s-Sequences and Monomial Modules

Abstract

In this paper we study a monomial module M generated by an s-sequence and the main algebraic and homological invariants of the symmetric algebra of M. We show that the first syzygy module of a finitely generated module M, over any commutative Noetherian ring with unit, has a specific initial module with respect to an admissible order, provided M is generated by an s-sequence. Significant examples complement the results.

1. Introduction

In this paper we consider finitely generated modules, over a Noetherian commutative ring with identity R, generated by an s-sequence, whose rank is greater or equal to one, that is the modules are not necessarily ideals.

In this direction, the modules that imitate the ideals are the direct sum modules $\oplus I_i{e}_i$, submodules of a free R-module with basis $\left\{e_i\right\},i=1,\dots ,n$, and $I_i$ ideals of R. Since the main idea in the use of Gröbner bases is to reduce all problems to questions of monomial ideals, we study the monomial submodules $\oplus I_i{e}_i$, where all $I_i$ are monomial ideals. Monomial modules were defined in [1] and were studied by many authors (see [2,3,4,5,6,7]). The aim of this paper is to investigate the symmetric algebra of a monomial module $M=\oplus I_i{e}_i$, a submodule of $R^n$, $R=K[x_1,\dots ,x_m]$, K a field, and $I_1,\dots ,I_n$ monomial ideals of R, via the theory of s-sequences [8,9,10]. the In Section 2, we review basic concepts of the theory of s-sequences and results about the main algebraic and homological invariants of the symmetric algebra of a finitely generated graded R-module M, generated by an s-sequence, provided R is a standard graded K-algebra and the generators of M are homogeneous sequence, or R is a polynomial ring in the field K. Then we introduce monomial modules and we recall several results and examples. After introducing a term order on the free module $M=I_i{e}_i$, $I_i\subset K[x_1,\dots ,x_m]$, which is induced by the order $x_1<x_2<\dots <x_m<e_1<\dots <e_n$, we formulate sufficient conditions to be a monomial module M generated by an s-sequence. As an application, we consider the special class of squarefree monomial S-modules $M=\oplus I^{\left(i\right)}{e}_i$, where each $I^{\left(i\right)}$ is the $(t_i-1)$-th squarefree Veronese ideal of the polynomial ring $S^{\left(i\right)}=K[x_1^{\left(i\right)},\dots ,x_{t_i}^{\left(i\right)}]$, $S=K[{\underline{x}}^{\left(1\right)},{\underline{x}}^{\left(2\right)},\dots ,{\underline{x}}^{\left(n\right)}]$, ${\underline{x}}^i=\{x_1^{\left(i\right)},x_2^{\left(i\right)},\dots ,x_{t_i}^{\left(i\right)}\},1\le i\le n$. In Section 3, inspired by [8], we introduce an admissible term order on the free module $R^n$, with basis $\left\{e_i\right\}$, $i=1,\dots ,n$, such that $e_1<e_2<\dots <e_n$, R a Noetherian ring with unit. We prove a remarkable result for the feature of the initial module, with respect to <, of the first syzygy module of a finitely generated R-module M generated by an s-sequence. Finally, we give an application to the first syzygy module of the class of mixed product ideals in two sets of variables [11,12], generated by an s-sequence [13,14,15].

Although the theory of s-sequences is defined in any field K, $char\left(K\right)=p\ge 0$, p a prime natural number, we fix the field $K=\mathbb{Q}$ if we use software CoCoA ([16]) to compute the Gröbner basis of the relation ideal of the symmetric algebra of a finitely generated $K[x_1,\dots ,x_m]$-module and the related algebraic invariants.

2. s-Sequences and Monomial Modules

The notion of s-sequences was given first in [8]. Let R be a Noetherian ring and let M be a finitely generated R-module with generators $f_1,f_2,\cdots ,f_n$. We denote by $\left(a_{ij}\right)$, $i=1,\cdots ,t,j=1,\dots ,n$, the presentation matrix of M and by $Sy{m}_R\left(M\right)={\oplus }_{i\ge 0}Sy{m}_i\left(M\right)$ the symmetric algebra of M, $Sy{m}_i\left(M\right)$ the i-th symmetric power of $Sy{m}_R\left(M\right)$. Note that $Sy{m}_R\left(M\right)=R[y_1,\dots ,y_n]/J$, where $J=(g_1,\dots ,g_t)$, and $g_i={\sum }_{j=1}^n{a}_{ij}{y}_j$, $i=1,\dots ,t$. We consider a graded ring $S=R[y_1,\dots ,y_n]$ by assigning to each variable $y_i$ the degree 1 and to the elements of R the degree 0. Then J is a graded ideal of S and the natural epimorphism $S\to Sy{m}_R\left(M\right)$ is a homomorphism of graded R-algebras. Now, we introduce a monomial order < on the monomials in $y_1,\dots ,y_n$ which is induced by the order on the variables $y_1<y_2<\dots <y_n$. We call such an order an admissible order. For any polynomial $f\in R[y_1,\dots ,y_n],f={\sum }_{\alpha }{a}_{\alpha }{y}^{\alpha }$, we put $in\left(f\right)=a_{\alpha }{y}^{\alpha }$ where $y^{\alpha }$ is the largest monomial in f with $a_{\alpha }\ne 0$, and we set $in\left(J\right)=\left(in\right(f):f\in J)$. For $i=1,\dots ,n$, we set $M_i={\sum }_{j=1}^{i}R{f}_j$, and let $I_i$ be the colon ideal $M_{i-1}:<f_i>$. For convenience we put $I_0=\left(0\right)$.

The colon ideals $I_i$ are called annihilator ideals of the sequence $f_1,\dots ,f_n$. It easy to see that $(I_1{y}_1,I_2{y}_2,\dots ,I_n{y}_n)\subseteq in\left(J\right)$ and the two ideals coincide in degree 1.

Definition 1.

The generators$f_1,\dots ,f_n$of M are called an s-sequence (with respect to an admissible order <) if $in\left(J\right)=(I_1{y}_1,I_2{y}_2,\dots ,I_n{y}_n)$.

If in addition$I_1\subset I_2\subset \cdots \subset I_n$, then $f_1,\dots ,f_n$is called a strong s-sequence.

In the case M is generated by an s-sequence, the theory of s-sequences leads to computations of invariants of $Sy{m}_R\left(M\right)$ quite efficiently, in particular the Krull dimension $dim\left(Sy{m}_R\left(M\right)\right)$, the multiplicity $e\left(Sy{m}_R\left(M\right)\right)$, the Castelnuovo Mumford regularity $reg\left(Sy{m}_R\left(M\right)\right)$ and the $depth\left(Sy{m}_R\left(M\right)\right)$, with respect to the graded maximal ideal, in terms of the invariants of quotients of R by the annihilators ideals of M (for more details on the invariants, see [17]).

