Some results on Betti numbers of Stanley-Reisner rings

Naoki Terai; Takayuki Hibi

doi:10.1016/S0012-365X(96)83021-4

Some results on Betti numbers of Stanley-Reisner rings

Naoki Terai, Takayuki Hibi

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

We study the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ring k[Δ] = A/I_Δ of a simplicial complex Δ over a field k. It is known that the second Betti number of k[Δ] is independent of the base field k. We show that, when the ideal I_Δ is generated by square-free monomials of degree two, the third and fourth Betti numbers are also independent of k. On the other hand, we prove that, if the geometric realization of Δ is homeomorphic to either the 3-sphere or the 3-ball, then all the Betti numbers of k[Δ] are independent of the base field k.

Original language	English
Pages (from-to)	311-320
Number of pages	10
Journal	Discrete Mathematics
Volume	157
Issue number	1-3
DOIs	https://doi.org/10.1016/S0012-365X(96)83021-4
Publication status	Published - Oct 1 1996
Externally published	Yes

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/S0012-365X(96)83021-4

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abstract = "We study the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ring k[Δ] = A/IΔ of a simplicial complex Δ over a field k. It is known that the second Betti number of k[Δ] is independent of the base field k. We show that, when the ideal IΔ is generated by square-free monomials of degree two, the third and fourth Betti numbers are also independent of k. On the other hand, we prove that, if the geometric realization of Δ is homeomorphic to either the 3-sphere or the 3-ball, then all the Betti numbers of k[Δ] are independent of the base field k.",

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AB - We study the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ring k[Δ] = A/IΔ of a simplicial complex Δ over a field k. It is known that the second Betti number of k[Δ] is independent of the base field k. We show that, when the ideal IΔ is generated by square-free monomials of degree two, the third and fourth Betti numbers are also independent of k. On the other hand, we prove that, if the geometric realization of Δ is homeomorphic to either the 3-sphere or the 3-ball, then all the Betti numbers of k[Δ] are independent of the base field k.

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Some results on Betti numbers of Stanley-Reisner rings

Abstract

ASJC Scopus subject areas

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