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arXiv · 2206.10872

Critical and injective modules over skew polynomial rings

Ken Brown·Paula A. A. B. Carvalho·Jerzy Matczuk
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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 $k[[X]]$.

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Ken Brown, Paula A. A. B. Carvalho, Jerzy Matczuk. 2022-06-22. Critical and injective modules over skew polynomial rings. https://arxiv.org/abs/2206.10872

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