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 [Reference Herzog and PopescuHP] and Gasharov [Reference GasharovGa]. In this paper, we
study a further generalization of these theorems.
Let 
$K$ be an infinite field and 
$S=K[x_{1},\ldots ,x_{n}]$ a standard graded polynomial ring. Recall that the Hilbert
function 
$H(M):\mathbb{Z}\rightarrow \mathbb{Z}$ of a finitely generated graded 
$S$-module 
$M$ is the numerical function defined by
$$\begin{eqnarray}H(M)(d)=\dim _{K}M_{d},\end{eqnarray}$$ where 
$M_{d}$ is the graded component of 
$M$ of degree 
$d$. A set 
$W$ of monomials of 
$S$ is said to be a lex-segment if, for all monomials 
$u,v\in S$ of the same degree, 
$u\in W$ and 
$v{>}_{\text{lex}}u$ imply 
$v\in W$, where 
${>}_{\text{lex}}$ 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-segment ideal if the set of monomials in 
$I$ is a lex-segment. The classical Macaulay’s theorem [Reference MacaulayMac] guarantees that, for any homogeneous ideal 
$I\subset S$, there exists a unique lex-segment ideal, denoted by 
$I^{\text{lex}}$, with the same Hilbert function as 
$I$. Green’s hyperplane section theorem [Reference GreenGr] states
Theorem 1.1. (Green’s hyperplane section theorem)
Let 
$I\subset S$ be a homogeneous ideal. For a general linear form 
$h\in S_{1}$,
$$\begin{eqnarray}H(I\cap (h))(d)\leqslant H(I^{\text{lex}}\cap (x_{n}))(d)\quad \text{for all }d\geqslant 0.\end{eqnarray}$$ Green’s hyperplane section theorem is known to be useful to prove several
important results on Hilbert functions such as Macaulay’s theorem [Reference MacaulayMac] and Gotzmann’s persistence theorem
[Reference GotzmannGo] (see [Reference GreenGr]). Herzog and Popescu [Reference Herzog and PopescuHP] (in
characteristic 
$0$) and Gasharov [Reference GasharovGa] (in positive
characteristic) generalized Green’s hyperplane section theorem in the
following form.
Theorem 1.2. (Herzog–Popescu, Gasharov)
Let 
$I\subset S$ be a homogeneous ideal. For a general homogeneous polynomial 
$h\in S$ of degree 
$a$,
$$\begin{eqnarray}H(I\cap (h))(d)\leqslant H(I^{\text{lex}}\cap (x_{n}^{a}))(d)\quad \text{for all }d\geqslant 0.\end{eqnarray}$$ We study a generalization of Theorems 1.1 and 1.2. Let 
${>}_{\text{oplex}}$ 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 an opposite lex-segment if, for all
monomials 
$u,v\in S$ of the same degree, 
$u\in W$ and 
$v{>}_{\text{oplex}}u$ imply 
$v\in W$. Also, we say that a monomial ideal 
$I\subset S$ is an opposite lex-segment ideal if the set of
monomials in 
$I$ is an opposite lex-segment. For a homogeneous ideal 
$I\subset S$, let 
$I^{\text{oplex}}$ be the opposite lex-segment ideal with the same Hilbert function as 
$I$ and let 
$\text{Gin}_{{\it\sigma}}(I)$ be the generic initial ideal [Reference EisenbudEi, Section 15.9] of 
$I$ with respect to a term order 
${>}_{{\it\sigma}}$.
In Section 3 we prove the following
Theorem 1.3. Suppose 
$\text{char}(K)=0$. Let 
$I\subset S$ and 
$J\subset S$ be homogeneous ideals such that 
$\text{Gin}_{\text{lex}}(J)$ is a lex-segment ideal. For a general change of coordinates 
$g$ of 
$S$,
$$\begin{eqnarray}H(I\cap g(J))(d)\leqslant H(I^{\text{lex}}\cap J^{\text{oplex}})(d)\quad \text{for all }d\geqslant 0.\end{eqnarray}$$ 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 a lex-segment ideal and 
$J$ is an opposite lex-segment ideal (Remark 3.5). Note also that if 
$\text{Gin}_{{\it\sigma}}(I)$ is a lex-segment ideal for some term order 
${>}_{{\it\sigma}}$ then 
$\text{Gin}_{\text{lex}}(J)$ must be a lex-segment ideal as well [Reference ConcaCo1, 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 
$\text{char}(K)=0$. Let 
$I\subset S$ and 
$J\subset S$ be homogeneous ideals. For a general change of coordinates 
$g$ of 
$S$,
$$\begin{eqnarray}H(Ig(J))(d)\geqslant H(I^{\text{lex}}J^{\text{oplex}})(d)\quad \text{for all }d\geqslant 0.\end{eqnarray}$$Inspired by Theorems 1.3 and 1.4, we suggest the following conjecture.
Conjecture 1.5. Suppose 
$\text{char}(K)=0.$ Let 
$I\subset S$ and 
$J\subset S$ be homogeneous ideals such that 
$\text{Gin}_{\text{lex}}(J)$ is a lex-segment ideal. For a general change of coordinates 
$g$ of 
$S$,
$$\begin{eqnarray}H(\text{Tor}_{i}(S/I,S/g(J)))(d)\leqslant H(\text{Tor}_{i}(S/I^{\text{lex}},S/J^{\text{oplex}}))(d)\quad \text{for all }d\geqslant 0.\end{eqnarray}$$ 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 a result of Conca [Reference ConcaCo2, Theorem 4.2]. Theorem 2.7, which we prove later, also provides
some evidence supporting the above inequality.
