Author: Eiko

Time: 2025-04-07 08:19:48 - 2025-04-07 08:19:48 (UTC)

Rigid Spaces

Let $A$ be an affinoid algebra over $k$ and $X=\mathrm{Sp}(A)$ be the associated affinoid space.

• A subset $R\subset X$ is rational if there are $(f_0,\dots ,f_s)=A$ such that

  $$R=\{x\in X\mid |f_i(x)|\le |f_0(x)|\text{ for all }i=1,\dots ,s\}.$$

  associated to $R$ is the affinoid algebra (requiring convergence of $f_i/f_0$)

  $$B=A⟨Z_1,\dots ,Z_s⟩/(f_0{Z}_1-f_1,\dots ,f_0{Z}_s-f_s).$$

• Let $\varphi :A\to B_{\{}{f}_i\}$ be the obvious morphism of affinoid algebras induced by the above definition. Geometrically there is a canonical map

  $$\mathrm{Sp}(\varphi ):\mathrm{Sp}(B)\to \mathrm{Sp}(A)=X$$

  We have

  • $\mathrm{Sp}(\varphi )\mathrm{Sp}(B)\subset R\subset X$, the image of $\mathrm{Sp}(B)$ lies inside $R$.