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All spaces in this question are adic spaces.

In the paper

Pierre Colmez, Sally Gilles, and Wiesława Nizioł. Arithmetic duality for p-adic pro-étale cohomology of analytic curves. arXiv:2308.07712, 2023.

the authors define a curious object called the "ghost circle".

Let $K$ be a finite extension of $\mathbf{Q}_p$ with ring of integers $\mathcal{O}_K$, and let $D$ be the open unit disk over $K$. Then the ghost circle is defined to be $$\partial D:=\varprojlim_n D\setminus D_n$$where $D_n$ is the closed disk of radius $p^{-1/n}$. In other words, $\partial D$ is the " boundary" of the open unit disk.

But I don't think I understand what this definition means. If one defines the open unit disk in the "usual" way, i.e., $$D=\bigcup_n D_n=\mathrm{Spa}(\mathcal{O}_K[[T]],\mathcal{O}_K[[T]])\times_{\mathrm{Spa}(\mathcal{O})}{\mathrm{Spa}(K)},$$ then it seems $\partial D$ should be empty.

My guess is that maybe I should view $\partial D$ only as a formal object. That is, $\partial D$ is probably just a pro-object whose cohomology is defined to be $$R\Gamma(\partial D,\mathcal{F}):=\varinjlim_n R\Gamma(D\setminus D_n, \mathcal{F})$$for any reasonable cohomology theory $R\Gamma(-,\mathcal{F})$ (e.g. $\mathcal{F}=\mathbf{Q}_p$).

Can someone explain to me if my guess is correct?

user393795
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    In $\S 6.4.1$, the authors say that the ring of regular functions on $D$ is the Robba ring (whose definition I forgot). Does this shed some light on your guess? – Aphelli Sep 27 '25 at 08:56

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Yes, you are correct. The "ghost circle" doesn't actually exist as a space, it's more like a notational device: you should understand $R\Gamma(\partial D, -)$ as the inductive limit you wrote down. This is basically forced upon us if we want to have an exact triangle $$ R\Gamma_c(D, -) \to R\Gamma(D, -) \to R\Gamma(\partial D, -) \to [+1]. $$

In particular, $\mathcal{O}(\partial D)$ has to be $\varinjlim \mathcal{O}(D \setminus D_n)$, and that is exactly the definition of the Robba ring, as @Aphelli pointed out in the comments.

David Loeffler
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