arXiv:1303.6030v1 [] 25 Mar 2013

ON HILBERT FUNCTIONS OF
GENERAL INTERSECTIONS OF IDEALS
GIULIO CAVIGLIA AND SATOSHI MURAI

Abstract. Let I and J be homogeneous ideals in a standard graded polynomial
ring. We study upper bounds of the Hilbert function of the intersection of I
and g(J), where g is a general change of coordinates. Our main result gives a
generalization of Green’s hyperplane section theorem.

1. Introduction
Hilbert functions of graded K-algebras are important invariants studied in several
areas of mathematics. In the theory of Hilbert functions, one of the most useful
tools is Green’s hyperplane section theorem, which gives a sharp upper bound for
the Hilbert function of R/hR, where R is a standard graded K-algebra and h is
a general linear form, in terms of the Hilbert function of R. This result of Green
has been extended to the case of general homogeneous polynomials by Herzog and
Popescu [HP] and Gasharov [Ga]. In this paper, we study a further generalization
of these theorems.
Let K be an infinite field and S = K[x1 , . . . , xn ] a standard graded polynomial
ring. Recall that the Hilbert function H(M, −) : Z → Z of a finitely generated
graded S-module M is the numerical function defined by
H(M, d) = dimK Md ,
where Md is the graded component of M of degree d. A set W of monomials of
S is said to be lex if, for all monomials u, v ∈ S of the same degree, u ∈ W and
v >lex u imply v ∈ W , where >lex is the lexicographic order induced by the ordering
x1 > · · · > xn . We say that a monomial ideal I ⊂ S is a lex ideal if the set of
monomials in I is lex. The classical Macaulay’s theorem [Mac] guarantees that, for
any homogeneous ideal I ⊂ S, there exists a unique lex ideal, denoted by I lex , with
the same Hilbert function as I. Green’s hyperplane section theorem [Gr] states
Theorem 1.1 (Green’s hyperplane section theorem). Let I ⊂ S be a homogeneous
ideal. For a general linear form h ∈ S1 ,
H(I ∩ (h), d) ≤ H(I lex ∩ (xn ), d) for all d ≥ 0.
2010 Mathematics Subject Classification. Primary 13P10, 13C12, Secondary 13A02.
The work of the first author was supported by a grant from the Simons Foundation (209661 to
G. C.). The work of the second author was supported by KAKENHI 22740018.
1

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GIULIO CAVIGLIA AND SATOSHI MURAI

Green’s hyperplane section theorem is known to be useful to prove several important results on Hilbert functions such as Macaulay’s theorem [Mac] and Gotzmann’s
persistence theorem [Go], see [Gr]. Herzog and Popescu [HP] (in characteristic 0)
and Gasharov [Ga] (in positive characteristic) generalized Green’s hyperplane section theorem in the following form.
Theorem 1.2 (Herzog–Popescu, Gasharov). Let I ⊂ S be a homogeneous ideal.
For a general homogeneous polynomial h ∈ S of degree a,
H(I ∩ (h), d) ≤ H(I lex ∩ (xan ), d) for all d ≥ 0.
We study a generalization of Theorems 1.1 and 1.2. Let >oplex be the lexicographic
order on S induced by the ordering xn > · · · > x1 . A set W of monomials of S is
said to be opposite lex if, for all monomials u, v ∈ S of the same degree, u ∈ W and
v >oplex u imply v ∈ W . Also, we say that a monomial ideal I ⊂ S is an opposite lex
ideal if the set of monomials in I is opposite lex. For a homogeneous ideal I ⊂ S, let
I oplex be the opposite lex ideal with the same Hilbert function as I and let Ginσ (I)
be the generic initial ideal ([Ei, §15.9]) of I with respect to a term order >σ .
In Section 3 we will prove the following
Theorem 1.3. Suppose char(K) = 0. Let I ⊂ S and J ⊂ S be homogeneous ideals
such that Ginlex (J) is lex. For a general change of coordinates g of S,
H(I ∩ g(J), d) ≤ H(I lex ∩ J oplex , d) for all d ≥ 0.
Theorems 1.1 and 1.2, assuming that the characteristic is zero, are special cases
of the above theorem when J is principal. Note that Theorem 1.3 is sharp since the
equality holds if I is lex and J is oplex (Remark 3.5). Note also that if Ginσ (I) is
lex for some term order >σ then Ginlex (J) must be lex as well ([Co1, Corollary 1.6]).
Unfortunately, the assumption on J, as well as the assumption on the characteristic of K, in Theorem 1.3 are essential (see Remark 3.6). However, we prove the
following result for the product of ideals.
Theorem 1.4. Suppose char(K) = 0. Let I ⊂ S and J ⊂ S be homogeneous ideals.
For a general change of coordinates g of S,
H(Ig(J), d) ≥ H(I lexJ oplex , d) for all d ≥ 0.
Inspired by Theorems 1.3 and 1.4, we suggest the following conjecture.
Conjecture 1.5. Suppose char(K) = 0. Let I ⊂ S and J ⊂ S be homogeneous
ideals such that Ginlex (J) is lex. For a general change of coordinates g of S,
dimK Tori (S/I, S/g(J))d ≤ dimK Tori (S/I lex , S/J oplex )d for all d ≥ 0.
Theorems 1.3 and 1.4 show that the conjecture is true if i = 0 or i = 1. The
conjecture is also known to be true when J is generated by linear forms by the result
of Conca [Co2, Theorem 4.2]. Theorem 2.8, which we prove later, also provides some
evidence supporting the above inequality.

