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Let $A$ be a Huber (i.e. f-adic) ring, meaning a topological ring with an open subring $A_0$ which is adic and has finitely generated ideal of definition.

Is there a good notion of closed disk of radius $r > 0$ over $\text{Spa } A$? If $A$ is a nonarchimedean field this is $$\text{Spa } \{ \sum_{n \geq 0} a_nx^n \ | \ |a_n|r^n \to 0 \}.$$ I'm hoping for a nice version of this in families, but I don't think we should just impose the condition above for all continuous valuations on $A$: it seems like nilpotents would cause problems. Ideally I would want functions on a disk of radius $0 < r < 1$ over any discrete ring $A$ to be $A[[x]]$.

And what about annuli? Again it would be nice if there was a uniform definition so that functions on an annulus with radii $0 \leq r_1 < r_2 < 1$ over $A$ discrete are $A((x))$.

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    $\begingroup$ When $A$ is an affinoid algebra over a non-archimedean field $k$ this is solved in Berkovich's theory, including the case of a field $k$ with trivial valuation, for which the result has underlying ring $k[\![x]\!]$ but the topology depends very much on $r$ (as it had better do to be compatible with base change). More conceptually, since value groups at points of adic spaces have nothing to do with each other without the crutch of a non-trivially valued ground field, how is a real number $r$ means to insert itself into those value groups? That is, define the functor you want to be represented. $\endgroup$
    – user74230
    Commented Feb 28, 2015 at 22:35

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