An absolute beginner level question on rigid analytic spaces & their interplay with formal schemes (after Raynaud). In this minicourse by Bosch on this topic he introduced as motivation following pictures whose meaning I not completely understand, see slide 37.
Question(s): Could somebody elaborate what is precisely depicted in the pictures on the "Motivating example" slide attached below? There I also specified the questions.
Firstly, the setting: $R=(R,m,v)$ is a complete (for sake of simplicity say discrete) valuation ring for non-Archimedean valuation $v$ of with field of fraction $K$ and residue field $k=R/m, m=(q)$ unique principal max ideal with generator $q \in R$ (especially $\vert q \vert_v <1$), so consists as Zariski spectrum of two points, generic $\eta$ corresp to $K$ and special pt $s$ corresp to residue $k$. (eg think of $p$-adic numbers $\Bbb Z_p$ or their extension)
A rigid space model $\Bbb G_{m,K}^{\text{an}}$ of multuplicative group scheme can be realized as union of affinoid subdomains $\Bbb G_{m,K}^{\text{an}}= \bigcup_n q^n \cdot (U_1 \cup U_2) $ where $U_1,U_2$ as open affinoid subdomains which are given as annuli of the unit disk $B:=B^1_K =\text{Sp} K\langle \zeta \rangle$ as follows:
$$ U_1 :=B(q\zeta^{-2})= \text{Sp} K\langle \zeta, q\zeta^{-2} \rangle=\{x \in B \ \vert \ \vert q \vert^{\frac{1}{2}} \le \vert \zeta(x) \vert \le 1 \} $$
and
$$ U_2 :=B(q\zeta^{-1}, q^{-1}\zeta^{2})= \text{Sp} K\langle \zeta, q\zeta^{-1}, q^{-1}\zeta^{2} \rangle=\{x \in B \ \vert \ \vert q \vert \le \vert \zeta(x) \vert \le \vert q \vert^{\frac{1}{2}} \} $$
Denote for annulus $U_1$ the obvious upper and lower boundaries
$$ U_1^+ :=B(\zeta^{-1})= \text{Sp} K\langle \zeta, \zeta^{-1} \rangle=\{x \in B \ \vert \ \vert \vert \zeta(x) \vert = 1 \} $$
and
$$ U_1^- :=B(q^{-1}\zeta^{2}, q^{1}\zeta^{-2})= \text{Sp} K\langle \zeta, q^{-1}\zeta^{2}, q^{1}\zeta^{-2} \rangle=\{x \in B \ \vert \ \vert \zeta(x) \vert = \vert q \vert^{\frac{1}{2}} \} $$
and analogously for $U_2 $ and their shifts $ q^nU_i$ by $q^n$. We identify $q^nU_i$ in the union canonically along their boundaries.
(All used notations can be looked up in linked or any other script on rigid geometry. ) Rmk.: The explicit construction of $\Bbb G_{m,K}^{\text{an}} $ I took from Bosch's book on Formal and Rigid Geometty, see page 218. Although I not completely grasped why we need to introduce $U_1,U_2$ as two separate annuli instead to consider directly one annulus $U_1 \cup U_2$ and to consider the union of their shifts by $q^n$. Is there a deeper reason for choice of this decomposing? Any ideas?)
Here finally the picture I'm refering to:
Concrete questions:
Firstly, in picture (1) the generic fibre over $\eta$ of $\Bbb G_{m,R}^{\text{an}}$ I guess the the vertical black bars (divided by horizontal bars) represent the annuli $q^n(U_1 \cup U_2)$, right? While the horizontal black bars represent the boudaries
$q^nU_i^{\pm}$, right?
Now to the special fibre in pictures (2) ($\Bbb G_{m,R}^{\text{an}}$ ), (3) (under $q^{\Bbb Z}$-action) & (4) (formal completion (...of what? $k[\zeta,\zeta^{-1}]$ with resp ideal $I=(q) R[\zeta] $?)) in the middle slide 37.
In (2) I guess the single red bar is representing the image of $\Bbb G_{m,R}^{\text{an}}$ under reduction of coefficients map, so isomorphic to $k[\zeta,\zeta^{-1}]$, right?
What about the grid of red bars in pictures (3) & (4) over the special point. What do these represent and how one should think about these?
Going maybe a step back, for instance, which space is depicted in 3rd picture indicated above (under $q^{\Bbb Z}$-action) description, having $\Bbb G_{m,K}^{\text{an}}$ as its generic fibre and such grid of red bars representing the special fibre?
Furthermore. if actually in (3) & (4) represent the formal scheme completion of $k[\zeta]$ at ideal $I=(q) R[\zeta] $, to what correspond all these red bars?
UPDATE #1: (a guess) Or, does the 3rd picture above marked as $q^{\Bbb Z}$-action depict an admissible formal blowup of the 2nd $\Bbb G_{m, R}$ giving all these new red bars? But then, why it is called "$q^{\Bbb Z}$-action"?
UPDATE #2 : (added later) Here I tried to reproduce pictorially how I understood Alex Youcis' comment on how to think about it [<- Deleted! The pictures turned out to give to be wrong interpretation. The attached pictures in the answer below should give right interpretation]
