v.
1

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SARA MALEC

We remark that the intersection algebra has connections to the double Rees algebra
R[Iu, Jv], although in practice they can be very different. This relationship is significant
due to the importance of the Rees algebra, but the two objects behave differently. The
source for the different behaviour lies in the obvious fact that the intersection I r ∩ J s is
harder to predict than I r J s as r and s vary. These differences in behavior are of great
interest and should be further explored.
In this paper, we produce an algorithm that gives a set of generators for the algebra for
two principal ideals in a UFD, and we implement the algorithm in Macaulay2 for the case
of principal monomial ideals in a polynomial ring over a field.
The finite generation of an N2 -graded algebra, such as the intersection algebra, can be
rephrased in the following way:
L
Proposition 1.3. Let B = r,s∈N Br,s be an N2 -graded algebra over a ring R. Then B is
finitely generated if and only if there exists an N ∈ N such that for every r, s ∈ N with
r, s > N,
Br,s =

N
X Y
I

j,k=0

i

jk
,
Bj,k

where I is the set of all N × N matrices I = (ijk ) such that
r=

N
X

jijk and s =

N
X

kijk .

j,k=0

j,k=0

Proof. If B is finitely generated over R, there exists an N ∈ N such that all the generators
for B come from components Br,s with r, s ≤ N. Let b ∈ Br,s with r, s > N. So b can be
written as a polynomial in the generators with coefficients in R, in other words
b∈

N
X Y

finite j,k=0

i

jk
Bj,k
.

Also, the right hand side must be in
Pthe (r, s)-graded
P component of B: i.e. the sum can
only run over all j, k such that r =
jijk and s =
kijk and j and k both go from 0 to
N. So b ∈ Br,s must be in the right hand side above as claimed. For the other inclusion,
note again that the degrees match up:
N
Y

j,k=0

P

i

jk
⊂ Br,s
Bj,k

P

as long as r =
jijk and s =
kijk , so obviously sums of such expressions are also
included in Br,s . The other direction is obvious.

2. The UFD Case
Theorem 2.1. If R is a UFD and I and J are principal ideals, then B is finitely generated
as an algebra over R.

ON THE INTERSECTION ALGEBRA OF PRINCIPAL IDEALS

3

The proof for the finite generation of B relies heavily on semigroup theory. The following
definitions and theorems provide the necessary framework for the proof.
Definition 2.2. A semigroup is a set together with a closed associative binary operation.
A semigroup generalizes a monoid in that it need not contain an identity element. We call
a semigroup an affine semigroup if it isomorphic to a subgroup of Zd for some d. An affine
semigroup is called pointed if it contains the identity, which is the only invertible element
of the semigroup.
Definition 2.3. A polyhedral cone in Rd is the intersection of finitely many closed linear
half-spaces in Rd , each of whose bounding hyperplanes contains the origin. Every polyhedral cone C is finitely generated, i.e. there exist c1 , . . . , cr ∈ Rd with
C = {λ1 c1 + · · · + λr cr |λ1 , . . . , λr ∈ R≥0 }.
We call the cone C rational if c1 , . . . , cr can be chosen to have rational coordinates, and
C is pointed if C ∩ (−C) = {0}.
We present the following two important results without proof: the full proofs are contained in [2] as Theorem 7.16 and 7.15, respectively.
Theorem 2.4. (Gordan’s Lemma) If C is a rational cone in Rd , then C ∩ A is an affine
semigroup for any subgroup A of Zd .
Theorem 2.5. Any pointed affine semigroup Q has a unique finite minimal generating set
HQ .
Definition 2.6. Let C be a rational pointed cone in Rd , and let Q = C ∩ Zd . Then HQ is
called the Hilbert Basis of the cone C.
We will use these results to provide a list of generators for B. Notice that for any two
strings of numbers
a = {a1 , . . . , an }, b = {b1 , . . . , bn } with ai , bi ∈ N,
we can associate to them a fan of pointed, rational cones in N2 .
Definition 2.7. We will call two such strings of numbers fan ordered if
ai
ai+1
≥
for all i = 1, . . . , n.
bi
bi+1
Assume a and b are fan ordered. Additionally, let an+1 = b0 = 0 and a0 = bn+1 = 1. Then
for all i = 0, . . . , n, let
Ci = {λ1 (bi , ai ) + λ2 (bi+1 , ai+1 )|λ1 , λ2 ∈ R≥0 }.
Let Σa,b be the fan formed by these cones and their faces, and call it the fan of a and b in
N2 . Hence
Σa,b = {Ci |i = 0, . . . , n}.

