### Abstract

Let S = K[x1, ⋯,xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let Δ be a simplicial complex on [n] = {1, ⋯,n} and I. Δ S its Stanley-Reisner ideal. We write.e for the exterior algebraic shifted complex of Δ and.c for a combinatorial shifted complex of. Let ßii+j (I.) = dimK Tori(K, I.)i+j denote the graded Betti numbers of I. In the present paper it will be proved that (i) βii%+j (I.e) = βii%+j (I.c) for all i and j, where the base field is infinite, and (ii) βii%+j (I.) = βii%+j (I.c) for all i and j, where the base field is arbitrary. Thus in particular one has βii%+j (I.) = βii%+j (I.lex) for all i and j, where.lex is the unique lexsegment simplicial complex with the same f-vector as Δ and where the base field is arbitrary.

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

Pages (from-to) | 1853-1865 |

Number of pages | 13 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 Apr 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*361*(4), 1853-1865. https://doi.org/10.1090/S0002-9947-08-04707-7

**Algebraic shifting and graded betti numbers.** / Murai, Satoshi; Hibi, Takayuki.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 361, no. 4, pp. 1853-1865. https://doi.org/10.1090/S0002-9947-08-04707-7

}

TY - JOUR

T1 - Algebraic shifting and graded betti numbers

AU - Murai, Satoshi

AU - Hibi, Takayuki

PY - 2009/4/1

Y1 - 2009/4/1

N2 - Let S = K[x1, ⋯,xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let Δ be a simplicial complex on [n] = {1, ⋯,n} and I. Δ S its Stanley-Reisner ideal. We write.e for the exterior algebraic shifted complex of Δ and.c for a combinatorial shifted complex of. Let ßii+j (I.) = dimK Tori(K, I.)i+j denote the graded Betti numbers of I. In the present paper it will be proved that (i) βii%+j (I.e) = βii%+j (I.c) for all i and j, where the base field is infinite, and (ii) βii%+j (I.) = βii%+j (I.c) for all i and j, where the base field is arbitrary. Thus in particular one has βii%+j (I.) = βii%+j (I.lex) for all i and j, where.lex is the unique lexsegment simplicial complex with the same f-vector as Δ and where the base field is arbitrary.

AB - Let S = K[x1, ⋯,xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let Δ be a simplicial complex on [n] = {1, ⋯,n} and I. Δ S its Stanley-Reisner ideal. We write.e for the exterior algebraic shifted complex of Δ and.c for a combinatorial shifted complex of. Let ßii+j (I.) = dimK Tori(K, I.)i+j denote the graded Betti numbers of I. In the present paper it will be proved that (i) βii%+j (I.e) = βii%+j (I.c) for all i and j, where the base field is infinite, and (ii) βii%+j (I.) = βii%+j (I.c) for all i and j, where the base field is arbitrary. Thus in particular one has βii%+j (I.) = βii%+j (I.lex) for all i and j, where.lex is the unique lexsegment simplicial complex with the same f-vector as Δ and where the base field is arbitrary.

UR - http://www.scopus.com/inward/record.url?scp=77950632770&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950632770&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-08-04707-7

DO - 10.1090/S0002-9947-08-04707-7

M3 - Article

AN - SCOPUS:77950632770

VL - 361

SP - 1853

EP - 1865

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 4

ER -