Inequalities of invariants on Stanley-Reisner rings of Cohen-Macaulay simplicial complexes
aa r X i v : . [ m a t h . A C ] J a n INEQUALITIES OF INVARIANTS ON STANLEY-REISNER RINGS OFCOHEN–MACAULAY SIMPLICIAL COMPLEXES
AKIHIRO HIGASHITANI, HIROJU KANNO, AND KAZUNORI MATSUDA
Abstract.
The goal of the present paper is the study of some algebraic invariantsof Stanley–Reisner rings of Cohen–Macaulay simplicial complexes of dimension d − d ≤ reg(∆) · type(∆) holds for any ( d − k [∆]. Moreover, for any given integers d, r, t satisfying r, t ≥ r ≤ d ≤ rt , we construct a Cohen–Macaulay simplicial complex ∆( G ) as anindependent complex of a graph G such that dim(∆( G )) = d −
1, reg(∆( G )) = r andtype(∆( G )) = t . Introduction
The theory of monomial ideals is one of the most well-studied topics in the area ofcombinatorial commutative algebra. Since any monomial ideal can be deduced into a squarefree monomial ideal by polarization (see [9, Section 1.6]), Stanley–Reisner rings andStanley–Reisner ideals play a crucial role. The goal of the present paper is the investigationof some algebraic invariants on Stanley–Reisner rings (or ideals).For the terminologies used throughout the present paper, see Section 2. One typicalclass of Stanley–Reisner ideals is the edge ideals I ( G ) of graphs G , which coincide with theStanley–Reisner ideals of the independent complexes ∆( G ) of G . Recently, the invariantson the Stanley–Reisner rings k [∆( G )] of ∆( G ) are intensively investigated ([3, 4, 7, 10, 11,12, 13, 14], and so on). What we would like to do in the present paper is to generalize theprevious studies on the edge ideals and initiate a new study for more general squarefreemonomial ideals than edge ideals.For the investigation of the invariants on Stanley–Reisner rings of simplicial complexes∆, we focus on the Castelnuovo–Mumford regularity reg(∆) and the Cohen–Macaulaytype type(∆). The first main result of the present paper is the following: Theorem 1.1.
Let ∆ be a Cohen–Macaulay simplicial complex of dimension d − satis-fying ∆ = core(∆) . Then we have d ≤ reg(∆) · type(∆) . (1.1)We can also see in the following proposition that a stronger inequality holds for thesimplicial complexes ∆ under some stronger assumption on k [∆]: Proposition 1.2.
Let ∆ be a Cohen–Macaulay simplicial complex of dimension d − satisfying ∆ = core(∆) . Assume that ∆ satisfies one of the following: (1) k [∆] has a -linear resolution; or Mathematics Subject Classification.
Primary 13F55; Secondary 13D02, 13D40, 05C70, 05E40 .
Key words and phrases.
Stanley–Reisner rings, Cohen–Macaulay, edge ideals, Castelnuovo–Mumfordregularity, Cohen–Macaulay type. a (∆) = 0 , where a (∆) denotes the a -invariant of k [∆] .Then the inequality d ≤ reg(∆) + type(∆) − holds. Note that the inequality (1.2) implies (1.1) since reg(∆) ≥ ≥
1. InSection 3, we give a proof of Theorem 1.1 and Proposition 1.2.If an inequality appears among the invariants, then it is quite natural to think of whetherthat is best possible or not. The following theorem, which is the second main result, showsthat the inequality (1.1) is best possible:
Theorem 1.3.
Let d, r, t be integers with r, t ≥ and assume that the inequalities r ≤ d ≤ rt hold. Then there exists a graph G having no isolated vertex such that ∆( G ) isCohen–Macaulay and dim(∆( G )) = d − , reg(∆( G )) = r and type(∆( G )) = t. We note that an isolated vertex of G corresponds to a vertex in V ( G ) \ core( V ( G )), sothe condition ∆ = core(∆) is equivalent to what G has no isolated vertex. Moreover, theinequality r ≤ d naturally holds (see Remark 2.1).In Section 4, we give a proof of Theorem 1.3. Acknowledgements.
