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Algebras with the Weak Lefschetz Property

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Homological and Computational Methods in Commutative Algebra

Part of the book series: Springer INdAM Series ((SINDAMS,volume 20))

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Abstract

This is a survey on some works in which the Weak Lefschetz Property (WLP) for Artinian standard graded algebras is investigated, see for instance (Ragusa and Zappalà, arXiv:1112.1498. To appear in Rend Circ Mat Palermo; Colloq Math 64:73–83, 2013; Favacchio et al, J Pure Appl Algebra 217:1955–1966, 2013). In particular, it is shown that the Hilbert function of an almost complete intersection Artinian standard graded algebra of codimension 3 is a Weak Lefschetz sequence, i.e. it is the Hilbert function of some Artinian algebra with WLP or equivalently it is unimodal and the positive part of their first differences is a O-sequence. Moreover we give both some numerical condition on the Hilbert function and other conditions on the graded Betti numbers in order to force Artinian Gorenstein standard graded algebras of codimension 3 to enjoy the WLP. For Artinian standard graded algebras with the WLP we study the behavior of their linear quotients both with respect to the Hilbert function and to the graded Betti numbers. From this we produce a new property denominated Betti Weak Lefschetz Property (β-WLP) which permits a good behavior of the grade Betti numbers for the linear quotients of Artinian standard graded algebras with the WLP. We find conditions on the generators’ degrees of a complete intersection Artinian graded algebra with the WLP which force the algebra to have the β-WLP.

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References

  1. A.M. Bigatti, Upper bounds for the Betti numbers Of A given Hilbert function. Commun. Algebra 21(7), 2317–2334 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Brenner, A. Kaid, Syzygy bundles on \(\mathbb{P}^{2}\) and the Weak Lefschetz property. Ill. J. Math. 51(4), 1299–1308 (2007)

    MathSciNet  MATH  Google Scholar 

  3. D.A. Buchsbaum, D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Am. J. Math. 99(1), 447–485 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Diesel, Irreducibility and dimension theorems for families of height 3 Gorenstein algebras. Pac. J. Math. 172(4), 365–397 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Favacchio, A. Ragusa, G. Zappalà, Linear quotient of Artinian weak Lefschetz algebras. J. Pure Appl. Algebra 217, 1955–1966 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Harima, J.C. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for Artinian K-algebras. J. Algebra 262, 99–126 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, J. Watanabe, The Lefschetz Properties. Lecture Notes in Mathematics, vol. 2080 (Springer, Heidelberg, 2013)

    Google Scholar 

  8. J. Herzog, D. Popescu, The strong Lefschetz property and simple extensions, preprint. Available on the arXiv at http://front.math.ucdavis.edu/0506.5537

  9. H. Hulett, Maximum Betti numbers for a given Hilbert function. Commun. Algebra 21(7), 2335–2350 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Ikeda, Results on Dilworth and Rees numbes of Artinian local rings. Jpn. J. Math. 22, 147–158 (1996)

    Article  MATH  Google Scholar 

  11. E. Mezzetti, R.M. Mirò Roig, G. Ottaviani, Laplace equations and the weak Lefschetz property. Can. J. Math. 65(3), 634–654 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Migliore, The geometry of the weak Lefschetz property and level sets of points. Can. J. Math. 60(2), 391–411 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Migliore, R. Miró-Roig, U. Nagel, Monomial ideals, almost complete intersections and the weak Lefschetz property. Trans. Am. Math. Soc. 363(1), 229–257 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Migliore, U. Nagel, A tour of the weak and strong Lefschetz properties. J. Commut. Algebra 5, 329–358 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Migliore, F. Zanello, The Hilbert functions which force the weak Lefschetz property. J. Pure Appl. Algebra 210(2), 465–471 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. R.M. Mirò Roig, Ordinay curves, webs and the ubiquity of the weak Lefschetz property. Algebr. Represent. Theory 17(5), 1587–1596 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Pardue, Deformation classes Of graded modules and maximal Betti numbers. Ill. J. Math. 40, 564–585 (1996)

    MathSciNet  MATH  Google Scholar 

  18. A. Ragusa, G. Zappalà, Properties of 3-codimensional Gorenstein schemes. Commun. Algebra 29(1), 303–318 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Ragusa, G. Zappalà, On the weak-Lefschetz property for Artinian Gorenstein algebras. arXiv:1112.1498. To appear in Rend. Circ. Mat. Palermo.

    Google Scholar 

  20. A. Ragusa, G. Zappalà, On complete intersections contained in Cohen-Macaulay and Gorenstein ideals. Algebra Colloq. 18(Spec 1), 857–872 (2011)

    Google Scholar 

  21. A. Ragusa, G. Zappalà, On the Weak Lefschetz property for Hilbert functions of almost complete intersections. Colloq. Math. 64, 73–83 (2013)

    Google Scholar 

  22. L. Reid, L. Roberts, M. Roitman, On complete intersections and their Hilbert functions. Can. Math. Bull. 34(4), 525–535 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Stanley, Hilbert functions of graded algebras. Adv. Math. 28, 57–83 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Algebr. Discrete Methods 1, 168–184 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Watanabe, The Dilworth number of Artinian rings and finite posets with rank function. Commutative Algebra Comb. Adv. Stud. Pure Math. 11, 303–312 (1987)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, Catania, 95125, Italy

    Alfio Ragusa

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  1. Alfio Ragusa
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Correspondence to Alfio Ragusa .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Genova, Genova, Italy

    Aldo Conca

  2. Department of Mathematics, San Francisco State University, San Francisco, California, USA

    Joseph Gubeladze

  3. FB Mathematik/Informatik, Universität Osnabrück, Osnabrück, Germany

    Tim Römer

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Ragusa, A. (2017). Algebras with the Weak Lefschetz Property. In: Conca, A., Gubeladze, J., Römer, T. (eds) Homological and Computational Methods in Commutative Algebra. Springer INdAM Series, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-61943-9_12

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