Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

Residually Reducible Representations of Algebras over Local Artinian Rings

In this paper we generalize a result of Urban on the structure of residually reducible representations on local Artinian rings from the case that the semi-simplification of the residual representation splits into 2 absolutely irreducible representations to the case where it splits into m ≥ 2 absolutely irreducible representations.

Artinian and Noetherian Rings

Artinian Band Sums of Rings

Band sums of associative rings were introduced by Weissglass in 1973. The main theorem claims that the support of every Artinian band sum of rings is finite. This result is analogous to the well-known theorem on Artinian semigroup rings. 1991 Mathematics subject classification (Amer. Math. Soc): primary 16P20, 16W50; secondary 20M25. Let B be a band, that is, a semigroup consisting of idempoten...

Lefschetz Elements of Artinian Gorenstein Algebras and Hessians of Homogeneous Polynomials

We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.

The strong Lefschetz property for Artinian algebras with non-standard grading

Let A = ⊕ i=0Ai be a graded Artinian K-algebra, where Ac 6= (0) and charK = 0. (The grading may not necessarily be standard.) Then A has the strong Lefschetz property if there exists an element g ∈ A1 such that the multiplication×g c−2i : Ai −→ Ac−i is bijective for every i = 0, 1, . . . , [c/2]. The main results obtained in this paper are as follows: 1. A has the strong Lefschetz property if a...