  This paper contains a complete proof of Fukaya's and Kato's
epsilon$-isomorphism conjecture in [23] for invertible \Lambda-modules (the
case of V = V_0(r) where V_0 is unramified of dimension 1). Our results rely
heavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms for
one dimensional representations of G_{Q_p} in [27], but apart from fixing some
sign-ambiguities in (loc.\ cit.) we use the theory of (\phi,\Gamma)-modules
instead of syntomic cohomology. Also, for the convenience of the reader we give
a slight modification or rather reformulation of it in the language of [23] and
extend it to the (slightly non-commutative) semi-global setting. Finally we
discuss some direct applications concerning the Iwasawa theory of CM elliptic
curves, in particular the local Iwasawa Main Conjecture for CM elliptic curves
E 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] on
noncommutative Main Conjectures for CM elliptic curves.
