Squarefree monomial ideals with maximal depth
Ahad Rahimi
math.AC math.CO 13C15, 05E40






[2010] 13C15, 05E40.


 Ahad Rahimi, Department of Mathematics, Razi University, Kermanshah, Iranahad.rahimi@razi.ac.ir


Let (R,) be a Noetherian local ring and M a finitely generated R-module.  We say M has maximal depth if there is an associated prime  of M  such that M= R/. In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graph with this property are classified.



Squarefree monomial ideals with maximal depth
     Ahad Rahimi
    December 30, 2023
=============================================





§ INTRODUCTION

Let K be a field and (R,) be a Noetherian local ring, or a standard graded K-algebra with graded
maximal ideal . Let M be a finitely generated R-module.
A basic fact in commutative algebra says that

    M≤min{ R/: ∈(M)}.

We set _R M=min{ R/: ∈(M)}. For simplicity, we write M instead of _RM. We say M has maximal depth if the equality holds, i.e., M = M. In other words, there is an associated prime  of M  such that M= R/. In this paper, we study squarefree monomial ideals with maximal depth.

Let I⊂ S=K[x_1,…,x_n] be a squarefree monomial ideal. We say I has maximal depth if S/I has maximal depth.  We observe that, I has maximal depth is equivalent to say that I^∨ is the maximum degree of the generators of I^∨.  This fact motivates us to work on squarefree monomial ideals with maximal depth. Here I^∨ is the Alexander dual of I and M denotes the regularity
of a finitely generated graded S-module M.

Several authors have been working on this topic and some known results in this regards are as follows: If I ⊂ S is a generic monomial ideal, then it has maximal depth, see <cit.>. If a monomial ideal I has maximal depth, then so does its polarization, see <cit.>. Algebraic properties and some classifications of modules with maximal depth are given in <cit.>.


In <cit.>,  the depth of the line graph L_n is explicitly computed. In Section 2, we compute the depth of the line graph L_n in a different way. Our proof relies on the fact that trees, and line graphs in particular, have maximal depth, see Proposition <ref>.

In <cit.>,  the depth of the cycle graph C_n of length n is also computed.  This number is independent
of the characteristic of the chosen field. By using this result, we classify all cycle graphs C_n which have maximal depth. In fact, C_n has maximal depth if and only if n≡ 0( 3) or n≡ 2( 3), see Proposition <ref>.


Adding a whisker to C_n at a vertex x_1  means adding a new vertex
x_n+1  and the edge {x_1, x_n+1} to C_n.   We denote by C_n∪ W(x_1)  the graph
obtained from C_n  by adding a whisker at x_1.  By using Proposition <ref>, we show that  C_n and C_n∪ W(x_1) have the same depth as well as  C_n∪ W(x_1) has maximal depth.






In Section 3, we consider the transversal polymatroidal ideals. A transversal polymatroidal ideal is an ideal I of the form
I = _F_1…_F_r  where F_1, …, F_r is a collection of non-empty subsets of [n] with r≥ 1. Here for a non-empty subset F  of [n], we denote by _F the monomial prime
ideal ({x_i: i ∈ F}). The depth of a transversal polymatroidal ideal is explicitly given in <cit.>. By applying this result, we classify all transversal polymatroidal ideals which have maximal depth. In fact, we prove the following: Let I⊂ S be a transversal polymatroidal ideal. Then,  I has maximal depth if and only  I is a product of monomial prime ideals such that at most one of the factors is not principal. In the following, we also classify ideals of Veronese type which have maximal depth.

In the final section, we consider G to be a connected bipartite graph and I its edge ideal. We show I^k has maximal depth for k≫ 0 if and only if G is a star graph.




§ PRELIMINARIES



Let K be a field and (R,) be a Noetherian local ring, or a standard graded K-algebra with graded
maximal ideal .
It is a classical fact that if M≠ 0 is an R-module, then

    M≤min{ R/: ∈(M)},

see <cit.>. We set _R M=min{ R/: ∈(M)}. For simplicity, we write M instead of _RM.
Thus M≤ M≤ M. Observe that (M)=0 if and only if (M)=0. Thus, if M=1, then M=1.
We say M has maximal depth if the equality holds, i.e.,

    M = M.

