Level algebras with bad properties

Authors:
Mats Boij and Fabrizio Zanello

Journal:
Proc. Amer. Math. Soc. **135** (2007), 2713-2722

MSC (2000):
Primary 13H10; Secondary 13D40, 13E10, 14M05

Published electronically:
May 4, 2007

MathSciNet review:
2317944

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949-956) of the second author, and those of Juan Migliore (math. AC/0508067). Our results are:

1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for , we will construct a codimension three, type two -vector of socle degree such that *all* the level algebras with that -vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each .

2). There exist reduced level sets of points in of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same -vectors we mentioned in 1).

3). For any integer , there exist non-unimodal monomial artinian level algebras of codimension . As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the above-mentioned preprint, Theorem 4.3) that, for any , there exist reduced level sets of points in whose artinian reductions are non-unimodal.

**[BI]**David Bernstein and Anthony Iarrobino,*A nonunimodal graded Gorenstein Artin algebra in codimension five*, Comm. Algebra**20**(1992), no. 8, 2323–2336. MR**1172667**, 10.1080/00927879208824466**[Bo1]**Mats Boij,*Graded Gorenstein Artin algebras whose Hilbert functions have a large number of valleys*, Comm. Algebra**23**(1995), no. 1, 97–103. MR**1311776**, 10.1080/00927879508825208**[Bo2]**Mats Boij,*Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function*, Pacific J. Math.**187**(1999), no. 1, 1–11. MR**1674301**, 10.2140/pjm.1999.187.1**[BL]**Mats Boij and Dan Laksov,*Nonunimodality of graded Gorenstein Artin algebras*, Proc. Amer. Math. Soc.**120**(1994), no. 4, 1083–1092. MR**1227512**, 10.1090/S0002-9939-1994-1227512-2**[FL]**R. Fröberg and D. Laksov,*Compressed algebras*, Complete intersections (Acireale, 1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 121–151. MR**775880**, 10.1007/BFb0099360**[Ge]**Anthony V. Geramita,*Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals*, The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995) Queen’s Papers in Pure and Appl. Math., vol. 102, Queen’s Univ., Kingston, ON, 1996, pp. 2–114. MR**1381732****[GHMS]**A.V. Geramita, T. Harima, J. Migliore and Y.S. Shin:*The Hilbert Function of a Level Algebra*, Memoirs of the Amer. Math. Soc., to appear.**[HMNW]**Tadahito Harima, Juan C. Migliore, Uwe Nagel, and Junzo Watanabe,*The weak and strong Lefschetz properties for Artinian 𝐾-algebras*, J. Algebra**262**(2003), no. 1, 99–126. MR**1970804**, 10.1016/S0021-8693(03)00038-3**[IK]**Anthony Iarrobino and Vassil Kanev,*Power sums, Gorenstein algebras, and determinantal loci*, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman. MR**1735271****[Ik]**Hidemi Ikeda,*Results on Dilworth and Rees numbers of Artinian local rings*, Japan. J. Math. (N.S.)**22**(1996), no. 1, 147–158. MR**1394376****[Macaulay2]**D.R. Grayson and M.E. Stillman:*Macaulay 2, a software system for research in algebraic geometry*, available at http://www.math.uiuc.edu/Macaulay2/.**[Mi]**J. Migliore:*The geometry of the Weak Lefschetz Property*, Canadian J. of Math., to appear (preprint: math.AC/0508067).**[MM]**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**, 10.1016/S0022-4049(02)00314-6**[St]**Richard P. Stanley,*Hilbert functions of graded algebras*, Advances in Math.**28**(1978), no. 1, 57–83. MR**0485835****[Za]**Fabrizio Zanello,*A non-unimodal codimension 3 level ℎ-vector*, J. Algebra**305**(2006), no. 2, 949–956. MR**2266862**, 10.1016/j.jalgebra.2006.07.009

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Additional Information

**Mats Boij**

Affiliation:
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Email:
boij@math.kth.se

**Fabrizio Zanello**

Affiliation:
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Email:
zanello@math.kth.se

DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08829-6

Keywords:
Type 2 level algebra,
Weak Lefschetz Property,
monomial algebra,
non-unimodality.

Received by editor(s):
December 15, 2005

Received by editor(s) in revised form:
May 20, 2006

Published electronically:
May 4, 2007

Additional Notes:
The second author is funded by the Göran Gustafsson Foundation

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.