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SARA MALEC

Then, since each Ci is a pointed rational cone, Qi = Ci ∩ Z2 has a Hilbert Basis, say
HQi = {(ri1 , si1 ), . . . , (rini , sini )}.

Note that any Σa,b partitions all of the first quadrant of R2 into cones, so the collection
{Qi |i = 0, . . . , n + 1} partitions all of N2 as well, so for any (r, s) ∈ N2 , (r, s) ∈ Qi for some
i = 0, . . . , n + 1.
In this paper, we are studying the intersection algebra when I and J are principal, so
the order of the exponents in their exponent vectors does not matter. In general, for any
two strings of numbers a and b, there is a unique way to rearrange them so that they are
fan ordered. So a unique fan can be associated to any two vectors. For the purposes of this
paper, we will assume without loss of generality that the exponent vectors are fan ordered.
Theorem 2.8. Let R be a UFD with principal ideals I = (pa11 · · · pann ) and J = (pb11 · · · pbnn ),
where pi , i = 0, . . . , n, and let Σa,b be the fan associated to a = (a1 , . . . , an ) and b =
(b1 , . . . , bn ). Then B is generated over R by the set
a rij

{p11

ar

b

i+1
· · · pi i ij pi+1

sij

· · · pbnn sij urij v sij |i = 0, . . . , n, j = 1, . . . , ni },

where (rij , sij ) run over the Hilbert basis for each Qi = Ci ∩ Z2 for every Ci ∈ Σa,b.
Proof. Since B has a natural N2 grading, it is enough to consider only homogeneous monomials b ∈ B with deg(b) = (r, s). Then (r, s) ∈ Qi = Ci ∩ Z2 for some Ci ∈ Σa,b . In other
words, r, s ∈ N2 and
ai
s
ai+1
≥ ≥
.
bi
r
bi+1
So ai r ≥ bi s, and by the ordering on the ai and the bi , aj r ≥ bj s for all j < i. Also,
ai+1 r ≤ bi+1 s, and again by the ordering, aj r ≤ bj s for all j > i. So
b ∈ I r ∩ J s ur v s = (pa11 · · · pann )r ∩ (pb11 · · · pbnn )s ur v s
b

s

i+1
· · · pbnn s )ur v s
= (pa11 r · · · piai r · pi+1

ON THE INTERSECTION ALGEBRA OF PRINCIPAL IDEALS
b

5

s

i+1
So b = f · p1a1 r · · · piai r · pi+1
· pbnn s ur v s for some monomial f ∈ R.

Since (r, s) ∈ Qi , P
the pair has a decomposition into a sum
Hilbert basis
Pnelements.
Pnof
ni
i
i
mj sij .
So we have (r, s) = j=1 mj (rij , sij ) with mj ∈ N, and r = j=1 mj rij , s = j=1
Therefore
b

s

i+1
b =f (pa11 r · · · piai r pi+1
· · · pbnn s ur v s )
ni
Y
m (a r )
m (a r ) mj (bi+1 sij )
=f
p1 j 1 ij · · · pi j i ij pi+1
· · · pnmj (bn sij ) umj (rij ) v mj (sij )

=f

j=1
ni
Y

a rij

(p11

j=1

ar

b

i+1
· · · pi i ij pi+1

sij

· · · pbnn sij urij v sij )mj

So b is generated over R by the given finite set as claimed.



