With the motivation of generalizing the corresponding geometrization of Tamagawa-Iwasawa theory after Kedlaya-Pottharst, and with motivation of establishing the corresponding equivariant version of the relative p-adic Hodge theory after Kedlaya-Liu aiming at the deformation of representations of profinite fundamental groups and the family of \'etale local systems, we initiate the corresponding Hodge-Iwasawa theory with deep point of view and philosophy in mind from early work of Kato and Fukaya-Kato. In this talk, we are going to discuss some foundational results on the Hodge-Iwasawa modules and Hodge-Iwasawa sheaves, as well as some interesting investigation towards the goal in our mind, which are taken from our first paper in this series project.