The authors would like to be grateful to Satoshi Murai for thecomments on the first version of the present paper. Thanks to his comments, the authorscould improve Theorem 1.1 of the first version and make the present paper more readable.The first named author is partially supported by JSPS Grant-in-Aid for Scientific Re-search (C) 20K03513. The third named author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 20K03550.2.
Notation and terminologies
In this section, we collect the notation used in the present paper. Please consult, e.g.,[2, Section 5] or [9], for the detailed information.Let V = { v , . . . , v n } be a finite set. We call a family ∆ of subsets of V an (abstract)simplicial complex on the vertex set V if it satisfies that(i) { v i } ∈ ∆ for any v i ∈ V ; and(ii) if F ∈ ∆ and F ′ ⊂ F , then F ′ ∈ ∆.Note that ∅ belongs to ∆ for any simplicial complex ∆. Let dim(∆) denote the dimensionof ∆, i.e., dim(∆) = max { dim( F ) : F ∈ ∆ } .Throughout this section, let ∆ be a simplicial complex of dimension d − V = { v , . . . , v n } .2.1. Terminologies on simplicial complexes.
Given F ∈ ∆, let∆ \ F = { G ∈ ∆ : G ⊂ V \ F } , star( F ) = { G ∈ ∆ : F ∪ G ∈ ∆ } , andlink( F ) = star( F ) \ F. When F = { v } , we use the notation ∆ \ v instead of ∆ \ { v } , and so on. iven a new vertex v , let ∆ ∗ v be a new simplicial complex on the vertex set V ⊔ { v } consisting of { F, F ∪{ v } : F ∈ ∆ } . We call ∆ ∗ v a cone of ∆. Note that star ∆ ∗ v ( v ) = ∆ ∗ v .For W ⊂ V , let ∆ W = { F ∈ ∆ : F ⊂ W } . Let core( V ) = { v ∈ V : star( v ) = ∆ } and letcore(∆) = ∆ core( V ) . In the sequel, we often assume that ∆ = core(∆), which is equivalentto core( V ) = V .We recall the definition of vertex decomposable simplicial complexes. A simplicialcomplex ∆ on the vertex set V is called vertex decomposable ([1]) if ∆ is a simplex or ifthere is a vertex v ∈ V , called a shedding vertex , such that • ∆ \ v and link( v ) are vertex decomposable; and • no face of link( v ) is a facet of ∆ \ v .It is well known that for a pure simplicial complex ∆, the vertex decomposability of ∆implies Cohen–Macaulayness of ∆. (Here, we say that ∆ is pure if all facets of ∆ have thesame dimension.)Let ∆ , ∆ be two simplicial complexes on disjoint vertex sets of dimension d −
1. Let R ∈ ∆ and R ∈ ∆ be ridges, which are faces of dimension d −
2, respectively. Wedefine a new simplicial complex, called the ridge sum of ∆ and ∆ , by identifying R and R in the union ∆ ∪ ∆ of simplicial complexes. The terminology of ridge sums wasintroduced in [15] and similar notions appear in [5].We say that a ( d − strongly connected if forany two facets F, F ′ of ∆, there exists a sequence of facets F , F , . . . , F p of ∆ such that F = F , F p = F ′ and dim( F i − ∩ F i ) = d − ≤ i ≤ p .2.2. Terminologies on Stanley–Reisner rings.
We refer the reader to [2] for the in-troduction to the theory of Stanley–Reisner rings.Let S = k [ x , . . . , x n ] be the polynomial ring with n (= | V | ) variables over a field k . Wedenote by I ∆ (resp. k [∆]) the Stanley–Reisner ideal (resp.
