Levelness versus almost Gorensteinness of edge rings of complete multipartite graphs
aa r X i v : . [ m a t h . A C ] F e b LEVELNESS VERSUS ALMOST GORENSTEINNESS OF EDGE RINGSOF COMPLETE MULTIPARTITE GRAPHS
AKIHIRO HIGASHITANI AND KOJI MATSUSHITA
Abstract.
Levelness and almost Gorensteinness are well-studied properties on gradedrings as a generalized notion of Gorensteinness. In the present paper, we study thoseproperties for the edge rings of the complete multipartite graphs, denoted by k [ K r ,...,r n ]with 1 ≤ r ≤ · · · ≤ r n . We give the complete characterization of which k [ K r ,...,r n ] islevel in terms of n and r , . . . , r n . Similarly, we also give the complete characterizationof which k [ K r ,...,r n ] is almost Gorenstein in terms of n and r , . . . , r n . Introduction
Backgrounds.
Cohen–Macaulay (local or graded) rings and Gorenstein (local orgraded) rings are definitely the most important properties and play the crucial roles inthe theory of commutative algebras. However, there are quite many examples which areCohen–Macaulay but not Gorenstein. Thus, many researchers working on commutativealgebras have been attempting the introduction of good “intermediate” classes of thosetwo properties. Thanks to those previous studies, many classes of Cohen–Macaulay gradedrings which are not Gorenstein have been defined and those theories have been developed.In the present paper, we concentrate on two well-studied properties, level rings and al-most Gorenstein homogeneous rings. For the precise definitions of level rings and almostGorenstein homogeneous rings, see Section 2. For example, the characterizations of bothlevelness and almost Gorensteinness of Hibi rings are given by Miyazaki ([12] and [13]).Those characterizations will be mentioned in Section 3 in detail.It is clear that level rings and almost Gorenstein rings are Gorenstein, but it is hardto determine how different between levelness and Gorensteinness as well as almost Goren-steinness and Gorensteinness. Even so, it is a naive problem to analyze the differences ofthose properties. Moreover, it is also natural to think of the difference of levelness andalmost Gorensteinness for homogeneous rings and compare these properties from pointsof view of an extension of Gorenstein homogeneous rings. For those purpose, we restrictthe objects of homogeneous rings. The central objects of the present paper are the edgerings of the complete multipartite graphs.1.2.
Edge polytopes and edge rings.
We recall the definition of edge rings. Note thatedge rings are homogeneous rings arising from graphs and regarded as toric rings associatedto some convex polytopes, which are called edge polytopes. See, e.g., [18, Section 10] or[7, Section 5] for the introduction to edge rings.For a positive integer d , let [ d ] := { , . . . , d } . Consider a finite simple graph G on thevertex set V ( G ) = [ d ] with the edge set E ( G ). Given an edge e = { i, j } ∈ E ( G ), let Mathematics Subject Classification.
Primary 13H10; Secondary 52B20, 13F65, 13A02, 05C25.
Key words and phrases. level, almost Gorenstein, edge rings, complete multipartite graphs. ( e ) := e i + e j , where e i denotes the i -th unit vector of R d for i = 1 , . . . , d . We define theconvex polytope associated to G as follows: P G := conv( { ρ ( e ) : e ∈ E ( G ) } ) ⊂ R d . We call P G the edge polytope of G . We also define the edge ring of G , denoted by k [ G ],as a subalgebra of the polynomial ring k [ t ] = k [ t , . . . , t d ] in d variables over a field k asfollows: k [ G ] := k [ t i t j : { i, j } ∈ E ( G )] . This is actually a monoid k -algebra associated to the monoid Z ≥ ( P G ∩ Z d ), i.e., the edgering is the toric ring (a.k.a. the polytopal monomial subring) of the edge polytope. Notethat dim P G = d − G is non-bipartite ([14, Proposition 1.3]). Thus, we conclude thatthe Krull dimension of k [ G ], denoted by dim k [ G ], is equial to d if G is non-bipartite.1.3. Edge rings of complete multipartite graphs.
We also recall what completemultipartite graphs are. Let K r ,...,r n be the graph on the vertex set F nk =1 V k , | V k | = r k for k = 1 , . . . , n and 1 ≤ r ≤ · · · ≤ r n , with the edge set {{ u, v } : u ∈ V i , v ∈ V j , ≤ i < j ≤ n } . This graph K r ,...,r n is called the complete multipartite graph with type ( r , . . . , r n ).We always denote the number of vertices of K r ,...,r n by d , i.e., d = P ni =1 r i .The edge polytope and the edge ring of K r ,...,r n were investigated in [15, Section 2]. Forexample, the Ehrhart polynomial of P K r ,...,rn (i.e., the Hilbert function of k [ K r ,...,r n ]) iscompletely determined in [15, Theorem 2.6]. Moreover, the Gorensteinness of k [ K r ,...,r n ]is proved as follows: Proposition 1.1 (Characterization of Gorensteinness, [15, Remark 2.8]) . Let ≤ r ≤· · · ≤ r n and let d = P ni =1 r i , where n ≥ . Then the edge ring of the complete multipartitegraph K r ,...,r n is Gorenstein if and only if • n = 2 and ( r , r ) ∈ { (1 , m ) , ( m, m ) : m ≥ } ; • n = 3 and ≤ r ≤ r ≤ r ≤ ; • n = 4 and r = · · · = r = 1 . This proposition is a direct consequence of [3].In [10], the authors of the present paper investigated k [ K r ,...,r n ] from different pointsof view. For example, it is proved that the class group of k [ K r ,...,r n ] is isomorphic to Z n if n = 3 with r ≥ n ≥
4. Moreover, the authors also discuss its conic divisorial idealsand construct non-commutative crepant resolutions for Gorenstein edge rings of K r ,...,r n .1.4. Main Results.
