Extremal Betti Numbers of t-Spread Strongly Stable Ideals

Abstract

Let K be a field and let $S=K[x_1,\cdots ,x_n]$ be a polynomial ring over K. We analyze the extremal Betti numbers of special squarefree monomial ideals of S known as the t-spread strongly stable ideals, where t is an integer $\ge 1$. A characterization of the extremal Betti numbers of such a class of ideals is given. Moreover, we determine the structure of the t-spread strongly stable ideals with the maximal number of extremal Betti numbers when $t=2$.

1. Introduction

Let us consider the polynomial ring $S=K[x_1,\dots ,x_n]$. A squarefree monomial ideal I of S is an ideal generated by squarefree monomials. The class of squarefree monomial ideals is a meaningful object in commutative algebra, due to its strong connections to combinatorics and topology. During the last years, many authors have focused their attention toward problems and questions involving such a class of ideals. Recently, Ene, Herzog, and Qureshi have introduced the notion of t-spread monomial ideal [1] (see also [2,3]), where t is a non–negative integer. More precisely, if $t\ge 0$ is an integer, a monomial $x_{i_1}{x}_{i_2}\cdots x_{i_d}$ with $1\le i_1\le i_2\le \cdots \le i_d\le n$ is called t-spread, if $i_j-i_{j-1}\ge t$ for $2\le j\le d$. A monomial ideal in S is called a t-spread monomial ideal, if it is generated by t-spread monomials. Such a notion generalizes the notion of (squarefree) monomial ideal. Indeed, it is clear that every monomial ideal of S is a 0-spread monomial ideal, whereas every squarefree monomial ideal of S is a 1-spread monomial ideal.

In this paper, we study the extremal Betti numbers [4] of t-spread strongly stable ideals for $t\ge 0$ (Definition 2). This class of ideals is a natural generalization of the class of strongly stable ideals [5]. Hence, one can use the same tools as in [6,7,8,9,10,11,12] for establishing a characterization of their extremal Betti numbers (Theorem 1).

The goal of this paper is to determine the maximal admissible number of extremal Betti numbers of a t-spread strongly stable ideal, for $t\ge 1$. At present it seems to be a too difficult combinatorial problem to determine such a number for all $t\ge 1$. Therefore we concentrate our attention to two-spread strongly stable ideals. Many surprising situations occur in such a case. The case when $t=1$ has been studied and solved in [7].

The plan of the paper is the following. Section 2 contains some preliminary notions that will be used in the paper. In Section 3, the behavior of the extremal Betti numbers of t-spread strongly stable ideals is examined (Theorem 1, Corollary 2). A fundamental tool is the Ene, Herzog, Qureshi formula [1] for computing the graded Betti numbers of such a class of monomial ideals. In Section 4, we analyze the extremal Betti numbers of two-spread strongly stable ideals in the polynomial ring $S=K[x_1,\dots ,x_n]$. More precisely, we face the following problem: let ${\mathcal{S}}_{2,n}$ be the set of all two-spread strongly stable ideals in S, what is the maximal number of extremal Betti numbers allowed for an ideal in ${\mathcal{S}}_{2,n}$? The study of this problem has led to distinguish the cases n odd and n even (Theorems 2 and 4). Moreover, if $n\ge 11$ is an odd integer, given $r=\frac{n-3}{2}$ pairs of positive integers $(k_1,{\ell }_1)$, …, $(k_r,{\ell }_r)$ such that $n-3\ge k_1>k_2>\cdots >k_r\ge 1$ and $2={\ell }_1<{\ell }_2<\cdots <{\ell }_r$, we determine the conditions under which there exists a two-spread strongly stable ideal I of $S=K[x_1,\cdots ,x_n]$ of initial degree ${\ell }_1=2$ with ${\beta }_{k_1,k_1+{\ell }_1}\left(I\right)$, …, ${\beta }_{k_r,k_r+{\ell }_r}\left(I\right)$ as extremal Betti numbers (Theorem 3). A similar result is proved for $n\ge 12$ even integer (Theorem 5). We provide some examples illustrating the main results in the section. Finally, Section 5 contains our conclusions and perspectives.

All the examples are constructed by means of Macaulay2 packages [13], some of which were developed by the authors of this article.

2. Preliminaries and Notation

Let us consider the polynomial ring $S=K[x_1,\dots ,x_n]$ as an $\mathbb{N}$-graded ring where $deg{x}_i=1$, $i=1,\dots ,n$. A monomial ideal I of S is an ideal generated by monomials. If I is a monomial ideal of S, we denote by $G\left(I\right)$ the unique minimal set of monomial generators of I, by $G{\left(I\right)}_{\ell }$ the set of monomials u of $G\left(I\right)$ such that $degu=\ell$, and by $G{\left(I\right)}_{\ge \ell }$ the set of monomials u of $G\left(I\right)$ such that $degu\ge \ell$. If $I={\oplus }_{j\ge 0}{I}_j$ is a graded ideal of the polynomial ring S, we denote by $indegI$ the initial degree of I, i.e., the minimum j such that $I_j\ne 0$.

For a monomial $1\ne u\in S$, we set

$$supp\left(u\right)=\{i:x_i\mathrm{divides}u\},$$

and we write

$$max\left(u\right)=max\{i:i\in supp(u\left)\right\}.$$

Moreover, we set $max\left(1\right)=0$.

Definition 1.

Let I be a monomial ideal of S. I is called a stable ideal if for all $u\in G\left(I\right)$ one has $\left(x_{j}u\right)/x_{max\left(u\right)}\in I$ for all $j<max\left(u\right)$.

I is called a strongly stable ideal if for all $u\in G\left(I\right)$ one has $\left(x_{j}u\right)/x_i\in I$ for all $i\in supp\left(u\right)$ and all $j<i$.

A monomial $x_{i_1}{x}_{i_2}\cdots x_{i_d}\in S$ is squarefree if $1\le i_1<i_2<\cdots <i_d\le n$.

A graded ideal I of S is a squarefree monomial ideal if I is generated by squarefree monomials.

Definition 2.

Let I be a squarefree monomial ideal of S. I is called a squarefree stable ideal if for all $u\in G\left(I\right)$ one has $\left(x_{j}u\right)/x_{max\left(u\right)}\in I$ for all $j<max\left(u\right),j\notin supp\left(u\right)$.

I is called a squarefree strongly stable ideal if for all $u\in G\left(I\right)$ one has $\left(x_{j}u\right)/x_i\in I$ for all $i\in supp\left(u\right)$ and all $j<i$, $j\notin supp\left(u\right)$.

In [1], the notion of a t-spread monomial ideal has been introduced.

Let $t\ge 0$ be an integer. A monomial $x_{i_1}{x}_{i_2}\cdots x_{i_d}$ with $1\le i_1\le i_2\le \cdots \le i_d\le n$ is called t-spread, if $i_j-i_{j-1}\ge t$ for $2\le j\le d$. Note that, any monomial is 0-spread, while the squarefree monomials are one-spread.

For example, the monomial $x_2{x}_5{x}_8\in K[x_1,\dots ,x_8]$ is three-spread.

Definition 3.

A monomial ideal in S is called a t-spread monomial ideal, if it is generated by t-spread monomials.

It is clear that if $t\ge 1$, then every t-spread monomial is a squarefree monomial ideal.

Definition 4.

A t-spread monomial ideal I of S is called t-spread stable, if for all t-spread monomials $u\in I$ and for all $i<max\left(u\right)$ such that $x_i(u/x_{max\left(u\right)})$ is a t-spread monomial, it follows that $x_i(u/x_{max\left(u\right)})\in I$.

The ideal I is called t-spread strongly stable, if for all t-spread monomials $u\in I$, all $j\in supp\left(u\right)$ and all $i<j$ such that $x_i(u/x_j)$ is t-spread, it follows that $x_i(u/x_j)\in I$.

Every t-spread strongly stable ideal is also t-spread stable.

One can notice that the notion of t-spread (strongly) stable ideal generalizes the notion of (strongly) stable ideal.

Remark 1.

The defining property of a t-spread strongly stable ideal needs to be checked only for the set of monomial generators. Indeed, if I is a t-spread monomial ideal of S, then I is t-spread strongly stable if and only if the ideal I satisfies the following condition: for $u\in G\left(I\right)$ and $j\in supp\left(u\right)$, if $i<j$ and $x_i(u/x_j)$ is a t-spread monomial, then $x_i(u/x_j)\in I$ ([1], Lemma 1.2).

