Seminar
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| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
Artinian Gorenstein Algebras: Lefschetz properties and Hilbert functions
The Hard Lefschetz Theorem on the cohomology rings of smooth complex projective varieties motivated Stanley's formulation of the Lefschetz properties for standard graded Artinian algebras. Graded Artinian Gorenstein algebras are algebraic analogs of the cohomology rings of smooth complex projective varieties. The question of whether Artinian Gorenstein algebras satisfy the Lefschetz properties is mostly open, even in low codimensions. While it is true that all Artinian Gorenstein algebras satisfy the Strong Lefschetz property in codimension two, this is unknown in codimension three; however, it is conjectured to be true. On the other hand, the classification of the Hilbert functions of Artinian Gorenstein algebras has been a longstanding open question in general. In this talk, we address these two problems and characterize the Hilbert functions of the Artinian Gorenstein algebras satisfying the Lefschetz properties. As an application of this result, we show that generically, Artinian Gorenstein algebras of codimension three have the Strong Lefschetz property.
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