The goal of the present paper is to determine when k [ K r ,...,r n ] islevel or almost Gorenstein, where 1 ≤ r ≤ · · · ≤ r n . In the case where n = 2 or n = 3with r = 1, Proposition 3.2 says that the edge ring k [ K r ,...,r n ] is isomorphic to a certainHibi ring. Thus, the characterizations can be obtained from the results on Hibi rings. SeeSection 3. Hence, our main concern is in the case where n = 3 with r ≥ n ≥ k [ K r ,...,r n ]: Theorem 1.2 (Characterization of levelness) . Let ≤ r ≤ · · · ≤ r n and let d = P ni =1 r i ,where n ≥ . Then the edge ring of the complete multipartite graph K r ,...,r n is level if andonly if n and ( r , . . . , r n ) satisfy one of the following: (i) n = 2 ; (ii) n = 3 and ( r , r , r ) ∈ { (1 , , m ) : m ≥ } ∪ { (1 , , m ) : m ≥ } ; (iii) n = 3 and ( r , r , r ) ∈ { (2 , , m ) : m ≥ } ∪ { (3 , , } ; (iv) n = 4 and ( r , r , r , r ) ∈ { (1 , , , m ) : m ≥ } ; v) n = 5 and r = · · · = r = 1 . Note that the first two cases come from the results on Hibi rings. See Proposition 3.5.The second main result is the characterization of the almost Gorensteinness of k [ K r ,...,r n ]: Theorem 1.3 (Characterization of almost Gorensteinness) . Let ≤ r ≤ · · · ≤ r n and let d = P ni =1 r i , where n ≥ . Then the edge ring of the complete multipartite graph K r ,...,r n is almost Gorenstein if and only if n and ( r , . . . , r n ) satisfy one of the following: (i) n = 2 and ( r , r ) ∈ { (1 , m ) , ( m, m ) : m ≥ } ∪ { (2 , m ) : m ≥ } ; (ii) n = 3 and ( r , r , r ) ∈ { (1 , , m ) , (1 , m, m ) : m ≥ } ; (iii) n = 3 and ( r , r , r ) = (2 , , ; (iv) n = 4 and ( r , r , r , r ) ∈ { (1 , , m, m ) : m ≥ } ; (v) n ≥ and ( r , . . . , r n − , r n ) = (1 , . . . , , n − . (vi) n is even with n ≥ and r = · · · = r n = 1 ; Note that the first two cases come from the result on Hibi rings. See Section 3 (belowProposition 3.5).As an immediate corollary of those theorems, we obtain the following:
Corollary 1.4.
The edge ring of the complete multipartite graph K r ,...,r n is level andalmost Gorenstein but not Gorenstein if and only if one of the following holds: • n = 2 and ( r , r ) ∈ { (2 , m ) : m ≥ } ; • n = 3 and ( r , r , r ) ∈ { (1 , , m ) : m ≥ } .In particular, in both cases, the edge rings are isomorphic to certain Hibi rings. Example 1.5 ( n = 2 or n = 3) . For K r ,r , we see that k [ K r ,r ] is always level. More-over, k [ K ,m ] and k [ K m,m ] are Gorenstein, while k [ K ,m ] is not Gorenstein but almostGorenstein if m ≥ K r ,r ,r , we see that k [ K r ,r ,r ] is • level but not Gorenstein for K , ,m , K , ,m , K , ,m with m ≥ K , , ; • almost Gorenstein but not Gorenstein if and only if K , ,m , K ,m,m with m ≥ • level and almost Gorenstein but not Gorenstein if and only if K , ,m with m ≥ Example 1.6 ( n = 4) . For K r ,r ,r ,r , we see that k [ K r ,r ,r ,r ] is • level for K , , ,m ; • almost Gorenstein for K , ,m,m ; • Gorenstein for K , , , . Example 1.7 ( n ≥ . In the case n ≥ k [ K r ,...,r n ] is never Gorenstein, and it islevel only for K . On the other hand, it is almost Gorenstein for K m with m ≥ K , . . . , | {z } n − ,n − .1.5. Organization.
First, in Section 2, we recall the precise definitions of level rings andalmost Gorenstein homogeneous rings and some known results on them, which we will usein the proofs of our main results. Next, in Section 3, we discuss the main theorems in thecase where n = 2 or n = 3 with r = 1, which can be deduced into the results on Hibirings. Thus, we also recall the results on Hibi rings. Finally, in Section 4, we give proofsof Theorems 1.2 and 1.3. Acknowledgement.
The first named author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 20K03513. . Levelness and almost Gorensteinness of homogeneous rings
Throughout this section, let R be a Cohen–Macaulay homogeneous ring of dimension d over an algebraically closed field k with characteristic 0. We recall the definitions oflevelness and almost Gorensteinness and some properties on those algebras.Before defining them, we recall some fundamental materials. (Consult, e.g., [2] for theintroduction to homogeneous rings.) • Let ω R denote a canonical module of R . Let a ( R ) denote the a -invariant of R , i.e., a ( R ) = − min { j : ( ω R ) j = 0 } . • For a graded R -module M , we use the following notation: – Let µ j ( M ) denote the number of elements of minimal generators of M withdegree j as an R -module, and let µ ( M ) = P j ∈ Z µ j ( M ), i.e., the number ofminimal generators. – Let e ( M ) denote the multiplicity of M . Then the inequality µ ( M ) ≤ e ( M )always holds. – Let M ( − ℓ ) denote the R -module whose grading is given by M ( − ℓ ) n = M n − ℓ for any n ∈ Z . • Let r ( R ) denote the Cohen–Macaulay type of R . Note that r ( R ) = µ ( ω R ). We seethat R is Gorenstein if and only if r ( R ) = 1. • Let H ( M, m ) denote the Hilbert function of M , i.e., H ( M, m ) = dim k M m , wheredim k stands for the dimension as a k -vector space. Note that H ( M, m ) can bedescribed by a polynomial in m of degree d − e ( M ) /d !. (See [2, Section 4].) • Write ( h , . . . , h s ) for the h -vector of R , which is the coefficients of the polynomialappearing in the numerator of the Hilbert series of R , i.e., ∞ X n =0 dim k R n t n = P si =0 h i t i (1 − t ) d , where h s = 0. We call the index s the socle degree of R . Note that h s ≤ r ( R )holds. Moreover, we see that d + a ( R ) = s . (See [2, Section 4.4].) Definition 2.1 (Level, [17]) . We say that R is level if all the degrees of the minimalgenerators of ω R are the same. In particular, what R is Gorenstein implies what R is levelsince ω R is generated by a unique element if R is Gorenstein. Remark . Let ( h , h , . . . , h s ) be the h -vector of R . Assume that R is level. In this case,if h s = 1, then R is Gorenstein. In fact, since what R is level is equivalent to h s = r ( R )(see, e.g., [2, Section 5]), we obtain that r ( R ) = 1.Regarding the levelness of homogeneous domains, we know the following: Theorem 2.3 ([19, Corollary 3.11]) . Let R be an almost Gorenstein homogeneous domainwith its socle degree s . If s = 2 , then R is level. Definition 2.4 (Almost Gorenstein, [5, Definition 1.5]) . We call R almost Gorenstein ifthere exists an exact sequence of graded R -modules0 → R → ω R ( − a ) → C → µ ( C ) = e ( C ). ote that there always exists a degree-preserving injection from R to ω R ( − a ) if R isa domain ([9, Proposition 2.2]). Moreover, we see that µ ( C ) = r ( R ) − Theorem 2.5 ([9, Theorem 4.7]) . Let R be an almost Gorenstein homogeneous domainand ( h , h , . . . , h s ) its h -vector with s ≥ . Then h s = 1 . In the case of Hibi rings
In this section, we recall Hibi rings and the characterization results on Gorensteinness,levelness and almost Gorensteinness of Hibi rings.Let Π = { p , . . . , p d − } be a finite partially ordered set (poset, for short) equipped witha partial order ≺ . For p, q ∈ Π, we say that p covers q if q ≺ p and there is no p ′ ∈ Π \{ p, q } with q ≺ p ′ ≺ p . For a subset I ⊂ Π, we say that I is a poset ideal of Π if p ∈ I and q ≺ p then q ∈ I . Let I (Π) denote the set of all poset ideals in Π. Note that ∅ ∈ I (Π).We define the k -algebra k [Π] as a subalgebra of the polynomial ring k [ x , . . . , x d − , t ]by setting k [Π] := k [ x I t : I ∈ I (Π)] , where x I = Q p i ∈ I x i for I ⊂ Π and each x I t is defined to be of degree 1. The standardgraded k -algebra k [Π] is called the Hibi ring of Π. The following fundamental propertieson Hibi rings were originally proved in [8]: • The Krull dimension of k [Π] is equal to | Π | + 1; • k [Π] is a Cohen–Macaulay normal domain; • k [Π] is an algebra with straightening laws on Π.We say that a poset Π is pure if each maximal chain contained in the poset has thesame length, where the length of the chain p i ≺ p i ≺ · · · ≺ p i ℓ is defined to be ℓ . Thefollowing is well known: Theorem 3.1 ([8]) . The Hibi ring k [Π] of Π is Gorenstein if and only if Π is pure. Given positive integers m and n , let Π m,n = { p , . . . , p m , p m +1 , . . . , p m + n } be the posetequipped with the partial orders p ≺ · · · ≺ p m and p m +1 ≺ · · · ≺ p m + n . Moreover,let Π ′ m,n be the poset having an additional relation p ≺ p m + n . Note that the Hibi ring k [Π m,n ] is isomorphic to the Segre product of the polynomial ring with ( m + 1) variablesand the polynomial ring with ( n + 1) variables (see, e.g., [11, Example 2.6]).Actually, the certain edge rings are isomorphic to the Hibi rings k [Π m,n ] and k [Π ′ m,n ]as follows: Proposition 3.2 ([10, Proposition 2.2]) . The edge ring k [ K m +1 ,n +1 ] (resp. k [ K ,m,n ] ) isisomorphic to the Hibi ring k [Π m,n ] (resp. k [Π ′ m,n ] ). Therefore, in the case where n = 2 or n = 3 with r = 1, the characterization oflevelness and almost Gorensteinness of k [ K r ,...,r n ] can be deduced into those of the Hibirings k [Π m,n ] and k [Π ′ m,n ].Hence, in what follows, we give the characterizations of those properties for k [Π m,n ] and k [Π ′ m,n ], which prove Theorems 1.2 and 1.3 in the case where n = 2 or n = 3 with r = 1.Miyazaki gave the characterizations of levelness [12] and almost Gorensteinness [13] ofHibi rings. For explaining those results, we introduce some notions. iven a poset Π, let b Π := Π ∪ { ˆ0 , ˆ1 } , where ˆ0 (resp. ˆ1) denotes a new unique minimal(maximal) element. • For x, y ∈ Π with x (cid:22) y , we set [ x, y ] Π := { z ∈ Π : x (cid:22) z (cid:22) y } . • We define rank P to be the maximal lenth of the chains in Π, and define rank[ x, y ] Π analogously. • Let y , x , y , x , . . . , y t , x t be a (possibly empty) sequence of elements in b Π. Wesay that the sequence y , x , y , x , . . . , y t , x t satisfies condition N if(1) x = ˆ0,(2) y ≻ x ≺ y ≻ x ≺ · · · ≺ y t ≻ x t , and(3) y i (cid:15) x j for any i, j with 1 ≤ i < j ≤ t . • Let r ( y , x , . . . , y t , x t ) := X i ∈ [ t ] (rank[ x i − , y i ] b Π − rank[ x i , y i ] b Π ) + rank[ x t , ˆ1] b Π , where we set an empty sum to be 0 and x = ˆ0. • Given x ∈ Π, let star Π ( x ) = { y ∈ Π : y (cid:22) x or x (cid:22) y } . Theorem 3.3 ([12, Theorem 3.9]) . The Hibi ring of Π is level if and only if r ( y , x , . . . , y t , x t ) ≤ rank b Π holds for any sequence of elements in b P with condition N. Corollary 3.4 ([12, Corollary 3.10]) . If [ x, ˆ1] b Π is pure for any x ∈ Π , then k [Π] is level. By using those results by Miyazaki, we can prove the following:
Proposition 3.5.
