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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.)

##### MSC:
 14G20 Local ground fields in algebraic geometry 14L05 Formal groups, $$p$$-divisible groups 11G25 Varieties over finite and local fields
##### Keywords:
rigid geometry; formal scheme
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##### References:
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