In other words, there is an associated prime  of M  such that M= R/.

Some examples of modules with maximal depth property are as follows:
 

  * Cohen–Macaulay modules have maximal depth because M= R/ for every associated prime of M, see <cit.>.

  * Sequentially Cohen-Macaulay modules have maximal depth, see <cit.>, see also <cit.> where the ring R is a polynomial ring.

  * If M is unmixed, then M has maximal depth if and only if M is Cohen–Macaulay.



Let I⊂ S=K[x_1,…,x_n] be a squarefree monomial ideal. Then I=_j=1^m _j where each of the _j  is a monomial prime ideal of I.  The ideal I^∨ which is minimally generated by the monomials u_j=∏_x_i∈_jx_i is called the Alexander dual of I. As usual we denote by M the regularity
of a finitely generated graded S-module M. We quote the following facts which for example can be found in <cit.>.



(Terai) I^∨= S/I.







(Auslander-Buchsbam formula) Let M be a finitely generated R-module with M<∞. Then

    M+ M= R.


The big height of an ideal J⊂ S, denoted by J, is the maximum height of the minimal primes of J.
The following simple fact motivates us to work on squarefree monomial ideals with maximal depth. We say I⊂ S has maximal depth if S/I has maximal depth.


Let I⊂ S be a squarefree monomial ideal. Then, I has maximal depth if and only if I^∨ is the maximum degree of the generators of I^∨.



Suppose I has maximal depth. Hence

    I^∨   =    S/I
       =   n- S/I
       =   n- S/I
       =    I.

Theorem <ref>  explains the first step in this sequence. Theorem <ref> provides the second step. Our assumption implies the third step. The forth step follows from that fact that when I is squarefree, the associated primes are the same as minimal primes containing I.  Notice that the I is the maximum degree of the generators of I^∨. Therefore, the conclusion follows.
Conversely, suppose I^∨ is the maximum degree of the generators of I^∨. By the same reasons as above, we have


    S/I=n- S/I
              =n- I^∨
              =n- I
              = S/I,


as desired.

We recall the following fact from <cit.>.

(Depth Lemma) If  0→ N → M → L → 0 is a short exact sequence of R-modules, then
 

  (a) If (M)<(L), then (N)=(M).

  (b) If (M)=(L), then (N)≥(M).

  (c) If (M)>(L), then (N)=(L)+1.








§ LINE AND CYCLE GRAPHS



Let G be a graph.  The vertex set of G will be denoted by V(G) and will be the set [n]={1,2,…,n}. We denote the set of edges of G by E(G). We consider the edge ideal I(G)  which is generated by all monomials x_ix_j with {i,j}∈ E(G). A subset C⊂ [n] is called a vertex cover of G  if C{i,j}≠∅ for all edges {i,j} of G.
A vertex cover C is called minimal if C is a
vertex of G, and no proper subset of C is a vertex cover of G. A minimal vertex cover of G is
called maximum if it has maximum cardinality among the minimal vertex covers of G. Thus I(G) is the cardinality of the maximum minimal vertex covers of G.

It is well known that the minimal vertex covers of G are the sets of generators of the minimal primes of I(G). In fact, a subset C={ i_1, …, i_r}⊂ [n]  is a minimal vertex cover of G if and only if _C=(x_i_1, …, x_i_r) is a
minimal prime ideal of I(G), see <cit.>.





The graph  G is called disconnected if V(G) is the disjoint union of W_1 and W_2 and there is no edge {i,j} of G with i∈ W_1 and j∈ W_2. The graph G is called connected if it is not disconnected.  A graph which has no cycle and which is connected is called a tree.

For n≥ 2, we let L_n denote the line graph on n vertices. This is the graph with vertices [n] and edges {j, j + 1} for all j = 1, …, n-1. Hence, its edge ideal is I(L_n)=(x_1x_2, x_2x_3, …, x_n-1x_n) in a polynomial ring with n variables.
In the following, we explicitly compute the depth of the line graph L_n. However, this is a known fact, see <cit.> but here we prove it in a different way.
   

Notation:
 For any graph G, we write G for the depth of S/I(G).




