  In this paper, we are going to establish a simultaneous generalization of the
relative Iwasawa theory proposed by Kedlaya-Pottharst and the relative $p$-adic
Hodge theory after Kedlaya-Liu. We call this Hodge-Iwasawa theory in the sense
that one could apply the theory to study noncommutative Iwasawa cohomology and
noncommutative Iwasawa theories in families and meanwhile one could apply the
theory to study the deformation theory of \'etale local systems or families of
representations of fundamental groups or the equivariant constructible $p$-adic
sheaves, with more sophisticated point of view coming from Kato, Fukaya-Kato.
We follow closely the approach of Kedlaya-Liu to study the corresponding
modules and sheaves over the corresponding deformed version of the period rings
and period sheaves.
