The Basic Geometry of Witt Vectors

This is a foundational account of the étale topology of generalized Witt vectors and of related constructions. The theory of the usual, “p-typical” Witt vectors of p-adic schemes of finite type is already reasonably well developed. The main point here is to generalize this theory in two different ways. We allow not just p-typical Witt vectors but also, for example, those taken with respect to any set of primes in any ring of integers in any global field. We also allow not just p-adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of the Greenberg transform. We investigate whether many standard geometric properties of spaces and maps are preserved by Witt vector functors.