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

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

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.

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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:
https://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.