Artinian algebras and Jordan type

TL;DR

This paper explores the properties and invariants of Jordan types in Artinian algebras, relating them to Hilbert functions and introducing new concepts like Jordan degree type, with applications to various algebraic structures.

Contribution

It develops foundational properties of Jordan types, introduces the Jordan degree type, and connects these invariants to Hilbert functions and advanced algebraic structures.

Findings

Established basic properties of Jordan type in Artinian algebras

Introduced and analyzed Jordan degree type as a finer invariant

Explored applications to Nagata idealizations, tensor products, and free extensions

Abstract

The Jordan type of an element $\ell$ of the maximal ideal of an Artinian k-algebra A acting on an A-module M of k-dimension n, is the partition of n given by the Jordan block decomposition of the multiplication map $m_\ell$ on M. In general the Jordan type has more information than whether the pair $(\ell,M)$ is strong or weak Lefschetz. We develop basic properties of the Jordan type and their loci for modules over graded or local Artinian algebras. We as well study the relation of generic Jordan type of $A$ to the Hilbert function of $A$. We introduce and study a finer invariant, the Jordan degree type. In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We as well propose open problems.