Duality for Cohen--Macaulay Complexes through Combinatorial Sheaves

Abstract

We prove a duality theorem for Cohen--Macaulay simplicial complexes. This is a generalisation of Poincaré 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.

45 pages. v4: Significantly shortened the paper and streamlined the exposition by combining the proofs of CM duality with ring coefficients and with coefficients in arbitrary (co)sheaves. See v3 for an explanation of the relationship between our work and classical sheaves and Verdier duality, a more elaborate treatment of naturality, and an example of the duality theorem in the setting of trees