Squarefree P-modules and the cd-index

Satoshi Murai, Kohji Yanagawa

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 languageEnglish
Pages (from-to)241-279
Number of pages39
JournalAdvances in Mathematics
Volume265
DOIs
Publication statusPublished - 2014 Nov 10
Externally publishedYes

Fingerprint

Poset
Cohen-Macaulay
Module
Barycentric Subdivision
Nonnegativity
Gorenstein
Stanley-Reisner Ring
H-vector
CW-complex
Commutative Algebra
Generating Function
Odd
Upper bound
Analogue
Concepts

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Advances in Mathematics, Vol. 265, 10.11.2014, p. 241-279.

Research output: Contribution to journalArticle

Murai, Satoshi ; Yanagawa, Kohji. / Squarefree P-modules and the cd-index. In: Advances in Mathematics. 2014 ; Vol. 265. pp. 241-279.
@article{e445114b7620418c87d813fd26c6416d,
title = "Squarefree P-modules and the cd-index",
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.",
keywords = "Barycentric subdivisions, Cd-Index, Flag f-vectors, Regular CW-complexes, Stanley-Reisner rings",
author = "Satoshi Murai and Kohji Yanagawa",
year = "2014",
month = "11",
day = "10",
doi = "10.1016/j.aim.2014.07.037",
language = "English",
volume = "265",
pages = "241--279",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",

}

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 -