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 eld. Using this map we investi- gate properties of the coecien t-wise multiplication on associated orders and modules of ideals. This technique allows to study the question of de- termining when the ring of integers is free over its associated order. We answer this question for an Abelian totally wildly ramied p-extension of complete discrete valuation elds whose dieren t is generated by an element of the base eld. We also determine when the ring of integers is free over a Hopf order as a Galois module.