## Lyubeznik resolutions and the arithmetical rank of monomial ideals

HTML articles powered by AMS MathViewer

- by Kyouko Kimura
- Proc. Amer. Math. Soc.
**137**(2009), 3627-3635 - DOI: https://doi.org/10.1090/S0002-9939-09-09950-X
- Published electronically: June 9, 2009
- PDF | Request permission

## 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.## References

- Margherita Barile,
*On the number of equations defining certain varieties*, Manuscripta Math.**91**(1996), no. 4, 483–494. MR**1421287**, DOI 10.1007/BF02567968 - Margherita Barile,
*On ideals whose radical is a monomial ideal*, Comm. Algebra**33**(2005), no. 12, 4479–4490. MR**2188323**, DOI 10.1080/00927870500274812 - Margherita Barile,
*A note on monomial ideals*, Arch. Math. (Basel)**87**(2006), no. 6, 516–521. MR**2283682**, DOI 10.1007/s00013-006-1834-3 - M. Barile,
*A note on the edge ideals of Ferrers graphs*, preprint, arXiv:math.AC/0606353. - Margherita Barile,
*On the arithmetical rank of the edge ideals of forests*, Comm. Algebra**36**(2008), no. 12, 4678–4703. MR**2473354**, DOI 10.1080/00927870802161220 - M. Barile,
*On the arithmetical rank of certain monomial ideals*, preprint, arXiv:math.AC/0611790. - Margherita Barile,
*Arithmetical ranks of Stanley-Reisner ideals via linear algebra*, Comm. Algebra**36**(2008), no. 12, 4540–4556. MR**2473347**, DOI 10.1080/00927870802182614 - M. Barile and N. Terai,
*Arithmetical ranks of Stanley–Reisner ideals of simplicial complexes with a cone*, preprint, arXiv:0809.2194. - K. Kimura, N. Terai, and K. Yoshida,
*Arithmetical rank of squarefree monomial ideals of small arithmetic degree*, J. Algebraic Combin.**29**(2009), 389–404. - K. Kimura, N. Terai, and K. Yoshida,
*Arithmetical rank of squarefree monomial ideals of deviation two*, submitted. - K. Kimura, N. Terai, and K. Yoshida,
*Arithmetical rank of squarefree monomial ideals whose Alexander duals have deviation two*, in preparation. - Gennady Lyubeznik,
*On the local cohomology modules $H^i_{{\mathfrak {a}}}(R)$ for ideals ${\mathfrak {a}}$ 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**, DOI 10.1007/BFb0099364 - 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**, DOI 10.1016/0022-4049(88)90088-6 - Isabella Novik,
*Lyubeznik’s resolution and rooted complexes*, J. Algebraic Combin.**16**(2002), no. 1, 97–101. MR**1941987**, DOI 10.1023/A:1020838732281 - 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**, DOI 10.1007/BF01673509 - D. Taylor,
*Ideals generated by monomials in an $R$-sequence*, Ph.D. Thesis, Chicago University (1960). - 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**, DOI 10.1016/S0022-4049(98)00128-5

## Bibliographic 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
- 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
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2529869