Let m ≤ n . (i) The Hibi ring k [Π m,n ] of Π m,n is level for any m, n . (ii) The Hibi ring k [Π ′ m,n ] of Π ′ m,n is level if and only if m = 1 or m = 2 .Proof. The assertion (i) directly follows from Corollary 3.4.Similarly, we see from Corollary 3.4 that k [Π ′ ,n ] and k [Π ′ ,n ] are level.Let y = p m + n and x = p and take the sequence y , x , which satisfies condition N.Then we see that r ( y , x ) = rank[ˆ0 , y ] b Π − rank[ x , y ] b Π + rank[ x , ˆ1] b Π = n − m. On the other hand, we have rank b Π = n + 1. If m ≥
3, then m − >
1, so we have n + m − r ( x , y ) > rank b Π = n + 1 . Hence, k [Π ′ m,n ] is not level when m ≥ (cid:3) Hence, in the case where n = 2 or n = 3 with r = 1, k [ K r ,...,r n ] is level if and only if K r ,...,r n satisfies (1) or (2) in Theorem 1.2.Note that k [ K ,n ] is isomorphic to a polynomial ring with n variables.Regarding the characterization of almost Gorenstein Hibi rings, see [13, Introduction].According to it, we see the following: • Let m ≤ n . Consider the poset Π m,n . – We see that Π ,n fits into the case of (1) of [13, Introduction]. Thus, k [Π ,n ]is almost Gorenstein. Since Π m,m is pure, we know that k [Π m,m ] is Gorenstein, in particular, almostGorenstein. – If 1 < m < n , then we see from the characterization that k [Π m,n ] is neveralmost Gorenstein. • Let m ≤ n . Consider the poset Π ′ m,n . – We have star Π ′ ,n ( p n +1 ) = Π ′ ,n and Π ′ ,n \ { p n +1 } fits into the case of (1) of[13, Introduction]. Thus, k [Π ,n ] is almost Gorenstein. – We see that Π ′ m,m fits into the case of (2) (ii) (with p = 0). Hence, k [Π ′ m,m ]is almost Gorenstein. – If 1 < m < n , then we see from the characterization that k [Π ′ m,n ] is neveralmost Gorenstein.4. Proofs of main theorems
The goal of this section is to complete the proofs of Theorems 1.2 and 1.3. As shown inSection 3, the case where n = 2 or n = 3 with r = 1 was already done. Thus, in principle,we discuss the case where n = 3 with r ≥ n ≥ Preliminaries for edge polytopes.
Before proving the assertions, we recall somegeometric information on edge polytopes.For i ∈ [ d ], let p i : R d → R be the i -th projection, and let p V k := P j ∈ V k p j for k ∈ [ n ].For k ∈ [ n ], let f k := X i ∈ [ d ] \ V k e i − X j ∈ V k e j . We define Ψ r , Ψ f and Ψ as follows:Ψ r := { e i : i ∈ [ d ] } , Ψ f := { f k : k ∈ [ n ] } , Ψ := Ψ r ∪ Ψ f . Note that for e i ∈ Ψ r (resp. f k ∈ Ψ f ), we have h− , e i i = p i ( − ) (resp. h− , f k i = (cid:18) X j ∈ [ n ] \{ k } p V j − p V k (cid:19) ( − )), where h− , −i : R d × R d → R stands for the usual inner product.It is proved in [14] that the hyperplanes H l := { x ∈ R d : h x, l i = 0 } for l ∈ Ψ definethe facets of P K r ,...,rn . Although those are not necessarily one-to-one, in the case where n = 3 with r ≥ n ≥
4, we see that it is one-to-one.Let G be a graph on the vertex set [ d ]. We say that G satisfies odd cycle condition if forany two distinct odd cycles which have no common vertex, there is a bridge between them.It is known that k [ G ] is normal if and only if G satisfies odd cycle condition. Note that K r ,...,r n satisfies odd cycle condition. Let ( h , h , . . . , h s ) be the h -vector of the edge ring k [ G ], where s is the scole degree, and let ℓ = min { m ∈ Z > : mP ◦ G ∩ Z d = ∅} , where P ◦ G denotes the relative interior of P G . Since the ideal generated by the monomials containedin the interior of P G is the canonical module of k [ G ], we see that ℓ = − a ( k [ G ]). Thus, d = ℓ + s holds if G is non-bipartite. In the case where n = 3 with r ≥ n ≥
4, onehas ι ∈ mP ◦ K r ,...,rn ∩ Z d if and only if h ι, l i > l ∈ Ψ.For ι ∈ ( ℓ + k ) P ◦ G ∩ Z d for k ∈ Z > , then we say that ι is a first appearing interiorpoint in ( ℓ + k ) P ◦ G ∩ Z d if it cannot be written as a sum of ι ′ ∈ ( ℓ + i ) P ◦ G ∩ Z d with0 ≤ i < k and ρ ( e )’s with e ∈ E ( G ), where the elements in ℓP ◦ G ∩ Z d are regarded as firstappearing interior points. Let µ k ( G ) denote the number of first appearing interior pointsin ( ℓ + k ) P ◦ G ∩ Z d for k ∈ Z ≥ . Note that µ ( G ) = h s holds. n what follows, for the study of k [ K r ,...,r n ] with 1 ≤ r ≤ · · · ≤ r n and d = P di =1 r i ,we divide into the following three cases on K r ,...,r n :(A) 2 r n < d and d is even;(B) 2 r n < d and d is odd;(C) 2 r n ≥ d .Given a graph G with the edge set E ( G ), we say that M ⊂ E ( G ) is a perfect matching (a.k.a. 1 -factor ) if every vertex of G is incident to exactly one edge of M . Lemma 4.1.
The complete multipartite graph K r ,...,r n has a perfect matching if and onlyif d is even and r n ≤ d .Proof. Tutte’s theorem (see, e.g., [4, Theorem 2.2.1]) claims that a graph G on the vertexset V ( G ) has a perfect matching if and only if q ( G − U ) ≤ | U | holds for any U ⊂ V ( G ),where q ( · ) denotes the number of connected components with odd cardinality and G − U denotes the induced subgraph of G by V ( G ) \ U .When U = V ( K r ,...,r n ), the inequality trivially holds. When U = ∅ , we can see that q ( G ) = 0 holds if and only if d is even.Consider ∅ 6 = U ( V ( K r ,...,r n ). If there are two verticies u, v in V ( K r ,...,r n ) \ U with u ∈ V i and v ∈ V j with i = j , the number of connected components of K r ,...,r n − U is equalto 1, so q ( K r ,...,r n − U ) ≤ | U | holds. If there exists k ∈ [ n ] with V ( K r ,...,r n ) \ U ⊂ V k ,then it follows from 1 ≤ r ≤ · · · ≤ r n and V i ⊂ U for i ∈ [ n ] \ { k } that q ( K r ,...,r n − U ) ≤ r k ≤ r n ≤ | U | . Therefore, by considering the case k = n , we conclude the following: q ( K r ,...,r n − U ) ≤ | U | for any U ⇐⇒ r n ≤ X k ∈ [ n − r k and d is even ⇐⇒ r n ≤ d and d is even. (cid:3) Proposition 4.2.
