Cohen-Macaulay and (S₂) Properties of the Second Power of Squarefree Monomial Ideals

Abstract

We show that Cohen-Macaulay and (S${}_2$) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal I such that $S/I^2$ satisfies the Serre condition (S${}_2$), but is not Cohen-Macaulay.

1. Introduction

Let K be a fixed field. Let $S=K[x_1,\dots ,x_n]$ be a polynomial ring with $deg{x}_i=1$ for all $i\in \left[n\right]=\{1,2,\dots ,n\}$. Let I be a squarefree monomial ideal.

For a Stanley-Reisner ring $S/I$, the Cohen-Macaulay and (S${}_2$) properties are different in general. For instance, consider the Stanley-Reisner ring of a non-Cohen-Macaulay manifold, e.g., a torus, which satisfies the (S${}_2$) condition. However, for some special classes of such rings, they are known to be equivalent. The quotient ring of the edge ideal of a very well-covered graph (see [1]) and a Stanley-Reisner ring with “large” multiplicity (see [2] for the precise statement) are such examples. What about the powers of squarefree monomial ideals?

As for the third and larger powers, the following is proven in [3]:

Theorem 1.

Let I be a squarefree monomial ideal. Then, the following conditions are equivalent for a fixed integer $m\ge 3$:

1. 

$S/I$ is a complete intersection.

2. 

$S/I^m$ is Cohen-Macaulay.

3. 

$S/I^m$ satisfies the Serre condition (S${}_2$).

Then, what about the second power of a squarefree monomial ideal? This is the theme of this article. If the second power $I^2$ is Cohen-Macaulay, I is not necessarily a complete intersection. Gorenstein ideals with height three give such examples.

In Section 3, we prove that the Cohen-Macaulay and (S${}_2$) properties are equivalent for the second power of a squarefree monomial ideal generated in degree two:

Theorem 2.

Let I be a squarefree monomial ideal generated in degree two. Then, the following conditions are equivalent:

1. 

$S/I^2$ is Cohen-Macaulay.

2. 

$S/I^2$ satisfies the Serre condition (S${}_2$).

In Section 4, we first give an upper bound of the number of variables in terms of the dimension of $S/I$ when I is a squarefree monomial ideal generated in degree two and $S/I^2$ has the Cohen-Macaulay (equivalently (S${}_2$)) property. Using a computer, we classify squarefree monomial ideals I generated in degree two with $dimS/I\le 4$ such that $S/I^2$ have the Cohen-Macaulay (equivalently (S${}_2$)) property. Since not many examples of squarefree monomial ideals I generated in degree two such that $S/I^2$ are Cohen-Macaulay are known, new examples might be useful. See [4,5] for the two- and three-dimensional cases, respectively, and [6,7] for the higher dimensional case. See also [6,8] for the fact that for a very well-covered graph G, the second power $I{\left(G\right)}^2$ is not Cohen-Macaulay if the edge ideal $I\left(G\right)$ of G is not a complete intersection.

In Section 5, we give an example of a Gorenstein squarefree monomial ideal I such that $S/I^2$ satisfies the Serre condition (S${}_2$), but is not Cohen-Macaulay. Hence, the Cohen-Macaulay and (S${}_2$) properties are different for the second power in general.

2. Preliminaries

2.1. Stanley-Reisner Ideals

We recall some notation on simplicial complexes and their Stanley-Reisner ideals. We refer the reader to [9,10,11] for the detailed information.

