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

Let Δ be a triangulated homology ball whose boundary complex is 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,Δ] of the pair (Δ,Δ). This result implies that an Artinian reduction of F[Δ,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.

triangulated manifolds, Buchsbaum rings, $h$-vectors, Stanley–Reisner rings
Mathematical Subject Classification 2010
Primary: 13F55
Secondary: 13H10, 05E40, 52B05, 57Q15
Received: 8 March 2016
Accepted: 8 October 2016
Published: 6 May 2017
Satoshi Murai
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology
Osaka University
Suita 565-0871
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