Let $u_1,\dots ,u_m$ be t-spread monomials in S. The unique t-spread strongly stable ideal containing $u_1,\dots ,u_m$ will be denoted by $B_t(u_1,\dots ,u_m)$ [1]. The monomials $u_1,\dots ,u_m$ are called t-spread Borel generators.

In the sequel, we refer to $B_t(u_1,\dots ,u_m)$ as the finitely generated t-spread Borel ideal.

Example 1.

The squarefree monomial ideal $I=(x_1{x}_3,$ $x_1{x}_4,$ $x_1{x}_5,$ $x_1{x}_6,$ $x_1{x}_7,$ $x_1{x}_8,$ $x_2{x}_4{x}_6,$ $x_2{x}_4{x}_7,$ $x_2{x}_4{x}_8,$ $x_2{x}_5{x}_7,$ $x_2{x}_5{x}_8,$ $x_2{x}_6{x}_8)$ of $K[x_1,\dots ,x_8]$ is a two-spread strongly stable ideal. We can observe that $I=B_2(x_1{x}_8,$ $x_2{x}_6{x}_8)$.

Now, let $M_{n,d,t}$ be the set of all t-spread monomials of degree d in S and let N be a non-empty subset of $M_{n,d,t}$. Let us define the following set:

$${Shad}_t\left(N\right)=\{x_{i}w:w\in N,i=1,\dots ,n\}\cap M_{n,d+1,t}.$$

It is clear that ${Shad}_t\left(N\right)$ could be empty. The set ${Shad}_t\left(N\right)\ne \varnothing$ will be called the t–shadow of N.

We recall the next definition from [7].

Definition 5.

Let $u=x_{i_1}\cdots x_{i_q}$ be a squarefree monomial of S of degree $q<n$. We say that u has a $j-gap$ if $i_{j+1}-i_j>1$ for some $1\le j<q$. The positive integer $i_{j+1}-i_j-1$ will be called the width of the $j-gap$.

The $j-gap$ of a squarefree monomial $u=x_{i_1}\cdots x_{i_q}\in S$ will be denoted by $j-gap\left(u\right)$, whereas its width will be denoted by $wd(j-gap(u\left)\right)$. Moreover, we define

$$Gap\left(u\right):=\{j\in [q]:\mathrm{there}\mathrm{exists}\mathrm{a}j-gap(u\left)\right\}.$$

One can observe that for $t\ge 1$, ${Shad}_t\left(N\right)\ne \varnothing$ if there exists a squarefree monomial $u=x_{i_1}\cdots x_{i_d}\in N$ satisfying at least one of the following conditions:

(i)

$i_1>t$;

(ii)

$wd(j-gap(u\left)\right)\ge 2t$, for $1\le j<d$;

(iii)

$i_d\le n-t$.

For instance, if $x_3{x}_5{x}_9\in M_{9,3,2}$, then ${Shad}_2\left(\left\{x_3{x}_5{x}_9\right\}\right)=\{x_1{x}_3{x}_5{x}_9,x_3{x}_5{x}_7{x}_9\}\subset M_{9,4,2}$; if $x_4{x}_9\in M_{12,2,3}$, then ${Shad}_3\left(\left\{x_4{x}_9\right\}\right)=\{x_1{x}_4{x}_9,x_4{x}_9{x}_{12}\}\subset M_{12,3,3}$; whereas, if $x_3{x}_6{x}_9\in M_{10,3,3}$, one has ${Shad}_3\left(\left\{x_3{x}_6{x}_9\right\}\right)=\varnothing$.

3. Extremal Betti Numbers

In this section, we study the extremal Betti numbers of t-spread strongly stable ideals.

For any graded ideal I of S, there is a minimal graded free S-resolution [14]

$${\mathbb{F}}_{•}:0\to F_s\to \cdots \to F_1\to F_0\to I\to 0,$$

where $F_i={\oplus }_{j\in \mathbb{Z}}S{(-j)}^{{\beta }_{i,j}}$. The integers ${\beta }_{i,j}={\beta }_{i,j}\left(I\right)={\dim }_K{Tor}_i{(K,I)}_j$ are called the graded Betti numbers of I.

If I is a t-spread strongly stable ideal, there exists a formula to compute the graded Betti numbers of I ([1], Corollary 1.12):

$${\beta }_{k,k+\ell }\left(I\right)=\sum _{u\in G{\left(I\right)}_{\ell }}\left(\genfrac{}{}{0pt}{}{max\left(u\right)-t(\ell -1)-1}{k}\right).$$

(1)

Note that the formula in (1) becomes the well–known formula of Eliahou and Kervaire [5] for (strongly) stable ideals for $t=0$; whereas, if $t=1$ then it coincides with the formula stated by Aramova, Herzog and Hibi for the class of squarefree (strongly) stable ideals [15,16].

Definition 6 ([4]).

A graded Betti number ${\beta }_{k,k+\ell }\left(I\right)\ne 0$ is called extremal if ${\beta }_{i,i+j}\left(I\right)=0$ for all $i\ge k$, $j\ge \ell$, $(i,j)\ne (k,\ell )$.

The pair $(k,\ell )$ is called a corner of I.

If $(k_1,{\ell }_1),\dots ,(k_r,{\ell }_r)$, with $n-1\ge k_1>k_2>\cdots >k_r\ge 1$ and $1\le {\ell }_1<{\ell }_2<\cdots <{\ell }_r$, are the corners of a graded ideal I, according to [8], the following notions can be introduced:

$$Corn\left(I\right)=\{(k_1,{\ell }_1),\dots ,(k_r,{\ell }_r)\},a\left(I\right)=({\beta }_{k_1,k_1+{\ell }_1}\left(I\right),\dots ,{\beta }_{k_r,k_r+{\ell }_r}\left(I\right)).$$

$Corn\left(I\right)$ is called the corner sequence of I, and $a\left(I\right)$ the corner values sequence of I.

The next results are quite similar to the ones in [10,11]. We include them in the paper for completeness of information.

Lemma 1.

Let I be a t-spread strongly stable ideal of S. If ${\beta }_{i,i+j}\left(I\right)\ne 0$, then ${\beta }_{k,k+j}\left(I\right)\ne 0$ for $k=0,\dots ,i$.

Proof. 

If ${\beta }_{i,i+j}\left(I\right)\ne 0$, by (1) there exists $u\in G{\left(I\right)}_j$ such that $max\left(u\right)-t(j-1)-1\ge i$, i.e., $max\left(u\right)\ge i+t(j-1)+1$. It follows that $max\left(u\right)\ge k+t(j-1)+1$, for $k=0,\cdots ,i$, and again from (1), the assertion follows. □

From Definition 6, it follows:

Corollary 1.

Let I be a t-spread strongly stable ideal. The following conditions are equivalent:

(a)

${\beta }_{k,k+\ell }\left(I\right)$ is extremal;

(b)

(b.1) ${\beta }_{k,k+\ell }\left(I\right)\ne 0$;

(b.2) ${\beta }_{k,k+j}\left(I\right)=0$, for $j>\ell$;

(b.3) ${\beta }_{i,i+\ell }\left(I\right)=0$, for $i>k$.

Lemmas 1 and Corollary 1 yield the following characterization.

Theorem 1.

Let I be a t-spread strongly stable ideal of S.

The following conditions are equivalent:

(1)

${\beta }_{k,k+\ell }\left(I\right)$ is extremal;

(2)

$k+t(\ell -1)+1=max\{max\left(u\right):u\in G{\left(I\right)}_{\ell }\}$ and $max\left(u\right)<k+t(j-1)+1$, for all $j>\ell$ and for all $u\in G{\left(I\right)}_j$.

Proof. 

(1)⇒ (2). By (1) ${\beta }_{k,k+\ell }\left(I\right)\ne 0$ if and only if there exists a monomial $u\in G{\left(I\right)}_{\ell }$ such that $max\left(u\right)\ge k+t(\ell -1)+1$. Hence $max\{max\left(u\right):u\in G{\left(I\right)}_{\ell }\}\ge k+t(\ell -1)+1$.

Suppose $j+t(\ell -1)+1:=max\{max\left(u\right):u\in G{\left(I\right)}_{\ell }\}>k+t(\ell -1)+1$. Hence ${\beta }_{j,j+\ell }\left(I\right)\ne 0$, for $j>k$. This is a contradiction from Corollary 1, (b.3). Hence

$$k+t(\ell -1)+1=max\{max\left(u\right):u\in G{\left(I\right)}_{\ell }\}.$$

Suppose there exist an integer $j>\ell$ and a monomial $u\in G{\left(I\right)}_j$ such that $max\left(u\right)\ge k+t(j-1)+1$. From (1), then ${\beta }_{k,k+j}\left(I\right)\ne 0$. Again a contradiction from Corollary 1, (b.2).