Let ( h , h , . . . , h s ) be the h -vector of k [ K r ,...,r n ] . (a) In the case of (A), we have ℓ = d/ and h s = 1 . (b) In the case of (B), we have ℓ = ( d + 1) / and h s ≥ . (c) In the case of (C), we have ℓ = r n + 1 and h s ≥ .Proof. In the case of (A), by Lemma 4.1, there exists a perfect matching
M ⊂ E ( K r ,...,r n ),and we obtain ρ ( M ) := P e ∈M ρ ( e ) = P i ∈ [ d ] e i . We can see that ρ ( M ) is the uniqueelement in |M| P ◦ K r ,...,rn ∩ Z d since h ρ ( M ) , l i > l ∈ Ψ, and it is clear that mP ◦ K r ,...,rn ∩ Z d = ∅ for m < |M| . Therefore, we have ℓ = |M| = d/ h s = 1.Next, assume the case of (B). We consider the induced subgraph K r ,...,r n − { v n } for v n ∈ V n . From the assumption, we observe that d − { r n − , r n − } ≤ r n ≤ d − M ′ of K r ,...,r n − { v n } byLemma 4.1. Take v ∈ V , add the edge { v , v n } to M ′ and write M ′′ for it. Then wehave ρ ( M ′′ ) = 2 e v + P i ∈ [ d ] \{ v } e i , and h ρ ( M ′′ ) , l i > l ∈ Ψ. In fact, for f , weobserve that h ρ ( M ′′ ) , f i = (cid:18) X i ∈ [ n ] \{ } p V i − p V (cid:19) ( ρ ( M ′′ )) = X k ∈ [ n ] \{ } r k − ( r + 1) ≥ X k ∈ [ n ] \{ , } r k − > . Thus, we have ρ ( M ′′ ) ∈ |M ′′ | P ◦ K r ,...,rn ∩ Z d and mP ◦ K r ,...,rn ∩ Z d = ∅ for m < |M ′′ | .Moreover, by exchanging v with another vertex of V or a vertex of V if r = 1, we obtain ( M ′′ ) ∈ |M ′′ | P ◦ K r ,...,rn ∩ Z d in the same way. Therefore, we have ℓ = |M ′′ | = ( d + 1) / h s ≥ r ′ n := P i ∈ [ n − r i and let r ′′ n := r n − r ′ n . We join r ′ n verticies of V n to verticies of [ d ] \ V n one-by-one, and join the remaining r ′′ n verticies of V n to v , and join a vertex v ∈ V to v . Let E be the set of those edges. Then we have ρ ( E ) = ( r ′′ n + 2) e v + 2 e v + P i ∈ [ d ] \{ v ,v } e i , and h ρ ( E ) , l i > l ∈ Ψ. In fact, for f ,we observe that h ρ ( E ) , f i = X k ∈ [ n ] \{ } r k + 1 − ( r + r ′′ n + 1) ≥ X k ∈ [ n ] \{ , ,n } r k + r ′ n > . Thus, we have ρ ( E ) ∈ |E| P ◦ K r ,...,rn ∩ Z d and mP ◦ K r ,...,rn ∩ Z d = ∅ for m < |E| since h ι, r i > ι ∈ ℓP ◦ K r ,...,rn ∩ Z d and r ∈ Ψ. Moreover, by exchanging v with another vertexof V or a vertex of V if r = 1, we obtain ρ ( E ) ∈ |E| P ◦ K r ,...,rn ∩ Z d in the same way.Therefore, we have ℓ = |E| = r n + 1 and h s ≥ (cid:3) Corollary 4.3.
Assume that n = 3 with r ≥ or n ≥ . If k [ K r ,...,r n ] is almostGorenstein, then K r ,...,r n is in the case of (A).Proof. In our assumption, we have d ≥
4. Note that s = d − ℓ .In the case of (B), since s = d − ℓ = d − ( d + 1) / d − / d is odd, we see that s ≥
2. In the case of (C), since s = d − ℓ = P i ∈ [ n ] r i − r n − P i ∈ [ n − r i − s ≥
2. Moreover, in both cases, we alsoknow that h s ≥
2. Therefore, these are never almost Gorenstein by Theorem 2.5. (cid:3)
Proof of Theorem 1.2.
This subsection is devoted to proving Theorem 1.2. Sincethe case where n = 2 or n = 3 with r = 1 has been already done in Section 3, we assumethat n = 3 with r ≥ n ≥
4. Under this assumption, we prove that k [ K r ,...,r n ] is levelif and only if one of the following holds: n = 3 with r = r = 2 or r = r = r = 3; n = 4 with r = r = r = 1; n = 5 with r = · · · = r = 1 . (4.1)Assume the case of (A). By Proposition 4.2, we have h s = 1. Thus, k [ K r ,...,r n ] isGorenstein if it is level (see Remark 2.2). Hence, k [ K r ,...,r n ] is level if and only if K r ,...,r n = K , , or K , , , by [15, Remark 2.8].Therefore, in what follows, we consider the cases (B) and (C). “Only if ” part :Assume the case of (B). Take M ′′ and v ∈ V as in the proof of Proposition 4.2. Thenthere exsists an edge { i, j } ∈ M ′′ such that i V and j V . Remove such edge from M ′′ and add { v , i } and { v , j } to M ′′ . Write N for it. Then we have ρ ( N ) = 4 e v + X i ∈ [ d ] \{ v } e i ∈ ( ℓ + 1) P K r ,...,rn ∩ Z d . Since there is only one entry which is more than 1, we see that ρ ( N ) cannot be writtenas a sum of ℓP ◦ K r ,...,rn ∩ Z d and ρ ( e ) for e ∈ E ( K r ,...,r n ). Hence, once we have ρ ( N ) ∈ ℓ + 1) P ◦ K r ,...,rn ∩ Z d , it is not level. Since we know h ρ ( N ) , r i > r ∈ Ψ r , we mayobserve those of Ψ f : h f , ρ ( N ) i = X i ∈ [ n ] \{ } r i − ( r + 3) >
