Maranda’s theorem for pure-injective modules and duality

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.