2 Dimension of 
$\text{Tor}$ and general change of coordinates
Let 
$\text{GL}_{n}(K)$ be the general linear group of invertible 
$n\times n$ matrixes over 
$K$. Throughout the paper, we identify each element 
$g=(g_{ij})\in \text{GL}_{n}(K)$ with the change of coordinates defined by 
$g(x_{j})=\sum _{i=1}^{n}g_{ij}x_{i}$ for all 
$j$.
We say that a property (P) holds for a general 
$g\in \text{GL}_{n}(K)$ if there is a nonempty Zariski open subset 
$U\subset \text{GL}_{n}(K)$ such that (P) holds for all 
$g\in U$.
We first present two lemmas which will allow us to reduce the proofs of the theorems in the introduction to combinatorial considerations regarding Borel-fixed ideals. The first lemma (Lemma 2.1) is probably clearly true to some experts, but we include its proof for the sake of the exposition. The ideas used in the second lemma (Lemma 2.3) are similar to that of [Reference CavigliaCa1, 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.
For two ideals 
$I\subset S$ and 
$J\subset S$, we define the functions 
$H(\cap ,I,J):\mathbb{Z}\rightarrow \mathbb{Z}$, 
$H(\cdot ,I,J):\mathbb{Z}\rightarrow \mathbb{Z}$ and 
$H(+,I,J):\mathbb{Z}\rightarrow \mathbb{Z}$ by
$$\begin{eqnarray}\displaystyle & H(\cap ,I,J)(d)=\min \{H(I\cap g(J))(d):g\in \text{GL}_{n}(K)\}, & \displaystyle \nonumber\\ \displaystyle & H(\cdot ,I,J)(d)=\max \{H(Ig(J))(d):g\in \text{GL}_{n}(K)\} & \displaystyle \nonumber\end{eqnarray}$$and
$$\begin{eqnarray}H(+,I,J)(d)=\max \{H(I+g(J))(d):g\in \text{GL}_{n}(K)\}\end{eqnarray}$$ for all 
$d\in \mathbb{Z}$.
Lemma 2.1. Let 
$I\subset S$ and 
$J\subset S$ be homogeneous ideals. There is a nonempty Zariski open subset 
$U\subset \text{GL}_{n}(K)$ such that 
$H(\cap ,I,J)=H(I\cap g(J))$, 
$H(\cdot ,I,J)=H(Ig(J))$ and 
$H(+,I,J)=H(I+g(J))$ for any 
$g\in U$.
Proof. We prove the statement for 
$I+g(J)$, which implies the desired statement for 
$I\cap g(J)$ (the proof for 
$Ig(J)$ is similar).
Let 
$t_{kl}$, where 
$1\leqslant k,l\leqslant n$, be indeterminates, 
$\tilde{K}=K(t_{kl}:1\leqslant k,l\leqslant n)$ the field of fractions of 
$K[t_{kl}:1\leqslant k,l\leqslant n]$ and 
$A=\tilde{K}[x_{1},\ldots ,x_{n}]$. Let 
${\it\rho}:S\rightarrow A$ be the ring map induced by 
${\it\rho}(x_{l})=\sum _{k=1}^{n}t_{kl}x_{k}$ for 
$l=1,2,\ldots ,n$, and 
$\tilde{L}=IA+{\it\rho}(J)A\subset A$. Let 
$L\subset S$ be the monomial ideal with the same monomial generators as 
$\text{in}_{\text{lex}}(\tilde{L})$.
We claim that
$$\begin{eqnarray}\displaystyle \dim _{K}L_{d}=\dim _{\tilde{K}}\tilde{L}_{d}\geqslant \dim _{K}(I+g(J))_{d} & & \displaystyle\end{eqnarray}$$ for any 
$d\in \mathbb{Z}$ and 
$g\in \text{GL}_{n}(K)$. The first equality is clear. To see the second inequality, let 
${\it\alpha}_{1},\ldots ,{\it\alpha}_{\ell }$ be a 
$K$-basis of 
$I_{d}$ and let 
${\it\beta}_{1},\ldots ,{\it\beta}_{m}$ be that of 
$J_{d}$. Then 
$\dim _{K}(I+g(J))_{d}$ is the rank of the matrix representing 
$({\it\alpha}_{1},\ldots ,{\it\alpha}_{\ell },g({\it\beta}_{1}),\ldots ,g({\it\beta}_{m}))$ with respect to the monomial basis of 
$S_{d}$. This rank is maximized when 
$g$ is sufficiently general. Also, this rank equals to 
$\dim _{\tilde{K}}\tilde{L}_{d}$ for a general 
$g$.
