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Formal and rigid geometry. IV: The reduced fibre theorem. (English) Zbl 0839.14014
[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.)

14G20 Local ground fields in algebraic geometry
14L05 Formal groups, \(p\)-divisible groups
11G25 Varieties over finite and local fields
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