Proposition 1

([8] (Proposition 2.4, Proposition 2.6)). Let M be a graded R-module, R a standard graded algebra, generated by a homogeneous s-sequence $f_1,\dots ,f_n$, where $f_1,\dots ,f_n$ have the same degree, with annihilator graded ideals $I_1,\dots ,I_n$. Then

$${\displaystyle d:=dim\left(Sy{m}_R\left(M\right)\right)=\underset{\begin{array}{c}0\le r\le n,\\ 1\le i_1<\dots <i_r\le n\end{array}}{max}\{dim(R/(I_{i_1}+\dots +I_{i_r}))+r\};}$$

$${\displaystyle e\left(Sy{m}_R\left(M\right)\right)=\sum _{\begin{array}{c}0\le r\le n,\\ 1\le i_1<\dots <i_r\le n,\\ dim(R/(I_{i_1}+\dots +I_{i_r}))=d-r\end{array}}e(R/(I_{i_1}+\dots +I_{i_r})).}$$

When $f_1,\dots ,f_n$ is a strong s-sequence, then

$${\displaystyle d=\underset{0\le r\le n}{max}\{dim(R/I_r)+r\};}$$

$$e\left(Sy{m}_R\left(M\right)\right)=\sum _{\begin{array}{c}0\le r\le n,\\ dim(R/I_r)=d-r\end{array}}e(R/I_r).$$

If $R=K[x_1,\dots ,x_m]$ and $f_1,f_2,\dots ,f_n$ is a strong s-sequence:

$$reg\left(Sy{m}_R\left(M\right)\right)\le max\{reg\left(I_i\right):i=1,\dots ,n\};$$

$$depth\left(Sy{m}_R\left(M\right)\right)\ge min\{depth(R/I_i)+i:i=0,1,\dots ,n\}.$$

We recall fundamental results on monomial sequences.

Consider $R=K[x_1,x_2,\dots ,x_m]$, where K is a field, and let $I=(f_1,\dots ,f_n)$ be, where $f_1,\dots ,f_n$ are monomials. Set $f_{ij}=\frac{f_i}{gcd(f_i,f_j)}$, $i\ne j$. Then J is generated by $g_{ij}:=f_{ij}{y}_j-f_{ji}{y}_i,1\le i<j\le n$, and the annihilator ideals of the sequence $f_1,\dots ,f_n$ are the ideals $I_i=(f_{1i},f_{2i},\dots ,f_{(i-1)i})$. As a consequence, a monomial sequence is an s-sequence if and only if the set $\{g_{ij}$, $1\le i<j\le n\}$, is a Gröbner basis for J for any term order on the monomials of $R[y_1,\dots ,y_n]$ which extends an admissible term order on the monomials in the $y_i$. Let us now fix such a term order.

Proposition 2

([8] (Proposition 1.7)). Let $I=(f_1,\dots ,f_n)\subset$ $K[x_1,x_2,\dots ,x_m]$ be a monomial ideal. Suppose that for all $i,j,k,l\in \{1,\dots ,n\}$, with $i<j$, $k<l$, $i\ne k$ and $j\ne l$, we have $gcd(f_{ij},f_{kl})=1$. Then $f_1,\dots ,f_n$ is an s-sequence.

Now let $R=K[x_1,x_2,\dots ,x_m]$ be and let F be the finite free R-module $F=R{e}_1\oplus \dots \oplus R{e}_n$ with basis $e_1,\dots ,e_n$. We refer to [1] (Ch.15, 15.2) for definitions and results on monomial modules.

Definition 2.

An element $m\in F$ is a monomial if m has the form $u{e}_i$, for some i, where u is a monomial of R. A submodule $U\subset F$ is a monomial module if it is generated by monomials of F.

One can observe that if U be a submodule of the free R-module $F={\oplus }_{i=1}^{n}R{e}_i$, then U is a monomial module if and only if for each i there exists a monomial ideal $I_i$ such that $U=I_1{e}_1\oplus I_2{e}_2\oplus \dots \oplus I_n{e}_n$. In particular, U is finitely generated.

Theorem 1.

Let $M={\oplus }_{i=1}^n{I}_i{e}_i$ be a monomial R-module, $M_i=I_i{e}_i$, $I_i=(m_{i1},\dots ,m_{i{r}_i})$, a monomial ideal of $R=K[x_1,\dots ,x_n]$ then

(i)

${Syz}_1\left(M_i\right)\cong Sy{z}_1\left(I_i\right)$,

(ii)

${Syz}_1\left(M\right)\cong Sy{z}_1\left(I_1\right)\oplus Sy{z}_1\left(I_2\right)\oplus \dots \oplus Sy{z}_1\left(I_n\right)$

Proof. 

(i) Write $M_i=\langle m_{i1}{e}_i,\dots ,m_{i{r}_i}{e}_i\rangle$ and let

$$\begin{array}{ccc}\hfill 0\to Sy{z}_1\left(M_i\right)\to & R^{r_i}\to M_i& \to 0\hfill \end{array}$$

(1)

be a presentation of $M_i$. Consider the R-linear homomorphism $R^{r_i}\to M_i$ such that $g_j\to m_{ij}{e}_i$, $R^{r_i}=R{g}_1\oplus \dots \oplus R{g}_{r_i}$, and a syzygy of $M_i$, $a\in R^{r_i}$, $a=({\lambda }_{i1},\dots ,{\lambda }_{i{r}_i})$. Then

$$\sum _{j=1}^{r_i}{\lambda }_{ij}{m}_{ij}=0,$$

and a is a syzygy of $I_i$.

(ii) It follows by (i). □

Let M be a monomial R-module defined as in Theorem 1. We will prove a criterion for a monomial module to be generated by an s-sequence. Set

$$m_{ij,lk}=\frac{m_{ij}}{gcd(m_{ij},m_{lk})},m_{ij}\in I_i,m_{lk}\in I_l,$$

$$1\le i,j\le n,1\le j\le r_i,1\le k\le r_l.$$

Theorem 2.

Let $M={\oplus }_{i=1}^n{I}_i{e}_i$ be a monomial module, $I_i=(m_{i1},\dots ,m_{i{r}_i})$, $i=1,\dots ,n$. Suppose $gcd(m_{ij,ik},m_{tu,tv})=1$, $j<k$, $u<v$, with $i=t$ and $j\ne u$, $k\ne v$ or with $i\ne t$ and $1\le j,k\le r_i,1\le u,v\le r_t$. Then M is generated by the s-sequence $m_{11}{e}_1,\dots ,m_{1r_1}{e}_1,\dots$, $m_{n1}{e}_n,\dots ,m_{n{r}_n}{e}_n$.

Proof. 

For each $i=1,\dots ,n$, $Sy{z}_1\left(M_i\right)$ is generated by the binomials:

$$m_{ij,ik}{g}_{ik}-m_{ik,ij}{g}_{ij}$$

since i is fixed, $1\le j,k\le r_i$, being $g_{ik},g_{ij}$ the free basis of $R^{r_i}$. Thanks to the hypothesis, we have $gcd(m_{ij,ik},m_{iu,iv})=1$, $j<k$, $u<v$, $j\ne u$, $k\ne v$, $\forall i=1,\dots ,n$, and we conclude, by Proposition 2, that $M_i$ is generated by an s-sequence.

