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Mathematische Annalen
Article . 1995 . Peer-reviewed
License: Springer TDM
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Points and topologies in rigid geometry

Authors: Schneider, P.; Put, M. van der;

Points and topologies in rigid geometry

Abstract

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.

Countries
Netherlands, Germany
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Keywords

Local ground fields in algebraic geometry, 510.mathematics, SPACES, sheaves, rigid space, affinoid space over a complete nonarchimedean field, Article, Arithmetic algebraic geometry (Diophantine geometry), points

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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
27
Top 10%
Top 10%
Average
Green
bronze