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arXiv · 2307.00957

Jordan type of an Artinian algebra, a survey

Nasrin Altafi·Anthony Iarrobino·Pedro Macias Marques
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Abstract

We consider Artinian algebras $A$ over a field $\mathsf{k}$, both graded and local algebras. The Lefschetz properties of graded Artinian algebras have been long studied, but more recently the Jordan type invariant of a pair $(\ell,A)$ where $\ell$ is an element of the maximal ideal of $A$, has been introduced. The Jordan type gives the sizes of the Jordan blocks for multiplication by $\ell$ on $A$, and it is a finer invariant than the pair $(\ell,A)$ being strong or weak Lefschetz. The Jordan degree type for a graded Artinian algebra adds to the Jordan type the initial degree of ``strings'' in the decomposition of $A$ as a $\mathsf{k}[\ell]$ module. We here give a brief survey of Jordan type for Artinian algebras, Jordan degree type for graded Artinian algebras, and related invariants for local Artinian algebras, with a focus on recent work and open problems.

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Nasrin Altafi, Anthony Iarrobino, Pedro Macias Marques. 2023-07-03. Jordan type of an Artinian algebra, a survey. https://arxiv.org/abs/2307.00957

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