Set $V=\left[n\right]=\{1,2,\dots ,n\}$. A nonempty subset $\Delta$ of the power set $2^V$ of V is called a simplicial complex on V if the following two conditions are satisfied: (i) $\left\{v\right\}\in \Delta$ for all $v\in V$, and (ii)$F\in \Delta$, $H\subseteq F$ imply $H\in \Delta$. An element $F\in \Delta$ is called a face of $\Delta$. The dimension of F, denoted by $dimF$, is defined by $dimF=\left|F\right|-1$. The dimension of $\Delta$ is defined by $dim\Delta =max\{dimF:F\in \Delta \}$. We call a maximal face of $\Delta$ a facet of $\Delta$. Let $\mathcal{F}(\Delta )$ denote the set of all facets of $\Delta$. We call $\Delta$ pure if all its facets have the same dimension. We call $\Delta$connected if for any pair $(p,q)$, $p\ne q$, of vertices of $\Delta$, there is a chain $p=p_0,p_1,p_2,\dots ,p_k=q$ of vertices of $\Delta$ such that $\{p_{i-1},p_i\}\in \Delta$ for $i=1,2,\dots ,k$.

The Stanley-Reisner ideal $I_{\Delta }$ of $\Delta$ is defined by:

$$I_{\Delta }=(x_{i_1}{x}_{i_2}\dots x_{i_p}:1\le i_1<\dots <i_p\le n,\{x_{i_1},\dots ,x_{i_p}\}\notin \Delta ).$$

The quotient ring $K[\Delta ]=K[x_1,\dots ,x_n]/I_{\Delta }$ is called the Stanley-Reisner ring of $\Delta$.

We say that $\Delta$ is a Cohen-Macaulay (resp. Gorenstein) complex if $K[\Delta ]$ is a Cohen-Macaulay (resp. Gorenstein) ring. A Gorenstein complex $\Delta$ is called Gorenstein* if $x_i$ divides some minimal monomial generator of $I_{\Delta }$ for each i.

For a face $F\in \Delta$, the link and star of F are defined by:

$$\begin{array}{ccc}\hfill {link}_{\Delta }F& =& \{H\in \Delta :H\cup F\in \Delta ,H\cap F=\varnothing \},\hfill \\ \hfill {star}_{\Delta }F& =& \{H\in \Delta :H\cup F\in \Delta \}.\hfill \end{array}$$

The Stanley-Reisner ideal $I_{\Delta }$ of $\Delta$ has the minimal prime decomposition:

$$I_{\Delta }=\bigcap _{F\in \mathcal{F}(\Delta )}{P}_F,$$

where $P_F=(x\in \left[n\right]\setminus F)$ for each $F\in \mathcal{F}(\Delta )$. We call $I_{\Delta }$unmixed if all $P_F$ have the same height for $F\in \mathcal{F}(\Delta )$. Note that $\Delta$ is pure if and only if $I_{\Delta }$ is unmixed. We define the ${\ell }^{\mathrm{th}}$ symbolic power of $I_{\Delta }$ by:

$$I_{\Delta }^{(\ell )}=\bigcap _{F\in \mathcal{F}(\Delta )}{P}_F^{\ell }.$$

For a Noetherian ring A, the following condition (S${}_i$) for $i=1,2,\dots$ is called Serre’s condition:

$$\left(S_i\right)depth{A}_P\ge min\{heightP,i\}\mathrm{for}\mathrm{all}P\in Spec\left(A\right).$$

See [12] for more information for Stanley-Reisner rings satisfying Serre’s condition (S${}_i$).

To introduce a characterization of the (S${}_2$) property for the second symbolic power of a Stanley-Reisner ideal, we first define the diameter of a simplicial complex. Let $\Delta$ be a connected simplicial complex. For p, q being two vertices of $\Delta$, the distance between p and q is the minimal length k of chains $p=p_0,p_1,p_2,\dots ,p_k=q$ of vertices of $\Delta$ such that $\{p_{i-1},p_i\}\in \Delta$ for $i=1,2,\dots ,k$. The diameter, denoted by $diam\Delta$, is the maximal distance between two vertices in $\Delta$. We set $diam\Delta =\infty$ if $\Delta$ is disconnected. The (S${}_2$) property of the second symbolic power of a Stanley-Reisner ideal is characterized as follows:

Theorem 3.

