Nonarchimedean geometry of Witt vectors

Abstract

Let $R$ be a perfect ${\mathbb{F}}_p$-algebra equipped with the trivial norm. Let $W\left(R\right)$ 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 $W\left(R\right)$ and the polynomial ring $R\left[T\right]$ equipped with the Gauss norm, in which the role of the structure morphism from $R$ to $R\left[T\right]$ is played by the Teichmüller map. For instance, we show that the analytic space associated to $R$ is a strong deformation retract of the space associated to $W\left(R\right)$. We also show that each fiber forms a tree under the relation of pointwise comparison, and we classify the points of fibers in the manner of Berkovich’s classification of points of a nonarchimedean disk. Some results pertain to the study of $p$-adic representations of étale fundamental groups of nonarchimedean analytic spaces (i.e., relative $p$-adic Hodge theory).