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W . Lütkebohmert

Definition 1.4, Let (^, (P^^ ) be an admissible formal i?-scheme and let »/ be a coherent sheaf of open ideals of Фд^. The formal blowing-up of .^ in ^ is a morphism of admissible formal schemes f :^'-♦^ which is defined as follows:

For any open affine subscheme Tr = Spf(^)of ^and/=Г(1Г, j^^),let F'bethe blowing-up of/in F=Spec(^). Then let ^' be the completion of V along its special fibre. Since the construction of 1^ 4s local in nature, these constructions fit together to build an admissible formal scheme ^\Ж'^Ж,

Such blowing-ups are referred to as admissible formal ones. By the center of the admissible blowing-up ^'-*^ we mean the locus of ^/бй^^ in ^q.

Such a blowing-up has the following properties; a summary on the results of blowing-ups can be found in [Rl, Sect. 5.1]:

( a ) The ideal ./(5^^ is invertible on ^'.

( b ) If g : J^ -► ^ is a morphism of any formal schemes such that У 6?^ is invertible on JT, then there exists a unique morphism \) : Ж-^Ж' factoring g; i.e. g = f° ï).

( c ) Let N be an integer and let 4ù^ be the open subscheme of Ж^ defined as the complement of the center. In particular, f^^ is an isomorphism over 41^^. \{(&^ с^^д^ is a closed immersion and lî^i^-^^^ is the blowing-up of the ideal ^%^, the canonical map ^^--^Ж^ identifies ^^ with the schematic closure ^^ of %n% in ^^^; cf [Rl,5.1.2(iv)].

There is a functorial way to associate to an admissible formal scheme Ж over Spf (Л) a rigid-analytic space over K, For an open formal subscheme % = Spf (^) of Ж, one associates to ^ the spectrum of maximal ideals и=8р(Ак) of Aj^ with its canonical rigid-analytic structure [T]. One calls Xthe generic fibre of Ж* and ^o the special fibre of ^. We refer to Ж* as a formel i?-model of X. An admissible open subvariety (resp. an admissible covering) of Zis said to be formal with respect to the model ^ if it is induced by an open subscheme (resp. by an open covering) of Ж.

An admissible formal blowing-up Ж'-^Ж gives rise to a rigid-analytic phism Z'::^Jif of their associated rigid spaces. In this paper we will make use of the following theorems which are mainly due to Raynaud; proofs have been worked out by Mehlmann [M].

Theorem 1.5 [M, 4.3.14]. Let Xbea quasi-compact rigid-analytic space with a finite covering {Ui}by open affinoidsubvarieties t/^. Then there exists an R-model3CofX such that the covering {U^ is induced by a formal covering of Ж.

Furthermore , any morphism between quasi-compact rigid-analytic spaces can be regarded as a formal morphism of suitable jR-models.

Июогет 1.6 [M, 4.3.13,4.3.8, 2.7.4]. Let Ж and ^ be quasi-compact admissible formal schemes. Let f : Jf-^ Y be a rigid-analytic morphism of their associated rigid spaces. Then there exists an admissible formal blowing-up Ж'-^Ж such that f is inducedby a formal morphism f : Ж*-^^, Furthermore, if fis an isomorphism, one can choose f to be an admissible formal blowing-up.

More precisely, if Ж is an open subscheme of an admissible formal scheme Ж and if f is an open imrmrsion, there exist admissibkformal blowing-ups 3t'-^3£ and^ '-^^ such that there exists an open immersion f ' : Ж'-*&'inducing f where Ж'=^^' x ^Ж.