The basic geometry of Witt vectors, II: Spaces

TL;DR

This paper extends the algebraic geometry of Witt vectors and arithmetic jet spaces to more general settings involving arbitrary primes, algebraic spaces, and global fields, providing geometric descriptions and property analyses.

Contribution

It generalizes Witt vector theory and arithmetic jet spaces beyond p-typical cases to include arbitrary primes and algebraic spaces over global fields.

Findings

Generalized Witt vectors to arbitrary primes and global fields

Provided concrete geometric descriptions of Witt spaces

Analyzed property preservation under functors

Abstract

This is an account of the algebraic geometry of Witt vectors and arithmetic jet spaces. The usual, "p-typical" Witt vectors of p-adic schemes of finite type are already reasonably well understood. The main point here is to generalize this theory in two ways. We allow not just p-typical Witt vectors but those taken with respect to any set of primes in any ring of integers in any global field, for example. This includes the "big" Witt vectors. 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 Buium's formal arithmetic jet functor, which is dual to the Witt functor. We also give concrete geometric descriptions of Witt spaces and arithmetic jet spaces and investigate whether a number of standard geometric properties are preserved by these functors.