The Weak and Strong Lefschetz properties for Artinian K-algebras

By T. Harima, Juan Migliore, Uwe Nagel, Junzo Watanabe. 2002

236 citations recorded by the source corpus.

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 :A i →A i+1 has maximal rank, for every i, and A has the Strong Lefschetz property if ×l 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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/be86689da6a4c3525a3d8740011da1a0d48cb827

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Citations in the corpus

On higher Hessians and the Lefschetz properties (2015, background): “see that for a standard graded Artinian Gorenstein algebra A = Q/Ann(f) the notion of SLP and SLP in the narrow sense coincide. For N = 1 (codimension 2) all Artinian graded algebras satisfy the SLP ([HMNW]). Thus all polynomials of degree d in two variables have non-vanishing kth Hessian for all k ”
The strong Lefschetz property for Artinian algebras with non-standard grading (2007, background): “In section 8 we show m ore exam ples where the consideration ofa non-standard grading workse ectively.”
Line arrangements and configurations of points with an unexpected geometric property (2016, background): “ 5.22. 6. The strongLefschetzproperty We now relate the existence of an unexpected curve to the failure of a Lefschetz property. Lefschetz properties of graded algebras were formalized and studied in [HMNW], although as the name suggests, the idea behind it has been a fundamental concept for at leas”
Syzygy Bundles on P^2 and the Weak Lefschetz Property (2006, unknown): “Moreover, Syz(X(3), Y (3), Z(3), XY Z)(4)|L ∼= OL(1) ⊕OL ⊕OL(−1), hence by Theorem 2.”
Unimodality problems in Ehrhart theory (2015, background): “3 of [34], if A has a weak Lefschetz element, then the Hilbert series H(A;z) has unimodal coefficients.”
Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials (2009, background): “Q/AnnQF for the Fermat type polynomial F = Pn i=1x n i −n(n−1)s Qn i=1xi. When one of the higher Hessians of F is identically zero, the algebra Q/AnnQF does not have the strong Lefschetz property. In [8], [10] and [17], examples of Artinian Gorenstein algebras which do not have the strong Lefschetz ”
Ideals of general forms and the ubiquity of the Weak Lefschetz property (2002, background): “ral remarks. First, in the introduction it was asked whether all Gorenstein k-algebras in k[x 1,x 2,x 3] have the Weak (or Strong) Lefschetz property, as was recently shown for complete intersections [10]. A natural way that one might hope to prove this result is by liaison. If one could show that t”
The weak Lefschetz property, monomial ideals, and lozenges (2009, methodology): “If an algebra A has the weak Lefschetz property, then its Hilbert function is unimodal as shown in [7].”
On the shape of a pure O-sequence (2010, background): “Indeed, basically the same argument proves, more generally, the following purely algebraic result which also follows from [33], Proposition 3.”
The Hilbert Function of a Level Algebra (2007, background): “Consider the sequence h′ = (hi) where h ′ i = max{hi − hi−1, 0} Recall from [35] the fact that if A has the WLP then h′ must again be an Osequence (equivalently, if t is the last index for which ht ≥ hi for every i, then (1, h1, .”
On the Weak Lefschetz Property for Powers of Linear Forms (2010, background): “, a5 equals one, then A has the SLP by [Harima et al. 2003], so the claim is true.”