Topological Cohen-Macaulay criteria for monomial ideals

TL;DR

This survey consolidates recent topological criteria for Cohen-Macaulayness of monomial ideals, highlighting combinatorial and topological connections in a comprehensive manner.

Contribution

It provides a unified, self-contained overview of topological Cohen-Macaulay criteria for monomial ideals, including direct combinatorial-topological links.

Findings

Multiple topological characterizations of Cohen-Macaulay monomial ideals

Connections between simplicial complex topology and algebraic properties

Comprehensive, self-contained proofs of key criteria

Abstract

Scattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. The purpose of this survey is to gather the developments into one location, with self-contained proofs, including direct combinatorial topological connections between them.