A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the h-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial d-polytopes, that is simplicial d-polytopes whose underlying graphs are d-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. The aim of this article is to prove this conjecture of Klee and Novik. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes and thereby obtain new inequalities for their h-numbers.

In this paper, we show that two balanced triangulations of a closed surface are not necessarily connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2-sphere are connected by a sequence of pentagon contractions and their inverses if none of them are the octahedral sphere.

We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension d >= 4, if Delta is a tight connected closed homology d-manifold whose ith homology vanishes for 1 < i < d - 1, then Delta is a stacked triangulation of a manifold.These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.

We resolve a conjecture of Kalai asserting that the g(2)-number of any (finite) simplicial complex Delta that represents a normal pseudomanifold of dimension d >= 3 is at least as large as ((d+2)(2))m(Delta), where m(Delta) denotes the minimum number of generators of the fundamental group of Delta. Furthermore, we prove that a weaker bound, h(2)(Delta) >= ((d+1)(2))m(Delta), applies to any d-dimensional pure simplicial poset Delta all of whose faces of co-dimension >= 2 have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset Psi all of whose vertex links satisfy Serre's condition (S-r), we establish lower bounds on h(1)(Psi) ,..., h(r)(Psi) in terms of the mu-numbers introduced by Bagchi and Datta.

We study face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. Specifically, we establish a sharp lower bound on the number of interior edges of a simplicial normal pseudomanifold with boundary in terms of the number of interior vertices and relative Betti numbers. Moreover, for triangulations of manifolds with boundary all of whose vertex links have the weak Lefschetz property, we extend this result to sharp lower bounds on the number of higher-dimensional interior faces. Along the way we develop a version of Bagchi and Datta's sand mu-numbers for the case of relative simplicial complexes and prove stronger versions of the above statements with the Betti numbers replaced by the mu-numbers. Our results provide natural generalizations of known theorems and conjectures for closed manifolds and appear to be new even for the case of a ball.

Let 1 be a triangulated homology ball whose boundary complex is partial derivative Delta. A result of Hochster asserts that the canonical module of the Stanley-Reisner ring F[Delta] of Delta is isomorphic to the Stanley-Reisner module F[Delta, partial derivative Delta] of the pair (Delta, partial derivative Delta]. This result implies that an Artinian reduction of F[Delta, partial derivative Delta] is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of F[Delta]. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the h ''-numbers of Buchsbaum complexes and use it to prove the monotonicity of h ''-numbers for pairs of Buchsbaum complexes as well as the unimodality of h ''-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold g-conjecture.

In this paper, we study simplicial complexes whose Stanley-Reisner rings are almost Gorenstein and have a-invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen-Macaulay simplicial complexes. A d-dimensional simplicial complex Delta is said to be uniformly Cohen-Macaulay if it is Cohen-Macaulay and, for any facet F of Delta, the simplicial complex Delta \ {F} is Cohen-Macaulay of dimension d. We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen-Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension <= 2. (C) 2016 Elsevier Inc. All rights reserved.

Let I and J be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of I and g(J), where g is a general change of coordinates. Our main result gives a generalization of Green's hyperplane section theorem.

In this paper, we study the conjecture of Kuhnel and Lutz, who state that a combinatorial triangulation of the product of two spheres with is tight if and only if it has exactly vertices. To approach this conjecture, we use graded Betti numbers of Stanley-Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.

We study the Castelnuovo-Mumford regularity of the module of Koszul cycles Z(t)(I, M) of a homogeneous ideal I in a polynomial ring S with respect to a graded module M in the homological position t is an element of N. Under mild assumptions on the base field we prove that reg Z(t)(I, S) is a subadditive function of t when dim S/I = 0. For Borel-fixed ideals I, J we prove that reg Z(t)(I, S/J) <= t(1 + reg I) + reg S/J, a result already announced by Bruns, Conca and Romer.

