Chordality of Clutters with Vertex Decomposable Dual and Ascent of Clutters
Ashkan Nikseresht







Chordality of Clutters with Vertex Decomposable Dual and Ascent of Clutters
    Ashkan Nikseresht[While this paper was under review, the author's affiliation has been
 changed to: Department of Mathematics, Shiraz University, 71457-13565, Shiraz, Iran] 

Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS),

P.O. Box 45195-1159, Zanjan, Iran.

E-mail: ashkan_nikseresht@yahoo.com


    
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In this paper, we consider the generalization of chordal graphs to clutters proposed by Bigdeli, et al in  J. Combin.
Theory, Series A (2017). Assume that  is a d-dimensional uniform clutter. It is known that if  is
chordal, then I() has a linear resolution over all fields. The converse has recently been rejected, but the
following question which poses a weaker version of the converse is still open: “if I() has linear quotients,
is  necessarily chordal?”. Here, by introducing the concept of the ascent of a clutter, we split this question
into two simpler questions and present some clues in support of an affirmative answer. In particular, we show that if
I() is the Stanley-Reisner ideal of a simplicial complex with a vertex decomposable Alexander dual, then 
is chordal.


Keywords and Phrases:  chordal clutter, linear resolution, vertex decomposable simplicial
complex, ascent of a clutter, squarefree monomial ideal.

2010 Mathematical Subject Classification: 13F55, 05E40, 05C65.


                                        

§ INTRODUCTION


In this paper all rings are commutative with identity, K is a field and S=K[x_1,…, x_n] is the polynomial ring
in n indeterminates over K. Given an arbitrary ideal I of S, we can get a monomial ideal by taking its initial
ideal or its generic initial ideal with respect to some monomial order. Many of the properties of I is similar or at
least related to its (generic) initial ideal, especially if I is graded (see, for example, <cit.>). Also if I is a monomial ideal of S then its polarization is a squarefree monomial ideal,
again with algebraic properties similar to I (see <cit.>). Therefore, if we know the algebraic
properties of squarefree monomial ideals well, then we can understand many of the algebraic properties of much larger
classes of ideals.

On the other hand, squarefree monomial ideals have a combinatorial nature and correspond to combinatorial
objects such as simplicial complexes, graphs, clutters and posets. Many researchers have tried to
characterize algebraic properties of squarefree monomial ideals, using their combinatorial counterparts, see
for example <cit.> and Part III of <cit.> and their references. In this regard, two
important objectives are to classify combinatorially squarefree monomial ideals which are Cohen-Macaulay or
those which have a linear resolution (either over every field or over a fixed field). Indeed these two tasks
are equivalent under taking Alexander dual.

A well-known theorem of Fröberg states that if I is a squarefree monomial ideal generated in degree two, then I
has a linear resolution I is the edge ideal of the complement of a choral graph. Motivated by this, many have
tried to generalize the concept of chordality to clutters or simplicial complexes in such a way that at least one side
of Fröberg's theorem stays true in degrees >2 (see, for instance, <cit.>). One of the most
promising such generalizations seems to be the concept of chordal clutters presented in <cit.>. In
<cit.> it is shown that many other classes of “chordal clutters” defined by other researchers, including the
class defined in <cit.> which we call W-chordal clutters, is strictly contained in the class defined in
<cit.>. They also show that if a clutter is chordal, then the circuit ideal of its complement has a linear
resolution over every field. Several clues were presented in <cit.> supporting the correctness of the
converse (see <cit.>). But recently a counterexample to the converse was presented by Eric Babson
(see Example <ref>). Despite this it is still unknown whether the following statement which is a weaker
version of the converse is true or not: “if the circuit ideal of the complement of a clutter has linear quotients,
then that clutter is chordal.” Moreover, in <cit.>, it is shown that many numerical invariants of the ideal
corresponding to a chordal clutter can be combinatorially read off the clutter.

Here, after presenting a brief review of the main concepts and setting the notations, in Section 3 we prove that if the
Alexander dual of the clique complex of a clutter  is vertex decomposable, then  is chordal. As clique
complexes of W-chordal clutters are vertex decomposable, this generalizes the results of Subsection 3.1 of
<cit.>. Then in Section 4, we define the notion of the ascent of a clutter and show how we can use this concept
to divide the question “is  chordal, given that the circuit ideal of the complement of  has  linear
quotients?” into two simpler questions.

                                        

§ PRELIMINARIES, NOTATIONS AND A COUNTEREXAMPLE





  
Algebraic background.
Suppose that I is a graded ideal of S considered with the standard grading. This grading induces a natural grading
on _i^S(K,I). We denote the degree j part of _i^S(K,I) by _i^S(K,I)_j and its dimension over K is
denoted by β_ij^S(I)= β_ij(I). These β_ij's are called the graded Betti numbers of I. If
there is a d≥ 0 such that β_ij(I)=0 for j ≠ i+d, it is said that I has a linear (or
d-linear) resolution. Also we say that I has linear quotients with respect to an ordered
system of homogenous generators f_1, …, f_t, if for each i the ideal f_1, …, f_i-1: f_i is
generated by linear forms. It is known that if I is generated in degree d and has linear quotients with respect to
some system of homogenous generators, then I has a d-linear resolution (see <cit.>).

Now assume that I is a squarefree monomial ideal, that is, I is generated by some squarefree monomials. Then I
has a unique smallest generating set consisting of squarefree monomials, say x_F_1, …, x_F_t where
x_F=∏_i∈ F x_i for F [n]={1,…,n}. In this case, if I has linear quotients with respect to a
permutation of this minimal system of generators, we simply say that I has linear quotients. Also a specific order of
minimal generators of I, for which the aforementioned colon ideals are linear, is called an admissible order.
According to <cit.>, x_F_1, …, x_F_t is an admissible order for the ideal they
generate, for each i and all j<i, there is a l∈ F_j F_i and a k<i such that F_k F_i={l}. For
more details on these algebraic concepts see <cit.>.



  
Clutters.
A clutter  on the vertex set V= () is a family of subsets of V which are pairwise incomparable
under inclusion. We call the elements of  circuits. For any subset F of V we set F=|F|-1. If all
circuits of  have the same dimension d, we say that  is a d-dimensional uniform clutter or a
d-clutter for short. If v∈(), then by -v we mean the clutter on (){v} with
circuits {F∈|v∉ F}. For simplicity, we write for example ab, Ex or Eab instead of {a,b},
E∪{x} or E∪{a,b} for a,b,x∈ V and E V. If we assume that V=[n], then I() is defined as
the ideal of S generated by {x_F|F∈}.

