Lyubeznik resolutions and the arithmetical rank of monomial ideals

Author:
Kyouko Kimura

Journal:
Proc. Amer. Math. Soc. **137** (2009), 3627-3635

MSC (2000):
Primary 13E15; Secondary 13D02

DOI:
https://doi.org/10.1090/S0002-9939-09-09950-X

Published electronically:
June 9, 2009

MathSciNet review:
2529869

Full-text PDF Free Access

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**, https://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**, https://doi.org/10.1080/00927870500274812**3.**Margherita Barile,*A note on monomial ideals*, Arch. Math. (Basel)**87**(2006), no. 6, 516–521. MR**2283682**, https://doi.org/10.1007/s00013-006-1834-3**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**, https://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 Stanley-Reisner ideals via linear algebra*, Comm. Algebra**36**(2008), no. 12, 4540–4556. MR**2473347**, https://doi.org/10.1080/00927870802182614**8.**M. Barile and N. Terai,*Arithmetical ranks of Stanley-Reisner 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), 389-404.**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 ℜ-sequence*, Complete intersections (Acireale, 1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 214–220. MR**775884**, https://doi.org/10.1007/BFb0099364**13.**Gennady Lyubeznik,*A new explicit finite free resolution of ideals generated by monomials in an 𝑅-sequence*, J. Pure Appl. Algebra**51**(1988), no. 1-2, 193–195. MR**941900**, https://doi.org/10.1016/0022-4049(88)90088-6**14.**Isabella Novik,*Lyubeznik’s resolution and rooted complexes*, J. Algebraic Combin.**16**(2002), no. 1, 97–101. MR**1941987**, https://doi.org/10.1023/A:1020838732281**15.**Thomas Schmitt and Wolfgang Vogel,*Note on set-theoretic intersections of subvarieties of projective space*, Math. Ann.**245**(1979), no. 3, 247–253. MR**553343**, https://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 Goresky-MacPherson formula for subspace arrangements*, J. Pure Appl. Algebra**146**(2000), no. 3, 305–318. MR**1742346**, https://doi.org/10.1016/S0022-4049(98)00128-5

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 464-8602, Japan

Address at time of publication:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan

Email:
m04012w@math.nagoya-u.ac.jp, kimura@math.sci.osaka-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-09-09950-X

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.