The depth of the line graph L_n is independent
of the characteristic of the chosen field and is

    L_n= {[            n/3  if  n≡ 0( 3),;          n+2/3 if   n≡ 1( 3),;          n+1/3 if   n≡ 2( 3). ].



Notice that the line graph is a tree.  Trees are sequentially Cohen-Macaulay, see <cit.>. As sequentially Cohen-Macaulay modules have maximal depth, all trees have maximal depth. In particular, L_n has maximal depth for all n.  Let I=I(L_n) be the edge ideal of L_n in a polynomial ring S with n variables. We consider the following cases:

Case 1: n≡ 0( 3).  We claim that the set
 
    C={1, 3, 4, 6,  7, 9, 10, …, n-3, n-2, n}

  is a maximum minimal vertex cover of L_n. A minimal vertex cover of L_n cannot contain 3 consecutive vertices because of minimality. This implies that if we divide the vertices of L_n into blocks of 3 vertices, then each block can have at most 2 vertices in the cover. Therefore the cardinality of a minimal vertex cover can be at most 2n/3.  Thus
  
    _C=(x_1, x_3, x_4, x_6,  x_7, x_9, x_10,…, x_n-3, x_n-2, x_n)

  is a minimal prime ideal of I with maximum height and so I= 2n/3. It follows that L_n=n-2n/3=n/3 and hence L_n=n/3.

  Case 2:  n≡ 1( 3).  Hence n-1≡ 0( 3). We claim that the set
   
    C={ 2, 3, 5, 6, 8, 9,…, n-2, n-1}

is a maximum minimal vertex cover of L_n. In fact, the vertices of L_n can be divided into blocks with 3 vertices as well as one block with only 1 vertex. Then each block of 3 vertices can have at most 2 vertices in the cover. The vertex in the block with one vertex need not to be in the cover. Therefore the cardinality of a minimal vertex cover can be at most 2(n-1)/3. Hence

    _C=(x_2, x_3, x_5, x_6, x_8, x_9,…, x_n-2, x_n-1)

 is a minimal prime ideal of I with maximum height and so I= 2(n-1)/3. Consequently, L_n=n-2(n-1)/3=(n+2)/3 and so L_n=(n+2)/3.

  Case 3:  n≡ 2( 3).  Hence, n-2≡ 0( 3).  The set

    C={ 2, 3, 5, 6, 8, 9,…, n-3, n-2, n }

is a maximum minimal vertex cover of L_n. Indeed, the vertices of L_n can be divided into blocks with 3 vertices as well as only one block with 2 vertex. Then each block of 3 vertices can have at most 2 vertices and the block of 2 vertices can have at most 1 vertex in the cover.  Therefore the cardinality of a minimal vertex cover can be at most 2(n-2)/3+1=(2n-1)/3. Hence

    _C=(x_2, x_3, x_5, x_6, x_8, x_9,…, x_n-3, x_n-2, x_n)

is a minimal prime ideal of I with maximum height and so I= 2(n-1)/3. Consequently,  L_n=n-(2n-1)/3=(n+1)/3 and so L_n=(n+1)/3. We remark that the proof of proposition does not depend on the characteristic of the field K.


Let C_n be a cycle graph of length n.
 We recall the following result from <cit.>.


The depth of the cycle graph is independent
of the characteristic of the chosen field and is

    C_n= {[            n/3  if  n≡ 0( 3),;          n-1/3 if   n≡ 1( 3),;          n+1/3 if   n≡ 2( 3). ].

 
In the following, we classify all cycle graphs which have maximal depth.


The cycle graph C_n has maximal depth if and only if n≡ 0( 3) or n≡ 2( 3).


Let I=I(C_n) be the edge ideal of C_n in a polynomial ring S with n variables. We need to consider the following three cases.

Case 1:
 n≡ 0( 3). For the maximum
minimal vertex covers of cycles one can use the line graphs. A similar argument as in the proof of Proposition <ref> shows that,  the cardinality of a minimal vertex cover of C_n in this case can be at most 2n/3. The set
 
    C={i, i+1, i+3, i+4,  i+6, i+7,…, n-i-1, n-i }

 is a maximum minimal vertex cover of C_n for all i. Thus,
  
    _C=(x_i, x_i+1, x_i+3, x_i+4,  x_i+6, x_i+7,…, x_n-i-1, x_n-i)

  is a minimal prime ideal of I with maximum height and so I=2n/3. Hence C_n=n-2n/3=n/3= C_n. Fact <ref> provides the last equality. Therefore, C_n has maximal depth.

