Property $(\diamond)$ for Ore extensions of small Krull dimension
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918035497943040 |
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| author | Brown, Ken Carvalho, Paula A. A. B. Matczuk, Jerzy |
| author_facet | Brown, Ken Carvalho, Paula A. A. B. Matczuk, Jerzy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09239 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Property $(\diamond)$ for Ore extensions of small Krull dimension Brown, Ken Carvalho, Paula A. A. B. Matczuk, Jerzy Rings and Algebras 16D50, 16P40, 16S35 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. |
| title | Property $(\diamond)$ for Ore extensions of small Krull dimension |
| topic | Rings and Algebras 16D50, 16P40, 16S35 |
| url | https://arxiv.org/abs/2403.09239 |