The basic geometry of Witt vectors, I: The affine case

TL;DR

This paper provides a detailed description of the category of etale algebras over Witt vectors, extending classical theory to generalized Witt vectors over various fields, laying foundations for future geometric and arithmetic applications.

Contribution

It introduces a new approach to generalized Witt vectors using commuting Frobenius lifts and develops their basic theory for affine algebraic geometry.

Findings

Concrete description of etale algebras over Witt vectors

Development of generalized Witt vector theory via Frobenius lifts

Foundations for Witt schemes and arithmetic jet spaces

Abstract

We give a concrete description of the category of etale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also for variants of these functors which are in a certain sense their analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for the classical Witt vectors. The larger purpose of this paper is to provide the affine foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces. So the basics here are developed somewhat fully, with an eye toward future applications.