([7], Corollary 3.3) Let Δ be a pure simplicial complex. Then, the following conditions are equivalent:

1. 

$S/I_{\Delta }^{\left(2\right)}$ satisfies $\left(S_2\right)$.

2. 

$diam\left({link}_{\Delta }F\right)\le 2$ for any face $F\in \Delta$ with $dim{link}_{\Delta }F\ge 1$.

2.2. Edge Ideals

Let G be a graph, which means a finite simple graph, which has no loops and multiple edges. We denote by $V\left(G\right)$ (resp. $E\left(G\right)$) the set of vertices (resp. edges) of G. We call $F\subseteq V\left(G\right)$ an independent set of G if any $e\in E\left(G\right)$ is not contained in F. The independence complex $\Delta \left(G\right)$ of G is defined by:

$$\Delta \left(G\right)=\{F\subset V(G):e\subseteq F\mathrm{for}\mathrm{any}e\in E(G\left)\right\},$$

which is a simplicial complex on the vertex set $V\left(G\right)$. We define $\alpha \left(G\right)$ by:

$$\alpha \left(G\right)=dim\Delta \left(G\right)+1.$$

We define the neighbor set $N_G\left(a\right)$ of a vertex a of G by:

$$N_G\left(a\right)=\{b\in V:ab\in E\left(G\right)\}.$$

Set $N_G\left[a\right]:=\left\{a\right\}\cup N_G\left(a\right)$, which is called the closed neighbor set of a vertex a of G. For $S\subseteq V\left(G\right)$, we denote by $G\setminus S$ the induced subgraph on the vertex set $V\left(G\right)\setminus S$. Set $G_S:=G\setminus N_G\left[S\right]$, where $N_G\left[S\right]:={\cup }_{x\in S}{N}_G\left[x\right].$ If $S\in \Delta \left(G\right)$, then:

$${link}_{\Delta \left(G\right)}\left(S\right)=\Delta \left(G_S\right).$$

See ([11], Lemma 7.4.3). For $ab\in E\left(G\right)$, set $G_{ab}:=G\setminus (N_G\left(a\right)\cup N_G\left(b\right))$.

Set $V\left(G\right)=\{1,\dots ,n\}$. Then, the edge ideal of G, denoted by $I\left(G\right)$, is a squarefree monomial ideal of $S=K[x_1,\dots ,x_n]$ defined by:

$$I\left(G\right)=(x_i{x}_j:\{x_i,x_j\}\in E\left(G\right)).$$

Note that $I\left(G\right)=I_{\Delta \left(G\right)}$. We call Gwell-covered (or unmixed) if $I\left(G\right)$ is unmixed.

Theorem 4.

([13,14]). Let G be a graph. Then, the following conditions are equivalent:

1. 

G is triangle-free.

2. 

$I{\left(G\right)}^{\left(2\right)}=I{\left(G\right)}^2$.

Theorem 5.

([15]). Let G be a graph. Then, the following conditions are equivalent:

1. 

G is triangle-free, and $I\left(G\right)$ is Gorenstein.

2. 

$S/I{\left(G\right)}^2$ is Cohen-Macaulay.

3. The Second Power of Edge Ideals

In this section, we show that the Cohen-Macaulay and (S${}_2$) properties are equivalent for the second power of an edge ideal.

Lemma 1.

Let G be a graph with $\alpha \left(G\right)\ge 2$. The following conditions are equivalent:

1. 

$S/I{\left(G\right)}^{\left(2\right)}$ satisfies the ($S_2$) property,

2. 

G is a well-covered graph and satisfies $diam\Delta \left(G_F\right)\le 2$ for all the independent sets F of G such that $\left|F\right|\le \alpha \left(G\right)-2$,

3. 

$G_{ab}$ is well-covered and satisfies $\alpha \left(G_{ab}\right)=\alpha \left(G\right)-1$ for all $ab\in E\left(G\right)$.

Proof. 

