Stanley depth of powers of the edge ideal of a forest
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- by M. R. Pournaki, S. A. Seyed Fakhari and S. Yassemi PDF
- Proc. Amer. Math. Soc. 141 (2013), 3327-3336 Request permission
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 $\textrm {depth}(S/I^t)$. In this paper, we show that for every $t\geq 1$, the mentioned quantity is also a lower bound for $\textrm {sdepth}(S/I^t)$. 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.References
- M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 1, 35–39. MR 530808, DOI 10.1017/S030500410000061X
- Lindsay Burch, Codimension and analytic spread, Proc. Cambridge Philos. Soc. 72 (1972), 369–373. MR 304377, DOI 10.1017/s0305004100047198
- Mircea Cimpoeas, Some remarks on the Stanley depth for multigraded modules, Matematiche (Catania) 63 (2008), no. 2, 165–171 (2009). MR 2531658
- Susan Morey, Depths of powers of the edge ideal of a tree, Comm. Algebra 38 (2010), no. 11, 4042–4055. MR 2764849, DOI 10.1080/00927870903286900
- M. R. Pournaki, S. A. Seyed Fakhari, M. Tousi, and S. Yassemi, What is $\dots$ Stanley depth?, Notices Amer. Math. Soc. 56 (2009), no. 9, 1106–1108. MR 2568497
- Asia Rauf, Stanley decompositions, pretty clean filtrations and reductions modulo regular elements, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 50(98) (2007), no. 4, 347–354. MR 2370321
- Asia Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra 38 (2010), no. 2, 773–784. MR 2598911, DOI 10.1080/00927870902829056
- Aron Simis, Wolmer V. Vasconcelos, and Rafael H. Villarreal, On the ideal theory of graphs, J. Algebra 167 (1994), no. 2, 389–416. MR 1283294, DOI 10.1006/jabr.1994.1192
- Richard P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175–193. MR 666158, DOI 10.1007/BF01394054
- Wolmer Vasconcelos, Integral closure, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. Rees algebras, multiplicities, algorithms. MR 2153889
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
- MR Author ID: 881160
- 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
- MR Author ID: 352988
- Email: yassemi@ipm.ir
- 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
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 141 (2013), 3327-3336
- MSC (2010): Primary 13C15, 05E99; Secondary 13C13
- DOI: https://doi.org/10.1090/S0002-9939-2013-11594-7
- MathSciNet review: 3080155