Property $(\diamond )$ for Ore extensions of small Krull dimension

Abstract

This paper is a continuation of a project to determine which skew polynomial algebras $S=R[\theta ;\alpha ]$ satisfy property $(\diamond )$, namely that the injective hull of every simple $S$-module is locally artinian, where $k$ is a field, $R$ is a commutative noetherian $k$-algebra, and $\alpha$ is a $k$-algebra automorphism of $R$. Earlier work (which we review) and further analysis done here leads us to focus on the case where $S$ is a primitive domain and $R$ has Krull dimension 1 and contains an uncountable field. Then we show first that if $|\mathrm{Spec}(R)|$ is infinite then $S$ does not satisfy $(\diamond )$. Secondly we show that when $R=k[X{]}_{<X>}$ and $\alpha (X)=qX$ where $q\in k\setminus \{0\}$ is not a root of unity then $S$ does not satisfy $(\diamond )$. This is in complete contrast to our earlier result that, when $R=k[[X]]$ and $\alpha$ is an arbitrary $k$-algebra automorphism of infinite order, $S$ satisfies $(\diamond )$. A number of open questions are stated.

15pages, comments welcome