Maranda's Theorem for Pure-Injective Modules and Duality

Abstract

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/M π 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-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-torsionfree and H(M ) is the pure-injective hull of M then H(M )/H(M )π k is the pure-injective hull of M/M π 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.