A Duality in Buchsbaum Rings and Triangulated Manifolds

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