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 âetale Z[subscript p]-local systems and Q[subscript p]-local systems on the algebraic and analytic spaces associated to the base ring, and the relationship between (pro-)âetale cohomology and [phi]-cohomology. We also make a critical link to mixed characteristic by exhibiting an equivalence of tensor categories between the finite âetale algebras over an arbitrary perfect Banach algebra over a nontrivially normed complete field of characteristic p and the finite âetale algebras over a corresponding Banach Q[subscript p]-algebra. This recovers the homeomorphism between the absolute Galois groups of F[subscript p](([pi])) and Q[subscriptp] ([mu] [subscript p][infinity]) 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 âetale local systems on analytic spaces over p-adic fields. One of these descriptions uses a relative version of the Fargues-Fontaine curve."