From now on, we always assume that  is a d-clutter with |()|=n.  We call  complete, when all
d-dimensional subsets of V=() are in . A clique of  is a subset A of V such that
_A is complete, where _A={F∈|F A} is the induced d-clutter on the vertex set A. The
complement  of  is the d-clutter with the same vertex set as  and circuits {F V|  F=d, F
∉}.

The set {e V| e=d-1, ∃ F∈   e F} is called the set of maximal subcircuits of 
and is denoted by (). For a (d-1)-dimensional subset e of V, the closed neighborhood N_[e] is
defined as e∪{v∈ V| ev∈}. If e∈(), then -e means the clutter on the vertex set V with
circuits {F∈|eF}.

A maximal subcircuit e of  is called a simplicial maximal subcircuit, when N_[e] is a clique. We
denote the set of all simplicial maximal subcircuits of  by (). If for each 1≤ i< t there is an
e_i∈(_i-1), where _0= and _i= _i-1-e_i, such that _t has no circuits, then 
is called chordal (see <cit.>). In the case that d=1 (that is,  is graph) this notion
coincides with the usual notion of chordal graphs. Theorem 3.3 of <cit.> states that if  is chordal and
not complete, then I() has a linear resolution over every field. There is an example showing that the converse
is not true, that is, there is a non-chordal clutter  with I() having a linear resolution over every field
(see Example <ref>). But it is still unknown whether there is a non-chordal clutter  with I()
having linear quotients. For further reference, we label this statement:

enumi



  *  If I() has linear quotients, then  is chordal.

enumi




  
Simplicial complexes.
A simplicial complex Δ on the vertex set V=(Δ) is a family of subsets of V (called faces
of Δ) such that if A B∈Δ, then A∈Δ. The dimension of Δ is defined as Δ= max_F∈Δ F. The set of maximal faces of Δ which are called facets is denoted by
(Δ). If |(Δ)|=1, then Δ is called a simplex.

If all facets of Δ have the same dimension, we say that Δ is pure. In this case
(Δ) is a d-dimensional uniform clutter. Also if  is a clutter, then  denotes the
simplicial complex Δ with (Δ)=. It should be clear that (Δ) = Δ and
() =, for any simplicial complex Δ and any (not necessarily uniform) clutter .
Another simplicial complex associated to a d-clutter  is the clique complex Δ() of
 defined as the family of all subsets L of () with the property that L is a clique in .
Note that all subsets of () with size ≤ d are cliques by assumption.

For a face F of Δ, we define _Δ F= {G F|F G∈Δ}, which is a simplicial complex on
the vertex set V F. Also if v∈ V, Δ-v is the simplicial complex on the vertex set V{v} with
faces {F∈Δ|v∉ F}.

Assuming that (Δ)=[n], the ideal of S generated by {x_F|F is a minimal non-face of Δ} is called
the Stanley-Reisner ideal of Δ and is denoted by I_Δ. When SI_Δ is a Cohen-Macaulay
ring, Δ is said to be Cohen-Macaulay over K.
 



Let A be a commutative ring with identity and -1≤ d≤Δ and denote by C_d(Δ)=
C_d(Δ, A) the free A-module whose basis is the set of all d-dimensional faces of Δ. Consider the
A-homomorphism _d: C_d(Δ) →C_d-1(Δ) defined by
 
    _d({v_0, …, v_d}) =∑_i=0^d (-1)^i {v_0, …, v_i-1,v_i+1,…, v_d},

where v_0<⋯ <v_d for a fixed total order < on (Δ). Then (C_, _) is a complex of
free A-modules and A-homomorphisms called the augmented oriented chain complex of Δ over A. We
denote the i-th homology of this complex by H_i(Δ; A).

By the Alexander dual of a simplicial complex Δ we mean Δ^∨= {(Δ)  F| F(Δ)
, F∉Δ} and also we set ^∨={() F| F∈}. Then it follows from the
Eagon-Reiner theorem (<cit.>) and the lemma below that I() has a linear resolution over
K, ^∨ is Cohen-Macaulay over K. For more details on simplicial complexes and related
algebraic concepts the reader is referred to <cit.>. We frequently use the following lemma in the sequel without
any further mention.


Let  be a d-clutter. Then


  * I()=I_Δ();

  * ^∨= (Δ())^∨.



When working with both clutters and simplicial complexes, one should notice the differences between the
similar concepts and notations defined for these objects. For example, we say that a d-dimensional
simplicial complex Δ is i-complete to mean that Δ has all possible faces of dimension
i, where i≤ d. But when we are talking about a d-clutter , there is no need to mention the
dimension and say that  is i-complete, since we must have i=d which is implicit in  and it is
completely meaningless to say that  is i-complete for i<d. Therefore we just say that  is
complete. Another difference arises from the concept of Alexander dual which is illustrated in the following
example.

Let ={125,235,345} be a 2-clutter on [5], Γ= and Δ=Δ(). Note that
14 ∉Γ because it is not contained in any facet of Γ. But by definition, 14 is a clique of
 and hence 14∈Δ={125,235,345,13,14,24}. Now for example 245∈ and hence
[5] 245=13∈^∨. Indeed, ^∨={13,15,23,24,25,35,45}, which by <ref> is equal
to the set of facets of Δ^∨. But as 14∉Γ, we see that [5] 14= 235∈Γ^∨.
In fact, ^∨= Γ^∨= {235,245,135}≠^∨.



Recall that Δ_W={F∈Δ|F W} where W V.

The ideal I_Δ has a t-linear resolution over K, H_i(Δ_W; K)=0 for all W(Δ) and
i≠ t-2.

Note that in the case that Δ= Δ(), then since all possible faces of dimension less than d
are in Δ, it follows that H_i(Δ_W; K)=0 for all W and i<d-1. Hence I()
has a (d+1)-linear resolution over K, H_i(Δ_W; K)=0 for all W(Δ) and i≥
d. Using this we can state an example of a non-chordal clutter  with I() having a linear
resolution over every field. This example is due to Eric Babson who visited IASBS as a lecturer in the first
Research School on Commutative Algebra and Algebraic Geometry in 2017.

[Babson]
Let  be the 2-clutter shown in Fig-Dunce which is a triangulation of the dunce hat. Then Δ=
Δ() is obtained by adding the missing edges (1-dimensional subsets of vertices) to Δ'=,
since  has no cliques on more that 3 vertices. Therefore, H_t(Δ; K)=  H_t(Δ'; K) for every
t≥ 2. It is known that the dunce hat is a contractible space and hence 0=  H_t(Δ'; K) for all t≥ 2.
Also it is easy to verify that for any W [8], _W is chordal (indeed, any proper subclutter of  has an
edge which is contained in exactly one triangle and hence is simplicial) and thus has a linear resolution over every
field. Therefore, H_t(Δ_W; K)=  H_t(Δ(_W); K)=0 for all t≥ 2. Consequently, by Föberg's
theorem I_Δ= I() has a linear resolution over every field.




