## 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 $dim\Delta =max\{dimF\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}F\in \Delta \}$. 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:

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:

The Stanley-Reisner ideal

${I}_{\Delta}$ of

$\Delta $ has the minimal prime decomposition:

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:

For a Noetherian ring

A, the following condition (S

${}_{i}$) for

$i=1,2,\dots $ is called

Serre’s condition:

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:

which is a simplicial complex on the vertex set

$V\left(G\right)$. We define

$\alpha \left(G\right)$ by:

We define the

neighbor set ${N}_{G}\left(a\right)$ of a vertex

a of

G by:

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:

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:

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:

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:

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 ${G}_{ab}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}G\setminus ({N}_{G}\left(a\right)\cup {N}_{G}\left(b\right))$ 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,

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:

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.

## 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: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.