Relative p-adic Hodge theory, I: Foundations

We initiate 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 Berkovich. In this paper, we give a thorough development of φ-modules over a relative Robba ring associated to a perfect Banach ring of characteristic p, including the relationship between these objects and étale Zp-local systems and Qp-local systems on the algebraic and analytic spaces associated to the base ring, and the relationship between étale cohomology and φ-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 Qp-algebra. This recovers the homeomorphism between the absolute Galois groups of Fp((π)) and Qp(μp∞) given by the field of norms construction of Fontaine and Wintenberger, as well as generalizations considered by Andreatta, Brinon, Faltings, and Scholl. Applications to the description of étale local systems on nonarchimedean analytic spaces over p-adic fields will be described in subsequent papers.