#### Vol. 11, No. 3, 2017

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A duality in Buchsbaum rings and triangulated manifolds

### Satoshi Murai, Isabella Novik and Ken-ichi Yoshida

Vol. 11 (2017), No. 3, 635–656
##### 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 $\mathbb{F}\left[\Delta \right]$ of $\Delta$ is isomorphic to the Stanley–Reisner module $\mathbb{F}\left[\Delta ,\partial \Delta \right]$ of the pair $\left(\Delta ,\partial \Delta \right)$. This result implies that an Artinian reduction of $\mathbb{F}\left[\Delta ,\partial \Delta \right]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $\mathbb{F}\left[\Delta \right]$. 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 ${h}^{\prime \prime }$-numbers of Buchsbaum complexes and use it to prove the monotonicity of ${h}^{\prime \prime }$-numbers for pairs of Buchsbaum complexes as well as the unimodality of ${h}^{\prime \prime }$-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold $g$-conjecture.

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##### Keywords
triangulated manifolds, Buchsbaum rings, $h$-vectors, Stanley–Reisner rings
##### Mathematical Subject Classification 2010
Primary: 13F55
Secondary: 13H10, 05E40, 52B05, 57Q15
##### Milestones
Received: 8 March 2016
Accepted: 8 October 2016
Published: 6 May 2017
##### Authors
 Satoshi Murai Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology Osaka University Suita 565-0871 Japan Isabella Novik Department of Mathematics University of Washington Box 354350 Seattle, WA 98195-4350 United States Ken-ichi Yoshida Department of Mathematics, College of Humanities and Sciences Nihon University Setagaya-ku, Tokyo 156-8550 Japan