Duality for Cohen--Macaulay Complexes through Combinatorial Sheaves

TL;DR

This paper establishes a duality theorem for Cohen--Macaulay simplicial complexes using combinatorial sheaves, extending Poincaré Duality and connecting to Bieri-Eckmann duality for groups.

Contribution

It introduces a new duality framework for Cohen--Macaulay complexes via combinatorial sheaves, generalizing classical duality theorems.

Findings

Proves a duality theorem for Cohen--Macaulay complexes

Links combinatorial sheaves with classical duality theories

Provides a self-contained, accessible treatment

Abstract

We prove a duality theorem for Cohen--Macaulay simplicial complexes. This is a generalisation of Poincar\'e Duality, framed in the language of combinatorial sheaves. Our treatment is self-contained and accessible for readers with a working knowledge of simplicial complexes and (co)homology. The main motivation is a link with Bieri-Eckmann duality for discrete groups, which is explored in a companion paper.