(2)⇒ (1). Since $k+t(\ell -1)+1=max\{max\left(u\right):u\in G{\left(I\right)}_{\ell }\}$, then ${\beta }_{k,k+\ell }\left(I\right)\ne 0$ and ${\beta }_{i,i+\ell }\left(I\right)=0$, for all $i>k$. On the other hand $max\left(u\right)<k+t(j-1)+1$, for all $j>\ell$ and for all $u\in G{\left(I\right)}_j$, implies ${\beta }_{k,k+j}\left(I\right)=0$. Hence from Corollary 1, we get the assertion. □

As a consequence we obtain the following:

Corollary 2.

Let I be a t-spread strongly stable ideal of S and let ${\beta }_{k,k+\ell }\left(I\right)$ an extremal Betti number of I. Then

$${\beta }_{k,k+\ell }\left(I\right)=\left|\{u\in G{\left(I\right)}_{\ell }:max\left(u\right)=k+t(\ell -1)+1\}\right|.$$

Now, let $t\ge 1$ and let $M_{n,\ell ,t}$ be the set of all t-spread monomials of degree ℓ in S. From [3] (see also [1], Theorem 2.3), one has

$$|M_{n,\ell ,t}|=\left(\genfrac{}{}{0pt}{}{n-(\ell -1)(t-1)}{\ell }\right).$$

(2)

Hence, if $(k,\ell )$ is a pair of positive integers such that $k+t(\ell -1)+1\le n$, one has

$$\begin{array}{cc}\hfill \left|\right\{u\in M_{n,\ell ,t}:max\left(u\right)=k+t(\ell -1)+1\left\}\right|& =\left(\genfrac{}{}{0pt}{}{k+t(\ell -1)+1-(\ell -1)(t-1)-1}{\ell -1}\right)\hfill \\ & =\left(\genfrac{}{}{0pt}{}{k+\ell -1}{\ell -1}\right).\hfill \end{array}$$

As a consequence, if I is a t-spread strongly stable ideal of S and ${\beta }_{k,k+\ell }\left(I\right)$ is an extremal Betti number of I, then from Theorem 1, we have the following bound:

$$1\le {\beta }_{k,k+\ell }\left(I\right)\le \left(\genfrac{}{}{0pt}{}{k+\ell -1}{\ell -1}\right).$$

(3)

4. Corners of Two-Spread Strongly Stable Ideals

In this Section, we analyze the extremal Betti numbers of two-spread strongly stable ideals in the polynomial ring $S=K[x_1,\dots ,x_n]$. If ${\mathcal{S}}_{2,n}$ is the set of all two-spread strongly stable ideals in S, we determine the largest number of corners allowed for an ideal $I\in {\mathcal{S}}_{2,n}$. It is worth to point out that if $t\ge 2$, then a t-spread strongly stable ideal has initial degree $\ge 2$.

For our purpose, we focus our attention on the ideals $I\in {\mathcal{S}}_{2,n}$ such that all the entries of their corner values sequence $a\left(I\right)$ are equal to 1, i.e., every extremal Betti numbers of I equals 1. The subset of ${\mathcal{S}}_{2,n}$ consisting of such two-spread strongly stable ideals will be denoted by ${\mathcal{S}}_{2,n,\mathbf{1}}$.

The study of this problem has shown that one has to consider two cases:

n odd, n even.

Firstly, we analyze the odd case.

Discussion 1.

Let us consider $S=K[x_1,\dots ,x_n]$, with $n\ge 3$ odd integer.

For $n=3$, the only two-spread strongly stable ideal $I\in {\mathcal{S}}_{2,n,\mathbf{1}}$ is $I=\left(x_1{x}_3\right)$ with $Corn\left(I\right)=\left\{\right(0,2\left)\right\}$.

For $n=5$, the only two-spread strongly stable ideal $I\in {\mathcal{S}}_{2,n,\mathbf{1}}$ is $I=B_2\left(x_1{x}_5\right)$ with $Corn\left(I\right)=\left\{\right(2,2\left)\right\}$.

For $n=7,9,11$, the monomials which determine the largest number of admissible corners of a two-spread strongly stable ideal in ${\mathcal{S}}_{2,n,\mathbf{1}}$ with a corner in degree two are the bold highlighted ones in Figure 1:

In each of these cases, the finitely generated two-spread Borel ideal with the bold highlighted monomials as generators is the ideal we are looking for.

For every $1\le d\le n$, let us denote by ${Mon}_d^s\left(S\right)$ the set of all squarefree monomials of degree d of S. We can order ${Mon}_d^s\left(S\right)$ with the squarefree lexicographic order${\ge }_{slex}$ [15]. More precisely, let

$$u=x_{i_1}{x}_{i_2}\cdots x_{i_d},v=x_{j_1}{x}_{j_2}\cdots x_{j_d},$$

with $1\le i_1<i_2<\cdots <i_d\le n$, $1\le j_1<j_2<\cdots <j_d\le n$, be squarefree monomials of degree d in S, then

$$u{>}_{\mathrm{slex}}vif{i}_1=j_1,\dots ,i_{s-1}=j_{s-1}\mathrm{and}{i}_s<j_s,$$

for some $1\le s\le d$.

From now on, we assume that the sets ${Mon}_d^s\left(S\right)$ ($1\le d\le n$) are endowed with the ordering ${\ge }_{slex}$.

Theorem 2.

Let $n\ge 11$ be odd. A two-spread strongly stable ideal $S=K[x_1,\dots ,x_n]$ of initial degree two and with a corner in degree two can have at most $\frac{n-3}{2}$ corners.

Proof. 

We will prove the existence of a two-spread strongly stable ideal I of S generated in degrees $2,3,\dots ,\frac{n-1}{2}$ such that $|Corn\left(I\right)|=\frac{n-3}{2}$ and $a\left(I\right)=(1,1,\dots ,1)$.

Set $G{\left(I\right)}_2=B_2\left(x_1{x}_n\right)$. One can observe that the monomial

$$x_3{x}_5\cdots x_{n-2}{x}_n$$

is a two-spread monomial of the largest degree which is not multiple of $x_1{x}_n$. Moreover it is also the smallest two-spread monomial of $M_{n,\frac{n-1}{2},2}$.

Claim 1. There exist two-spread monomials $w_i\in S$, $i=1,\dots ,\frac{n-7}{2}=\frac{n-3}{2}-2$, such that

$$I=B_2(x_1{x}_n,w_1,\dots ,w_{\frac{n-7}{2}},x_3{x}_5\cdots x_{n-2}{x}_n).$$

Proof. 

We will verify that $wd(1-gap\left(x_1{x}_n\right))=n-2\ge 9$ allows us to prove the existence of the desired $w_i$’s.

The greatest two-spread monomial not belonging to ${Shad}_2\left(B_2\left(x_1{x}_n\right)\right)$ is $x_2{x}_4{x}_n$. Hence, we choose $w_1=x_2{x}_4{x}_n$. Therefore, since $wd(2-gap\left(w_1\right))=n-5\ge 6$, we set $w_2=x_2{x}_5{x}_7{x}_n$. It is the greatest two-spread monomial not belonging to ${Shad}_2\left(B_2\left(x_2{x}_4{x}_n\right)\right)$.

Now, $wd(3$$-gap\left(w_2\right))=n-8\ge 3$.

Let us distinguish the following cases:

$$n=11,13,15\mathrm{and}n\ge 17.$$

If $n=11$, then $x_3{x}_5{x}_7{x}_9{x}_{11}$ is the greatest two-spread monomial of degree $5=\frac{11-1}{2}$ not belonging to ${Shad}_2\left(B_2\left(x_2{x}_5{x}_7{x}_{11}\right)\right)$ and the smallest two-spread monomial in $M_{11,5,2}$. Hence, $I=B_2(x_1{x}_{11},w_1,w_2,x_3{x}_5{x}_7{x}_9{x}_{11})$ is the two-spread strongly stable ideal we are looking for.

If $n=13$, then $w_3=x_2{x}_5{x}_8{x}_{10}{x}_{13}$ is the greatest two-spread monomial of degree five not belonging to ${Shad}_2\left(B_2\left(x_2{x}_5{x}_7{x}_{13}\right)\right)$. On the other hand, the greatest two-spread monomial of degree $6=\frac{13-1}{2}$ not belonging to ${Shad}_2\left(B_2\left(w_3\right)\right)$ is $x_3{x}_5{x}_7{x}_9{x}_{11}{x}_{13}$. Hence, $I=B_2(x_1{x}_{13},w_1,w_2,w_3,x_3{x}_5{x}_7{x}_9{x}_{11}{x}_{13})$ is the wished two-spread strongly stable ideal.

