Cohomology of formal group moduli and deeply ramified extensions

Abstract

The aim of this paper is to answer a question of Coates and Greenberg: let $F$ be a commutative $m$-dimensional formal group over the ring of integers of a local field $k$, and let $K$ be an algebraic extension of $k$ with infinite ramification index. Denote by ${\cal M}_{\Mbar}$ the maximal ideal in the ring of integers of the separable closure of $K$. Suppose that the height of $F$ is greater than $m$. Does $H^1 (K, F ({\cal M}{}^m_{\Mbar}))=0$ imply that $K$ is deeply ramified? The answer is positive.