Homological invariants of the Stanley-Reisner ring of a $k$-decomposable simplicial complex

TL;DR

This paper investigates the homological invariants of Stanley-Reisner rings associated with $k$-decomposable simplicial complexes, providing recursive formulas and bounds for regularity and projective dimension.

Contribution

It introduces a recursive approach to compute invariants of $k$-decomposable complexes and Betti numbers of their monomial ideals, with applications to chordal clutters.

Findings

Recursive formulas for regularity and projective dimension.

An inductive formula for Betti numbers of $k$-decomposable monomial ideals.

Upper bounds for regularity of edge ideals in chordal clutters.

Abstract

We study the regularity and the projective dimension of the Stanley-Reisner ring of a $k$-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of $k$-decomposable monomial ideals which is the dual concept for $k$-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a chordal clutter $\mathcal{H}$, an upper bound for $reg(I(\mathcal{H}))$ is given in terms of the regularities of edge ideals of some chordal clutters which are minors of $\mathcal{H}$.