Dimensions in non-Archimedean geometries

Abstract

Let $K$ be an algebraically closed non-Archimedean field. Leonard Lipshitz has introduced a manageable notion of subanalytic sets of the unit polydisc. This class contains the class of affinoid sets and is stable under projection. We associate to a subanalytic set its counterpart in the Berkovich polydisc. This allows us to give a new insight to the dimension of subanalytic sets using the degrees of the completed residual fields. With these methods we obtain new results, such as the invariance of the dimension under subanalytic bijection in any characteristic. Then we study more generally subsets $S$ of $K^m\times \Gamma^n$ and of $K^m\times \Gamma^n \times k^p$ where $\Gamma$ is the value group and $k$ the residue field. We allow $S$ to be either definable in ACVF, or definable in the analytic language of L. Lipshitz. We define a dimension for such sets $S$. In the case when $S \subset K^n$ (resp. $S\subset \Gamma^n$, $S\subset k^n$), it coincides with the above dimension (resp. the o-minimal dimension, the Zariski dimension). We prove that this dimension is invariant under definable bijection and decreases under projection. This allows us to generalize previous results on tropicalization of Berkovich spaces and to place them in a general framework.