## Algebra & Number Theory

### A duality in Buchsbaum rings and triangulated manifolds

#### Abstract

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.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 3 (2017), 635-656.

Dates
Accepted: 8 October 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.ant/1508431775

Digital Object Identifier
doi:10.2140/ant.2017.11.635

Mathematical Reviews number (MathSciNet)
MR3649363

Zentralblatt MATH identifier
1370.13019

#### Citation

Murai, Satoshi; Novik, Isabella; Yoshida, Ken-ichi. A duality in Buchsbaum rings and triangulated manifolds. Algebra Number Theory 11 (2017), no. 3, 635--656. doi:10.2140/ant.2017.11.635. https://projecteuclid.org/euclid.ant/1508431775

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