aa r X i v : . [ m a t h . A C ] N ov THE FACE IDEAL OF A SIMPLICIAL COMPLEX
J ¨URGEN HERZOG AND TAKAYUKI HIBI
Abstract.
Given a simplicial complex we associate to it a squarefree monomialideal which we call the face ideal of the simplicial complex, and show that ithas linear quotients. It turns out that its Alexander dual is a whisker complex.We apply this construction in particular to chain and antichain ideals of a finitepartially ordered set. We also introduce so-called higher dimensional whiskercomplexes and show that their independence complexes are shellable.
Introduction
Let S = K [ x , . . . , x n , y , . . . , y n ] denote the polynomial ring in 2 n variables overthe field K . In general, given any subset F of { x , . . . , x n } , we define the squarefreemonomial u F of S of degree n by setting u F = Y x i ∈ F x i Y x j ∈{ x ,...,x n }\ F y j . Given a collection S of subsets of { x , . . . , x n } one defines the ideal I S generated bythe monomials u F with F ∈ S . Of particular interest are collections of sets whichnaturally arise in combinatorics.The first example of this kind which appeared in the literature is the following:let P = { x , . . . , x n } be a finite partially ordered set. A poset ideal of P is a subset α of P with the property that if x i ∈ α and x j ≤ x i , then x j ∈ α . In particular theempty set as well as P itself is a poset ideal of P . The squarefree monomial ideal I ( P ) ⊂ S which is generated by those monomials u α for which α is a poset ideal of P has played an important role in combinatorial and computational commutativealgebra ([8], [5]). The ideal I ( P ) has the remarkable property that it has a linearreolution [9, Theorem 9.1.8].Now let I A ( P ) ⊂ S be the squarefree monomial ideal which is generated by thosemonomials u β for which β is an antichain of P . (Recall that an antichain of P isa subset β ⊂ P for which any two elements x i and x j with i = j belonging to β are incomparable in P .) The toric ring generated by u α with u α ∈ I ( P ) and thetoric ring generated by u β with u β ∈ I A ( P ) have similar properties. In particular,both are algebras with straighening laws on suitable distributive lattices, see [10]and [12]. Thus one may expect that I ( P ) and I A ( P ) have similar properties as well.Similarly we may also consider the squarefree monomial ideal I C ( P ) ⊂ S which isgenerated by those monomials u γ for which γ is a chain of P . (Recall that a chain of P is a totally ordered subset of P .) As expected, each of the antichain ideal I A ( P )and the chain ideal I C ( P ) has linear quotients will be shown in Theorem 3.1. owever, far beyond the study of antichain ideals and chain ideals of partiallyordered sets, the study of the present paper will be done in a much more generalsituation. Since the set of antichains of P as well as the set of chains of P is clearlya simplicial complex on the vertex set { x , . . . , x n } , it is natural to study moregenerally ideals I S where S is the set of faces of any simplicial complex. Thus weintroduce the face ideal J ∆ of a simplicial ∆ on { x , . . . , x n } . In other words, theface ideal J ∆ is the ideal of S which is generated by the monomials u F with F ∈ ∆.Theorem 1.1 gives the structure of the Alexander dual ( J ∆ ) ∨ of J ∆ . Somewhatsurprisingly, it turns out that ( J ∆ ) ∨ is a whisker complex, which is a generalization ofwhisker graphs, first introduced by Villarreal [15] and further studied and generalizedin [1], [7], [11] and [14]. A simple polarization argument shows that ( J ∆ ) ∨ is Cohen–Macaulay. Hence, again. J ∆ has a linear resolution. We also describe the explicitminimal free resolution of J ∆ (Theorem 1.3) and compute the Betti numbers of J ∆ (Corollary 1.4). We would like to mention that whisker complexes are special classesof grafted complexes as introduced by Faridi [6].In Section 2, we show that the face ideal J ∆ of a simplicial complex ∆ has linearquotients. This fact implies that the independence complex of the whisker complexof an arbitrary simplicial complex is shellable (Corollary 2.2). Recall that if Γ isa simplicial complex and I (Γ) its facet ideal, then the simplicial complex ∆ with I ∆ = I (Γ) is called the independence complex of Γ. Here I ∆ denotes the Stanley–Reisner ideal of ∆. The faces of ∆ are those subsets of the vertex set of Γ which donot contain any facet of Γ.Dochtermann and Engstr¨om [3] even showed that the independence complex ofwhisker graphs are pure and vertex decomposable, which in particular implies thatthe independence complex of any whisker graph is shellable, see also [11].Finally, in Section 4, we introduce the concept of higher dimensional whiskercomplexes, which is a generalization of whisker complexes introduced in Section 1.Theorem 4.1 guarantees that hte independence complex of the higher dimensionalwhisker complex of an arbitrary simplicial complex is shellable. Thus our Theo-rem 4.1 generalizes the result of Dochtermann and Engstr¨om regarding shellability.A. Engstr¨om kindly informed us that his student Lauri Loiskekoski in his MasterThesis [13] has also introduced what we call the face ideal of a simplicial complex,and, among other results, has shown that face ideals have linear resolutions, cf. ourCorollary 1.2. 1. Face ideals and whisker complexes
Let S = K [ x , . . . , x n , y , . . . , y n ] denote the polynomial ring in 2 n variables overthe field K and ∆ a simplicial complex on the vertex set { x , . . . , x n } .The whisker complex of ∆ is the simplicial complex W (∆) on the vertex set { x , . . . , x n , y , . . . , y n } which is obtained from ∆ by adding the facets { x i , y i } , calledwhiskers, for i = 1 , . . . , n .For each face F ∈ ∆ we associate the monomial u F ∈ S defined by u F = x F y F c , here x F = Y x i ∈ F x i and y F c = Y x j ∈{ x ,...,x n }\ F y j . The ideal of S generate by those squarefree monomials u F with F ∈ ∆ is called the face ideal of ∆ and is denoted by J ∆ . As usual we write I ∆ ( ⊂ K [ x , . . . , x n ]) forthe Stanley–Reisner ideal ([9, p. 16]) of ∆ and I (∆) ( ⊂ K [ x , . . . , x n ]) for the facetideal of ∆.Let, in general, I ⊂ S be a squarefree monomial ideal of S with I = T mk =1 P k wherethe ideals P k are the minimal prime ideals of I . Each P k is generated by variables.The Alexander dual I ∨ of I is defined to be the ideal of S generated by the squarefreemonomials u , . . . , u m , where u k = ( Q x i ∈ P k x i )( Q y j ∈ P k y j ). In particular, ( I ∆ ) ∨ = I ∆ ∨ where ∆ ∨ is the Alexander dual of ∆. Theorem 1.1.
