  In Part I we review some specific properties of the $\Lambda$-modules in
Iwasawa theory, which add structure to the general properties of Noetherian
$\Lambda$-torsion modules. Part II deals with Kummer theory and gives a
detailed construction of the Iwasawa linear space. This provides a new, simpler
proof of the conjectures of Leopoldt and Gross for CM extensions. Using a
construction of Thaine, we then prove that $\lambda^+ = 0$ in these fields,
thus proving a part of Greenberg's conjecture - the fact $\mu^+ = 0$ still has
to be shown. In the Appendices we give some elementary partial proofs of the
main facts proved using Iwasawa's linear space. These two papers do not give
proofs for non CM fields, and the reader interested in methods for dealing with
this case is referred to the "$T and T^*$" paper on this arxive. This methods
will be integrated in the Snoqit series.
