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The Lefschetz property for componentwise linear ideals and Gotzmann ideals. (English) Zbl 1089.13500
From the paper: Let \(S=K[x_1,\dots,x_n]\) be the polynomial ring over an infinite field \(K\). The maximal ideal \((x_1,\dots,x_n)\) will be denoted by \({\mathfrak m}\). Let \(I\subset S\) be an \({\mathfrak m}\)-primary graded ideal and set \(A=S/I\). One says that \(A\) has the weak Lefschetz property, if there is a linear form \(l\in A_1\) which satisfies the following condition: The multiplication map \(A_1\to A_{i+1}\), \(f \mapsto lf\), has maximal rank (that means, is injective or surjective) for all \(i\in\mathbb{N}\).
If there exists an element \(l\in A_1\) such that the multiplication map \(A_i\to A_{i+k}\), \(f\mapsto l^kf\), has maximal rank for all \(i\in\mathbb{N}\) and all \(k\geq 1\), one says that \(A\) has the strong Lefschetz property.
Under which conditions does a standard graded Artinian \(K\)-algebra \(A\) admit the weak (respectively strong) Lefschetz property? Since more than twenty years, this question has challenged several authors, and many interesting results have been obtained.
In this paper, we give an answer to the above question in the following special cases: Let \(I\) be a componentwise linear ideal (respectively a Gotzmann ideal) in the polynomial ring \(S=K[x_1,\dots,x_n]\). In the case that \(I\) is componentwise linear, we give a necessary and sufficient condition for \(S/I\) to have the weak Lefschetz property in terms of the graded Betti numbers of \(I\). Under the stronger assumption that \(I\) is even a Gotzmann ideal, we give a necessary and sufficient condition for \(S/I\) to have the weak Lefschetz property in terms of the Hilbert function of \(I\).

MSC:
13A02 Graded rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
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