On the Behavior of Minimal Free Resolutions of Trivariate Generic Monomial Ideals

TL;DR

This paper investigates the structure of minimal free resolutions of trivariate monomial ideals, especially primary to the maximal ideal, highlighting differences between generic and non-generic cases and providing criteria to identify generic ideals.

Contribution

It characterizes the last matrix of minimal free resolutions for primary trivariate monomial ideals and offers a method to determine if an ideal is generic based on this matrix.

Findings

Differences in resolution structures between generic and non-generic ideals.

Identification criteria for generic ideals from the last matrix.

Properties of the last matrix in minimal free resolutions.

Abstract

We will explore some properties of minimal graded free resolutions of $R/I$, where $R$ is a trivariate polynomial ring over a field and $I$ is a monomial ideal. Our focus will be to consider a specific form of the resolutions when $I$ is primary to the homogeneous maximal ideal. We will identify certain characteristics of the last matrix of these resolutions, and observe differences in the resolutions for generic ideals in comparison to non-generic ideals. Finally, we learn how to identify whether $I$ is generic by knowing the structure of the last matrix in the minimal free resolution of $R/I$.