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 is isomorphic to the Stanley–Reisner module of the pair . This result implies that an Artinian reduction of is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of . 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 -numbers of Buchsbaum complexes and use it to prove the monotonicity of -numbers for pairs of Buchsbaum complexes as well as the unimodality of -vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold -conjecture.
"A duality in Buchsbaum rings and triangulated manifolds." Algebra Number Theory 11 (3) 635 - 656, 2017. https://doi.org/10.2140/ant.2017.11.635