(1) ⇔ (2): By [12], Theorem 8.3, $I\left(G\right)$ satisfies the ($S_2$) property if so does $S/I{\left(G\right)}^{\left(2\right)}$. Using [12], Corollary 5.4, we obtain that $\Delta \left(G\right)$ is pure. This means that G is well-covered, and thus:

$$dim{link}_{\Delta \left(G\right)}\left(F\right)=dim\Delta \left(G\right)-\left|F\right|$$

and ${link}_{\Delta \left(G\right)}\left(F\right)=\Delta \left(G_F\right)$. The result is implied by Theorem 3.

(2) ⇒ (3): For all $ab\in E\left(G\right)$, we have:

$$\alpha \left(G_{ab}\right)\le \alpha \left(G\right)-1.$$

Let F be an independent set of $G_{ab}$. If $\left|F\right|<\alpha \left(G\right)-1$, then $\left|F\right|\le \alpha \left(G\right)-2$. Recall that $G_{ab}=G\setminus (N_G\left(a\right)\cup N_G\left(b\right))$ and $F\subseteq V\left(G_{ab}\right)$. This implies that $a,b\notin N_G\left[F\right]$. Hence, we obtain that $\{a,b\}$ is an edge of $G_F$. In other words, $\{a,b\}$ is not an independent set of $G_F$. By the assumption, $diam\Delta \left(G_F\right)\le 2$, there is a vertex $c\in V\left(G_F\right)$ such that $\{a,c\},\{c,b\}$ are independent sets of $G_F$. Thus, $ac,bc\notin E\left(G_F\right)$. Hence, $c\in V\left(G_{ab}\right)$. Therefore, $F\cup \left\{c\right\}$ is an independent of $G_{ab}$. Then, $G_{ab}$ is well-covered, and moreover, $\alpha \left(G_{ab}\right)=\alpha \left(G\right)-1$.

(3) ⇒ (2): By [15], Lemma 4.1 (2), G is a well-covered graph. We will prove that $diam\Delta \left(G_F\right)\le 2$ for all independent set F with $\left|F\right|\le \alpha \left(G\right)-2$ by induction on $\alpha \left(G\right)$.

If $\alpha \left(G\right)=2$, then we must prove $diam\Delta \left(G\right)\le 2$. For all $a,b\in V\left(G\right)$, we assume $\{a,b\}\notin \Delta \left(G\right)$. Then, $ab\in E\left(G\right)$. By the assumption, $\alpha \left(G_{ab}\right)=\alpha \left(G\right)-1=1>0$. Therefore, we can take a vertex c in $G_{ab}$, and thus, $ac,bc\notin E\left(G\right)$. Hence, $\{a,c\},\{b,c\}\in \Delta \left(G\right)$. Therefore, we conclude that $diam\Delta \left(G\right)\le 2$.

Let $\alpha \left(G\right)>2$, and suppose that the assertion is true for all graphs $G^{\prime }$ with the same structure as G satisfying the condition “$G_{ab}$ is well-covered and satisfies $\alpha \left(G_{ab}\right)=\alpha \left(G\right)-1$ for all $ab\in E\left(G\right)$” with $\alpha \left(G^{\prime }\right)<\alpha \left(G\right)$. For all independent set F of G such that $\left|F\right|\le \alpha \left(G\right)-2$, we divide the proof into the following two cases:

Case 1:$F=\varnothing$. In this case, we need to prove that $diam\Delta \left(G\right)\le 2$. In fact, using the same argument as above, we obtain $diam\Delta \left(G\right)\le 2$.

