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

TL;DR

This paper introduces a new class of chordal clutters that generalize chordal graphs, demonstrating their circuit ideals have linear resolutions and are stable under certain reduction processes, advancing understanding of algebraic and combinatorial properties.

Contribution

It defines the class of chordal clutters, extending chordal graphs, and proves their circuit ideals have linear resolutions over any field, with stability under specific reduction operations.

Findings

Chordal clutters have circuit ideals with linear resolutions.

Betti numbers are stable under certain reduction processes.

The class includes many previously known chordal clutter families.

Abstract

For a given clutter $\mathcal{C}$, let $I:=I ( \bar{\mathcal{C}} )$ be the circuit ideal in the polynomial ring $S$. In this paper, we show that the Betti numbers of $I$ and $I + ( \textbf{x}_F )$ are the same in their non-linear strands, for some suitable $F \in \mathcal{C}$. 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.