Generic initial ideals and graded Artinian-level algebras not having the Weak-Lefschetz Property

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 H = ( h 0 , h 1 , … , h d − 1 > h d = h d + 1 ) cannot be level if h d ≤ 2 d + 3 , and that there exists a level O-sequence of codimension 3 of type H for h d ≥ 2 d + k for k ≥ 4 . Furthermore, we show that H is not level if β 1 , d + 2 ( I lex ) = β 2 , d + 2 ( I lex ) , and also prove that any codimension 3 Artinian graded algebra A = R / I cannot be level if β 1 , d + 2 ( Gin ( I ) ) = β 2 , d + 2 ( Gin ( I ) ) . In this case, the Hilbert function of A does not have to satisfy the condition h d − 1 > h d = h d + 1 . Moreover, we show that every codimension n graded Artinian level algebra having the Weak-Lefschetz Property has a strictly unimodal Hilbert function having a growth condition on ( h d − 1 − h d ) ≤ ( n − 1 ) ( h d − h d + 1 ) for every d > θ where h 0 < h 1 < ⋯ < h α = ⋯ = h θ > ⋯ > h s − 1 > h s . In particular, we show that if A is of codimension 3, then ( h d − 1 − h d ) < 2 ( h d − h d + 1 ) for every θ < d < s and h s − 1 ≤ 3 h s , and prove that if A is a codimension 3 Artinian algebra with an h -vector ( 1 , 3 , h 2 , … , h s ) such that h d − 1 − h d = 2 ( h d − h d + 1 ) > 0 and soc ( A ) d − 1 = 0 for some r 1 ( A ) < d < s , then ( I ≤ d + 1 ) is ( d + 1 ) -regular and dim k soc ( A ) d = h d − h d + 1 .