One of the most challenging problems in combinatorial commutative algebra is to give in combinatorial terms a
characterization of those monomial ideals which have a linear resolution independent of the characteristic of the
base field. In trying to solve this problem one may, by using polarization, focus on squarefree monomial ideals.
Squarefree monomial ideals generated in degree 2 may be interpreted as edge ideals of graphs. In this particular
case, the famous theorem of Fröberg [1] gives a complete answer to the above problem. Fröberg's theorem says
that the edge ideal I(G) of a graph G has a linear resolution if and only if the complementary graph of G is
chordal. In particular, for the edge ideal of a graph it does not depend on the characteristic of the base field
whether or not it has a linear resolution. One would like to have a similar theorem for squarefree monomial ideals
generated in degree d, where d is any integer greater or equal 2.
The d-uniform clutters generalize simple graphs, which are just the 2-uniform clutters. A d-uniform clutter C on the
vertex set [n] = {1, 2, ..., n } is nothing but a collection of d-subsets of [n]. The complementary clutter of C is the
set of all d-element subsets of [n] which do not belong to C. The circuit ideal of C is the ideal generated by the
monomials xF where F belongs to complementary clutter of C. The analogue of Fröberg's theorem would then say
that the circuit ideal of C has a linear resolution, independent of the characteristic of the base field, if and only if C
is chordal. But what does it mean that C is chordal? There have been several other attempts to define chordal
clutters, see [2-5]. Here we consider the concept of chordality of clutters as it was introduced in [6]. It was shown
in [7] that all previously defined chordal clutters are chordal in this new sense, and that the circuit ideal of a
chordal clutter has a linear resolution. However, the converse is still an open question. The definition of chordality
given in [6] imitates Dirac's characterization of chordal graphs. It is required that the clutter admits a simplicial
order. Roughly speaking, a simplicial order for a d-uniform clutter C is a sequence e= e1, ... , er of (d-1)-sets
such that for each i, the set ei has a clique as its closed neighborhood in C\e1\ ... \ er, where C\e1\ ... \ er is
obtained from C by 'deleting' e= e1, ... , ei-1. Furthermore, it is required that C\e1\ ... \ er =Ø. The precise
definition can be found here.
The cardinality of the open neighborhood belonging to ei, that is to say, the cardinality of the set of all elements c ϵ
[n] for which ei U c ϵ C, is denoted by Ni. Thus, with each simplicial order e of a chordal clutter there is
associated the multiset {N1, ... , Nr }. Simple examples show that a chordal clutter may have different simplicial
orders. However, as shown in [8, Corollary 2.2] that, all simplicial orders have the same multisets. Therefore we
can talk of the multiset of a chordal clutter. In fact it is shown in [8, Proposition 2.1] that, the f-vector of the clique
complex Δ(C) of a chordal clutter C is determined by the multiset of any of its simplicial orders. From this result it
can be also derived that the multiset of C determines the h-vector of Δ(C) as well as the Betti sequence of the
circuit ideal of C.. More important is the result, stated in [8, Theorem 3.3], which says that the Betti sequence of
any ideal with linear resolution is the Betti sequence of the circuit ideal of a chordal clutter. So numerically, the
circuit ideal of chordal clutters coincide with the ideals with linear resolution. This fact supports our expectation
that the circuit ideal of the chordal clutters are precisely the monomial ideals which admit a linear resolution,
independent of the characteristic of the base field.
Since each chordal clutter determines a multiset N={N1, ... , Nr }, and since this multiset determines all the
algebraic and homological data of the clutter, it is of interest to know which finite multisets of positive integers
occur as the multiset of a chordal clutter. With the multiset N we associate a sequence λ(N)=λ1(N), λ2(N), ...,
where λi(N) is the number of elements Nj with Nj = i. It is obvious that any finite multiset of positive integers is in
bijection to a sequence of integers λ1, λ2, ..., which is eventually constant zero. We denote by λ(C) the λ-
sequence of the multiset associated with C. It is shown in [8] that this sequence is intimately related to the Hilbert
function of a suitable K-algebra with dimK (Ri) =0 , for all i>n-d+2. By using this characterization, a strict upper
bound of By using this characterization a strict upper bound of λi(C) for each 0<i<n-d is given in [8, Corollary
4.3]. Also in [8, Proposition 4.5], a precise formula for the λ-sequence of the complete d-clutter on [n] is given.
A computer program has been provided here in order to check chordality of a given d-uniform clutter. A modified
version of this program, provides some numerical data about circuit ideal of chordal clutters. More precisely, this
program first checks whether a given d-uniform clutter C is chordal. Then, it computes the simplicial multiset of C,
its λ-sequence, the h-vector of Δ(C) and the Betti sequence of the ideal IΔ(C).