Case 2:$F\ne \varnothing$. Let $x\in F$. Recall that G is a well-covered graph, and thus, we have $\alpha \left(G_x\right)=\alpha \left(G\right)-1$. Hence, $|F\setminus \left\{x\right\}|=\left|F\right|-1\le \alpha \left(G\right)-3=\alpha \left(G_x\right)-2$. Note that for all $ab\in E\left(G_x\right)$, we have that ${\left(G_x\right)}_{ab}$ and ${\left(G_{ab}\right)}_x$ are two induced subgraphs of G on vertex set $V\left(G\right)\setminus (N_G\left[x\right]\cup N_G\left(a\right)\cup N_G\left(b\right))$. Thus, ${\left(G_x\right)}_{ab}={\left(G_{ab}\right)}_x$. By the assumption and [15], Lemma 4.1 (1), ${\left(G_{ab}\right)}_x$ is a well-covered graph with $\alpha \left({\left(G_{ab}\right)}_x\right)=\alpha \left(G_{ab}\right)-1$. Therefore, ${\left(G_x\right)}_{ab}$ is also a well-covered graph. Moreover,

$$\alpha \left({\left(G_x\right)}_{ab}\right)=\alpha \left({\left(G_{ab}\right)}_x\right)=\alpha \left(G_{ab}\right)-1=\alpha \left(G\right)-2=\alpha \left(G_x\right)-1.$$

Thus, $G_x$ has the same structure as G satisfying the condition “$G_{ab}$ is well-covered and satisfies $\alpha \left(G_{ab}\right)=\alpha \left(G\right)-1$ for all $ab\in E\left(G\right)$” with $\alpha \left(G_x\right)<\alpha \left(G\right)$. By the induction hypothesis, we obtain $diam\Delta \left({\left(G_x\right)}_{F\setminus \left\{x\right\}}\right)\le 2$. Note that:

$${\left(G_x\right)}_{F\setminus \left\{x\right\}}=G_x\setminus N_G[F\setminus \left\{x\right\}]=G\setminus (N_G\left[x\right]\cup N_G[F\setminus \left\{x\right\}])=G\setminus \left(N_G\left[F\right]\right)=G_F.$$

Therefore, $\Delta \left(G_F\right)=\Delta \left({\left(G_x\right)}_{F\setminus \left\{x\right\}}\right).$ Therefore, we conclude that $diam\Delta \left(G_F\right)\le 2$. □

Then, we get the following theorem.

Theorem 6.

Let G be a graph. The following conditions are equivalent:

1. 

$S/I{\left(G\right)}^2$ satisfies the ($S_2$) property,

2. 

$S/I{\left(G\right)}^2$ is Cohen-Macaulay,

3. 

G is triangle-free, and $G_{ab}$ is a well-covered graph with $\alpha \left(G_{ab}\right)=\alpha \left(G\right)-1$ for all $ab\in E\left(G\right)$.

Proof. 

By the statements of Conditions (1), (2) and (3), without loss of generality, we can assume that G contains no isolated vertices.

(2) ⇔ (3): By [15], Theorem 4.4, $S/I{\left(G\right)}^2$ is Cohen-Macaulay if and only if G is triangle-free and in $W_2$, which is a well-covered graph such that the removal of any vertex of G leaves a well-covered graph with the same independence number as G. By [15], Lemma 4.2, this is equivalent to the condition that G is triangle-free and $G_{ab}$ is a well-covered graph with $\alpha \left(G_{ab}\right)=\alpha \left(G\right)-1$ for all $ab\in E\left(G\right)$.

(2) ⇒ (1): It is obvious.

(1) ⇒ (3): If $\alpha \left(G\right)=1$, then G is a complete graph. By the assumption, G is one edge. Therefore, the statement holds true. Now, we assume $\alpha \left(G\right)\ge 2$. We know that $S/I{\left(G\right)}^2$ satisfies that ($S_2$) property if and only if $S/I{\left(G\right)}^{\left(2\right)}$ satisfies the ($S_2$) property and $I{\left(G\right)}^2$ has no embedded associated prime, which means $I{\left(G\right)}^2=I{\left(G\right)}^{\left(2\right)}$. By Theorem 4 and Lemma 1, G is triangle-free, and $G_{ab}$ is well-covered with $\alpha \left(G_{ab}\right)=\alpha \left(G\right)-1$ for all $ab\in E\left(G\right)$. □

Question. 