Let 
$f_{1},\ldots ,f_{s}$ be generators of 
$I$ and 
$g_{1},\ldots ,g_{t}$ those of 
$J$. Then the polynomials 
$f_{1},\ldots ,f_{s},{\it\rho}(g_{1}),\ldots ,{\it\rho}(g_{t})$ are generators of 
$\tilde{L}$. By the Buchberger algorithm, one can compute a
Gröbner basis of 
$\tilde{L}$ from 
$f_{1},\ldots ,f_{s},{\it\rho}(g_{1}),\ldots ,{\it\rho}(g_{t})$ by finite steps. Consider all elements 
$h_{1},\ldots ,h_{m}\in K(t_{kl}:1\leqslant k,l\leqslant n)$ which are the coefficients 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. Let 
$U=\{g\in \text{GL}_{n}(K):h_{i}(g)\neq 0\text{ for }i=1,2,\ldots ,m\}$, where 
$h_{i}(g)$ is an element obtained from 
$h_{i}$ by substituting 
$t_{kl}$ with the entries of 
$g$. By construction 
$U$ is a nonempty Zariski open subset and 
$\text{in}_{\text{lex}}(I+g(J))=L$ for every 
$g\in U$. This fact and (1) prove the desired statement.◻
The method used to prove the above lemma can be easily generalized to a number of
situations. For a finitely generated graded 
$S$-module 
$M$ and for a homogeneous ideal 
$J\subset S$, define the function 
$H(\text{Tor}_{i},M,J):\mathbb{Z}\rightarrow \mathbb{Z}$ by
$$\begin{eqnarray}H(\text{Tor}_{i},M,J)(d)=\min \{H(\text{Tor}_{i}(M,S/g(J)))(d):g\in \text{GL}_{n}(K)\}\end{eqnarray}$$ for all 
$d\in \mathbb{Z}$.
Proposition 2.2. With the same notation as above, there is a nonempty Zariski open subset 
$U\subset \text{GL}_{n}(K)$ such that 
$H(\text{Tor}_{i},M,J)=H(\text{Tor}_{i}(M,S/g(J)))$ for any 
$g\in U$.
Proof. Let 
$\mathbb{F}:0\stackrel{{\it\varphi}_{p+1}}{\longrightarrow }\mathbb{F}_{p}\stackrel{{\it\varphi}_{p}}{\longrightarrow }\cdots \longrightarrow \mathbb{F}_{1}\stackrel{{\it\varphi}_{1}}{\longrightarrow }\mathbb{F}_{0}\stackrel{{\it\varphi}_{0}}{\longrightarrow }0$ be a graded free resolution of 
$M.$ Given a change of coordinates 
$g$, one first notes that for every 
$i=0,1,\ldots ,p$, the Hilbert function 
$H(\text{Tor}_{i}(M,S/g(J)))$ is equal to the difference between the Hilbert function of 
$\text{Ker}({\it\pi}_{i-1}\circ {\it\varphi}_{i})$ and that of 
${\it\varphi}_{i+1}(F_{i+1})+F_{i}\otimes _{S}g(J)$ where 
${\it\pi}_{i-1}:F_{i-1}\rightarrow F_{i-1}\otimes _{S}S/g(J)$ is the canonical projection. Hence we have
$$\begin{eqnarray}\displaystyle & & \displaystyle H(\text{Tor}_{i}(M,S/g(J)))(d)\nonumber\\ \displaystyle & & \displaystyle \qquad \;\;\quad =H(F_{i})(d)-H({\it\varphi}_{i}(F_{i})+g(J)F_{i-1})(d)+H(g(J)F_{i-1})(d)\nonumber\\ \displaystyle & & \displaystyle \qquad \;\;\qquad -\,H({\it\varphi}_{i+1}(F_{i+1})+g(J)F_{i})(d)\nonumber\end{eqnarray}$$ for all 
$d\in \mathbb{Z}$. Clearly 
$H(F_{i})$ and 
$H(g(J)F_{i-1})$ do not depend on 
$g.$ Then, in the same way as in the proof of Lemma 2.1, one can prove that there is a
nonempty Zariski open subset 
$U\subset \text{GL}_{n}(K)$ such that the Hilbert function of 
${\it\varphi}_{i}(F_{i})+g(J)F_{i-1}$ is maximal for any 
$g\in U$.◻
Note that Lemma 2.1 can be
considered as a special case of the above proposition since 
$\text{Tor}_{0}(S/I,S/J)\cong S/(I+J)$ and 
$\text{Tor}_{0}(I,S/J)\cong I/IJ$.
For a vector 
$\mathbf{w}=(w_{1},\ldots ,w_{n})\in \mathbb{Z}_{{\geqslant}0}^{n}$, let 
$\text{in}_{\mathbf{w}}(I)$ be the initial ideal of a homogeneous ideal 
$I$ with respect to the weight order 
${>}_{\mathbf{w}}$ (see [Reference EisenbudEi, p. 345]).
