The Stacks project

Remark 86.18.2. The condition as formulated in Definition 86.18.1 is not right even for morphisms of finite type of locally adic* formal algebraic spaces. For example, if $A = (\bigcup _{n \geq 1} k[t^{1/n}])^\wedge $ where the completion is the $t$-adic completion, then there are no adic morphisms $\text{Spf}(R) \to \text{Spf}(A)$ where $R$ is a complete discrete valuation ring. Thus any morphism $X \to \text{Spf}(A)$ would be rig-surjective, but since $A$ is a domain and $t \in A$ is not zero, we want to think of $A$ as having at least one “rig-point”, and we do not want to allow $X = \emptyset $. To cover this particular case, one can consider adic morphisms

\[ \text{Spf}(R) \longrightarrow Y \]

where $R$ is a valuation ring complete with respect to a principal ideal $J$ whose radical is $\mathfrak m_ R = \sqrt{J}$. In this case the value group of $R$ can be embedded into $(\mathbf{R}, +)$ and one obtains the point of view used by Berkovich in defining an analytic space associated to $Y$, see [Berkovich]. Another approach is championed by Huber. In his theory, one drops the hypothesis that $\mathop{\mathrm{Spec}}(R/J)$ is a singleton, see [Huber-continuous-valuations].


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