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, 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 conditionMoreover, 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 onIn 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 ≤ 3h 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 .

On the shape of a pure 𝑂-sequence

“…This is true even for level algebras, since, for instance, it can be shown that h = (1, 13, 13, 14) is a level h-vector, but h clearly fails to be differentiable from degree 2 to degree 3. (To construct this level h-vector, one can start with Stanley's well-known non-unimodal Gorenstein h-vector (1,13,12,13,1) and add the fourth power of a general linear form in the same variables to the inverse system to get (1,13,13,14,2), and then truncate.) Theorem 3.4 also carries a consequence concerning the WLP, since it provides an indirect way to construct algebras without the WLP.…”

“…(We have already seen an example in socle degree 4 above, but we give a different one now.) (1,8,16,24,36). Since the first difference is (1,7,8,8,12), which is not an O-sequence (since 12 > (8 (3) ) 1 1 = 10), we obtain our desired example.…”

“…(1,8,16,24,36). Since the first difference is (1,7,8,8,12), which is not an O-sequence (since 12 > (8 (3) ) 1 1 = 10), we obtain our desired example. For socle degree 5 we make a similar construction using the pure O-sequences h = (1,3,6,10,15,21) and h = (1,3,5,5,3,1) to obtain the pure O-sequence h = (1,6,11,15,18,22).…”

“…For socle degree 5 we make a similar construction using the pure O-sequences h = (1,3,6,10,15,21) and h = (1,3,5,5,3,1) to obtain the pure O-sequence h = (1,6,11,15,18,22). Hence h is not differentiable, since its first difference is (1,5,5,4,3,4), and 4 > (3 (4) (which is the Hilbert function of the truncation of the algebra R/(x 3 ) after degree a, where R is a polynomial ring in 3 variables) and a pure O-sequence (. .…”

“…. 6,3,1) arising from a single monomial (in three new variables) of degree a with each variable having an exponent at least equal to 2. These combine to give the pure O-sequence (.…”