Let 
$T$ be a new indeterminate and 
$R=S[T]$. For 
$\mathbf{a}=(a_{1},\ldots ,a_{n})\in \mathbb{Z}_{{\geqslant}0}^{n}$, let 
$x^{\mathbf{a}}=x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}$ and 
$(\mathbf{a},\mathbf{w})=a_{1}w_{1}+\cdots +a_{n}w_{n}$. For a polynomial 
$f=\sum _{\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{n}}c_{\mathbf{a}}x^{\mathbf{a}}$, where 
$c_{\mathbf{a}}\in K$, let 
$b=\max \{(\mathbf{a},\mathbf{w}):c_{\mathbf{a}}\neq 0\}$ and
$$\begin{eqnarray}\tilde{f}=T^{b}\left(\mathop{\sum }_{\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{n}}T^{-(\mathbf{a},\mathbf{w})}c_{\boldsymbol{ a}}x^{\mathbf{a}}\right)\in R.\end{eqnarray}$$ Note that 
$\tilde{f}$ can be written as 
$\tilde{f}=\text{in}_{\mathbf{w}}(f)+Tg$ where 
$g\in R$. For an ideal 
$I\subset S$, let 
$\tilde{I} =(\tilde{f}:f\in I)\subset R$. For 
${\it\lambda}\in K\setminus \{0\}$, let 
$D_{{\it\lambda},\mathbf{w}}$ be the diagonal change of coordinates defined by 
$D_{{\it\lambda},\mathbf{w}}(x_{i})={\it\lambda}^{-w_{i}}x_{i}$. From the definition, we have
$$\begin{eqnarray}R/\big(\tilde{I} +(T)\big)\cong S/\text{in}_{\mathbf{w}}(I)\end{eqnarray}$$and
$$\begin{eqnarray}R/\big(\tilde{I} +(T-{\it\lambda})\big)\cong S/D_{{\it\lambda},\mathbf{w}}(I)\end{eqnarray}$$ where 
${\it\lambda}\in K\setminus \{0\}$. Moreover, 
$(T-{\it\lambda})$ is a nonzero divisor of 
$R/\tilde{I}$ for any 
${\it\lambda}\in K$. See [Reference EisenbudEi,
Section 15.8].
Lemma 2.3. Fix an integer 
$j$. Let 
$\mathbf{w}\in \mathbb{Z}_{{\geqslant}0}^{n}$, 
$M$ a finitely generated graded 
$S$-module and 
$J\subset S$ a homogeneous ideal. For a general 
${\it\lambda}\in K$, one has
$$\begin{eqnarray}H\big(\text{Tor}_{i}\big(M,S/\text{in}_{\mathbf{w}}(J)\big)\big)(j)\geqslant H\big(\text{Tor}_{i}\big(M,S/D_{{\it\lambda},\mathbf{w}}(J)\big)\big)(j)\quad \text{for all }i.\end{eqnarray}$$Proof. Consider the ideal 
$\tilde{J}\subset R$ defined as above. Let 
$\tilde{M}=M\otimes _{S}R$ and 
$T_{i}=\text{Tor}_{i}^{R}(\tilde{M},R/\tilde{J})$. By the structure theorem for modules over a PID (see [Reference LangLa, p. 149]), we have
$$\begin{eqnarray}(T_{i})_{j}\cong K[T]^{a_{ij}}\bigoplus A_{ij}\end{eqnarray}$$ as a finitely generated 
$K[T]$-module, where 
$a_{ij}\in \mathbb{Z}_{{\geqslant}0}$ and where 
$A_{ij}$ is the torsion submodule. Moreover, 
$A_{ij}$ is a module of the form
$$\begin{eqnarray}A_{ij}\cong \bigoplus _{h=1}^{b_{ij}}K[T]/(P_{h}^{i,j}),\end{eqnarray}$$ where 
$P_{h}^{i,j}$ is a nonzero polynomial in 
$K[T]$. Set 
$l_{{\it\lambda}}=T-{\it\lambda}$. Consider the exact sequence
By considering the long exact sequence induced by 
$\text{Tor}_{i}^{R}(\tilde{M},-)$, we have the following exact sequence
$$\begin{eqnarray}0\longrightarrow T_{i}/l_{{\it\lambda}}T_{i}\longrightarrow \text{Tor}_{i}^{R}\big(\tilde{M},R/\big((l_{{\it\lambda}})+\tilde{J}\big)\big)\longrightarrow K_{i-1}\longrightarrow 0,\end{eqnarray}$$ where 
$K_{i-1}$ is the kernel of the map 
$T_{i-1}\xrightarrow[]{\cdot l_{{\it\lambda}}}T_{i-1}$. Since 
$l_{{\it\lambda}}$ is a regular element for 
$R$ and 
$\tilde{M}$, the middle term in (3) is isomorphic to
$$\begin{eqnarray}\displaystyle \text{Tor}_{i}^{R/(l_{{\it\lambda}})}\big(\tilde{M}/l_{{\it\lambda}}\tilde{M},R/\big((l_{{\it\lambda}})+\tilde{J}\big)\big)=\left\{\begin{array}{@{}ll@{}}\text{Tor}_{i}^{S}\big(M,S/\text{in}_{\boldsymbol{ w}}(J)\big) & \text{ if }{\it\lambda}=0,\\ \text{Tor}_{i}^{S}\big(M,S/D_{{\it\lambda},\mathbf{w}}(J)\big) & \text{ if }{\it\lambda}\neq 0,\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$ (see [Reference MatsumuraMat,
p. 140]). By taking the graded component of degree 
$j$ in (3), we
obtain
$$\begin{eqnarray}\displaystyle \begin{array}{@{}rcl@{}}\dim _{K}\text{Tor}_{i}^{S}\big(M,S/\text{in}_{\mathbf{w}}(J)\big)_{j}\ & =\ & a_{ij}+\#\{P_{h}^{ij}:P_{h}^{i,j}(0)=0\}\\ \ & \ & +\#\{P_{h}^{i-1,j}:P_{h}^{i-1,j}(0)=0\},\end{array} & & \displaystyle\end{eqnarray}$$ where 
