Systems with the integer rounding property in normal monomial subrings

MATHEMATICAL SCIENCES

Systems with the integer rounding property in normal monomial subrings

Luis A. Dupont^I; Carlos Rentería-Márquez^II; Rafael H. Villarreal^I

^IDepartamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14–740, 07000 Mexico City, D.F

^IIDepartamento de Matemáticas, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Unidad Adolfo López Mateos, Edif. 9, 07738 México City, D.F

^{Correspondence to}

ABSTRACT

Let C be a clutter and let A be its incidence matrix. If the linear system x > 0; x A < 1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the clutter is a connected graph, we describe when the aforementioned linear system has the integer rounding property in combinatorial and algebraic terms using graph theory and the theory of Rees algebras. As a consequence we show that the extended Rees algebra of the edge ideal of a bipartite graph is Gorenstein if and only if the graph is unmixed.

Key words: canonical module, a-invariant, normal ideal, perfect graph, maximal cliques, Rees algebra, Ehrhart ring, integer rounding property.

RESUMO

Seja C uma desordem (família de Sperner) e seja A sua matriz de incidência. Se o sistema linear x > 0; x A < 1 tem a propriedade do arredondamento inteiro, fornecemos a descrição do módulo canônico e do a-invariante de certos subaneis monomiais associados a C. Se a desordem é um grafo conexo, descreve-se quando o supra-mencionado sistema linear tem a propriedade do arredondamento inteiro em termos combinatórios e algébricos, usando a teoria dos grafos e a teoria das álgebras de Rees. Como consequência, mostra-se que a álgebra de Rees estendida do ideal de arestas de um grafo bipartido é um anel de Gorenstein se e somente se o grafo é de altura pura.

Palavras-chave: módulo canônico, a-invariante, ideal normal, grafo perfeito, cliques maximais, álgebra de Rees, anel de Erhart, propriedade do arredondamento inteiro.

INTRODUCTION

A clutter C with finite vertex set X is a family of subsets of X called edges, and none of which is included in another. The set of vertices and edges of C are denoted by V(C) and E(C), respectively. A basic example of a clutter is a graph.

Let C be a clutter with finite vertex set X = {x₁,..., x_n}. We shall always assume that C has no isolated vertices, i.e., each vertex occurs in at least one edge. Let f₁,..., f_q be the edges of C, and let be the characteristic vector of f_k, where e_i is the ith unit vector in . The incidence matrix A of C is the matrix with column vectors . If and are vectors in , then means that for all . Thus, means that for all . The system has the integer rounding property if

for each integral vector for which is finite. Here denotes the vector with all its entries equal to , and denotes the standard inner product. For a thorough study of this property see (Schrijver 1986).

Let be a polynomial ring over a field , and let be the set of all integral vectors such that for some . We will examine the integer rounding property using the monomial subring:

where is a new variable. As usual, we use the notation , where . The subring is called normal if is integrally closed, i.e., , where is the integral closure of in its field of fractions; see (Vasconcelos 2005).

The contents of this paper are as follows. One of the results in (Brennan et al. 2008) shows that is normal if and only if the system has the integer rounding property (see Theorem 2.1). As a consequence, we show that if all edges of C have the same number of elements and either linear system or has the integer rounding property, then, the subring is normal (see Corollary 2.4).

Let be a connected graph and let be its incidence matrix. The main results of Section 3 show that the following conditions are equivalent (see Theorems 3.2 and 3.3):

The most interesting part of this result is the equivalence between (a), which is a linear optimization property, and (f), which is a graph theoretical property. Edge ideals are defined in Section 2. We prove that the ring in (e) is isomorphic to the extended Rees algebra of the edge ideal of (see Proposition 3.1). If is bipartite and is its edge ideal, we are able to prove that the extended Rees algebra of is a Gorenstein standard -algebra if and only if is unmixed (see Corollary 4.3). If we work in the more general context of clutters, none of the conditions (a) to (e) are equivalent. Some of these conditions are equivalent under certain assumptions (Gitler et al. 2009).

