Multigraded minimal free resolutions of simplicial subclutters

TL;DR

This paper investigates the algebraic properties of simplicial subclutters, providing formulas for Betti numbers, and characterizes ideals with linear resolutions in terms of these structures.

Contribution

It introduces a formula for Betti numbers of ideals associated with simplicial subclutters and characterizes ideals with linear resolutions using these structures.

Findings

Betti numbers of subclutter ideals can be computed from those of the original ideal and combinatorial data.

Ideals with linear resolutions are related to simplicial subclutters of complete clutters.

Not all equigenerated square-free monomial ideals with linear quotients are associated to simplicial subclutters.

Abstract

This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter $\mathcal{C}$ and its simplicial subclutter $\mathcal{D}$, we compare some algebraic properties and invariants of the ideals $I, J$ associated to these two clutters, respectively. We give a formula for computing the (multi)graded Betti numbers of $J$ in terms of those of $I$ and some combinatorial data about $\mathcal{D}$. As a result, we see that if $\mathcal{C}$ admits a simplicial subclutter, then there exists a monomial $u \notin I$ such that the (multi)graded Betti numbers of $I+(u)$ can be computed through those of $I$. It is proved that the Betti sequence of any graded ideal with linear resolution is the Betti sequence of an ideal associated to a simplicial subclutter of the complete clutter. These ideals turn out to have linear quotients. However, they do not form all the equigenerated square-free monomial ideals with linear quotients. If $\mathcal{C}$ admits $\varnothing$ as a simplicial subclutter, then $I$ has linear resolution over all fields. Examples show that the converse is not true.