Level algebras with bad properties
Authors:
Mats Boij and Fabrizio Zanello
Journal:
Proc. Amer. Math. Soc. 135 (2007), 27132722
MSC (2000):
Primary 13H10; Secondary 13D40, 13E10, 14M05
Published electronically:
May 4, 2007
MathSciNet review:
2317944
Fulltext PDF Free Access
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), 949956) 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 nonunimodal 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 abovementioned preprint, Theorem 4.3) that, for any , there exist reduced level sets of points in whose artinian reductions are nonunimodal.
 [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
(93i:13012), http://dx.doi.org/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
(96h:13040), http://dx.doi.org/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
(2000j:14006), http://dx.doi.org/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
(94g:13008), http://dx.doi.org/10.1090/S00029939199412275122
 [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
(86f:13012), http://dx.doi.org/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
(97h:13012)
 [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
𝐾algebras, J. Algebra 262 (2003),
no. 1, 99–126. MR 1970804
(2004b:13001), http://dx.doi.org/10.1016/S00218693(03)000383
 [IK]
Anthony
Iarrobino and Vassil
Kanev, Power sums, Gorenstein algebras, and determinantal
loci, Lecture Notes in Mathematics, vol. 1721, SpringerVerlag,
Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman. MR 1735271
(2001d:14056)
 [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
(97g:13034)
 [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. MiróRoig, Ideals of general forms and the ubiquity of
the weak Lefschetz property, J. Pure Appl. Algebra
182 (2003), no. 1, 79–107. MR 1978001
(2004c:13027), http://dx.doi.org/10.1016/S00224049(02)003146
 [St]
Richard
P. Stanley, Hilbert functions of graded algebras, Advances in
Math. 28 (1978), no. 1, 57–83. MR 0485835
(58 #5637)
 [Za]
Fabrizio
Zanello, A nonunimodal codimension 3 level ℎvector,
J. Algebra 305 (2006), no. 2, 949–956. MR 2266862
(2007h:13026), http://dx.doi.org/10.1016/j.jalgebra.2006.07.009
 [BI]
 D. Bernstein and A. Iarrobino: A nonunimodal graded Gorenstein Artin algebra in codimension five, Comm. in Algebra 20 (1992), No. 8, 23232336. MR 1172667 (93i:13012)
 [Bo1]
 M. Boij: Graded Gorenstein Artin algebras whose Hilbert functions have a large number of valleys, Comm. in Algebra 23 (1995), No. 1, 97103. MR 1311776 (96h:13040)
 [Bo2]
 M. Boij: Components of the space parameterizing graded Gorenstein Artinian algebras with a given Hilbert function, Pacific J. Math. 187 (1999), 111. MR 1674301 (2000j:14006)
 [BL]
 M. Boij and D. Laksov: Nonunimodality of graded Gorenstein Artin algebras, Proc. Amer. Math. Soc. 120 (1994), 10831092. MR 1227512 (94g:13008)
 [FL]
 R. Fröberg and D. Laksov: Compressed Algebras, Conference on Complete Intersections in Acireale, Lecture Notes in Mathematics, No. 1092 (1984), 121151, SpringerVerlag. MR 0775880 (86f:13012)
 [Ge]
 A.V. Geramita: Inverse Systems of Fat Points: Waring's Problem, Secant Varieties and Veronese Varieties and Parametric Spaces of Gorenstein Ideals, Queen's Papers in Pure and Applied Mathematics, No. 102, The Curves Seminar at Queen's (1996), Vol. X, 3114. MR 1381732 (97h:13012)
 [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]
 T. Harima, J. Migliore, U. Nagel and J. Watanabe: The Weak and Strong Lefschetz Properties for Artinian Algebras, J. of Algebra 262 (2003), 99126. MR 1970804 (2004b:13001)
 [IK]
 A. Iarrobino and V. Kanev: Power Sums, Gorenstein Algebras, and Determinantal Loci, Springer Lecture Notes in Mathematics (1999), No. 1721, Springer, Heidelberg. MR 1735271 (2001d:14056)
 [Ik]
 H. Ikeda: Results on Dilworth and Rees numbers of Artinian local rings, Japan. J. of Math. 22 (1996), 147158. MR 1394376 (97g:13034)
 [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. MiróRoig: Ideals of general forms and the ubiquity of the Weak Lefschetz property, J. of Pure and Applied Algebra 182 (2003), 79107. MR 1978001 (2004c:13027)
 [St]
 R. Stanley: Hilbert functions of graded algebras, Adv. Math. 28 (1978), 5783. MR 0485835 (58:5637)
 [Za]
 F. Zanello: A nonunimodal codimension 3 level vector, J. Alg. 305 (2006), 949956. MR 2266862
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
13H10,
13D40,
13E10,
14M05
Retrieve articles in all journals
with MSC (2000):
13H10,
13D40,
13E10,
14M05
Additional Information
Mats Boij
Affiliation:
Department of Mathematics, Royal Institute of Technology, S100 44 Stockholm, Sweden
Email:
boij@math.kth.se
Fabrizio Zanello
Affiliation:
Department of Mathematics, Royal Institute of Technology, S100 44 Stockholm, Sweden
Email:
zanello@math.kth.se
DOI:
http://dx.doi.org/10.1090/S0002993907088296
PII:
S 00029939(07)088296
Keywords:
Type 2 level algebra,
Weak Lefschetz Property,
monomial algebra,
nonunimodality.
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.
