Cohen-Macaulayness for symbolic power ideals of edge ideals

Giancarlo Rinaldo; Naoki Terai; Ken ichi Yoshida

doi:10.1016/j.jalgebra.2011.09.007

Cohen-Macaulayness for symbolic power ideals of edge ideals

Giancarlo Rinaldo, Naoki Terai, Ken ichi Yoshida

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

18 Citations (Scopus)

Abstract

Let S=K[x₁,..., x_n] be a polynomial ring over a field K. Let I(G)⊆S denote the edge ideal of a graph G. We show that the ℓth symbolic power I(G)^(ℓ) is a Cohen-Macaulay ideal (i.e., S/I(G)^(ℓ) is Cohen-Macaulay) for some integer ℓ≥3 if and only if G is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers I(G)^(ℓ) are Cohen-Macaulay ideals. Similarly, we characterize graphs G for which S/I(G)^(ℓ) has (FLC).As an application, we show that an edge ideal I(G) is complete intersection provided that S/I(G)^ℓ is Cohen-Macaulay for some integer ℓ≥3. This strengthens the main theorem in Crupi et al. (2010) [3].

Original language	English
Pages (from-to)	1-22
Number of pages	22
Journal	Journal of Algebra
Volume	347
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2011.09.007
Publication status	Published - Dec 1 2011
Externally published	Yes

Keywords

Cohen-Macaulay
Complete intersection
Edge ideal
FLC
Polarization
Simplicial complex
Symbolic powers

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2011.09.007

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title = "Cohen-Macaulayness for symbolic power ideals of edge ideals",

abstract = "Let S=K[x1,..., xn] be a polynomial ring over a field K. Let I(G)⊆S denote the edge ideal of a graph G. We show that the ℓth symbolic power I(G)(ℓ) is a Cohen-Macaulay ideal (i.e., S/I(G)(ℓ) is Cohen-Macaulay) for some integer ℓ≥3 if and only if G is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers I(G)(ℓ) are Cohen-Macaulay ideals. Similarly, we characterize graphs G for which S/I(G)(ℓ) has (FLC).As an application, we show that an edge ideal I(G) is complete intersection provided that S/I(G)ℓ is Cohen-Macaulay for some integer ℓ≥3. This strengthens the main theorem in Crupi et al. (2010) [3].",

keywords = "Cohen-Macaulay, Complete intersection, Edge ideal, FLC, Polarization, Simplicial complex, Symbolic powers",

author = "Giancarlo Rinaldo and Naoki Terai and Yoshida, {Ken ichi}",

note = "Funding Information: The second author was supported by JSPS 20540047. The third author was supported by JSPS 19340005. The authors would like to express their gratitude to the referee for his careful reading.",

year = "2011",

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doi = "10.1016/j.jalgebra.2011.09.007",

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AU - Rinaldo, Giancarlo

AU - Terai, Naoki

AU - Yoshida, Ken ichi

N1 - Funding Information: The second author was supported by JSPS 20540047. The third author was supported by JSPS 19340005. The authors would like to express their gratitude to the referee for his careful reading.

PY - 2011/12/1

Y1 - 2011/12/1

N2 - Let S=K[x1,..., xn] be a polynomial ring over a field K. Let I(G)⊆S denote the edge ideal of a graph G. We show that the ℓth symbolic power I(G)(ℓ) is a Cohen-Macaulay ideal (i.e., S/I(G)(ℓ) is Cohen-Macaulay) for some integer ℓ≥3 if and only if G is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers I(G)(ℓ) are Cohen-Macaulay ideals. Similarly, we characterize graphs G for which S/I(G)(ℓ) has (FLC).As an application, we show that an edge ideal I(G) is complete intersection provided that S/I(G)ℓ is Cohen-Macaulay for some integer ℓ≥3. This strengthens the main theorem in Crupi et al. (2010) [3].

AB - Let S=K[x1,..., xn] be a polynomial ring over a field K. Let I(G)⊆S denote the edge ideal of a graph G. We show that the ℓth symbolic power I(G)(ℓ) is a Cohen-Macaulay ideal (i.e., S/I(G)(ℓ) is Cohen-Macaulay) for some integer ℓ≥3 if and only if G is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers I(G)(ℓ) are Cohen-Macaulay ideals. Similarly, we characterize graphs G for which S/I(G)(ℓ) has (FLC).As an application, we show that an edge ideal I(G) is complete intersection provided that S/I(G)ℓ is Cohen-Macaulay for some integer ℓ≥3. This strengthens the main theorem in Crupi et al. (2010) [3].

KW - Cohen-Macaulay

KW - Complete intersection

KW - Edge ideal

KW - FLC

KW - Polarization

KW - Simplicial complex

KW - Symbolic powers

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Cohen-Macaulayness for symbolic power ideals of edge ideals

Abstract

Keywords

ASJC Scopus subject areas

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