$\#X$ denotes the cardinality of a finite set 
$X$, and
$$\begin{eqnarray}\displaystyle \dim _{K}\text{Tor}_{i}^{S}\big(M,S/D_{{\it\lambda},\mathbf{w}}(J)\big)_{j} & = & \displaystyle a_{ij}\end{eqnarray}$$ for a general 
${\it\lambda}\in K$. This proves the desired inequality.◻
Corollary 2.4. With the same notation as in Lemma 2.3, for a general 
${\it\lambda}\in K$,
$$\begin{eqnarray}H\big(\text{Tor}_{i}\big(M,\text{in}_{\mathbf{w}}(J)\big)\big)(j)\geqslant H\big(\text{Tor}_{i}\big(M,D_{{\it\lambda},\mathbf{w}}(J)\big)\big)(j)\quad \text{for all }i.\end{eqnarray}$$Proof. For any homogeneous ideal 
$I\subset S$, by considering the long exact sequence induced by 
$\text{Tor}_{i}(M,-)$ from the short exact sequence 
$0\longrightarrow I\longrightarrow S\longrightarrow S/I\longrightarrow 0$, we have
$$\begin{eqnarray}\text{Tor}_{i}(M,I)\cong \text{Tor}_{i+1}(M,S/I)\quad \text{for }i\geqslant 1\end{eqnarray}$$and
$$\begin{eqnarray}\dim _{K}\!\text{Tor}_{0}(M,I)_{j}=\dim _{K}\!\text{Tor}_{1}(M,S/I)_{j}+\dim _{K}\!M_{j}-\dim _{K}\!\text{Tor}_{0}(M,S/I)_{j}.\end{eqnarray}$$Thus, by Lemma 2.3 it is enough to prove that
$$\begin{eqnarray}\displaystyle \dim _{K} & & \displaystyle \!\!\text{Tor}_{1}\big(M,S/\text{in}_{\mathbf{w}}(J)\big)_{j}-\dim _{K}\text{Tor}_{1}\big(M,S/D_{{\it\lambda},\mathbf{w}}(J)\big)_{j}\nonumber\\ \displaystyle & & \displaystyle \geqslant \dim _{K}\text{Tor}_{0}\big(M,S/\text{in}_{\mathbf{w}}(J)\big)_{j}-\dim _{K}\text{Tor}_{0}\big(M,S/D_{{\it\lambda},\mathbf{w}}(J)\big)_{j}.\nonumber\end{eqnarray}$$Proposition 2.5. Fix an integer 
$j$. Let 
$I\subset S$ and 
$J\subset S$ be homogeneous ideals. Let 
$\mathbf{w},\mathbf{w}^{\prime }\in \mathbb{Z}_{{\geqslant}0}^{n}$. Then
(i)
$H(\text{Tor}_{i},S/I,J)(j)\leqslant H(\text{Tor}_{i}(S/\text{in}_{\mathbf{w}}(I),S/\text{in}_{\mathbf{w}^{\prime }}(J)))(j)\text{ for all }i.$(ii)
$H(\text{Tor}_{i},I,J)(j)\leqslant H(\text{Tor}_{i}(\text{in}_{\mathbf{w}}(I),S/\text{in}_{\mathbf{w}^{\prime }}(J)))(j)\text{ for all }i.$
Proof. We prove (ii) (the proof for (i) is similar). By Lemma 2.3 and Corollary 2.4, we have
$$\begin{eqnarray}\displaystyle H\big(\text{Tor}_{i}\big(\text{in}_{\mathbf{w}}(I),S/\text{in}_{\mathbf{w}^{\prime }}(J)\big)\big)(j) & {\geqslant} & \displaystyle H\big(\text{Tor}_{i}\big(D_{{\it\lambda}_{1},\mathbf{w}}(I),S/D_{{\it\lambda}_{2},\mathbf{w}^{\prime }}(J)\big)\big)(j)\nonumber\\ \displaystyle & = & \displaystyle H(\text{Tor}_{i}\big(I,S/D_{{\it\lambda}_{1},\mathbf{w}}^{-1}\big(D_{{\it\lambda}_{2},\mathbf{w}^{\prime }}(J)\big)\big))(j)\nonumber\\ \displaystyle & {\geqslant} & \displaystyle H(\text{Tor}_{i},S/I,J)(j),\nonumber\end{eqnarray}$$ as desired, where 
${\it\lambda}_{1},{\it\lambda}_{2}$ are general elements in 
$K$.◻
Remark 2.6. Let 
$\mathbf{w}^{\prime }=(1,1,\ldots ,1)$ and note that the composite of two general changes of
coordinates is still general. By replacing 
$J$ by 
$g(J)$ for a general change of coordinates 
$g,$ from Proposition 2.5(i) it follows that
$$\begin{eqnarray}H(\text{Tor}_{i}(S/I,S/g(J)))(j)\leqslant H(\text{Tor}_{i}(S/\text{in}_{{>}_{{\it\sigma}}}(I),S/g(J)))(j)\end{eqnarray}$$ for any term order 
${>}_{{\it\sigma}}$.
The above fact gives, as a special case, an affirmative answer to [Reference ConcaCo2, Question 6.1]. This was originally proved in the thesis of the first author [Reference CavigliaCa2]. We mention it here because there seem to be no published article which includes the proof of this fact.
Theorem 2.7. Fix an integer 
$j$. Let 
$I\subset S$ and 
$J\subset S$ be homogeneous ideals, and let 
${>}_{{\it\sigma}}$ and 
${>}_{{\it\tau}}$ be term orders. Then
(i)
$H(\text{Tor}_{i},S/I,J)(j)\leqslant H(\text{Tor}_{i}(S/\text{Gin}_{{\it\sigma}}(I),S/\text{Gin}_{{\it\tau}}(J)))(j)\text{ for all }i$.(ii)
$H(\text{Tor}_{i},I,J)(j)\leqslant H(\text{Tor}_{i}(\text{Gin}_{{\it\sigma}}(I),S/\text{Gin}_{{\it\tau}}(J)))(j)\text{ for all }i$.
