2024/06/25 by Mats Boij, Juan Migliore, Boij, Mats +5
Mathematics · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2406.17943
openalex publication_date 2024/06/25 · openalex created_date 2024/06/28 · openalex updated_date 2026/07/28
For artinian Gorenstein algebras in codimension four and higher, it is well known that the Weak Lefschetz Property (WLP) does not need to hold. For Gorenstein algebras in codimension three, it is still open whether all artinian Gorenstein algebras satisfy the WLP when the socle degree and the Sperner number are both higher than six. We here show that all artinian Gorenstein algebras with socle degree d and Sperner number at most d+1 satisfy the WLP, independent of the codimension. This is a sharp bound in general since there are examples of artinian Gorenstein algebras with socle degree d and Sperner number d+2 that do not satisfy the WLP for all d≥ 3.