Remark 2.9. This theorem extends and refines the main result in [5]
√
Remark 2.10. For any two ideals I and J in R with J ⊂ I, where I is not nilpotent and
∩k I k = (0), define vI (J, m) to be the largest integer n such that J m ⊆ I n and wJ (I, n) to be
the smallest m such that J m ⊆ I n . The two sequences {vI (J, m)/m}m and {wJ (I, n), n}n
have limits lI (J) and LJ (I), respectively. See [6, 7] for related work.
Given two principal ideals I and J in a UFD R whose radicals are equal (i.e. the factorizations of their generators use the same irreducible elements), our procedure to determine
generators also shows that the vectors (b1 , a1 ) and (bn , an ) are related to the pairs of points
(r, s) where I r ⊆ J s (respectively J s ⊆ I r ): notice that C0 is the cone between the y-axis
and the line through the origin with slope a0 /b0 , and for all (r, s) ∈ C0 ∩ N2 , I r ⊆ J s .
Therefore lJ (I) = a0 /b0 . Similarly, Cn , the cone between the x-axis and the line through
the origin with slope an /bn , contains all (r, s) ∈ N2 where J s ⊆ I r , so lJ (I) = an /bn . Then,
since lI (J)LJ (I) = 1, this gives that LJ (I) = a1 /b1 and LI (J) = bn /an as well. This agrees
with the observations of Samuel and Nagata as mentioned in [1].
3. The Polynomial Case
In this section, we will show that in the special case where R is a polynomial ring in
finitely many variables over a field, then the intersection algebra of two principal monomial
ideals is a semigroup ring whose generators can be algorithmically computed.
Definition 3.1. Let k be a field. The semigroup ring k[Q] of a semigroup Q is the k-algebra
with k-basis {ta |a ∈ Q} and multiplication defined by ta · tb = ta+b .
Note that when F = {f1 , . . . , fq } is a collection of monomials in R, k[F ] is equal to
the semigroup ring k[Q], where Q = Nlog(f1 ) + · · · + Nlog(fq ) is the subsemigroup of Nq
generated by log(F ). It is easy to see that multiplying monomials in the semigroup ring
amounts to adding exponent vectors in the semigroup, as in the following example:
We can consider B both as an R-algebra and as a k-algebra, and it is important to keep
in mind which structure one is considering when proving results. While there are important

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SARA MALEC

distinctions between the two, finite generation as an algebra over R is equivalent to finite
generation as an algebra over k.
Theorem 3.2. Let R be a ring that is finitely generated as an algebra over a field k. Then
B is finitely generated as an algebra over R if and only if it is finitely generated as an
algebra over k.
Proof. Let B be finitely generated over k. Then since k ⊂ R, B is automatically finitely
generated over R. Now let
generated over R, say by elements b1 , . . . , bn ∈ B.
PqB be finitely
αi
Then for any b ∈ B, b = i=1 ri bi with ri ∈ R. But R is finitely generated over k, say by
P
P P
β
β
elements k1 , . . . , km , so ri = pj=1 aij kj ij , with aij ∈ k. So b = qi ( pj aij kj ij )bαi i , and B is
finitely generated as an algebra over k by {b1 , . . . , bn , k1 , . . . , km }.

A few definitions are required before stating the main results of this section.

Definition 3.3. Let R = k[x] = k[x1 , . . . , xn ] be the polynomial ring over a field k in n
variables. Let F = {f1 , . . . , fq } be a finite set of distinct monomials in R such that fi 6= 1
for all i. The monomial subring spanned by F is the k-subalgebra
k[F ] = k[f1 , . . . , fq ] ⊂ R.

Definition 3.4. For c ∈ Nn , we set xc = xc11 · · · xcnn . Let f be a monomial in R. The
exponent vector of f = xα is denoted by log(f ) = α ∈ Nn . If F is a collection of monomials
in R, log(F ) denotes the set of exponent vectors of the monomials in F .
Theorem 3.5. If R is a polynomial ring in n variables over k, and I and J are ideals
generated by monomials (i.e. monic products of variables) in R, then B is a semigroup
ring.
Proof. Since I and J are monomial ideals, I r ∩ J s is as well for all r and s. So each (r, s)
component of B is generated by monomials, therefore B is a subring of k[x1 , . . . , xn , u, v]
generated over k by a list of monomials {bi |i ∈ Λ}. Let Q be the semigroup generated by
{log(bi )|i ∈ Λ}. Then B = k[Q], and B is a semigroup ring over k.


