Maranda’s theorem for pure-injective modules and duality

Gregory, Lorna 

Canadian journal of mathematics, Tome 75 (2023) no. 2, pp. 581-607

Résumé

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.

Détail

DOI : 10.4153/S0008414X22000098

Mots-clés : Order over a Dedekind domain, pure-injective, Ziegler spectrum