Algebra & Number Theory

A duality in Buchsbaum rings and triangulated manifolds

Satoshi Murai, Isabella Novik, and Ken-ichi Yoshida

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

Permanent link to this document
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

Subjects
Primary: 13F55: Stanley-Reisner face rings; simplicial complexes [See also 55U10]
Secondary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 05E40: Combinatorial aspects of commutative algebra 52B05: Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx] 57Q15: Triangulating manifolds

Keywords
triangulated manifolds Buchsbaum rings $h$-vectors Stanley–Reisner rings

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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