Let ∆ be a simplicial complex on { x , . . . , x n } and J ∆ ⊂ S the faceideal of ∆ . Then one has ( J ∆ ) ∨ = I ( W (Γ)) , where Γ is the simplicial complex on { y , . . . , y n } with I ∆ ′ = I (Γ) , where ∆ ′ is the copy of ∆ on { y , . . . , y n } , i.e., ∆ ′ = { { y i : x i ∈ F } : F ∈ ∆ } .Proof. Let ∆ ♯ denote the simplicial complex on { x , . . . , x n , y , . . . , y n } with J ∆ = I (∆ ♯ ). It then follows that a squarefree monomial x i · · · x i s y j · · · y j t belongs to theminimal system of monomial generators of J ∨ ∆ if and only if { x i , . . . , x i s , y j , . . . , y j t } is a minimal vertex cover ([9, p. 156]) of ∆ ♯ .Since each facet of ∆ ♯ contains either x i or y i for 1 ≤ i ≤ n , it follows that { x i , y i } is a minimal vertex cover of ∆ ♯ for 1 ≤ i ≤ n .We claim F = { y j : j ∈ B } , where B ⊂ [ n ] = { , . . . , n } , is a vertex cover of ∆ ♯ if and only if F ∆ ′ . In fact, if F ∈ ∆ ′ , then { x j : j ∈ B } ∪ { y i : i ∈ [ n ] \ B } is a facet of ∆ ♯ . Hence F cannot be a vertex cover of ∆ ♯ . Conversely suppose that F ∆ ′ . Let { x j : j ∈ C } ∪ { y i : i ∈ [ n ] \ C } be a facet of ∆ ♯ . Then { x j : j ∈ C } is a face of ∆. Since F ∆ ′ , it follows that B C . Thus B ∩ ([ n ] \ C ) = ∅ . Hence F is a vertex cover of ∆ ♯ .Finally, suppose that G = { x i : i ∈ A } ∪ { y j : j ∈ B } is a minimal vertex coverof ∆ ′ , where A ⊂ [ n ] , B ⊂ [ n ] with A ∩ B = ∅ . We claim A = ∅ . In fact, if A = ∅ ,then F = { y j : j ∈ B } cannot be a vertex cover of ∆ ♯ . Hence F must be a face of∆ ′ . Then { x i : i ∈ B } ∪ { y j : j ∈ [ n ] \ B } is a facet of ∆ ♯ . Since A ∩ B = ∅ , itfollows that G cannot be a vertex cover of ∆ ′ .In consequence, the minimal vertex covers of ∆ ♯ are either { x i , y i } for 1 ≤ i ≤ n or the minimal nonfaces of ∆ ′ . Since I ∆ ′ = I (Γ), the the minimal nonfaces of ∆ ′ coincides with the facets of Γ. It then follows that J ∨ ∆ = I ( W (Γ)), as desired. (cid:3) Corollary 1.2.
Let ∆ be a simplicial complex. Then J ∆ has a linear resolution.Proof. By applying the Eagon-Reiner Theorem [4] (see also [9, Theorem 8.1.9]) itsuffices to show that I ( W (Γ)) is a Cohen–Macaulay ideal. This is a well-knownfact: Notice that I ( W (Γ)) is the polarization of L = ( I (Γ) , y , . . . , y n ). Since im K [ y , . . . , y n ] /L = 0. it follows that L is a Cohen-Macaulay ideal. Hence by [9,Corollary 1.6.3], I ( W (Γ)) is a Cohen–Macaulay ideal as well. (cid:3) The next result describes the precise structure of the resolution of
S/J ∆ . For asimplicial complex ∆ on [ n ] = { , . . . , n } we let ∆ = { [ n ] \ F : F ∈ ∆ } . Theorem 1.3.
Let ∆ be a simplicial complex of dimension d − on the vertex set [ n ] . For each integer j ≥ , let F j be the free S -module with basis elements e G,H indexed by G ∈ ∆ and H ∈ ∆ satisfying the condition that | G ∩ H | = j − and G ∪ H = [ n ] . Furthermore, we set F = S . For each j = 2 , . . . , d we define the S -linear map ∂ j : F j → F j − , with ∂ j ( e G,H ) = X i ∈ G ∩ H ( − σ ( G ∩ H,i ) ( x i e G \{ i } ,H − y i e G,H \{ i } ) , where σ ( G ∩ H, i ) = |{ j ∈ G ∩ H : j < i }| . Then F ∆ : 0 −−−→ F d ∂ d −−−→ F d − ∂ d − −−−→ · · · ∂ −−−→ F ∂ −−−→ F −−−→ , is the graded free resolution