A Duality in Buchsbaum rings and triangulated manifolds

Let $Delta$ be a triangulated homology ball whose boundary complex is $partialDelta$. 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 $(Delta, partialDelta)$, $mathbb F[Delta,partial Delta]$. 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.