---
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.'
address:
- ' Department of Mathematics, Purdue University, West Lafayette, IN 47901, USA. '
- ' Satoshi Murai, Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan. '
author:
- Giulio Caviglia
- Satoshi Murai
title: |
    On Hilbert functions of\
    general intersections of ideals
---

[^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 $S=K[x_1,\dots,x_n]$ a standard graded polynomial ring. Recall that the *Hilbert function* $H(M,-) : \mathbb{Z} \to \mathbb{Z}$ of a finitely generated graded $S$-module $M$ is the numerical function defined by $$H(M,d)=\dim_K M_d,$$ where $M_d$ 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 $u,v \in S$ of the same degree, $u \in W$ and $v>_{{\mathrm{lex}}}u$ imply $v \in W$, where $>_{{\mathrm{lex}}}$ is the lexicographic order induced by the ordering $x_1> \cdots > x_n$. We say that a monomial ideal $I \subset S$ is a *lex ideal* if the set of monomials in $I$ is lex. The classical Macaulay’s theorem [@Ma] guarantees that, for any homogeneous ideal $I \subset S$, there exists a unique lex ideal, denoted by $I^{{{\mathrm{lex}}}}$, with the same Hilbert function as $I$. Green’s hyperplane section theorem [@Gr] states

\[green\] Let $I \subset S$ be a homogeneous ideal. For a general linear form $h \in S_1$, $$H(I \cap (h),d) \leq  H(I^{{\mathrm{lex}}}\cap (x_n),d) \ \ \mbox{for all } d \geq 0.$$

Green’s hyperplane section theorem is known to be useful to prove several important results on Hilbert functions such as Macaulay’s theorem [@Ma] 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.

\[hpg\] Let $I \subset S$ be a homogeneous ideal. For a general homogeneous polynomial $h \in S$ of degree $a$, $$H(I \cap (h),d) \leq  H(I^{{\mathrm{lex}}}\cap(x_n^a),d) \ \ \mbox{for all } d \geq 0.$$

We study a generalization of Theorems \[green\] and \[hpg\]. Let $>_{{\mathrm{{oplex}}}}$ be the lexicographic order on $S$ induced by the ordering $x_n> \cdots > x_1$. A set $W$ of monomials of $S$ is said to be *opposite lex* if, for all monomials $u,v \in S$ of the same degree, $u \in W$ and $v>_{{\mathrm{{oplex}}}}u$ imply $v \in W$. Also, we say that a monomial ideal $I \subset S$ is an *opposite lex ideal* if the set of monomials in $I$ is opposite lex. For a homogeneous ideal $I \subset S$, let $I^{{\mathrm{{oplex}}}}$ be the opposite lex ideal with the same Hilbert function as $I$ and let ${\ensuremath{\mathrm{Gin}}}_\sigma(I)$ be the generic initial ideal ([@Ei §15.9]) of $I$ with respect to a term order $>_\sigma$.

In Section 3 we will prove the following

\[intersection\] Suppose $\mathrm{char}(K)=0$. Let $I\subset S$ and $J \subset S$ be homogeneous ideals such that ${\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(J)$ is lex. For a general change of coordinates $g$ of $S$, $$H(I \cap g(J),d) \leq H(I^{{\mathrm{lex}}}\cap J^{{\mathrm{{oplex}}}},d)
\ \ \mbox{for all } d\geq 0.$$

Theorems \[green\] and \[hpg\], assuming that the characteristic is zero, are special cases of the above theorem when $J$ is principal. Note that Theorem \[intersection\] is sharp since the equality holds if $I$ is lex and $J$ is oplex (Remark \[rem1\]). Note also that if ${\ensuremath{\mathrm{Gin}}}_\sigma(I)$ is lex for some term order $>_\sigma$ then ${\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(J)$ 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 \[intersection\] are essential (see Remark \[example\]). However, we prove the following result for the product of ideals.

\[product\] Suppose $\mathrm{char}(K)=0$. Let $I\subset S$ and $J \subset S$ be homogeneous ideals. For a general change of coordinates $g$ of $S$, $$H(I  g(J),d) \geq H(I^{{\mathrm{lex}}}J^{{\mathrm{{oplex}}}},d)
\ \ \mbox{for all } d\geq 0.$$

Inspired by Theorems \[intersection\] and \[product\], we suggest the following conjecture.

\[conj\] Suppose $\mathrm{char}(K)=0.$ Let $I\subset S$ and $J \subset S$ be homogeneous ideals such that ${\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(J)$ is lex. For a general change of coordinates $g$ of $S$, $$\dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I,S/g(J))_d \leq  \dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I^{{\mathrm{lex}}},S/J^{{\mathrm{{oplex}}}})_d
\ \ \mbox{for all } d\geq 0.$$

Theorems \[intersection\] and \[product\] 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 [@Co Theorem 4.2]. Theorem \[2.5\], which we prove later, also provides some evidence supporting the above inequality.