If $S/I{\left(G\right)}^{\left(2\right)}$ satisfies the (S${}_2$) property, then is it Cohen-Macaulay?

The question is affirmative if G is a triangle-free graph by Theorems 4 and 6.

4. Classification

The purpose of the section is to classify all graphs G such that $S/I{\left(G\right)}^2$ is Cohen-Macaulay with dimension less than five. First, we give an upper bound of the number of vertices of a graph G such that $S/I{\left(G\right)}^2$ is Cohen-Macaulay.

4.1. Upper Bound of the Number of Vertices

Theorem 7.

(Upper bound).Let G be a graph with the vertex set $\left[n\right]$. Suppose G has no isolate vertex. If $S/I{\left(G\right)}^2$ is d-dimensional Cohen-Macaulay, where $d\ge 3$, then we have $n\le \frac{d^2+3d-2}{2}$.

Proof. 

We prove this by induction on d. For $d=3$, we have $n\le 8$ by [5] (see Proposition 3). Set $N\left(d\right)=\frac{d^2+3d-2}{2}$. Let n be the number of vertices of G such that $S/I{\left(G\right)}^2$ is d-dimensional and Cohen-Macaulay. Let $i\in \left[n\right].$ Then, we have $n=|V\left({star}_{\Delta \left(G\right)}\left\{i\right\}\right)|+|(\left[n\right]\setminus V\left({star}_{\Delta \left(G\right)}\left\{i\right\}\right)|$. Since G is triangle-free by Theorem 5, an edge among $\{i,p\}$, $\{i,q\}$ and $\{p,q\}$ belongs to $\Delta \left(G\right)$ for any $p,q\in (\left[n\right]\setminus V\left({star}_{\Delta \left(G\right)}\left\{i\right\}\right)$, where $p\ne q$. By the definition of ${star}_{\Delta \left(G\right)}\left\{i\right\}$, we have $\{i,p\}$, $\{i,q\}\notin \Delta \left(G\right)$. Then, we have $\{p,q\}\in \Delta \left(G\right)$. By the fact that $I\left(G\right)$ is generated in degree two, all minimal non-faces of $\Delta \left(G\right)$ have cardinality two. Now, we know that $\{p,q\}\in \Delta \left(G\right)$ for any $p,q\in (\left[n\right]\setminus V\left({star}_{\Delta \left(G\right)}\left\{i\right\}\right)$; hence, we have $\left[n\right]\setminus V\left({star}_{\Delta \left(G\right)}\left\{i\right\}\right)\in \Delta \left(G\right)$. By the assumption that $S/I{\left(G\right)}^2$ is d-dimensional, we have $|\left[n\right]\setminus V({star}_{\Delta \left(G\right)}\left\{i\right\}\left)\right|\le d$. Since $\Delta \left(G\right)$ is Gorenstein*, so is ${link}_{\Delta \left(G\right)}\left\{i\right\}$ by [10], Theorem 5.1. By Theorem 5, $I_{{link}_{\Delta \left(G\right)}\left\{i\right\}}^2$ is Cohen-Macaulay. Hence, $|V\left({star}_{\Delta \left(G\right)}\left\{i\right\}\right)|=|V\left({link}_{\Delta \left(G\right)}\left\{i\right\}\right)|+1\le N(d-1)+1$ by the induction hypothesis. Therefore, $n\le N(d-1)+d+1=\frac{{(d-1)}^2+3(d-1)-2}{2}+d+1=\frac{d^2+3d-2}{2}=N\left(d\right)$. □

4.2. Classification

In this subsection, we classify all graphs G such that $S/I{\left(G\right)}^2$ is Cohen-Macaulay with dimension less than five.

Proposition 1.

