The Weak and Strong Lefschetz Properties for Artinian K-Algebras

TL;DR

This paper investigates the Weak and Strong Lefschetz properties in Artinian K-algebras, proving their prevalence in certain cases, characterizing possible Hilbert functions, and establishing bounds on Betti numbers.

Contribution

It provides new results on the Weak and Strong Lefschetz properties, including proofs for height three complete intersections and a complete characterization of Hilbert functions.

Findings

All height three complete intersections have the Weak Lefschetz property.

A complete characterization of Hilbert functions for K-algebras with Lefschetz properties.

A sharp bound on Betti numbers for algebras with these properties.

Abstract

Let A = bigoplus_{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 ell of degree 1 such that the multiplication times ell : A_i --> A_{i+1} has maximal rank, for every i, and A has the Strong Lefschetz property if times ell^d : A_i --> A_{i+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 P^2 and the Grauert-Mulich 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.