                         

§ CLUTTERS WITH VERTEX DECOMPOSABLE DUAL ARE CHORDAL



A vertex v of a nonempty simplicial complex Δ is called a shedding vertex, when no face (or
equivalently facet) of _Δ(v) is a facet of Δ-v.  Recall that a nonempty simplicial complex
Δ is called vertex decomposable, when either it is a simplex or there is a shedding vertex
v∈ V(Δ) such that both _Δ v and Δ- v are vertex decomposable. This concept was
first introduced in <cit.> in connection with the Hirsch conjecture which has applications in the
analysis of the simplex method in linear programming.


Suppose that Δ is a pure d-dimensional simplicial complex. Then v is a shedding vertex Δ-v is pure with (Δ-v)=d.


(): Assume that Δ-v is not pure or (Δ-v)≠ d. Since every facet of Δ-v has
dimension d-1 or d, there is a facet F of Δ-v such that F= d-1.  As F∈Δ and
Δ is pure, we see that F F' for some F'∈(Δ). Hence F'=Fv∈Δ and F∈_Δ(v), which contradicts the shedding property for v.

(): Every face of _Δ(v) has dimension ≤ d-1 and by assumption Δ-v is pure and
of dimension d. Thus _Δ(v) ∩(Δ-v)= and v is a shedding vertex.



Suppose that Γ is as in Example dual-exam. By choosing 1 and then 4 as the shedding vertex,
we see that Γ is vertex decomposable. Here Γ-5= 12, 23, 34 is pure with dimension
1= (Γ)-1 and Γ-2= 15, 345 is not pure. So by the previous lemma, neither 5 is a
shedding vertex of Γ nor 2. Also Γ^∨ which is indeed isomorphic to Γ is vertex
decomposable. Now assume that Δ= 12,34 which is a 1-dimensional simplicial complex. Then
Δ-1= 2, 34 is not pure, so 1 is not a shedding vertex of Δ. Similarly we see that
Δ has no shedding vertex and is not vertex decomposable.


It is well-known that if Δ is pure and vertex decomposable, then it is shellable and Cohen-Macaulay (see
<cit.>), hence I_Δ^∨ has a linear resolution and in fact linear quotients. So if  is the
clutter with I()=I_Δ^∨ and if statement <ref> is true,  should be chordal. In this section, we
prove that this is indeed the case. By <ref>, in the above situation we have Δ= (Δ()) ^∨=
 ^∨. First we need some lemmas. In the sequel, we assume that  is a d-clutter and |()|=n.
Also recall that the pure i-skeleton of a simplicial complex Δ, is the simplicial complex whose facets
are i-dimensional faces of Δ.


Let  be a d-clutter. Suppose that Δ= and Γ=^∨. Also assume that v∈(Γ) (=(Δ)) and Γ-v is pure of dimension =(Γ) (that is, v is a shedding vertex of
Γ) and set = (_Δ(v)). Then


  *  _Δ(v) is the pure (d-1)-skeleton of (Γ-v)^∨;

  *  ^∨= Γ-v;

  *  e∈() ev∈().



<ref> Suppose that F∈ (Γ-v)^∨ and F=d-1. Then A=(Γ-v) F∉Γ-v and hence A∉Γ and A∉^∨. But (A)=(n-1)- (F)-2= n-d-2= 
(Γ)= (^∨). Thus Fv= (Δ) A∈, that is, F is a facet of _Δ(v).
So the pure (d-1)-skeleton of (Γ-v)^∨ is contained in _Δ(v). The proof of the reverse
inclusion is similar.

<ref> Noting that ((Γ-v))^∨ is exactly the set of (d-1)-dimensional faces of
(Γ-v)^∨, we see that part <ref> indeed states = ((Γ-v))^∨, which is
equivalent to <ref>.

<ref> Assume that e∈(). Then for x∈() e we have ex∈ exv∈(Δ)=. Therefore, _[e]=_[ev]{v}. If ev∈() and A is a
(d-1)-dimensional subset of _[e], then Av_[ev] which is a clique and hence Av∈,
that is, A∈. So _[e] is a clique and e∈().

Conversely, assume that e∈() and A is a d-dimensional subset of _[ev]. We must show
A∈. If v∈ A, then A{v}_[e] and hence A{v}∈, which means A∈. Thus suppose v∉ A. If A∉, then v∈ B=() A∈^∨=(Γ).
Consequently, B{v}∈Γ-v and there is a F∈(Γ-v)=^∨ containing B{v}. So
A'= () F∉. Since (Γ-v)=Γ= n-d-2= |F|-1 and ()=(){v}, we see that |A'|=d. Also A' A_[ev]{v}=_[e], and as e is simplicial, we
get A'∈, a contradiction from which the result follows.



Suppose that  is a 2-clutter on [6] with circuits 123, 124, 134, 234, 345, 346, 126 and Δ,
Γ and  are defined as in <ref>. Then

    Γ= ^∨  =     136, 146, 236, 246, 346,
      135, 235, 145, 245,
      123, 124,  134,  234  andΓ-6=    135, 235, 145, 245,
      123, 124,  134,  234 .

So 6 is a shedding vertex of Γ. Note that (Γ-6) is a 3-clutter on [5], thus
(Γ-6)^∨ =  12, 34 which is exactly _Δ(6). Also = {12,34} is indeed a
graph and ()={1,2,3,4}. As claimed in the previous lemma, simplicial maximal subcircuits of 
which contain the vertex 6 are 16, 26, 36,46.



Let  be a d-clutter. Assume that Δ= and Γ=^∨. Let v be a shedding
vertex of Γ and v∈ e∈(). Then v is a shedding vertex of (-e)^∨=
(Δ(-e))^∨. Furthermore, if Δ'=-e, then (_Δ' v)= (_Δ v)-e',
where e'=e{v}.