Now, suppose $i<t$. If $T_{ik}$ and $T_{tv}$ are the variables that correspond to $g_{ik}$ and $g_{tv}$, then $T_{ik}\ne T_{tv}$. We have $gcd(m_{ij,ik}{T}_{ik},m_{tu,tv}{T}_{tv})=gcd(m_{ij,ik},m_{tu,tv})=1$ by hypothesis. In conclusion, the S-pair $S(b_{ijk},b_{tuv})$ reduces to zero, where $b_{ijk}=m_{ij,ik}{T}_{ik}-m_{ik,ij}{T}_{ij}$ and $b_{tuv}=m_{tu,tv}{T}_{tv}-m_{tv,tu}{T}_{tu}$. Then the assertion follows. □

Example 1.

Let $M=I_1{e}_1\oplus I_2{e}_2$, $I_1=(x^2,y^2,z)$ and $I_2=(z^2,zy)$ be ideals of $K[x,y,z]$. We have $m_{11,12}=m_{11,13}=x^2,m_{12,13}=y^2,m_{21,22}=z$. Since $gcd\left(m_{11,12},m_{12,13}\right)=gcd\left(m_{11,12},m_{21,22}\right)=gcd\left(m_{11,13},m_{21,22}\right)=1$, then M is generated by the s-sequence $x^2{e}_1,y^2{e}_1,z{e}_1,z^2{e}_2,zy{e}_2$.

The next example considers a monomial module M not generated by an s-sequence, even if each addend is generated by an s-sequence.

Example 2.

Let $M=(x,y)e_1\oplus (x,y)e_2$ be, $I_1=I_2=(x,y)$ ideals of $R=K[x,y]$. Write $Sy{m}_R\left(M\right)=R[T_1,T_2,T_3,T_4]/J$, where $J=(y{T}_1-x{T}_2,y{T}_3-x{T}_4)$ We compute the S-pair $S(y{T}_1-x{T}_2,y{T}_3-x{T}_4)=-y(T_1{T}_4-T_2{T}_3)$, with $T_4>T_3>T_2>T_1$. If $T_1{T}_4>T_2{T}_3$, ${in}_{<}J=(x{T}_2,x{T}_4,y{T}_1{T}_4)$ and if $T_1{T}_4<T_2{T}_3$, ${in}_{<}J=(x{T}_2,x{T}_4,y{T}_2{T}_3)$. In any case, J does not have a Gröbner basis which is linear in the variables $T_i$.

Now we quote a statement on computation of the annihilator ideals of $M={\oplus }_{i=1}^n{I}_i{e}_i$, that is to say the annihilator ideals of the generating sequence of M

$$m_{11}{e}_1,m_{12}{e}_1,\dots ,m_{1r_1}{e}_1,m_{21}{e}_2,\dots ,m_{2r_2}{e}_2,\dots ,m_{n1}{e}_n,\dots ,m_{n{r}_n}{e}_n.$$

Proposition 3.

Let $K_{i1},K_{i2},\dots ,K_{i{r}_i}$ be the annihilator ideals of $M_i=I_i{e}_i$, Set $J_1,\dots ,J_{r_1},J_{r_1+1},J_{r_1+2},\dots ,J_{r_1+r_2},J_{r_1+r_2+1},\dots ,J_{r_1+r_2+\dots +r_n}$ the annihilator ideals of the sequence. Then we have:

$$\begin{array}{c}{J}_1=K_{11}=\left(0\right),J_2=K_{12},\dots ,J_{r_1}=K_{1r_1},J_{r_1+1}=K_{21}=\left(0\right),J_{r_1+2}=K_{22},\dots ,\\ J_{r_1+r_2}=K_{2r_2},\dots ,J_{r_1+r_2+\dots +r_{n-1}+1}=K_{n1}=\left(0\right),J_{r_1+r_2+\dots +r_{n-1}+2}=K_{n2},\\ \dots ,J_{r_1+r_2+\dots +r_n}=K_{n{r}_n}.\end{array}$$

Proof. 

An elementary computation gives:

$$\langle 0\rangle :\langle m_{11}{e}_1\rangle =K_{11}=\left(0\right)$$

$$\langle m_{11}{e}_1\rangle :\langle m_{12}{e}_1\rangle =K_{12}$$

$$\langle m_{11}{e}_1,m_{12}{e}_1\rangle :\langle m_{13}{e}_1\rangle =K_{13}$$

$$\dots \dots \dots$$

$$\langle m_{11}{e}_1,m_{12}{e}_2,\dots ,m_{1r_{1-1}}{e}_1\rangle :\langle m_{1r_1}{e}_1\rangle =K_{1r_1}$$

$$\begin{array}{c}\hfill \langle m_{11}{e}_1,m_{12}{e}_1,\dots ,m_{1r_{1-1}}{e}_1,m_{1r_1}{e}_1\rangle :\langle m_{21}{e}_2\rangle =I_1{e}_1:\langle m_{21}{e}_2\rangle +\left(0\right):\langle m_{21}{e}_2\rangle =\\ \hfill =\left(0\right)+K_{21}=\left(0\right)\\ \hfill \langle m_{11}{e}_1,m_{12}{e}_1,\dots ,m_{1r_{1-1}}{e}_1,m_{1r_1}{e}_1,m_{21}{e}_2\rangle :\langle m_{22}{e}_2\rangle =\langle I_1{e}_1,m_{21}{e}_2\rangle :\langle m_{22}{e}_2\rangle =\\ \hfill =I_1{e}_1:\langle m_{22}{e}_2\rangle +K_{22}=\left(0\right)+K_{22}=K_{22}.\end{array}$$

The proof goes on by a routine computation. □

Example 3.

Let $M=I_1{e}_1\oplus I_2{e}_2$ be a monomial module on $R=K[x,y,z]$, where $I_1=(x^2,y^2,xy)$, $I_2=(z^2,zy)$.Then M is generated by the s-sequence $x^2{e}_1,y^2{e}_1,xy{e}_1,z^2{e}_2,zy{e}_2$ with $x<y<z<e_1<e_2$. The s-sequence has the following annihilator ideals:

$$\begin{array}{c}\hfill J_1=\langle 0\rangle :\langle x^2{e}_1\rangle =K_{11}=\left(0\right)\\ \hfill J_2=\langle x^2{e}_1\rangle :\langle y^2{e}_1\rangle =K_{12}=\left(x^2\right)\\ \hfill J_3=\langle x^2{e}_1,y^2{e}_1\rangle :\langle xy{e}_1\rangle =K_{13}=(x,y)\\ \hfill J_4=\langle x^2{e}_1,y^2{e}_1,xy{e}_1\rangle :\langle z^2{e}_2\rangle =K_{21}=\left(0\right)\\ \hfill J_5=\langle x^2{e}_1,y^2{e}_1,xy{e}_1,z^2{e}_2\rangle :\langle zy{e}_2\rangle =\left(0\right)+K_{22}=\left(z\right)\end{array}$$

