Global applications of relative (phi, Gamma)-modules I

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