In this paper, given a smooth proper
scheme X over a p-adic DVR and a p-power torsion \''etale
local system L on it, we study a family of sheaves
associated to the cohomology of local, relative
(ϕ,Γ)-modules of L and their cohomology. As
applications we derive descriptions of the \''etale cohomology
groups on the geometric generic fiber of X with values in L, as well as of their classical (ϕ,Γ)-modules, in
terms of cohomology of the above mentioned sheaves