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

Abstract

This paper is a continuation of a project to determine which skew polynomial algebras $S = R[θ; α]$ 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 $α$ 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 $α(X) = qX$ 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 $α$ is an arbitrary $k$-algebra automorphism of infinite order, $S$ satisfies $(\diamond)$. A number of open questions are stated.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• Theorem 2.7
• Theorem 2.8
• Theorem 3.1
• Proposition 3.2
• proof