Jordan type and Lefschetz Properties for Artinian algebras

Abstract: The Jordan type of a pair $(A,x),$ where $x$ is in the maximum ideal of a standard graded Artinian algebra A, is the partition P giving the Jordan block decomposition of the multiplication map by $x$ on $A.$ When $A$ is Artinian Gorenstein, we say that $(A,x)$ is weak Lefschetz if the number of parts in the Jordan type $P_x$ is the Sperner number of $A$ – the highest value of the Hilbert function H(A). We say that $(A,x)$ is strong Lefschetz if $P_x$ is the conjugate of the Hilbert function.

Weak and strong Lefschetz properties of $A$ for a generic choice of $x$ have been studied, due to the connection with topology and geometry, where A is the cohomology ring of a

topological space or a variety $X.$ We discuss some of the properties of Jordan type, and its

use as an invariant of $A,$ its behavior for tensor products and free extensions (defined by

T. Harima and J. Watanabe).

Mathematics

Audience: advanced learners

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IIT Bombay Virtual Commutative Algebra Seminar

Series comments: Note: To get meeting ID,fill the Google form on the web-site of the series sites.google.com/view/virtual-comm-algebra-seminar/home

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