of S/J ∆ , where ∂ ( e G,H ) = x G y H for all e G,H ∈ F .Proof. We first show that F is a complex. One immediately verifies that ∂ ◦ ∂ = 0.Now let j > e G,H ∈ F j . Set L = G ∩ H . Then ∂ j ( ∂ j − ( e G,H )) = ∂ j − ( X i ∈ L ( − σ ( L,i ) ( x i e G \{ i } ,H − y i e G,H \{ i } ))= X i ∈ L ( − σ ( L,i ) [ x i ( X k ∈ L \{ i } ( − σ ( L \{ i } ,k ) ( x k e G \{ i,k } ,H − y k e G \{ i } ,H \{ k } )) − y i ( X k ∈ L \{ i } ( − σ ( L \{ i } ,k ) ( x k e G \{ k } ,H \{ i } − y k e G,H \{ i,k } ))]= X i,k ∈ L,i
1, and let F be a facet of ∆. We set Γ = ∆ \ { F } .We may assume that Γ is a simplicial complex on the vertex set [ n ′ ] where n ′ ≤ n .By induction hypothesis, F Γ is a graded minimal free S ′ -resolution of S ′ /J Γ , where S ′ = K [ x , . . . , x n ′ , y , . . . , y n ′ ]. Thus G = F Γ ⊗ S ′ S is a graded minimal free S -resolution of S/J Γ S , and we obtain an exact sequence of complexes0 −−−→ G ϕ −−−→ F ∆ ε −−−→ K → . (1)For all H ⊂ [ n ′ ] we set H ′ = H ∪ { n ′ + 1 , . . . , n } . Then ϕ : G → F ∆ is defined by ϕ ( e G,H ) = e G,H ′ for all e G,H ∈ G . Furthermore, K = F ∆ /ϕ ( G ).One verifies that K j is a free module admitting the basis ¯ e F,H = ε ( e F,H ) with H ⊂ [ n ] such that | F ∩ H | = j − F ∪ H = [ n ]. Denote by δ the differential of K . Then δ j (¯ e F,H ) = − X i ∈ F ∩ H ( − σ ( F ∩ H,i ) y i ¯ e F,H \{ i } . hus we see that K is isomorphic to the Koszul complex attached to the sequence( y i ) i ∈ F , homologically shifted by 1. In particular, it follows that H j ( K ) = 0 for j = 1, while H ( K ) = S/ ( y i : i ∈ F ).Thus from the long exact sequence attached to (1) we obtain that H j ( F ∆ ) = 0for j > G is acyclic by our induction hypothesis. Furthermore, weobtain the exact sequence0 −→ H ( F C ) −→ H ( K ) −→ H ( G ) −→ H ( F C ) −→ . Since H ( G ) = S/J Γ S and H ( F ∆ ) = S/J ∆ we see that Ker( H ( G ) −→ H ( F C )) = J ∆ /J Γ S . Now J ∆ /J Γ S ∼ = S/ ( J Γ S : x F y [ n ] \ F ) = S/ ( y i : i ∈ F ) = H ( K ). This provesthat H ( F C ) = 0 and completes the proof of the theorem. (cid:3) Corollary 1.4.
Let ∆ be a simplicial complex of dimension d − with f -vector ( f − , f , . . . , f d − ) . Then β j ( J ∆ ) = d − X i = − i + 1 j ! f i . In particular, proj dim J ∆ = dim ∆ + 1 . Whisker complexes and shellability
In this section we show that the face ideal of a simplicial complex does not onlyhave a linear resolution but even linear quotients.
Theorem 2.1.
Let ∆ be a simplicial complex. Then J ∆ has linear quotients.Proof. We choose any total order of the generators of J ∆ with the property that u G > u F if G ⊂ F , and claim that J ∆ has linear quotients with respect to this orderof the generators.Indeed, let F ∈ F (∆), and let J ( F ) be the ideal generated by all G with u G > u F .For any G ∈ F (∆) one has( u G ) : u F = u G gcd( u G , u F ) = x G \ F y F \ G . Now let u G ∈ J ( F ). Then F \ G = ∅ . Let x j ∈ F \ G , and let H = F \ { x j } .Then u H > u F and ( u H ) : u F = ( y j ). Since y j divides y F \ G it follows that y j divides( u G ) : u F , as desired. (cid:3) Corollary 2.2.
Let Γ be a simplicial complex. Then the independence complex of W (Γ) is shellable. Theorem 2.1 can be generalized as follows.