Similarly, if $n=15$, then $w_3=x_2{x}_5{x}_8{x}_{10}{x}_{15}$ is the greatest two-spread monomial of degree five not belonging to ${Shad}_2\left(B_2\left(x_2{x}_5{x}_7{x}_{15}\right)\right)$. Moreover, the greatest two-spread monomial of degree six not belonging to ${Shad}_2\left(B_2\left(w_3\right)\right)$ is $w_4=x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{15}$. Finally, the greatest two-spread monomial of degree $7=\frac{15-1}{2}$ not belonging to ${Shad}_2\left(B_2\left(w_4\right)\right)$ is $x_3{x}_5{x}_7{x}_9{x}_{11}{x}_{13}{x}_{15}$. Therefore, $I=B_2(x_1{x}_{15},w_1,w_2,w_3,w_4,x_3{x}_5{x}_7{x}_9{x}_{11}{x}_{13}{x}_{15})$ is the two-spread strongly stable ideal we are looking for.

Now, it is worth to point out that in the case when $n=15$ a monomial generator with $x_2{x}_5{x}_8{x}_{11}$ as divisor appears for the first time. Such a monomial will play a crucial role for the proof of the claim.

Let $n\ge 17$. First of all, we set $w_3=x_2{x}_5{x}_8{x}_{10}{x}_n$ and $w_4=x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_n$. Then, one can observe that the number q of all two-spread monomials z with $max\left(z\right)=n$ and $x_2{x}_5{x}_8{x}_{11}$ as a divisor depends on the integer $n-11$. Indeed, one can quickly verify that q is bounded by the integer $m=\lfloor \frac{n-11-2-1}{2}\rfloor$. We will prove that $q=m$.

Since $n-14\ge 3$ is odd, there exists a $5-gap\left(w_4\right)$ which allows us to get the smallest monomial of ${Shad}_2\left(B_2\left(w_4\right)\right)$, i.e., $x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{n-2}{x}_n$.

Let us distinguish two cases: $n=17$, $n>17$.

Let $n=17$. Then $n-14=3$ and $\lfloor \frac{n-14}{2}\rfloor =1$. Indeed the greatest monomial not belonging to ${Shad}_2\left(B_2\left(w_4\right)\right)$ is $w_5=x_2{x}_5{x}_9{x}_{11}{x}_{13}{x}_{15}{x}_{17}$. There exists only $w_4$ which is divisible by u.

Now, let $n>17$. In such a case, the greatest monomial not belonging to ${Shad}_2\left(B_2\left(w_4\right)\right)$ is $w_5=x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{16}{x}_n$. Since $wd(6-gap\left(w_5\right))=n-17=n-14-3=wd(5-gap\left(w_4\right))-3$ and $wd(4-gap\left(w_5\right))=14-11-1=2=wd(4-gap\left(w_4\right))+1$, then $\lfloor \frac{n-14-3+1}{2}\rfloor =m-1$. Hence, if $m-1>1$ one obtains $w_6=x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{17}{x}_{19}{x}_n$.

After $m-1$ iterations, we have $\lfloor \frac{n-14-2m+2}{2}\rfloor =\lfloor \frac{n-12-(n-15)}{2}\rfloor =\lfloor \frac{3}{2}\rfloor =1$. This assures that we can construct another (the last) two-spread monomial not belonging to ${Shad}_2\left(B_2\left(w_{3+m-1}\right)\right)$. It is $w_{3+m}=x_2{x}_5{x}_9{x}_{11}{x}_{13}{x}_{15}\cdots x_n$. It is the greatest two-spread monomial not belonging to ${Shad}_2\left(B_2\left(w_{3+m-1}\right)\right)$.

Finally, we can observe that the monomial $x_3{x}_5{x}_7{x}_9\cdots x_n$ is the greatest two-spread monomial not belonging to the ${Shad}_2\left(B_2(x_2{x}_5{x}_9{x}_{11}{x}_{13}{x}_{15}\cdots x_n)\right)$ and the smallest two-spread monomial in $M_{n,\frac{n-1}{2},2}$.

Proceeding in this way at the end we get

$$1+3+m+2=6+\lfloor \frac{n-14}{2}\rfloor =6+\frac{n-15}{2}=\frac{n-3}{2}$$

suitable monomials. The claim follows. □

The construction of these monomials together with Theorem 1 guarantees that there exists an ideal $I\in {\mathcal{S}}_{2,n,\mathbf{1}}$ with a corner in degree two in S with $|Corn\left(I\right)|=deg(x_3{x}_5{x}_7{x}_9\cdots x_n)-2+1$$=\frac{n-1}{2}-2+1=\frac{n-3}{2}$.

More precisely,

$$\begin{array}{cc}Corn\left(I\right)\hfill & =\left\{(k_i,{\ell }_i):k_i=n-2({\ell }_i-1)-1,{\ell }_i=2+(i-1),i=1,\dots ,\frac{n-3}{2}\right\}=\hfill \\ & =\left\{(n-3,2),(n-5,3),\dots ,(2,\frac{n-1}{2})\right\}.\hfill \end{array}$$

The proof points out that there exist $\frac{n-3}{2}$ monomials of S of degrees $2,3,\dots ,\frac{n-1}{2}$ each of which determines a corner. Moreover, the structure of $x_3{x}_5\cdots x_n$ assures that there does not exist a two-spread monomial of degree $deg(x_3{x}_5\cdots x_n)+1$ that gives a contribution for the computation of a corner. Hence, $\frac{n-3}{2}$ is the maximal admissible number of corners of a two-spread strongly stable ideal of S of initial degree two. □

The monomial generators $x_1{x}_n$, $w_1$, …, $w_{\frac{n-7}{2}}$, $x_3{x}_5\cdots x_{n-2}{x}_n$ will be called 2–spread basic monomials.

For later use, we need to define a partial order ⪰ on the set ${Mon}^s\left(S\right)$ of all squarefree monomials of S. More precisely, let $u,v$ be two squarefree monomials of S, we say that

$$u⪰v$$

-

if $degu=degv$ and $u{\ge }_{slex}v$, or

-

if $degu<degv$ and $u=x_{i}v/w$, with $i\notin supp\left(v\right)$, w divides v and $i<r$, for some $r\in supp\left(w\right)$.

We set $u\succ v$ if $u⪰v$ and $u\ne v$.

For instance, if one considers the two-spread monomials $x_1{x}_8,x_2{x}_4{x}_6,x_2{x}_6{x}_8\in K[x_1,\dots ,x_8]$, then $x_1{x}_8\succ x_2{x}_6{x}_8$. Indeed, $deg{x}_1{x}_8<deg{x}_2{x}_6{x}_8$ and $x_1{x}_8=$$x_1\left(x_2{x}_6{x}_8\right)/x_2{x}_6$.

On the contrary, $x_1{x}_8⋡x_2{x}_4{x}_6$ and $x_2{x}_4{x}_6⋡x_1{x}_8$; whereas $x_2{x}_4{x}_6\succ x_2{x}_6{x}_8$.

With the same notation as in Theorem 2, setting

$$A=\{x_1{x}_n,w_1,\dots ,w_{\frac{n-7}{2}},x_3{x}_5\cdots x_{n-2}{x}_n\},$$

then

$$x_1{x}_n\succ w_1\succ \dots \succ w_{\frac{n-7}{2}}\succ x_3{x}_5\cdots x_{n-2}{x}_n.$$

Now, we give a nice explicit description of the finitely generated Borel two-spread ideal of ${\mathcal{S}}_{2,n,\mathbf{1}}$ of initial degree two with the maximal number of corners, for all odd $n\ge 5$. We will denote it by $B_{2,n,\mathbf{1}}$.

Discussion 2.

Let $S=K[x_1,\dots ,x_n]$ be a polynomial ring, with $n\ge 5$ odd integer. In what follows both Discussion 1 and Theorem 2 (proof) will be crucial.

