Linearity of Free Resolutions of Monomial Ideals

TL;DR

This paper investigates the properties of monomial ideals with linear or near-linear resolutions, providing combinatorial characterizations, bounds on invariants, and constructing fractal examples related to simplicial complexes.

Contribution

It offers new combinatorial criteria for linear presentation of square-free ideals and almost linear resolutions, along with bounds on regularity and generator counts, and introduces fractal examples linked to topology.

Findings

Characterization of linear presentation for degree 3 square-free ideals

Bounds on Castelnuovo-Mumford regularity and generator numbers

Construction of fractal ideals related to the Sierpiński triangle

Abstract

We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except for the last step (the "almost linear" case). We also give sharp bounds on Castelnuovo-Mumford regularity and numbers of generators in some cases. It is a basic observation that linearity properties are inherited by the restriction of an ideal to a subset of variables, and we study when the converse holds. We construct fractal examples of almost linear primary ideals with relatively few generators related to the Sierpi\'nski triangle. Our results also lead to classes of highly connected simplicial complexes $\Delta$ that can not be extended to the complete $\dim \Delta$-skeleton of the simplex on the same variables by shelling.