Nonarchimedean geometry of Witt vectors

TL;DR

This paper explores the nonarchimedean analytic geometry of Witt vectors over perfect F_p-algebras, revealing deep analogies with polynomial rings and providing insights into p-adic Hodge theory.

Contribution

It establishes a geometric analogy between Witt vectors and polynomial rings with the Gauss norm, and analyzes the structure of associated Berkovich spaces.

Findings

The space of R is a strong deformation retract of the space of W(R).

Fibres form trees under pointwise comparison.

Classifies points of fibres similarly to Berkovich's disc classification.

Abstract

Let R be a perfect F_p-algebra, equipped with the trivial norm. Let W(R) be the ring of p-typical Witt vectors over R, equipped with the p-adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy between W(R) and the polynomial ring R[T] equipped with the Gauss norm, in which the role of the structure morphism from R to R[T] is played by the Teichmuller map. For instance, we show that the analytic space associated to R is a strong deformation retract of the space associated to W(R). We also show that each fibre forms a tree under the relation of pointwise comparison, and classify the points of fibres in the manner of Berkovich's classification of points of a nonarchimedean disc. Some results pertain to the study of p-adic representations of etale fundamental groups of nonarchimedean analytic spaces (i.e., relative p-adic Hodge theory).