Proof. Without loss of generality, we may assume 
$\text{in}_{{\it\sigma}}(I)=\text{Gin}_{{\it\sigma}}(I)$ and that 
$\text{in}_{{\it\tau}}(J)=\text{Gin}_{{\it\tau}}(J)$. It follows from [Reference EisenbudEi,
Proposition 15.16] that there are vectors 
$\mathbf{w},\mathbf{w}^{\prime }\in \mathbb{Z}_{{\geqslant}0}^{n}$ such that 
$\text{in}_{\mathbf{w}}(I)=\text{in}_{{\it\sigma}}(I)$ and 
$\text{in}_{\mathbf{w}^{\prime }}(J)=\text{Gin}_{{\it\tau}}(J)$. Then the desired inequality follows from
Proposition 2.5.◻
We later use the following special case of Theorem 2.7.
Corollary 2.8. Let 
$I\subset S$ and 
$J\subset S$ be homogeneous ideals. Then
(i)
$H(\cap ,I,J)(d)\leqslant H(\text{Gin}_{\text{lex}}(I)\cap \text{Gin}_{\text{oplex}}(J))(d)$ for all $d\geqslant 0$.(ii)
$H(\cdot ,I,J)(d)\geqslant H(\text{Gin}_{\text{lex}}(I)\text{Gin}_{\text{oplex}}(J))(d)$ for all $d\geqslant 0$.
We conclude this section with a result regarding the Krull dimension of certain Tor
modules. We show how Theorem 2.7 can be used to give a quick proof of Proposition 2.9, which is a special case (for the
variety 
$X=\mathbb{P}^{n-1}$ and the algebraic group 
$SL_{n}$) of the main Theorem of [Reference Miller and SpeyerMSp].
Let 
$B_{+}\subset \text{GL}_{n}(K)$ (resp. 
$B_{-}\subset \text{GL}_{n}(K)$) be the set of the nonsingular upper triangular
(resp. lower triangular) matrixes. For a group 
$G\subset \text{GL}_{n}(K)$, a homogeneous ideal 
$I\subset S$ is said to be 
$G$-fixed if 
$b(I)=I$ for any 
$b\in G$. Recall that, for an ideal 
$I$ of 
$S$, 
$\text{Gin}_{\text{lex}}(I)$ is 
$B_{+}$-fixed and 
$\text{Gin}_{\text{oplex}}(I)$ is 
$B_{-}$-fixed. See [Reference EisenbudEi,
Section 15.9] for more details on the combinatorial properties of 
$B_{+}$-fixed ideals.
Let 
$I$ and 
$J$ be ideals generated by linear forms. If we assume that 
$I$ is 
$B_{+}$-fixed and that 
$J$ is 
$B_{-}$-fixed, then there exist 
$1\leqslant i,j\leqslant n$ such that 
$I=(x_{1},\ldots ,x_{i})$ and 
$J=(x_{j},\ldots ,x_{n}).$ An easy computation shows that the Krull dimension of 
$\text{Tor}_{i}(S/I,S/J)$ is always zero when 
$i>0.$
More generally one has
Proposition 2.9. (Miller–Speyer)
Let 
$M$ be a finitely generated graded 
$S$-module, and let 
$J$ be a homogeneous ideal of 
$S.$ For a general change of coordinates 
$g$, the Krull dimension of 
$\text{Tor}_{i}(M,S/g(J))$ is zero for all 
$i>0.$
Proof. By considering a filtration 
$M=M_{k}\supset M_{k-1}\supset \cdots \supset M_{1}\supset 0$ such that 
$M_{k}/M_{k-1}\cong S/P_{k}$ for some prime ideal 
$P_{k}$ (see [Reference EisenbudEi, Proposition
3.7]), it is enough to consider the case when 
$M=S/I$ for some homogeneous ideal 
$I$. By Theorem 2.7, we may assume that 
$I$ is 
$B_{+}$-fixed and 
$J$ is 
$B_{-}$-fixed. Since an associated prime ideal of 
$B_{+}$-fixed ideal is an ideal of the form 
$(x_{1},\ldots ,x_{a})$ [Reference EisenbudEi, Corollary 15.25], by
considering filtrations of 
$S/I$ and 
$S/J$, one may assume 
$I=(x_{1},\ldots ,x_{a})$ and 
$J=(x_{b},\ldots ,x_{n})$. This proves the desired property.◻
3 General intersections and general products
In this section, we prove Theorems 1.3 and 1.4. We assume throughout
the rest of the paper 
$\text{char}(K)=0.$
A monomial ideal 
$I\subset S$ is said to be 0-Borel (or strongly
stable) if, for every monomial 
$ux_{j}\in I$ and for every 
$1\leqslant i<j$ one has 
$ux_{i}\in I$. Note that
$0$-Borel ideals are precisely all the possible 
$B_{+}$-fixed ideals in characteristic 
$0$. In general, the 
$B_{+}$-fixed property depends on the characteristic of the field and we refer
the readers to [Reference EisenbudEi, Section 15.9] for
the details. A set 
$W\subset S$ of monomials in 
$S$ is said to be 0-Borel if for every monomial 
$ux_{j}\in W$ and for every 
$1\leqslant i<j$ 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 
$j<i\leqslant n$ one has 
$ux_{i}\in J$.