Firstly, let $n=5,7,9$. In such cases, the finitely generated two-spread Borel ideals $B_{2,n,\mathbf{1}}$ of ${\mathcal{S}}_{2,n,\mathbf{1}}$ are described in

Table 1. Corner sequences for $n=5,7,9$.

n	Corner Sequence	2-Spread Strongly Stable Ideal
5	$\left\{\right(2,2\left)\right\}$	$B_{2,5,\mathbf{1}}=B_2\left(x_1{x}_5\right)=(x_1{x}_3,x_1{x}_4,x_1{x}_5)$
7	$\left\{\right(4,2),(2,3\left)\right\}$	$B_{2,7,\mathbf{1}}=B_2(x_1{x}_7,x_2{x}_4{x}_7)$
9	$\left\{\right(6,2),(4,3),(2,4\left)\right\}$	$B_{2,9,\mathbf{1}}=B_2(x_1{x}_9,x_2{x}_4{x}_9,x_2{x}_5{x}_7{x}_9)$

.

One can observe that the monomial of the type $x_1{x}_n$ ($n=5,7,9$) appears as two-spread Borel generator in all three ideals $B_{2,5,\mathbf{1}}$, $B_{2,7,\mathbf{1}}$ and $B_{2,9,\mathbf{1}}$; the monomial $x_2{x}_4{x}_n$ ($n=7,9$) appears as two-spread Borel generator in the ideals $B_{2,7,\mathbf{1}}$ and $B_{2,9,\mathbf{1}}$; whereas the monomial $x_2{x}_5{x}_7{x}_9$ appears only in the ideal $B_{2,9,\mathbf{1}}$ as a two-spread Borel generator.

For $n\ge 11$, the monomials in the following set

$$\{x_1{x}_n,x_2{x}_4{x}_n,x_2{x}_5{x}_7{x}_n\}$$

(4)

will be always two-spread Borel generators for $B_{2,n,\mathbf{1}}$.

Let us consider the case $n=11$. From Theorem 2 (proof) we have to introduce the monomial $x_3{x}_5{x}_7{x}_9{x}_{11}$ to complete the minimal system of monomial generators of $B_{2,11,\mathbf{1}}$.

Moreover, if $n\equiv 11\left(mod6\right)$, we need to add $r_1=\frac{n-11}{6}+1$ monomials of the type

$$\prod _{i=0}^{2k-1}{x}_{2+3i}{x}_{j+2}{x}_{j+4}\cdots x_{n-2}{x}_n,j=6k+1,k=0,\dots ,r_1-1.$$

(5)

to the set in (4) to get the minimal system of monomial generators of $B_{2,n,\mathbf{1}}$. We refer to them as the right-form basic monomials.

Note that, setting ${\prod }_{i=0}^{2k-1}{x}_{2+3i}=1$ for $k=0$, then ${\prod }_{i=0}^{2k-1}{x}_{2+3i}{x}_{j+2}{x}_{j+4}\cdots x_{n-2}{x}_n=x_3{x}_5{x}_7{x}_9\cdots x_{n-2}{x}_n$.

Hence, the monomials in

$$\{x_1{x}_n,x_2{x}_4{x}_n,x_2{x}_5{x}_7{x}_n,\prod _{i=0}^{2k-1}{x}_{2+3i}{x}_{j+2}\cdots x_{n-2}{x}_n,j=6k+1,k=0,\dots ,r_1-1\}$$

(6)

will belong to the minimal set of monomial generators of $B_{2,n,\mathbf{1}}$, for all odd integer $n\ge 11$.

Let us consider $n=13$. In such a case one has the monomial $x_2{x}_5{x}_8{x}_{10}{x}_{13}$. Such a monomial is smaller than all monomials in (4) and greater than the right-form ones, with respect to ⪰.

In general, if $n\equiv 13\left(mod6\right)$, we need to add $r_2=\frac{n-13}{6}+1$ monomials of the type

$$x_2{x}_5{x}_8\prod _{i=0}^{2k-1}{x}_{11+3i}{x}_j{x}_n,j=6k+10,k=0,\dots ,r_2-1$$

(7)

to the set in (6) to get the minimal system of monomial generators of $B_{2,n,\mathbf{1}}$. We refer to them as the first-left-form basic monomials.

Note that, setting ${\prod }_{i=0}^{2k-1}{x}_{11+3i}=1$ for $k=0$, then $x_2{x}_5{x}_8{\prod }_{i=0}^{2k-1}{x}_{11+3i}{x}_j{x}_n$ = $x_2{x}_5{x}_8{x}_{10}{x}_n$. Hence, the monomials in (7) together with the ones in (6) will belong to the minimal set of monomial generators of $B_{2,n,\mathbf{1}}$, for all odd $n\ge 13$.

Now, let us consider $n=15$. In such a case one has the monomial $x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{15}$. Such a monomial is greater than the right-form ones.

In general, if $n\equiv 15\left(mod6\right)$ then $r_3=\frac{n-15}{6}+1$ monomials of the type

$$x_2{x}_5{x}_8{x}_{11}\prod _{i=0}^{2k-1}{x}_{14+3i}{x}_j{x}_n,j=6k+13,k=0,\dots ,r_3-1$$

(8)

will belong to the minimal set of monomial generators of $B_{2,n,\mathbf{1}}$. We refer to them as the second-left-form basic monomials.

Note that, setting ${\prod }_{i=0}^{2k-1}{x}_{14+3i}=1$ for $k=0$, then $x_2{x}_5{x}_8{x}_{11}{\prod }_{i=0}^{2k-1}{x}_{14+3i}{x}_j{x}_n=x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_n$.

Finally, the monomials in (8) together with the ones in (7) and the ones in (6) will determine the minimal set of monomial generators of $B_{2,n,\mathbf{1}}$, for all odd $n\ge 15$.

Remark 2.

One can notice that given an odd integer $n\ge 15$ in order to determine the set A of the two-spread basic monomials, one can firstly consider the values n, $n-2$ and $n-4$. Then, if one writes down all the monomials (divisible by $x_n$) described in (8), (7), (5) via the integers n, $n-2$ and $n-4$ respectively, together with the monomials in (4), one gets:

$$\left|A\right|=(\frac{n-15}{6}+1)+(\frac{n-2-13}{6}+1)+(\frac{n-4-11}{6}+1)+3=6+\frac{n-15}{2}=\frac{n-3}{2}$$

which is the number of the desired generators.

The next example will illustrate Remark 2.

Example 2.

Let us consider the polynomial ring $S=K[x_1,\dots ,x_{21}]$. We want to construct the finitely generated two-spread strongly stable ideal $I\in {\mathcal{S}}_{2,21,\mathbf{1}}$ with the greatest number of corners and such that $indeg\left(I\right)=2$, i.e., $I=B_{2,21,\mathbf{1}}$.

One has $|Corn\left(I\right)|=\frac{n-3}{2}=9$ and

$$Corn\left(I\right)=\left\{\right(18,2),(16,3),(14,4),(12,5),(10,6),(8,7),(6,8),(4,9),(2,10\left)\right\}.$$

In order to determine the two-spread basic monomials that determine the minimal system of monomial generators $G\left(I\right)$ we proceed as follows.

Step 1.

At first, we consider the two-spread basic monomials $x_1{x}_{21}$, $x_2{x}_4{x}_{21}$ and $x_2{x}_5{x}_7{x}_{21}$.

Step 2.

Since $n=21\equiv 15\left(mod6\right)$, we have $\frac{21-15}{6}+1=2$ second-left-form basic monomials of the type

$$x_2{x}_5{x}_8{x}_{11}\prod _{i=0}^{2k-1}{x}_{14+3i}{x}_j{x}_n,$$

with $j=6k+13$ for $k=0,1$. They are $x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{21}$ ($k=0$, $j=13$) and $x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{17}{x}_{19}{x}_{21}$ ($k=1$, $j=19$).

Step 3.

Let us consider $n-2=19$. Since $n-2=19\equiv 13\left(mod6\right)$, we have $\frac{19-13}{6}+1=2$ first-left-form basic monomials of the type

$$x_2{x}_5{x}_8\prod _{i=0}^{2k-1}{x}_{11+3i}{x}_j{x}_n,$$

with $j=6k+10$ for $k=0,1$. They are $x_2{x}_5{x}_8{x}_{10}{x}_{21}$ ($k=0$, $j=10$) and $x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{16}{x}_{21}$ ($k=1$, $j=16$).

Step 4.

Let us consider $n-4=17$. Since $n-4=17\equiv 11\left(mod6\right)$, then we have $\frac{17-11}{6}+1=2$ right-form basic monomials of the type

$$\prod _{i=0}^{2k-1}{x}_{2+3i}{x}_{j+2}{x}_{j+4}\cdots x_{n-2}{x}_n,$$

with $j=6k+1$ for $k=0,1$. They are $x_3{x}_5{x}_7{x}_9{x}_{11}{x}_{13}{x}_{15}{x}_{17}{x}_{19}{x}_{21}$ ($k=0$, $j=1$) and $x_2{x}_5{x}_9{x}_{11}{x}_{13}{x}_{15}{x}_{17}{x}_{19}{x}_{21}$ ($k=1$, $j=7$).