In Section 4 we introduce the canonical module and the -invariant of . This invariant plays a key role in the theory of Hilbert fun-ctions (Bruns and Herzog 1997). There are some methods, based on combinatorial optimization, that have been used to study canonical modules of edge subrings of bipartite graphs (Valencia and Villarreal 2003). Our approach to study canonical modules is inspired by these methods. If is a normal domain, we express the canonical module of and its -invariant in terms of the vertices of the polytope

(see Theorem 4.1). We are able to give an explicit description of the canonical module of and its -invariant when C is the clutter of maximal cliques of a perfect graph (Theorem 4.2).

For unexplained terminology and notation on commutative algebra and integer programming, we refer to (Bruns and Herzog 1997, Vasconcelos 2005, Villarreal 2001) and (Schrijver 1986), respectively.

2 INTEGER ROUNDING AND NORMALITY

We continue using the definitions and terms from the Introduction. In what follows, denotes the set of non-negative integers and denotes the set of non-negative real numbers. Let . The convex hull of is denoted by , and the cone generated by is denoted by .

Theorem (Brennan et al. 2008). round-up-char Let C be a clutter and let be the columns of the incidence matrix A of C . If is the set of all such that for some , then the system has the integer rounding property if and only if the subring is normal.

Next we give an application of this result, but first we need to introduce some more terminology and notation. We have already defined in the Introduction when the linear system has the integer rounding property. The following is a dual notion.

Definition . Let A be a matrix with entries in . The system has the integer rounding property if

for each integral vector for which the right hand side is finite.

Let be a set of points in , let be the convex hull of , and let be a polynomial ring over a field . The Ehrhart ring of the lattice polytope is the monomial subring

where is a new variable. A nice property of is that it is always a normal domain (Bruns and Herzog 1997). Let C be a clutter with vertex set and edge set . For use below recall that C is called uniform if all its edges have the same number of elements. The edge ideal of C , denoted by , is the ideal of generated by all monomials such that . The Rees algebra of , denoted by , is given by

see (Vasconcelos 2005, Chapter 1) for a nice presentation of Rees algebras.

The next result holds for arbitrary monomial ideals (not necessarily square-free). This is the only place in the paper where a result is stated for arbitrary monomial ideals.

THEOREM (Dupont and Villarreal 2010).poset-main-normal Let be a monomial ideal, and letA be the matrix with column vectors . Then, the system has the integer rounding property if and only if is normal.

COROLLARY march6-08 Let C be a uniform clutter, let A be its incidence matrix, and let be the columns of A . If either system or has the integer rounding property and , then

In general, the subring is contained in . Assume that has the integer rounding property, and that every edge of C has elements. Let be the set of all such that for some . Then by Theorem 2.1, the subring is normal. Using that is the set of , with , it is not hard to see that is contained in .

Assume that has the integer rounding property. Let be the edge ideal of C , and let be its Rees algebra. By Theorem 2.3, is a normal domain. Since the clutter C is uniform, the required equality follows at once (Escobar et al. 2003, Theorem 3.15).

The converse of Corollary 2.4 fails as the following example shows.

EXAMPLE Let C be the uniform clutter with vertex set and edge set

The characteristic vectors of the edges of C are

Let A be the incidence matrix of C with column vectors , and let be the convex hull of . It is not hard to verify that the set

is a Hilbert basis in the sense of (Schrijver 1986). Therefore, we have the equality

Using Theorem 2.3 and (Brennan et al. 2008, Theorem 2.12) it is seen that none of the two systems and have the integer rounding property.

3 INCIDENCE MATRICES OF GRAPHS

Let be a connected graph with vertex set , and let be the column vectors of the incidence matrix of . The main result here is a purely combinatorial description of the integer rounding property of the system . Other equivalent algebraic conditions of this property will be presented.

