Cellular resolutions of Cohen-Macaulay monomial quotient rings

TL;DR

This paper studies cellular resolutions of Cohen-Macaulay monomial quotient rings, classifying maximal monomial labellings on various cell complexes to understand their algebraic and combinatorial properties.

Contribution

It introduces the concept of maximal monomial labellings, providing classifications for trees, certain polygon subdivisions, and some selfdual polytopes.

Findings

Finite number of maximal labellings for each cell complex

Classification results for trees and polygon subdivisions

Insights into the structure of Cohen-Macaulay monomial quotients

Abstract

We investigate monomial labellings on cell complexes, giving a minimal cellular resolution of the ideal generated by these monomials, and such that the associated quotient ring is Cohen-Macaulay. We introduce a notion of such a labelling being maximal. There is only a finite number of maximal labellings for each cell complex, and we classify these for trees, partly for subdivisions of polygons, and for some classes of selfdual polytopes.