The Weak and Strong Lefschetz properties for Artinian K-algebras

Abstract

Let A = i≥0 A i be a standard graded Artinian K-algebra, where char K = 0. Then A has the Weak Lefschetz property if there is an element ℓ of degree 1 such that the multiplication ×ℓ : A i → A i+1 has maximal rank, for every i, and A has the Strong Lefschetz property if ×ℓ d : A i → A i+d has maximal rank for every i and d.The main results obtained in this paper are the following.

On the shape of a pure 𝑂-sequence

“…Note that in the previous proof the only way that we have really used Hausel's theorem is that it guarantees the existence of a g-element for the algebra A, but we have never used that A is monomial, or level, or (at least in an essential way) that it has socle degree 3. Indeed, basically the same argument proves, more generally, the following purely algebraic result which also follows from [33] Also, notice the following fact: the very existence of a g-element for A is essential for the conclusion of Theorem 3.4, in the sense that this assumption cannot be relaxed by simply requiring, combinatorially, that the h-vector of A be differentiable up to degree e − 1. 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.…”

“…Part of the great interest in the WLP stems from the fact that its presence puts severe constraints on the possible Hilbert functions, which can appear in various disguises (see, e.g., [69] and [56]). Specifically, if R/I has the WLP then its Hilbert function is unimodal in the following strong sense: it is differentiable in the first interval (in particular it is strictly increasing), it is constant in the second (possibly trivial) interval, and then it is non-increasing in the third ( [33], Remark 3.3). Furthermore, the fact that R/I is level imposes the further condition that in the third interval (once there has been a strict decrease), it is actually strictly decreasing until it reaches 0 ([1], Theorem 3.6).…”

Lefschetz properties and basic constructions on simplicial spheres

“…Examples of Gorenstein algebras admitting the weak-Lefschetz property but not the strong-Lefschetz property were found in [5,Example 4.3]. For Gorenstein algebras arising as face rings of homology spheres the SL property is conjectured to hold.…”

Hilbert Function and Betti Numbers of Algebras with Lefschetz Property of Orderm

“…The m-times WLP is just a very natural generalization of it. For an overview of the main results achieved so far regarding this topic see [8], [11]. One interesting problem is the description of the Hilbert function of Artinian algebras having the WLP.…”

“…One interesting problem is the description of the Hilbert function of Artinian algebras having the WLP. In [8] the authors give a complete characterization of these Hilbert functions. First they make the remark that if and Artinian algebra has the WLP, then its Hilbert function must be a weak Lefschetz O-sequence in the sense of definition 1.2. and then they construct an Artinian algebra with the WLP for each weak Lefschetz O-sequence.…”