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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: 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 $ e\gg 0$, we will construct a codimension three, type two $ h$-vector of socle degree $ e$ such that all the level algebras with that $ h$-vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each $ e\gg 0$.

2). There exist reduced level sets of points in $ {\mathbf P}^3$ of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same $ h$-vectors we mentioned in 1).

3). For any integer $ r\geq 3$, there exist non-unimodal monomial artinian level algebras of codimension $ r$. 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 $ r\geq 3$, there exist reduced level sets of points in $ {\mathbf P}^r$ whose artinian reductions are non-unimodal.

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Additional Information

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

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

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.

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