Relative p-adic Hodge theory: Foundations

We describe a new approach to relative p-adic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean 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 etale Z_p-local systems and Q_p-local systems on the algebraic and analytic spaces associated to the base ring, and the relationship between etale cohomology and phi-cohomology. We also make a critical link to mixed characteristic by exhibiting an equivalence of tensor categories between the finite etale algebras over an arbitrary perfect Banach algebra over a nontrivially normed complete field of characteristic p and the finite etale algebras over a corresponding Banach Q_p-algebra. This recovers the homeomorphism between the absolute Galois groups of F_p((pi)) and Q_p(mu_{p^infty}) 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 etale local systems on analytic spaces over p-adic fields. One of these descriptions uses a relative version of the Fargues-Fontaine curve.