Lyubeznik resolutions and the arithmetical rank of monomial ideals
Author:
Kyouko Kimura
Journal:
Proc. Amer. Math. Soc. 137 (2009), 36273635
MSC (2000):
Primary 13E15; Secondary 13D02
Published electronically:
June 9, 2009
MathSciNet review:
2529869
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper, we prove that the length of a Lyubeznik resolution of a monomial ideal gives an upper bound for the arithmetical rank of the ideal.
 1.
Margherita
Barile, On the number of equations defining certain varieties,
Manuscripta Math. 91 (1996), no. 4, 483–494. MR 1421287
(97m:13041), http://dx.doi.org/10.1007/BF02567968
 2.
Margherita
Barile, On ideals whose radical is a monomial ideal, Comm.
Algebra 33 (2005), no. 12, 4479–4490. MR 2188323
(2006g:13039), http://dx.doi.org/10.1080/00927870500274812
 3.
Margherita
Barile, A note on monomial ideals, Arch. Math. (Basel)
87 (2006), no. 6, 516–521. MR 2283682
(2007h:13004), http://dx.doi.org/10.1007/s0001300618343
 4.
M. Barile, A note on the edge ideals of Ferrers graphs, preprint, arXiv:math.AC/0606353.
 5.
Margherita
Barile, On the arithmetical rank of the edge ideals of
forests, Comm. Algebra 36 (2008), no. 12,
4678–4703. MR 2473354
(2009j:13028), http://dx.doi.org/10.1080/00927870802161220
 6.
M. Barile, On the arithmetical rank of certain monomial ideals, preprint, arXiv:math.AC/0611790.
 7.
Margherita
Barile, Arithmetical ranks of StanleyReisner ideals via linear
algebra, Comm. Algebra 36 (2008), no. 12,
4540–4556. MR 2473347
(2009h:13030), http://dx.doi.org/10.1080/00927870802182614
 8.
M. Barile and N. Terai, Arithmetical ranks of StanleyReisner ideals of simplicial complexes with a cone, preprint, arXiv:0809.2194.
 9.
K. Kimura, N. Terai, and K. Yoshida, Arithmetical rank of squarefree monomial ideals of small arithmetic degree, J. Algebraic Combin. 29 (2009), 389404.
 10.
K. Kimura, N. Terai, and K. Yoshida, Arithmetical rank of squarefree monomial ideals of deviation two, submitted.
 11.
K. Kimura, N. Terai, and K. Yoshida, Arithmetical rank of squarefree monomial ideals whose Alexander duals have deviation two, in preparation.
 12.
Gennady
Lyubeznik, On the local cohomology modules
𝐻ⁱ_{𝔞}(ℜ) for ideals 𝔞 generated by
monomials in an ℜsequence, Complete intersections (Acireale,
1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984,
pp. 214–220. MR 775884
(86f:14002), http://dx.doi.org/10.1007/BFb0099364
 13.
Gennady
Lyubeznik, A new explicit finite free resolution of ideals
generated by monomials in an 𝑅sequence, J. Pure Appl. Algebra
51 (1988), no. 12, 193–195. MR 941900
(89c:13020), http://dx.doi.org/10.1016/00224049(88)900886
 14.
Isabella
Novik, Lyubeznik’s resolution and rooted complexes, J.
Algebraic Combin. 16 (2002), no. 1, 97–101. MR 1941987
(2003j:13021), http://dx.doi.org/10.1023/A:1020838732281
 15.
Thomas
Schmitt and Wolfgang
Vogel, Note on settheoretic intersections of subvarieties of
projective space, Math. Ann. 245 (1979), no. 3,
247–253. MR
553343 (81a:14025), http://dx.doi.org/10.1007/BF01673509
 16.
D. Taylor, Ideals generated by monomials in an sequence, Ph.D. Thesis, Chicago University (1960).
 17.
Zhao
Yan, An étale analog of the GoreskyMacPherson formula for
subspace arrangements, J. Pure Appl. Algebra 146
(2000), no. 3, 305–318. MR 1742346
(2000k:14041), http://dx.doi.org/10.1016/S00224049(98)001285
 1.
 M. Barile, On the number of equations defining certain varieties, Manuscripta Math. 91 (1996), 483494. MR 1421287 (97m:13041)
 2.
 M. Barile, On ideals whose radical is a monomial ideal, Comm. Algebra 33 (2005), 44794490. MR 2188323 (2006g:13039)
 3.
 M. Barile, A note on monomial ideals, Arch. Math. (Basel) 87 (2006), 516521. MR 2283682 (2007h:13004)
 4.
 M. Barile, A note on the edge ideals of Ferrers graphs, preprint, arXiv:math.AC/0606353.
 5.
 M. Barile, On the arithmetical rank of the edge ideals of forests, Comm. Algebra 36 (2008), 46784703. MR 2473354
 6.
 M. Barile, On the arithmetical rank of certain monomial ideals, preprint, arXiv:math.AC/0611790.
 7.
 M. Barile, Arithmetical ranks of StanleyReisner ideals via linear algebra, Comm. Algebra 36 (2008), 45404556. MR 2473347
 8.
 M. Barile and N. Terai, Arithmetical ranks of StanleyReisner ideals of simplicial complexes with a cone, preprint, arXiv:0809.2194.
 9.
 K. Kimura, N. Terai, and K. Yoshida, Arithmetical rank of squarefree monomial ideals of small arithmetic degree, J. Algebraic Combin. 29 (2009), 389404.
 10.
 K. Kimura, N. Terai, and K. Yoshida, Arithmetical rank of squarefree monomial ideals of deviation two, submitted.
 11.
 K. Kimura, N. Terai, and K. Yoshida, Arithmetical rank of squarefree monomial ideals whose Alexander duals have deviation two, in preparation.
 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.
 G. Lyubeznik, A new explicit finite free resolution of ideals generated by monomials in an sequence, J. Pure Appl. Algebra 51 (1988), 193195. MR 941900 (89c:13020)
 14.
 I. Novik, Lyubeznik's resolution and rooted complexes, J. Algebraic Combin. 16 (2002), 97101. MR 1941987 (2003j:13021)
 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.
 D. Taylor, Ideals generated by monomials in an sequence, Ph.D. Thesis, Chicago University (1960).
 17.
 Z. Yan, An étale analog of the GoreskyMacPherson formula for subspace arrangements, J. Pure Appl. Algebra 146 (2000), 305318. MR 1742346 (2000k:14041)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
13E15,
13D02
Retrieve articles in all journals
with MSC (2000):
13E15,
13D02
Additional Information
Kyouko Kimura
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya 4648602, Japan
Address at time of publication:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 5600043, Japan
Email:
m04012w@math.nagoyau.ac.jp, kimura@math.sci.osakau.ac.jp
DOI:
http://dx.doi.org/10.1090/S000299390909950X
PII:
S 00029939(09)09950X
Keywords:
Lyubeznik resolution,
$L$admissible,
$L$length,
arithmetical rank
Received by editor(s):
December 1, 2008
Received by editor(s) in revised form:
February 26, 2009
Published electronically:
June 9, 2009
Communicated by:
Bernd Ulrich
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