Dimension of ${\ensuremath{\mathrm{Tor}}}$ and general change of coordinates
============================================================================

Let ${{GL}}_n(K)$ be the general linear group of invertible $n \times n$ matrices over $K$. Throughout the paper, we identify each element $h=(a_{ij}) \in {{GL}}_n(K)$ with the change of coordinates defined by $h(x_i)=\sum_{j=1}^n a_{ji}x_j$ for all $i$.

We say that a property (P) holds for a general $g \in {{GL}}_n(K)$ if there is a non-empty Zariski open subset $U \subset {{GL}}_n(K)$ such that (P) holds for all $g \in U$.

We first prove that, for two homogeneous ideals $I \subset S$ and $J \subset S$, the Hilbert function of $I \cap g(J)$ and that of $I g(J)$ are well defined for a general $g \in {{GL}}_n (K)$, i.e. there exists a non-empty Zariski open subset of ${{GL}}_n(K)$ on which the Hilbert function of $I \cap g(J)$ and that of $I g(J)$ are constant.

\[2-0\] Let $I \subset S$ and $J \subset S$ be homogeneous ideals. For a general change of coordinates $g \in {{GL}}_n(K)$, the function $H(I \cap g(J),-)$ and $H(I g(J),-)$ are well defined.

We prove the statement for $I \cap g(J)$ (the proof for $Ig(J)$ is similar). It is enough to prove the same statement for $I+g(J)$. We prove that ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I+g(J))$ is constant for a general $g \in {{GL}}_n(K)$.

Let $t_{kl}$, where $1 \leq k,l \leq n$, be indeterminates, $\tilde K=K(t_{kl}: 1 \leq k,l \leq n)$ the field of fractions of $K[t_{kl}: 1 \leq k,l \leq n]$ and $A=\tilde K [x_1,\dots,x_n]$. Let $\rho: S \to A$ be the ring map induced by $\rho(x_k)= \sum_{l=1}^n t_{lk} x_l$ for $k=1,2,\dots,n$, and $\tilde L= I A + \rho(J)A \subset A$. Let $L \subset S$ be the monomial ideal with the same monomial generators as ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(\tilde L)$. We prove ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I+g(J))=L$ for a general $g \in {{GL}}_n(K)$.

Let $f_1,\dots,f_s$ be generators of $I$ and $g_1,\dots,g_t$ those of $J$. Then the polynomials $f_1,\dots,f_s,\rho(g_1),\dots,\rho(g_t)$ are generators of $\tilde L$. By the Buchberger algorithm, one can compute a Gröbner basis of $\tilde L$ from $f_1,\dots,f_s,\rho(g_1),\dots,\rho(g_t)$ by finite steps. Consider all elements $h_1,\dots,h_m \in K(t_{kl}:1 \leq k,l \leq n)$ which are the coefficient of polynomials (including numerators and denominators of rational functions) that appear in the process of computing a Gröbner basis of $\tilde L$ by the Buchberger algorithm. Consider a non-empty Zariski open subset $U \subset {{GL}}_n(K)$ such that $h_i(g) \in K \setminus \{0\}$ for any $g \in U$, where $h_i(g)$ is an element obtained from $h_i$ by substituting $t_{kl}$ with entries of $g$. By construction ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I+g(J))=L$ for every $g \in U$.

