On the Koszul Algebra for Trivariate Monomial Ideals

TL;DR

This paper classifies the structure of the Koszul algebra for trivariate monomial ideals using minimal free resolutions, building on Avramov's classification, and identifies new algebraic structures within this context.

Contribution

It provides a complete classification of the Koszul algebra for generic monomial ideals and introduces a previously unknown class of ideals with specific Koszul algebra structures.

Findings

Complete classification for generic monomial ideals

Identification of a new class of ideals with unique Koszul algebra structure

Method to determine Koszul algebra from minimal free resolution

Abstract

We will describe how we can identify the structure of the Koszul algebra for trivariate monomial ideals from minimal free resolutions. We use recent work of L. Avramov, where he classifies the behavior of Bass numbers of embedding codepth 3 commutative local rings. His classification relies on a corresponding classification of their respective Koszul algebras, which is comprised of 5 categories. Using Avramov's classification of the Koszul algebra, along with their respective Bass series we will learn how to identify the Koszul algebra structure by inspecting the minimal free resolution of the quotient ring. We give a complete classification of the Koszul algebra for generic monomial ideals and offer several examples. In addition we will describe a class of ideals with a specific Koszul algebra structure which was previously unknown.