Let Γ'= (-e)^∨ and suppose that F∈(_Γ' v). We have to show that F is not a
facet of Γ'-v. Note that ^∨ (-e)^∨ and hence Γ-vΓ'-v. Therefore, it suffices
to show that there is a G∈(Γ)=^∨ with v∉ G such that F G. We know that Fv∈(Γ')=(-e)^∨, so Fv∉-e where  =(-e)=(). As v∈ e, e Fv
and thus Fv∉. It follows that Fv∈Γ, that is, F∈_Γ v. No face of _Γ
v is a facet of Γ-v, because v is a shedding vertex of Γ. Consequently, F is strictly contained in
the facet G of Γ-v, as claimed. The proof of the “furthermore” statement, which follows from definitions, is
left to the reader.



Let , Δ, Γ and v=6 be as in Example vdec-ex and set e=26, Γ'=
(-e)^∨ and Δ'=-e. Then -e={126} and (-e)^∨= ^∨∪{345} and the facets of Γ' (resp. Γ'-6) are obtained by adding the face 345 to the set of
facets of Γ (resp. Γ-6). Thus Γ'-6 is pure and of dimension 2 and hence 6 is still a
shedding vertex in Γ'. Also the only facet of _Δ(6) which does not contain the vertex 2=
e{v} is 34 and _Δ'(6)= 34.


Suppose that v∈() has not appeared in any circuit of . Then chordality (and many other properties) of
 and -v are equivalent, although ()≠(-v). In this case, we misuse the notation and write
=-v.


Let  be a d-clutter. Assume that Δ=, v is a shedding vertex of Γ=^∨ and
=(_Δ v). If  is chordal, then there is a sequence e_1, …, e_t with e_i∈(_i-1), where _0= and _i=_i-1-e_i, such that _t=-v.


We prove the statement by induction on ||. If =, then =-v and the claim holds trivially. So
assume ≠ and e'∈() be such that -e is chordal. By <ref>, e=e'v∈(). Let Δ'=-e and '=(_Δ'v), then according to <ref>,
'=-e' is chordal and v is a shedding vertex of (-e)^∨. Thus by applying the induction
hypothesis on -e, the assertion follows.



Let's use the notations of Examples vdec-ex and vdec-ex2. We saw that ={12, 34},
which is a chordal graph with 2∈() and 4∈(-2). Now we see that 26∈() and
46∈(-26) and in -26-46 the vertex 6 has not appeared in any circuit. In our notations, this
means that -26-46=-6, as asserted in the previous lemma.



Suppose that  is obtained from a complete clutter by deleting exactly one circuit. Then  is chordal.


Let =_0-F, where _0 is a complete clutter and F∈_0. By <cit.>, _0 is
chordal and there is a sequence of simplicial maximal subcircuits e_1,…, e_t which sends _0 to the empty
clutter. By symmetry of _0, e_1 can be any maximal subcircuit of _0 and we can assume e_1 F. Hence
-e= _0-e is chordal and it is easy to see that e∈(), that is,  is also chordal.


Now we are ready to state the main result of this section.

Assume that  is a d-clutter and ^∨ is vertex decomposable. Then  is chordal.


We use induction on the number of vertices of . Let Δ= and Γ= ^∨. If
Γ is a simplex, then  is obtained from a complete clutter by deleting one circuit and by
<ref> is chordal. Thus assume that Γ is not a simplex. So there is a shedding vertex
v of Γ such that both Γ-v and _Γ v are vertex decomposable. Setting
=(_Δ v), it follows from <ref> that ^∨= Γ-v and is
vertex decomposable. Therefore,  is chordal by induction hypothesis and by <ref>, there is a
sequence e_1, …, e_t with e_i∈(_i-1), where _0= and _i=_i-1-e_i, such
that _t=-v. Consequently, we just need to show that -v is chordal. For this we show that (-v)^∨=_Γ v, which is vertex decomposable and the result follows by the induction
hypothesis:

    F∈(_Γ v)     Fv∈(Γ)=^∨ Fv∉    ({v}) F∉-v  F∈ (-v)^∨.





With notations as in Example vdec-ex3, we saw that -26-46=-6 and 26∈() and
46∈(-26). Now in (-6)^∨=  13,14,23,24,34 the vertices 1,2,3,4 are
shedding vertices. Applying <ref> with -6 instead of  and with v=1, we find for example
12∈(-6) and 13∈(-6-12) with =(-6)-12-13= -6-1= {234,345}. Applying
<ref> again we can find say 23∈() and finally 35∈(-23) to get that  is
chordal.


In <cit.>, a vertex of a not necessarily uniform clutter,  is called a simplicial vertex if for
every e_1,e_2∈ with v∈ e_1,e_2, there is an e_3∈ such that e_3 (e_1∪ e_2){v}. Also
the contraction /v is defined as the clutter of minimal sets of {e{v}| e∈}. Now if every
clutter obtained from  by a sequence of deletions or contractions of vertices has a simplicial vertex, then
Woodroofe in <cit.> calls  chordal and we call it W-chordal. In <cit.> it
is proved that every W-chordal clutter which is uniform  is chordal. As a corollary of the above theorem, we can get
the following slightly stronger version of Corollary 3.7 of <cit.>.


Suppose that  is a (not necessarily uniform) W-chordal clutter, d=minł{|F|| F∈}̊ and =ł{
F∈| |F|=d }̊. Then  is chordal.


Theorem 6.9 of <cit.> states that the Alexander dual of the independence complex of  (for the
definition of independence complex, see <cit.>) is vertex decomposable. But independence complex of
 is exactly Δ() and hence the result follows from Theorem vdec main.



                                     

§ CHORDALITY AND ASCENT OF CLUTTERS



In <cit.>, the concept of chorded simplicial complexes is defined and it is proved that I_Δ has a
(d+1)-linear resolution over a field of characteristic 2, Δ is chorded and it is the clique complex of
a d-clutter. We briefly recall this concept. Let Δ be a simplicial complex. We say that Δ is
d-path connected, when it is pure and for each pair F and G of d-faces of Δ there is a sequence
F_0,…, F_k of d-faces of Δ, with F_0=F, F_k=G and |F_i∩ F_i-1|=d. Also a pure
d-dimensional simplicial complex Δ is called a d-cycle, when Δ is d-path connected and every
(d-1)-face of Δ is contained in an even number of d-faces of Δ. A d-cycle Δ is called
face-minimal, if there is no d-cycle on a strict subset of the d-faces of Δ. Finally, we say that a
pure d-dimensional simplicial complex Δ is d-chorded, when for every face-minimal d-cycle
ΩΔ which is not d-complete, there exists a family {Ω_1, …,Ω_k} of 2≤ k
d-cycles with each Ω_iΔ such that
enumi


  *  Ω∪_i=1^k Ω_i,

  *  each d-face of Ω is contained in an odd number of Ω_i's,

  *  each d-face in ∪_i=1^k Ω_iΩ is contained in an even number of Ω_i's,

  *  (Ω_i)(Ω).

enumi
We refer the reader to <cit.>, for several examples of these concepts.

