Cellular Resolutions of Ideals Defined by Simplicial Homomorphisms

TL;DR

This paper introduces ordered homomorphism ideals and their minimal cellular resolutions via homomorphism complexes, explores cointerval simplicial complexes, and studies nonnesting monomial ideals with combinatorial significance.

Contribution

It presents a new class of ideals with cellular resolutions, introduces cointerval complexes, and analyzes nonnesting monomial ideals, advancing combinatorial and algebraic understanding.

Findings

Ordered homomorphism ideals admit minimal cellular resolutions.

Cointerval simplicial complexes have notable combinatorial and topological properties.

Nonnesting monomial ideals form a new family of interesting combinatorial ideals.

Abstract

In this paper we introduce the class of ordered homomorphism ideals and prove that these ideals admit minimal cellular resolutions constructed as homomorphism complexes. As a key ingredient of our work, we introduce the class of cointerval simplicial complexes and investigate their combinatorial and topological properties. As a concrete illustration of these structural results, we introduce and study nonnesting monomial ideals, an interesting family of combinatorially defined ideals.