Maranda’s theorem for pure-injective modules and duality

Gregory, Lorna ORCID: (2023) Maranda’s theorem for pure-injective modules and duality. Canadian Journal of Mathematics, 75 (2). pp. 581-607. ISSN 0008-414X

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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 ∈ N such that for all Λ-lattices L and M, if L/Lπ k ≃ M/Mπ k, then L ≃ 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-torsionfree part of the left Ziegler spectrum of Λ. 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)π k is the pure-injective hull of M/Mπ k. We use this result to give a characterization of R-torsion-free pure-injective Λ-modules and describe the pure-injective hulls of certain R-torsion-free Λ-modules.

Item Type: Article
Uncontrolled Keywords: order over a dedekind domain,ziegler spectrum,pure-injective,mathematics(all) ,/dk/atira/pure/subjectarea/asjc/2600
Faculty \ School: Faculty of Science > School of Mathematics
Related URLs:
Depositing User: LivePure Connector
Date Deposited: 19 Oct 2022 11:30
Last Modified: 17 Apr 2023 09:30
DOI: 10.4153/S0008414X22000098

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