A duality in Buchsbaum rings and triangulated manifolds

Murai, Satoshi; Novik, Isabella; Yoshida, Ken-ichi

doi:10.2140/ant.2017.11.635

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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

DOI: 10.2140/ant.2017.11.635

Abstract

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

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

Vol. 11, No. 3, 2017