The Weak and Strong Lefschetz properties for Artinian K-algebras

Abstract

Let A=⊕i⩾0Ai be a standard graded Artinian K-algebra, where charK=0. Then A has the Weak Lefschetz property if there is an element ℓ of degree 1 such that the multiplication ×ℓ:Ai→Ai+1 has maximal rank, for every i, and A has the Strong Lefschetz property if ×ℓd:Ai→Ai+d has maximal rank for every i and d. The main results obtained in this paper are the following. (1) Every height-three complete intersection has the Weak Lefschetz property. (Our method, surprisingly, uses rank-two vector bundles on P2 and the Grauert–Mülich theorem.) (2) We give a complete characterization (including a concrete construction) of the Hilbert functions that can occur for K-algebras with the Weak or Strong Lefschetz property (and the characterization is the same one!). (3) We give a sharp bound on the graded Betti numbers (achieved by our construction) of Artinian K-algebras with the Weak or Strong Lefschetz property and fixed Hilbert function. This bound is again the same for both properties! Some Hilbert functions in fact force the algebra to have the maximal Betti numbers. (4) Every Artinian ideal in K[x,y] possesses the Strong Lefschetz property. This is false in higher codimension.