2006/07/03 by Jeaman Ahn, Ahn, Jea-Man, Yong Su Shin +1 · 1 citation
Mathematics · #Commutative Algebra and Its Applications #Algebraic structures and combinatorial models #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.math/0607035
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 \H=(h0,h1,..., hd-1>hd=hd+1) cannot be level if hd≤ 2d+3, and that there exists a level O-sequence of codimension 3 of type \H for hd ≥ 2d+k for k≥ 4. Furthermore, we show that \H is not level if β1,d+2(I\rm lex)=β2,d+2(I\rm lex), and also prove that any codimension 3 Artinian graded algebra A=R/I cannot be level if β1,d+2(\Gin(I))=β2,d+2(\Gin(I)). In this case, the Hilbert function of A does not have to satisfy the condition hd-1>hd=hd+1. Moreover, we show that every codimension n graded Artinian level algebra having the Weak-Lefschetz Property has the strictly unimodal Hilbert function having a growth condition on (hd-1-hd) ≤ (n-1)(hd-hd+1) for every d > θ where h0...gt;hs-1gt;hs. In particular, we find that if A is of codimension 3, then (hd-1-hd) < 2(hd-hd+1) for every θ< d 0 and \soc(A)d-1=0 for some r1(A)