A Bott manifold is the total space of some iterated CP1 -bundles over a point. We prove that any graded ring isomorphism between the cohomology rings of two Bott manifolds preserves their Pontrjagin classes. Moreover, we prove that such an isomorphism is induced from a diffeomorphism if the Bott manifolds are Z/2 -trivial, where a Bott manifold is called Z/2 -trivial if its cohomology ring with Z/2 -coefficients is isomorphic to that of a product of copies of CP1.

In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen-Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley-Reisner ring of the barycentric subdivision of an odd dimensional Cohen Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function. (C) 2014 Elsevier Inc. All rights reserved.

We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f-vectors of order complexes triangulating rational homology 3-spheres, and show they are also the characterization of the flag f-vectors of the subfamily of regular CW-complexes homeomorphic to the 3-sphere. As a corollary, we characterize the f-vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization gives rise to a speculated intimate connection between the f-vectors of flag homology spheres and the f-vectors of Gorenstein* order complexes.

The notion of r-stackedness for simplicial polytopes was introduced by McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this paper, we define the r-stackedness for triangulated homology manifolds and study its basic properties. In addition, we find a new necessary condition for face vectors of triangulated manifolds when all the vertex links are polytopal.

A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal is said to be -full if for some , where is the graded maximal ideal of . It was proved by one of the authors that -full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not -full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.

In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the characterization of face vectors of simplicial posets whose geometric realizations are odd-dimensional manifolds without boundary.

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h (0), h (1), aEuro broken vertical bar, h (d) ) satisfies . Moreover, if h (r-1) = h (r) for some then P can be triangulated without introducing simplices of dimension a parts per thousand currency signd - r.
The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.

We show that the Castelnuovo-Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices. © 2013 Rocky Mountain Mathematics Consortium.

We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.

A simplicial cell ball is a simplicial poset whose geometric realization is homeomorphic to a ball. Recently, Samuel Kolins gave a series of necessary conditions and sufficient conditions on h-vectors of simplicial cell balls, and characterized them up to dimension 6. In this paper, we extend Kolins' results. We characterize all possible h-vectors of simplicial cell balls in arbitrary dimension. © 2012 American Mathematical Society.

We show that the Hilbert scheme, that parameterizes all ideals with the same Hilbert function over a Clements-Lindstrom ring W, is connected. More precisely, we prove that every graded ideal is connected by a sequence of deformations to the lex-plus-powers ideal with the same Hilbert function. This is an analogue of Hartshorne's theorem that Grothendieck's Hilbert scheme is connected. We also prove a conjecture by Gasharov, Hibi, and Peeva that the lex ideal attains maximal Betti numbers among all graded ideals in W with a fixed Hilbert function.

We show that the gamma-vector of the order complex of any polytope is the f-vector of a balanced simplicial complex. This is done by proving this statement for a subclass of Stanley's S-shellable CW-spheres which includes all polytopes. The proof shows that certain parts of the cd-index, when specializing c = 1 and considering the resulted polynomial in d, are the f-polynomials of simplicial complexes that can be colored with "few" colors. We conjecture that the cd-index of a regular CW-sphere is itself the flag f-vector of a colored simplicial complex in a certain sense.

In this paper, we investigate Lefschetz properties of Veronese subalgebras. We show that, for a sufficiently large r, the rth Veronese subalgebra of a Cohen-Macaulay standard graded K-algebra has properties similar to the weak and strong Lefschetz properties, which we call the 'quasi-weak' and 'almost strong' Lefschetz properties. By using this result, we obtain new results on h- and g-polynomials of Veronese subalgebras.

In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex Delta on the vertex set V with Delta not equal 2(v) the deleted join of Delta with its Alexander dual Delta(v) is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable. which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that these spheres are related to polarizations and Alexander duality for monomial ideals which appear in commutative algebra theory. (C) 2011 Elsevier Inc. All rights reserved.

We prove Evans' Lex-Plus-Powers Conjecture for ideals containing a monomial regular sequence. (C) 2010 Elsevier Inc. All rights reserved.

