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This paper contains a complete proof of Fukaya's and Kato'sepsilon$-isomorphism conjecture in [23] for invertible \Lambda-modules (thecase of V = V_0(r) where V_0 is unramified of dimension 1). Our results relyheavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms forone dimensional representations of G_{Q_p} in [27], but apart from fixing somesign-ambiguities in (loc.\ cit.) we use the theory of (\phi,\Gamma)-modulesinstead of syntomic cohomology. Also, for the convenience of the reader we givea slight modification or rather reformulation of it in the language of [23] andextend it to the (slightly non-commutative) semi-global setting. Finally wediscuss some direct applications concerning the Iwasawa theory of CM ellipticcurves, in particular the local Iwasawa Main Conjecture for CM elliptic curvesE over the extension of Q_p which trivialises the p-power division points E(p)of E. In this sense the paper is complimentary to the joint work [7] onnoncommutative Main Conjectures for CM elliptic curves.

On Kato's local epsilon-isomorphism Conjecture for rank one Iwasawa modules