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3

2. Dimension of Tor and general change of coordinates
Let GLn (K) be the general linear group of invertible n × n matrices over K.
Throughout the paper, we identify each P
element h = (aij ) ∈ GLn (K) with the
change of coordinates defined by h(xi ) = nj=1 aji xj for all i.
We say that a property (P) holds for a general g ∈ GLn (K) if there is a non-empty
Zariski open subset U ⊂ GLn (K) such that (P) holds for all g ∈ U.
We first prove that, for two homogeneous ideals I ⊂ S and J ⊂ S, the Hilbert
function of I ∩ g(J) and that of Ig(J) are well defined for a general g ∈ GLn (K),
i.e. there exists a non-empty Zariski open subset of GLn (K) on which the Hilbert
function of I ∩ g(J) and that of Ig(J) are constant.
Lemma 2.1. Let I ⊂ S and J ⊂ S be homogeneous ideals. For a general change
of coordinates g ∈ GLn (K), the function H(I ∩ g(J), −) and H(Ig(J), −) are well
defined.
Proof. We prove the statement for I ∩ g(J) (the proof for Ig(J) is similar). It is
enough to prove the same statement for I + g(J). We prove that inlex (I + g(J)) is
constant for a general g ∈ GLn (K).
Let tkl , where 1 ≤ k, l ≤ n, be indeterminates, K̃ = K(tkl : 1 ≤ k, l ≤ n) the field
of fractions of K[tkl : 1 ≤ k, l ≤ n] and A = K̃[x1 , . . . , xn ]. Let ρ : S → A be the ring
P
map induced by ρ(xk ) = nl=1 tlk xl for k = 1, 2, . . . , n, and L̃ = IA + ρ(J)A ⊂ A.
Let L ⊂ S be the monomial ideal with the same monomial generators as inlex (L̃).
We prove inlex (I + g(J)) = L for a general g ∈ GLn (K).
Let f1 , . . . , fs be generators of I and g1 , . . . , gt those of J. Then the polynomials
f1 , . . . , fs , ρ(g1 ), . . . , ρ(gt ) are generators of L̃. By the Buchberger algorithm, one
can compute a Gröbner basis of L̃ from f1 , . . . , fs , ρ(g1 ), . . . , ρ(gt ) by finite steps.
Consider all elements h1 , . . . , hm ∈ K(tkl : 1 ≤ k, l ≤ n) which are the coefficient of
polynomials (including numerators and denominators of rational functions) that appear in the process of computing a Gröbner basis of L̃ by the Buchberger algorithm.
Consider a non-empty Zariski open subset U ⊂ GLn (K) such that hi (g) ∈ K \ {0}
for any g ∈ U, where hi (g) is an element obtained from hi by substituting tkl with
entries of g. By construction inlex (I + g(J)) = L for every g ∈ U.

