Monomial ideals, almost complete intersections and the Weak Lefschetz property
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- by Juan C. Migliore, Rosa M. Miró-Roig and Uwe Nagel PDF
- Trans. Amer. Math. Soc. 363 (2011), 229-257 Request permission
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.References
- David J. Anick, Thin algebras of embedding dimension three, J. Algebra 100 (1986), no. 1, 235–259. MR 839581, DOI 10.1016/0021-8693(86)90076-1
- Mark B. Beintema, A note on Artinian Gorenstein algebras defined by monomials, Rocky Mountain J. Math. 23 (1993), no. 1, 1–3. MR 1212726, DOI 10.1216/rmjm/1181072606
- Holger Brenner, Looking out for stable syzygy bundles, Adv. Math. 219 (2008), no. 2, 401–427. With an appendix by Georg Hein. MR 2435644, DOI 10.1016/j.aim.2008.04.009
- Holger Brenner and Almar Kaid, Syzygy bundles on $\Bbb P^2$ and the weak Lefschetz property, Illinois J. Math. 51 (2007), no. 4, 1299–1308. MR 2417428
- CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it.
- 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, DOI 10.1016/S0021-8693(03)00038-3
- J. Herzog and D. Popescu, The strong Lefschetz property and simple extensions, arXiv:math.AC/0506537.
- 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, DOI 10.1090/memo/0732
- Juan C. Migliore, Introduction to liaison theory and deficiency modules, Progress in Mathematics, vol. 165, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1712469, DOI 10.1007/978-1-4612-1794-7
- 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, DOI 10.1016/S0022-4049(02)00314-6
- 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, DOI 10.4310/MRL.2008.v15.n2.a11
- 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, DOI 10.1137/0601021
- Richard P. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925, DOI 10.1016/0001-8708(80)90050-X
- 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, DOI 10.2969/aspm/01110303
Additional Information
- Juan C. Migliore
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- MR Author ID: 124490
- ORCID: 0000-0001-5528-4520
- 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
- MR Author ID: 125375
- ORCID: 0000-0003-1375-6547
- Email: miro@ub.edu
- Uwe Nagel
- Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027
- MR Author ID: 248652
- Email: uwenagel@ms.uky.edu
- 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. - © Copyright 2010 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 363 (2011), 229-257
- MSC (2010): Primary 13D40, 13E10, 13C13; Secondary 13C40, 13D02, 14J60
- DOI: https://doi.org/10.1090/S0002-9947-2010-05127-X
- MathSciNet review: 2719680