A Duality in Buchsbaum rings and triangulated manifolds

Abstract

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.

19 pages, minor changes

References in corpus (4)

• Thirty-five years and counting
• Balanced generalized lower bound inequality for simplicial polytopes
• Face numbers of manifolds with boundary
• Weak Lefschetz Property and Stellar Subdivisions of Gorenstein Complexes

Cited by in corpus (3)

• Combinatorial Lefschetz theorems beyond positivity
• $g$-vectors of manifolds with boundary
• The face numbers of homology spheres