Remark 2.2. The method used to prove the above lemma can be easily generalized
to a number of situations. For instance for a general g ∈ GLn (K) and a finitely
generated graded S-module M, the Hilbert function of Tori (M, S/g(J)) is well deϕp+1
ϕp
ϕ0
ϕ1
fined for every i. Let F : 0 −→ Fp −→ · · · −→ F1 −→ F0 −→ 0 be a graded
free resolution of M. Given a change of coordinates g, one first notes that for every
i = 0, . . . , p, the Hilbert function H(Tori (M, S/g(J)), −) is equal to the difference
between the Hilbert function of Ker(πi−1 ◦ ϕi ) and the one of ϕi+1 (Fi+1 ) + Fi ⊗S g(J)

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GIULIO CAVIGLIA AND SATOSHI MURAI

where πi−1 : Fi−1 → Fi−1 ⊗S S/g(J) is the canonical projection. Hence we have
H(Tori (M, S/g(J)), −) =
(1)

H(Fi , −) − H(ϕi (Fi ) + g(J)Fi−1, −) + H(g(J)Fi−1, −)
− H(ϕi+1 (Fi+1 ) + g(J)Fi , −).

Clearly H(Fi , −) and H(g(J)Fi−1, −) do not depend on g. Thus it is enough to show
that, for a general g, the Hilbert functions of ϕi (Fi ) + g(J)Fi−1 are well defined for
all i = 0, . . . , p + 1. This can be seen as in Lemma 2.1.
Next, we present two lemmas which will allow us to reduce the proofs of the
theorems in the third section to combinatorial considerations regarding Borel-fixed
ideals.
The first Lemma is probably clearly true to some experts, but we include its proof
for the sake of the exposition. The ideas used in Lemma 2.4 are similar to that of
[Ca1, Lemma 2.1] and they rely on the construction of a flat family and on the use
of the structure theorem for finitely generated modules over principal ideal domains.
Lemma 2.3. Let M be a finitely generated graded S-module and J ⊂ S a homogeneous ideal. For a general change of coordinates g ∈ GLn (K) we have that
dimK Tori (M, S/g(J))j ≤ dimK Tori (M, S/J)j for all i and for all j.
Proof. Let F be a resolution of M, as in Remark 2.2. Let i, 0 ≤ i ≤ p + 1 and
notice that, by equation (1), it is sufficient to show: H(ϕi (Fi ) + g(J)Fi−1 , −) ≥
H(ϕi (Fi ) + JFi−1 , −). We fix a degree d and consider the monomial basis of (Fi−1 )d .
Given a change of coordinates h = (akl ) ∈ GLn (K) we present the vector space
Vd = (ϕi (Fi ) + h(J)Fi−1 )d with respect to this basis. The dimension of Vd equals
the rank of a matrix whose entries are polynomials in the akl ’s with coefficients in
K. Such a rank is maximal when the change of coordinates h is general.

For a vector w = (w1 , . . . , wn ) ∈ Zn≥0 , let inw (I) be the initial ideal of a homogeneous ideal I with respect to the weight order >w (see [Ei, p. 345]). Let T be a new
indeterminate and R = S[T ]. For a = (a1 , . . . , an ) ∈ P
Zn≥0 , let xa = xa11 xa22 · · · xann
and (a, w) = a1 w1 + · · · + an wn . For a polynomial f = a∈Zn ca xa , where ca ∈ K,
≥0
let b = max{(a, w) : ca 6= 0} and


X
T −(a,w) ca xa  ∈ R.
f˜ = T b 
a∈Zn
≥0

Note that f˜ can be written as f˜ = inw (f ) + T g where g ∈ R. For an ideal I ⊂ S,
let I˜ = (f˜ : f ∈ I) ⊂ R. For λ ∈ K \ {0}, let Dλ,w be the diagonal change of
coordinates defined by Dλ,w (xi ) = λ−wi xi . From the definition, we have

R/ I˜ + (T ) ∼
= S/inw (I)

and


R/ I˜ + (T − λ) ∼
= S/Dλ,w (I)

HILBERT FUNCTIONS OF GENERAL INTERSECTIONS

5

where λ ∈ K \ {0}. Moreover (T − λ) is a non-zero divisor of R/I˜ for any λ ∈ K.
See [Ei, §15.8].
Lemma 2.4. Fix an integer j. Let w ∈ Zn≥0 , M a finitely generated graded S-module
and J ⊂ S a homogeneous ideal. For a general λ ∈ K, one has


dimK Tori M, S/inw (J) j ≥ dimK Tori M, S/Dλ,w (J) j for all i.