0; (4.2) h f k , ρ ( N ) i = X i ∈ [ n ] \{ k } r i + 3 − r k > k ∈ [ n ] \ { } . (4.3)The inequality (4.3) always holds by the assumption (B). The inequality (4.2) holds if n ≥ ,n = 5 with r ≥ ,n = 4 with r ≥ , or n = 3 with r ≥ . Therefore, in the case of (B), k [ K r ,...,r n ] are not level except for K , , , , , K , , , , K , , , and K , , .Note that we can confirm that k [ K , , , ] is not level by using Macaulay2 ([6]).Assume the case of (C). Take E , v ∈ V , and v ∈ V as in the proof of Proposition 4.2.In E , let v n ∈ V n be the vertex adjacent to v , and let v ′ n be the vertex adjacent to a vertex v ′ = v of V or a vertex v ∈ V if r = 1. Remove { v , v n } and { v ′ , v ′ n } from E and add { v , v n } , { v , v ′ } and { v , v ′ n } to E . Write N ′ for it. Then we have ρ ( N ′ ) = ( r ′′ n + 5) e v + X i ∈ [ d ] \{ v } e i ∈ ( ℓ + 1) P K r ,...,rn ∩ Z d . Then we see that ρ ( N ′ ) cannot be written as a sum of ℓP ◦ K r ,...,rn ∩ Z d and ρ ( e ) for e ∈ E ( K r ,...,r n ). Hence, once we have ρ ( N ′ ) ∈ ( ℓ + 1) P ◦ K r ,...,rn ∩ Z d , it is not level. Sincewe know h ρ ( N ′ ) , r i > r ∈ Ψ r , we may observe those of Ψ f : h f , ρ ( N ′ ) i = X i ∈ [ n ] \{ } r i − ( r + r ′′ n + 4) >
0; (4.4) h f k , ρ ( N ′ ) i = X i ∈ [ n ] \{ k } r i + r ′′ n + 4 − r k > k ∈ [ n ] \ { } . (4.5)The inequality (4.5) always holds by (C). The inequality (4.4) holds if n ≥ ,n = 4 with r ≥ , or n = 3 with r ≥ . Thus, in the case of (C), k [ K r ,...,r n ] is not level except for K , ,r with r ≥ K , , ,r with r ≥ k [ K r ,...,r n ] is not level if not in the case (4.1). “If ” part :Our remaining task is to show that the edge rings of (4.1) are level.( K , ,r with r ≥ r = 2, k [ K , , ] is Gorenstein, and if r = 3, k [ K , , ] is level by using Macaulay2 . ence, let us assume that r ≥
4. Then K r ,...,r n satisfies (C). Thus, we have ℓ = r + 1.It is enough to show that for any k ≥ ι ∈ ( ℓ + k ) P ◦ K , ,r ∩ Z d , ι can be written asa sum of an element of ℓP ◦ K , ,r ∩ Z d and k elements of P K , ,r ∩ Z d , i.e., ρ ( e ) , . . . , ρ ( e k )with e , . . . , e k ∈ E ( K , ,r ). We show this by induction on k . The case k = 0 triviallyholds.We have (cid:18)P i ∈ [ d ] p i (cid:19) ( ι ) = 2( ℓ + k ) = 2 r + 2 k + 2 ≥ r + 4, p i ( ι ) > i ∈ [ d ], p V ( ι ), p V ( ι ) ≥
2, and p V ( ι ) ≥ r . In the case p V ( ι ) = r , we can see that h ι, f j i > p V j ( ι ) ≥ j = 1 ,
2, and there exsist a v ∈ V and a v ∈ V such that p v j ( ι ) ≥ j = 1 ,
2. Let ι ′ := ι − ρ ( { v , v } ). If h ι ′ , l i > l ∈ Ψ, we have ι ′ ∈ ( ℓ + k − P ◦ K , ,r ∩ Z d . It is enough to discuss that of f : h ι ′ , f i = (cid:18) X k ∈{ , } p V k (cid:19) ( ι ) − − p V ( ι ) = ( r + 2 k + 2) − − r > . In the case p V ( ι ) ≥ r + 1, there exists a v ∈ V such that p v ( ι ) = 2. We may assumethat p V ( ι ) ≤ p V ( ι ). Then there is a v ′ ∈ V such that p v ′ ( ι ) ≥
2. Let ι ′ := ι − ρ ( { v ′ , v } ).If we have h ι, l i > l ∈ Ψ, we obtain ι ′ ∈ ( ℓ + k − P ◦ K , ,r ∩ Z d . It is enough todiscuss that of f : h ι ′ , f i = (cid:18) X k ∈{ , } p V k (cid:19) ( ι ) − − p V ( ι ) ≥ p V ( ι ) − > . Therefore, we obtain the desired result.( K , , )In the same way as above, it is enough to show that for any k ≥ ι ∈ ( ℓ + k ) P ◦ K , , ∩ Z d , ι can be written as a sum of an element of ℓP ◦ K , , ∩ Z d and ρ ( e ) , . . . , ρ ( e k )with e , . . . , e k ∈ E ( K , , ).By ℓ = 5, we have (cid:18)P i ∈ [ d ] p i (cid:19) ( ι ) = 2( ℓ + k ) = 2 k + 10 ≥
12. We may assume that p V ( ι ) ≤ p V ( ι ) ≤ p V ( ι ). If we have p V ( ι ) = p V ( ι ) = 3, we obtain p V ( ι ) = 2 k + 4 ≥ h ι ′ , f i ≤
0. This is a contradiction. Thus, we have 4 ≤ p V ( ι ) ≤ p V ( ι ). Hence, thereexsist a v ∈ V and v ∈ V such that p v j ( ι ) ≥ j = 1 ,
2. Let ι ′ := ι − ρ ( { v , v } ). If h ι ′ , l i > l ∈ Ψ, we obtain ι ′ ∈ ( ℓ + k − P ◦ K , , ∩ Z d . It is enough to discussthat of f : h ι ′ , f i = (cid:18) X k ∈{ , } p V k (cid:19) ( ι ) − − p V ( ι ) = (cid:16) p V − p V (cid:17) ( ι ) + (cid:16) p V ( ι ) − (cid:17) > . Therefore, we obtain the desired result.( K , , ,r with r ≥ k [ K , , , ] is Gorenstein, and we can check by Macaulay2 that k [ K , , , ]is level. For r ≥
3, by Proposition 4.2 (c), K , , ,r is in the case of (C) and s = d − r − k [ K , , , , ])We can check by Macaulay2 that k [ K , , , , ] is level. .3. Proof of Theorem 1.3.
