## Splittings of monomial ideals

HTML articles powered by AMS MathViewer

- by Christopher A. Francisco, Huy Tài Hà and Adam Van Tuyl
- Proc. Amer. Math. Soc.
**137**(2009), 3271-3282 - DOI: https://doi.org/10.1090/S0002-9939-09-09929-8
- Published electronically: May 7, 2009
- PDF | Request permission

## Abstract:

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire’s splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.## References

- CoCoATeam, CoCoA: A system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it
- John A. Eagon and Victor Reiner,
*Resolutions of Stanley-Reisner rings and Alexander duality*, J. Pure Appl. Algebra**130**(1998), no. 3, 265–275. MR**1633767**, DOI 10.1016/S0022-4049(97)00097-2 - Shalom Eliahou and Michel Kervaire,
*Minimal resolutions of some monomial ideals*, J. Algebra**129**(1990), no. 1, 1–25. MR**1037391**, DOI 10.1016/0021-8693(90)90237-I - G. Fatabbi,
*On the resolution of ideals of fat points*, J. Algebra**242**(2001), no. 1, 92–108. MR**1844699**, DOI 10.1006/jabr.2001.8798 - Christopher A. Francisco,
*Resolutions of small sets of fat points*, J. Pure Appl. Algebra**203**(2005), no. 1-3, 220–236. MR**2176661**, DOI 10.1016/j.jpaa.2005.03.004 - Christopher A. Francisco and Huy Tài Hà,
*Whiskers and sequentially Cohen-Macaulay graphs*, J. Combin. Theory Ser. A**115**(2008), no. 2, 304–316. MR**2382518**, DOI 10.1016/j.jcta.2007.06.004 - Christopher A. Francisco and Adam Van Tuyl,
*Sequentially Cohen-Macaulay edge ideals*, Proc. Amer. Math. Soc.**135**(2007), no. 8, 2327–2337. MR**2302553**, DOI 10.1090/S0002-9939-07-08841-7 - D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/
- Huy Tài Hà and Adam Van Tuyl,
*Splittable ideals and the resolutions of monomial ideals*, J. Algebra**309**(2007), no. 1, 405–425. MR**2301246**, DOI 10.1016/j.jalgebra.2006.08.022 - Huy Tài Hà and Adam Van Tuyl,
*Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers*, J. Algebraic Combin.**27**(2008), no. 2, 215–245. MR**2375493**, DOI 10.1007/s10801-007-0079-y - Jürgen Herzog and Takayuki Hibi,
*Distributive lattices, bipartite graphs and Alexander duality*, J. Algebraic Combin.**22**(2005), no. 3, 289–302. MR**2181367**, DOI 10.1007/s10801-005-4528-1 - Mordechai Katzman,
*Characteristic-independence of Betti numbers of graph ideals*, J. Combin. Theory Ser. A**113**(2006), no. 3, 435–454. MR**2209703**, DOI 10.1016/j.jcta.2005.04.005 - M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals. Preprint, 2009. arXiv:0902.0437
- F. Mohammadi and S. Moradi, Resolution of unmixed bipartite graphs. Preprint, 2009. arXiv.0901.3015v1
- Giuseppe Valla,
*Betti numbers of some monomial ideals*, Proc. Amer. Math. Soc.**133**(2005), no. 1, 57–63. MR**2085153**, DOI 10.1090/S0002-9939-04-07557-4 - Rafael H. Villarreal,
*Cohen-Macaulay graphs*, Manuscripta Math.**66**(1990), no. 3, 277–293. MR**1031197**, DOI 10.1007/BF02568497 - Xinxian Zheng,
*Resolutions of facet ideals*, Comm. Algebra**32**(2004), no. 6, 2301–2324. MR**2100472**, DOI 10.1081/AGB-120037222

## Bibliographic Information

**Christopher A. Francisco**- Affiliation: Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, Oklahoma 74078
- MR Author ID: 719806
- Email: chris@math.okstate.edu
**Huy Tài Hà**- Affiliation: Department of Mathematics, Tulane University, 6823 St. Charles Avenue, New Orleans, Louisiana 70118
- ORCID: 0000-0002-6002-3453
- Email: tai@math.tulane.edu
**Adam Van Tuyl**- Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada
- MR Author ID: 649491
- ORCID: 0000-0002-6799-6653
- Email: avantuyl@lakeheadu.ca
- Received by editor(s): July 14, 2008
- Received by editor(s) in revised form: February 13, 2009
- Published electronically: May 7, 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), 3271-3282 - MSC (2000): Primary 13D02, 13P10, 13F55, 05C99
- DOI: https://doi.org/10.1090/S0002-9939-09-09929-8
- MathSciNet review: 2515396