Advances in Applied Mathematics· 2026Q2
The weak Lefschetz property for artinian Gorenstein algebras of small Sperner number
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- Q2SCImago
- 2026year
Short summary
All artinian Gorenstein algebras with socle degree 'd' and Sperner number at most 'd+1' satisfy the Weak Lefschetz Property (WLP), regardless of codimension.
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Key points
- Artinian Gorenstein algebras with socle degree 'd' and Sperner number ≤ d+1 satisfy the WLP.
- This WLP property holds independent of the algebra's codimension.
- The bound is sharp: algebras with Sperner number d+2 can fail the WLP for d ≥ 3.
- This resolves a question for Gorenstein algebras in codimension three with high socle degree and Sperner number.
AI-generated from the title and abstract; the full text is not read.
Abstract
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 .
The authors' abstract, as published at the source. Advances in Applied Mathematics, 2026 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics