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The weak and strong Lefschetz properties for Artinian $$K$$-algebras. (English) Zbl 1018.13001
Summary: Let $$A=\bigoplus_{i\geq 0}A_i$$ be a standard graded Artinian $$K$$-algebra, where $$\text{char} K=0$$. Then $$A$$ has the “weak Lefschetz property” if there is an element $$\ell$$ of degree 1 such that the multiplication $$\times\ell: A_i\to A_{i+1}$$ has maximal rank, for every $$i$$, and $$A$$ has the “strong Lefschetz property” if $$\times\ell^d :A_i\to A_{i+d}$$ 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 $$\mathbb{P}^2$$ 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.

##### MSC:
 13A02 Graded rings 13E10 Commutative Artinian rings and modules, finite-dimensional algebras 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Macaulay2
Full Text:
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