(One-dimensional case) Let G be a graph with the vertex set $\left[n\right]$. Suppose G has no isolate vertex. Then, $S/I{\left(G\right)}^2$ is one-dimensional Cohen-Macaulay if and only if $n=2$ and $I\left(G\right)=\left(x_1{x}_2\right)$.

Proposition 2.

([4]). (Two-dimensional case) Let G be a graph with the vertex set $\left[n\right]$. Suppose G has no isolate vertex. Then, $S/I{\left(G\right)}^2$ is two-dimensional Cohen-Macaulay if and only if $I\left(G\right)$ is one of the following up to the permutation of variables:

1. 

If $n=4$, then $(x_1{x}_3,x_2{x}_4)$.

2. 

If $n=5$, then $(x_1{x}_3,x_1{x}_4,x_2{x}_3,x_2{x}_5,x_4{x}_5)$.

Proposition 3.

([5]). (Three-dimensional case) Let G be a graph with the vertex set $\left[n\right]$. Suppose G has no isolate vertex. Then, $S/I{\left(G\right)}^2$ is three-dimensional Cohen-Macaulay if and only if $I\left(G\right)$ is one of the following up to the permutation of variables:

1. 

If $n=6$, then $(x_1{x}_4,x_2{x}_5,x_3{x}_6)$.

2. 

If $n=7$, then $(x_1{x}_5,x_1{x}_6,x_2{x}_5,x_2{x}_7,x_3{x}_4,x_6{x}_7)$.

3. 

If $n=8$, then $(x_1{x}_2,x_1{x}_5,x_1{x}_8,x_2{x}_3,x_3{x}_4,x_4{x}_5,x_4{x}_8,x_5{x}_6,x_6{x}_7,x_7{x}_8)$.

Using a computer with Nauty [16] and CoCoA [17], we classify four-dimensional case: By Theorem 7, it is enough to search for them up to $n=13$.

Theorem 8.

(Four-dimensional case) Let G be a graph with the vertex set $\left[n\right]$. Suppose G has no isolate vertex. Then, $S/I{\left(G\right)}^2$ is four-dimensional Cohen-Macaulay if and only if $I\left(G\right)$ is one of the following up to the permutation of variables:

1. 

If $n=8$, then $(x_1{x}_5,x_2{x}_6,x_3{x}_7,x_4{x}_8)$.

2. 

If $n=9$, then $(x_1{x}_5,x_2{x}_6,x_3{x}_7,x_1{x}_8,x_4{x}_8,x_4{x}_9,x_5{x}_9)$.

3. 

If $n=10$, then

 (a) $(x_1{x}_5,x_2{x}_6,x_3{x}_7,x_1{x}_8,x_4{x}_8,x_2{x}_9,x_4{x}_9,x_5{x}_9,x_4{x}_{10},x_5{x}_{10},x_6{x}_{10})$.

 (b) $(x_1{x}_5,x_2{x}_6,x_1{x}_7,x_3{x}_7,x_3{x}_8,x_5{x}_8,x_2{x}_9,x_4{x}_9,x_4{x}_{10},x_6{x}_{10})$.

3. 

If $n=11$, then

 (a) $(x_1{x}_5,x_2{x}_6,x_3{x}_7,x_1{x}_8,x_4{x}_8,x_2{x}_9,x_4{x}_9,x_5{x}_9,x_3{x}_{10},x_4{x}_{10},x_5{x}_{10},x_6{x}_{10},x_4{x}_{11},x_5{x}_{11},x_6{x}_{11},x_7{x}_{11})$.

 (b) $(x_1{x}_5,x_2{x}_6,x_1{x}_7,x_3{x}_7,x_3{x}_8,x_5{x}_8,x_2{x}_9,x_4{x}_9,x_1{x}_{10},x_4{x}_{10},x_6{x}_{10},x_4{x}_{11},x_5{x}_{11},x_6{x}_{11},x_7{x}_{11})$.