\[ConstantHF\] The method used to prove the above lemma can be easily generalized to a number of situations. For instance for a general $g \in {{GL}}_n(K)$ and a finitely generated graded $S$-module $M,$ the Hilbert function of ${\ensuremath{\mathrm{Tor}}}_i(M,S/g(J))$ is well defined for every $i$. Let $\mathbb F: 0 \stackrel{\varphi_{p+1}}{\longrightarrow} 
\mathbb F_p \stackrel{\varphi_p}{\longrightarrow} 
\cdots 
\longrightarrow
\mathbb F_1 \stackrel{\varphi_1}{\longrightarrow} 
\mathbb F_0
\stackrel{\varphi_0}{\longrightarrow}0$ be a graded free resolution of $M.$ Given a change of coordinates $g$, one first notes that for every $i=0,\dots,p$, the Hilbert function $H({\ensuremath{\mathrm{Tor}}}_i(M,S/g(J)),-)$ is equal to the difference between the Hilbert function of $\rm{Ker}(\pi_{i-1} \circ \varphi_i)$ and the one of $\varphi_{i+1}(F_{i+1}) + F_i \otimes_S g(J)$ where $\pi_{i-1}: F_{i-1} \rightarrow F_{i-1} \otimes_S S/g(J)$ is the canonical projection. Hence we have $$\begin{aligned}
\label{H-TOR}
\nonumber H({\ensuremath{\mathrm{Tor}}}_i & (M,S/g(J)),-)= \\
 &H(F_i, -) -H(\varphi_i(F_i)+    g(J) F_{i-1},-)
 + H(g(J) F_{i-1},-)\\
\nonumber &- H(\varphi_{i+1}(F_{i+1}) + g(J) F_i,-).\end{aligned}$$ Clearly $H(F_i,-)$ and $H(g(J) F_{i-1},-)$ do not depend on $g.$ Thus it is enough to show that, for a general $g$, the Hilbert functions of $\varphi_i(F_i)+g(J) F_{i-1}$ are well defined for all $i=0,\dots,p+1.$ This can be seen as in Lemma \[2-0\].

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 \[lemma2\] 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.

\[lemma1\] Let $M$ be a finitely generated graded $S$-module and $J \subset S$ a homogeneous ideal. For a general change of coordinates $g \in {{GL}}_n(K)$ we have that $\dim_K {\ensuremath{\mathrm{Tor}}}_i(M,S/g(J))_j \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i(M,S/J)_j$ for all $i$ and for all $j.$

Let $\mathbb F$ be a resolution of $M,$ as in Remark \[ConstantHF\]. Let $i$, $0\leq i \leq p+1$ and notice that, by equation , it is sufficient to show: $H(\varphi_i(F_i)+g(J) F_{i-1},-)\geq 
H(\varphi_i(F_i)+JF_{i-1},-).$ We fix a degree $d$ and consider the monomial basis of $ (F_{i-1})_d.$ Given a change of coordinates $h=(a_{kl}) \in {{GL}}_n(K)$ we present the vector space $V_d=(\varphi_i(F_i)+h(J)F_{i-1})_d$ with respect to this basis. The dimension of $V_d$ equals the rank of a matrix whose entries are polynomials in the $a_{kl}$’s with coefficients in $K.$ Such a rank is maximal when the change of coordinates $h$ is general.

For a vector ${\mathbf w}=(w_1,\ldots,w_n) \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$, let ${\ensuremath{\mathrm{in}}}_{\mathbf w}(I)$ be the initial ideal of a homogeneous ideal $I$ with respect to the weight order $>_{\mathbf w}$ (see [@Ei p. 345]). Let $T$ be a new indeterminate and $R=S[T]$. For $\mathbf a=(a_1,\dots,a_n) \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$, let $x^{\mathbf a}=x_1^{a_1} x_2^{a_2} \cdots x_n^{a_n}$ and $(\mathbf a, {\mathbf w})= a_1w_1 + \cdots + a_n w_n$. For a polynomial $f= \sum_{\mathbf a \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n} c_{\mathbf a} x^{\mathbf a}$, where $c_{\mathbf a} \in K$, let $b= \max \{ (\mathbf a,{\mathbf w}) : c_{\mathbf a} \ne 0\}$ and $$\tilde f = T^b \left(\sum_{\mathbf a \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n} T^{-(\mathbf a,{\mathbf w})}c_{\mathbf a} x^{\mathbf a}\right) \in R.$$ Note that $\tilde f$ can be written as $\tilde f={\ensuremath{\mathrm{in}}}_{\mathbf w}(f) + T g$ where $g \in R$. For an ideal $I \subset S$, let $\tilde I =(\tilde f :f \in I) \subset R$. For $\lambda \in K \setminus\{0\}$, let $D_{\lambda,{\mathbf w}}$ be the diagonal change of coordinates defined by $D_{\lambda,{\mathbf w}}(x_i)=\lambda^{-w_i} x_i$. From the definition, we have $$R/\big(\tilde I +(T)\big) \cong S/ {\ensuremath{\mathrm{in}}}_{\mathbf w}(I)$$ and $$R/\big(\tilde I +(T-\lambda)\big) \cong S/D_{\lambda,{\mathbf w}}(I)$$ where $\lambda \in K \setminus \{0\}$. Moreover $(T-\lambda)$ is a non-zero divisor of $R/\tilde I$ for any $\lambda \in K$. See [@Ei §15.8].