Stanley–Reisner ring ) of ∆over a field k , i.e., I ∆ = ℓ Y j =1 x i j : { v i , . . . , v i ℓ } 6∈ ∆ ⊂ S and k [∆] = S/I ∆ . The invariants of ∆ or k [∆] or I ∆ discussed in the present paper are listed below: • Let dim( k [∆]) denote the Krull dimension of k [∆]. Then dim( k [∆]) = dim(∆) + 1. • We say that ∆ is Cohen–Macaulay over k if k [∆] is Cohen–Macaulay. In the sequel,we often omit “over k ”. By Reisner’s criterion ([2, Corollary 5.3.9]), ∆ is Cohen–Macaulay if and only if e H i (link( F )) = 0 for all F ∈ ∆ and i < dim(link( F )), where e H i (Γ) denotes the i -th reduced simplicial homology of a simplicial complex Γ withvalues in k . • Let ω ∆ denote the canonical module of k [∆]. • On the Tor modules, we use the notation Tor Si (∆) instead of Tor Si ( k , k [∆]). • Let β ij (∆) = dim k (Tor Si (∆) i + j ) for i, j ∈ Z , where dim k stands for the dimen-sion as a k -vector space, and let β i (∆) = P j ∈ Z β ij (∆). We call β i (∆) the i -thgraded Betti number of k [∆]. Note that by Auslander–Buchsbaum formula (see [2,Theorem 1.3.3]), we see that ∆ is Cohen–Macaulay if and only if the projectivedimension of k [∆] is n − d , which is equivalent to β i (∆) = 0 for any i > n − d and β n − d (∆) = 0 (see [2, Corollary 1.3.2]). • Let reg( k [∆]) denote the Castelnuovo–Mumford regularity of k [∆], i.e., reg( k [∆]) =max { j − i : β ij (∆) = 0 } . We use the notation reg(∆) instead of reg( k [∆]). Assume that ∆ is Cohen–Macaulay over k . Let type( k [∆]) denote the Cohen–Macaulay type of k [∆]. Namely, type( k [∆]) = β c (∆), where c = n − dim( k [∆]).Similarly to the above, we use the notation type(∆) instead of type( k [∆]). • Moreover, when ∆ is Cohen–Macaulay over k , let a ( k [∆]) = a (∆) denote the a -invariant (see [8]) of k [∆]. Namely, we have a (∆) = − min { j : ( ω ∆ ) j = 0 } . It iswell known that reg(∆) = a (∆) + dim( k [∆]) . • Let Indeg( I ∆ ) denote the maximal degree of the minimal system of generators of I ∆ . Let MNF(∆) = { G ⊂ V : G ∆ , G ′ ∈ ∆ for any G ′ ( G } . Namely, MNF(∆) is the set of minimal non-faces of ∆. Note that Indeg( I ∆ ) =max {| G | : G ∈ MNF(∆) } . • We say that k [∆] has a p -linear resolution if β ij (∆) = 0 unless j − i = p . Inparticular, reg(∆) = p in this case. Note that if k [∆] has a p -linear resolution,then Indeg( I ∆ ) = p + 1, but the converse is not true in general. Remark . It is known that a (∆) = reg(∆) − dim( k [∆]) ≤ ≤ dim(∆)+1 always holds for any Cohen–Macaulay simplicialcomplex ∆. Remark . Any Cohen–Macaulay simplicial complex ∆ is always strongly connected. Infact, if a ( d − F, F ′ such that dim( F ∩ F ′ ) < d − F ∩ F ′ ) is disconnected.When ∆ is pure, we have dim(link( F ∩ F ′ )) = d − − dim( F ∩ F ′ ) >
0. Hence, link( F ∩ F ′ )is not Cohen–Macaulay by Reisner’s criterion (see also [2, Exercise 5.1.26]). On the otherhand, link( G ) must be Cohen–Macaulay for any G ∈ ∆ if ∆ is Cohen–Macaulay.2.3. Terminologies on graphs.
