  In this paper we study the corresponding categories and the corresponding
cohomologies of the Hodge-Iwasawa modules we developed in our series papers on
Hodge-Iwasawa theory. The corresponding cohomologies will be essential in the
corresponding development of the contact with the corresponding Iwasawa
theoretic consideration, while they are as well very crucial in the
corresponding study of the corresponding deformations of local systems over
general analytic spaces. We contact with some applications in analytic geometry
and arithmetic geometry which all have their own interests and deserve further
study for us in the future, including local systems over general analytic
spaces after Kedlaya-Liu, arithmetic Riemann-Hilbert correspondence in families
after Liu-Zhu, and equivariant Iwasawa theory and geometrization of equivariant
Iwasawa theory after Berger-Fourquaux and Nakamura.