Suppose that Δ is the clique complex of a d-clutter. Connon and Faridi, first in <cit.> prove that if
I_Δ has a (d+1)-linear resolution over a field of characteristic 2, then Δ is d-chorded. Then in
<cit.>, the same authors show that I_Δ has a (d+1)-linear resolution over a field of
characteristic 2, Δ is d-chorded, and moreover Δ^[m] is m-chorded for all m>d
(Δ^[m] is the pure m-skeleton of Δ). Now if I_Δ=I() for a d-clutter , then
Δ=Δ() and hence (Δ^[m]) is just the set of cliques of  with dimension m. Therefore,
the aforementioned results show that to study when I() has a linear resolution, it may be useful to consider the
higher dimensional clutters (Δ()^[m]). Motivated by this observation, we define the ascent of a clutter
as follows.


Let  be a d-clutter. We call the family of all  (d+1)-dimensional cliques of  (that is, cliques on d+2
vertices), the ascent of  and denote it by ^+. In other words, ^+=Δ()^[d+1].



Let  be as in Example counter Ex. Then =() is a 1-clutter (that is, a graph) on
[8]. The circuits of  are the edges in Fig-Dunce. Now ^+ is the set of cliques of size 3
of , in particular, all of the triangles in Fig-Dunce are in ^+. Thus ^+.
Note that all of the edges 12, 13, 23, 24, 34 are in , so 123 and 234 are in ^+ and ^+≠. This shows that  is not of the form (')^+ for some 1-clutter ' (else ' should
contain ()= and ^+'^+, a contradiction). Now ^+ means the set of all 4-subsets of
[8] such as A, with the property that all 3-subsets of A are in . But there is no such
4-subset of [8] and hence ^+=. Moreover, one could check that for example

    ^+⊋   ∪{123, 23a,12a,13a|a=4,5,7,8}^++=     (^+)^+ ⊋{123a, 136a|a=4,5,7,8}^+++=    {123ab, 136ab|ab=45,78,48},  ^++++=


Our first result, considers how the property of having a linear resolution behaves under ascension. Recall
that throughout the paper, we assume that  is a d-clutter on vertex set V with |V|=n and K is an
arbitrary field.


Let Δ=Δ(). Then I() has a linear resolution over K, I(^̱+̱) has a linear
resolution over K and H_d(Δ_W;K)=0 for all W V.


Note that if Δ'= Δ(^+), then Δ'_W and Δ_W differ only in d-dimensional faces
and hence H_t(Δ'_W; K)=  H_t(Δ_W; K) for each t> d and W V. Thus the result follows
directly form Fröberg's theorem <ref>. Note that Δ^[d]=, thus Δ is
d-chorded  is so.


The previous simple result shows why the concept of ascent of a clutter can be useful. For example, we show
that a main theorem of <cit.> can simply follow <ref> and the results of <cit.>. First we
state the needed results of <cit.> as a lemma.


Let  be a d-clutter. Suppose that K is a field of characteristic 2 and Δ=Δ(). Then H_d(Δ_W;K)=0 for all W V  is d-chorded.


() is <cit.> and the proof for () is the proof of part 2 of <cit.>.


We inductively define a CF-chordal clutter. We consider , CF-chordal and we say that a non empty
d-clutter  is CF-chordal, when  is d-chorded and ^+ is CF-chordal. It is easy to check that
 is CF-chordal Δ()^[m] is m-chorded for all m≥ d. In <cit.> simplicial complexes with
this property are called chorded. Thus the next result is indeed Theorem 18 of <cit.>, which is one of the
two main theorems of that paper. Before stating this result, it should be mentioned that an example of a clutter 
with  d-chorded but ^+ not (d+1)-chorded, is presented in <cit.>. This
example shows that  can be d-chorded while  is not CF-chordal.


Let  be a d-clutter. Suppose that K is a field of characteristic 2. Then I() has a linear resolution
over K,  is CF-chordal.


This follows from <ref> and <ref> and a simple induction on n-d.


As the following result shows, similar to having a linear resolution, chordality is also preserved under ascension.

Let  be a d-clutter. If  is chordal, then so is ^+.


We use induction on ||. The result is clear for ||=0,1, where ^+=. Choose e∈() such
that -e is chordal. If e is contained in only one circuit, then it is not contained in any clique on more than
d+1 vertices. Thus ^+=(-e)^+ and the claim follows the induction hypothesis. So we assume that _[e] is
a clique on at least d+2 vertices. Let F be any d+1 subset of _[e] with e F. If v∈_^+[F] F, then Fv is a clique in  and hence ev∈. This shows that _^+[F]_[e]. Let {v_1, …, v_t}= _[e] e, F_i=ev_i and _i= - F_1- ⋯- F_i. Note that
_i^+= ^+ -F_1- ⋯- F_i. We show that F_i∈(^+_i-1), if F_i∈(_i-1^+). Then as
_t=-e is chordal it follows by the induction hypothesis that _t^+ and hence ^+ are chordal.

Suppose that G__i-1^+[F_i] and |G|=d+2. Then by the above argument G_[e] and hence G∈^+. If v_j∈ G for some j<i, then F_iv_j∈_i-1^+. But F_j= ev_j F_iv_j and F_j∉_i-1, a contradiction. Therefore, no v_j∈ G for j<i and hence no F_j G for such a j. Thus G is a
clique in _i-1, that is, G∈_i-1^+ and hence F_i∈(_i-1^+).



Suppose that  is the 2-clutter shown in Fig octa. The circuits of  are the faces of the
hollow octahedron and the four triangles shown in a darker color. It is not hard to check that  is
chordal, for example, the sequence 12, 14, 64, 62, 13, 23, 63, 43, is a sequence of consecutive simplicial
maximal subcircuits, by deletion of which, we reach to the empty clutter. Hence ^+={1235,1345, 2356,
3456} is also chordal by <ref>. Indeed, all maximal subcircuits of ^+ are simplicial except
for the four darker ones.




Next we state a theorem that shows some connections between deleting elements of (^+) and having a linear
resolution.

Let  be a d-clutter. Suppose that ^+≠. Then, considering the following statements, we have
<ref>  <ref>  <ref>. Moreover, if (^+)≠, then
(<ref>, <ref> and <ref> are equivalent.

  
  *  There is a F∈(^+) such that I(-̱F̱) has a linear resolution
      over K and also I(^̱+̱) and I(-̱v̱) have linear resolutions over K for each v∈
      V.
  