Throughout the present paper, we only treat finite sim-ple graphs (i.e., a finite graph with no loops and no multiple edges), and we omit “finitesimple”.Let G be a graph on the vertex set V ( G ) with the edge set E ( G ). We call a subset W of V ( G ) an independent set if { v, w } 6∈ E ( G ) for any v, w ∈ W . Then we can associate anabstract simplicial complex ∆( G ) on V ( G ) as follows:∆( G ) = { W ⊂ V ( G ) : W is an independent set of G } , which is called an independent complex of G . • For a vertex v ∈ V ( G ), let N G ( v ) = { w ∈ V ( G ) : { v, w } ∈ E ( G ) } and let N G [ v ] = N G ( v ) ∪ { v } . • For a subset S ⊂ V ( G ), we denote by G | S the subgraph of G on the vertex set S with the edge set {{ v, w } : v, w ∈ S } ∩ E ( G ). We use the notation G \ S insteadof G | V ( G ) \ S . • Let M ⊂ E ( G ) be a subset of the edge set. We say that M is an induced matching of G if M satisfies that – e ∩ e ′ = ∅ for any e, e ′ ∈ M with e = e ′ ; and – there is no e ′′ ∈ E ( G ) such that e ∩ e ′′ = ∅ and e ′ ∩ e ′′ = ∅ .Moreover, let im( G ) denote the maximal cardinality among induced matchings of G , called the induced matching number of G . • For a subset C ⊂ V ( G ), we say that C is a vertex cover of G if C ∩ e = ∅ for any e ∈ E ( G ). et V ( G ) = { v , . . . , v n } and consider the polynomial ring S = k [ x , . . . , x n ]. The edgeideal I ( G ) of a graph G is defined by I ( G ) = ( x i x j : { v i , v j } ∈ E ( G )) ⊂ S . Edge idealsare very well studied around the area of combinatorial commutative algebra. We noticethat the edge ideal of G coincides with the Stanley–Reisner ideal of ∆( G ).3. Proofs of Theorem 1.1 and Proposition 1.2
The goal of this section is to give proofs of Theorem 1.1 and Proposition 1.2.Let ∆ be a Cohen–Macaulay simplicial complex of dimension d − X (∆) = { F : e H dim(link( F )) (link( F )) = 0 } . We define M (∆) by setting the set of minimal faces of X (∆) with respect to inclusion.We claim the following lemma: Lemma 3.1.
Work with the same notation as above. Then we have the following: (1) [ F ∈ M (∆) star( F ) = ∆;(2) d − | F | ≤ reg(∆) for each F ∈ M (∆);(3) | M (∆) | ≤ type(∆) . Proof. (1) It is enough to show that H ∈ S F ∈ M (∆) star( F ) for each facet H of ∆. Givenany facet H ∈ ∆, since link( H ) = {∅} satisfies that dim k ( e H − (link( H )) = 1, we see that H ∈ X (∆). Therefore, H ∈ star( F ) for some F ∈ M (∆).(2) By [2, Exercise 5.6.6], we see that( ω ∆ ) j = 0 = ⇒ j ≥ | F | for F ∈ X (∆) . (3.1)Hence, for each F ∈ M (∆). we have − a (∆) = min { j : ( ω ∆ ) j = 0 } ≤ | F | . Therefore, we obtain that reg(∆) = a (∆) + d ≥ d − | F | .(3) We see from (3.1) that the elements of X (∆) correspond to the non-vanishing squarefree Z n -degree components of ω ∆ , where n is the number of vertices of ∆. Since type(∆) isequal to the number of minimal generators of ω ∆ and each F in M (∆) should correspondto a minimal generator of ω ∆ , we obtain that | M (∆) | ≤ type(∆). (cid:3) Proof of Theorem 1.1.
