Let be a perfect -algebra equipped with the trivial norm. Let be the ring of -typical Witt vectors over equipped with the -adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy between and the polynomial ring equipped with the Gauss norm, in which the role of the structure morphism from to is played by the Teichmüller map. For instance, we show that the analytic space associated to is a strong deformation retract of the space associated to . 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 -adic representations of étale fundamental groups of nonarchimedean analytic spaces (i.e., relative -adic Hodge theory).
"Nonarchimedean geometry of Witt vectors." Nagoya Math. J. 209 111 - 165, March 2013. https://doi.org/10.1215/00277630-1959469