Abstract
We deal with the weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension 3. We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence \(H\) of codimension 3 we found several Gorenstein Betti sequences compatible with \(H\) which again force the WLP. Finally we show that for every Gorenstein Betti sequence the general Artinian standard graded Gorenstein algebra with such Betti sequence has the WLP.
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Buchsbaum, D.A., Eisenbud, D.: Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Am. J. Math. 99(1), 447–485 (1977)
Brenner, H., Kaid, A.: Syzygy bundles on \(\mathbb{P}^{2}\) and the Weak Lefschetz property. Ill. J. Math. 51(4), 1299–1308 (2007)
Diesel, S.: Irreducibility and dimension theorems for families of height 3 Gorenstein algebras. Pac. J. Math. 172(4), 365–397 (1996)
Gaeta, F.: Quelques progrès récents dans la classification des variétés algébriques d’un espace projectif, pp. 145–183. Liége, Deuxiéme Colloque de Géométrie Algébrique (1952)
Harima, T.: A note on Artinian Gorenstein algebras of codimension three. J. Pure Appl. Algebra. 135(1), 45–56 (1999)
Harima, T.: Characterization of Hilbert functions of Gorenstein Artin algebras with the weak Stanley property. Proc. Am. Math. Soc. 123(12), 3631–3638 (1995)
Harima, T., Migliore, J., Nagel, U., Watanabe, J.: The Weak and Strong Lefschetz properties for Artinian K-algebras. J. Algebra. 262, 99–126 (2003)
Herzog, J., Popescu, D.: The strong Lefschetz property and simple extensions, preprint. Available on the arXiv at http://front.math.ucdavis.edu/0506.5537
Ikeda, H.: Results on Dilworth and Rees numbers of Artinian local rings. Jpn. J. Math. 22, 147–158 (1996)
Migliore, J., Miró-Roig, R., Nagel, U.: Monomial ideals, almost complete intersections and the weak Lefschetz property. Trans. Am. Math. Soc. 363(1), 229–257 (2011)
Migliore, J., Nagel, U.: A tour of the Weak and Strong Lefschetz Properties, arXiv:1109.5718
Migliore, J., Zanello, F.: The strength of the weak Lefschetz property. Ill. J. Math. 52(4), 1417–1433 (2008)
Ragusa, A., Zappalà, G.: Properties of \(3\)-codimensional Gorenstein schemes. Commun. Algebra 29(1), 303–318 (2001)
Ragusa, A., Zappalà, G.: Gorenstein schemes on general surfaces. Nagoya Math. J. 162, 111–125 (2001)
Reid, L., Roberts, L., Roitman, M.: On complete intersections and their Hilbert functions. Can. Math. Bull. 34(4), 525–535 (1991)
Stanley, R.: Hilbert functions of graded algebras. Adv. Math. 28, 57–83 (1978)
Stanley, R.: Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Algebraic Discrete Methods 1, 168–184 (1980)
Watanabe, J.: The Dilworth number of Artinian rings and finite posets with rank function. Commut. Algebra Comb. Adv. Stud. Pure Math. 11, 303–312 (1987)
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Ragusa, A., Zappalà, G. On the Weak-Lefschetz property for Artinian Gorenstein algebras. Rend. Circ. Mat. Palermo 62, 199–206 (2013). https://doi.org/10.1007/s12215-012-0102-6
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DOI: https://doi.org/10.1007/s12215-012-0102-6