By our assumption ∆ = core(∆), there are facets
H, H ′ of ∆ with H ∩ H ′ = ∅ . Moreover, the Cohen–Macaulayness of ∆ implies that ∆ is strongly connected(see Remark 2.2). Hence, there exists a sequence of facets F , . . . , F p such that F = H , F p = H ′ and dim( F i ∩ F i +1 ) = d − ≤ i ≤ p −
1. By Lemma 3.1 (1), thereexists a sequence of elements S , . . . , S q ∈ M (∆) such that F i j − +1 , . . . , F i j ∈ star( S j ) foreach 1 ≤ j ≤ q , where 1 = i < i < · · · < i q = p . Then we may assume thatstar( S i ) and star( S i +1 ) contain a common facet for each i . (3.2)In fact, by choice of a sequence S , . . . , S q , star( S i ) and star( S i +1 ) contain a common face H with dim( H ) = d −
2. Since there are at least two facets containing H , i.e., link( H )consists of at least two vertices, we have H ∈ X (∆). Hence, there is T ∈ M (∆) with T ⊂ H . If either T = S i or T = S i +1 holds, then we see that star( S i ) and star( S i +1 )contain a common facet. Even if T = S i and T = S i +1 , we may add T between S i and i +1 , i.e., we replace the sequence by S , . . . , S i , T, S i +1 , . . . , S q . Then star( S i ) and star( T )contain a common facet, and so do star( T ) and star( S i +1 ).Since we may also assume that S , . . . , S q are all distinct, we can take a required sequence S , . . . , S q satisfying (3.2). Then we can see that d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j \ i =1 S i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ j X i =1 ( d − | S i | ) for each 1 ≤ j ≤ q (3.3)by induction. The case j = 1 is trivial. Let j > j − G of star( S j − ) and star( S j ). Then we have | ( T j − i =1 S i ) ∪ S j | ≤| S j − ∪ S j | ≤ | G | = d . Thus, by the hypothesis of induction, we conclude that d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j − \ i =1 S i ! ∩ S j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j − \ i =1 S i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | S j | − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j − \ i =1 S i ! ∪ S j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! ≤ j − X i =1 ( d − | S i | ) − | S j | + d = j X i =1 ( d − | S i | ) . Here, we have that S ⊂ H and S q ⊂ H ′ . Hence, S ∩ S q ⊂ H ∩ H ′ = ∅ . In particular, T qi =1 S i = ∅ holds. Therefore, by applying (3.3) with j = q , we obtain that d = d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q \ i =1 S i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q X i =1 ( d − | S i | ) ≤ max { d − | S i | : 1 ≤ i ≤ q } · | M (∆) |≤ reg(∆) · type(∆)by Lemma 3.1 (2) and (3), as desired. (cid:3) Next, we prove Proposition 1.2. Here, we know that for a simplicial complex ∆,Indeg( I ∆ ) = 2 holds if and only if there is a graph G such that ∆ = ∆( G ). SinceIndeg( I ) = 2 holds if S/I has a 1-linear resolution and I contains no linear polynomial,we may discuss the edge ideals of graphs in the case (1).The following theorem is used for the proof of Proposition 1.2 (1). Theorem 3.2 ([7, Theorem 3.3]) . Let G be a graph with n vertices all of which are non-isolated, and let d − G )) . Suppose that ∆( G ) is Cohen–Macaulay. Then wehave n − d ≥ d .Proof of Proposition 1.2. (1) Let n be the number of vertices of G and let c = n − d , where d − G )). Here, Betti numbers of homogeneous Cohen–Macaulay k -algebraswith p -linear resolutions over the polynomial ring are completely determined by Eisenbud–Goto ([6, Proposition 1.7 (c)]) and we know that type(∆) = β c (∆) = (cid:0) c + p − p (cid:1) for a Cohen–Macaulay simplicial complex whose Stanley–Reisner ring has a p -linear resolution. Fromthis, we obtain that type(∆( G )) = c . Since reg(∆( G )) = 1, it follows from Proposition 3.2that reg(∆( G )) + type(∆( G )) − c ≥ d , as required.(2) Let ∆ be a Cohen–Macaulay simplicial complex of dimension d − a (∆) = 0, since a (∆) = reg(∆) − dim( k [∆]), the inequality (1.2) triviallyholds. Note that if ∆ is Gorenstein and ∆ = core(∆), then a (∆) = 0 (see [2, Exercise5.6.8]). Thus, the inequality (1.2) holds, as required. (cid:3) . Proof of Theorem 1.3
The goal of this section is to construct examples of simplicial complexes ∆ such thatthe triple (dim(∆) , reg(∆) , type(∆)) satisfies the required inequalities. We construct suchsimplicial complexes as independent complexes.Before the construction of examples, let us recall two constructions of graphs from agiven graph G ; whiskered graph W ( G ) and S -suspension G S . Let G be a graph on thevertex set V ( G ) = { v , . . . , v d } with the edge set E ( G ).4.1. Whiskered graphs.