Proof. Consider the ideal J˜ ⊂ R defined as above. Let M̃ = M ⊗S R and Ti =
˜
TorR
i (M̃ , R/J). By the structure theorem for modules over a PID (see [La, p. 149]),
we have
M
Aij
(Ti )j ∼
= K[T ]aij

as a finitely generated K[T ]-module, where aij ∈ Z≥0 and where Aij is the torsion
submodule. Moreover Aij is a module of the form
Aij ∼
=

bij
M

K[T ]/(Phi,j ),

h=1

where Phi,j is a non-zero polynomial in K[T ]. Set lλ = T − λ. Consider the exact
sequence

·lλ
(2)
→ R/J˜ −−−→ R/ (lλ ) + J˜ −−−→ 0.
0 −−−→ R/J˜ −−−

By considering the long exact sequence induced by TorR
i (M̃ , −), we have the following exact sequence

˜ −→ Ki−1 −→ 0,
(3)
0 −→ Ti /lλ Ti −→ TorR
i M̃ , R/ (lλ ) + J
·l

λ
Ti−1 . Since lλ is a regular element for
where Ki−1 is the kernel of the map Ti−1 −→
R and M̃ , the middle term in (3) is isomorphic to



TorSi M, S/inw (J) , if λ = 0,
R/(lλ )
˜
Tori
M̃ /lλ M̃ , R/ (lλ ) + J =
TorSi M, S/Dλ,w (J) , if λ 6= 0

(see [Mat, p. 140]). By taking the graded component of degree j in (3), we obtain

dimK TorSi M, S/inw (J) j = aij + #{Phij : Phi,j (0) = 0}
(4)
+#{Phi−1,j : Phi−1,j (0) = 0},
where #X denotes the cardinality of a finite set X, and

dimK TorSi M, S/Dλ,w (J) j = aij
(5)
for a general λ ∈ K. This proves the desired inequality.

Corollary 2.5. With the same notation as in Lemma 2.4, for a general λ ∈ K,


dimK Tori M, inw (J) j ≥ dimK Tori M, Dλ,w (J) j for all i.



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GIULIO CAVIGLIA AND SATOSHI MURAI

Proof. For any homogeneous ideal I ⊂ S, by considering the long exact sequence
induced by Tori (M, −) from the short exact sequence 0 −→ I −→ S −→ S/I −→ 0
we have
Tori (M, I) ∼
= Tori+1 (M, S/I) for i ≥ 1
and
dimK Tor0 (M, I)j = dimK Tor1 (M, S/I)j + dimK Mj − dimK Tor0 (M, S/I)j .
Thus by Lemma 2.4 it is enough to prove that


dimK Tor1 M, S/inw (J) j − dimK Tor1 M, S/Dλ,w (J) j


≥ dimK Tor0 M, S/inw (J) j − dimK Tor0 M, S/Dλ,w (J) j .
This inequality follows from (4) and (5).



Proposition 2.6. Fix an integer j. Let I ⊂ S and J ⊂ S be homogeneous ideals.
Let w, w′ ∈ Zn≥0 . For a general change of coordinates g ∈ GLn (K),
(i) dimK Tori (S/I, S/g(J))j ≤ dimK Tori (S/inw (I), S/inw′ (J))j for all i.
(ii) dimK Tori (I, S/g(J))j ≤ dimK Tori (inw (I), S/inw′ (J))j for all i.
Proof. We prove (ii) (the proof for (i) is similar). By Lemmas 2.3 and 2.4 and
Corollary 2.5, we have


dimK Tori inw (I), S/inw′ (J) j ≥ dimK Tori Dλ1 ,w (I), S/Dλ2 ,w′ (J) j

= dimK Tori I, S/Dλ−1
Dλ2 ,w′ (J) j
1 ,w

≥ dimK Tori I, S/g(J) j ,

as desired, where λ1 , λ2 are general elements in K.



