Gorenstein and Sr path ideals of cycles

Dariush Kiani; Sara Seedi Madani; Naoki Terai

doi:10.1017/S0017089514000111

Gorenstein and S_r path ideals of cycles

Dariush Kiani, Sara Seedi Madani, Naoki Terai

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

1 Citation (Scopus)

Abstract

Let R = k[x 1,..,x n], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cn be an n-cycle. We show that R/It(Cn) is Sr if and only if it is Cohen-Macaulay or n n-t-1 r+3. In addition, we prove that R/It(Cn) is Gorenstein if and only if n = t or 2t + 1. Also, we determine all ordinary and symbolic powers of It(Cn) which are Cohen-Macaulay. Finally, we prove that It(Cn) has a linear resolution if and only if t ≥ n-2.

Original language	English
Pages (from-to)	7-15
Number of pages	9
Journal	Glasgow Mathematical Journal
Volume	57
Issue number	1
DOIs	https://doi.org/10.1017/S0017089514000111
Publication status	Published - Jan 25 2015
Externally published	Yes

Keywords

05C38
05C75
13F55
2010 Mathematics Subject Classification 13D02

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1017/S0017089514000111

Cite this

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title = "Gorenstein and Sr path ideals of cycles",

abstract = "Let R = k[x 1,..,x n], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cn be an n-cycle. We show that R/It(Cn) is Sr if and only if it is Cohen-Macaulay or n n-t-1 r+3. In addition, we prove that R/It(Cn) is Gorenstein if and only if n = t or 2t + 1. Also, we determine all ordinary and symbolic powers of It(Cn) which are Cohen-Macaulay. Finally, we prove that It(Cn) has a linear resolution if and only if t ≥ n-2.",

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T1 - Gorenstein and Sr path ideals of cycles

AU - Kiani, Dariush

AU - Madani, Sara Seedi

AU - Terai, Naoki

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N2 - Let R = k[x 1,..,x n], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cn be an n-cycle. We show that R/It(Cn) is Sr if and only if it is Cohen-Macaulay or n n-t-1 r+3. In addition, we prove that R/It(Cn) is Gorenstein if and only if n = t or 2t + 1. Also, we determine all ordinary and symbolic powers of It(Cn) which are Cohen-Macaulay. Finally, we prove that It(Cn) has a linear resolution if and only if t ≥ n-2.

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