We define the whiskered graph of G , denoted by W ( G ), bysetting V ( W ( G )) = V ( G ) ⊔ { w , . . . , w d } and E ( W ( G )) = E ( G ) ⊔ {{ v i , w i } : i = 1 , . . . , d } , (4.1)where w , . . . , w d are new vertices. Lemma 4.1 (cf. [3], [10, Theorem 1.1]) . For any graph G with d vertices, ∆( W ( G )) ispure and vertex decomposable. In particular, ∆( W ( G )) is a Cohen–Macaulay simplicialcomplex of dimension d − . S -suspensions. Given an independent set S of G , we define the S -suspension of G ,denoted by G S , by setting V ( G S ) = V ( G ) ⊔ { w } and E ( G S ) = E ( G ) ⊔ {{ v, w } : v S } , where w is a new vertex. Note that S ⊔ { w } becomes an independent set of G S . Hence,in the language of the independent complex, ∆( G S ) is the ridge sum of ∆( G ) and a newsimplex S ⊔ { w } along the ridge S . Lemma 4.2 ([11, Lemma 1.2], [5, Theorem 2.9], [12, Lemma 1.5]) . Let G be a graph andlet S be an independent set of G .(1) Assume that ∆( G ) is Cohen–Macaulay. If | S | = dim(∆( G )) , then ∆( G S ) is alsoCohen–Macaulay of the same dimension as ∆( G ) .(2) Assume that | S | = dim(∆( G )) . Then type(∆( G S )) = type(∆( G )) + 1 .(3) Assume that G has no isolated vertex. Then reg(∆( G S )) = reg(∆( G )) .Proof. The statement (1) (resp. (3)) is a direct consequence of [11, Lemma 1.2] (resp. [12,Lemma 1.5]). The statement (2) follows from [5, Theorem 2.9]. (cid:3)
Whiskered graphs of complete multi-partite graphs.
For the construction ofour desired graph, we consider the whiskered graphs of complete multi-partite graphs,which play the essential role in the proof of Theorem 1.3.
Proposition 4.3.
Let r , . . . , r t be integers with r ≥ · · · ≥ r t ≥ . Consider G = W ( K r ,...,r t ) . Then dim(∆( G )) = t X i =1 r i − , reg(∆( G )) = r and type(∆( G )) = t. For the proof of Proposition 4.3, we recall a result from [4]. Let G be a graph on thevertex set V ( G ) with 2 d vertices all of which are non-isolated vertices. We consider the ondition that V ( G ) = X ⊔ Y, where X = { x , . . . , x d } is a minimal vertex cover of G , and Y = { y , . . . , y d } is a maximal independent set of G , such that {{ x i , y i } : i = 1 , . . . , n } ⊂ E ( G ) . (4.2)Consider the graph satisfying (4.2). Let E i = { k ∈ { , . . . , d } : { x k , y i } ∈ E ( G ) } \ { i } for i = 1 , . . . , d , and let O [ d ] ( G ) be the graph on the vertex set V ( O [ d ] ( G )) = V ( G ) withthe edge set E ( O [ d ] ( G )) = E ( G ) \ d [ i =1 {{ x k , y i } : k ∈ E i } ! ∪ d [ i =1 {{ x k , x i } : k ∈ E i } . Proposition 4.4 ([4, Corollary 4.4]) . Let G be a graph with d vertices all of which arenon-isolated vertices and let dim(∆( G )) = d − . Assume that ∆( G ) is Cohen–Macaulayand G satisfies (4.2) . Then type(∆( G )) = υ ( O [ n ] ( G ) | X ) , where υ ( H ) denotes the number of minimal vertex covers of a graph H . Before giving the proof of Proposition 4.3, we fix the notation on K r ,...,r t . Let V i = { x ( i )1 , . . . , x ( i ) r i } for i = 1 , . . . , t and let V ( K r ,...,r t ) = F ti =1 V i . Let G = W ( K r ,...,r t ), let V ( G ) = { x ( i ) j , y ( i ) j : 1 ≤ i ≤ t, ≤ j ≤ r i } , and let E ( G ) = E ( K r ,...,r t ) ⊔ {{ x ( i ) j , y ( i ) j } : 1 ≤ i ≤ t, ≤ j ≤ r i } . Proof of Proposition 4.3.
