For sets, in $A\subset B$ equality is not excluded and $A\setminus B$ denotes the complement of $B$ in $A$. ${\mathbb{N}}$ includes $0$. All the rings and algebras are commutative with unity. For a ring $A$, the group of units is denoted by $A^{\times}$. If $V$ is a topological space, for a set $U\subset V$ we denote by $U^{\circ}$ the topological interior of $U$ in $V$. A variety over a field $k$ is an irreducible and reduced scheme which is separated and of finite type over $k$.

For the rest of the paper we fix a non-archimedean field $K$. This means here that the field $K$ is equipped with a non-archimedean absolute value $|\ |:K\to{\mathbb{R}}_{+}$ which is complete and non-trivial. Let $v:=-\log|\phantom{a}|$ be the corresponding valuation. We have a valuation ring $K^{\circ}:=\{x\in K\mid v(x)\geq 0\}$ with maximal ideal $K^{\circ\circ}:=\{x\in K\mid v(x)>0\}$ and residue field $\widetilde{K}:=K^{\circ}/K^{\circ\circ}$. We denote by $\overline{K}$ an algebraic closure of $K$ and we set ${\mathbb{C}}_{K}\coloneqq\widehat{\overline{K}}$ for the completion of $\overline{K}$.