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The EGH Conjecture and the Sperner property of complete intersections


Authors: Tadahito Harima, Akihito Wachi and Junzo Watanabe
Journal: Proc. Amer. Math. Soc. 145 (2017), 1497-1503
MSC (2010): Primary 13M10; Secondary 13C40
DOI: https://doi.org/10.1090/proc/13347
Published electronically: October 26, 2016
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A$ be a graded complete intersection over a field and $ B$ the monomial complete intersection with the generators of the same degrees as $ A$. The EGH conjecture says that if $ I$ is a graded ideal in $ A$, then there should be an ideal $ J$ in $ B$ such that $ B/J$ and $ A/I$ have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property.


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  • [1] Abed Abedelfatah, On the Eisenbud-Green-Harris conjecture, Proc. Amer. Math. Soc. 143 (2015), no. 1, 105-115. MR 3272735, https://doi.org/10.1090/S0002-9939-2014-12216-7
  • [2] Ian Anderson, Combinatorics of finite sets, Dover Publications, Inc., Mineola, NY, 2002. Corrected reprint of the 1989 edition. MR 1902962
  • [3] Martin Aigner, Combinatorial theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 234, Springer-Verlag, Berlin-New York, 1979. MR 542445
  • [4] Béla Bollobás, Combinatorics, Cambridge University Press, Cambridge, 1986. Set systems, hypergraphs, families of vectors and combinatorial probability. MR 866142
  • [5] N. G. de Bruijn, Ca. van Ebbenhorst Tengbergen, and D. Kruyswijk, On the set of divisors of a number, Nieuw Arch. Wiskunde (2) 23 (1951), 191-193. MR 0043115
  • [6] David Cook II, The Lefschetz properties of monomial complete intersections in positive characteristic, J. Algebra 369 (2012), 42-58. MR 2959785, https://doi.org/10.1016/j.jalgebra.2012.07.015
  • [7] David Cook II and Uwe Nagel, The weak Lefschetz property, monomial ideals, and lozenges, Illinois J. Math. 55 (2011), no. 1, 377-395 (2012). MR 3006693
  • [8] David Eisenbud, Mark Green, and Joe Harris, Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 295-324. MR 1376653, https://doi.org/10.1090/S0273-0979-96-00666-0
  • [9] Curtis Greene and Daniel J. Kleitman, Proof techniques in the theory of finite sets, Studies in combinatorics, MAA Stud. Math., vol. 17, Math. Assoc. America, Washington, D.C., 1978, pp. 22-79. MR 513002
  • [10] Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, and Junzo Watanabe, The Lefschetz properties, Lecture Notes in Mathematics, vol. 2080, Springer, Heidelberg, 2013. MR 3112920
  • [11] Hidemi Ikeda and Junzo Watanabe, The Dilworth lattice of Artinian rings, J. Commut. Algebra 1 (2009), no. 2, 315-326. MR 2504938, https://doi.org/10.1216/JCA-2009-1-2-315
  • [12] Chris McDaniel, The strong Lefschetz property for coinvariant rings of finite reflection groups, J. Algebra 331 (2011), 68-95. MR 2774648, https://doi.org/10.1016/j.jalgebra.2010.11.007
  • [13] Juan Migliore, Rosa M. Miró-Roig, Satoshi Murai, Uwe Nagel, and Junzo Watanabe, On ideals with the Rees property, Arch. Math. (Basel) 101 (2013), no. 5, 445-454. MR 3125561, https://doi.org/10.1007/s00013-013-0565-5
  • [14] 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, https://doi.org/10.1016/S0022-4049(02)00314-6
  • [15] Juan Migliore and Uwe Nagel, Survey article: a tour of the weak and strong Lefschetz properties, J. Commut. Algebra 5 (2013), no. 3, 329-358. MR 3161738, https://doi.org/10.1216/JCA-2013-5-3-329
  • [16] Emanuel Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1928), no. 1, 544-548 (German). MR 1544925, https://doi.org/10.1007/BF01171114
  • [17] 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
  • [18] J. Watanabe, $ \mathfrak{m}$-full ideals, Nagoya Math. J., 106, (1987), 101-111.

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Additional Information

Tadahito Harima
Affiliation: Department of Mathematics Education, Niigata University, Niigata, 950-2181 Japan
Email: harima@ed.niigata-u.ac.jp

Akihito Wachi
Affiliation: Department of Mathematics, Hokkaido University of Education, Kushiro, 085-8580 Japan
Email: wachi.akihito@k.hokkyodai.ac.jp

Junzo Watanabe
Affiliation: Department of Mathematics, Tokai University, Hiratsuka, 259-1201 Japan
Email: watanabe.juzno@tokai-u.jp

DOI: https://doi.org/10.1090/proc/13347
Received by editor(s): November 16, 2015
Received by editor(s) in revised form: June 21, 2016
Published electronically: October 26, 2016
Additional Notes: This work was supported by JSPS KAKENHI Grant numbers (C) (15K04812), (C) (23540179).
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society

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