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GORENSTEIN AND Sr PATH IDEALS OF CYCLES

Published online by Cambridge University Press:  26 August 2014

DARIUSH KIANI ,
SARA SAEEDI MADANI  and
NAOKI TERAI
DARIUSH KIANI
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Ave., Tehran 15914, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran e-mail: dkiani@aut.ac.ir, dkiani7@gmail.com
SARA SAEEDI MADANI
Affiliation:
e-mail: sarasaeedi@aut.ac.ir
NAOKI TERAI
Affiliation:
Naoki Terai, Department of Mathematics, Faculty of Culture and Education, Saga University, Saga 840-8502, Japan e-mail: terai@cc.saga-u.ac.jp
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Abstract

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Let R = k[x1,…,xn], 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 $\lceil \frac{n}{n-t-1}\rceil\geq r+3$. 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.

Keywords

13D0213F5505C7505C38

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 1 , January 2015 , pp. 7 - 15
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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