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Maranda’s theorem for pure-injective modules and duality

Part of: Model theory Representation theory of rings and algebras

Published online by Cambridge University Press:  17 March 2022

Lorna Gregory Open the ORCID record for Lorna Gregory [Opens in a new window]
Lorna Gregory*
Affiliation:
Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli,” Viale Abramo Lincoln, 5, 81100 Caserta CE, Italy
*
e-mail: [email protected]
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Abstract

Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ . Let $\Lambda $ be an R-order such that $Q\Lambda $ is a separable Q-algebra. Maranda showed that there exists $k\in \mathbb {N}$ such that for all $\Lambda $ -lattices L and M, if $L/L\pi ^k\simeq M/M\pi ^k$ , then $L\simeq M$ . Moreover, if R is complete and L is an indecomposable $\Lambda $ -lattice, then $L/L\pi ^k$ is also indecomposable. We extend Maranda’s theorem to the class of R-reduced R-torsion-free pure-injective $\Lambda $ -modules.

As an application of this extension, we show that if $\Lambda $ is an order over a Dedekind domain R with field of fractions Q such that $Q\Lambda $ is separable, then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of $\Lambda $ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of $\Lambda $ .

Furthermore, 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 $H(M)/H(M)\pi ^k$ is the pure-injective hull of $M/M\pi ^k$ . We use this result to give a characterization of R-torsion-free pure-injective $\Lambda $ -modules and describe the pure-injective hulls of certain R-torsion-free $\Lambda $ -modules.

Keywords

Order over a Dedekind domainpure-injectiveZiegler spectrum

MSC classification

Primary: 03C60: Model-theoretic algebra
Secondary: 03C98: Applications of model theory 16G30: Representations of orders, lattices, algebras over commutative rings 16H20: Lattices over orders
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 2 , April 2023 , pp. 581 - 607
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The majority of this work was completed while the author was employed by the University of Camerino.

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