Description

We describe a new approach to relative p-adic Hodge theory
based on systematic use of Witt vector constructions and nonar-chimedean
analytic geometry in the style of both Berkovich and
Huber. We give a thorough development of Phi-modules over a relative
Robba ring associated to a perfect Banach ring of characteristic
p, including the relationship between these objects and
étale Z_p-local systems and Q_p-local systems on the algebraic and
analytic spaces associated to the base ring, and the relationship
between (pro-)étale cohomology and Phi-cohomology. We also make
a critical link to mixed characteristic by exhibiting an equivalence
of tensor categories between the finite étale algebras over an arbitrary
perfect Banach algebra over a nontrivially normed complete
field of characteristic p and the finite étale algebras over a
corresponding Banach Q_p-algebra. This recovers the homeomorphism
between the absolute Galois groups of F_p((Pi)) and Q_p(µ_pInfini)
given by the field of norms construction of Fontaine and Wintenberger,
as well as generalizations considered by Andreatta, Brinon,
Faltings, Gabber, Ramero, Scholl, and most recently Scholze. Using
Huber's formalism of adic spaces and Scholze's formalism of
perfectoid spaces, we globalize the constructions to give several
descriptions of the étale local systems on analytic spaces over p-adic
fields. One of these descriptions uses a relative version of the
Fargues-Fontaine curve.