\[lemma2\] Fix an integer $j$. Let ${\mathbf w}\in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$, $M$ a finitely generated graded $S$-module and $J \subset S$ a homogeneous ideal. For a general $\lambda \in K$, one has $$\dim_K {\ensuremath{\mathrm{Tor}}}_i \big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big)_j\geq \dim_K {\ensuremath{\mathrm{Tor}}}_i\big(M, S/D_{\lambda,{\mathbf w}}(J)\big)_j
\ \mbox{ for all $i$.}$$

Consider the ideal $\tilde {J} \subset R$ defined as above. Let $\tilde M = M \otimes_S R$ and $T_i={\ensuremath{\mathrm{Tor}}}_i^{R}(\tilde M,R/\tilde{J})$. By the structure theorem for modules over a PID (see [@La p. 149]), we have $$(T_i)_j\cong K[T]^{a_{ij}}
\bigoplus A_{ij}$$ as a finitely generated $K[T]$-module, where $a_{ij} \in { \ensuremath{\mathbb{Z}}}_{\geq 0}$ and where $A_{ij}$ is the torsion submodule. Moreover $A_{ij}$ is a module of the form $$A_{ij}\cong \bigoplus_{h=1}^{b_{ij}} K [T]/(P^{i,j}_{h}),$$ where $P^{i,j}_h$ is a non-zero polynomial in $K[T]$. Set $l_{\lambda}=T-\lambda$. Consider the exact sequence $$\begin{aligned}
\label{aa}
\begin{CD} 0 @>>> R/\tilde{J} @>\cdot l_{\lambda}>>
R/\tilde{J} @>>> R/\big((l_{\lambda})+\tilde{J} \big) @>>> 0.
\end{CD}\end{aligned}$$ By considering the long exact sequence induced by ${\ensuremath{\mathrm{Tor}}}^R_i(\tilde M,-),$ we have the following exact sequence $$\label{bo} 0\longrightarrow T_i/l_{\lambda} T_i \longrightarrow
{\ensuremath{\mathrm{Tor}}}_i^{R}\big(\tilde M,R/\big((l_{\lambda})+\tilde{J}\big)\big) \longrightarrow
K_{i-1} \longrightarrow 0,
$$ where $K_{i-1}$ is the kernel of the map $T_{i-1} \xrightarrow{\cdot l_{\lambda}} T_{i-1}$. Since $l_{\lambda}$ is a regular element for $R$ and $\tilde M$, the middle term in (\[bo\]) is isomorphic to $$\begin{aligned}
{\ensuremath{\mathrm{Tor}}}_i^{R/(l_\lambda)} \big(\tilde M /l_\lambda \tilde M, R/\big((l_{\lambda})+\tilde J \big)\big)
=\left\{
\begin{array}{lll}
{\ensuremath{\mathrm{Tor}}}_i^S \big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big), & \mbox{ if } \lambda=0,\\
{\ensuremath{\mathrm{Tor}}}_i^S \big(M,S/D_{\lambda,{\mathbf w}}(J)\big), & \mbox{ if } \lambda\ne0
\end{array}
\right.\end{aligned}$$ (see [@Mat p. 140]). By taking the graded component of degree $j$ in (\[bo\]), we obtain $$\begin{aligned}
\label{banngou} 
\begin{array}{lll}
\dim_K
{\ensuremath{\mathrm{Tor}}}_i^{S}\big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J) \big)_j &=& a_{ij} + 
\# \{P^{ij}_h : P^{i,j}_h(0)=0\}\\
&& +  \# \{P^{i-1,j}_h : P^{i-1,j}_h(0)=0\},
\end{array}\end{aligned}$$ where $\# X$ denotes the cardinality of a finite set $X$, and $$\begin{aligned}
\label{yon}
\dim_K
{\ensuremath{\mathrm{Tor}}}_i^{S}\big(M,S/D_{\lambda,{\mathbf w}}(J) \big)_j &=& a_{ij}\end{aligned}$$ for a general $\lambda \in K$. This proves the desired inequality.