′

Remark 2.7. Let w = (1, 1, . . . , 1) and note that the composite of two general
changes of coordinates is still general. By replacing J by h(J) for a general change
of coordinates h, from Proposition 2.6(i) it follows that
dimK Tori (S/I, S/h(J))j ≤ dimK Tori S/in>σ (I), S/h(J))j
for any term order >σ .
The above fact gives, as a special case, an affirmative answer to [Co2, Question
6.1]. This was originally proved in the thesis of the first author [Ca2]. We mention
it here because there seem to be no published article which includes the proof of
this fact.
Theorem 2.8. Fix an integer j. Let I ⊂ S and J ⊂ S be homogeneous ideals. For
a general change of coordinates g ∈ GLn (K),
(i) dimK Tori (S/I, S/g(J))j ≤ dimK Tori (S/Ginlex (I), S/Ginoplex (J))j for all i.
(ii) dimK Tori (I, S/g(J))j ≤ dimK Tori (Ginlex (I), S/Ginoplex (J))j for all i.
Proof. Without loss of generality, we may assume inlex (I) = Ginlex (I) and that
inoplex (J) = Ginoplex (J). It follows from [Ei, Propositin 15.16] that there are vectors
w, w′ ∈ Zn≥0 such that inw (I) = inlex (I) and inw′ (g(J)) = Ginoplex (J). Then the
desired inequality follows from Proposition 2.6.


HILBERT FUNCTIONS OF GENERAL INTERSECTIONS

7

Since Tor0 (S/I, S/J) ∼
= I/IJ, we have the next
= S/(I + J) and Tor0 (I, S/J) ∼
corollary.
Corollary 2.9. Let I ⊂ S and J ⊂ S be homogeneous ideals. For a general change
of coordinates g ∈ GLn (K),
(i) H(I ∩ g(J), d) ≤ H(Ginlex (I) ∩ Ginoplex (J), d) for all d ≥ 0.
(ii) H(Ig(J), d) ≥ H(Ginlex (I)Ginoplex (J), d) for all d ≥ 0.
We conclude this section with a result regarding the Krull dimension of certain
Tor modules. We show how Theorem 2.8 can be used to give a quick proof of
Proposition 2.10, which is a special case (for the variety X = Pn−1 and the algebraic
group SLn ) of the main Theorem of [MS].
Recall that generic initial ideals are Borel-fixed, that is they are fixed under the
action of the Borel subgroup of GLn (K) consisting of all the upper triangular invertible matrices. In particular for an ideal I of S and an upper triangular matrix
b ∈ GLn (K) one has b(Ginlex (I)) = Ginlex (I). Similarly, if we denote by op the
change of coordinates of S which sends xi to xn−i for all i = 1, . . . , n, we have that
b(op(Ginoplex (I))) = op(Ginoplex (I)).
We call opposite Borel-fixed an ideal J of S such that op(J) is Borel-fixed (see
[Ei, §15.9] for more details on the combinatorial properties of Borel-fixed ideals).
It is easy to see that if J is Borel-fixed, then so is (x1 , . . . , xi ) + J for every
i = 1, . . . , n. Furthermore if j is an integer equal to min{i : xi 6∈ J} then J : xj
is also Borel-fixed; in this case I has a minimal generator divisible by xj or I =
(x1 , . . . , xj−1 ). Analogous statements hold for opposite Borel-fixed ideals.
Let I and J be ideals generated by linear forms. If we assume that I is Borel
fixed and that J is opposite Borel fixed, then there exist 1 ≤ i, j ≤ n such that
I = (x1 , . . . , xi ) and J = (xj , . . . , xn ). An easy computation shows that the Krull
dimension of Tori (S/I, S/J) is always zero when i > 0.
More generally one has
Proposition 2.10 (Miller–Speyer). Let I and J be two homogeneous ideals of S.
For a general change of coordinates g, the Krull dimension of Tori (S/I, S/g(J)) is
zero for all i > 0.
Proof. When I or J are equal to (0) or to S the result is obvious. Recall that a
finitely generated graded module M has Krull dimension zero if and only if Md = 0
for all d sufficiently large. In virtue of Theorem 2.8 it is enough to show that
Tori (S/I, S/J) has Krull dimension zero whenever I is Borel-fixed, J opposite Borelfixed and i > 0. By contradiction, let the pair I, J be a maximal counterexample
(with respect to point-wise inclusion). By the above discussion, and by applying op
if necessary, we can assume that I has a minimal generator of degree greater than 1.
Let j = min{h : xh 6∈ I} and notice that both (I : xj ) and (I+(xj )) strictly contain I.
For every i > 0 the short exact sequence 0 → S/(I : xj ) → S/I → S/(I + (xj )) → 0
induces the exact sequence
Tori (S/(I : xj ), S/J) → Tori (S/I, S/J) → Tori (S/(I + (xj )), S/J).