Case 2:   n≡ 2( 3).  A similar argument as in the proof of Proposition <ref> shows that,  the cardinality of a minimal vertex cover of C_n in this case can be at most 2(n-2)/3+1=(2n-1)/3. One observes that the set

    C= { i, i+1, i+3, i+4, …, n+i-5, n+i-4, n+i-2}

is a maximum minimal vertex cover of C_n for all i. Hence

    _C=(x_i, x_i+1, x_i+3, x_i+4, …, x_n+i-5, x_n+i-4, x_n+i-2)

 is a minimal prime ideal of I with maximum height and so I=(2n-1)/3. Consequently,
  
    C_n=n-(2n-1)/3=(n+1)/3= C_n.

   Fact <ref> explains the last equality.

Case 3:  n≡ 1( 3).  In this case, one has that, the cardinality of a minimal vertex cover of C_n can be at most 2(n-1)/3 and the set
 
    C={ i, i+2, i+3, i+5, i+6, …, n+i-5, n+i-4, n+i-2}

is a maximum minimal vertex cover of C_n for all i. Hence

    _C=(x_i, x_i+2, x_i+3, x_i+5, x_i+6, …, x_n+i-5, x_n+i-4, x_n+i-2)

 is a minimal prime ideal of I with maximum height and so I=2(n-1)/3. Thus C_n=n-2(n-1)/3=(n+2)/3. Fact <ref> provides C_n=(n-1)/3. Thus, C_n has no maximal depth in this case.








 Adding a whisker to C_n at a vertex x_1  means adding a new vertex
x_n+1  and the edge {x_1, x_n+1} to C_n.   We denote by C_n∪ W(x_1)  the graph
obtained from C_n  by adding a whisker at x_1.  Thus I(C_n∪ W(x_1))=I(C_n)+(x_1x_n+1).
 In  the following, by using Proposition <ref>, we show that  C_n and C_n∪ W(x_1) have the same depth as well as  C_n∪ W(x_1) has maximal depth.


The following statements hold.

    C_n=  C_n∪ W(x_1),

 and C_n∪ W(x_1) has maximal depth.


We set I(C_n)=J and I(C_n∪ W(x_1))=I. Consider the exact sequence

    0→ R/(I: x_n+1) → R/I → R/(I+(x_n+1)) → 0,

where R=S[x_n+1].  One has

    R/(I: x_n+1)  K[x_2, …, x_n][x_n+1]/(x_2x_3, x_3x_4, …, x_n-1x_n),

and

    R/(I+(x_n+1))  S/J.

We consider the following three cases:


Case 1: n≡ 0( 3).  Thus n-1≡ 2( 3). By Proposition <ref>

    K[x_2, …, x_n]/(x_2x_3, x_3x_4, …, x_n-1x_n)=((n-1)+1)/3=n/3.

Hence R/(I: x_n+1)=n/3+1.
Fact <ref> provides S/J=n/3. Thus by using (1) we have

    R/I ≥min{n/3+1, n/3 }=n/3.


  For computing R/I in this case, a similar argument as in the proof of Proposition <ref> shows that,  the cardinality of a minimal vertex cover of C_n∪ W(x_1) can be at most (2n+3)/3. One observes that the set

    C={n+1, 2, 3, 5, 6, …, n-4, n-3, n-1, n}

is a maximum minimal vertex cover of C_n∪ W(x_1). Thus

    _C=(x_n+1, x_2, x_3, x_5, x_6, …, x_n-4, x_n-3, x_n-1, x_n)

 is a minimal prime ideal of I with _C=(2n+3)/3. Hence

    R/I=(n+1)-_C=(n+1)-(2n+3)/3=n/3.

Consequently,

    n/3 ≤ R/I≤ R/I=n/3.

 Thus,  the results follow in this case.

Case 2: n≡ 1( 3).  Thus n-1≡ 0( 3). By Proposition <ref>

    K[x_2, …, x_n]/(x_2x_3, x_3x_4, …, x_n-1x_n)=(n-1)/3.