  *  I() has a linear resolution over K.
  
  *  For all F∈(^+), I(-̱F̱) has a linear resolution over K and also
      I(^̱+̱) and I(-̱v̱) have linear resolutions over K for each v∈ V.



<ref>  <ref>: Set Δ=Δ(). According to <ref>, we have to show
that H_d(Δ_W; K)=0 for all W V. If W V, say v∈ V W, then Δ_W= Δ(_W)=
Δ((-v)_W). Thus by Fröberg's theorem, as I(-̱v̱) has a linear resolution over K, it follows that
H_d(Δ_W; K)=0. Consequently it remains to show that H_d(Δ; K)=0.

Assume that x= ∑_i=1^l a_iF_i∈ C_d(Δ; K) with _d(x)=0 where 0≠ a_i∈ K and F_i∈.
We have to show that x∈_d+1. Let F∈(^+) be such that I(-̱F̱) has a linear resolution
over K. Assume that for some i, we have F_i=F. As F∈(^+), there is a G∈^+ such that F G.
Now y= _d+1(a_iG) has a term ± a_iF_i. Hence in one of the two elements x∓ y, the coefficient of F is
zero. Call this element z. Then _d(z)=_d(x)∓_d(_d+1(a_iG))= 0. Thus we can assume that for no
i, F_i=F. Then x∈ C_d(Δ(-F); K) and as I(-F) has a linear resolution, H_d(Δ(-F);
K)=0. So there are G_i's in (-F)^+^+ and b_j's in K such that x=_d+1(∑ b_jG_j)∈_d+1.

<ref>  <ref>: As Δ(-v)_W = Δ()_W{v} and by Fröberg's
theorem, I(-̱v̱) has a linear resolution over K for each v∈ V. According to <ref>,
I(^̱+̱) also has a linear resolution over K. Now let F∈(^+) and also assume that V=[n]. Let
L= x_F and I=I(). We have to show I(-̱F̱)=I+L has a linear resolution over K.

First we compute the minimal generating set of I∩ L. Suppose that u is one of the minimal generators of the
squarefree monomial ideal I∩ L. Then as x_F|u, we should have u=x_F∪ A for some A [n] F. Since
u∈ I, there is a G_0∈ such that G_0 F∪ A. Assume |A|>1 and v∈ A. Then for no G∈
we have G Fv, else x_Fv∈ I∩ L which contradicts u being a minimal generator. This means that Fv is a
clique of  and hence v∈_^+[F]. Since v∈ A was arbitrary, we have F∪ A N_^+[F] which
is a clique, because F is simplicial. But this contradicts G_0 F∪ A and it follows that |A|=1, say
A={a}. As G_0 Fa, we get a∉_^+[F]. On the other hand, for an arbitrary a∈ B=[n]_^+[F], it is easy to see that x_Fa∈ I∩ L. Consequently, I∩ L= x_Fa| a∈ B =x_F   x_a|
a∈ B.

Since multiplying in x_F is an S-isomorphism of degree d+1 from x_a| a∈ B to I∩ L and by
<cit.>, I∩ L has a (d+2)-linear resolution over K. Also I and L have (d+1)-linear
resolutions over K. Now consider the exact sequence

    0→ I∩ L→ I L → I+L → 0.

Writing the (i+j)-th degree part of the long exact sequence of ^S(K,-) applied on the above sequence, we get
the exact sequence

    ⋯→_i^S(K, I L)_i+j→_i^S(K, I+L)_i+j→_i-1^S(K, I∩ L)_i-1+(j+1)→⋯.

If j≠ d+1 then both flanking terms are zero and hence the middle term is also zero. This means that I+L has a
(d+1)-linear resolution, as required.


In the following example, we show how the above theorem can be used.

Suppose that  is as in Example counter Ex. Let =∪{278}. Then ^+={1278},
278∈(^+) and ^+ is chordal. Hence I(^̱+̱) has a linear resolution over every field.
Also it is easy to verify that for any W [8], _W is chordal (indeed, if W≠{1,2,7,8}, then
either _W is empty or it has an edge which is contained in exactly one triangle and hence is simplicial
and if W= {1,2,7,8}, _W is complete) and thus has a linear resolution over every field. Thus, as
I(-̱2̱7̱8̱)= I() has a linear resolution over any field (see Example counter Ex) and
according to <ref>, we get that I() has a linear resolution over every field.


Note that  in <ref><ref>, against part <ref>, we cannot replace F∈(^+) with F∈(^+). For example, if  is as in Example chorded octa, then -135 is not 2-chorded, and hence I(-̱1̱3̱5̱) has not a linear resolution over _2, although
 is chordal and I() has a linear resolution over every field.

Also note that <ref><ref> does not necessarily imply <ref><ref>, since we may have a clutter  satisfying <ref><ref>, but with
(^+)=. Although, it should be mentioned that the author could not find such an example. Thus we
raise the following question.

Are the three statements of <ref> equivalent for a d-clutter  with a non-empty ascent?



In the proof of <ref>  <ref> of <ref>, we proved that if there
exists F∈(^+) such that I(-̱F̱) has a linear resolution over K, then H_d(Δ(); K)=0.

The following is also a corollary to the proof of <ref>. Recall that β_i(I)=∑_jβ_ij(I).

Let  be a d-clutter. Suppose that I= I() has a linear resolution over K, F∈(^+) and
J=I(-̱F̱). Then J has a linear resolution over K with Betti numbers β_i(J)= β_i(I)+t  i,
where t=n-|_^+[F]|.


Suppose that L= x_F and B=[n]_^+[F] as in the proof of <ref>  <ref>. So t=|B| and if L'=  x_a|a∈ B, then I∩ L=x_F L' and β_i(I∩ L)= β_i(L') which by
<cit.> is equal to ∑_k=1^t k-1  i= ∑_k=i^t-1k  i= t  i
. Also β_0(L)=1 and β_i(L)=0 for all i>0. Thus by taking _K of the long exact sequence of
_i^S(K,-) in the proof of <ref>, we get β_i(J)= β_i(I+L)= β_i(I)+β_i(L)+
β_i-1(I∩ L), where the last term is assumed to be zero if i=0.


If in <ref> we restrict to the case that char K=2 and by using <ref>, we get
the following.

Let  be a d-clutter. Suppose that ^+≠.


  *  If ^+ is CF-chordal and there is a F∈(^+) such that -F is CF-chordal and
    -v is CF-chordal for all v∈(), then  is CF-chordal.

  *  If  is CF-chordal, then for all F∈(^+) and all v∈(), -F and
    -v are CF-chordal.