By Proposition 1, we have $dim\left(Sy{m}_R\left(M\right)\right)=5$. The maximum of the dimensions is obtained by $dim(R/(J_1+J_2+J_3+J_4+J_5))+5=dim(R/(\left(x^2\right)+(x,y)+\left(z\right))+5=5$. For the multiplicity, we have $e\left(Sy{m}_R\left(M\right)\right)=e(R/(J_1+J_4))+e(R/(J_1+J_2+J_3+J_4$$+J_5\left)\right)=1$, since $e(R/(J_1+J_4))=e\left(K[x,y,z]\right)=1$ and $e(R/(J_1+J_2+J_3+J_4+J_5))=$$e\left(K\right)=0$. Concerning the depth and the Castelnuovo regularity, since it results $Sy{m}_R\left(M\right)=R[T_1,T_2,T_3,T_4,T_5]/J=$ $R[T_1,T_2,T_3,T_4,T_5]/(x{T}_2-y{T}_3,y{T}_1-x{T}_3,y{T}_4-z{T}_5)$, we compute $depth\left(Sy{m}_R\left(M\right)\right)=5$ and $reg\left(Sy{m}_R\left(M\right)\right)=3$ using software CoCoA ([16]).

We conclude the section yielding a class of monomial modules that would be of large interest in combinatorics, considering that they involve monomial squarefree ideals. Let $S=K[{\underline{x}}^{\left(1\right)},{\underline{x}}^{\left(2\right)},\dots ,{\underline{x}}^{\left(n\right)}]$ be a polynomial ring in n sets of variables ${\underline{x}}^i=\{x_1^{\left(i\right)},x_2^{\left(i\right)},\dots ,x_{t_i}^{\left(i\right)}\}$, $1\le i\le n.$ Let $I_s$ be the monomial ideal of S generated by all squarefree monomials of degree s (the s-th squarefree Veronese ideal of S). Consider the squarefree monomial ideal $I_{t_i-1}^{\left(i\right)},i=1,\dots ,n$, of $S^{\left(i\right)}=K\left[{\underline{x}}^{\left(i\right)}\right]$ generated by all squarefree monomials of degree $t_i-1$ (the $(t_i-1)$-th squarefree Veronese ideal) as a monomial ideal of S.

Theorem 3.

The monomial module $M={\oplus }_{i=1}^n{I}_{t_i-1}^{\left(i\right)}{e}_i$ on $S=K[{\underline{x}}^{\left(1\right)},$ ${\underline{x}}^{\left(2\right)},\dots ,{\underline{x}}^{\left(n\right)}]$ is generated by an s-sequence.

Proof. 

It is known that for each i, $I_{t_i-1}^{\left(i\right)}$ is generated by an s-sequence ([14] (Theorem 2.3)), being generated by $t_i$ squarefree monomials in $t_i-1$ variables in the polynomial ring in $t_i$ variables and that condition $1)$ of [14] (Theorem 1.3.2.) is satisfied. The ideals $I_{t_i-1}^{\left(i\right)}$ and $I_{t_j-1}^{\left(j\right)}$, for any $i\ne j$, $i,j=1,\dots ,n$, are generated in 2 disjoint sets of variables, then the condition of Theorem 2 is easily verified. □

The invariants of $Sy{m}_S\left(M\right)$ depend on the invariants of each addend of M.

Theorem 4.

Let $M={\oplus }_{i=1}^n{I}_{t_i-1}^{\left(i\right)}{e}_i$ be and let $Sy{m}_S\left(M\right)$ be its symmetric algebra. Then:

(1)

${dim}_S\left(Sy{m}_S\left(M\right)\right)={\sum }_{i=1}^n{t}_i+n={\sum }_{i=1}^n{dim}_{S^{\left(i\right)}}\left(Sy{m}_{S^{\left(i\right)}}\left(M_i\right)\right)$

(2)

$depth\left(Sy{m}_S\left(M\right)\right)={\sum }_{i=1}^n{t}_i+n={\sum }_{i=1}^n{depth}_{S^{\left(i\right)}}\left(Sy{m}_{S^{\left(i\right)}}\left(M_i\right)\right)$

(3)

$e\left(Sy{m}_S\left(M\right)\right)={\sum }_{j=1}^{\sum t_i-n-1}\left(\genfrac{}{}{0pt}{}{\sum t_i-n}{j}\right)+2$

(4)

$reg\left(Sy{m}_S\left(M\right)\right)={\sum }_{i=1}^n{t}_i-n$

Proof. 

We consider an admissible term order on the monomials of $S[T_1^{\left(1\right)},\dots ,T_{t_n}^{\left(n\right)}]$ such that $x_j^l<T_1^{\left(1\right)}<T_2^{\left(1\right)}<\dots <T_{t_n}^{\left(n\right)}$.

$\left(1\right)$ The annihilators ideals of the module $M_i=$$I_{t_i-1}^{\left(i\right)}{e}_i$ are the annihilators ideals $J_j^{\left(i\right)}$ of the sequence generating $I_{t_i-1}^{\left(i\right)}$, in the lexicographic order, for each $i=1,\dots ,n$, $j=1,\dots ,t_i$. By [14] (Proposition 3.1), we have $J_1^{\left(i\right)}=\left(0\right),J_2^{\left(i\right)}=\left(x_{t_i-1}^{\left(i\right)}\right),J_3^{\left(i\right)}=\left(x_{t_i-2}^{\left(i\right)}\right),\cdots ,J_{t_i}^{\left(i\right)}=\left(x_1^{\left(i\right)}\right)$. Then, if J is the relation ideal of $Sy{m}_S\left(M\right)$, we have:

$${in}_{<}\left(J\right)=(x_{t_1-1}^{\left(1\right)}{T}_2^{\left(1\right)},x_{t_1-2}^{\left(1\right)}{T}_3^{\left(1\right)},\dots ,x_1^{\left(1\right)}{T}_{t_1}^{\left(1\right)},\dots ,x_{t_n-1}^{\left(n\right)}{T}_2^{\left(n\right)},$$

$$x_{t_n-2}^{\left(n\right)}{T}_3^{\left(n\right)},\dots ,x_1^{\left(n\right)}{T}_{t_n}^{\left(n\right)})$$

and it is generated by a regular sequence. We obtain

$${dim}_S\left(Sy{m}_S\left(M\right)\right)=\sum _{i=1}^n{t}_i+\sum _{i=1}^n{t}_i-\left(\sum _{i=1}^n{t}_i-n\right)=\sum _{i=1}^n{t}_i+n.$$

$\left(2\right)$ Since $depth\left(Sy{m}_S\left(M\right)\right)\le {dim}_S\left(Sy{m}_S\left(M\right)\right)={\sum }_{i=1}^n{t}_i+n$ and $depth\left(Sy{m}_S\left(M\right)\right)\ge depth(S[T_1^{\left(1\right)},\dots ,T_{t_1}^{\left(1\right)},\dots ,T_1^{\left(n\right)},\dots ,T_{t_n}^{\left(n\right)}]/{in}_{<}\left(J\right))={\sum }_{i=1}^n{t}_i+n,$ the equality follows.

