2018/02/21 by Anthony Iarrobino, Iarrobino, Anthony, Pedro Macías Marques +3 · 1 citation
Mathematics · #13E10 (Primary) 13D40 #13H10 #14B07 #14C05 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1802.07383
openalex publication_date 2018/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Jordan type of an element ℓ 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_ℓ on M. In general the Jordan type has more information than whether the pair (ℓ,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.