Let 
${>}_{\text{rev}}$ be the reverse lexicographic order induced by the ordering 
$x_{1}>\cdots >x_{n}$. We recall the following result [Reference MuraiMu, Lemma 3.2].
Lemma 3.1. Let 
$V=\{v_{1},\ldots ,v_{s}\}\subset S_{d}$ be a 
$0$-Borel set of monomials and 
$W=\{w_{1},\ldots ,w_{s}\}\subset S_{d}$ the lex-segment set of monomials, where 
$v_{1}\geqslant _{\text{rev}}\cdots \geqslant _{\text{rev}}v_{s}$ and 
$w_{1}\geqslant _{\text{rev}}\cdots \geqslant _{\text{rev}}w_{s}$. Then 
$v_{i}\geqslant _{\text{rev}}w_{i}$ for all 
$i=1,2,\ldots ,s$.
Since generic initial ideals with respect to 
${>}_{\text{lex}}$ are 
$0$-Borel, the next lemma and Corollary 2.8(i) prove Theorem 1.3.
Lemma 3.2. Let 
$I\subset S$ be a 
$0$-Borel ideal and 
$P\subset S$ an opposite lex-segment ideal. Then 
$H(I\cap P)(d)\leqslant H(I^{\text{lex}}\cap P)(d)$ for all 
$d\geqslant 0$.
Proof. Fix a degree 
$d$. Let 
$V,W$ and 
$Q$ be the sets of monomials of degree 
$d$ in 
$I$, 
$I^{\text{lex}}$ and 
$P$, respectively. It is enough to prove that 
$\#V\cap Q\leqslant \#W\cap Q$.
Observe that 
$Q$ is the set of the smallest 
$\#Q$ monomials in 
$S_{d}$ with respect to 
${>}_{\text{rev}}$. Let 
$m=\max _{{>}_{\text{rev}}}Q$. Then by Lemma 3.1
$$\begin{eqnarray}\#V\cap Q=\#\{v\in V:v\leqslant _{\text{rev}}m\}\leqslant \#\{w\in W:w\leqslant _{\text{rev}}m\}=\#W\cap Q,\end{eqnarray}$$as desired. ◻
Next, we consider products of ideals. For a monomial 
$u\in S$, let 
$\max u$ (resp., 
$\min u$) be the maximal (resp., 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 
$I_{({\leqslant}k)}$ be the 
$K$-vector space spanned by all monomials 
$u\in I$ with 
$\max u\leqslant k$.
Lemma 3.3. Let 
$I\subset S$ be a 
$0$-Borel ideal and 
$P\subset S$ an opposite 
$0$-Borel ideal. Let 
$G(P)=\{u_{1},\ldots ,u_{s}\}$ be the set of the minimal monomial generators of 
$P$. As a 
$K$-vector space, 
$IP$ is the direct sum
$$\begin{eqnarray}IP=\bigoplus _{i=1}^{s}(I_{({\leqslant}\min u_{i})})u_{i}.\end{eqnarray}$$Proof. 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
(a)
$\max f(w)\leqslant \min g(w)$.(b)
$g(w)\in G(P)$.
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 
$w=f^{\prime }(w)g^{\prime }(w)$ satisfy conditions (a) and (b). Then, by (a), 
$g(w)$ divides 
$g^{\prime }(w)$ or 
$g^{\prime }(w)$ divides 
$g(w)$. Since 
$g(w)$ and 
$g^{\prime }(w)$ are generators of 
$P$, 
$g(w)=g^{\prime }(w)$. Hence the expression is unique.◻
Lemma 3.4. Let 
$I\subset S$ be a 
$0$-Borel ideal and 
$P\subset S$ an opposite 
$0$-Borel ideal. Then 
$H(IP)(d)\geqslant H(I^{\text{lex}}P)(d)$ for all 
$d\geqslant 0$.
Proof. Lemma 3.1 shows that 
$\dim _{K}{I_{({\leqslant}k)}}_{d}\geqslant \dim _{K}{I_{({\leqslant}k)}^{\text{lex}}}_{d}$ for all 
$k$ and 
$d\geqslant 0$. Then the statement follows from Lemma 3.3.◻
Finally we prove Theorem 1.4.
Proof of Theorem 1.4.
Let 
$I^{\prime }=\text{Gin}_{\text{lex}}(I)$ and 
$J^{\prime }=\text{Gin}_{\text{oplex}}(J)$. Since 
$I^{\prime }$ is 
$0$-Borel and 
$J^{\prime }$ is opposite 
$0$-Borel, by Corollary 2.8(ii) and Lemma 3.4
$$\begin{eqnarray}H(Ig(J))(d)\geqslant H(I^{\prime }J^{\prime })(d)\geqslant H(I^{\text{lex}}J^{\prime })(d)\geqslant H(I^{\text{lex}}J^{\text{oplex}})(d)\end{eqnarray}$$ for all 
$d\geqslant 0$.◻
Remark 3.5. Theorems 1.3
and 1.4 are sharp. Let 
$I\subset S$ be a 
$B_{+}$-fixed ideal and 
$J\subset S$ a 
$B_{-}$-fixed ideal. For a general 
$g\in \text{GL}_{n}(K)$, we have the LU decomposition 
$g=bb^{\prime }$ where 
$b\in B_{+}$ and 
$b^{\prime }\in B_{-}$. Then
$$\begin{eqnarray}H(\cap ,I,J)=H(I\cap g(J))=H(b^{-1}(I)\cap b^{\prime }(J))=H(I\cap J)\end{eqnarray}$$and similarly
$$\begin{eqnarray}H(\cdot ,IJ)=H(b(I)b^{\prime }(J))=H(IJ).\end{eqnarray}$$ Thus, if 
$I$ is a lex-segment ideal and 
$J$ is an opposite lex-segment ideal then we have equalities in
Theorems 1.3 and 1.4.
