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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Stanley depth of powers of the edge ideal of a forest


Authors: M. R. Pournaki, S. A. Seyed Fakhari and S. Yassemi
Journal: Proc. Amer. Math. Soc. 141 (2013), 3327-3336
MSC (2010): Primary 13C15, 05E99; Secondary 13C13
Published electronically: June 7, 2013
MathSciNet review: 3080155
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Abstract: Let $ \mathbb{K}$ be a field and $ S=\mathbb{K}[x_1,\dots ,x_n]$ be the polynomial ring in $ n$ variables over the field $ \mathbb{K}$. Let $ G$ be a forest with $ p$ connected components $ G_1,\ldots ,G_p$ and let $ I=I(G)$ be its edge ideal in $ S$. Suppose that $ d_i$ is the diameter of $ G_i$, $ 1\leq i\leq p$, and consider $ d =\max \hspace {0.04cm}\{d_i\mid 1\leq i\leq p\}$. Morey has shown that for every $ t\geq 1$, the quantity $ \max \{\lceil \frac {d-t+2}{3}\rceil +p-1,p\}$ is a lower bound for $ {\rm depth}(S/I^t)$. In this paper, we show that for every $ t\geq 1$, the mentioned quantity is also a lower bound for $ {\rm sdepth}(S/I^t)$. By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.


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Additional Information

M. R. Pournaki
Affiliation: Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
Email: pournaki@ipm.ir

S. A. Seyed Fakhari
Affiliation: Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran
Email: fakhari@ipm.ir

S. Yassemi
Affiliation: School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
Email: yassemi@ipm.ir

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11594-7
Keywords: Edge ideal, monomial ideal, Stanley depth, Stanley conjecture
Received by editor(s): August 25, 2011
Received by editor(s) in revised form: December 10, 2011
Published electronically: June 7, 2013
Additional Notes: The research of the first and third authors was partially supported by grants from IPM (No. 90130073 and No. 90130214)
Communicated by: Irena Peeva
Article copyright: © Copyright 2013 American Mathematical Society