#
Cohen-Macaulay and (S_{2}) Properties of the Second Power of Squarefree Monomial Ideals

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

- 1.
- $S/I$ is a complete intersection.
- 2.
- $S/{I}^{m}$ is Cohen-Macaulay.
- 3.
- $S/{I}^{m}$ satisfies the Serre condition (S${}_{2}$).

**Theorem**

**2.**

- 1.
- $S/{I}^{2}$ is Cohen-Macaulay.
- 2.
- $S/{I}^{2}$ satisfies the Serre condition (S${}_{2}$).

## 2. Preliminaries

#### 2.1. Stanley-Reisner Ideals

**Theorem**

**3.**

- 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

**Theorem**

**4.**

- 1.
- G is triangle-free.
- 2.
- $I{\left(G\right)}^{\left(2\right)}=I{\left(G\right)}^{2}$.

**Theorem**

**5.**

- 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

**Lemma**

**1.**

- 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.**

**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,

**Theorem**

**6.**

- 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.**

**Question.**

## 4. Classification

#### 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.**

#### 4.2. Classification

**Proposition**

**1.**

**Proposition**

**2.**

**.**(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.**

**.**(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 $({x}_{1}{x}_{2},{x}_{1}{x}_{5},{x}_{1}{x}_{8},{x}_{2}{x}_{3},{x}_{3}{x}_{4},{x}_{4}{x}_{5},{x}_{4}{x}_{8},{x}_{5}{x}_{6},{x}_{6}{x}_{7},{x}_{7}{x}_{8})$.

**Theorem**

**8.**

- 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$$\begin{array}{c}({x}_{1}{x}_{5},{x}_{2}{x}_{6},{x}_{1}{x}_{7},{x}_{3}{x}_{7},{x}_{2}{x}_{8},{x}_{4}{x}_{8},{x}_{2}{x}_{9},{x}_{3}{x}_{9},{x}_{5}{x}_{9},{x}_{1}{x}_{10},{x}_{3}{x}_{10},{x}_{4}{x}_{10},{x}_{6}{x}_{10},{x}_{3}{x}_{11},{x}_{5}{x}_{11},{x}_{6}{x}_{11},\hfill \\ {x}_{8}{x}_{11},{x}_{2}{x}_{12},{x}_{4}{x}_{12},{x}_{5}{x}_{12},{x}_{7}{x}_{12},{x}_{4}{x}_{13},{x}_{6}{x}_{13},{x}_{7}{x}_{13},{x}_{9}{x}_{13}).\hfill \end{array}$$

## 5. Example

**Theorem**

**9.**

**.**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.

**Question.**

- 1.
- Δ is Gorenstein.
- 2.
- $S/{I}_{\Delta}^{2}$ satisfies the Serre condition (S${}_{2}$).

**Example**

**1.**

- 1.
- Δ is Gorenstein.
- 2.
- $S/{I}_{\Delta}^{2}$ satisfies the Serre condition (S${}_{2}$).
- 3.
- $S/{I}_{\Delta}^{2}$ is not Cohen-Macaulay.

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Hoang, D.T.; Rinaldo, G.; Terai, N.
Cohen-Macaulay and (S_{2}) Properties of the Second Power of Squarefree Monomial Ideals. *Mathematics* **2019**, *7*, 684.
https://doi.org/10.3390/math7080684

**AMA Style**

Hoang DT, Rinaldo G, Terai N.
Cohen-Macaulay and (S_{2}) Properties of the Second Power of Squarefree Monomial Ideals. *Mathematics*. 2019; 7(8):684.
https://doi.org/10.3390/math7080684

**Chicago/Turabian Style**

Hoang, Do Trong, Giancarlo Rinaldo, and Naoki Terai.
2019. "Cohen-Macaulay and (S_{2}) Properties of the Second Power of Squarefree Monomial Ideals" *Mathematics* 7, no. 8: 684.
https://doi.org/10.3390/math7080684