In this paper we study the minimal free resolution of lex-ideals over a Koszul toric ring. In particular we study in which toric ring R all lex-ideals are compare twis linear. We give a certain necessity aid sufficiently condition for this property and show that lex-ideals in a strongly Koszul toric ring are componentwise linear. In addition, it is shown that, in the toric ring arising from the Segre product P(1) x ... x P(1), every Hilbert function of graded ideal is attained by a lex-ideals and that lex-ideals have the greatest graded Betti numbers among all ideals having the same Hilbert function.

Macaulay's Theorem (Macaulay in Proc. Lond Math Soc 26:531-555, 1927) characterizes the Hilbert functions of graded ideals in a polynomial ring over a field. We characterize the Hilbert functions of graded ideals in a Veronese ring R (the coordinate ring of a Veronese embedding of P (r-1)). We also prove that the Hilbert scheme, which parametrizes all graded ideals in R with a fixed Hilbert function, is connected; this is an analogue of Hartshorne's Theorem (Hartshorne in Math. IHES 29:5-48, 1966) that Hilbert schemes over a polynomial ring are connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the same Hilbert function.

We show that the Castelnuovo-Mumford regularity of the canonical or a deficiency module of the quotient of a polynomial ring by a monomial ideal is bounded by its dimension.

We prove that Gotzmann's Persistence Theorem holds over every Clements-Lindstrom ring. We also construct the infinite minimal free resolution of a square-free Borel ideal over such a ring. (C) 2010 Elsevier Inc. All rights reserved.

Let A=K[x (1),aEuro broken vertical bar,x (n) ] be a polynomial ring over a field K and M a monomial ideal of A. The quotient ring R=A/M is said to be Macaulay-Lex if every Hilbert function of a homogeneous ideal of R is attained by a lex ideal. In this paper, we introduce some new Macaulay-Lex rings and study the Betti numbers of lex ideals of those rings. In particular, we prove a refinement of the Frankl-Furedi-Kalai Theorem which characterizes the face vectors of colored complexes. Additionally, we disprove a conjecture of Mermin and Peeva that lex-plus-M ideals have maximal Betti numbers when A/M is Macaulay-Lex.

Let G he a chordal graph and I(G) its edge ideal. Let beta(I(G)) = (beta(0), beta(1).....beta(p)) denote the Betti sequence of I(G), where beta(1) stands for the ith total Betti number of I(G) and where p is the projective dimension of I(G). It will be shown that there exists a simplicial complex Delta of dimension p whose f-vector f(Delta) = (f(0). f(1).....f(p)) coincides with beta(I(G)). (C) 2010 Elsevier Inc, All rights reserved.

Recently, Nevo introduced the notion of strongly edge decomposable spheres. in this paper, we characterize algebraic shifted complexes of those spheres. Algebraically, this result yields the characterization of the generic initial ideal of the Stanley-Reisner ideal of Gorenstein* complexes having the strong Lefschetz property in characteristic 0. (C) 2009 Elsevier Inc. All rights reserved.

We study face vectors of barycentric subdivisions of simplicial homology manifolds. Recently, Kubitzke and Nevo proved that the g-vector of the barycentric subdivision of a Cohen-Macaulay simplicial complex is an M-vector, which in particular proves the g-conjecture for barycentric subdivisions of simplicial homology spheres. In this paper, we prove an analogue of this result for Buchsbaum simplicial posets and simplicial homology manifolds.

We study h-vectors of simplicial complexes which satisfy Serre's condition (S(r)). Let r be a positive integer. We say that a simplicial complex Delta satisfies Serre's condition (S(r)) if (H) over tilde (i)(lk(Delta)(F); K) = 0 for all F is an element of Delta and for all i < min {r-1, dim lk(Delta)(F)}, where lk(Delta) (F) is the link of Delta with respect to F and where <(H)over tilde>(i) (Delta; K) is the reduced homology groups of Delta over a field K. The main result of this paper is that if Delta satisfies Serre's condition (S(r)) then (i) h(k) (Delta) is non-negative for k = 0, 1, ... , r and (ii) Sigma(k >= r) h(k) (Delta) is non-negative.

In this paper, we characterize all possible h-vectors of 2-dimensional Buchsbaum simplicial complexes.