We still assume the condition n = 3 with r ≥ n ≥ P ⊂ R N be an integral convex poly-tope, which is a convex polytope all of whose vertices belong to Z N . For m ∈ Z > ,consider the number of integer points contained in mP ∩ Z N . Then it is known that suchnumber | mP ∩ Z N | can be described by a polynomial in m of degree dim P , denoted by i ( P, m ). The enumerating polynomial i ( P, m ) is called the
Ehrhart polynomial of P . Forthe introduction to the Ehrhart polynomials, see, e.g., [1].Throughout the remaining parts of this section, let R = k [ K r ,...,r n ]. Note that R isnormal since K r ,...,r n satisfies odd cycle condition. Regarding the definition of almostGorensteinness, let C be the cokernel of the injection R → ω R ( − a ). Note that C is aCohen–Macaulay R -module of dimension d −
1. Our goal is to characterize when e ( C ) = µ ( C ) holds. For this, we prepare the following two lemmas. Lemma 4.4.
Assume the case of (A). Then e ( C ) = X k ∈ [ n ] (cid:16) d − r k − (cid:17)(cid:18) d − r k − (cid:19) . Proof.
Let i ( P K r ,...,rn , m ) = c d − m d − + c d − m d − + · · · + 1 be the Ehrhart polynomialof P K r ,...,rn . Since R is normal, we see that H ( R, m ) = i ( P K r ,...,rn , m ). Note that i ( P ◦ K r ,...,rn , m ) = ( − d − i ( P K r ,...,rn , − m ) holds. (See, e.g., [1, Theorem 4.1].) Froman exact sequence (2.1), we can see that the Hilbert function H ( C, m ) of C coincides with i ( P ◦ K r ,...,rn , m + ℓ ) − i ( P K r ,...,rn , m ) , where ℓ = − a ( R ). This implies that the leading coefficient of H ( C, m ) coincides with( d − ℓc d − − c d − . Note that dim C = d −
1. Thus, e ( C ) = ( d − d − ℓc d − − c d − ) . Here, [15, Theorem 2.6] claims that i ( P K r ,...,rn , m ) = (cid:18) d + 2 m − d − (cid:19) − X k ∈ [ n ] X ≤ i ≤ j ≤ r k (cid:18) j − i + m − j − i (cid:19)(cid:18) d − j + m − d − j (cid:19) . Hence, a direct computation shows that( d − c d − = 2 d − − X k ∈ [ n ] X j ∈ [ r k ] (cid:18) d − j − (cid:19) (see [15, Corollary 2.7]), and( d − c d − = 2 d − d − X k ∈ [ n ] X j ∈ [ r k ] (cid:18) d − j − (cid:19) + d − (cid:18)(cid:18) d − j − (cid:19) + (cid:18) d − j − (cid:19)(cid:19)! . Remark ℓ = d/ e ( C ) = X k ∈ [ n ] X j ∈ [ r k ] (cid:18) d − j − (cid:19) + ( d − (cid:18)(cid:18) d − j − (cid:19) + (cid:18) d − j − (cid:19)(cid:19) − d (cid:18) d − j − (cid:19)! , where we set (cid:0) nr (cid:1) for r ∈ Z < to be 0. By using (cid:18) n − r − (cid:19) + (cid:18) n − r (cid:19) = (cid:18) nr (cid:19) for n, r ∈ Z , (4.6) e obtain that e ( C ) = X k ∈ [ n ] X j ∈ [ r k ] d (cid:18) d − j − (cid:19) − d − (cid:18) d − j − (cid:19) + 2( d − (cid:18) d − j − (cid:19)! . It is enough to prove that X j ∈ [ r k ] d (cid:18) d − j − (cid:19) − d − (cid:18) d − j − (cid:19) + 2( d − (cid:18) d − j − (cid:19)! = (cid:16) d − r k − (cid:17)(cid:18) d − r k − (cid:19) (4.7)holds. We prove this by induction on r k . We can directly see that (4.7) holds when r k = 1.Suppose that r k >
1. By the hypothesis of induction, we have X j ∈ [ r k +1] d (cid:18) d − j − (cid:19) − d − (cid:18) d − j − (cid:19) + 2( d − (cid:18) d − j − (cid:19)! = (cid:16) d − r k − (cid:17)(cid:18) d − r k − (cid:19) + d (cid:18) d − r k (cid:19) − d − (cid:18) d − r k (cid:19) + 2( d − (cid:18) d − r k (cid:19)! . By using ( d − r k − p ) (cid:18) d − pr k (cid:19) = ( d − p ) (cid:18) d − p − r k (cid:19) for p = 1 , (cid:16) d − r k − (cid:17)(cid:18)(cid:18) d − r k (cid:19) − (cid:18) d − r k (cid:19)(cid:19) + d (cid:18) d − r k (cid:19) − d − (cid:18) d − r k (cid:19) + 2( d − r k − (cid:18) d − r k (cid:19)! = (cid:16) d − r k − (cid:17)(cid:18) d − r k (cid:19) − (cid:16) d r k + 1 (cid:17)(cid:18) d − r k (cid:19) = (cid:0) d − (cid:1)(cid:18) d − r k (cid:19) − (cid:16) d r k + 1 (cid:17)(cid:18) d − r k (cid:19) = (cid:16) d − r k − (cid:17)(cid:18) d − r k (cid:19) . This completes the proof. (cid:3)
Lemma 4.5.
Assume the case of (A). Then µ ( C ) = X k ∈ [ n ] X j ∈ [ d − r k − (cid:18) r k − jr k − (cid:19) , where we let X j ∈ [ d − r k − (cid:18) r k − jr k − (cid:19) = 0 if r k = d − .Proof. Remark ℓ = d/ µ ( C ) = P j ≥ ℓ µ j ( ω K r ,...,rn ) − P j ≥ ℓ +1 µ j ( ω K r ,...,rn ) = P j ≥ µ j ( K r ,...,r n ),where we recall that µ j ( G ) is the number of first appearing interior points. We compute µ j ( K r ,...,r n ) for j ≥ rom Lemma 4.6 below, we see that ι ∈ ( ℓ + j ) P K r ,...,rn ∩ Z d ( j ≥
1) is a first appearinginterior point if and only if there exists k ∈ [ n ] such that ι satisfies p V j ( ι ) = r k + 2 j,p i ( ι ) = 1 for i ∈ [ d ] \ V k , and h ι, f k i = ( d − r k ) − ( r k + 2 j ) > , that is , ≤ j ≤ d/ − r k − . Hence, for j and k respectively, we observe that the number of first appearing interiorpoints is (cid:18) r k − jr k − (cid:19) , and so µ ( C ) = X j ≥ µ j ( K r ,...,r n ) = X j ≥ X k ∈ [ n ] (cid:18) r k − jr k − (cid:19) . (cid:3) Lemma 4.6.
