Average Behavior of Minimal Free Resolutions of Monomial Ideals

TL;DR

This paper investigates the typical homological properties of random monomial ideals, showing that they usually have maximal projective dimension and are rarely Cohen-Macaulay, with detailed characterizations of their genericity and Scarf properties.

Contribution

It provides a rigorous analysis of the homological behavior of random monomial ideals, including conditions for maximal projective dimension and the rarity of Cohen-Macaulayness.

Findings

Random monomial ideals almost always have maximal projective dimension.

Cohen-Macaulayness is a rare property among RMI's.

RMI's are Scarf only when they are generic, outside specific parameter ratios.

Abstract

We describe the typical homological properties of monomial ideals defined by random generating sets. We show that, under mild assumptions, random monomial ideals (RMI's) will almost always have resolutions of maximal length; that is, the projective dimension will almost always be $n$, where $n$ is the number of variables in the polynomial ring. We give a rigorous proof that Cohen-Macaulayness is a "rare" property. We characterize when an RMI is generic/strongly generic, and when it "is Scarf"---in other words, when the algebraic Scarf complex of $M\subset S=k[x_1,\ldots,x_n]$ gives a minimal free resolution of $S/M$. As a result we see that, outside of a very specific ratio of model parameters, RMI's are Scarf only when they are generic. We end with a discussion of the average magnitude of Betti numbers.