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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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##### References:
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