Binomial arithmetical rank of edge ideals of forests
Authors:
Kyouko Kimura and Naoki Terai
Journal:
Proc. Amer. Math. Soc. 141 (2013), 19251932
MSC (2010):
Primary 13F55, 05C05
Published electronically:
January 2, 2013
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We prove that the binomial arithmetical rank of the edge ideal of a forest coincides with its big height.
 1.
M. Barile, On the arithmetical rank of the edge ideals of forests, Comm. Algebra 36 (2008), 46784703. MR 2473354 (2009j:13028)
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M. Barile, D. Kiani, F. Mohammadi and S. Yassemi, Arithmetical rank of the cyclic and bicyclic graphs, J. Algebra Appl. 11 (2012), no. 2, 1250039, 14 pp. MR 2925452
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M. Barile and N. Terai, Arithmetical ranks of StanleyReisner ideals of simplicial complexes with a cone, Comm. Algebra 38 (2010), 36863698. MR 2760684 (2011j:13033)
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M. Barile and N. Terai, The StanleyReisner ideals of polygons as settheoretic complete intersections, Comm. Algebra 39 (2011), 621633. MR 2773327
 5.
V. Ene, O. Olteanu and N. Terai. Arithmetical rank of lexsegment edge ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53 (101) (2010), 315327. MR 2777678 (2011m:13037)
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J. He and A. Van Tuyl, Algebraic properties of the path ideals of a tree, Comm. Algebra 38 (2010), 17251742. MR 2642022 (2011e:13028)
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K. Kimura, Arithmetical rank of CohenMacaulay squarefree monomial ideals of height two, J. Commut. Algebra 3 (2011), 3146. MR 2782698
 8.
K. Kimura, G. Rinaldo and N. Terai, Arithmetical rank of squarefree monomial ideals generated by five elements or with arithmetic degree four, to appear in Comm. Algebra.
 9.
K. Kimura, N. Terai and K. Yoshida, Arithmetical rank of squarefree monomial ideals of small arithmetic degree, J. Algebraic Combin. 29 (2009), 389404. MR 2496313 (2009m:13030)
 10.
K. Kimura, N. Terai and K. Yoshida, Arithmetical rank of monomial ideals of deviation two, in Combinatorial Aspects of Commutative Algebra (V. Ene and E. Miller, eds.), Contemporary Mathematics, 502, AMS (2009), 73112. MR 2583275 (2011c:13043)
 11.
M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30 (2009), 429445. MR 2563135 (2010j:13040)
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G. Lyubeznik, On the local cohomology modules for ideals generated by monomials in an sequence, in Complete Intersections, Acireale, 1983 (S. Greco and R. Strano, eds.), Lecture Notes in Mathematics, No. 1092, SpringerVerlag, 1984, pp. 214220. MR 775884 (86f:14002)
 13.
M. Morales, Simplicial ideals, linear ideals and arithmetical rank, J. Algebra 324 (2010), 34313456. MR 2735392 (2011j:13028)
 14.
P. Mongelli, The arithmetical rank of a special class of monomial ideals, preprint, arXiv:1005.2586.
 15.
T. Schmitt and W. Vogel, Note on settheoretic intersections of subvarieties of projective space, Math. Ann. 245 (1979), 247253. MR 553343 (81a:14025)
 16.
M. Varbaro, Symbolic powers and matroids, Proc. Amer. Math. Soc. 139 (2011), 23572366. MR 2784800
 1.
 M. Barile, On the arithmetical rank of the edge ideals of forests, Comm. Algebra 36 (2008), 46784703. MR 2473354 (2009j:13028)
 2.
 M. Barile, D. Kiani, F. Mohammadi and S. Yassemi, Arithmetical rank of the cyclic and bicyclic graphs, J. Algebra Appl. 11 (2012), no. 2, 1250039, 14 pp. MR 2925452
 3.
 M. Barile and N. Terai, Arithmetical ranks of StanleyReisner ideals of simplicial complexes with a cone, Comm. Algebra 38 (2010), 36863698. MR 2760684 (2011j:13033)
 4.
 M. Barile and N. Terai, The StanleyReisner ideals of polygons as settheoretic complete intersections, Comm. Algebra 39 (2011), 621633. MR 2773327
 5.
 V. Ene, O. Olteanu and N. Terai. Arithmetical rank of lexsegment edge ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53 (101) (2010), 315327. MR 2777678 (2011m:13037)
 6.
 J. He and A. Van Tuyl, Algebraic properties of the path ideals of a tree, Comm. Algebra 38 (2010), 17251742. MR 2642022 (2011e:13028)
 7.
 K. Kimura, Arithmetical rank of CohenMacaulay squarefree monomial ideals of height two, J. Commut. Algebra 3 (2011), 3146. MR 2782698
 8.
 K. Kimura, G. Rinaldo and N. Terai, Arithmetical rank of squarefree monomial ideals generated by five elements or with arithmetic degree four, to appear in Comm. Algebra.
 9.
 K. Kimura, N. Terai and K. Yoshida, Arithmetical rank of squarefree monomial ideals of small arithmetic degree, J. Algebraic Combin. 29 (2009), 389404. MR 2496313 (2009m:13030)
 10.
 K. Kimura, N. Terai and K. Yoshida, Arithmetical rank of monomial ideals of deviation two, in Combinatorial Aspects of Commutative Algebra (V. Ene and E. Miller, eds.), Contemporary Mathematics, 502, AMS (2009), 73112. MR 2583275 (2011c:13043)
 11.
 M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30 (2009), 429445. MR 2563135 (2010j:13040)
 12.
 G. Lyubeznik, On the local cohomology modules for ideals generated by monomials in an sequence, in Complete Intersections, Acireale, 1983 (S. Greco and R. Strano, eds.), Lecture Notes in Mathematics, No. 1092, SpringerVerlag, 1984, pp. 214220. MR 775884 (86f:14002)
 13.
 M. Morales, Simplicial ideals, linear ideals and arithmetical rank, J. Algebra 324 (2010), 34313456. MR 2735392 (2011j:13028)
 14.
 P. Mongelli, The arithmetical rank of a special class of monomial ideals, preprint, arXiv:1005.2586.
 15.
 T. Schmitt and W. Vogel, Note on settheoretic intersections of subvarieties of projective space, Math. Ann. 245 (1979), 247253. MR 553343 (81a:14025)
 16.
 M. Varbaro, Symbolic powers and matroids, Proc. Amer. Math. Soc. 139 (2011), 23572366. MR 2784800
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Additional Information
Kyouko Kimura
Affiliation:
Department of Mathematics, Faculty of Science, Shizuoka University, 836 Ohya, Surugaku, Shizuoka 4228529, Japan
Email:
skkimur@ipc.shizuoka.ac.jp
Naoki Terai
Affiliation:
Department of Mathematics, Faculty of Culture and Education, Saga University, Saga 8408502, Japan
Email:
terai@cc.sagau.ac.jp
DOI:
http://dx.doi.org/10.1090/S000299392013114735
PII:
S 00029939(2013)114735
Keywords:
Binomial arithmetical rank,
primitive tree,
treelike system,
edge ideal
Received by editor(s):
June 27, 2011
Received by editor(s) in revised form:
September 26, 2011
Published electronically:
January 2, 2013
Communicated by:
Irena Peeva
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
