Stability of Betti numbers under reduction processes: Towards chordality of clutters

Abstract

For a given clutter C, let I := I C be the circuit ideal in the polynomial ring S. In this paper, we show that the Betti numbers of I and I + (xF ) are the same in their non-linear strands, for some suitable F ∈ C. Motivated by this result, we introduce a class of clutters that we call chordal. This class, is a natural extension of the class of chordal graphs and has the nice property that the circuit ideal associated to any member of this class has a linear resolution over any field. Finally we compare this class with all known families of clutters which generalize the notion of chordality, and show that our class contains several important previously defined classes of chordal clutters. We also show that in comparison with others, this class is possibly the best approximation to the class of d-uniform clutters with linear resolution over any field.2010 Mathematics Subject Classification. Primary 13D02, 13F55; Secondary 05E45, 05C65.

Chordality, d-collapsibility, and componentwise linear ideals

“…Definition 2.6 of a "d-chordal simplicial complex" is a Stanley-Reisner equivalent of "chordal (d + 1)-uniform clutters" in [7]. The following statement follows directly from the definitions, we include it for the sake of comparison.…”

“…In particular, as in the case of [7], our definition of chordality for simplicial complexes extends that of graphs. Given a simplicial complex Γ, its 1-closure ∆ 1 (Γ) is the clique complex of a graph G = Γ [1] .…”

“…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].…”

“…In this paper, we adapt the concept of chordal clutters from [7] and change the perspective from clutters to simplicial complexes. As a result, we show that chordality of the Stanley-Reisner complex of an ideal generated in degree d + 1 is equivalent to d-collapsibility, a notion well-known and well-used in algebraic topology and combinatorics which has specific homological consequences.…”

Decomposable clutters and a generalization of Simon's conjecture

“…Following the notation in [3], we use C d to denote the class of all d-uniform chordal clutters. Definition 1.6.…”