Cohen-Macaulay and (S 2 ) Properties of the Second Power of Squarefree Monomial Ideals

: 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.


Introduction
Let K be a fixed field. Let S = K[x 1 , . . . , x n ] be a polynomial ring with deg x i = 1 for all i ∈ [n] = {1, 2, . . . , 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 ≥ 3:

1.
S/I is a complete intersection.

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.
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 dim S/I ≤ 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(G) 2 is not Cohen-Macaulay if the edge ideal I(G) 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.

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 = [n] = {1, 2, . . . , n}. A nonempty subset ∆ of the power set 2 V of V is called a simplicial complex on V if the following two conditions are satisfied: We call a maximal face of ∆ a facet of ∆. Let F (∆) denote the set of all facets of ∆. We call ∆ pure if all its facets have the same dimension. We call ∆ connected if for any pair (p, q), p = q, of vertices of ∆, there is a chain p = p 0 , p 1 , p 2 , . . . , p k = q of vertices of ∆ such that {p i−1 , p i } ∈ ∆ for i = 1, 2, . . . , k.
The Stanley-Reisner ideal I ∆ of ∆ is defined by: The quotient ring K[∆] = K[x 1 , . . . , x n ]/I ∆ is called the Stanley-Reisner ring of ∆. We say that ∆ is a Cohen-Macaulay (resp. Gorenstein) complex if K[∆] is a Cohen-Macaulay (resp. Gorenstein) ring. A Gorenstein complex ∆ is called Gorenstein* if x i divides some minimal monomial generator of I ∆ for each i.
For a face F ∈ ∆, the link and star of F are defined by: The Stanley-Reisner ideal I ∆ of ∆ has the minimal prime decomposition: Note that ∆ is pure if and only if I ∆ is unmixed. We define the th symbolic power of I ∆ by: For a Noetherian ring A, the following condition (S i ) for i = 1, 2, . . . 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 ∆ be a connected simplicial complex. For p, q being two vertices of ∆, the distance between p and q is the minimal length k of chains p = p 0 , p 1 , p 2 , . . . , p k = q of vertices of ∆ such that {p i−1 , p i } ∈ ∆ for i = 1, 2, . . . , k. The diameter, denoted by diam ∆, is the maximal distance between two vertices in ∆. We set diam ∆ = ∞ if ∆ is disconnected. The (S 2 ) property of the second symbolic power of a Stanley-Reisner ideal is characterized as follows: , Corollary 3.3) Let ∆ be a pure simplicial complex. Then, the following conditions are equivalent:

Edge Ideals
Let G be a graph, which means a finite simple graph, which has no loops and multiple edges. We denote by V(G) (resp. E(G)) the set of vertices (resp. edges) of G. We call which is a simplicial complex on the vertex set V(G). We define α(G) by: We define the neighbor set N G (a) of a vertex a of G by: . . , n}. Then, the edge ideal of G, denoted by I(G), is a squarefree monomial ideal of S = K[x 1 , . . . , x n ] defined by: Note that I(G) = I ∆(G) . We call G well-covered (or unmixed) if I(G) is unmixed.
G is triangle-free.

Theorem 5 ([15]
). Let G be a graph. Then, the following conditions are equivalent: 1. G is triangle-free, and I(G) is Gorenstein. 2.