Theorem 3.6. Let I = (xa11 · · · xann ) and J = (xb11 · · · xbnn ) be principal ideals in R =
k[x1 , . . . , xn ], and let Σa,b be the fan associated to a = (a1 , . . . , an ) and b = (b1 , . . . , bn ).
Let
Qi = Ci ∩ Z2 for every Ci ∈ Σa,b
and HQi be its Hilbert basis of cardinality ni for all i = 0, . . . , n. Further, let Q be the
subsemigroup in N2 generated by
{(a1 rij , . . . , ai rij , bi+1 sij , . . . , bn sij , rij , sij )|i = 0, . . . , n, j = 1, . . . , ni } ∪ log(x1 , . . . , xn ),

where (rij , sij ) ∈ HQi for every i = 0, . . . n, j = 1, . . . , ni . Then B = k[Q].
Proof. Since R is a UFD, by Thm 2.8, B is generated over R by
a rij

{x11

ar

b

i+1
· · · xi i ij xi+1

sij

· · · xbnn sij urij v sij |i = 0, . . . , n, j = 1, . . . , ni }.

Then, since R is generated as an algebra over k by x1 , . . . , xn , it follows that B ⊂ k[x1 , . . . , xn , u, v]
is generated as an algebra over k by the set
a rij

P = {x1 , . . . , xn , x11

ar

b

i+1
· · · xi i ij xi+1

sij

· · · xbnn sij urij v sij |i = 0, . . . , n, j = 1, . . . , ni }.

ON THE INTERSECTION ALGEBRA OF PRINCIPAL IDEALS

7

This is a set of monomials in k[x1 , . . . , xn , u, v]. Now note that therefore
log(P ) ={(a1 rij , . . . , ai rij , bi+1 sij , . . . , bn sij , rij , sij )|i = 0, . . . , n, j = 1, . . . , ni }
∪ log(x1 , . . . , xn ).

In conclusion, log(P ) = Q and hence B = k[Q].



Example 3.7. Let I = (x5 y 2 ) and J = (x2 y 3). Then a1 = 5, a2 = 2 and b1 = 2, b2 = 3,
and 5/2 ≥ 2/3. Then we have the following cones:
C0 = {λ1 (0, 1) + λ2 (2, 5)|λi ∈ R≥0 }

C1 = {λ1 (2, 5) + λ2 (3, 2)|λi ∈ R≥0 }

C2 = {λ1 (3, 2) + λ2 (1, 0)|λi ∈ R≥0 }

C0 is the wedge of the first quadrant between the y-axis and the vector (2, 5), C1 is the
wedge between (2, 5) and (3, 2), and C3 is the wedge between (3, 2) and (1, 0). It is easy to
see that this fan fills the entire first quadrant. Intersecting these cones with Z2 is equivalent
to only considering the integer lattice points in these cones.
The Hilbert Basis of Q0 = C0 ∩ Z2 is {(0, 1), (1, 3), (2, 5)}, and their corresponding
monomials in B are given by the generators of Br,s for each (r, s):
(0, 1) : (I 0 ∩ J 1 )u = (x2 y 3 )v − generator is x2 y 3v

(1, 3) : (I 1 ∩ J 3 )uv 3 = ((x5 y 2 ) ∩ (x6 y 9))uv 3 = (x6 y 9)uv 3 − generator is x6 y 9 uv 3

(2, 5) : (I 2 ∩ J 5 )u2 v 5 = ((x10 y 4 ) ∩ (x10 y 15))u2 v 5 = (x10 y 15 )u2v 5 − generator is x10 y 15 u2 v 5
Notice that all the generator monomials are of the form xb1 s y b2 s ur v s , with b1 = 2, b2 = 3,
and (r, s) is a Hilbert Basis element, as shown earlier.
The Hilbert Basis of Q1 is {(1, 1), (1, 2), (3, 2), (2, 5)}. In the same way as above, their
monomials are x5 y 3 uv, x5y 6 uv 2 , x15 y 6u3 v 2 , x10 y 15 u2 v 5 , all of which have the form xa1 r y b2 s ur v s
with a1 = 5, b2 = 3 and (r, s) a basis element.
Lastly, the Hilbert Basis of Q2 is {(1, 0), (2, 1), (3, 2)}, which gives rise to generators
x5 y 2u, x10 y 4u2 , x15 y 6 u3 v 2 , all of which look like xa1 r y a2 r ur v s with a1 = 5, a2 = 2.
Notice there are a few redundant generators in this list: those arise from lattice points
that lie on the boundaries of the cones. So B is generated over R by
{x5 y 2 u, x10 y 4 u2 , x15 y 6u3 v 2 , x5 y 3 uv, x5y 6 uv 2 , x2 y 3 v, x6 y 9 uv 3, x10 y 15 u2 v 5 }.