3. 

If $n=12$, then

$$\begin{array}{c}(x_1{x}_5,x_2{x}_6,x_1{x}_7,x_3{x}_7,x_2{x}_8,x_4{x}_8,x_2{x}_9,x_3{x}_9,x_5{x}_9,x_1{x}_{10},x_4{x}_{10},x_6{x}_{10},x_4{x}_{11},x_5{x}_{11},x_6{x}_{11},\hfill \\ x_7{x}_{11},x_3{x}_{12},x_5{x}_{12},x_6{x}_{12},x_8{x}_{12}).\hfill \end{array}$$

3. 

If $n=13$, then

$$\begin{array}{c}(x_1{x}_5,x_2{x}_6,x_1{x}_7,x_3{x}_7,x_2{x}_8,x_4{x}_8,x_2{x}_9,x_3{x}_9,x_5{x}_9,x_1{x}_{10},x_3{x}_{10},x_4{x}_{10},x_6{x}_{10},x_3{x}_{11},x_5{x}_{11},x_6{x}_{11},\hfill \\ x_8{x}_{11},x_2{x}_{12},x_4{x}_{12},x_5{x}_{12},x_7{x}_{12},x_4{x}_{13},x_6{x}_{13},x_7{x}_{13},x_9{x}_{13}).\hfill \end{array}$$

See [18] for the concrete algorithm we used. By Theorem 6 in this case, the Cohen-Macaulay property is equivalent to the $\left(S_2\right)$ property, which is independent of the base field K.

5. Example

In this section, we give an example of a Gorenstein squarefree monomial ideal I such that $S/I^2$ satisfies the Serre condition (S${}_2$), but it is not Cohen-Macaulay.

The Cohen-Macaulay property of $I_{\Delta }^2$ implies the “Gorenstein” property of $I_{\Delta }$. More precisely:

Theorem 9.

([7]). Let Δ be a simplicial complex on $\left[n\right]$. Suppose that $S/I_{\Delta }^2$ is Cohen-Macaulay over any field K. Then, Δ is Gorenstein for any field K.

In [7], the authors asked the following question:

Question. 

Let Δ be a simplicial complex on $\left[n\right]$. Let $S=K[x_1,\dots ,x_n]$ be a polynomial ring for a fixed field K. Suppose Δ satisfies the following conditions:

1. 

Δ is Gorenstein.

2. 

$S/I_{\Delta }^2$ satisfies the Serre condition (S${}_2$).

Then, is it true that $S/I_{\Delta }^2$ is Cohen-Macaulay?

Using a list in [19] and CoCoA, we have the following counter-example:

Example 1.

Let K be a field of characteristic zero. Set:

$$I_{\Delta }=(x_1{x}_{10},x_3{x}_9,x_2{x}_9,x_7{x}_8,x_2{x}_8,x_4{x}_7,x_5{x}_6,x_3{x}_6,x_4{x}_5,x_6{x}_8{x}_{10},x_2{x}_5{x}_{10},x_1{x}_4{x}_9,x_1{x}_3{x}_7).$$

Then, the following conditions hold:

1. 

Δ is Gorenstein.

2. 

$S/I_{\Delta }^2$ satisfies the Serre condition (S${}_2$).

3. 

$S/I_{\Delta }^2$ is not Cohen-Macaulay.

We explain how to find the example. The manifold page of Lutz [19] gives a classification of all triangulations $\Delta$ of the three-sphere with 10 vertices, which shows that there are 247,882 types. Using Theorem 3, we checked the Serre condition (S${}_2$) for them, and there were only nine types such that $S/I_{\Delta }^2$ satisfies the Serre condition (S${}_2$). Among the nine types, there was only one simplicial complex $\Delta$ such that $S/I_{\Delta }^2$ is not Cohen-Macaulay, which is the above example. Note that a triangulation $\Delta$ of a sphere is always Gorenstein. See [18] for more information.