\[add\] With the same notation as in Lemma \[lemma2\], for a general $\lambda \in K$, $$\dim_K {\ensuremath{\mathrm{Tor}}}_i \big(M,{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big)_j \geq \dim_K {\ensuremath{\mathrm{Tor}}}_i \big(M, D_{\lambda,{\mathbf w}}(J) \big)_j
\mbox{ for all }i.$$

For any homogeneous ideal $I \subset S$, by considering the long exact sequence induced by ${\ensuremath{\mathrm{Tor}}}_i(M,-)$ from the short exact sequence $0 \longrightarrow I \longrightarrow S \longrightarrow S/I \longrightarrow 0$ we have $${\ensuremath{\mathrm{Tor}}}_i(M,I) \cong {\ensuremath{\mathrm{Tor}}}_{i+1}(M,S/I)
\mbox{ for }i \geq 1$$ and $$\dim_K {\ensuremath{\mathrm{Tor}}}_0(M,I)_j = \dim_K {\ensuremath{\mathrm{Tor}}}_1(M,S/I)_j + \dim_K M_j - \dim_K {\ensuremath{\mathrm{Tor}}}_0(M,S/I)_j.$$ Thus by Lemma \[lemma2\] it is enough to prove that $$\begin{aligned}
&&\dim_K {\ensuremath{\mathrm{Tor}}}_1\big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big)_j -\dim_K {\ensuremath{\mathrm{Tor}}}_1\big(M,S/D_{\lambda,{\mathbf w}}(J)\big)_j\\
&&\geq \dim_K {\ensuremath{\mathrm{Tor}}}_0\big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big)_j -\dim_K {\ensuremath{\mathrm{Tor}}}_0\big(M,S/D_{\lambda,{\mathbf w}}(J)\big)_j.\end{aligned}$$ This inequality follows from (\[banngou\]) and (\[yon\]).

\[2.3\] Fix an integer $j$. Let $I \subset S$ and $J \subset S$ be homogeneous ideals. Let ${\mathbf w},{\mathbf w}' \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$. For a general change of coordinates $g \in {{GL}}_n(K)$,

-   $\dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I,S/g(J))_j
    \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i (S/{\ensuremath{\mathrm{in}}}_{{\mathbf w}}(I), S/{{\ensuremath{\mathrm{in}}}_{{\mathbf w}'}}(J))_j
    \ \mbox{ for all }i.$

-   $\dim_K {\ensuremath{\mathrm{Tor}}}_i(I,S/g(J))_j
    \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i ({\ensuremath{\mathrm{in}}}_{{\mathbf w}}(I), S/{{\ensuremath{\mathrm{in}}}_{{\mathbf w}'}}(J))_j
    \ \mbox{ for all }i.$

We prove (ii) (the proof for (i) is similar). By Lemmas \[lemma1\] and \[lemma2\] and Corollary \[add\], we have $$\begin{aligned}
\dim_K {\ensuremath{\mathrm{Tor}}}_i \big({\ensuremath{\mathrm{in}}}_{{\mathbf w}}(I), S/{\ensuremath{\mathrm{in}}}_{{\mathbf w}'}(J)\big)_j 
&\geq& \dim_K {\ensuremath{\mathrm{Tor}}}_i \big(D_{\lambda_1,{\mathbf w}}(I), S/D_{\lambda_2,{\mathbf w}'}(J)\big)_j \\
&=& \dim_K {\ensuremath{\mathrm{Tor}}}_i \big(I, S/D^{-1}_{\lambda_1,{\mathbf w}} \big(D_{\lambda_2,{\mathbf w}'}(J)\big)\big)_j\\
&\geq& \dim_K {\ensuremath{\mathrm{Tor}}}_i\big(I,S/g(J)\big)_j,\end{aligned}$$ as desired, where $\lambda_1,\lambda_2$ are general elements in $K$.