Let S = K[x(1),...,x(n),] denote the polynomial ring in n variables over a field K with each deg x(i) = 1. Let Delta be a simplicial complex on [n] = {1,...,n} and I(Delta) subset of S its Stanley-Reisner ideal. We write Delta(e) for the exterior algebraic shifted complex of Delta and Delta(c) for a combinatorial shifted complex of Delta. Let beta(ii+j) (I(Delta)) = dim(K) Tor(i) (K, I(Delta))(i+j) denote the graded Betti numbers of I(Delta). In the present paper it will be proved that (i) beta(ii+j) (I(Delta e)) <= beta(ii+j) (I(Delta c)) for all i and j, where the base field is infinite, and (ii) beta(ii+j) (I(Delta)) <= beta(ii+j) (I(Delta c)) for all i and j, where the base field is arbitrary. Thus in particular one has beta(ii+j) (I(Delta)) <= beta(ii+j) (I(Delta lex)) for all i and j, where Delta(lex) is the unique lexsegment simplicial complex with the same f-vector as Delta and where the base field is arbitrary.

Let A = K[x(1),..., x (n)] denote the polynomial ring in n variables over a field K. We will classify all the Gotzmann ideals of A with at most n generators. In addition, we will study Hilbert functions H for which all homogeneous ideals of A with the Hilbert function H have the same graded Betti numbers. These Hilbert functions will be called inflexible Hilbert functions. We introduce the notion of segmentwise critical Hilbert functions and show that segmentwise critical Hilbert functions are inflexible.

Let S = K[x(1), ..., x(n)] be a standard graded polynomial ring over a field K. In this paper, we show that the lex-plus-powers ideal has the largest graded Betti numbers among all Borel-plus-powers monomial ideals with the same Hilbert function. In addition p in the case of characteristic 0, by using this result, we prove the lex-plus-powers conjecture for graded ideals containing x(1)(p), ..., x(n)(p), where p is a prime number. (C) 2007 Elsevier B.V. All rights reserved.

Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers beta(S)(ii+k)(S/>I) = beta(S)(ii+k)(S/Gin(I)) for all q >= i, where I subset of S is a graded ideal. Second, we show that if beta(E)(ii+k)(E/I) = beta(E)(ii+k)(E/Gin(I)) for all q >= 1, where I subset of E is a graded ideal. In addition, it will be shown that the graded Betti numbers beta(R)(ii+k)(R/I) = beta(R)(ii+k)(R/Gin(I)) for all i >= 1 if and only if I-< k > and I < k+1 > have a linear resolution. Here (< d >)is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra.

In this paper, we will show that the color-squarefree operation does not change the graded Betti numbers of strongly color-stable ideals. In addition, we will give an example of a nonpure balanced complex which shows that colored algebraic shifting, which was introduced by Babson and Novik, does not always preserve the dimension of reduced homology groups of balanced simplicial complexes.

We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals.

Gotzmann proved the persistence for minimal growth of Hilbert functions of homogeneous ideals. His theorem is called Gotzmann's persistence theorem. In this paper, based on the combinatorics of binomial coefficients, a simple combinatorial proof of Gotzmann's persistence theorem in the special case of monomial ideals is given. (c) 2006 Elsevier Ltd. All rights reserved.

Let A = K[x(1),...,x(n)] denote the polynomial ring in n variables over a field K with each deg x(i) = 1. Let I be a homogeneous ideal of A with I not equal A and H-A/I the Hilbert function of the quotient algebra A/I. Given a numerical function H:N -> N satisfying H = H-A/I for some homogeneous ideal I of A, we write A(H) for the set of those integers 0 <= r <= n such that there exists a homogeneous ideal I of A with H-A/I=H and with depth A/I = r. It will be proved that one has either AH = {0, 1,...,b} for some 0 <= b <= n or |A(H)| = 1.

A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal-Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented.

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d, where d is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let p >= 0 and d > 0 be integers. If the base field is a field of characteristic 0 and there is a graded ideal I whose projective dimension proj dirn(I) is smaller than or equal to p and whose regularity reg(I) is smaller than or equal to d. then there exists a monomial ideal L having the maximal graded Belli numbers among graded ideals J which have the same Hilbert function as 1 and which satisfy proj dim(J) <= p and reg(J) <=, d. We also prove the same fact for squarefree monomial ideals. The main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals. (c) 2007 Elsevier Inc. All rights reserved.