Assume the case of (A). Given ι ∈ ( ℓ + j ) P ◦ K r ,...,rn ∩ Z d for each j ≥ , ι isa first appearing interior point if and only if there exists k ∈ [ n ] such that ( p V j ( ι ) = r k + 2 j,p i ( ι ) = 1 for i ∈ [ d ] \ V k . (4.8) Proof. “If ” part : By the condition on ι , we see that ι cannot be written as a sum ofan element of ( ℓ + j ′ ) P ◦ K r ,...,rn ∩ Z d with j ′ < j and ρ ( e ) , . . . , ρ ( e j ′ ) with e , . . . , e j ′ ∈ E ( K r ,...,r n ). Thus, we obtain the desired result. “Only if ” part : We prove the assertion by induction on j ≥
0. If j = 0, then P i ∈ [ d ] e i is the unique first appearing interior point in ℓP ◦ K r ,...,rn ∩ Z d .Let j ≥
1. Let r ′ k := p V k ( ι ) − r k for k ∈ [ n ]. By the hypothesis of induction, there are atleast two k ’s with r ′ k = 0. Take these r ′ k ≥ r ′ k ≥ · · · ≥ r ′ k s > k p > k q if r ′ k p = r ′ k q .Remark s ≥
2. Then there are v k ∈ V k and v k ∈ V k such that p v k , p v k ≥ ι ′ = ι − ρ ( { v k , v k } ) ∈ ( ℓ + j − P ◦ K r ,...,rn ∩ Z d , then ι is not a firstappearing interior point. Since h ι ′ , r i > r ∈ Ψ r holds, we may show that h ι ′ , f k i > k = max { p V j ( ι ) : j ∈ [ n ] } . We see that k should be one of k , k and n .( k = k ) We have h ι ′ , f k i = h ι, f k i > k = k ) We see that p V k ( ι ) = p V k ( ι ) = p V k ( ι ). Remark p V k ( ι ) = r k + r ′ k . Then h ι ′ , f k i = (cid:18) X i ∈ [ n ] \{ k } p V i (cid:19) ( ι ) − − p V k ( ι )= (cid:18) X i ∈ [ n ] \{ k ,k ,k } p V i (cid:19) ( ι ) + ( r k −
1) + ( r ′ k − > . ( k = n ) If r ′ n ≥ r ′ k , then we have n = k or k , so we can deduce the case k = k or k .Hence, we may assume that r ′ n < r ′ k . Then we see that h ι ′ , f n i = (cid:18) X i ∈ [ n − p V i (cid:19) ( ι ) − − p V n ( ι ) ≥ (cid:18) X i ∈ [ n − r i − r n (cid:19) + ( r ′ k −
1) + ( r ′ k − r ′ n − > . Therefore, we obtain the desired result. (cid:3)
Lemma 4.7.
Let r , . . . , r n be positive integers with ≤ r ≤ · · · ≤ r n , let d = P i ∈ [ n ] r i be even. Assume that n = 3 with r ≥ or n ≥ . Then n and ( r , . . . , r n ) satisfy one f the conditions (iii)—(vi) in Theorem 1.3 if and only if r i ∈ { , d/ − } holds for any i ∈ [ n ] .Proof. Since “only if” part is easy to see, we prove “if” part.Assume that r i ∈ { , d/ − } holds for any i ∈ [ n ]. Note that d ≥ n by definition. Let α be the number of r i ’s with r i = d/ −
1. Then r = · · · = r n − α = 1. Thus, we have d = ( n − α ) + ( d/ − α .The case α = 0 is nothing but the case (vi). Note that n should be even by d = n . If α = 1, then d = ( n −
1) + ( d/ −
1) holds by definition, which implies that d = 2 n − α ≥
2. When n ≥
5, we see that d = ( d/ − α + n ≥ d − n > d , acontradiction. Hence, n = 3 or n = 4. • Let n = 3. Since we assume r ≥
2, we may discuss the case α = 3. Then d = 3( d/ −
1) holds, i.e., d = 6. Hence, we obtain that r = r = r = 2, which isthe case (iii). • Let n = 4. If α = 2, then we see that r (= r ) can be arbitrary, which is the case(iv). If α = 3 (resp. α = 4), then d = 1 + 3( d/ −
1) (resp. d = 4( d/ − d = 4. Hence, we obtain that r = · · · = r = 1, which is included in (iv). (cid:3) Now, we are ready to give a proof of Theorem 1.3. Since the case where n = 2 or n = 3with r = 1 has been already done in Section 3, we assume that n = 3 with r ≥ n ≥
4. Under this assumption, it suffices to prove that k [ K r ,...,r n ] is almost Gorensteinif and only if d is even and r i ∈ { , d/ − } for any i ∈ [ n ] by Lemma 4.7. Then wemay assume the case (A) by Corollary 4.3. Remark that 1 ≤ r k ≤ d/ − e ( C ) = X k ∈ [ n ] e k ( C ) and µ ( C ) = X k ∈ [ n ] µ k ( C ), where e k ( C ) := (cid:16) d − r k − (cid:17)(cid:18) d − r k − (cid:19) and µ k ( C ) = X j ∈ [ d − r k − (cid:18) r k − jr k − (cid:19) for each k ∈ [ n ] . • If r k = 1, then we have e k ( C ) = µ k ( C ) = d/ − • If r k = d/ −
1, we have e k ( C ) = µ k ( C ) = 0. • If 1 < r k < d/ −
1, then we have µ k ( C ) ≤ X j ∈ [ d − r k − (cid:18) d − r k − r k − (cid:19) = (cid:18) d − r k − (cid:19) (cid:18) d − r k − r k − (cid:19) < e k ( C ) . Hence, if there is k ∈ [ n ] with 1 < r k < d/ −
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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] (K. Matsushita) 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]@ist.osaka-u.ac.jp