Let ${\mathbf w}'=(1,1,\dots,1)$ 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 \[2.3\](i) it follows that $$\dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I,S/h(J))_j \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i\big(S/{\ensuremath{\mathrm{in}}}_{>_{\sigma}}(I),S/h(J))_j$$ for any term order $>_\sigma$.

The above fact gives, as a special case, an affirmative answer to [@Co 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.

\[2.5\] Fix an integer $j$. Let $I \subset S$ and $J \subset S$ be homogeneous ideals. For a general change of coordinates $g \in {{GL}}_n(K)$,

-   $\dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I,S/g(J))_j \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i(S/{\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I),S/{\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J))_j
    \ \ \mbox{for all }i.$

-   $\dim_K {\ensuremath{\mathrm{Tor}}}_i(I,S/g(J))_j \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i({\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I),S/{\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J))_j
    \ \ \mbox{for all }i.$

Without loss of generality, we may assume ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I)={\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I)$ and that ${\ensuremath{\mathrm{in}}}_{{\mathrm{{oplex}}}}(J)={\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J)$. It follows from [@Ei Propositin 15.16] that there are vectors ${\mathbf w}, {\mathbf w}' \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$ such that ${\ensuremath{\mathrm{in}}}_{\mathbf w}(I)={\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I)$ and ${\ensuremath{\mathrm{in}}}_{{\mathbf w}'}(g(J))={\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J)$. Then the desired inequality follows from Proposition \[2.3\].

Since ${\ensuremath{\mathrm{Tor}}}_0(S/I,S/J)\cong S/(I+J)$ and ${\ensuremath{\mathrm{Tor}}}_0(I,S/J)\cong I/IJ$, we have the next corollary.

\[2.6\] Let $I \subset S$ and $J \subset S$ be homogeneous ideals. For a general change of coordinates $g \in {{GL}}_n(K)$,

-   $H(I \cap g(J) ,d) \leq H({\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I)\cap {\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J),d)$ for all $d \geq 0$.

-   $H(Ig(J),d) \geq H({\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I){\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J),d)$ for all $d \geq 0$.

We conclude this section with a result regarding the Krull dimension of certain Tor modules. We show how Theorem \[2.5\] can be used to give a quick proof of Proposition \[MiSp\], which is a special case (for the variety $X=\mathbb{P}^{n-1}$ and the algebraic group ${{SL}}_n$) 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 ${{GL}}_n(K)$ consisting of all the upper triangular invertible matrices. In particular for an ideal $I$ of $S$ and an upper triangular matrix $b\in {{GL}}_n(K)$ one has $b({\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I))= {\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I).$ Similarly, if we denote by $op$ the change of coordinates of $S$ which sends $x_i$ to $x_{n-i}$ for all $i=1,\dots,n,$ we have that $b( op ({\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(I)))= op ({\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(I)).$

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 $(x_1,\dots,x_i)+J$ for every $i=1,\dots,n.$ Furthermore if $j$ is an integer equal to $\min \{i : x_i\not \in J \}$ then $J:x_j$ is also Borel-fixed; in this case $I$ has a minimal generator divisible by $x_j$ or $I=(x_1,\dots,x_{j-1}).$ 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\leq i,j \leq n $ such that $I=(x_1,\dots,x_i)$ and $J=(x_j,\dots,x_n).$ An easy computation shows that the Krull dimension of ${\ensuremath{\mathrm{Tor}}}_i(S/I,S/J)$ is always zero when $i>0.$

More generally one has

\[MiSp\] Let $I$ and $J$ be two homogeneous ideals of $S.$ For a general change of coordinates $g$, the Krull dimension of ${\ensuremath{\mathrm{Tor}}}_i(S/I,S/g(J))$ is zero for all $i>0.$

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 $M_d=0$ for all $d$ sufficiently large. In virtue of Theorem \[2.5\] it is enough to show that ${\ensuremath{\mathrm{Tor}}}_i(S/I,S/J)$ has Krull dimension zero whenever $I$ is Borel-fixed, $J$ opposite Borel-fixed 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 $j=\min \{h : x_h\not \in I \}$ and notice that both $(I:x_j)$ and $(I+(x_j))$ strictly contain $I.$ For every $i>0$ the short exact sequence $ 0 \rightarrow S/(I:x_j) \rightarrow S/I \rightarrow S/(I+(x_j)) \rightarrow 0$ induces the exact sequence $${\ensuremath{\mathrm{Tor}}}_i(S/(I:x_j),S/J)
\rightarrow 
{\ensuremath{\mathrm{Tor}}}_i(S/I,S/J)
 \rightarrow 
{\ensuremath{\mathrm{Tor}}}_i(S/(I+(x_j)),S/J).$$ 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.