Hence R/(I: x_n+1)=(n+2)/3.
  Fact <ref> explains S/J=(n-1)/3. Thus by using (1) we have
  
    R/I ≥min{(n+2)/3, (n-1)/3 }=(n-1)/3.

 One observes that the cardinality of a minimal vertex cover of C_n∪ W(x_1) in this case can be at most 2(n+2)/3 and the set
 
    C={n+1, 2, 3, 5, 6, …, n-5, n-4, n-2, n}

is a maximum minimal vertex cover of C_n∪ W(x_1). Thus

    _C=(x_n+1, x_2, x_3, x_5, x_6, …, x_n-5, x_n-4, x_n-2, x_n)

 is a minimal prime ideal of I with _C=2(n+2)/3. Hence
 
    R/I=(n+1)-(2n+4)/3=(n-1)/3.

 We conclude that
 
    (n-1)/3 ≤ R/I≤ R/I=(n-1)/3.

 Therefore, the desired conclusions follow in this case.

Case 3: n≡ 2( 3).  Thus n-1≡ 1( 3). By Proposition <ref>

    K[x_2, …, x_n]/(x_2x_3, x_3x_4, …, x_n-1x_n)=((n-1)+2)/3=(n+1)/3.

Hence R/(I: x_n+1)=(n+4)/3.
 Fact <ref> provides  S/J=(n+1)/3. Thus by using (1) we have
 
    R/I≥min{(n+4)/3, (n+1)/3 }=(n+1)/3.

 One observes that the cardinality of a minimal vertex cover of C_n∪ W(x_1) can be at most 2(n+1)/3 and the set
 
    C={n+1, 2, 3, 5, 6, …, n-3, n-2, n}

is a maximum minimal vertex cover of C_n∪ W(x_1). Thus

    _C=(x_n+1, x_2, x_3, x_5, x_6, …,  x_n-3, x_n-2, x_n)

 is a minimal prime ideal  of I with _C=2(n+1)/3. Hence
 
    R/I=(n+1)-_C=(n+1)-(2n+2)/3=(n+1)/3.


Consequently,

    (n+1)/3 ≤ R/I≤ R/I=(n+1)/3.

 Therefore, the desired conclusions follow in this case too.













We remark that the second part of Proposition <ref> also follows from <cit.> in a different way.












§ TRANSVERSAL POLYMATROIDS AND IDEALS OF VERONESE TYPE

In this section, we classify all transversal polymatroidal ideals and all ideals of Veronese type which have maximal depth.
Let F be a non-empty subset of [n]. We denote by _F the monomial prime
ideal ({x_i: i ∈ F}). A transversal polymatroidal ideal is an ideal I of the form
I = _F_1…_F_r  where F_1, …, F_r is a collection of non-empty subsets of [n] with r≥ 1.
Let G_I be the graph with vertex set {1, …, r} and for which {i, j}  is an edge of G_I if and only if F_i∩ F_j ≠∅.
We recall the following fact from <cit.>.

 Let I=_F_1…_F_r⊂ S be a transversal polymatroidal ideal. Then

    S/I=c(G_I)-1+n-|∪_i=1^r F_i|,

where by c(G_I) we denote the number of connected components of the graph G_I.

Let ℋ be a subgraph of G_I. We associate the prime ideal _ℋ=∑_i∈𝒱(ℋ)_F_i. We denote by (I)  the set
of associated prime ideals of R/I. The set associated primes of R/I is explicitly described in <cit.> as follows.


Let I⊂ S be a transversal polymatroidal ideal. Then

    (I)={_𝒯: 𝒯  is a tree in   G_I}.


In the following we characterize all transversal polymatroidal ideals which have maximal depth.

Let I=_F_1…_F_r⊂ S be a transversal polymatroidal ideal. The following conditions are equivalent:
 

  (a)  I has maximal depth;

  (b) I is a product of monomial prime ideals such that at most one of the factors is not principal.



