Critical and injective modules over skew polynomial rings

TL;DR

This paper characterizes when skew polynomial rings over certain local algebras have all simple modules with injective hulls that are locally Artinian, extending known results to infinite order automorphisms and classifying critical modules.

Contribution

It provides a new criterion for property $(ullet)$ in skew polynomial rings with infinite order automorphisms, based on classifying cyclic critical modules.

Findings

Skew polynomial rings with automorphisms of finite order satisfy property $(ullet)$.

Classification of cyclic critical modules for automorphisms of infinite order.

The ring $k[[X]][ heta, ext{aut}]$ satisfies property $(ullet)$ for all automorphisms.

Abstract

Let $R$ be a commutative local $k$-algebra of Krull dimension one, where $k$ is a field. Let $\alpha$ be a $k$-algebra automorphism of $R$, and define $S$ to be the skew polynomial algebra $R[\theta; \alpha]$. We offer, under some additional assumptions on $R$, a criterion for $S$ to have injective hulls of all simple $S$-modules locally Artinian - that is, for $S$ to satisfy property $(\diamond)$. It is easy and well known that if $\alpha$ is of finite order, then $S$ has this property, but in order to get the criterion when $\alpha$ has infinite order we found it necessary to classify all cyclic (Krull) critical $S$-modules in this case, a result which may be of independent interest. With the help of the above we show that $\hat{S}=k[[X]][\theta, \alpha]$ satisfies $(\diamond)$ for all $k$-algebra automorphisms $\alpha$ of $k[[X]]$.