A Duality in Buchsbaum rings and triangulated manifolds

Abstract

Let $Δ$ be a triangulated homology ball whose boundary complex is $\partialΔ$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $Δ$, $\mathbb F[Δ]$, is isomorphic to the Stanley--Reisner module of the pair $(Δ, \partialΔ)$, $\mathbb F[Δ,\partial Δ]$. This result implies that an Artinian reduction of $\mathbb F[Δ,\partial Δ]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $\mathbb F[Δ]$. 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.