Artinian algebras and Jordan type

Abstract

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 $\mathfrak{𝔪}$ of $A$, acting on $M$. Here, the Jordan type of $\ell \in {\mathfrak{𝔪}}_A$ is the partition giving the Jordan blocks of the multiplication map $m_{\ell }:M\to M$. In particular, we consider the Jordan type of a generic linear element $\ell$ in $A_1$, or in the case of a local ring $\mathcal{𝒜}$, that of a generic element $\ell \in {\mathfrak{𝔪}}_{\mathcal{𝒜}}$, the maximum ideal.

We often take $M=A$, the graded algebra, or $M=\mathcal{𝒜}$ 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 $\ell \in {\mathfrak{𝔪}}_{\mathcal{𝒜}}$ 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.