Theorem 2.3.
Let S be a non-empty collection of subsets of { x , . . . , x n } satisfyingthe following conditions: (i) if F, G ∈ S , then F ∩ G ∈ S ; (ii) for F, G ∈ S with G ⊂ F there exists x i ∈ F \ G such that F \ { x i } ∈ S .Then I S has linear quotients. roof. We choose any total order of the generators of J ∆ with the property that u G > u F if G ⊂ F , and claim that J S has linear quotients with respect to this orderof the generators. Let u G > u F , and let N = { i : F \ { x i } ∈ S} . Then, by (ii), N 6 = ∅ . We claim that ( u G : u G > u F ) = ( y i : i ∈ N ) . To see why this is true, let u G > u F . Then, by (i), H = G ∩ F belongs to S , andby (ii) there exists i ∈ N such that x i ∈ F \ H . Then x i G , and hence y i divides( u G ) : u F . Since ( u F \ { x i } ) : u F = ( y i ), we are done. (cid:3) Chain ideals and antichain ideals
In this section we consider special classes of face ideals arising from finite partiallyordered sets (posets, for short). Let P = { p , . . . , p n } be a finite poset, and S = K [ x , . . . , x n , y , . . . , y n ] the polynomial ring in 2 n variables over the field K . Foreach chain α of P we define the monomial u α ∈ S by setting u α = ( Y p i ∈ α x i )( Y p j α y j ) , and let I C ( P ) be the ideal generated by the monomials u α where α is a chain of P .We call I C ( P ) the chain ideal of P .Similarly, for each antichain β of P we define the monomial u β ∈ S by setting u β = ( Y p i ∈ β x i )( Y p j β y j ) , and let I A ( P ) be the ideal generated by the monomials u β where β is an antichainof P . We call I A ( P ) the antichain ideal of P .Recall that the comparability graph of P is a finite simple graph C ( P ) on [ n ] whoseedges are those subsets { i, j } such that p i and p j are comparable in P , i.e., { p i , p j } isa chain of P . Similarly, the incomparability graph of P is a finite simple graph A ( P )on [ n ] whose edges are those subsets { i, j } such that p i and p j are incomparable in P , i.e., { p i , p j } is an antichain of P . Theorem 3.1.
Let P be a finite poset. (a) The Alexander dual of the chain ideal I C ( P ) is the edge ideal of the whiskergraph of the incomparability graph of P . (b) The Alexander dual of the antichain ideal I A ( P ) is the edge ideal of the whiskergraph of the comparability graph of P .Proof. (a) Let ∆ be a simplicial complex on { y , . . . , y n } whose faces are those F ⊂ { y , . . . , y n } with { p i : y i ∈ F } is a chain of P . Theorem 1.1 says that theAlexander dual I C ( P ) ∨ of I C ( P ) coincides with I ( W (Γ)), where Γ is a simplicialcomplex on { y , . . . , y n } with I ∆ = I (Γ). A subset F ⊂ { y , . . . , y n } is nonface of∆ if and only if an antichain of P is contained in F . Thus the minimal nonfaces of∆, which coincides with the facets of Γ are those subset { y i , y j } such that { p i , p j } isan antichain of P . Thus I (Γ) is the edge ideal of the incomparagraph of P . Hence I ( W (Γ)) is the edge ideal of the whisker graph of the incomparability graph of P ,as desired. b) Let ∆ be a simplicial complex on { y , . . . , y n } whose faces are those F ⊂{ y , . . . , y n } with { p i : y i ∈ F } is an antichain of P . Theorem 1.1 says that theAlexander dual I A ( P ) ∨ of I A ( P ) coincides with I ( W (Γ)), where Γ is a simplicialcomplex on { y , . . . , y n } with I ∆ = I (Γ). A subset F ⊂ { y , . . . , y n } is nonface of ∆if and only if a chain of P is contained in F . Thus the minimal nonfaces of ∆, whichcoincides with the facets of Γ are those subset { y i , y j } such that { p i , p j } is a chainof P . Thus I (Γ) is the edge ideal of the comparagraph of P . Hence I ( W (Γ)) is theedge ideal of the whisker graph of the comparability graph of P , as desired. (cid:3) The
Dilworth number of a finite poset P is the least number of chains into which P can be partitioned. Dilworth’s theorem [2] guarantees that the Dilworth numberof P is equal to the maximal cardinality of the antichains of P . Corollary 3.2.