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GIULIO CAVIGLIA AND SATOSHI MURAI

By the maximality of I, J, the first and the last term have Krull dimension zero.
Hence the middle term must have dimension zero as well, contradicting our assumption.

3. General intersections and general products
In this section, we prove Theorems 1.3 and 1.4. We will assume throughout the
rest of the paper char(K) = 0.
A monomial ideal I ⊂ S is said to be 0-Borel (or strongly stable) if, for every
monomial uxj ∈ I and for every 1 ≤ i < j one has uxi ∈ I. Note that 0-Borel
ideals are precisely all the possible Borel-fixed ideals in characteristic 0. In general,
the Borel-fixed property depends on the characteristic of the field and we refer the
readers to [Ei, §15.9] for the details. A set W ⊂ S of monomials in S is said to be
0-Borel if the ideal they generate is 0-Borel, or equivalently if for every monomial
uxj ∈ W and for every 1 ≤ i < j one has uxi ∈ W . Similarly we say that a
monomial ideal J ⊂ S is opposite 0-Borel if for every monomial uxj ∈ J and for
every j < i ≤ n one has uxi ∈ J.
Let >rev be the reverse lexicographic order induced by the ordering x1 > · · · > xn .
We recall the following result [Mu1, Lemma 3.2].
Lemma 3.1. Let V = {v1 , . . . , vs } ⊂ Sd be a 0-Borel set of monomials and W =
{w1 , . . . , ws } ⊂ Sd the lex set of monomials, where v1 ≥rev · · · ≥rev vs and w1 ≥rev
· · · ≥rev ws . Then vi ≥rev wi for all i = 1, 2, . . . , s.
Since generic initial ideals with respect to >lex are 0-Borel, the next lemma and
Corollary 2.9(i) prove Theorem 1.3.
Lemma 3.2. Let I ⊂ S be a 0-Borel ideal and P ⊂ S an opposite lex ideal. Then
dimK (I ∩ P )d ≤ dimK (I lex ∩ P )d for all d ≥ 0.
Proof. Fix a degree d. Let V, W and Q be the sets of monomials of degree d in I,
I lex and P respectively. It is enough to prove that #V ∩ Q ≤ #W ∩ Q.
Observe that Q is the set of the smallest #Q monomials in Sd with respect to
>rev . Let m = max>rev Q. Then by Lemma 3.1
#V ∩ Q = #{v ∈ V : v ≤rev m} ≤ #{w ∈ W : w ≤rev m} = #W ∩ Q,
as desired.



Next, we consider products of ideals. For a monomial u ∈ S, let max u (respectively, min u) be the maximal (respectively, minimal) integer i such that xi divides
u, where we set max 1 = 1 and min 1 = n. For a monomial ideal I ⊂ S, let I(≤k) be
the K-vector space spanned by all monomials u ∈ I with max u ≤ k.
Lemma 3.3. Let I ⊂ S be a 0-Borel ideal and P ⊂ S an opposite 0-Borel ideal.
Let G(P ) = {u1 , . . . , us } be the set of the minimal monomial generators of P . As a
K-vector space, IP is the direct sum
s
M
IP =
(I(≤min ui ) )ui .
i=1

