  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.