Using this technique, we have written a program in Macaulay2 that will provide the list
of generators of B for any I and J. First it fan orders the exponent vectors, then finds

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SARA MALEC

the Hilbert Basis for each cone that arises from those vectors. Finally, it computes the
corresponding monomial for each basis element. The code is below:
loadPackage "Polyhedra"
--function to get a list of exponent vectors from an ideal I
expList=(I) ->(
flatten exponents first flatten entries gens I
)
algGens=(I,J)->(
B:=(expList(J))_(positions(expList(J),i->i!=0));
A:=(expList(I))_(positions(expList(J),i->i!=0));
L:=sort apply(A,B,(i,j)->i/j);
C:=flatten {0,apply(L,i->numerator i),1};
D:=flatten {1, apply(L,i->denominator i),0};
M:=matrix{C,D};
G:=unique flatten apply (#C-1, i-> hilbertBasis
(posHull submatrix(M,{i,i+1})));
S:=ring I[u,v];
flatten apply(#G,i->((first flatten entries gens
intersect(I^(G#i_(1,0)),J^(G#i_(0,0)))))*u^(G#i_(1,0))*v^(G#i_(0,0)))
)
4. Fan Algebras
The intersection algebra is in fact a specific case of a more general class of algebras that
can be naturally associated to a fan of cones. We will call such objects fan algebras, and
the first result in this section shows that they are finitely generated. First a definition:
Definition 4.1. Given a fan of cones Σa,b , a function f : N2 → N is called fan-linear if it is
nonnegative and linear on each subgroup Qi = Ci ∩ Z2 for each Ci ∈ Σa,b , and subadditive
on all of N2 , i.e.
f (r, s) + f (r ′ , s′ ) ≥ f (r + r ′ , s + s′ ) for all (r, s), (r ′, s′ ) ∈ N2 .

In other words, f (r, s) is a piecewise linear function where

f (r, s) = gi (r, s) when (r, s) ∈ Ci ∩N2 for each i = 0, . . . n, and each gi is linear on Ci ∩N2 .
Note that each piece of f agrees on the faces of the cones, that is gi = gj for every
(r, s) ∈ Ci ∩ Cj ∩ N2 .
Example 4.2. Let a = {1} = b, so Σa,b is the fan defined by

C0 = {λ1 (0, 1) + λ2 (1, 1)|λi ∈ R≥0 }

C1 = {λ1 (1, 1) + λ2 (1, 0)|λi ∈ R≥0 },

and set Qi = Ci ∩ Z2 . Also let

g0 (r, s) = r + 2s if (r, s) ∈ Q0
f=
g1 (r, s) = 2r + s if (r, s) ∈ Q1

ON THE INTERSECTION ALGEBRA OF PRINCIPAL IDEALS

9

Then f is a fan-linear function. It is clearly nonnegative and linear on both Q0 and Q1 .
The function is also subadditive on all of N2 : Let (r, s) ∈ Q0 and (r ′ , s′ ) ∈ Q1 , and say that
(r + r ′ , s + s′ ) ∈ Q0 . Then
f (r, s) + f (r ′, s′ ) = g0 (r, s) + g1 (r ′, s′ ) = r + 2s + 2r ′ + s
f (r + r ′ , s + s′ ) = g0 (r + r ′ , s + s′ ) = r + r ′ + 2(s + s′ )
Comparing the two, we see that
f (r, s) + f (r ′, s′ ) ≥ f (r + r ′ , s + s′ ) whenever r + 2s + 2r ′ + s′ ≥ r + r ′ + 2(s + s′ ),
or equivalently when r ′ ≥ s′ . But that is true, since (r ′ , s′ ) ∈ Q1 . The proof for (r + r ′ , s +
s′ ) ∈ Q1 is similar. The two pieces of f also agree on the boundary between Q0 and Q1 ,
since the intersection of Q0 and Q1 is the ray in N2 where r = s, and
g0 (r, r) = 3r = g1 (r, r).
So f is a fan-linear function.
Theorem 4.3. Let I1 . . . , In be ideals in a domain R and Σa,b be a fan of cones in N2 . Let
f1 , . . . , fn be fan-linear functions. Then the algebra
M f (r,s)
B=
I1 1
· · · Infn (r,s) ur v s
r,s