In 1988 Kalai constructed a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjecture about generic initial ideals of Stanley-Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai's conjecture, based on the fact that every squeezed (d-1)-sphere is the boundary of a certain d-ball, called a squeezed d-ball, generic initial ideals of Stanley-Reisner ideals of squeezed balls will be determined. In addition, generic initial ideals of exterior face ideals of squeezed balls are determined. On the other hand, we study the squeezing operation, which assigns to each Gorenstein* complex Gamma having the weak Lefschetz property a squeezed sphere Sq(Gamma), and show that this operation increases graded Betti numbers. (c) 2007 Elsevier Inc. All rights reserved.

In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let sigma and tau be simplicial complexes and sigma * tau be their join. Let J(sigma) be the exterior face ideal of sigma and Delta(sigma) the exterior algebraic shifted complex of sigma. Assume that sigma * tau is a simplicial complex on [n] = {1, 2,..., n}. For any d-subset S subset of [n], let m <=(rev) s (sigma) denote the number of d-subsets R is an element of sigma which are equal to or smaller than S with respect to the reverse lexicographic order. We will prove that m <=(rev)s (Delta (sigma*tau))> m <= S-rev (Delta (Delta(sigma)* Delta (tau))) for all S subset of[n]. To prove this fact, we also prove that m <=(rev)s(Delta(sigma))>-m <= S-rev(Delta(Delta(phi)(sigma))) for all S subset of[n] and for all nonsingular matrices phi, where Delta(phi)(sigma) is the simplicial complex defined by J(Delta phi)(sigma)=in(phi(J sigma)).

Gil Kalai introduced the shifting-theoretic upper bound relation as a method to generalize the g-theorem for simplicial spheres by using algebraic shifting. We will study the connection between the shifting-theoretic upper bound relation and combinatorial shifting. Also, we will compute the exterior algebraic shifted complex of the boundary complex of the cyclic d-polytope as well as of a stacked d-polytope. It will turn out that, in both cases, the exterior algebraic shifted complex coincides with the symmetric algebraic shifted complex. (C) 2006 Elsevier B.V. All rights reserved.

Given the f-vector f = (f(0), f(1),...) of a Cohen-Macaulay simplicial complex, it will be proved that there exists a shellable simplicial complex Delta(f) with f (Delta(f)) = f such that, for any Cohen-Macaulay simplicial complex A with f(Delta) = f, one has beta(ij)(I Delta) <= beta(ij)(I-Delta f) for all i and j, where f(Delta) is the f-vector of A and where beta(ij)(IA) are graded Betti numbers of the Stanley-Reisner ideal I-Delta of Delta.

In the present article, for bipartite graphs and chordal graphs, their exterior algebraic shifted graph and their symmetric algebraic shifted graph are studied. First, we will determine the symmetric algebraic shifted graph of complete bipartite graphs. It turns out that for a >= 3 and b >= 3, the exterior algebraic shifted graph of the complete bipartite graph K-a,K-b of size a, b is different from the symmetric algebraic shifted graph of K-a,K-b. Second, we will show that the exterior algebraic shifted graph of any chordal graph G coincides with the symmetric algebraic shifted graph of G. In addition, it will be shown that the exterior algebraic shifted graph of any chordal graph G is equal to some combinatorial shifted graph of G.

Let A = K[x(1),..., x(n)] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg x(i) = 1. Given arbitrary integers i and j with 2 <= i <= n and 3 <= j <= n, we will construct a monomial ideal I subset of A such that (i) beta k(I) < beta(k)(Gin(I)) for all k < i, (ii) beta(i) (I) = beta(i) (Gin (I)), (iii) beta(l) (Gin (I)) < beta(l) (Lex (I)) for all l < j and (iv) beta(j) (Gin (I)) = beta(j) (Lex (I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x(1) >... > x(n) and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.