  We consider more general framework than the corresponding one considered in
our previous work on the Hodge-Iwasawa theory. In our current consideration we
consider the corresponding more general base spaces, namely the analytic adic
spaces and analytic perfectoid spaces in Kedlaya's AWS Lecture notes. We hope
our discussion will also shed some light on further generalization to even more
general spaces such as those considered by Gabber-Ramero namely one just
considers certain topological rings which satisfy the Fontaine-Wintenberger
idempotent correspondence and calls them perfectoid generalizing the notions
from Scholze, Fontaine, Kedlaya-Liu and Kedlaya (AWS Lecture notes). Actually
some of the discussion we presented here is already in some more general form
for this purpose (although we have not made enough efforts to write all the
things).