(a)(b):
We may assume that ∪_i=1^r F_i=[n]. Let k=c(G_I) and G_1, …, G_k be the connected components of G_I. Fact <ref> provides
S/I=k-1. We denote by I_1, …, I_k  the transversal polymatroidal ideals for which the associated graphs are the connected components of G_I. Hence I=I_1… I_k=I_1∩…∩ I_k, since
the ideals I_j are generated in pairwise disjoint sets of variables. Thus (I)=⋃_i=1^k (I_i).  We may assume that 1≤ l_1≤…≤ l_k where for all j we have l_j=|⋃_i∈𝒱(G_j)F_i|. Note that l_1+ … + ł_k=n. In view of Fact <ref>, we have S/I=n-l_k.  Since I has maximal depth, it follows that n-l_k=k-1, and hence l_1+… +l_k-1=k-1. Consequently,  I=(x_1)… (x_k-1)(x_k, …, x_n), as desired.

(b)(a): If I is a product of monomial prime ideals such that all the factors are principal, then S/I is Cohen–Macaulay and hence I has maximal depth. Thus, we may assume that  I=(x_1)… (x_k-1)(x_k, …, x_n).  As 
    (I)={ (x_1), …,  (x_k-1), (x_k, …, x_n)},

 we have S/I=n-(n-k+1)=k-1. The ideal I is a transversal polymatroidal ideal. It follows from Theorem <ref> that S/I=k-1. Here c(G_I)=k and |∪_i=1^r F_i|=n. Therefore, I has maximal depth.

As a consequence one has

Let I⊂ S be the intersection of monomial prime ideals in pairwise disjoint sets of variables.  Then I has maximal depth if and only if I is a product of monomial prime ideals such that at most one of the factors is not principal.



One of the most distinguished polymatroidal ideals is the ideal of Veronese type. Let
S = K[x_1, …, x_n] and fix positive integers d and a_1, …, a_n with 1≤ a_1≤…≤ a_n≤ d.
The ideal of Veronese type of S indexed by d and (a_1, …,a_n) is the ideal I_d;a_1, …, a_n
which is generated by those monomials u=x_1^u_1… x_n^u_n of S of degree d with u_i≤ a_i for each 1≤ i≤ n.



 The set of associated prime ideals and depth of the ideal of Veronese type are described in <cit.> as follows

    (S/I) = {_F: F⊂ [n], ∑_i=1^na_i≥ d-1+|F|  and  ∑_i∉Fa_i≤ d-1},

and

    S/I=max{0, d+n-1-∑_i=1^na_i}.




The ideal of Veronese type has maximal depth if and only if there exists a _F ∈(S/I) where |F|=∑_i=1^na_i-(d-1).


In view of (<ref>), _F has the maximum height if |F|=∑_i=1^na_i-(d-1). Thus S/I=n-|F|=n+d-1-∑_i=1^na_i which is the same as S/I by (<ref>).
Therefore, the conclusion follows.




Here is an example
Consider I=I_5; 1, 2, 3⊂ S=K[x_1, x_2, x_3]. Then I=(x_1^2x_2^2x_3, x_1^3x_2x_3, x_1^3x_2^2). Formula (<ref>) yields

    (S/I)={(x_1), (x_2), (x_1, x_2), (x_1, x_3), (x_2, x_3) }.

As _F ∈(S/I) with |F|=2, I has maximal depth and S/I= S/I=1.  




§ POWERS OF IDEALS


A subset D⊂ [n] is called an independent set of G if D contains no set {i,j} which is an edge of G. The graph G is called bipartite if V(G) is  the disjoint union of V_1 and V_2 such that V_1 and V_2 are independent sets.
The bipartite graph G is called a complete bipartite graph if {i, j }∈ E(G) for all i∈ V_1 and j ∈ V_2.


Let G be a complete bipartite graph on the vertex set V with bipartition V=V_1 V_2 where  V_1={v_1,…,v_n} and V_2={w_1,…,w_m} with 1≤ n≤ m. Then G has maximal depth if and only if n=1, i.e., G is a star graph.


The edge ideal of G is _1 ∩_2 where _1=(x_1, …, x_n) and _2=(y_1, …, y_m). We set S=K[x_1, …, x_n, y_1, …, y_m] and R=S/(_1 ∩_2). Consider the exact sequence 0→ S/(_1 ∩_2) → S/_1   S/_2 → S/(_1+_2) → 0. Since S/(_1+ _2)=0, it follows from Lemma <ref>(Depth lemma) that
 R=1. On the other hand, (R)={_1, _2}. It follows that R=n.
Consequently, R- R=n-1. Therefore, the conclusion follows.


