Points and topologies in rigid geometry
Let \(X\) be an affinoid space over a complete nonarchimedean field. The usual points of \(X\) are not sufficient to investigate sheaves on \(X\); for instance, there exist sheaves of \({\mathcal O}_{\mathcal X}\)-modules with all stalks equal to (0) and which are not (0). A main question is ``to add points'' to obtain a topoï with sufficiently many points. Preceeding works in that direction have been made by \textit{M. van der Put} [Compos. Math. 45, 165-198 (1982; Zbl 0491.14014)], \textit{V. G. Berkovich} [Math. Surveys Monographs 33, Am. Math. Soc. (1990; Zbl 0715.14013) and Publ. Math., Inst. Hautes Étud. Sci. 78, 5-161 (1993; Zbl 0804.32019)], \textit{R. Huber} [Math. Z. 212, 455-477 (1993; Zbl 0788.13010)] and \textit{P. Schneider} [J. Reine Angew. Math. 434, 127-157 (1993; Zbl 0774.14021)]. The purpose of this paper is to clarify the various concepts of points. Indeed, a link between Huber's approach and Berkovich's is given, which works for points and sheaves.
affinoid space over a complete nonarchimedean field
scientific article; zbMATH DE number 769470
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