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.