A non-unimodal codimension 3 level $h$-vector

Details A non-unimodal codimension 3 level $h$-vector A non-unimodal codimension 3 level $h$-vector

2005-05-31

arxiv.org/abs/math/0505678v3

J. of Algebra 305 (2006), No. 2, 949-956.

Mathematics

Commutative Algebra

12 pages, a few minor changes. To appear in the J. of Algebra

Scientific paper

$(1,3,6,10,15,21,28,27,27,28)$ is a level $h$-vector! This example answers negatively the open question as to whether all codimension 3 level $h$-vectors are unimodal. Moreover, using the same (simple) technique, we are able to construct level algebras of codimension 3 whose $h$-vectors have exactly $N$ ` ` maxima", for any positive integer $N$. These non-unimodal $h$-vectors, in particular, provide examples of codimension 3 level algebras not enjoying the Weak Lefschetz Property (WLP). Their existence was also an open problem before. In the second part of the paper we further investigate this fundamental property, and show that there even exist codimension 3 level algebras of type 3 without the WLP.

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