Multiplicity and Castelnuovo–Mumford Regularity of Stanley–Reisner Rings

Naoki Terai; Ken ichi Yoshida

doi:10.1007/s40306-014-0103-y

Multiplicity and Castelnuovo–Mumford Regularity of Stanley–Reisner Rings

Naoki Terai, Ken ichi Yoshida

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

Abstract

In this paper, we pose the following conjecture and give a positive answer to the case dimΔ≤2: Let Δ be a (d−1)-dimensional simplicial complex on [n]. Fix an integer ℓ with 0≤ℓ≤n−d−1. If e(K[Δ])≤(ℓ+1)d−ℓ and β_ℓ,ℓ+d(K[Δ])=0, then reg K[Δ]≤d−1. Moreover, we discuss the relationship between the above conjecture and the lower bound theorem.

Original language	English
Pages (from-to)	61-69
Number of pages	9
Journal	Acta Mathematica Vietnamica
Volume	40
Issue number	1
DOIs	https://doi.org/10.1007/s40306-014-0103-y
Publication status	Published - Mar 1 2015
Externally published	Yes

Keywords

Arithmetic degree
Castelnuovo–Mumford regularity
Lower bound theorem
Multiplicity
Stanley–Reisner ring

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s40306-014-0103-y

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AU - Terai, Naoki

AU - Yoshida, Ken ichi

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N2 - In this paper, we pose the following conjecture and give a positive answer to the case dimΔ≤2: Let Δ be a (d−1)-dimensional simplicial complex on [n]. Fix an integer ℓ with 0≤ℓ≤n−d−1. If e(K[Δ])≤(ℓ+1)d−ℓ and βℓ,ℓ+d(K[Δ])=0, then reg K[Δ]≤d−1. Moreover, we discuss the relationship between the above conjecture and the lower bound theorem.

AB - In this paper, we pose the following conjecture and give a positive answer to the case dimΔ≤2: Let Δ be a (d−1)-dimensional simplicial complex on [n]. Fix an integer ℓ with 0≤ℓ≤n−d−1. If e(K[Δ])≤(ℓ+1)d−ℓ and βℓ,ℓ+d(K[Δ])=0, then reg K[Δ]≤d−1. Moreover, we discuss the relationship between the above conjecture and the lower bound theorem.

KW - Arithmetic degree

KW - Castelnuovo–Mumford regularity

KW - Lower bound theorem

KW - Multiplicity

KW - Stanley–Reisner ring

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