Building monomial ideals with fixed betti numbers

TL;DR

This paper constructs infinite families of monomial ideals with a fixed set of Betti numbers, using discrete homotopy theory, and shows that Cohen-Macaulay properties are preserved during expansion.

Contribution

It introduces a novel method to build monomial ideals with predictable Betti numbers and maintains Cohen-Macaulay properties as the number of generators increases.

Findings

Constructed infinitely many monomial ideals with fixed Betti numbers.

Demonstrated preservation of Cohen-Macaulay property during ideal expansion.

Used elementary collapses from discrete homotopy theory for ideal construction.

Abstract

Motivated by the fact that as the number of generators of an ideal grows so does the complexity of calculating relations among the generators, this paper identifies collections of monomial ideals with a growing number of generators which have predictable free resolutions. We use elementary collapses from discrete homotopy theory to construct infinitely many monomial ideals, with an arbitrary number of generators, which have similar or the same betti numbers. We show that the Cohen-Macaulay property in each unmixed (pure) component of the ideal is preserved as the ideal is expanded.