Seminar
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Seminar in Algebraic Geometry and Number Theory

Nancy Abdallah, Högskolan i Borås: Lefschetz properties of Artin Gorenstein algebras in codimensions 3 and 4

Overview

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  • Date:Starts 27 September 2023, 15:15Ends 27 September 2023, 16:15
  • Location:
    Pascal, Hörsalsvägen 1
  • Language:English

Abstract: The weak and strong Lefschetz properties (WLP and SLP) has been much studied for Artinian algebras. Codimension two Artinian algebras over a field of characteristic zero have the strong Lefschetz property. It is open whether Artin Gorenstein (AG) algebras of codimension c = 3 satisfy the weak or strong Lefschetz properties. Harima, Migliore, Nagel, and Watanabe proved that complete intersection algebras of codimension three satisfy the weak Lefschetz property. For c = 4, Gondim showed that WLP always holds for regularity r ≤ 4 and gives a family where WLP fails for any r ≥ 7, building on an earlier example of Ikeda of failure for
r = 5.

In this talk we go through an overview of Lefschetz properties over AG algebras and we give some recent results in codimensions 3 and 4. In codimension 4 we explore relations between Lefschetz properties of an AG algebra A and the free resolutions of A.