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The closed adic disc is defined as $Spa(\mathbb{Q}_p\langle T\rangle,\mathbb{Z}_p\langle T\rangle)$, and the open adic disc is defined to be the fiber $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ over the generic point $\eta\in Spa(\mathbb{Z}_p,\mathbb{Z}_p)$ of $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])\rightarrow Spa(\mathbb{Z}_p,\mathbb{Z}_p)$.

Presumably, at least some of the motivation for calling them closed and open discs, is that their respective functors of points are easy to describe:

  1. The functor of points of $Spa(\mathbb{Q}_p\langle T\rangle,\mathbb{Z}_p\langle T\rangle)$ takes $(R,R^+)$ to $R^+$.
  2. The functor of points of $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ takes $(R,R^+)$ to $R^{oo}$, the ring of topologically nilpotent elements.

But why should we expect the functor of points of an "adic closed disc", whatever that means, to be that? Same question for open.

For schemes, $Spec(A)$ is, heuristically, the space over which the ring of functions is $A$. So one when sets forth to define a "small open disc", one asks oneself what functions are defined on an infinitesimally small open disc. If one looks at the open disc at $x=0$ of the affine line over $\mathbb{C}$, then every function defined on a tiny open disc around $0$ can be expressed as a power series. So it makes sense to define an infinitesimally open disc around $0$ as $Spec(\mathbb{C}[[x]])$. (Well... I guess this intuition breaks down a little bit: there are power series that have radius of convergence $0$. But that's roughly the idea.)

But for adic spaces, I'm less clear on what open or closed discs are even supposed to aspire to be. After all $Spa(A,A^+)$ has a slighly more complicated intuition than schemes: it's heuristically the space over which the ring of functions is $A$, and all of the valuations on $A$ satisfy that $A^+$ goes into the unit closed disc around the origin. So how does this intuition make it any clearer that $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ and $Spa(\mathbb{Q}_p\langle T\rangle,\mathbb{Z}_p\langle T\rangle)$ are open and cloesd discs?

Is the adic closed disc the "closure" of the adic open disc in some sense...? Do they relate to one another in any way? Why does the open disc deal with $\mathbb{Z}_p[[T]]$, but the closed disc deal with $\mathbb{Z}_p\langle T \rangle$?

What does "open disc" and "closed disc" even mean at an intuitive level? In the scheme situation that I described, I took a one-dimensional space $\mathbb{C}[x]$ and looked locally near $0$. Is there a similarly nice geometric picture associated with adic open and closed discs, or are these definitions purely technical?

Clarifying edit (taken from my own comment): Let's take for a minute the open disc of radius 1 in C with the complex topology. Then in the category of complex manifolds, it doesn't make any sense that the functor of points of the open disc applied to some complex manifold is "the open disc of radius 1" of that manifold, because complex manifolds don't have a unique open disc. It just seems so far removed from the complex situation that either "open/closed disc" is disorientingly misleading, or I'm missing some hidden intuition...

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  • $\begingroup$ If you compute the $\mathbf{C}_p$ (or $\overline{\mathbf{Q}}_p$) points of these two spaces, then you might get some intuition for why these are natural candidates for closed and open discs. Note that these are supposed to be discs of radius one -- infinitesimally small discs are a red herring here. $\endgroup$
    – David Loeffler
    Commented Aug 31, 2018 at 12:25
  • $\begingroup$ Let's take for a minute the open disc of radius 1 in $\mathbb{C}$ with the complex topology. Then in the category of complex manifolds, it doesn't make any sense that the functor of points of the open disc applied to some complex manifold is "the open disc of radius 1" of that manifold, because complex manifolds don't have a unique open disc. It just seems so far removed from the complex situation that either "open/closed disc" is disorientingly misleading, or I'm missing some hidden intuition... I'm hoping for the latter. $\endgroup$
    – Andrew NC
    Commented Sep 1, 2018 at 5:23
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    $\begingroup$ May I suggest that you set aside for a moment all of this abstract nonsense with functors of points, analogies to $\mathbf{C}$, etc, and actually do the exercise I outlined in my previous comment? $\endgroup$
    – David Loeffler
    Commented Sep 1, 2018 at 8:15
  • $\begingroup$ It sounds like what you're saying is that the functor of points applied to C_p produces the open/closed discs of radius 1. So it sounds like your point is that the reason that it's called the open/closed discs is primarily and perhaps solely because they represent functors of points that are nice. $\endgroup$
    – Andrew NC
    Commented Sep 3, 2018 at 4:47
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    $\begingroup$ One way to think about it is that the ring $\mathbb Q_p\langle T \rangle$ consists of power series which converge if you plug in numbers in the closed unit disk in $\mathbb C_p$. But if you allow all power series in $\mathbb Z_p[[T]]$ (or $\mathbb Z_p[[T]] \otimes_{\mathbb Z_p} \mathbb Q_p$ for that matter), these series only converge when you plug in numbers in the interior of the disk. This should explain the "functions on a space" motivation $\endgroup$
    – Ashwin Iyengar
    Commented Apr 14, 2021 at 12:02

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