$\left(3\right)$ In the following, we often use methods and tools of [14] (Theorem 3.6). For each i, $1\le i\le n$, with $S^{\left(i\right)}=K\left[{\underline{x}}^{\left(i\right)}\right]$, we have

$$e\left(Sy{m}_{S^{\left(i\right)}}\left(I_{t_i-1}^{\left(i\right)}{e}_i\right)\right)=\sum _{1\le i_1<\dots <i_r\le t_i}e\left(S^{\left(i\right)}/(J_{i_1}^{\left(i\right)},\dots ,J_{i_r}^{\left(i\right)})\right)$$

with $dim\left(S^{\left(i\right)}/(J_{i_1}^{\left(i\right)},\dots ,J_{i_r}^{\left(i\right)})\right)=d-r$, $d=dim\left(Sy{m}_{S^{\left(i\right)}}\left(I_{t_i-1}^{\left(i\right)}{e}_i\right)\right)=t_i+1$ and $1\le r\le t_i$, being $J_{i_1}^{\left(i\right)},\dots ,J_{t_i}^{\left(i\right)}$ the annihilators ideals of $I_{t_i-1}^{\left(i\right)}$. It results, by the structure of the annihilators ideals, $H^{\left(i\right)}=(J_{i_1}^{\left(i\right)},\dots ,J_{i_r}^{\left(i\right)})=(x_{i_1}^{\left(i\right)},\dots ,x_{i_r}^{\left(i\right)})$. Put $H=(H^{\left(1\right)},H^{\left(2\right)},\dots ,H^{\left(n\right)})=(x_{i_1}^{\left(1\right)},\dots ,x_{i_r}^{\left(1\right)},x_{i_1}^{\left(2\right)},\dots ,$ $x_{i_r}^{\left(2\right)},\dots ,x_{i_1}^{\left(n\right)},\dots ,x_{i_r}^{\left(n\right)})$. Then $e(S/H)=1$ since $S/H$ is a polinomial ring on a field k. Let

$$d^{\prime }=dim(S/(J_{i_1}^{\left(i\right)},\dots ,J_{i_r}^{\left(i\right)}))=\sum _{i=1}^n{t}_i+n-r,1\le i\le n,1\le r\le \sum _{i=1}^n{t}_i,$$

then $e\left(Sy{m}_S\left(M\right)\right)$ is given by the sum of the following addends:

$$e(S/(0\left)\right)=1$$

for $r=1,d^{\prime }={\sum }_{i=1}^n{t}_i+n-1$.

$$\sum _{j=2}^{\sum t_i}e(S/J_j^{\left(i\right)})=\underset{\sum t_i-n}{\underbrace{1+\dots +1}}$$

for $r=2,d^{\prime }={\sum }_{i=1}^n{t}_i+n-2$.

$$\sum _{2\le k_1\le t_k,2\le l_1\le t_l}e(S/(J_{k_1}^{\left(k\right)},J_{l_1}^{\left(l\right)}))=\underset{\left(\genfrac{}{}{0pt}{}{\sum t_i-n}{2}\right)}{\underbrace{1+\dots +1}}$$

for $r=3,d^{\prime }={\sum }_{i=1}^n{t}_i+n-3,1\le k,l\le n$

$$\sum _{2\le k_1\le t_k,2\le l_1\le t_l,2\le m_1\le t_m}e(S/(J_{k_1}^{\left(k\right)},J_{l_1}^{\left(l\right)},J_{m_1}^{\left(m\right)}))=\underset{\left(\genfrac{}{}{0pt}{}{\sum t_i-n}{3}\right)}{\underbrace{1+\dots +1}}$$

for $r=4,d^{\prime }=\sum t_i+n-4,1\le k,l,m\le n$⋮

$$\sum _{2\le u_1<\cdots <u_r\le t_1,\dots ,2\le s_1<\cdots <s_r\le t_n}e(S/(J_{u_1}^{\left(1\right)},\dots ,J_{u_r}^{\left(1\right)},\dots ,J_{s_1}^{\left(n\right)},\dots ,J_{s_r}^{\left(n\right)})=\underset{\left(\genfrac{}{}{0pt}{}{\sum t_i-n}{\sum t_i-n-1}\right)}{\underbrace{1+\dots +1}}$$

for $r=\sum t_i-1,d^{\prime }=n+1.$

$$e\left(S/(J_2^{\left(1\right)},\dots ,J_{t_1}^{\left(1\right)},J_2^{\left(2\right)},\dots ,J_{t_2}^{\left(2\right)},J_2^{\left(n\right)},\dots ,J_{t_n}^{\left(2\right)})\right)=1$$

for $r={\sum }_{i=1}^n{t}_i,d^{\prime }=n.$ Thus,

$$e\left(Sy{m}_S\left(M\right)\right)=\sum _{j=1}^{\sum t_i-n-1}\left(\genfrac{}{}{0pt}{}{\sum t_i-n}{j}\right)+2.$$

$\left(4\right) reg\left(Sy{m}_S\left(M\right)\right)=reg(S[{\underline{T}}^{\left(1\right)},\dots ,{\underline{T}}^{\left(n\right)}]/J)\le reg(S[{\underline{T}}^{\left(1\right)},\dots ,{\underline{T}}^{\left(n\right)}]/{in}_{<}\left(J\right))$, ${\underline{T}}^{\left(i\right)}=\{T_1^{\left(i\right)}\dots T_{t_i}^{\left(i\right)}\}$, for $1\le i\le n$. The ideal

$${in}_{<}\left(J\right)=(x_{t_1-1}^{\left(1\right)}{T}_2^{\left(1\right)},\dots ,x_1^{\left(1\right)}{T}_{t_1}^{\left(1\right)},x_{t_2-1}^{\left(2\right)}{T}_2^{\left(2\right)},\dots ,x_1^{\left(2\right)}{T}_{t_2}^{\left(1\right)},\dots x_{t_n-1}^{\left(n\right)}{T}_2^{\left(n\right)},\dots ,x_1^{\left(n\right)}{T}_{t_n}^{\left(n\right)})$$

is generated by a regular sequence of length $\sum t_i-n$ of monomials of degree 2. The ring $S[{\underline{T}}^{\left(1\right)},\dots ,{\underline{T}}^{\left(n\right)}]/{in}_{<}\left(J\right)$ has a resolution of length ${\sum }_{i=1}^n{t}_i-n$, equal to the number of generators of ${in}_{<}\left(J\right)$, given by the Koszul complex of ${in}_{<}\left(J\right)$. Then $reg\left(Sy{m}_S\left(M\right)\right)\le {\sum }_{i=1}^n{t}_i-n$. Since J is Cohen-Macaulay and

$$dim\left(Sy{m}_S\left(M\right)\right)=\sum _{i=1}^n{t}_i+n,dimS[{\underline{T}}^{\left(1\right)},\dots ,{\underline{T}}^{\left(n\right)}]/J=\sum _{i=1}^n{t}_i+\sum _{i=1}^n{t}_i-ht\left(J\right),$$

then $ht\left(J\right)=grad\left(J\right)=2{\sum }_{i=1}^n{t}_i-({\sum }_{i=1}^n{t}_i+n)={\sum }_{i=1}^n{t}_i-n$. Since J is a graded ideal [17] (Proposition 1.5.12), we can choose the regular sequence in J inside the binomials of degree two generating J. So the Koszul complex on the regular sequence gives a 2-linear resolution of J. It follows

$$reg(S[{\underline{T}}^{\left(1\right)},\dots ,{\underline{T}}^{\left(n\right)}]/J)\ge 2\left(\sum _{i=1}^n{t}_i-n\right)-\left(\sum _{i=1}^n{t}_i-n\right)=\sum _{i=1}^n{t}_i-n.$$

