Monomial ideals, almost complete intersections and the Weak Lefschetz property

Authors:
Juan C. Migliore, Rosa M. Miró-Roig and Uwe Nagel

Journal:
Trans. Amer. Math. Soc. **363** (2011), 229-257

MSC (2010):
Primary 13D40, 13E10, 13C13; Secondary 13C40, 13D02, 14J60

Published electronically:
August 17, 2010

MathSciNet review:
2719680

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Many algebras are expected to have the Weak Lefschetz property, although this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field and on arithmetic properties of the exponent vectors of the monomials.

**1.**David J. Anick,*Thin algebras of embedding dimension three*, J. Algebra**100**(1986), no. 1, 235–259. MR**839581**, 10.1016/0021-8693(86)90076-1**2.**Mark B. Beintema,*A note on Artinian Gorenstein algebras defined by monomials*, Rocky Mountain J. Math.**23**(1993), no. 1, 1–3. MR**1212726**, 10.1216/rmjm/1181072606**3.**Holger Brenner,*Looking out for stable syzygy bundles*, Adv. Math.**219**(2008), no. 2, 401–427. With an appendix by Georg Hein. MR**2435644**, 10.1016/j.aim.2008.04.009**4.**Holger Brenner and Almar Kaid,*Syzygy bundles on ℙ² and the weak Lefschetz property*, Illinois J. Math.**51**(2007), no. 4, 1299–1308. MR**2417428****5.**CoCoA: a system for doing Computations in Commutative Algebra, Available at`http://cocoa.dima.unige.it`.**6.**Tadahito Harima, Juan C. Migliore, Uwe Nagel, and Junzo Watanabe,*The weak and strong Lefschetz properties for Artinian 𝐾-algebras*, J. Algebra**262**(2003), no. 1, 99–126. MR**1970804**, 10.1016/S0021-8693(03)00038-3**7.**J. Herzog and D. Popescu,*The strong Lefschetz property and simple extensions*, arXiv:math.AC/0506537.**8.**Jan O. Kleppe, Juan C. Migliore, Rosa Miró-Roig, Uwe Nagel, and Chris Peterson,*Gorenstein liaison, complete intersection liaison invariants and unobstructedness*, Mem. Amer. Math. Soc.**154**(2001), no. 732, viii+116. MR**1848976**, 10.1090/memo/0732**9.**Juan C. Migliore,*Introduction to liaison theory and deficiency modules*, Progress in Mathematics, vol. 165, Birkhäuser Boston, Inc., Boston, MA, 1998. MR**1712469****10.**J. Migliore and R. M. Miró-Roig,*Ideals of general forms and the ubiquity of the weak Lefschetz property*, J. Pure Appl. Algebra**182**(2003), no. 1, 79–107. MR**1978001**, 10.1016/S0022-4049(02)00314-6**11.**Juan Migliore, Uwe Nagel, and Fabrizio Zanello,*A characterization of Gorenstein Hilbert functions in codimension four with small initial degree*, Math. Res. Lett.**15**(2008), no. 2, 331–349. MR**2385645**, 10.4310/MRL.2008.v15.n2.a11**12.**Richard P. Stanley,*Weyl groups, the hard Lefschetz theorem, and the Sperner property*, SIAM J. Algebraic Discrete Methods**1**(1980), no. 2, 168–184. MR**578321**, 10.1137/0601021**13.**Richard P. Stanley,*The number of faces of a simplicial convex polytope*, Adv. in Math.**35**(1980), no. 3, 236–238. MR**563925**, 10.1016/0001-8708(80)90050-X**14.**Junzo Watanabe,*The Dilworth number of Artinian rings and finite posets with rank function*, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 303–312. MR**951211**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
13D40,
13E10,
13C13,
13C40,
13D02,
14J60

Retrieve articles in all journals with MSC (2010): 13D40, 13E10, 13C13, 13C40, 13D02, 14J60

Additional Information

**Juan C. Migliore**

Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Email:
migliore.1@nd.edu

**Rosa M. Miró-Roig**

Affiliation:
Facultat de Matemàtiques, Department d’Algebra i Geometria, University of Barce- lona, Gran Via des les Corts Catalanes 585, 08007 Barcelona, Spain

Email:
miro@ub.edu

**Uwe Nagel**

Affiliation:
Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027

Email:
uwenagel@ms.uky.edu

DOI:
https://doi.org/10.1090/S0002-9947-2010-05127-X

Received by editor(s):
January 13, 2009

Published electronically:
August 17, 2010

Additional Notes:
Part of the work for this paper was done while the first author was sponsored by the National Security Agency under Grant Number H98230-07-1-0036.

Part of the work for this paper was done while the second author was partially supported by MTM2007-61104.

Part of the work for this paper was done while the third author was sponsored by the National Security Agency under Grant Number H98230-07-1-0065. The authors thank Fabrizio Zanello for useful and enjoyable conversations related to some of this material. They also thank David Cook II for useful comments.

Article copyright:
© Copyright 2010
American Mathematical Society