Fall 2022 Artinian algebras and Jordan type
Anthony Iarrobino, Pedro Macias Marques, Chris McDaniel
J. Commut. Algebra 14(3): 365-414 (Fall 2022). DOI: 10.1216/jca.2022.14.365


There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras A, especially those that are Artinian Gorenstein. A more general invariant of an Artinian algebra A or finite A-module M that we consider here is the set of Jordan types of elements of the maximal ideal 𝔪 of A, acting on M. Here, the Jordan type of 𝔪A is the partition giving the Jordan blocks of the multiplication map m:MM. In particular, we consider the Jordan type of a generic linear element in A1, or in the case of a local ring 𝒜, that of a generic element 𝔪𝒜, the maximum ideal.

We often take M=A, the graded algebra, or M=𝒜 a local algebra. The strong Lefschetz property of an element, as well as the weak Lefschetz property can be expressed simply in terms of its Jordan type and the Hilbert function of M. However, there has not been until recently a systematic study of the set of possible Jordan types for a given Artinian algebra A or A-module M, except, importantly, in modular invariant theory, or in the study of commuting Jordan types.

We first show some basic properties of the Jordan type. In a main result we show an inequality between the Jordan type of 𝔪𝒜 and a certain local Hilbert function. 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 also propose open problems.


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Anthony Iarrobino. Pedro Macias Marques. Chris McDaniel. "Artinian algebras and Jordan type." J. Commut. Algebra 14 (3) 365 - 414, Fall 2022. https://doi.org/10.1216/jca.2022.14.365


Received: 23 December 2019; Revised: 25 June 2020; Accepted: 26 June 2020; Published: Fall 2022
First available in Project Euclid: 7 October 2022

MathSciNet: MR4492997
zbMATH: 1509.13024
Digital Object Identifier: 10.1216/jca.2022.14.365

Primary: 13E10
Secondary: 13D40 , 13H10 , 14B07 , 14C05

Keywords: artinian algebra , Hilbert function , Jordan type , Lefschetz property , tensor product

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium


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Vol.14 • No. 3 • Fall 2022
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