Finally, ordering the monomials in steps 1–4 with respect to ⪰, we have the ideal

$$\begin{array}{cc}I=\hfill & B_2(x_1{x}_{21},x_2{x}_4{x}_{21},x_2{x}_5{x}_7{x}_{21},x_2{x}_5{x}_8{x}_{10}{x}_{21},x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{21},x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{16}{x}_{21},\hfill \\ & x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{17}{x}_{19}{x}_{21},x_2{x}_5{x}_9{x}_{11}{x}_{13}{x}_{15}{x}_{17}{x}_{19}{x}_{21},x_3{x}_5{x}_7{x}_9{x}_{11}{x}_{13}{x}_{15}{x}_{17}{x}_{19}{x}_{21}).\hfill \end{array}$$

From Theorem 2 and Discussion 2, the next result follows.

Theorem 3.

Let $n\ge 11$ be an odd integer and ${\ell }_1=2$. Given $\frac{n-3}{2}$ pairs of positive integers

$$(k_1,{\ell }_1),(k_2,{\ell }_2),\dots ,(k_{\frac{n-3}{2}},{\ell }_{\frac{n-3}{2}}),$$

(9)

with $1\le k_{\frac{n-3}{2}}<k_{\frac{n-3}{2}-1}<\cdots <k_1\le n-3$ and $2={\ell }_1<{\ell }_2<\cdots <{\ell }_{\frac{n-3}{2}}\le \frac{n-1}{2}$, then there exists a two-spread strongly stable ideal I of S of initial degree ${\ell }_1$ and with the pairs in (9) as corners if and only if $k_i+2({\ell }_i-1)+1=n$, for $i=1,\dots ,\frac{n-3}{2}$.

Remark 3.

For an arbitrary monomial ideal I, let $I_j$ be the j–th graded component of I. Following [3], we call the set of t-spread monomials in $I_j$, the t-spread part of $I_j$ and denote it by ${\left[I_j\right]}_t$. A special class of t-spread strongly stable ideals consists of t-spread lex ideals, which are defined as follows [3].

A subset L of $M_{n,d,t}$ is called a t-spread lex set, if for all $u\in L$ and for all $v\in M_{n,d,t}$ with $v{>}_{lex}u$, it follows that $v\in L$. A t-spread monomial ideal I is called a t-spread lex ideal, if ${\left[I_j\right]}_t$ is a t-spread lex set for all j.

It is clear that the two-spread strongly stable ideal in Theorem 2 is a two-spread lex ideal.

Now, we analyze the even case. The development will be very similar to the odd case. We include it for completeness and for highlighting the differences with the odd case.

Discussion 3.

Let us consider $S=K[x_1,\dots ,x_n]$, with $n\ge 4$ even.

For $n=4$, the only two-spread strongly stable ideal $I\in {\mathcal{S}}_{2,n}$ in S is $I=B_2\left(x_1{x}_4\right)$ with $Corn\left(I\right)=\left\{\right(1,2\left)\right\}$. For $n=6,8,10,12,14$, the monomials which determine the maximal number of admissible corners of a two-spread strongly stable ideal in ${\mathcal{S}}_{2,n,\mathbf{1}}$ with a corner in degree two are the bold highlighted ones in Figure 2:

In each of the cases described in Figure 2, the finitely generated two-Borel ideal with the bold highlighted monomials as generators is the wished ideal.

Theorem 4.

Let $n\ge 14$ be even. A two-spread strongly stable ideal $S=K[x_1,\dots ,x_n]$ of initial degree two and with a corner in degree two can have at most $\frac{n-4}{2}$ corners.

Proof. 

The proof is verbatim the same of Theorem 2.

We prove the existence of a two-spread strongly stable ideal I of S generated in degrees $2,3,\dots ,\frac{n-2}{2}$ such that $|Corn\left(I\right)|=\frac{n-4}{2}$ and $a\left(I\right)=(1,1,\dots ,1)$.

Set $G{\left(I\right)}_2=B_2\left(x_1{x}_n\right)$. One can observe that $wd(1$$-gap\left(x_1{x}_n\right))=n-2\ge 12$. The greatest two-spread monomial not belonging to ${Shad}_2\left(B_2\left(x_1{x}_n\right)\right)$ is $x_2{x}_4{x}_n$. Hence, we set $w_1=x_2{x}_4{x}_n$.

Note that the monomial

$$x_2{x}_6{x}_8\cdots x_{n-2}{x}_n,$$

of degree $\frac{n-2}{2}$ is a two-spread monomial of ${Mon}^s\left(S\right)$ of the largest degree which is not multiple both of $x_1{x}_n$ and of $w_1$.

Claim 2. We prove the existence of certain two-spread monomials $w_i\in S$, for $i=2,\dots ,\frac{n-8}{2}=\frac{n-4}{2}-2$, such that

$$I=B_2(x_1{x}_n,w_1,\dots ,w_{\frac{n-8}{2}},x_2{x}_6{x}_8\cdots x_{n-2}{x}_n).$$

Proof. 

Firstly, since $wd(2$$-gap\left(w_1\right))=n-5\ge 9$, we set $w_2=x_2{x}_5{x}_7{x}_n$. On the other hand, $wd(3$$-gap\left(w_2\right))=n-8\ge 6$. Then we set $w_3=x_2{x}_5{x}_8{x}_{10}{x}_n$ and $wd(4$$-gap\left(w_3\right))=n-11\ge 3$. Let us distinguish the following cases:

$$n=14,16,18\mathrm{and}n\ge 20.$$

If $n=14$, then $x_2{x}_6{x}_8{x}_{10}{x}_{12}{x}_{14}$ is the greatest two-spread monomial of degree $6=\frac{14-2}{2}$ not belonging to ${Shad}_2\left(B_2\left(x_2{x}_5{x}_8{x}_{10}{x}_{14}\right)\right)$. Hence, $I=B_2(x_1{x}_{14},w_1,w_2,w_3,x_2{x}_6{x}_8{x}_{10}{x}_{12}{x}_{14})$ is the two-spread strongly stable ideal we are looking for.

If $n=16$, then $w_4=x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{16}$ is the greatest two-spread monomial of degree six not belonging to ${Shad}_2\left(B_2\left(x_2{x}_5{x}_8{x}_{10}{x}_{16}\right)\right)$. Finally, we can construct the greatest two-spread monomial of degree $7=\frac{16-2}{2}$ not belonging to ${Shad}_2\left(B_2\left(w_4\right)\right)$. It is $x_2{x}_6{x}_8{x}_{10}{x}_{12}{x}_{14}{x}_{16}$. Hence, the two-spread strongly stable ideal we are looking for is $I=B_2(x_1{x}_{16},w_1,w_2,w_3,w_4,x_2{x}_6{x}_8{x}_{10}{x}_{12}{x}_{14}{x}_{16})$.

If $n=18$, then $w_4=x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{18}$ is the greatest two-spread monomial of degree six not belonging to ${Shad}_2\left(B_2\left(x_2{x}_5{x}_8{x}_{10}{x}_{18}\right)\right)$. Moreover, the greatest two-spread monomial of degree seven not belonging to ${Shad}_2\left(B_2\left(w_4\right)\right)$ is $w_5=x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{16}{x}_{18}$. Finally, the greatest two-spread monomial of degree $8=\frac{18-2}{2}$ not belonging to ${Shad}_2\left(B_2\left(w_5\right)\right)$ is $x_2{x}_6{x}_8{x}_{10}{x}_{12}{x}_{14}{x}_{16}{x}_{18}$. Hence, the desired two-spread strongly stable ideal is $I=B_2(x_1{x}_{18},w_1,w_2,w_3,w_4,w_5,x_2{x}_6{x}_8{x}_{10}{x}_{12}{x}_{14}{x}_{16}{x}_{18})$.

Also in this case, the monomial generators with $u=x_2{x}_5{x}_8{x}_{11}$ as divisor will play a crucial role in the proof. We note that when $n=18$ u does not divide any monomial generators.

