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Formal and rigid geometry. III: The relative maximum principle. (English) Zbl 0839.14013
[For part I and II of this paper see ibid. 295, No. 2, 291-317 and 296, No. 3, 403-429 (1993; Zbl 0808.14017 and 14018)].
This paper on formal and rigid geometry is the third part in a series. The classical maximum principle for affinoid spaces can be found in most treatises on rigid geometry. The aim of this paper is to prove a kind of maximum principle in a relative situation. The framework of admissible formal schemes, such as developed in the parts I and II of the series, is used to state the results.
Consider an admissible formal base scheme \(S\), a flat admissible formal \(S\)-scheme \(X\) and a global function \(f\) on \(X\). Then there exists a rig-flat and rig-quasi finite cover \(S'\to S\). There exists a finite rig-isomorphism \(Y' \to X \times_S S'\) and there exists a rig-invertible function \(\beta\) on \(S'\) and a function \(g\) on \(Y'\) which is invertible on a non empty open part of \(Y'\) and such that \(f = \beta \cdot g\). – If the rigid fibres of \(X/S\) are geometrically reduced then \(S' \to S\) may be assumed to be a rig-étale cover. – If the geometric fibres of \(X/S\) are reduced and irreducible then \(S' \to S\) may be assumed to be an admissible formal blowing up. (In this terminology a formal situation is rig-true if it is true in the associated rigid situation.)
[See also the following review of part IV of this paper].

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