Cohen-Macaulayness of large powers of Stanley-Reisner ideals

Naoki Terai; Ngo Viet Trung

doi:10.1016/j.aim.2011.10.004

Cohen-Macaulayness of large powers of Stanley-Reisner ideals

Naoki Terai, Ngo Viet Trung

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

31 Citations (Scopus)

Abstract

We prove that for m≥3, the symbolic power Iδ(m) of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex δ is a matroid. Similarly, the ordinary power Iδm is Cohen-Macaulay for some m≥3 if and only if I_δ is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.

Original language	English
Pages (from-to)	711-730
Number of pages	20
Journal	Advances in Mathematics
Volume	229
Issue number	2
DOIs	https://doi.org/10.1016/j.aim.2011.10.004
Publication status	Published - Jan 30 2012
Externally published	Yes

Keywords

Buchsbaum
Cohen-Macaulay
Complete intersection
Cover ideal
Facet ideal
Generalized Cohen-Macaulay
Matroid
Power
Serre condition (S)
Stanley-Reisner ideal

ASJC Scopus subject areas

Mathematics(all)

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title = "Cohen-Macaulayness of large powers of Stanley-Reisner ideals",

abstract = "We prove that for m≥3, the symbolic power Iδ(m) of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex δ is a matroid. Similarly, the ordinary power Iδm is Cohen-Macaulay for some m≥3 if and only if Iδ is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.",

keywords = "Buchsbaum, Cohen-Macaulay, Complete intersection, Cover ideal, Facet ideal, Generalized Cohen-Macaulay, Matroid, Power, Serre condition (S), Stanley-Reisner ideal",

author = "Naoki Terai and Trung, {Ngo Viet}",

note = "Funding Information: The first author is supported by JSPS 20540047 and the second author by the National Foundation of Science and Technology Development . ",

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AU - Terai, Naoki

AU - Trung, Ngo Viet

N1 - Funding Information: The first author is supported by JSPS 20540047 and the second author by the National Foundation of Science and Technology Development .

PY - 2012/1/30

Y1 - 2012/1/30

N2 - We prove that for m≥3, the symbolic power Iδ(m) of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex δ is a matroid. Similarly, the ordinary power Iδm is Cohen-Macaulay for some m≥3 if and only if Iδ is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.

AB - We prove that for m≥3, the symbolic power Iδ(m) of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex δ is a matroid. Similarly, the ordinary power Iδm is Cohen-Macaulay for some m≥3 if and only if Iδ is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.

KW - Buchsbaum

KW - Cohen-Macaulay

KW - Complete intersection

KW - Cover ideal

KW - Facet ideal

KW - Generalized Cohen-Macaulay

KW - Matroid

KW - Power

KW - Serre condition (S)

KW - Stanley-Reisner ideal

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Cohen-Macaulayness of large powers of Stanley-Reisner ideals

Abstract

Keywords

ASJC Scopus subject areas

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