HILBERT FUNCTIONS OF GENERAL INTERSECTIONS

9

Proof. It is enough to prove that, for any monomial w ∈ IP , there is the unique
expression w = f (w)g(w) with f (w) ∈ I and g(w) ∈ P satisfying
(a) max f (w) ≤ min g(w).
(b) g(w) ∈ G(P ).
Given any expression w = f g such that f ∈ I and g ∈ P , since I is 0-Borel and
xmin g
P is opposite 0-Borel, if max f > min g then we may replace f by f xmax
∈ I and
f
xmax f
replace g by g xmin g ∈ P . This fact shows that there is an expression satisfying (a)
and (b).
Suppose that the expressions w = f (w)g(w) and w = f ′ (w)g ′(w) satisfy conditions
(a) and (b). Then, by (a), g(w) divides g ′ (w) or g ′(w) divides g(w). Since g(w) and
g ′ (w) are generators of P , g(w) = g ′ (w). Hence the expression is unique.

Lemma 3.4. Let I ⊂ S be a 0-Borel ideal and P ⊂ S an opposite 0-Borel ideal.
Then dimK (IP )d ≥ dimK (I lex P )d for all d ≥ 0.
lex
Proof. Lemma 3.1 shows that dimK I(≤k) d ≥ dimK I(≤k)
for all k and d ≥ 0. Then
d
the statement follows from Lemma 3.3.


Finally we prove Theorem 1.4.
Proof of Theorem 1.4. Let I ′ = Ginlex (I) and J ′ = Ginoplex (J). Since I ′ is 0-Borel
and J ′ is opposite 0-Borel, by Corollary 2.9(ii) and Lemmas 3.4
H(Ig(J), d) ≥ H(I ′ J ′ , d) ≥ H(I lex J ′ , d) ≥ H(I lex J oplex , d)
for all d ≥ 0.



Remark 3.5. Theorems 1.3 and 1.4 are sharp. Let I ⊂ S be a Borel-fixed ideal and
J ⊂ S an ideal satisfying that h(J) = J for any lower triangular matrix h ∈ GLn (K).
For a general g ∈ GLn (K), we have the LU decomposition g = bh where h ∈ GLn (K)
is a lower triangular matrix and b ∈ GLn (K) is an upper triangular matrix. Then
as K-vector spaces
I ∩ g(J) ∼
= b−1 (I) ∩ h(J) = I ∩ J and Ig(J) ∼
= b−1 (I)h(J) = IJ.
Thus if I is lex and J is opposite lex then H(I ∩ g(J), d) = H(I ∩ J, d) and
H(Ig(J), d) = H(IJ, d) for all d ≥ 0.
Remark 3.6. The assumption on Ginlex (J) in Theorem 1.3 is necessary. Let I =
(x31 , x21 x2 , x1 x22 , x32 ) ⊂ K[x1 , x2 , x3 ] and J = (x23 , x23 x2 , x3 x22 , x32 ) ⊂ K[x1 , x2 , x3 ]. Then
the set of monomials of degree 3 in I lex is {x31 , x21 x2 , x21 x3 , x1 x22 } and that of J oplex is
{x33 , x23 x2 , x23 x1 , x3 x22 }. Hence H(I lex ∩ J oplex , 3) = 0. On the other hand, as we see
in Remark 3.5, H(I ∩ g(J), 3) = H(I ∩ J, 3) = 1. Similarly, the assumption on the
characteristic of K is needed as one can easily see by considering char(K) = p > 0,
I = (xp1 , xp2 ) ⊂ K[x1 , x2 ] and J = xp2 . In this case we have H(I lex ∩ J oplex , p) = 0,
while H(I ∩ g(J), p) = H(g −1(I) ∩ J, p) = 1 since I is fixed under any change of
coordinates.

10

GIULIO CAVIGLIA AND SATOSHI MURAI

Since Tor0 (S/I, S/J) ∼
= (I ∩ J)/IJ for all homo= S/(I + J) and Tor1 (S/I, S/J) ∼
geneous ideals I ⊂ S and J ⊂ S, Theorems 1.3 and 1.4 show the next statement.
Remark 3.7. Conjecture 1.5 is true if i = 0 or i = 1.
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Department of Mathematics, Purdue University, West Lafayette, IN 47901, USA.
E-mail address: gcavigli@math.purdue.edu
Satoshi Murai, Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan.
E-mail address: murai@yamaguchi-u.ac.jp