First, dim(∆( G )) = P ti =1 r i − G )) = r . Here, it follows from [14, Theorem 2.4] thatreg(∆( H )) = im( H ) if a graph H contains no induced cycle of length 5 and ∆( H ) isvertex decomposable. By the structure of G , we see that G contains no induced cycle oflength 5. Moreover, Lemma 4.1 implies that ∆( G ) is vertex decomposable. Since we seethat {{ x ( i ) j , y ( i ) j } : j = 1 , . . . , r } forms an induced matching of G for each i and those aremaximal ones, we conclude that im( G ) = r .Finally, we show that type(∆( G )) = t . Here, we can check that G satisfies (4.2) bysetting X = V ( K r ,...,r t ) and Y = V ( G ) \ X . Under this setting, we see that O [ d ] ( G ) = E ( K r ,...,r t ), i.e., all vertices in Y become isolated in O [ d ] ( G ). Hence, we may count thenumber of minimal vertex covers of K r ,...,r t . Let V = V ( K r ,...,r t ). Then C ⊂ V is aminimal vertex cover if and only if C = V \ V i for some i . In fact, if there are i and i ′ with i = i ′ such that V i \ C = ∅ and V i ′ \ C = ∅ , since we consider a complete multi-partitegraph, there must be an edge between V i \ C and V i ′ \ C , a contradiction. Therefore,one has υ ( O [ d ] ( G | X )) = υ ( O [ d ] ( K r ,...,r t )) = t , so we conclude that type(∆( G )) = t byProposition 4.4, as required. (cid:3) Now, we are ready to prove Theorem 1.3.
Proof of Theorem 1.3.
Let d, r, t be integers with r, t ≥ r ≤ d ≤ rt .Write d = pr + q , where 0 ≤ q < r , i.e., p (resp. q ) is the quotient (resp. the remainder)of d divided by r . Note that we have 1 ≤ p ≤ t and q = 0 if p = t by our assumption. et G = W ( Kr, . . . , r | {z } p ,q ) if q = 0 and let G = W ( Kr, . . . , r | {z } p ) if q = 0. Note that | V ( Kr, . . . , r | {z } p ,q ) | = pr + q = d . Thus, it follows from Lemma 4.1 that ∆( G ) is Cohen–Macaulay of dimension d −
1. Moreover, by Proposition 4.3, we know thatreg(∆( G )) = r and type(∆( H )) = p + 1 . If p + 1 < t , by applying certain S -suspensions of ( t − p −
1) times, we obtain the desiredgraph by Lemma 4.2. (cid:3)
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Department of Pure and Applied Mathematics, Graduate School of Infor-mation Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
Email address : [email protected] (H. Kanno) Department of Pure and Applied Mathematics, Graduate School of InformationScience and Technology, Osaka University, Suita, Osaka 565-0871, Japan
Email address : [email protected] (K. Matsuda) Kitami Institute of Technology, Kitami, Hokkaido 090-8507, Japan
Email address : [email protected]@mail.kitami-it.ac.jp