Local Leopoldt's problem for rings of integers in Abelian $p$-extensions of complete discrete valuation fields

Abstract

Using the standard duality we construct a linear embedding of an associated module for a pair of ideals in an extension of a Dedekind ring into a tensor square of its fraction field. Using this map we investigate properties of the coefficient-wise multiplication on associated orders and modules of ideals. This technique allows to study the question of determining when the ring of integers is free over its associated order. We answer this question for an Abelian totally wildly ramified $p$-extension of complete discrete valuation fields whose different is generated by an element of the base field. We also determine when the ring of integers is free over a Hopf order as a Galois module.