The chain ideal and the antichain ideal of a finite poset have alinear resolution. Moreover, proj dim I C ( P ) = rank P + 1 while proj dim I A ( P ) isthe Dilworth number of P . Higher dimensional whiskers
The purpose of this section is to generalize Corollary 2.2. Let ∆ be a simplicialcomplex on { x , . . . , x n } . Given positive integers k , . . . , k n and d , . . . , d n with d i ≤ k i for all i , we define the higher dimensional whisker complex W d ,...,d n k ,...,k n (∆) of∆ to be the simplicial complex on the vertex set x , x (1)1 , · · · , x ( k )1 , x , x (1)2 , · · · , x ( k )2 , · · · , x n , x (1) n , · · · , x ( k n ) n , whose facets are the facets of ∆ together with all subsets of cardinality d i + 1 of { x i , x (1) i , . . . , x ( k i ) i } for i = 1 , . . . , n . These subsets are called the whiskers of ∆.Note that the wisker complex of ∆ as defined in Section 1 is just the complex W ,..., ,..., (∆). See Figure 1 for an example of a higher whisker complex. Theorem 4.1.
The independence complex of a higher whisker complex is shellable.Proof.
Let W d ,...,d n k ,...,k n (Γ) be the whisker complex, ∆ its independence complex and I = I ∆ ∨ . We will show that I has linear quotients. This is equivalent to say that∆ is shellable. Note that the generators of I correspond bijectively to the vertexcovers of W d ,...,d n k ,...,k n (Γ). Indeed, if C ⊂ { x , x (1)1 , . . . , x ( k )1 , x , x (1)2 , . . . , x ( k )2 , . . . , x n , x (1) n , . . . , x ( k n ) n } is a minimal vertex cover of W d ,...,d n k ,...,k n (Γ), then the product of the elements in C isthe corresponding generator of I .We claim, that C is a minimal vertex cover of W d ,...,d n k ,...,k n (Γ) if and only if thefollowing conditions are satisfied:(1) C ∩ { x , . . . , x n } is a vertex cover of Γ;(2) C ∩ { x i , x ( i )1 , . . . , x ( i ) k i } is a minimal vertex cover of the d i -skeleton of thesimplex on { x i , x i , . . . , x ik i } . •• • x (1)3 •••• •• •• • x x x x x (2)1 x (1)1 x (3)1 x (1)4 x (2)4 x (1)2 x (2)2 x (3)2 Figure 1.
