Simplicial orders and chordality

TL;DR

This paper explores the properties of chordal clutters defined via simplicial orders, showing their circuit ideals have linear resolutions and characterizing all possible Betti sequences and related invariants.

Contribution

It introduces the $ ext{lambda}$-sequence for chordal clutters, characterizes all such sequences, and links them to algebraic invariants like the Hilbert function and Betti numbers.

Findings

All Betti sequences of ideals with linear resolution can be realized by chordal clutter circuit ideals.

The $ ext{lambda}$-sequence characterizes the Hilbert function and other invariants of the circuit ideal.

The $ ext{lambda}$-sequence provides a new numerical invariant for chordal clutters.

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 $\lambda$-sequence of the chordal clutter. All possible $\lambda$-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 $\lambda$-sequence of a chordal clutter we determine other numerical invariants of the circuit ideal, such as the $\textbf{h}$-vector and the Betti numbers.