Let K be an algebraically closed field of characteristic zero and let I = (f1, . . . , fn) be a homogeneous R+-primary ideal in R := K[X, Y, Z]. If the corresponding syzygy bundle Syz(f1, . . . , fn) on the projective plane is semistable, we show that the Artinian algebra R/I has the Weak Lefschetz property if and only if the syzygy bundle has a special generic splitting type. As a corollary we...

The rank rk(R) of a ring R is the supremum of minimal cardinalities of generating sets of I as I ranges over ideals of R. Matson showed that every n ∈ Z+ occurs as the rank of some ring R. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension 0 or 1, we give four different constructions of rings of rank n (for all n ∈ Z+). Two constructions use one-dimension...

Let I be a homogeneous Artinian ideal in a polynomial ring R = k[x1, . . . , xn] over a field k of characteristic 0. We study an equivalent condition for the generic initial ideal gin(I) with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that Moreno-Socias conjecture implies Fröberg conjecture. And for the case codimI ≤ 3, we show that R/I has t...

In this paper, we study Lefschetz properties of Artinian reductions of Stanley–Reisner rings of balanced simplicial 3-polytopes. A (d − 1)-dimensional simplicial complex is said to be balanced if its graph is d-colorable. If a simplicial complex is balanced, then its Stanley–Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stan...