In Proposition <ref>, we showed R- R=n-1. Thus, the difference between  and  can be any number.
 





 









In the following we classify all connected bipartite graph such that I^k has maximal depth for all k≫ 0.


Let G be a connected bipartite graph and I=I(G) its edge ideal. Then I^k has maximal depth for k≫ 0 if and only if G is a star graph.


Suppose I^k has maximal depth for k≫ 0. By <cit.>, we have  S/I^k=1 for k≫ 0. Hence S/I^k=1 for k≫ 0. As G is bipartite, we have (I)=(I^k) for all k, see <cit.>. It follows that S/I=1 and hence S/I=1.
Thus there exists a minimal vertex cover F of G such that |[n]∖ F|=1. Therefore, G is a star graph.

Now, suppose G is a star graph and J is its edge ideal. By Proposition <ref> we have S/J= S/J=1. As G is bipartite, we have S/J^k=1 for all k. It follows that S/J^k=1 for all k and hence J^k has maximal depth for k≫ 0.

Let I be an ideal in a Noetherian ring R.  Brodmann [3] showed that (I^k)=(I^k+1)
for all k≫ 0. The ideal I for which (I^k)⊂(I^k+1) for all k≥ 1, is said to satisfy the persistence property. Edge ideals of graphs and  polymatroidal ideals have persistence property, see <cit.>, <cit.>.
In this case, we have  S/I^k+1≤ S/I^k for all k and say the ideal I has non-increasing mdepth.



Acknowledgment




I would like to thank Jürgen Herzog for helpful discussions on this work. I would also like to thank the referee for helpful comments on this article.



99

BH W. Bruns and J. Herzog, "Cohen–Macaulay rings" (Revised edition), Cambridge Studies
in  Advanced Mathematics 39, Cambridge University Press, 1998. Zbl 0788.13005, MR1251956

CO A.Capani, G.Niesi, L.Robbiano:  CoCoA, a system for doing Computations in Commutative Algebra. http://cocoa.dima.unige.it. /research/publications.html, 1995.

F S.  Faridi, Simplicial trees are sequentially Cohen-Macaulay,  J. Pure Appl. Algebra, 190 (2004), 121-136. Zbl 1045.05029, MR2043324

FH C.A. Francisco, H. T. Ha, Whiskers and sequentially Cohen–Macaulay graphs,
J. Combin. Theory Ser. A 115 (2008) 304-316. Zbl 1142.13021,  MR2382518


FT A. Frübis-Kruger and N. Terai, Bounds for the regularity of monomial ideals, Mathematiche
(Catania) 53 (Suppl.) (1998) 83-97.

HH J. Herzog and T. Hibi, Monomial Ideals. GTM 260. Springer 2010. Zbl 1206.13001, MR 2724673

HRV J. Herzog, A. Rauf, M. Vladoiu, The stable set of associated prime ideals of a polymatroidal ideal, J. Algebr. Comb. 37 (2) (2013), 289-312. Zbl 1258.13014, MR3011344

  SJ S.  Jacques, Betti Numbers of Graph Ideals, Ph.D. Thesis, University of Sheffield, 2004.

LT    H.  M.  Lam, N.  V. Trung, Associated primes of powers of edge ideals and ear decompositions of graphs, Trans. Amer. Math. Soc., to appear.

MMV J. Martines-Bernal, S. Morey, R. H. Vilarreal, Associated primes of powers of edge ideals, Collect. Math. 63(2012), 361-374. Zbl 06175355, MR2957976

MSY E. Miller, B. Sturmfels, and K. Yanagawa, Generic and cogeneric monomial ideals, J.
Symb. Comput., 29 (2000), 691-708. Zbl 0955.13008, MR1769661

R A. Rahimi, Maximal depth property of finitely generated modules, J. Algebra Appl., 17, No. 11 (2018) 185020–12.

V R. H. Villarreal, Monomial Algebras, Second Edition, Monographs and Research Notes in Mathematics, Chapman and Hall/CRC, 2015. Zbl 1325.13004, MR 3362802