Let $n\ge 20$. We set $w_4=x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_n$ and $w_5=x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{16}{x}_n$. We observe that the number q of all two-spread monomials z with $max\left(z\right)=n$ and $x_2{x}_5{x}_8{x}_{11}$ as a divisor depends on the integer $n-11$. Indeed, in this case q is bounded by the integer $m=\lfloor \frac{n-11-2-2}{2}\rfloor$. We will prove that $q=m$. We can observe that $n-15\ge 5$ is odd. This assures the existence of a 6$-gap\left(w_5\right)$ which allows us to obtain the smallest monomial of ${Shad}_2\left(B_2\left(w_5\right)\right)$, i.e., $x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{16}{x}_{n-2}{x}_n$.

Let $n=20$. Then $n-15=5$ and $\lfloor \frac{n-15}{2}\rfloor =2$. Indeed the greatest monomial not belonging to ${Shad}_2\left(B_2\left(w_5\right)\right)$ is $w_6=x_2{x}_5{x}_8{x}_{12}{x}_{14}{x}_{16}{x}_{18}{x}_{20}$. Hence there exist two monomials, $w_4$ and $w_5$, that are divisible by u.

Now, let us consider $n>20$. Hence the greatest monomial not belonging to ${Shad}_2\left(B_2\left(w_5\right)\right)$ is $w_6=x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{17}{x}_{19}{x}_n$. One can observe that $wd(7$$-gap\left(w_6\right))=n-20=n-17-3=wd(6$$-gap\left(w_5\right))-3$ and $wd(5$$-gap\left(w_5\right))=17-14-1=2=wd(5$$-gap\left(w_4\right))+1$. This leads that $\lfloor \frac{n-15-3+1}{2}\rfloor =m-1$.

Hence, if $m-1>1$ we obtain the two-spread monomial $w_7=$$x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{17}{x}_{20}{x}_{22}{x}_n$.

After $m-1$ iterations we have $\lfloor \frac{n-15-2m+2}{2}\rfloor =\lfloor \frac{n-13-(n-16)}{2}\rfloor =\lfloor \frac{3}{2}\rfloor =1$. This assures that we can construct the last two-spread monomial not belonging to ${Shad}_2\left(B_2\left(w_{3+m-1}\right)\right)$. It is $w_{3+m}=x_2{x}_5{x}_8{x}_{12}{x}_{14}{x}_{16}\cdots x_n$ which is the greatest two-spread monomial not belonging to this shadow.

Finally, we can observe that the monomial $x_2{x}_6{x}_8{x}_{10}\cdots x_{n-2}{x}_n$ is the greatest two-spread monomial not belonging to the ${Shad}_2\left(B_2(x_2{x}_5{x}_8{x}_{12}{x}_{14}{x}_{16}\cdots x_n)\right)$.

Proceedings in this way we are able to identify

$$1+3+m+2=6+\lfloor \frac{n-15}{2}\rfloor =6+\frac{n-16}{2}=\frac{n-4}{2}$$

monomials which are the ones we are looking for. □

The construction of these monomials together with Theorem 1 leads to the existence of an ideal $I\in {\mathcal{S}}_{2,n,\mathbf{1}}$ of initial degree two in S with $|Corn\left(I\right)|=\frac{n-2}{2}-2+1=\frac{n-4}{2}$. More in details,

$$\begin{array}{cc}Corn\left(I\right)\hfill & =\left\{(k_i,{\ell }_i):k_i=n-2({\ell }_i-1)-1,{\ell }_i=2+(i-1),i=1,\dots ,\frac{n-4}{2}\right\}=\hfill \\ & =\left\{(n-3,2),(n-5,3),\dots ,(3,\frac{n-2}{2})\right\}\hfill \end{array}$$

The proof points out that there exist $\frac{n-4}{2}$ monomials of S of degrees $2,3,\dots ,\frac{n-2}{2}$ each of which determines a corner. Furthermore, the structure of $x_2{x}_6\cdots x_n$ assures that there does not exist a two-spread monomial of degree $deg(x_2{x}_6\cdots x_n)+1$ that gives a contribution for the computation of a corner. Hence, $\frac{n-4}{2}$ is the maximal admissible number of corners of a two-spread strongly stable ideal of S of initial degree two. □

Now, we give an explicit description of the finitely generated Borel two-spread ideal of ${\mathcal{S}}_{2,n,\mathbf{1}}$ of initial degree two with the maximal number of corners, for all even integer $n\ge 4$. We will denote it by $B_{2,n,\mathbf{1}}$ as in the case when n is odd.

Discussion 4.

Let $S=K[x_1,\dots ,x_n]$ a polynomial ring, with $n\ge 4$ even integer.

Firstly, let us consider $n=4,6,8,10$. In such cases, the ideals $B_{2,n,\mathbf{1}}$ are described in

Table 2. Corner sequences for $n=4,6,8,10$.

n	Corner Sequence	2-Spread Strongly Stable Ideal
4	$\left\{\right(1,2\left)\right\}$	$B_{2,4,\mathbf{1}}=B_2\left(x_1{x}_4\right)=(x_1{x}_3,x_1{x}_4)$
6	$\left\{\right(3,2),(1,3\left)\right\}$	$B_{2,6,\mathbf{1}}=B_2(x_1{x}_6,x_2{x}_4{x}_6)$
8	$\left\{\right(5,2),(3,3\left)\right\}$	$B_{2,8,\mathbf{1}}=B_2(x_1{x}_8,x_2{x}_4{x}_8)$
10	$\left\{\right(7,2),(5,3),(3,4\left)\right\}$	$B_{2,10,\mathbf{1}}=B_2(x_1{x}_{10},x_2{x}_4{x}_{10},x_2{x}_5{x}_7{x}_{10})$

.

One can observe that the monomial of the type $x_1{x}_n$ ($n=4,6,8,10$) appears as two-spread Borel generators in all four ideals $B_{2,4,\mathbf{1}}$, $B_{2,6,\mathbf{1}}$, $B_{2,8,\mathbf{1}}$ and $B_{2,10,\mathbf{1}}$; the monomial $x_2{x}_4{x}_n$ ($n=6,8,10$) appears as two-spread Borel generators in the ideals $B_{2,6,\mathbf{1}}$, $B_{2,8,\mathbf{1}}$ and $B_{2,10,\mathbf{1}}$; whereas the monomial $x_2{x}_5{x}_7{x}_{10}$ appears only in the ideal $B_{2,10,\mathbf{1}}$, as two-spread Borel generator.

It is worth to underline a difference from the n odd case. Indeed for two consecutive even values of n ($n=6,8$), one has the same type of Borel generators.

For $n\ge 12$, the monomials in the following set

$$\{x_1{x}_n,x_2{x}_4{x}_n,x_2{x}_5{x}_7{x}_n\}$$

(10)

will be always two-spread Borel generators for $B_{2,n,\mathbf{1}}$.

Let us consider $n=12$. From the Theorem 4 (proof) we have to introduce the monomial $x_2{x}_5{x}_8{x}_{10}{x}_{12}$ to complete the minimal system of monomial generators of $B_{2,12,\mathbf{1}}$. Such a monomial is smaller than the monomials in (10), with respect to ⪰.

Furthermore, if $n\equiv 12\left(mod6\right)$, we need to add $r_1=\frac{n-12}{6}+1$ monomials of the type

$$x_2{x}_5{x}_8\prod _{i=0}^{2k-1}{x}_{11+3i}{x}_j{x}_n,j=6k+10,k=0,\dots ,r_1-1$$

to the set in (10) to get the minimal system of monomial generators of $B_{2,n,\mathbf{1}}$, for all even integer $n\ge 12$. We refer to them as the first-left-form basic monomials.

Note that, setting ${\prod }_{i=0}^{2k-1}{x}_{11+3i}=1$ for $k=0$, then $x_2{x}_5{x}_8{\prod }_{i=0}^{2k-1}{x}_{11+3i}{x}_j{x}_n=x_2{x}_5{x}_8{x}_{10}{x}_n$.

Therefore, the monomials in

$$\{x_1{x}_n,x_2{x}_4{x}_n,x_2{x}_5{x}_7{x}_n,x_2{x}_5{x}_8\prod _{i=0}^{2k-1}{x}_{11+3i}{x}_j{x}_n,j=6k+10,k=0,\dots ,r_1-1\}$$

(11)

will belong to the minimal set of monomial generators of $B_{2,n,\mathbf{1}}$, for all even integer $n\ge 12$.

Let us consider $n=14$. In such a case we introduce the monomial $x_2{x}_6{x}_8{x}_{10}{x}_{12}{x}_{14}$ as Borel generator. For $n\ge 14$, the monomial of the type $x_2{x}_6{x}_8{x}_{10}\cdots x_{n-2}{x}_n$ is the smallest generator of the ideal, with respect to ⪰.

