  In this article, we present the first half of our project on the Iwasawa
theory of higher rank Galois deformations over deformations rings of arbitrary
dimension. We develop a theory of Coleman maps for a very general class of
coefficient rings, devise a dimension reduction procedure for locally
restricted Euler systems and finally, put these into use in order to prove a
divisibility in a 3-variable main conjecture for nearly ordinary families of
Rankin-Selberg convolutions.