The equality follows. □

3. Groebner Bases of Syzygy Modules and $\mathbf{s}$-Sequences

Let R be a Noetherian commutative ring with unit. Let N be a finitely generated R-module submodule of a free R-module $R^n=R{e}_1\oplus \dots \oplus R{e}_n$, $N=R{g}_1+\dots +R{g}_m$, $g_i=a_{i1}{e}_1+\dots a_{in}{e}_n$, $i=1,\dots ,m$. Consider an order on the standard vectors $e_1,\dots ,e_n$ of $R^n$ such that $e_n>\dots >e_1$. We may view N as a graded module by assigning to each vector $e_i$ the degree 1 and to the elements of R the degree 0. For any vector $h\in R{e}_1+\dots R{e}_n$, $h={\sum }_{i=1}^n{a}_i{e}_i$, we put $in\left(h\right)=a_j{e}_j$, where $e_j$ is the largest vector in h with $a_j\ne 0$. Such an order will be called admissible. Set $in\left(N\right)=<in\left(h\right),h\in N>$. We say that $g_1,\dots ,g_m$ is a initial basis for N if $in\left(N\right)=<K_1{e}_1,\dots ,K_n{e}_n>=\oplus K_i{e}_i$, where $K_j$ are ideals of R.

Take $N={Syz}_1\left(M\right)$ the first syzygy module of a finitely generated R-module M. We have:

Theorem 5.

Let M be a finitely R-module generated by an s-sequence $f_1,\dots ,f_n$ and let $N={Syz}_1\left(M\right)$. Then $in\left(N\right)=<I_1{e}_1,\dots ,I_n{e}_n>$, where $I_1,\dots ,I_n$ are the annihilator ideals of the sequence $f_1,\dots ,f_n$.

Proof. 

Let us introduce an admissible order in $R^n={\oplus }_{i=1}^{n}R{e}_i$, with $e_1<e_2<\dots <e_n$. Then ${in}_{<}\left(N\right)=<{in}_{<}\left(f\right),f\in N>=<K_1{e}_1,\dots ,K_n{e}_n>$, with $K_j$ ideals of R. Passing to the symmetric algebras $Sy{m}_R\left(M\right)$, the relation ideal J is generated linearly in the variables $T_j$, $j=1,\dots ,n$, corresponding to the vectors $e_1<e_2<\dots <e_n$, with the order $T_1<T_2<\dots <T_n$, and ${in}_{<}\left(J\right)=(I_1{T}_1,\dots ,I_n{T}_n)$. Let $G\left(J\right)$ be the finite set of linear forms in $T_1,T_2,\dots ,T_n$, which generate J and such that ${in}_{<}\left(J\right)=({in}_{<}f,f\in G\left(J\right))$ and let $\tilde{G}\left(J\right)=G\left(N\right)$ be the set of generators $\tilde{f}$ of $N={Syz}_1\left(M\right)$ corresponding to f under the substitution $T_i\to e_i$, $i=1,\dots ,n$. Then we have ${in}_{<}\left(N\right)=<{in}_{<}\left(\tilde{f}\right),\tilde{f}\in G\left(N\right)>$. We deduce that $K_j=I_j$ for $j=1,\dots ,n$. Hence the assertion follows. □

Example 4.

Let $I=(X^2,Y^2,XY)$ be an ideal of $R=K[X,Y]$. The relation ideal J of $Sy{m}_R\left(I\right)$ is $J=(X{T}_3-Y{T}_1,Y{T}_3-X{T}_2)$. The Gröbner basis of J is $G\left(J\right)=\{X{T}_3-Y{T}_1,Y{T}_3-X{T}_2,X^2{T}_2-Y^2{T}_1\}$ which is linear in the variables $T_1,T_2,T_3$ and I is generated by the s-sequence $X^2,Y^2,XY$. Consider ${Syz}_1\left(I\right)=<X{e}_3-Y{e}_1,Y{e}_3-X{e}_2>$. Then $\tilde{G}\left(J\right)=G\left(N\right)=\{X{e}_3-Y{e}_1,Y{e}_3-X{e}_2,X^2{e}_2-Y^2{e}_1\}$ and ${in}_{<}J=(\left(X^2\right)T_2,(X,Y)T_3)$, ${in}_{<}\left(N\right)=<\left(X^2\right)e_2,(X,Y)e_3>$.

Notice that $X^2,XY,Y^2$ is not an s-sequence for I. In fact, in such case, the relation ideal is $J=(X{T}_2-Y{T}_2,Y{T}_2-X{T}_3)$ and $G\left(J\right)=\{X{T}_2-Y{T}_1,Y{T}_2-X{T}_3,X^2{T}_3-Y^2{T}_1,T_2^2-T_1{T}_3\}$ not linear in the variables $T_1,T_2,T_3$, in both cases $T^2>T_1{T}_3$ or $T_1{T}_3>T^2$. We have $G\left(N\right)=\{X{e}_2-Y{e}_1,Y{e}_2-X{e}_3,X^2{e}_3-Y^2{e}_1\}$, but the generators of $G\left(N\right)$ are not obtained by the substitution of $T_i$ with $e_i$, in the elements of the Gröbner basis of J.

Now, let $R=K[X_1,\dots ,X_t]$ be a polynomial ring over the field K, and let < be a term order on the monomials of $R^n=K[X_1,\dots ,X_t]e_1\oplus \dots \oplus K[X_1,\dots ,X_t]e_n$ with $e_1<\dots <e_n$ and $X_j<e_i$, for all i and j. The excellent book of D. Eisenbud ([1] (Ch.15,15.2)) covers all background for free modules on polynomial rings and Gröbner bases for their submodules. It is easy to prove:

• For any Gröbner basis G of N (with respect to the order <) that exists finite, we have $in\left(N\right)=<in\left(f\right),f\in G>$.
• If M is a monomial module, ${in}_{<}\left(M\right)=in\left(M\right)$.

