Generic Initial Ideals And Graded Artinian Level Algebras Not Having The Weak-Lefschetz Property
arXiv:math/0607035
Abstract
We find a sufficient condition that $\H$ is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function cannot be level if , and that there exists a level O-sequence of codimension 3 of type $\H$ for for . Furthermore, we show that $\H$ is not level if , and also prove that any codimension 3 Artinian graded algebra cannot be level if $β_{1,d+2}(\Gin(I))=β_{2,d+2}(\Gin(I))$. In this case, the Hilbert function of does not have to satisfy the condition . Moreover, we show that every codimension graded Artinian level algebra having the Weak-Lefschetz Property has the strictly unimodal Hilbert function having a growth condition on for every where In particular, we find that if is of codimension 3, then for every and , and prove that if is a codimension 3 Artinian algebra with an -vector such that $$ h_{d-1}-h_d=2(h_d-h_{d+1})>0 \quad \text{and} \quad \soc(A)_{d-1}=0 $$ for some , then is -regular and $\dim_k\soc(A)_d=h_d-h_{d+1}$.
25 pages