### Abstract

In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen-Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen-Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley-Reisner ring of the barycentric subdivision of an odd dimensional Cohen-Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function.

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

Pages (from-to) | 241-279 |

Number of pages | 39 |

Journal | Advances in Mathematics |

Volume | 265 |

DOIs | |

Publication status | Published - 2014 Nov 10 |

Externally published | Yes |

### Fingerprint

### Keywords

- Barycentric subdivisions
- Cd-Index
- Flag f-vectors
- Regular CW-complexes
- Stanley-Reisner rings

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*265*, 241-279. https://doi.org/10.1016/j.aim.2014.07.037

**Squarefree P-modules and the cd-index.** / Murai, Satoshi; Yanagawa, Kohji.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 265, pp. 241-279. https://doi.org/10.1016/j.aim.2014.07.037

}

TY - JOUR

T1 - Squarefree P-modules and the cd-index

AU - Murai, Satoshi

AU - Yanagawa, Kohji

PY - 2014/11/10

Y1 - 2014/11/10

N2 - In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen-Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen-Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley-Reisner ring of the barycentric subdivision of an odd dimensional Cohen-Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function.

AB - In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen-Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen-Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley-Reisner ring of the barycentric subdivision of an odd dimensional Cohen-Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function.

KW - Barycentric subdivisions

KW - Cd-Index

KW - Flag f-vectors

KW - Regular CW-complexes

KW - Stanley-Reisner rings

UR - http://www.scopus.com/inward/record.url?scp=84906344792&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906344792&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2014.07.037

DO - 10.1016/j.aim.2014.07.037

M3 - Article

VL - 265

SP - 241

EP - 279

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -