Monomial ideals, almost complete intersections and the Weak Lefschetz property
Authors:
Juan C. Migliore, Rosa M. MiróRoig and Uwe Nagel
Journal:
Trans. Amer. Math. Soc. 363 (2011), 229257
MSC (2010):
Primary 13D40, 13E10, 13C13; Secondary 13C40, 13D02, 14J60
Published electronically:
August 17, 2010
MathSciNet review:
2719680
Fulltext 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
(88d:13016a), 10.1016/00218693(86)900761
 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
(94h:13016), 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 (2009g:14049), 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
(2009j:13012)
 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
𝐾algebras, J. Algebra 262 (2003),
no. 1, 99–126. MR 1970804
(2004b:13001), 10.1016/S00218693(03)000383
 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
MiróRoig, 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
(2002e:14083), 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
(2000g:14058)
 10.
J.
Migliore and R.
M. MiróRoig, Ideals of general forms and the ubiquity of
the weak Lefschetz property, J. Pure Appl. Algebra
182 (2003), no. 1, 79–107. MR 1978001
(2004c:13027), 10.1016/S00224049(02)003146
 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
(2009b:13037), 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
(82j:20083), 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 (81f:52014), 10.1016/00018708(80)90050X
 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, NorthHolland, Amsterdam,
1987, pp. 303–312. MR 951211
(89k:13015)
 1.
 D. Anick, Thin algebras of embedding dimension three, J. Algebra 100 (1986), 235259. MR 839581 (88d:13016a)
 2.
 M. Beintema, A note on Artinian Gorenstein algebras defined by monomials, Rocky Mountain J. Math. 23 (1993), 13. MR 1212726 (94h:13016)
 3.
 H. Brenner, Looking out for stable syzygy bundles, Adv. Math. 219 (2008), 401427. MR 2435644 (2009g:14049)
 4.
 H. Brenner and A. Kaid, Syzygy bundles on and the Weak Lefschetz Property, Illinois J. Math. 51 (2007), 12991308. MR 2417428 (2009j:13012)
 5.
 CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it.
 6.
 T. Harima, J. Migliore, U. Nagel and J. Watanabe, The weak and strong Lefschetz properties for Artinian algebras, J. Algebra 262 (2003), 99126. MR 1970804 (2004b:13001)
 7.
 J. Herzog and D. Popescu, The strong Lefschetz property and simple extensions, arXiv:math.AC/0506537.
 8.
 J. Kleppe, R. MiróRoig, J. Migliore, and C. Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 154 (2001), no. 732. MR 1848976 (2002e:14083)
 9.
 J. Migliore, ``Introduction to Liaison Theory and Deficiency Modules,'' Progress in Mathematics 165, Birkhäuser, 1998. MR 1712469 (2000g:14058)
 10.
 J. Migliore and R. MiróRoig, Ideals of general forms and the ubiquity of the Weak Lefschetz Property, J. Pure Appl. Algebra 182 (2003), 79107. MR 1978001 (2004c:13027)
 11.
 J. Migliore, U. Nagel and F. Zanello, A characterization of Gorenstein Hilbert functions in codimension four with small initial degree, Math. Res. Lett. 15 (2008), 331349. MR 2385645 (2009b:13037)
 12.
 R. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), 168184. MR 578321 (82j:20083)
 13.
 R. Stanley, The number of faces of a simplicial convex polytope, Adv. Math. 35 (1980), 236238. MR 563925 (81f:52014)
 14.
 J. Watanabe, The Dilworth number of Artinian rings and finite posets with rank function, Commutative Algebra and Combinatorics, Advanced Studies in Pure Math., Vol. 11, North Holland, Amsterdam (1987), 303312. MR 951211 (89k:13015)
Similar Articles
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. MiróRoig
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 405060027
Email:
uwenagel@ms.uky.edu
DOI:
http://dx.doi.org/10.1090/S00029947201005127X
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 H982300710036.
Part of the work for this paper was done while the second author was partially supported by MTM200761104.
Part of the work for this paper was done while the third author was sponsored by the National Security Agency under Grant Number H982300710065. 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
