Artinian algebras and Jordan type
Anthony Iarrobino, Pedro Macias Marques, Chris McDaniel

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 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 on M. In general the Jordan type has more information than whether the pair 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 to the Hilbert function of . 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.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Commutative Algebra and Its Applications · Advanced Topics in Algebra