General intersections and general products
==========================================

In this section, we prove Theorems \[intersection\] and \[product\]. We will assume throughout the rest of the paper $\mathrm{char}(K)=0.$

A monomial ideal $I \subset S$ is said to be *$0$-Borel* (or *strongly stable*) if, for every monomial $u x_j \in I$ and for every $1 \leq i <j$ one has $ux_i \in I$. 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 $W \subset S$ of monomials in $S$ is said to be *$0$-Borel* if the ideal they generate is $0$-Borel, or equivalently if for every monomial $u x_j \in W$ and for every $1 \leq i <j$ one has $ux_i \in W$. Similarly we say that a monomial ideal $J \subset S$ is *opposite $0$-Borel* if for every monomial $ux_j \in J$ and for every $j < i \leq n$ one has $ux_i \in J$.

Let $>_{{\mathrm{{rev}}}}$ be the reverse lexicographic order induced by the ordering $x_1 > \cdots >x_n$. We recall the following result [@Mu Lemma 3.2].

\[3-1\] Let $V=\{v_1,\dots,v_s\} \subset S_d$ be a $0$-Borel set of monomials and $W =\{w_1,\dots,w_s\} \subset S_d$ the lex set of monomials, where $v_1 \geq_{{{\mathrm{{rev}}}}} \cdots \geq_{{{\mathrm{{rev}}}}} v_s$ and $w_1 \geq_{{{\mathrm{{rev}}}}} \cdots \geq _{{{\mathrm{{rev}}}}} w_s$. Then $v_i \geq_{{{\mathrm{{rev}}}}} w_i$ for all $i=1,2,\dots,s$.

Since generic initial ideals with respect to $>_{{\mathrm{lex}}}$ are $0$-Borel, the next lemma and Corollary \[2.6\](i) prove Theorem \[intersection\].

\[3-2\] Let $I \subset S$ be a $0$-Borel ideal and $P \subset S$ an opposite lex ideal. Then $\dim_K(I\cap P)_d \leq \dim_K (I^{{\mathrm{lex}}}\cap P)_d$ for all $d\geq 0$.

Fix a degree $d$. Let $V,W$ and $Q$ be the sets of monomials of degree $d$ in $I$, $I^{{\mathrm{lex}}}$ and $P$ respectively. It is enough to prove that $\# V \cap Q \leq \# W \cap Q$.

Observe that $Q$ is the set of the smallest $\#Q$ monomials in $S_d$ with respect to $>_{{\mathrm{{rev}}}}$. Let $m=\max_{>_{{\mathrm{{rev}}}}} Q$. Then by Lemma \[3-1\] $$\# V \cap Q = \# \{ v \in V: v \leq_{{{\mathrm{{rev}}}}} m\} 
\leq \# \{ w \in W: w \leq_{{{\mathrm{{rev}}}}} m\} = \# W \cap Q,$$ as desired.

Next, we consider products of ideals. For a monomial $u \in S$, let $\max u$ (respectively, $\min u$) be the maximal (respectively, minimal) integer $i$ such that $x_i$ divides $u$, where we set $\max 1 = 1$ and $\min 1 = n$. For a monomial ideal $I \subset S$, let $I_{(\leq k)}$ be the K-vector space spanned by all monomials $u \in I$ with $\max u \leq k$.

\[3-4\] Let $I \subset S$ be a $0$-Borel ideal and $P \subset S$ an opposite $0$-Borel ideal. Let $G(P)=\{u_1,\dots,u_s\}$ be the set of the minimal monomial generators of $P$. As a $K$-vector space, $IP$ is the direct sum $$IP=\bigoplus_{i=1}^s (I_{(\leq \min u_i)})u_i.$$

It is enough to prove that, for any monomial $w \in IP$, there is the unique expression $w=f(w)g(w)$ with $f(w) \in I$ and $g(w) \in P$ satisfying

-   $\max f(w) \leq \min g(w)$.

-   $g(w) \in G(P)$.