is finitely generated.
Proof. First notice that the subadditivity of the functions fi guarantees that B is a subalgebra of R[u, v] with the natural grading. Since B has a natural N2 -grading, it is
enough to consider only homogeneous monomials b ∈ B with deg(b) = (r, s). Then
(r, s) ∈ Qi = Ci ∩ Z2 for some Ci ∈ Σa,b . Since Qi is a pointed rational cone, it has
a Hilbert basis
HQi = {(ri1 , si1 ), . . . , (rini , sini )}.
So we can write
ni
X

(r, s) =

mj (rij , sij ).

j=1

Then, since each fk is nonnegative and linear on Qi , we have
fk (r, s) =

ni
X

mj fk (rij , sij )

j=1

for each k = 1, . . . , n. Since R is Noetherian, for each i, there exists a finite set Λi,j,k ⊂ R
such that
f (rij ,sij )

Ik k

= (xk |xk ∈ Λi,j,k ).

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SARA MALEC

So
f (r,s)

b ∈ Br,s = I1 1

Pni

= I1

j=1

· · · Infn (r,s) ur v s

mj f1 (rij ,sij )

m1 f1 (ri1 ,si1 )

Pni

· · · In

j=1

Pni
mj fn (rij ,sij ) Pni m r
j ij
j=1
j=1 mj sij

u

v

mn f1 (rin ,sin )

mn fn (rin ,sin )

i
i
i
i
· · · I1 i
· · · Inm1 fn (ri1 ,si1 ) · · · In i
um1 ri1 · · · umni rini v m1 si1 · · · v mni sini

m1


f1 (rini ,sini )
fn (rini ,sini ) rin sin mni
f1 (ri1 ,si1 )
fn (ri1 ,si1 ) ri1 si1
i
i
.
= I1
· · · In
u v
· · · I1
· · · In
u v

= I1

So B is generated as an algebra over R by the set

{x1 · · · xn urij v sij |(rij , sij ) ∈ HQi , xk ∈ Λi,j,k }.



This result justifies the following definition.
Definition 4.4. Given ideals I1 , . . . , In in a domain R, Σa,b a fan of cones in N2 , and
f1 , . . . , fn are fan-linear functions, we define
M f (r,s)
B(Σa,b , f ) =
I1 1
· · · Infn (r,s) ur v s
r,s

to be the fan algebra of f on Σa,b , where f = (f1 , . . . , fn ).
Remark 4.5. The intersection algebra of two principal ideals I = (pa11 · · · pann ) and J =
(pb11 · · · pbnn ) in a UFD is a special case of a fan algebra. Let Ii = (pi ) and fi = max(rai , sbi )
for each i = 1, . . . , n, and define the fan Σa,b to be the fan associated to a = (a1 , . . . , an )
and b = (b1 , . . . , bn ). Then
M
B(I, J) =
(p1 )max(ra1 ,sb1 ) · · · (pn )max(ran ,sbn ) ur v s .
r,s

This is a fan algebra since the max function is fan-linear: it’s subadditive on all of N2 ,
and linear and nonnegative on each cone, since the faces of each cone in Σa,b are defined
by lines through the origin with slopes ai /bi for each i = 0, . . . , n. So, as in the proof of
Theorem 2.8, for any pair (r, s) ∈ Qi = Ci ∩ Z2 for every Ci ∈ Σa,b , we have that
s
ai+1
ai
≥ ≥
.
bi
r
bi+1
So ai r ≥ bi s, and by the ordering on the ai and the bi , aj r ≥ bj s for all j < i. Also,
ai+1 r ≤ bi+1 s, and again by the ordering, aj r ≤ bj s for all j > i. Since fk = max(rak , sbk )
for all k = 1, . . . , n, we have that
fk = rak for all k ≤ i and fk = sbk for all k > i.

So each fk is linear on each cone, and the above theorem applies.