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 α(G) ≥ 2. The following conditions are equivalent: 1.
G is a well-covered graph and satisfies diam ∆(G F ) ≤ 2 for all the independent sets F of G such that |F| ≤ α(G) − 2, 3.
(2) ⇒ (3): For all ab ∈ E(G), we have: Let F be an independent set of G ab . If |F| < α(G) − 1, then |F| ≤ α(G) − 2. Recall that G ab = G\(N G (a) ∪ N G (b)) and F ⊆ V(G ab ). This implies that a, b / ∈ N G [F]. 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 ∆(G F ) ≤ 2, there is a vertex c ∈ V(G F ) such that {a, c}, {c, b} are independent sets of G F . Thus, ac, bc / ∈ E(G F ). Hence, c ∈ V(G ab ). Therefore, F ∪ {c} is an independent of G ab . Then, G ab is well-covered, and moreover, α(G ab ) = α(G) − 1.
If α(G) = 2, then we must prove diam ∆(G) ≤ 2. For all a, b ∈ V(G), we assume {a, b} / ∈ ∆(G). Then, ab ∈ E(G). By the assumption, α(G ab ) = α(G) − 1 = 1 > 0. Therefore, we can take a vertex c in G ab , and thus, ac, bc / ∈ E(G). Hence, {a, c}, {b, c} ∈ ∆(G). Therefore, we conclude that diam ∆(G) ≤ 2. Let α(G) > 2, and suppose that the assertion is true for all graphs G with the same structure as G satisfying the condition "G ab is well-covered and satisfies α(G ab ) = α(G) − 1 for all ab ∈ E(G)" with α(G ) < α(G). For all independent set F of G such that |F| ≤ α(G) − 2, we divide the proof into the following two cases: In this case, we need to prove that diam ∆(G) ≤ 2. In fact, using the same argument as above, we obtain diam ∆(G) ≤ 2.
Case 2: F = ∅. Let x ∈ F. Recall that G is a well-covered graph, and thus, we have By the assumption and [15], Lemma 4.1 (1), (G ab ) x is a well-covered graph with α((G ab ) x ) = α(G ab ) − 1. Therefore, (G x ) 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 α(G ab ) = α(G) − 1 for all ab ∈ E(G)" with α(G x ) < α(G). By the induction hypothesis, we obtain diam ∆((G x ) F\{x} ) ≤ 2. Note that: Then, we get the following theorem. Theorem 6. Let G be a graph. The following conditions are equivalent: 1.
G is triangle-free, and G ab is a well-covered graph with α(G ab ) = α(G) − 1 for all ab ∈ E(G). (1), (2) and (3), without loss of generality, we can assume that G contains no isolated vertices.

Proof. By the statements of Conditions
(2) ⇔ (3): By [15], Theorem 4.4, S/I(G) 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 α(G ab ) = α(G) − 1 for all ab ∈ E(G).
(1) ⇒ (3): If α(G) = 1, then G is a complete graph. By the assumption, G is one edge. Therefore, the statement holds true. Now, we assume α(G) ≥ 2. We know that S/I(G) 2 satisfies that (S 2 ) property if and only if S/I(G) (2) satisfies the (S 2 ) property and I(G) 2 has no embedded associated prime, which means I(G) 2 = I(G) (2) . By Theorem 4 and Lemma 1, G is triangle-free, and G ab is well-covered with α(G ab ) = α(G) − 1 for all ab ∈ E(G).
The question is affirmative if G is a triangle-free graph by Theorems 4 and 6.

Classification
The purpose of the section is to classify all graphs G such that S/I(G) 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(G) 2 is Cohen-Macaulay.

Classification
In this subsection, we classify all graphs G such that S/I(G) 2 is Cohen-Macaulay with dimension less than five.
See [18] for the concrete algorithm we used. By Theorem 6 in this case, the Cohen-Macaulay property is equivalent to the (S 2 ) property, which is independent of the base field K.

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 2 ∆ implies the "Gorenstein" property of I ∆ . More precisely: ). Let ∆ be a simplicial complex on [n]. Suppose that S/I 2 ∆ is Cohen-Macaulay over any field K. Then, ∆ is Gorenstein for any field K.
In [7], the authors asked the following question: Question. Let ∆ be a simplicial complex on [n]. Let S = K[x 1 , . . . , x n ] be a polynomial ring for a fixed field K. Suppose ∆ satisfies the following conditions:
Then, is it true that S/I 2 ∆ is Cohen-Macaulay?
S/I 2 ∆ is not Cohen-Macaulay.
We explain how to find the example. The manifold page of Lutz [19] gives a classification of all triangulations ∆ 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 2 ∆ satisfies the Serre condition (S 2 ). Among the nine types, there was only one simplicial complex ∆ such that S/I 2 ∆ is not Cohen-Macaulay, which is the above example. Note that a triangulation ∆ of a sphere is always Gorenstein. See [18] for more information.