Level algebras with bad properties

Abstract

Abstract. This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949-956) of the second author, and those of Juan Migliore (math. AC/0508067). Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for e 0, we will construct a codimension three, type two h-vector of socle degree e such that all the level algebras with that h-vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each e 0. 2). There exist reduced level sets of points in P 3 of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same h-vectors we mentioned in 1).3). For any integer r ≥ 3, there exist non-unimodal monomial artinian level algebras of codimension r. As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the abovementioned preprint, Theorem 4.3) that, for any r ≥ 3, there exist reduced level sets of points in P r whose artinian reductions are non-unimodal.

On the shape of a pure 𝑂-sequence

“…If we allow the socle degree to be large, we now show that pure O-sequences of any codimension can be non-unimodal with arbitrarily many peaks. This result answers positively Question iii) (section 6) of [9]. Proof.…”

“…Another challenge is to find the smallest socle degree for which a non-unimodal example exists. The best example of which we are aware is Example 5.1 of [9], where it is shown that h = (1,3,6,10,15,21,28,33,36,37,36,36,37) is a pure O-sequence; notice that this has socle degree 12.…”

“…(By this we mean that there is at most one "valley" in the sense of [7].) In [9], a monomial example is constructed (which can be lifted to reduced sets of points -cf. for instance [55]), but again it is only done for two maxima.…”

“…The first and last authors [9] showed that without the condition of being monomial, there do exist type two level algebras failing the WLP, even when the inverse system is generated by one monomial and one binomial. We close this chapter with a short discussion of the characteristic.…”

“…R/I has Hilbert function (1,3,6,10,13,13,10,6,2), while for a general linear form L, R/(I, (L 3 )) has Hilbert function (1,3,6,9,10,7,1). Thus the multiplication by L 3 from degree 3 to degree 6 fails to be an isomorphism, and the SLP fails.…”