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Let $k$ be a field with a non-archimedean complete valuation $|\ |$, $X$ a reduced rigid analytic space over $k$. The presheaf $\mathcal{O}^0$ which to an affinoid $U$ of $X$ attaches the ring $\mathcal{O}(U)^0$ of power-bounded elements of $\mathcal{O}(U)$ is a sheaf for the $G$-topology (isn't it?) for the condition for an $f \in \mathcal{O}(U)$ of being power bounded is equivalent to $|f(x)| \leq 1$ for all $x \in U$, a clearly local condition.

Is there a reference for this statement, and more generally a theory of "coherent" $\mathcal{O}^0$-modules, analog to Kiehl's theory of coherent $\mathcal{O}$-modules?

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Dear Joel: The sheaf property doesn't seem to need a reference (it's obvious, as you say). For disc.-valued $k$, if $\mathfrak{X}$ is a normal flat top. lft formal scheme over $O_k$ and $X = \mathfrak{X}^{\rm{rig}}$ then ring of global fns on $\mathfrak{X}$ maps isomorphically onto ring of power-bounded fns on $X$ (see Thm. 7.4.1 deJong's IHES paper "Crystalline Dieudonne theory..."). Also, by 6.4.1/6 in BGR, $A^0$ is top. f.type over $O_k$ for any $k$-affinoid $A$ in discr.-valued case, but formation is bad for surjections (see BGR, 6.4.3), so a "useful" Kiehl-type theory seems unlikely. –  BCnrd Dec 11 '10 at 18:15
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