Interpretation functors which are full on pure-injective modules with applications to $R$-torsion-free modules over $R$-orders
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| Format: | Preprint |
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2024
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| _version_ | 1866917871789015040 |
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| author | Gregory, Lorna |
| author_facet | Gregory, Lorna |
| contents | Let $R,S$ be rings, $\mathcal{X}\subseteq \text{mod}$-$R$ a covariantly finite subcategory, $\mathcal{C}$ the smallest definable subcategory of $\text{Mod}$-$R$ containing $\mathcal{X}$ and $\mathcal{D}$ a definable subcategory of $\text{Mod}$-$S$. We show that if $I:\mathcal{C}\rightarrow \mathcal{D}$ is an interpretation functor such that $I\mathcal{X}\subseteq \text{mod}$-$S$ and whose restriction to $\mathcal{X}$ is full then $I$ is full on pure-injective modules. We apply this theorem to an extension of a functor introduced by Ringel and Roggenkamp which, in particular, allows us to describe the torsion-free part of the Ziegler spectra of tame Bäckström orders. We also introduce the notion of a pseudogeneric module over an order which is intended to play the same role for lattices over orders as generic modules do for finite-dimensional modules over finite-dimensional algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13396 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interpretation functors which are full on pure-injective modules with applications to $R$-torsion-free modules over $R$-orders Gregory, Lorna Representation Theory Logic 03C60, 16G30 (primary), 03C98, 16H2 Let $R,S$ be rings, $\mathcal{X}\subseteq \text{mod}$-$R$ a covariantly finite subcategory, $\mathcal{C}$ the smallest definable subcategory of $\text{Mod}$-$R$ containing $\mathcal{X}$ and $\mathcal{D}$ a definable subcategory of $\text{Mod}$-$S$. We show that if $I:\mathcal{C}\rightarrow \mathcal{D}$ is an interpretation functor such that $I\mathcal{X}\subseteq \text{mod}$-$S$ and whose restriction to $\mathcal{X}$ is full then $I$ is full on pure-injective modules. We apply this theorem to an extension of a functor introduced by Ringel and Roggenkamp which, in particular, allows us to describe the torsion-free part of the Ziegler spectra of tame Bäckström orders. We also introduce the notion of a pseudogeneric module over an order which is intended to play the same role for lattices over orders as generic modules do for finite-dimensional modules over finite-dimensional algebras. |
| title | Interpretation functors which are full on pure-injective modules with applications to $R$-torsion-free modules over $R$-orders |
| topic | Representation Theory Logic 03C60, 16G30 (primary), 03C98, 16H2 |
| url | https://arxiv.org/abs/2412.13396 |