A Duality in Buchsbaum rings and triangulated manifolds

TL;DR

This paper generalizes a duality in Buchsbaum rings related to triangulated homology manifolds with boundary, providing algebraic interpretations of h''-numbers and applications to the manifold g-conjecture.

Contribution

It extends Hochster's duality to all triangulations of connected orientable homology manifolds with boundary and interprets h''-numbers algebraically.

Findings

Established a duality for all triangulations of orientable homology manifolds with boundary.

Proved monotonicity of h''-numbers for pairs of Buchsbaum complexes.

Demonstrated unimodality of h''-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes.

Abstract

Let $\Delta$ be a triangulated homology ball whose boundary complex is $\partial\Delta$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $\Delta$, $\mathbb F[\Delta]$, is isomorphic to the Stanley--Reisner module of the pair $(\Delta, \partial\Delta)$, $\mathbb F[\Delta,\partial \Delta]$. This result implies that an Artinian reduction of $\mathbb F[\Delta,\partial \Delta]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $\mathbb F[\Delta]$. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the $h"$-numbers of Buchsbaum complexes and use it to prove the monotonicity of $h"$-numbers for pairs of Buchsbaum complexes as well as the unimodality of $h"$-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold $g$-conjecture.