It is important to note that the intersection algebra is not always Noetherian. One such
example, given in [3], is constructed by taking an ideal P in R such that the algebra
R ⊕ P (1) ⊕ P (2) ⊕ · · ·

ON THE INTERSECTION ALGEBRA OF PRINCIPAL IDEALS

11

is not finitely generated. Then, it is shown that there exists an f ∈ R such that (P a : f a ) =
P (a) for all a. It follows that the intersection algebra of P and (f ) is not finitely generated.
One important question that should be considered is what conditions on f and P are
necessary to ensure that the intersection algebra of f and P is Noetherian.
Proposition 4.6. Let R be a standard N2 -graded ring with maximal ideal m and f a
homogeneous element in m. Then
M
(f )r ∩ ms ur v s
B = BR (f, m) =
r,s∈N2

is finitely generated as an R-algebra.
Proof. Say deg f = a, and let x ∈ (f )r ∩ ms . Then x = f r · y ∈ ms , so y ∈ (ms : f r ).
Decompose y into its homogeneous pieces, y = y0 + · · · + ym . Then f r (y0 + · · · + ym ) ∈ ms ,
so
ra + deg yi ≥ s for all i = 0, . . . , n,
or, equivalently deg yi ≥ s − ra. Therefore y ∈ ms−ra , and so
(f )r ∩ (ms ) = f r · ms−ra .

Then
B=

X

r,s∈N2

f r · ms−ra ur v s =

Let
B̃ =

X

X

ms−ra (f u)r v s .

r,s∈N2

ms−ra w r v s ,

r,s∈N2

and ϕ : B̃ → B be the map that sends w to f u and is the identity on R and v. This map
is obviously surjective, therefore, if B̃ is finitely generated, so is B. But B̃ is a fan algebra
with I1 = m and

s − ra if s/r ≥ a
,
f1 (r, s) =
0
if s/r < a
which is certainly fan-linear on the fan formed by two cones
C0 = {λ1 (0, 1) + λ2 (1, a))|λi ∈ R≥0 } and C1 = {λ1 (1, a) + λ2 (1, 0)|λi ∈ R≥0 },

since C0 contains the collection of all (r, s) ∈ N2 where s/r ≥ a and C1 contains all (r, s) ∈ N
where s/r < a.
So by Theorem 4.3, if we define Λij ⊂ R to be a finite subset where msij −rij a = (x|x ∈ Λij ),
then B̃ is generated over R by the set
{xur0j v s0j |(r0j , s0j ) ∈ HQ0 , x ∈ Λij } ∪ {ur1j v s1j |(r1j , s1j ) ∈ HQ1 },

and therefore B is finitely generated as an R-algebra.



Remark 4.7. We are grateful to Mel Hochster, who noticed that the above proof works
the same in the case where R is regular local by replacing the degree of f with its order.
When (R, m) is a regular local ring, the order defines a valuation on R because grm (R) is a
polynomial ring over R/m (and hence a domain). Therefore, when (R, m) is a regular local
ring and f ∈ R, B(f, m) is a finitely generated R-algebra.

12

SARA MALEC

Acknowledgements. The author thanks her advisor, Florian Enescu, for many fruitful
discussions, as well as Yongwei Yao for his useful suggestions.
References
[1] C. Ciupercă, F. Enescu, and S. Spiroff. Asymptotic growth of powers of ideals. Illinois Journal of
Mathematics, 51(1):29–39, 2007.
[2] B. Sturmfels E. Miller. Combinatorial Commutative Algebra. Springer, 2005.
[3] J. B. Fields. Length functions determined by killing powers of several ideals in a local ring. Ph.D.
Dissertation, University of Michigan, Ann Arbor, Michigan, 2000.
[4] J. B. Fields. Lengths of Tors determined by killing powers of ideals in a local ring. Journal of Algebra,
247(1):104–133, 2002.
[5] Sara Malec. Noetherian filtrations and finite intersection algebras. Master’s thesis, Georgia State University, 2008.
[6] M. Nagata. Note on a paper of Samuel concerning asymptotic properties of ideals. Mem. College Sci.
Univ. Kyoto Ser. A Math, 30(2):165–175, 1957.
[7] P. Samuel. Some asymptotic properties of powers of ideals. Annals of Mathematis (2), 56(1):11–21,
1952.
Department of Mathematics, University of the Pacific, Stockton, CA 95207
E-mail address: smalec@pacific.edu