Clearly any set C of vertices satisfying (1) and (2) is a minimal vertex cover.We denote by u C ∈ I the monomial corresponding to the vertex cover C of W d ,...,d n k ,...,k n (Γ). Then u C = x C x C · · · x C n , where x C = Y x j ∈ C x i and x C j = Y { i : x ( i ) j ∈ C } x ( i ) j for j = 1 , . . . , n .We now define a total order of the monomial generators of I as follows: let u C = x C x C · · · x C n and u D = x D x D · · · x D n . Then u C > u D if either x C > x D withrespect to the degree lexicographic order, or x C = x D and x C · · · x C n > x D · · · x D n with respect to the lexicographic order induced by x (1)1 > . . . > x ( k )1 > x (1)2 > . . . > x ( k )2 > . . . > x (1) n > . . . > x ( k n ) n . We claim that with this ordering of the generators, I has linear quotients. We firstnote that all generators of I have the same degree, namely, P ni =1 ( k i − d i ) + n . Wemust show that for all u C the colon ideal ( u D : u D > u C ) : u C is generated byvariables. Let u D > u C with x D = x C . Then x D > x C , and hence there exists x j such that x j divides x D but does not divide x C . Let u E be the generator with x E = x C x j , x E i = x C i for i = j and x E j = x C j /x ( l ) j where x ( l ) j divides x C j . Then u E > u C and ( u E ) : u C = ( x j ). Since x j divides ( u D ) : u C , and we are done in thiscase. e now consider the case x D = x C . Let A = { u E : x E = x C } . For each i ≥ A i = { x E i : u E ∈ A} . If x i divides x C , then A i is the set of all monomials ofdegree k i − d i in the variables x ( i )1 , . . . , x ( i ) k i , and if x i does not divide x C , then A i isthe set of all monomials of degree k i − d i + 1 in the same set of variables. Note that A i generates a matroidal ideal. Moreover the product of matroidal ideals in pairwisedisjoint sets of variables is again a matroidal ideal. It is known, see [9, Theorem12.6.2], that matroidal ideals have linear quotients with respect to the lexicographicorder of the generators. This completes the proof of the theorem. (cid:3) References [1] D. Cook II and U. Nagel, Cohen–Macaulay graphs and face vectors of flag complexes,
SIAMJ. Discrete Math. (2012), 89–101.[2] R. P. Dilworth, A decomposition theorem for partially ordered sets, Annals of Mathematics (1950), 161–166.[3] A. Dochtermann and A. Engstr¨om, Algebraic properties of edge ideals via combinatorial topol-ogy,
Electron. J. Combin. (2009), Special volume in honor of Anders Bj¨orner, Research Paper2, 24 pp.[4] J. A. Eagon and V. Reiner, Resolutions of Stanley-Reisner rings and Alexander duality, J. Pureand Appl. Algebra (1998), 265–275.[5] V. Ene, J. Herzog and F. Mohammadi, Monomial ideals and toric rings of Hibi type arisingfrom a finite poset,
European J. Combin. (2011), 404–421.[6] S. Faridi, Cohen-Macaulay Properties of Square-Free Monomial Ideals, J. Combin. TheorySer. A (2005), 299–329.[7] C. A. Francisco and H. T. H`a, Whiskers and sequentially Cohen–Macaulay graphs,
J. Combin.Theory Ser. A (2008), 304–316.[8] J. Herzog and T. Hibi, Distributive lattices, bipartite graphs and Alexander duality,
J. AlgebraicCombin. (2005), 289–302.[9] J. Herzog and T. Hibi, “Monomial ideals,” Graduate Texts in Mathematics , Springer,London, 2010.[10] T. Hibi, Distributive lattices, affine semigroup rings and algebras with straightening laws, in “Commutative Algebra and Combinatorics” (M. Nagata and H. Matsumura, Eds.), AdvancedStudies in Pure Math., Volume 11, North–Holland, Amsterdam, 1987, pp. 93 – 109.[11] T. Hibi, A. Higashitani, K. Kimura and A. B. O’Keefe, Algebraic study on Cameron–Walkergraphs, J. Algebra (2015), 257–269.[12] T. Hibi and N. Li, Chain polytopes and algebras with straightening laws, arXiv:1207.2538.[13] L. Loiskekoski, Resolutions and associated primes of powers of ideals, Master Thesis, AaltoUniversity, School of Science, 2013.[14] A. Van Tuyl and R. H. Villarreal, Shellable graphs and sequentially Cohen–Macaulay bipartitegraphs,
J. Combin. Theory Ser. A (2008), 799–814.[15] R. H. Villarreal, Cohen-Macaulay graphs,
Manuscripta Math. (1990), 277–293. J¨urgen Herzog, Fachbereich Mathematik, Universit¨at Duisburg-Essen, CampusEssen, 45117 Essen, Germany
E-mail address : [email protected] Takayuki Hibi, Department of Pure and Applied Mathematics, Graduate Schoolof Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
E-mail address : [email protected]@math.sci.osaka-u.ac.jp