Given any expression $w=fg$ such that $f \in I$ and $g \in P$, since $I$ is $0$-Borel and $P$ is opposite $0$-Borel, if $\max f > \min g$ then we may replace $f$ by $f \frac{x_{\min g}} {x_{\max f}} \in I$ and replace $g$ by $g \frac{x_{\max f}} {x_{\min g}} \in P$. 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.

\[3-5\] Let $I \subset S$ be a $0$-Borel ideal and $P \subset S$ an opposite $0$-Borel ideal. Then $\dim_K(IP)_d \geq \dim_K (I^{{{\mathrm{lex}}}}P)_d$ for all $d\geq 0$.

Lemma \[3-1\] shows that $\dim_K {I_{(\leq k)}}_d \geq \dim_K {I^{{\mathrm{lex}}}_{(\leq k)}}_d$ for all $k$ and $d \geq 0$. Then the statement follows from Lemma \[3-4\].

Finally we prove Theorem \[product\].

Let $I'={\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I)$ and $J'={\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J)$. Since $I'$ is $0$-Borel and $J'$ is opposite $0$-Borel, by Corollary \[2.6\](ii) and Lemmas \[3-5\] $$H(Ig(J),d) \geq H(I'J',d) \geq H(I^{{{\mathrm{lex}}}} J',d) \geq H(I^{{\mathrm{lex}}}J^{{\mathrm{{oplex}}}},d)$$ for all $d \geq 0$.

\[rem1\] Theorems \[intersection\] and \[product\] are sharp. Let $I \subset S$ be a Borel-fixed ideal and $J \subset S$ an ideal satisfying that $h(J)=J$ for any lower triangular matrix $h \in {{GL}}_n(K)$. For a general $g \in {{GL}}_n(K)$, we have the LU decomposition $g=bh$ where $h \in {{GL}}_n(K)$ is a lower triangular matrix and $b \in {{GL}}_n(K)$ is an upper triangular matrix. Then as $K$-vector spaces $$I \cap g(J) \cong b^{-1}(I) \cap h(J)= I\cap J
\mbox{ and }
I g(J) \cong b^{-1}(I) h(J)= I J.$$ Thus if $I$ is lex and $J$ is opposite lex then $H(I\cap g(J),d)=H(I\cap J,d)$ and $H(Ig(J),d)=H(I J,d)$ for all $d\geq 0$.

\[example\] The assumption on ${\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(J)$ in Theorem \[intersection\] is necessary. Let $I=(x_1^3,x_1^2x_2,x_1x_2^2,x_2^3) \subset K[x_1,x_2,x_3]$ and $J=(x_3^2,x_3^2x_2,x_3x_2^2,x_2^3)\subset K[x_1,x_2,x_3]$. Then the set of monomials of degree $3$ in $I^{{\mathrm{lex}}}$ is $\{x_1^3,x_1^2x_2,x_1^2x_3,x_1x_2^2\}$ and that of $J^{{\mathrm{{oplex}}}}$ is $\{x_3^3,x_3^2x_2,x_3^2x_1,x_3x_2^2\}$. Hence $H(I^{{\mathrm{lex}}}\cap J^{{\mathrm{{oplex}}}},3)=0$. On the other hand, as we see in Remark \[rem1\], $H(I\cap g(J),3)=H(I\cap J,3)=1$. Similarly, the assumption on the characteristic of $K$ is needed as one can easily see by considering $\mathrm{char}(K)=p>0$, $I=(x_1^p,x_2^p)\subset K[x_1,x_2]$ and $J=x_2^p.$ In this case we have $H(I^{{\mathrm{lex}}}\cap J^{{\mathrm{{oplex}}}},p)=0$, while $H(I\cap g(J),p)=H(g^{-1}(I)\cap J,p)=1$ since $I$ is fixed under any change of coordinates.

Since ${\ensuremath{\mathrm{Tor}}}_0(S/I,S/J) \cong S/(I+J)$ and ${\ensuremath{\mathrm{Tor}}}_1(S/I,S/J) \cong (I\cap J)/ IJ$ for all homogeneous ideals $I \subset S$ and $J \subset S$, Theorems \[intersection\] and \[product\] show the next statement.

\[cor\] Conjecture \[conj\] is true if $i=0$ or $i=1.$

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[^1]: 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.