Now we recall the definition of monomial mixed product ideals which were first introduced in [11], since some classes of such ideals are generated by an s-sequence. To be precise, in the polynomial ring $R=K[X_1,\dots ,X_n;Y_1,\dots ,$ $Y_m]$ in two set of variables on a field K, the squarefree monomial ideals $I_k{J}_r+I_s{J}_t$, with $k+r=s+t$, are called ideals of mixed products, where $I_k$ (resp. $J_r$) is the squarefree ideal of $K[X_1,\dots ,X_n]$ (resp. $K[Y_1,\dots ,Y_m]$) generated by all squarefree monomials of degree k(resp. degree r). In the same way $I_s$ and $J_t$ are defined. Setting $I_0=J_0=R$, in [14] we find the following classification:

• $I_k+J_k$, $1\le k\le inf\{n,m\}$
• $I_k{J}_r$, $1\le k\le n,1\le r\le m$
• $I_k{J}_r+I_{k+1}{J}_{r-1}$, $1\le k\le n,2\le r\le m$
• $J_r+I_s{J}_t$, with $r=s+t$, $1\le s\le n,1\le r\le m$, $t\ge 1$
• $I_k{J}_r+I_s{J}_t$, with $k+r=s+t$, $1\le k\le n,1\le r\le m$

Theorem 6

([14] (Theorem 2.8, Theorem 2.11, Theorem 2.14)). Let the ideal $L_i$ be one of the following mixed product ideals

1.

$L_1=I_{n-1}{J}_m$

2.

$L_2=I_n{J}_{m-1}$

3.

$L_3=I_1{J}_m$

4.

$L_4=I_n{J}_1$

5.

$L_5=I_n{J}_{m-1}+I_{n-1}{J}_m$

6.

$L_6=I_n{J}_1+J_m,n+1=m$.

Then $L_i$ is generated by an s-sequence.

We premise the following:

Proposition 4.

Let $I_{n-1}$ be the Veronese squarefree $(n-1)$-th ideal of $R=K[X_1,\dots ,X_n]$. Let $N=Syz\left(I_{n-1}\right)$ and G be the Gröbner basis of N. Then

1.

$G=\{X_n{e}_1-X_{n-1}{e}_2,X_{n-1}{e}_2-X_{n-2}{e}_3,\dots ,X_2{e}_{n-1}-X_1{e}_n\}$

2.

${in}_{<}N=\left(X_{n-1}\right)e_2\oplus \left(X_{n-2}\right)e_3\oplus \dots \oplus \left(X_1\right)e_n\cong$

$\underset{(n-1)-\mathrm{times}}{\underbrace{R\left(n\right)\oplus R\left(n\right)\oplus \dots \oplus R\left(n\right)}}$ as graded R-modules.

3.

${in}_{<}N$ is generated by a s-sequence.

Proof. 

Let < be an admissible term order introduced on the monomials of $R^n=\oplus R{e}_i$, with $X_1<X_2<\dots <X_n<e_1<e_2<\dots <e_n$, $R=K[X_1,\dots ,X_n]$. The ideal $I_{n-1}=(X_1\cdots X_{n-1},\dots ,X_2\cdots X_{n-1}{X}_n)$ is generated by an s-sequence ([14] (Theorem 2.3)), then

$${in}_{<}\left(J\right)=(K_2{T}_2,\dots ,K_n{T}_n),$$

where J is the relation ideal of $Sy{m}_R\left(I_{n-1}\right)$ and $K_i=\left(X_{n-i+1}\right),i=2,\dots ,n$, are the annihilator ideals of $I_{n-1}$ (See [14] (Proposition 3.1)). Let $N={Syz}_1\left(I_{n-1}\right)$ be. Then $N=<X_n{e}_1-X_{n-1}{e}_2,X_{n-1}{e}_2-X_{n-2}{e}_3,\dots ,X_2{e}_{n-1},X_2{e}_{n-1}-X_1{e}_n>$ is generated by a Gröbner basis, being J generated by a Gröbner basis, $J=(X_n{T}_1-X_{n-1}{T}_2,X_{n-1}{T}_2-X_{n-2}{T}_3,\dots ,X_2{T}_{n-1},X_2{T}_{n-1}-X_1{T}_n)$, with $X_1<X_2<\dots <X_n<T_1<T_2<\dots <T_n$ ([13] (Theorem 2.13)) and

$${in}_{<}N=<\left(X_{n-1}\right)e_2,\left(X_{n-2}\right)e_3,\dots ,\left(X_1\right)e_n>$$

and it is trivially generated by an s-sequence or it follows by Theorem 2. □

For each $L_i$, $i=1,\dots ,6$, as in Theorem 6, we assume that $f_1<f_2<\dots <f_{s_i}$ in the lexicografic order and $X_1<X_2<\dots <X_n<Y_1<Y_2<\dots <Y_m$ in the ring $R=K[X_1,\dots X_n;Y_1,\dots ,Y_m]$.

Theorem 7.

Let $N_i=Syz\left(L_i\right)$ be the first syzygy module of $L_i$ defined in Theorem 6 and let $G\left(N_i\right)$ be the Gröbner basis of $N_i$. Then we have:

1.

$G\left(N_1\right)=\{X_n{e}_1-X_{n-1}{e}_2,X_{n-1}{e}_2-X_{n-2}{e}_3,\dots ,X_2{e}_{n-1}-X_1{e}_n\}$ and

$${in}_{<}\left(N_1\right)=K_2{e}_2\oplus \dots \oplus K_n{e}_n,K_i=\left(X_{n-i+1}\right),i=2,\dots ,n$$

2.

$G\left(N_2\right)=\{Y_m{e}_1-Y_{m-1}{e}_2,Y_{m-1}{e}_2-Y_{m-2}{e}_3,\dots ,Y_2{e}_{m-1}-Y_1{e}_m\}$ and

$${in}_{<}\left(N_2\right)=K_2{e}_2\oplus \dots \oplus K_m{e}_m,K_i=\left(Y_{m-i+1}\right),i=2,\dots ,m$$

3.

$G\left(N_3\right)=\{X_1{e}_2-X_2{e}_1,X_2{e}_3-X_3{e}_2,\dots ,X_{n-1}{e}_n-X_n{e}_{n-1}\}$ and

$${in}_{<}\left(N_3\right)=K_2{e}_2\oplus \dots \oplus K_n{e}_n,K_i=(X_1,\dots ,X_{i-1}),i=2,\dots ,n$$

4.

$G\left(N_4\right)=\{Y_1{e}_2-Y_2{e}_1,Y_2{e}_3-Y_3{e}_2,\dots ,Y_{m-1}{e}_m-Y_m{e}_{m-1}\}$

$${in}_{<}\left(N_4\right)=K_2{e}_2\oplus \dots \oplus K_m{e}_m,K_i=(Y_1,\dots ,Y_{i-1}),i=2,\dots ,m$$

5.

$G\left(N_5\right)=\{Y_m{e}_1-Y_{m-1}{e}_2,\dots ,Y_2{e}_{m-1}-Y_1{e}_m,Y_1{e}_m-X_n{e}_{m+1},X_n{e}_{m+1}-X_{n-1}{e}_{m+2},\dots ,X_2{e}_{m+n-1}-X_1{e}_{m+n}\}$

$$\mathrm{and}{in}_{<}\left(N_5\right)=K_2{e}_2\oplus \dots \oplus K_m{e}_m\oplus K_{m+1}{e}_{m+1}\oplus \dots \oplus K_{m+n}{e}_{m+n}$$

with$K_i=\left(Y_{m-i+1}\right),i=2,\dots ,m$, and$K_i=\left(X_{n+m-i+1}\right),i=m+1,\dots ,m+n$

6.