Moreover, if (^+)≠, then the converses of both <ref> and <ref> hold.


In the rest of the paper, we study if we can get some results similar to <ref> for having linear quotients
or being chordal instead of having a linear resolution. For having linear quotients we have:

Let  be a d-clutter. Suppose that I() has linear quotients. Then I(^̱+̱), I(-̱F̱)  and
I(-̱v̱) have linear quotients for each F∈(^+) and v∈().


If we denote the ideal generated by all squarefree monomials in I with degree t by I_[t], then it is easy to
see that I(^̱+̱)= I()_[d+2]. Hence the fact that I(^̱+̱) has linear quotients follows from
<cit.>. But for the convenience of the readers, we present a direct shorter proof for this.
Assume that x_F_1, …, x_F_t is an admissible order of I().  Then ^̱+̱={F_iv|v∈ V F_i,
1≤ i≤ t}. For each 1≤ i≤ t, let _i= {F_iv|v∈ V F_i} (∪_j<i_j) and suppose that
_i={F_i1, …, F_ir_i}. We prove that F_11, …, F_1r_1, …, F_t1, …, F_tr_t
corresponds to an admissible order for I(^̱+̱).

Consider two circuits F_ij, F_i'j' of ^̱+̱ with i≤ i'. If i=i', then |F_ij F_i'j'|=1 and
hence the admissibility condition holds trivially for them. Thus we assume i<i'. Therefore by assumption there is a
l∈ F_i F_i' and a k< i' such that F_k F_i'= {l}. Let F_i'j' F_i'={v}. Note that as
F_i'j'∈_i', there is no j<i' with F_i'j'∈_j by the definition of _i's. So v≠ l, else
F_i'j'= F_k∪ (F_i' F_k) and as |F_i' F_k|=1, we have F_i'j'∈_j for some j≤ k, a
contradiction. Consequently, v∉ F_k and F_kv which is a circuit of ^̱+̱, should appear in some _j
with j≤ k. Noting that F_kv F_i'j'= {l}, the proof is concluded.

Now assume that F∈(^+). We show that if we add x_F to an admissible order of I(), we get an
admissible order of I(-̱F̱)= I()+  x_F. We just need to show if G∈, then there is a G'∈ and an l∈ G such that G F={l}. Since _^+[F] is a clique of , G_^+[F],
say l∈ G_^+[F]. Then Fl∉^+ and hence there is a G' Fl with G'∈. So G'
F={l}, as required.

Finally, note that if x_F_1, …, x_F_t is an admissible order for I(), then by deleting x_F_i's
with v∈ F_i, we get an admissible order for I(-̱v̱) and I(-̱v̱) has linear quotients.



Let  be the clutter of Example chorded octa. It is not hard to check that the following is an admissible
order for I(): 162, 163, 164, 165, 124, 624, 245, 234. Thus by the previous result, I(^̱+̱) has linear
quotients. Indeed, by the proof of <ref>, we get the following admissible order for I(^̱+̱): 1623, 1624,
1625, 1634, 1635, 1645, 1243, 1245, 6243, 6245, 2453. Also note that the assumption  that F is simplicial, is crucial
in <ref>. For example, I(-̱1̱3̱5̱) has not a linear resolution and hence has not linear quotients.


We do not know whether the converse of the above theorem is correct or if a statement similar to <ref>
 <ref> of <ref> holds for having linear quotients.

Next consider the “chordal version” of <ref>, that is, consider the statements obtained by replacing
“I() has a linear resolution over K” with “ is chordal” in the three parts of <ref>, where
the replacement occurs for all clutters  appearing in these assertions. Then clearly <ref> <ref> holds for the chordal version. Also <ref> shows that part of <ref> <ref> is true for chordality. We will also show that if  is chordal and F∈(^+), then
-F is d-chorded, which is weaker than being chordal. But before proving this, we utilize <ref>
to show that if the chordal version of <ref>  <ref> (or <ref> <ref>) holds, then we can reduce proving statement <ref> to verifying it only for clutters with empty
ascent.

In what follows, by a free maximal subcircuit of , we mean a maximal subcircuit which is
contained in exactly one circuit of . In the particular case that =1, that is  is a
graph, free maximal subcircuits are exactly leaves (or free vertices) of the graph. Also a simplicial complex
Δ is called extendably shellable, if any shelling of a subcomplex of Δ could be
continued to a shelling of Δ. To see a brief literature review of this concept and some related
results consult <cit.>.

Consider the following statements and also statement <ref> of Section 2 on a uniform clutter .
enumi



  *  If ≠^+ is chordal and -F and -v are chordal for each F∈(^+) and
    v∈(), then ()≠.





  *  If ^+= and I() has linear quotients, then  has a free maximal subcircuit.



  *  Simon's Conjecture (<cit.>): Every d-skeleton of a simplex is extendably
    shellable.

enumi
Then: <ref> + <ref>  <ref>  <ref> + <ref>.


Assume that <ref> and <ref> hold. We claim that if I() has linear quotients, then ()≠.
Then it follows from <cit.> that <ref> is correct. To prove the claim, we use induction on
(n-d, ||) considered with lexicographical order. If ^+=, then as every free maximal subcircuit is
simplicial, the claim holds by <ref>. If ^+≠, then applying <ref> and using the induction
hypothesis, we see that ^+, -F and -v are chordal for every F∈(^+) and v∈().
Consequently, the claim follows from <ref>.

Now suppose that statement <ref> is correct. If ^+=, then |_[e]|=d+1  for every e∈() and
hence every simplicial maximal subcircuit of  is a free maximal subcircuit. Therefore, <ref> holds as a special
case of <ref>.  Finally <ref>  <ref> follows from Theorem 2.5 and Corollary 3.7 of <cit.>.





The “chordal version” of <ref>  <ref> in <ref>, partly states that if
 is chordal, then for every F∈(^+), -F should be chordal. The author could neither prove nor
reject this in the general case, but we prove a weaker result. In particular, we will show in the case that =1, this is true.




We need the following lemmas which present an equivalent condition for being d-chorded.


The set of facets of a d-cycle is a disjoint union of the set of facets of a family of face-minimal d-cycles.


