Simplicial orders and chordality

Abstract

Chordal clutters in the sense of [14] and [3] are defined via simplicial orders. Their circuit ideal has a linear resolution, independent of the characteristic of the base field. We show that any Betti sequence of an ideal with linear resolution appears as the Betti sequence of the circuit ideal of such a chordal clutter. Associated with any simplicial order is a sequence of integers which we call the λ-sequence of the chordal clutter. All possible λ-sequences are characterized. They are intimately related to the Hilbert function of a suitable standard graded K-algebra attached to the chordal clutter. By the λ-sequence of a chordal clutter we determine other numerical invariants of the circuit ideal, such as the h-vector and the Betti numbers.2010 Mathematics Subject Classification. Primary 13D02, 13P20; Secondary 05E45, 05C65.

Chordality, d-collapsibility, and componentwise linear ideals

“…We now show that the study of the Betti tables of componentwise linear ideals reduces to the study of the Betti tables of Stanley-Reisner ideals of chordal complexes, generalizing a similar result of Bigdeli and coauthors in the case of equigenerated ideals [5,Theorem 3.3].…”

“…However, it was shown by Bigdeli, Herzog, Yazdanpour and Zaare-Nahandi [5] that every Betti table of a graded ideal with linear resolution is the Betti table of an ideal coming from a chordal clutter, as defined in [7].…”

Decomposable clutters and a generalization of Simon's conjecture

“…It follows from [2,Proposition 4.6] that all simplicial orders of C n,d are of length l = Σ n−d+1 i=1 n−1−i d−2 = Σ n−2 i=0 i d−2 . Therefore, if a squarefree monomial ideal I generated in degree d − 1 has at least l minimal generators, the ideal I has linear quotients if and only if there exist l elements u 1 , .…”

Chordality of clutters with vertex decomposable dual and ascent of clutters

“…They also show that if a clutter is chordal, then the circuit ideal of its complement has a linear resolution over every field. Several clues were presented in [1,2] supporting the correctness of the converse (see [2,Question 1]). But recently a counterexample to the converse was presented by Eric Babson (see Example 2.4).…”