Maranda's Theorem for Pure-Injective Modules and Duality
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866913620040876032 |
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| author | Gregory, Lorna |
| author_facet | Gregory, Lorna |
| contents | Let $R$ be a discrete valuation domain with field of fractions $Q$ and maximal ideal generated by $π$. Let $Λ$ be an $R$-order such that $QΛ$ is a separable $Q$-algebra. Maranda showed that there exists $k\in\mathbb{N}$ such that for all $Λ$-lattices $L$ and $M$, if $L/Lπ^k\simeq M/Mπ^k$ then $L\simeq M$. Moreover, if $R$ is complete and $L$ is an indecomposable $Λ$-lattice, then $L/Lπ^k$ is also indecomposable. We extend Maranda's theorem to the class of $R$-reduced $R$-torsion-free pure-injective $Λ$-modules.
As an application of this extension, we show that if $Λ$ is an order over a Dedekind domain $R$ with field of fractions $Q$ such that $QΛ$ is separable then the lattice of open subsets of the $R$-torsion-free part of the right Ziegler spectrum of $Λ$ is isomorphic to the lattice of open subsets of the $R$-torsion-free part of the left Ziegler spectrum of $Λ$.
Finally, with $k$ as in Maranda's theorem, we show that if $M$ is $R$-torsion-free and $H(M)$ is the pure-injective hull of $M$ then $H(M)/H(M)π^k$ is the pure-injective hull of $M/Mπ^k$. We use this result to give a characterisation of $R$-torsion-free pure-injective $Λ$-modules and describe the pure-injective hulls of certain $R$-torsion-free $Λ$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_07802 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Maranda's Theorem for Pure-Injective Modules and Duality Gregory, Lorna Representation Theory Logic 03C60 03C98 16G30 16H20 Let $R$ be a discrete valuation domain with field of fractions $Q$ and maximal ideal generated by $π$. Let $Λ$ be an $R$-order such that $QΛ$ is a separable $Q$-algebra. Maranda showed that there exists $k\in\mathbb{N}$ such that for all $Λ$-lattices $L$ and $M$, if $L/Lπ^k\simeq M/Mπ^k$ then $L\simeq M$. Moreover, if $R$ is complete and $L$ is an indecomposable $Λ$-lattice, then $L/Lπ^k$ is also indecomposable. We extend Maranda's theorem to the class of $R$-reduced $R$-torsion-free pure-injective $Λ$-modules. As an application of this extension, we show that if $Λ$ is an order over a Dedekind domain $R$ with field of fractions $Q$ such that $QΛ$ is separable then the lattice of open subsets of the $R$-torsion-free part of the right Ziegler spectrum of $Λ$ is isomorphic to the lattice of open subsets of the $R$-torsion-free part of the left Ziegler spectrum of $Λ$. Finally, with $k$ as in Maranda's theorem, we show that if $M$ is $R$-torsion-free and $H(M)$ is the pure-injective hull of $M$ then $H(M)/H(M)π^k$ is the pure-injective hull of $M/Mπ^k$. We use this result to give a characterisation of $R$-torsion-free pure-injective $Λ$-modules and describe the pure-injective hulls of certain $R$-torsion-free $Λ$-modules. |
| title | Maranda's Theorem for Pure-Injective Modules and Duality |
| topic | Representation Theory Logic 03C60 03C98 16G30 16H20 |
| url | https://arxiv.org/abs/1812.07802 |