Remark 3.6. The assumption on 
$\text{Gin}_{\text{lex}}(J)$ in Theorem 1.3 is necessary. Let 
$I=(x_{1}^{3},x_{1}^{2}x_{2},x_{1}x_{2}^{2},x_{2}^{3})\subset K[x_{1},x_{2},x_{3}]$ and 
$J=(x_{3}^{2},x_{3}^{2}x_{2},x_{3}x_{2}^{2},x_{2}^{3})\subset K[x_{1},x_{2},x_{3}]$. Then
$$\begin{eqnarray}I^{\text{lex}}=(x_{1}^{3},x_{1}^{2}x_{2},x_{1}^{2}x_{3},x_{1}x_{2}^{2},x_{1}x_{2}x_{3}^{2},x_{1}x_{3}^{4},x_{2}^{6})\end{eqnarray}$$ and 
$J^{\text{oplex}}$ is obtained from 
$I^{\text{lex}}$ by exchanging 
$x_{1}$ and 
$x_{3}$. Then 
$H(I^{\text{lex}}\cap J^{\text{oplex}})(3)=0$. On the other hand, as we see in Remark 3.5, 
$H(I\cap g(J))(3)=H(I\cap J)(3)=1$. Similarly, the assumption on the characteristic of 
$K$ is needed as one can easily see by considering 
$\text{char}(K)=p>0$, 
$I=(x_{1}^{p},x_{2}^{p})\subset K[x_{1},x_{2}]$ and 
$J=(x_{2}^{p}).$ In this case we have 
$H((I^{\prime })^{\text{lex}}\cap (J^{\prime })^{\text{oplex}})(p)=0$, while 
$H(I^{\prime }\cap g(J^{\prime }))(p)=H(g^{-1}(I^{\prime })\cap J^{\prime })(p)=1$ since 
$I$ is fixed under any change of coordinates.
Remark 3.7. Let 
$h:\mathbb{Z}\rightarrow \mathbb{Z}$ and 
$h^{\prime }:\mathbb{Z}\rightarrow \mathbb{Z}$ be Hilbert functions of homogeneous ideals of 
$S$. One may ask if, there are ideals 
$L$ and 
$L^{\prime }$ such that 
$H(L)=h$, 
$H(L^{\prime })=h^{\prime }$ and
$$\begin{eqnarray}H(\cap ,L,L^{\prime })(d)=\max \{H(\cap ,I,J)(d):H(I)=h,~H(J)=h^{\prime }\}\end{eqnarray}$$ for all 
$d$. Corollary 2.8 and Remark 3.5 say that to study this question, one may assume that 
$L$ is 
$B_{+}$-fixed,
$L^{\prime }$ is 
$B_{-}$-fixed and 
$H(\cap ,L,L^{\prime })=H(L\cap L^{\prime })$.
Unfortunately, not all Hilbert functions 
$h$ and 
$h^{\prime }$ satisfy this property. Let 
$I$ and 
$J$ be ideals given in Remark 3.6 and 
$h=H(I)=H(J)$. Then
$$\begin{eqnarray}\max \{H(\cap ,I^{\prime },J^{\prime })(3):H(I^{\prime })=H(J^{\prime })=h\}=1.\end{eqnarray}$$ Also, if 
$L$ is 
$B_{+}$-fixed, 
$L^{\prime }$ is 
$B_{-}$-fixed and 
$H(L\cap L^{\prime })(3)=1$, then we have 
$L=I$ and 
$L^{\prime }=J$. However,
$$\begin{eqnarray}H(I\cap J)(5)=9<10=H(I^{\text{lex}}\cap J^{\text{lex}})(5).\end{eqnarray}$$ Since 
$\text{Tor}_{0}(S/I,S/J)\cong S/(I+J)$ and 
$\text{Tor}_{1}(S/I,S/J)\cong (I\cap J)/IJ$ for all homogeneous ideals 
$I\subset S$ and 
$J\subset S$, Theorems 1.3
and 1.4 show the next
statement.
Remark 3.8. Conjecture 1.5 is true if 
$i=0$ or 
$i=1.$
It would be of interest to study lower bounds of the Hilbert functions of the modules 
$\text{Tor}_{i}(S/I,S/J)$ and 
$\text{Ext}^{i}(S/I,S/J)$. For example, it was asked in [Reference Miller and SturmfelsMSt, Problem 18.35] which monomial ideal 
$I$ minimize the 
$K$-dimension of 
$\text{Hom}(I,S/I)$ among all monomial ideals 
$I\subset S$ of co-length 
$n$. At the moment, we are not sure if the techniques used in this paper
is applicable to this problem.
Acknowledgments
We thank the anonymous referee for many valuable suggestions. In particular, we thank the referee for providing a short proof of Proposition 2.9.
