Minimal Free Resolutions of Certain Equigenerated Monomial Ideals

TL;DR

This paper investigates specific equigenerated monomial ideals in polynomial rings, using advanced algebraic techniques to explicitly construct their minimal free resolutions and analyze their Betti numbers.

Contribution

It introduces a method to compute minimal free resolutions for certain monomial ideals via iterated trimming complexes and splitting mapping cones.

Findings

Explicit minimal free resolutions constructed for the studied ideals.

Betti numbers determined for these classes of ideals.

Questions posed about additional structures on the complexes.

Abstract

Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the generators. We then use iterated trimming complexes to deduce Betti numbers for such ideals. Furthermore, using a result on splitting mapping cones by Miller and Rahmati, we construct the minimal free resolutions for all ideals under consideration explicitly and conclude with questions about extra structure on these complexes.