Vol. 6, No. 2, 2012

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Uniformly rigid spaces

Christian Kappen

Vol. 6 (2012), No. 2, 341–388

We define a new category of nonarchimedean analytic spaces over a complete discretely valued field, which we call uniformly rigid. It extends the category of rigid spaces, and it can be described in terms of bounded functions on products of open and closed polydiscs. We relate uniformly rigid spaces to their associated classical rigid spaces, and we transfer various constructions and results from rigid geometry to the uniformly rigid setting. In particular, we prove an analog of Kiehl’s patching theorem for coherent ideals, and we define the uniformly rigid generic fiber of a formal scheme of formally finite type. This uniformly rigid generic fiber is more intimately linked to its model than the classical rigid generic fiber obtained via Berthelot’s construction.

semiaffinoid, uniformly rigid, formally finite type, rigid geometry, formal geometry, Berthelot construction
Mathematical Subject Classification 2010
Primary: 14G22
Secondary: 14K15
Received: 6 September 2010
Revised: 22 February 2011
Accepted: 25 March 2011
Published: 24 June 2012
Christian Kappen
Institut für Experimentelle Mathematik
Universität Duisburg-Essen
Ellernstrasse 29
D-45326 Essen