Generic Initial Ideals and Graded Artinian Level Algebras Not Having the Weak-lefschetz Property

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 Osequence 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 ) = β2,d+2(I ), 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 WeakLefschetz 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 < h1 < · · · < hα = · · · = hθ > · · · > hs−1 > hs. In particular, we find that if A is of codimension 3, then (hd−1 − hd) < 2(hd − hd+1) for every θ < d < s and hs−1 ≤ 3hs, and prove that if A is a codimension 3 Artinian algebra with an h-vector (1, 3, h2, . . . , hs) such that hd−1 − hd = 2(hd − hd+1) > 0 and soc(A)d−1 = 0 for some r1(A) < d < s, then (I≤d+1) is (d+ 1)-regular and dimk soc(A)d = hd − hd+1.