Suppose that Ω is a d-cycle. We use induction on |(Ω)|. If Ω is face-minimal, then the result
is trivial. Thus we can assume that there is a d-cycle Ω' whose facets form a strict subset of (Ω).
Let Γ= (Ω)(Ω'). Note that each (d-1)-face of Γ is contained in an even number
of facets of Ω and an even number of facets of Ω'. Hence each (d-1)-face of Γ is contained in an
even number of facets of Γ. Suppose that Γ_1, …, Γ_t are d-path connected components of
Γ, that is, the maximal pure d-dimensional subcomplexes of Γ which are d-path connected. Note that if
i≠ j, then Γ_i and Γ_j do not share a (d-1)-face, else Γ_i ∪Γ_j is d-path
connected, contradicting maximality of both Γ_i and Γ_j. Consequently, for every i, each (d-1)-face
of Γ_i is contained in an even number of d-faces of Γ_i and Γ_i is a d-cycle. Noting that
(Γ_i) are mutually disjoint, and by applying induction hypothesis on Γ_i's, it follows that
(Γ) is a disjoint union of facets of a family  of face-minimal d-cycles. So (Ω) is the
disjoint union of facets of the family ∪{Ω'} of face-minimal d-cycles, as required.


 
Let  be a d-clutter. The simplicial complex  is d-chorded the facets of every d-cycle
Ω of  is the symmetric difference of a family of complete subclutters of , each on a
(d+2)-subset of (Ω).


(): By induction on |(Ω)|. Because of <ref>, we can assume that Ω is
face-minimal. If Ω is d-complete, then by face-minimality |(Ω)|=d+2 and we are done. Thus we assume
that Ω is not d-complete. Suppose that Ω_i's are d-cycles in  satisfying
<ref>–<ref> in the definition of a d-chorded complex. Let _i=(Ω_i). Since each d-face in
∪_i=1^k _i(Ω) appears in an even number of _i's by <ref>, such d-faces are not in the
symmetric difference of _i's. Also by <ref> and <ref>, we see that each element of (Ω_i) is in an odd
number of _i's and hence is in their symmetric difference. Therefore, (Ω)=_1 ⊕⋯⊕_k, where ⊕ denotes symmetric difference. Also by <ref> of the definition of a d-chorded complex, each
_i is a cycle on an smaller number of vertices in (Ω). Thus by applying the induction hypothesis
on _i's we get a decomposition of Ω as the symmetric difference of a set of complete subclutters of
 on (d+2)-subsets of (Ω).


(): Suppose that Ω is a face-minimal cycle of  which is not d-complete. Then
(Ω)= _1 ⊕⋯⊕_t where _i's are complete subclutters of  with size
d+2 and (_i)(Ω). If we set Ω_i=_i, then Ω_i's clearly satisfy
<ref>–<ref> of the definition of a d-chorded clutter. This means that  is d-chorded.



Let  be a d-clutter. Suppose that  is d-chorded (for example, if  is chordal) and F∈(^+). Then -F is d-chorded.


Suppose that Ω is a d-cycle of -F and = (Ω). Since  is d-chorded and
according to <ref>, there are complete subclutters _1, …, _t of  on
(d+2)-subsets of () such that = ⊕_i=1^t _i. We choose _i's in a way that m=
|{i|F∈_i}| is minimum possible. Assume that m≥ 1. Because F∉, m is at least 2.
Thus there are two _i, say _1,_2, containing F. Therefore there exist v_1 ≠ v_2∈()
such that (_i)= Fv_i for i=1,2. Let W=Fv_1v_2 and '= _W. Then since F∈(^+)
and W_^+(F), ' is complete. Now if '= _1 ⊕_2, then ' is a
d-cycle in '-F. By <ref>, '-F is chordal and hence has a linear
resolution over every field. Thus by <cit.> '-F is d-chorded. Consequently
according to <ref>, '='_1 ⋯'_t', where '_i's are complete
subclutters of '-F (and hence ) on (d+2)-subsets of (')(). By replacing
⊕_i=1^t''_i instead of _1 ⊕_2 in the decomposition of , we get a
decomposition with less terms containing F. This contradicts the choice of the decomposition of  and
hence m=0.  Therefore, the decomposition = ⊕_i=1^t _i is in -F and the result follows
by <ref>.


It should be mentioned that the previous result is not correct for arbitrary F∈(^+). This can be
seen by noting that I(-̱1̱3̱5̱) is not 2-chorded, where  is as in Example chorded octa.

If  is a graph, we call a F∈(^+) a simplicial edge of .

Suppose that  is a chordal graph and F is a simplicial edge of . Then -F is chordal. Hence there is an
ordering F_1, …, F_t, F_t+1, …, F_m of edges of  such that for 1≤ i≤ t, F_i is a
simplicial edge of the chordal graph _i= -F_1-⋯-F_i-1 and for t<i≤ m, _i is a tree and F_i
is a leaf edge of _i.


The first statement is just <ref> in the case that =1. Now if  is chordal and ^+≠, then (^+)≠ by <ref>. So starting with , we can delete simplicial edges until
we reach a chordal graph ' with (')^+=. But a chordal graph without any cliques on more than two
vertices is a tree and hence we can delete leaf edges from ' until there is no more edges and the statement is
established.


Let  be the graph in Fig. Then ^+ and -F are chordal and also I(^̱+̱) and
I(-̱F̱) have linear quotients for each F∈(^+). Despite this  is not chordal, since
-e is a cycle of length 5. Thus the converses of <ref> and <ref>  do not hold.
Also edges of  can be ordered as in <ref>, so the converse of <ref> is not true, either.



Clearly Statement <ref> holds when =1. Using the concept of d-cycles, we show that <ref>
holds when ^∨≤ 1 or equivalently, if n-d≤ 3. As in <cit.>, by a
CF-tree we mean a d-clutter  with the property that  has no d-cycles.

Assume that  is a d-clutter on n vertices with n≤ d+3. If ^+= and I() has a linear
resolution over every field, then  is chordal. In particular, statement <ref> of <ref> holds for .


Suppose that ^+=  and I() has a linear resolution over every field. According to
<cit.>,  is d-chorded. But since  has no cliques of size d+2, it
follows from <ref> that  has no d-cycles, that is,  is a CF-tree. Now the
result follows from <cit.>.

The proof of Corollary 3.7 of <cit.> uses Alexander dual. We end this paper mentioning that more generally,
one can get a statement equivalent to <ref> by passing to the Alexander dual of . Indeed, by arguments
quite similar to <cit.> one can see that <ref> holds for all d-clutters on n vertices,
statement (ii) of <cit.> holds, when we replace “Cohen-Macaulay over _2” in that
statement with “shellable”.



  
Acknowledgements
The author would like to thank Rashid Zaare-Nahandi and Ali Akbar Yazdan Pour for their helpful discussions
and comments. Also the author is thankful to the reviewer whose comments led to great improvements in the
presentation of the paper.



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