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

We will explore some properties of minimal graded free resolutions of $R\mathrm{/}I$, 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 $R\mathrm{/}I$.