Formal and rigid geometry. IV: The reduced fibre theorem
type: Item
instance of: scholarly article
title: Formal and rigid geometry. IV: The reduced fibre theorem
zbMATH Open document ID: 0839.14014
DOI: 10.1007/BF01245187
author: Siegfried Bosch
author: Werner Lütkebohmert
author: Michel Raynaud
published in: Inventiones Mathematicae
publication date: 1996-06-23T00:00:00Z
full work available at URL: https://eudml.org/doc/144261
review text: [For part III of this paper see Math. Ann. 302, No. 1, 1-29 (1995; see the preceding review).] In the classical theory of rigid geometry one has the following result: Let \(R\) be a complete height 1 valuation ring with field of fractions \(K\) and with residue field \(k\). Let \(A\) be a \(K\)-affinoid algebra and let \(A^0\) be the corresponding algebra of elements having supremum norm \(\leq 1\). Assume that \(A\) is geometrically reduced in the sense that it remains reduced after tensoring with a finite extension of \(K\). Then there exists a finite extension \(K'\) of \(K\) such that, writing \(R'\) for the algebraic closure of \(R\) in \(K'\) and \(k'\) for the residue field of \(R'\) and \(A' = A \otimes_K K'\), \(A'{}^0\) is topologically of finite type and \(A'{}^0 \otimes_{R'} k'\) is also geometrically reduced. The aim of this paper is to prove the following relative version of the result. Let \(S\) be a noetherian formal scheme and let \({\mathcal I}\) be a coherent ideal on \(S\) such that \(S\) is complete for the \({\mathcal I}\)-adic topology. Let \(U\) be the associated rigid space on \(S - V ({\mathcal I})\). Let \(f : X \to S\) be a morphism of \({\mathcal I}\)-adic formal schemes, topologically of finite type and such that, over \(U\), \(f\) is flat and has geometrically reduced fibres. Then there exists a rig-étale cover \(S' \to S\) and a finite morphism of formal schemes \(X' \to X \times_SS'\) which is an isomorphism over \(U\) and such that \(X'\) is flat over \(S'\) and has geometrically reduced fibres. (In this terminology a formal situation is rig-true if it is true in the associated rigid situation.)
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 11G25
zbMATH DE Number: 783993
zbMATH Keywords: rigid geometry
zbMATH Keywords: formal scheme
reviewed by: Guido Van Steen
MaRDI profile type: Publication
cites work: Q3963102
cites work: Q3325851
cites work: Formal and rigid geometry. I: Rigid spaces
cites work: Formal and rigid geometry. II: Flattening techniques
cites work: Formal and rigid geometry. III: The relative maximum principle
cites work: Über die Methode der diskret bewerteten Rigne in der nicht- archimedischen Analysis
cites work: Q4143433
cites work: Toroidal embeddings. I
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: Rigid analytic spaces
OpenAlex ID: W2994972836
description: scientific article; zbMATH DE number 783993