Topological Cohen–Macaulay criteria for monomial ideals

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. It is unclear whether researchers thinking about this topic have, to this point, been aware of the full spectrum of related developments. Therefore, the purpose of this survey is to gather the developments into one location, with self-contained proofs, including direct combinatorial topological connections between them. Four families of simplicial complexes are defined in reverse chronological order: via distraction, the Čech complex, Alexander duality, and then the Koszul complex. Each comes with historical remarks and context, including forays into Stanley decompositions, standard pairs, A-hypergeometric systems, cellular resolutions, the Čech hull, polarization, local duality, and duality for Z-graded resolutions. Results or definitions appearing here for the first time include the general categorical definition of cellular complex in Definition 3.2, as well as the statements and proofs of Lemmas 3.9 and 3.10, though these are very easy. The characterization of exponent simplicial complexes in Corollary 2.9 might be considered new; certainly the consequent connection in Theorem 4.1 to the Čech simplicial complexes is new, as is the duality in Theorem 6.3 between these and the dual Čech simplicial complexes. The geometric connection between distraction and local cohomology in Theorem 4.7, and its consequences in Section 5, generalize and refine results from [BeMa08], in addition to providing commutative proofs. Finally, the connection between Čech and Koszul simplicial complexes in Lemma 7.2 and Corollary 7.8, as well as its general Z-graded version in Theorem 7.7, appear to be new.