Let $V$ be a paracompact strictly $K$-analytic space. We use here the analytic spaces and the terminology introduced by Berkovich in [Ber93, Section 1]. Then a formal ${K^{\circ}}$-model is an admissible formal scheme ${\mathfrak{V}}$ over ${K^{\circ}}$ [Bos14, §7.4] with a fixed isomorphism ${\mathfrak{V}}_{\eta}\cong V$ on the generic fiber ${\mathfrak{V}}_{\eta}$ which we again use for identification. Note that we have a canonical reduction map $\pi:V\to{\mathfrak{V}}_{s}$ to the special fiber ${\mathfrak{V}}_{s}$ (see [GRW15, Section 2]). If $\zeta_{Y}$ is the generic point of an irreducible component $Y$ of ${\mathfrak{V}}_{s}$, then $x_{Y}\coloneqq\pi^{-1}(\zeta_{Y})$ is finite and the points in this preimage are called divisorial points of $V$.

The category of paracompact strictly $K$-analytic spaces is equivalent to the category of quasiseparated rigid analytic varieties over $K$ with a strictly $K$-affinoid ${\rm G}$-covering of finite type (see [Ber93, §1.6]) and hence we may apply Raynaud’s theorem from [Bos14, Theorem 8.4.4]. In particular, we see that a formal ${K^{\circ}}$-model of $V$ exists and that the set of isomorphism classes of formal ${K^{\circ}}$-models is again directed. Some of the references in the following require that $V$ is compact, because the original formulation of Raynaud’s theorem in [BL93a, Theorem 4.1] used that the underlying rigid space is quasicompact and quasiseparated. This will be bypassed by using the more general version in [Bos14, Theorem 8.4.4] for paracompact $V$ (remember that paracompact includes Hausdorff).

Let $L$ be a line bundle on $V$ which means that $L$ is a locally free sheaf of rank $1$ on the ${\rm G}$-topology. We always consider the ${\rm G}$-topology induced by the strictly $K$-affinoid domains in $V$. A formal ${K^{\circ}}$-model $({\mathfrak{V}},{\mathfrak{L}})$ of $(V,L)$ consists of a formal ${K^{\circ}}$-model ${\mathfrak{V}}$ of $V$ and a line bundle ${\mathfrak{L}}$ on ${\mathfrak{V}}$ with a fixed isomorphism from ${\mathfrak{L}}|_{V}$ to $L$ which we use for identification. The argument in [Gub98, Lemma 7.6] shows that $(V,L)$ always has a formal ${K^{\circ}}$-model.