## Journal of Commutative Algebra

### The strong Lefschetz property in codimension two

David Cook, II

#### Abstract

Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of monomial ideals. Using this we classify when both bounds are sharp. Moreover, we prove that the artinian quotient of a monomial ideal in $K[x,y]$ always has the strong Lefschetz property, regardless of the characteristic of the field, exactly when the ideal is lexsegment. As a consequence, we describe a family of non-monomial complete intersections that always have the strong Lefschetz property.

#### Article information

Source
J. Commut. Algebra, Volume 6, Number 3 (2014), 323-344.

Dates
First available in Project Euclid: 17 November 2014

https://projecteuclid.org/euclid.jca/1416233321

Digital Object Identifier
doi:10.1216/JCA-2014-6-3-323

Mathematical Reviews number (MathSciNet)
MR3278807

Zentralblatt MATH identifier
1303.13019

#### Citation

Cook, David. The strong Lefschetz property in codimension two. J. Commut. Algebra 6 (2014), no. 3, 323--344. doi:10.1216/JCA-2014-6-3-323. https://projecteuclid.org/euclid.jca/1416233321

#### References

• R. Basili and A. Iarrobino, Pairs of commuting nilpotent matrices, and Hilbert function, J. Alg. 320 (2008), 1235–1254.
• M. Boij, J. Migliore, R. Miró-Roig, U. Nagel and F. Zanello, On the shape of a pure $O$-sequence, Mem. Amer. Math. Soc. 218 (2012), vii+78 pp.
• J. Briançon, Description de $\mbox{Hilb}^n \mathbb{C}\{x,y\}$, Invent. Math. 41 (1977), 45–89.
• K. Chandler, The geometric interpretation of Fröberg-Iarrobino conjectures on infinitesimal neighbourhoods of points in projective space, J. Alg. 286 (2005), 421–455.
• M. Ciucu, T. Eisenkölbl, C. Krattenthaler and D. Zare, Enumerations of lozenge tilings of hexagons with a central triangular hole, J. Combin. Theor. 95 (2001), 251–334.
• D. Cook II, The Lefschetz properties of monomial complete intersections in positive characteristic, J. Alg. 369 (2012), 42–58; doi:10.1016/j.jalgebra.2012.07.015.
• D. Cook II and U. Nagel, Enumerations of lozenge tilings, lattice paths, and perfect matchings and the weak Lefschetz property, preprint (2013); available at arXiv:1305.1314.
• I. Gessel and X.G. Viennot, Determinants, paths and plane partitions, preprint (1989); available at http://people.brandeis.edu/$\sim$gessel/homepage/papers/.
• T. Harima, J. Migliore, U. Nagel and J. Watanabe, The weak and strong Lefschetz properties for Artinian $K$-algebras, J. Alg. 262 (2003), 99–126.
• J. Herzog and T. Hibi, Monomial ideals, Grad. Texts Math. 260, Springer-Verlag London, Ltd., London, 2011.
• A. Iarrobino, Associated graded algebra of a Gorenstein Artin algebra, Mem. Amer. Math. Soc. 107 (1994), viii+115 pp.
• J. Li and F. Zanello, Monomial complete intersections, the weak Lefschetz property and plane partitions, Discr. Math. 310 (2010), 3558–3570.
• B. Lindström, On the vector representations of induced matroids, Bull. Lond. Math. Soc. 5 (1973), 85–90.
• J. Migliore and R. Miró-Roig, Ideals of general forms and the ubiquity of the weak Lefschetz property, J. Pure Appl. Alg. 182 (2003), 79–107.
• J. Migliore, R. Miró-Roig and U. Nagel, Monomial ideals, almost complete intersections and the weak Lefschetz property, Trans. Amer. Math. Soc. 363 (2011), 229–257.
• ––––, On the weak Lefschetz property for powers of linear forms, Alg. Num. Theory 6 (2012), 487–526.
• J. Migliore and U. Nagel, Survey article: A tour of the weak and strong Lefschetz properties, J. Comm. Alg. 5 (2013), 329–358.
• J. Migliore and F. Zanello, The Hilbert functions which force the weak Lefschetz property, J. Pure Appl. Alg. 210 (2007), 465–471.
• A. Wiebe, The Lefschetz property for componentwise linear ideals and Gotzmann ideals, Comm. Alg. 32 (2004), 4601–4611.
• F. Zanello and J. Zylinski, Forcing the strong Lefschetz and the maximal rank properties, J. Pure Appl. Alg. 213 (2009), 1026–1030.