Nonarchimedean geometry of Witt vectors

Abstract

AbstractLetRbe a perfect 𝔽-algebra equipped with the trivial norm. LetW(R) be the ring ofp-typical Witt vectors overRequipped with thep-adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy betweenW(R) and the polynomial ringR[T] equipped with the Gauss norm, in which the role of the structure morphism fromRtoR[T] is played by the Teichmüller map. For instance, we show that the analytic space associated toRis a strong deformation retract of the space associated toW(R). 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 ofp-adic representations of étale fundamental groups of nonarchimedean analytic spaces (i.e., relativep-adic Hodge theory).