Let be a polynomial ring over a field , and let be the edge ideal of . Recall that the extended Rees algebra of is the subring

where is the Rees algebra of . Rees algebras of edge ideals of graphs were first studied in (Simis et al. 1994).

PROPOSITION

.

PROOF. We set . Note that and are both integral domains of the same Krull dimension; this follows from the dimension formula given in (Sturmfels 1996,Lemma 4.2). Thus, it suffices to prove that there is an epimorphism of -algebras.

Let be a new set of variables, and let , be the maps of -algebras defined by the diagram -10mm

To complete the proof, we will show that there is an epimorphism of -algebras that makes this diagram commutative, i.e., . To show the existence of , we need only to show the inclusion . As being a toric ideal is generated by binomials (Sturmfels 1996), it suffices to prove that any binomial of belongs to . Let

be a binomial in . Then,

Taking degrees in and , we obtain

Thus , and we obtain the equality

i.e., , as required.

We come to the main result of this section.

Theorem round-for-conn-graphs Let be a connected graph and let A be its incidence matrix. Then, the system

has the integer rounding property if and only if the induced subgraph of the vertices of any two vertex disjoint odd cycles of is connected.

PROOF. Let be the column vectors of A . According to (Simis et al. 1998, Theorem 1.1, cf. Villarreal 2005, Corollary 3.10), the subring is normal if and only if any two vertex disjoint odd cycles of can be connected by at least one edge of . Thus, we need only to show that is normal if and only the system has the integer rounding property. Let be the edge ideal of . Since is connected, the subring is normal if and only if the Rees algebra of is normal (Simis et al. 1998, Corollary 2.8). By a result of (Herzog et al. 1991), is normal if and only if is normal. By Proposition 3.1, is normal if and only if the subring

is normal. Thus, we can apply Theorem 2.1 to conclude that is normal if and only if the system has the integer rounding property.

Theorem roundup-iff-rounddown Let be a connected graph and let A be its incidence matrix. Then, the system has the integer rounding property if and only if any of the following equivalent conditions hold

PROOF. According to THEOREM 2.3, the system has the integer rounding property if and only if the Rees algebra is normal. Thus, the result follows from the proof of THEOREM 3.2.

4 THE CANONICAL MODULE AND THE a-INVARIANT

In this section, we give a description of the canonical module and the -invariant for subrings arising from systems with the integer rounding property.

Let C be a clutter with vertex set , and let be the columns of the incidence matrix of C . For use below, consider the set of all such that for some . Let be a polynomial ring over a field and let

be the subring of generated by , where is a new variable. As lies in the hyperplane for all , is a standard -algebra. Thus, a monomial in has degree . In what follows, we assume that has this gading. Recall that the -invariant of , denoted , is the degree as a rational function of the Hilbert series of , see for instance (Vilrlarreal 2001, p. 99). If is Cohen-Macaulay and is the canonical module of , then

see (Bruns and Herzog 1997, p. 141) and (Villarreal 2001, Proposition 4.2.3). This formula applies if is normal because normal monomial subrings are Cohen-Macaulay (Hochster 1972). If is normal, then by a formula of Danilov-Stanley, see (Bruns and Herzog 1997, Theorem 6.3.5) and (Danilov 1978), the canonical module of is the ideal given by

where and is the interior of relative to , the affine hull of . In our case, .

The next theorem complements a result of (Brennan et al. 2008). In loc. cit. a somewhat different expression for the canonical module and -invariant are shown. Our expressions are simpler because they only involve the vertices of a certain polytope, while in (Brennan et al. 2008) some other parameters are involved.

THEOREM 4.1 can-mod-intr Let C be a clutter with incidence matrix A, let be the columns of A, and let be the set of all such that for some . If the system has the integer rounding property and are the non-zero vertices of , then the subring is normal, the canonical module of is given by

and the -invariant of is equal to . Here .

