### 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 hʺ-numbers of Buchsbaum complexes and use it to prove the monotonicity of hʺ-numbers for pairs of Buchsbaum complexes as well as the unimodality of hʺ-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold g-conjecture.

Original language | English |
---|---|

Pages (from-to) | 635-656 |

Number of pages | 22 |

Journal | Algebra and Number Theory |

Volume | 11 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Buchsbaum rings
- H-vectors
- Stanley–Reisner rings
- Triangulated manifolds

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Algebra and Number Theory*,

*11*(3), 635-656. https://doi.org/10.2140/ant.2017.11.635

**A duality in Buchsbaum rings and triangulated manifolds.** / Murai, Satoshi; Novik, Isabella; Yoshida, Ken Ichi.

Research output: Contribution to journal › Article

*Algebra and Number Theory*, vol. 11, no. 3, pp. 635-656. https://doi.org/10.2140/ant.2017.11.635

}

TY - JOUR

T1 - A duality in Buchsbaum rings and triangulated manifolds

AU - Murai, Satoshi

AU - Novik, Isabella

AU - Yoshida, Ken Ichi

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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 hʺ-numbers of Buchsbaum complexes and use it to prove the monotonicity of hʺ-numbers for pairs of Buchsbaum complexes as well as the unimodality of hʺ-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold g-conjecture.

AB - 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 hʺ-numbers of Buchsbaum complexes and use it to prove the monotonicity of hʺ-numbers for pairs of Buchsbaum complexes as well as the unimodality of hʺ-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold g-conjecture.

KW - Buchsbaum rings

KW - H-vectors

KW - Stanley–Reisner rings

KW - Triangulated manifolds

UR - http://www.scopus.com/inward/record.url?scp=85019209986&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019209986&partnerID=8YFLogxK

U2 - 10.2140/ant.2017.11.635

DO - 10.2140/ant.2017.11.635

M3 - Article

AN - SCOPUS:85019209986

VL - 11

SP - 635

EP - 656

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 3

ER -