In general, if $n\equiv 14\left(mod6\right)$ then we need to add $r_2=\frac{n-14}{6}+1$ monomials of the type

$$x_2\prod _{i=0}^{2k-1}{x}_{5+3i}{x}_{j+2}{x}_{j+4}\cdots x_{n-2}{x}_n,j=6k+4,k=0,\dots ,r_2-1$$

(12)

to the set in (11) to get the minimal system of monomial generators of $B_{2,n,\mathbf{1}}$, for all even integer $n\ge 14$. We refer to them as the right-form basic monomials.

Note that, setting ${\prod }_{i=0}^{2k-1}{x}_{5+3i}=1$ for $k=0$, then $x_2{\prod }_{i=0}^{2k-1}{x}_{5+3i}{x}_{j+2}{x}_{j+4}\cdots x_{n-2}{x}_n=x_2{x}_6{x}_8{x}_{10}\cdots x_{n-2}{x}_n$.

Hence, the monomials in (12) together with the ones in (11) will belong to the minimal set of monomial generators of $B_{2,n,\mathbf{1}}$, for all even $n\ge 14$.

Now, let us consider $n=16$. In such a case we need the monomial $x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{16}$. Such a monomial is greater than the right-form ones. In general, if $n\equiv 16\left(mod6\right)$ then $r_3=\frac{n-16}{6}+1$ monomials of the type

$$x_2{x}_5{x}_8{x}_{11}\prod _{i=0}^{2k-1}{x}_{14+3i}{x}_j{x}_n,j=6k+13,k=0,\dots ,r_3-1$$

(13)

will belong to the minimal set of monomial generators of $B_{2,n,\mathbf{1}}$, for all even $n\ge 16$. We refer to them as the second-left-form basic monomials.

Note that, setting ${\prod }_{i=0}^{2k-1}{x}_{14+3i}=1$ for $k=0$, then $x_2{x}_5{x}_8{x}_{11}{\prod }_{i=0}^{2k-1}{x}_{14+3i}{x}_j{x}_n=x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_n$.

Finally, the monomials in (13) together with the ones in (12) and the ones in (11) will determine the minimal set of monomial generators of $B_{2,n,\mathbf{1}}$, for all even $n\ge 16$.

The next example illustrates how given an even integer $n\ge 16$ in order to get the set of the generators of $B_{2,n,\mathbf{1}}$, one has to fix the the integers n, $n-2$, $n-4$. Reasoning as in Remark 2, the number of the monomials we need is given by

$$3+\frac{n-16}{6}+1+\frac{n-14-2}{6}+1+\frac{n-12-4}{6}+1=6+\frac{n-16}{2}=\frac{n-4}{2}.$$

and the two-spread basic monomials can be obtained by (13), (12) and (11) via n, $n-2$, $n-4$, respectively.

Example 3.

Let us consider the polynomial ring $S=K[x_1,\dots ,x_{20}]$. We want to construct the two-spread strongly stable ideal $B_{2,20,\mathbf{1}}$ of S. Setting $I=B_{2,20,\mathbf{1}}$, one has $|Corn\left(I\right)|=\frac{n-4}{2}=8$ and

$$Corn\left(I\right)=\left\{\right(17,2),(15,3),(13,4),(11,5),(9,6),(7,7),(5,8),(3,9\left)\right\}.$$

In order to get the two-spread basic monomials that determine the minimal system of monomial generators $G\left(I\right)$ we proceed as follows.

Step 1.

Consider the three two-spread basic monomials $x_1{x}_{20}$, $x_2{x}_4{x}_{20}$ and $x_2{x}_5{x}_7{x}_{20}$.

Step 2.

Since $n=20\equiv 14\left(mod6\right)$, we have $\frac{20-14}{6}+1=2$ right-form basic monomials of the type

$$x_2\prod _{i=0}^{2k-1}{x}_{5+3i}{x}_{j+2}{x}_{j+4}\cdots x_{n-2}{x}_n,$$

with $j=6k+4$ for $k=0,1$. They are $x_2{x}_6{x}_8{x}_{10}{x}_{12}{x}_{14}{x}_{16}{x}_{18}{x}_{20}$ ($k=0$, $j=4$) and $x_2{x}_5{x}_8{x}_{12}{x}_{14}{x}_{16}{x}_{18}{x}_{20}$ ($k=1$, $j=10$).

Step 3.

Let us consider $n-2=18$. Since $n-2=18\equiv 12\left(mod6\right)$, we have $\frac{18-12}{6}+1=2$ first-left-form basic monomials of the type

$$x_2{x}_5{x}_8\prod _{i=0}^{2k-1}{x}_{11+3i}{x}_j{x}_n,$$

with $j=6k+10$ for $k=0,1$. They are $x_2{x}_5{x}_8{x}_{10}{x}_{20}$ ($k=0$, $j=10$) and $x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{16}{x}_{20}$ ($k=1$, $j=16$).

Step 4.

Let us consider $n-4=16$. Since $n-4=16\equiv 16\left(mod6\right)$, we have $\frac{16-16}{6}+1=1$ second-left-form basic monomial of the type

$$x_2{x}_5{x}_8{x}_{11}\prod _{i=0}^{2k-1}{x}_{14+3i}{x}_j{x}_n,$$

with $j=13$ and $k=0$. It is $x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{20}$.

Finally, ordering the monomials in Steps 1–4 with respect to ⪰, we have the ideal

$$\begin{array}{cc}\hfill I=& B_2(x_1{x}_{20},x_2{x}_4{x}_{20},x_2{x}_5{x}_7{x}_{20},x_2{x}_5{x}_8{x}_{10}{x}_{20},x_2{x}_5{x}_8{x}_{11}{x}_{13}{x}_{20},x_2{x}_5{x}_8{x}_{11}{x}_{14}{x}_{16}{x}_{20},\hfill \\ & x_2{x}_5{x}_8{x}_{12}{x}_{14}{x}_{16}{x}_{18}{x}_{20},x_2{x}_6{x}_8{x}_{10}{x}_{12}{x}_{14}{x}_{16}{x}_{18}{x}_{20}).\hfill \end{array}$$

From the Theorem 4 and Discussion 4, the next result follows.

Theorem 5.

Let $n\ge 12$ an even integer and ${\ell }_1=2$. Given $\frac{n-4}{2}$ pairs of positive integers

$$(k_1,{\ell }_1),(k_2,{\ell }_2),\dots ,(k_{\frac{n-4}{2}},{\ell }_{\frac{n-4}{2}}),$$

(14)

with $1\le k_{\frac{n-4}{2}}<k_{\frac{n-4}{2}-1}<\cdots <k_1\le n-3$ and $2={\ell }_1<{\ell }_2<\cdots <{\ell }_{\frac{n-4}{2}}\le \frac{n-2}{2}$, then there exists a two-spread strongly stable ideal I of S of initial degree ${\ell }_1$ and with the pairs in (9) as corners if and only if $k_i+2({\ell }_i-1)+1=n$, for $i=1,\dots ,\frac{n-4}{2}$.

Also in such a case the two-spread strongly stable ideal in Theorem 5 is a two-spread lex ideal.

5. Conclusions and Perspectives

In this paper, we have discussed the extremal Betti numbers of t-spread strongly stable ideals and we have determined the maximal number of the admissible corners of two-spread strongly stable ideals. It would be nice to generalize the results in Section 4 to t-spread strongly stable ideals for all $t\ge 2$. The following questions are currently under investigation.

Open 1.

Given an integer $t\ge 2$, let ${\mathcal{S}}_{t,n}$ be the set of all t-spread strongly stable ideals in S. What is the largest number of corners allowed for an ideal of ${\mathcal{S}}_{t,n}$?

Open 2.

Given three positive integers $t\ge 2$, n and $r<n$, r pairs of positive integers $(k_1,{\ell }_1)$, …, $(k_r,{\ell }_r)$ such that $n-3\ge k_1>k_2>\cdots >k_r\ge 2$ and $2\le {\ell }_1<{\ell }_2<\cdots <{\ell }_r$, and r positive integers $a_1,\dots ,a_r$, under which conditions does there exist a t-spread strongly stable ideal I of $S=K[x_1,\cdots ,x_n]$ such that ${\beta }_{k_1,k_1+{\ell }_1}\left(I\right)=a_1$, …, ${\beta }_{k_r,k_r+{\ell }_r}\left(I\right)=a_r$ are its extremal Betti numbers?

A positive answer to the previous questions can be found in [7] when $t=1$ (see also [17,18]).