Let \( \mathbb{K} \) be a field.
In an Artinian \( \mathbb{K} \)-algebra, every element is a zero-divisor. So, the WLP can be seen as a way to measure how well-behaved the multiplication is in the algebra. In a broader scheme, the WLP is a special case of Fröberg's conjecture which gives a concrete formula for the Hilbert series of a quotient ring \( \mathbb{K}[x_1,\dots,x_n] / (f_1, \dots, f_k) \) using the degrees of the \( f_i \). Determining if an Artinian algebra \( A \) has the WLP seems like an innocent problem, but it turns out to be extremely hard to answer. Even in the case of Artinian Gorenstein algebras, this is still a very difficult question. To make things harder, the WLP is also characteristic dependent.
Although the WLP is phrased algebraically, this turns out to be a very powerful tool for combinatorialists as well. Of interest to many combinatorialists is if some sequence is unimodal (or log-concave). It turns out that the \( h \)-vector of an Artinian algebra with the WLP must be unimodal. So, a strategy for a combinatorialist to show that their sequence is unimodal is to construct an Artinian algebra and show that this algebra has the WLP. This plan of attack has been notably by Stanley to prove necessity of the \( g \)-theorem, and more recently by Adiprasito-Huh-Katz in solving long outstanding conjectures of Heron-Rota-Welsh and also Mason-Welsh.
Unless specified otherwise, \( A \) will be a standard-graded Artinian \( \mathbb{K} \)-algebra.
Motivated by the geometry of points in projective space, we show that if enough points lie on a unique hypersurface, and enough points lie off as well, then their Artinian reduction does not have the WLP. From here, we noticed that these Artinian reductions had a similar pattern in their Betti tables.
Definition. A Betti table \( B \) has an \( (n,d) \)-Koszul tail if it has an upper-left principal block of the form \[ \begin{array}{l|cccccccc} & 0 & 1 & 2 & 3 & \dots & n-2 & n-1 & n \\ \hline 0 & 1 & . & . & . & \dots & . & . & . \\ 1 & . & . & . & . & \dots & . & . & . \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ d-1 & . & . & . & . & \dots & . & . & . \\ d & . & n & \binom{n}{2} & \binom{n}{3} & \dots & \binom{n}{n-2} & n & 1 \end{array}. \] If \( B \) has an \( (n,d) \)-Koszul tail for an Artinian ring \( \mathbb{K}[x_1,\dots,x_n] / I \), then we say \( B \) has a maximal \( (n,d) \)-Koszul tail.
We then also show that if a Betti table has a maximal Koszul tail, then the algebra does not have the WLP. However, this is not a complete characterization of the WLP, so its open as to what is needed for sufficiency.
Analogous to the connected sum from topology of gluing together two spaces by excising and gluing along open sets, Ananthnarayan-Avramov-Moore introduced the connected sum construction for Gorenstein local rings. The connected sum construction gives a way to build new Gorenstein rings that have the WLP. We study the Betti numbers of the connected sum (and fiber product) of Artinian Gorenstein algebras with any number of summands. We also give explicit formulas for computing these Betti numbers in terms of the Betti numbers of the summands, as well as similar formulas for other invariants such as the Poincaré polynomials.