Note that in Eq. (3) we regard the ’s and as column vectors. The normality of follows from Theorem 2.1. Let and let be its antiblocking polyhedron

By the finite basis theorem (Schrijver 1986), we can write

where are the vertices of and . Notice that the vertices of are in . From Eq. (4), we readily get the equality

Using Eq. (4) again and noticing that for all , we get

Hence, using this equality and (Schrijver 1986, Theorem 9.4), we obtain

By (Fulkerson 1971, Theorem 8), we have the equality

Therefore, using Eqs. (5) and (6), we conclude the following duality:

We set . Note that . From Eq. (7), it is seen that

Here denotes the closed halfspace and stands for the hyperplane through the origin with normal vector . Notice that

are proper faces of . Hence, from Eq. (8), we get that a vector , with , , is in the relative interior of if and only if the entries of are positive and for all . Thus, the required expression for , i.e., Eq. (3), follows using the normality of and the Danilov-Stanley formula given in Eq. (2).

It remains to prove the formula for , the -invariant of . Consider the vector , where . Using Eq. (3), it is not hard to see (by direct substitution of ) that the monomial is in . Thus, from Eq. (1), we get . Conversely, if the monomial is in , then again from Eq. (3) we get for all and for all , where . Hence,

Since is an integer, we obtain for all . Therefore, , i.e., . As was an arbitrary monomial in , by the formula for the -invariant of given in Eq. (1), we obtain that . Altogether one has , as required.

MONOMIAL SUBRINGS OF CLIQUES OF PERFECT GRAPHS

Let be a set of vertices of a graph , the induced subgraph is the maximal subgraph of with vertex set . A clique of a graph is a subset of the set of vertices that induces a complete subgraph. Let be a graph with vertex set . A colouring of the vertices of is an assignment of colours to the vertices of in such a way that adjacent vertices have distinct colours. The chromatic number of is the minimal number of colours in a colouring of . A graph is perfect if, for every induced subgraph , the chromatic number of equals the size of the largest complete subgraph of . Let be a subset of the vertices of . The set is called independent if no two vertices of are adjacent.

For use below we consider the empty set as a clique whose vertex set is empty. The support of a monomial is given by . Note that if and only if .

Theorem .perfect-canon-ainv Let be a perfect graph and let be the subring generated by all square-free monomials such that is a clique of . Then, the canonical module of is given by

where are the characteristic vectors of the maximal independent sets of , and the -invariant of is equal to .

PROOF. Let be the set of characteristic vectors of the maximal cliques of . Note that is the set of all such that for some . Since is a perfect graph, by (Korte and Vygen 2000, Theorem 16.14) we have the equality

where and are the characteristic vectors of the independent sets of . We may assume that correspond to the maximal independent sets of . Furthermore, since has only integral vertices, by a result of (Lovász 1972), the system is totally dual integral, i.e., the minimum in the LP-duality equation

has an integral optimum solution for each integral vector with finite minimum. In particular the system satisfies the integer rounding property. Therefore, the result follows readily from Theorem 4.1.

For use below recall that a graph is called unmixed if all maximal independent sets of have the same cardinality. Unmixed bipartite graphs have been nicely characterized in (Villarreal 2007).

Corollary .extended-gorenstein Let be a connected bipartite graph, and let be its edge ideal. Then, the extended Rees algebra is a Gorenstein standard -algebra if and only if is unmixed.

Let be the canonical module of . Recall that is Gorenstein if and only if is a principal ideal (Bruns and Herzog 1997). Since any bipartite graph is a perfect graph, the result follows using Proposition 3.1 together with the description of the canonical module given in Theorem 4.2.

ACKNOWLEDGMENTS

This work was partially supported by CONACyT Grant 49251-F and SNI. The second author was supported by COFAA-IPN.

Manuscript received on September 9, 2009; accepted for publication on May 5, 2010

AMS Classification: Primary 13B22; Secondary 13H10, 13F20, 52B20.

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Publication Dates

• Publication in this collection
  28 Feb 2011
• Date of issue
  Dec 2010

History

• Received
  09 Sept 2009
• Accepted
  05 May 2010