$G\left(N_6\right)=\{Y_1{e}_2-Y_2{e}_1,Y_2{e}_3-Y_3{e}_2,\dots ,Y_{m-1}{e}_m-Y_m{e}_{m-1},(X_1\cdots X_n)e_{m+1}-(Y_2\cdots Y_m)e_1\}$ and

$${in}_{<}\left(N_6\right)=K_2{e}_2\oplus \dots \oplus K_m{e}_m\oplus (X_1\cdots X_n)e_{m+1},K_i=(Y_1,\dots ,Y_{i-1}),i=2,\dots ,m.$$

Proof. 

For each $i=1,\dots ,6$, the relation ideal $J_i$ of $Sy{m}_R\left(L_i\right)$ is generated by a Gröbner basis $G\left(J\right)$, then we apply Theorem 5 and we obtain the Gröbner basis $G\left(N_i\right)$, by the substitution of the vector $e_i$ to the variable $T_i$ in the forms of the set $G\left(J_i\right)$. For the structure of ${in}_{<}\left(N_i\right)$, $i=1,\dots ,6$, we have:

• The ideal $I_{n-1}{J}_m$ has annihilator ideals $K_i=\left(X_{n-i+1}\right)$, $i=2,\dots ,n$ (See [14] (Proposition 3.3)). Then

  $${in}_{<}{N}_1=<\left(X_{n-1}\right)e_2,\left(X_{n-2}\right)e_3,\dots ,\left(X_1\right)e_n>=\left(X_{n-1}\right)e_2\oplus \left(X_{n-2}\right)e_3\oplus \dots \oplus \left(X_1\right)e_n\cong$$

  $$\cong \underset{(n-1)-\mathrm{times}}{\underbrace{R(m+n)\oplus \dots \oplus R(m+n)}}$$

  as graded R-modules.
• In this case the the annihilator ideals of $I_n{J}_{m-1}$ are $K_i=\left(Y_{m-i+1}\right),i=2,\dots ,m$. The proof is analogue to the case of $I_{n-1}{J}_m$.
• The ideal $I_1{J}_m=(X_1,\dots ,X_n)(Y_1\cdots Y_m)$ is generated by an s-sequence and ${in}_{<}\left(J\right)=(K_2{T}_2,\dots ,K_n{T}_n),$ where $K_i=(X_1,\dots ,X_{i-1}),i=2,\dots ,n$, are the annihilator ideals (See [13] (Proposition 3.7)). Let $N_3={Syz}_1\left(I_1{J}_m\right)$ be. Then

  $${in}_{<}{N}_3=<\left(X_1\right)e_2,(X_1,X_2)e_3,\dots ,(X_1,\dots ,X_{n-1})e_n>\cong \underset{i=2}{\overset{n}{⨁}}{K}_i(m+2)$$

  as graded R-modules.
• The annihilator ideals of $I_n{J}_1$ are $K_i=(Y_1,\dots ,Y_{i-1}),i=2,\dots ,m$ (See [13] (Proposition 3.7)). The proof is analogue to the case of $I_1{J}_m$ and ${\displaystyle {in}_{<}{N}_4\cong \underset{i=2}{\overset{m}{⨁}}{K}_i(n+1)}$ as graded R-modules.
• The annihilator ideals of $I_n{J}_{m-1}+I_{n-1}{J}_m$ are $K_i=\left(Y_{m-i+1}\right)$ for $i=2,\dots ,m$ and $K_i=\left(X_{n+m-i+1}\right)$ for $i=m+1,\dots ,m+n$ by [13] (Proposition 3.11). The assertion follows and we have

  $${in}_{<}{N}_5=\underset{i=2}{\overset{m+n}{⨁}}{K}_i{e}_i\cong \underset{(m+n)-\mathrm{times}}{\underbrace{R(m+n-1)\oplus \dots \oplus R(m+n)}}$$

  as graded R-modules.
• The annihilator ideals of $I_n{J}_1+J_m$ are $K_i=(Y_1,\dots ,Y_{i-1}),i=2,\dots ,m$ (See [13] (Proposition 3.7)) and $K_{m+1}=(X_1{X}_2\cdots X_n)$, generated by the monomial $X_1{X}_2\cdots X_n$. The assertion follows and we have

  $${in}_{<}{N}_6\cong \underset{i=2}{\overset{m}{⨁}}{K}_i{e}_i\oplus (X_1\cdots X_n)e_{m+1}\cong \underset{i=2}{\overset{m}{⨁}}{K}_i(n+2)\oplus R(m+n)$$

  as graded R-modules.

□

Proposition 5.

The modules ${in}_{<}{N}_1,{in}_{<}{N}_2,{in}_{<}{N}_5$ are generated by an s-sequence.

Proof. 

The assertion follows by Theorem 2. □

Theorem 8.

The modules ${in}_{<}{N}_3$, ${in}_{<}{N}_4$ and ${in}_{<}{N}_6$ are not generated by an s-sequence.

Proof. 

Let ${in}_{<}{N}_3=<\left(X_1\right)e_2,(X_1,X_2)e_3,\dots ,(X_1,\dots ,X_{n-1})e_n>$ be and with generating sequence $X_1{e}_2,X_1{e}_3,X_2{e}_3,X_1{e}_4,X_2{e}_4,X_3{e}_4,\dots ,X_{n-2}{e}_n,X_{n-1}{e}_n$. The corresponding symmetric algebra is

$$Sy{m}_R\left({in}_{<}{N}_3\right)=R[T_{12},T_{13},T_{23},T_{14},T_{24},T_{34},\dots T_{(n-2)n},T_{(n-1)n}]/J,$$

with $T_{12}<T_{13}<T_{23}<T_{14}<T_{24}<T_{34}<\dots <T_{(n-2)n}<T_{(n-1)n}$. Consider the relations $g_1=X_1{T}_{23}-X_2{T}_{13}$, $g_2=X_1{T}_{24}-X_2{T}_{14}$ and the S-pair $S(g_1,g_2)=-X_2(T_{23}{T}_{14}-T_{24}{T}_{13})$. Then we have:

$${in}_{<}J=(X_1{T}_{23},X_1{T}_{24},X_2{T}_{23}{T}_{14},\dots )\mathrm{if}{T}_{23}{T}_{14}>T_{24}{T}_{13}$$

or

$${in}_{<}J=(X_1{T}_{23},X_1{T}_{24},X_2{T}_{24}{T}_{13},\dots )\mathrm{if}{T}_{23}{T}_{14}<T_{24}{T}_{13},$$

where < is a term order on all monomials in the variables $X_i$, $T_{jk}$.

Since all initial terms of J are of the form $X_1{T}_{2j}$, $3\le j\le n$, the Gröbner basis of J is never linear in the variables $T_{jk}$.

The same argument can be applied to ${in}_{<}{N}_4$ and ${in}_{<}{N}_6$. □