Sheaf of power-bounded elements in rigid analytic geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:41:42Z http://mathoverflow.net/feeds/question/49057 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49057/sheaf-of-power-bounded-elements-in-rigid-analytic-geometry Sheaf of power-bounded elements in rigid analytic geometry Joël 2010-12-11T17:23:14Z 2010-12-11T17:23:14Z <p>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.</p> <blockquote> <p>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?</p> </blockquote>