On arithmetic families of filtered φ-modules and crystalline representations [Elektronische Ressource] / Eugen Hellmann

Eugen Hellmann

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On arithmetic families of filteredϕ-modules and crystalline representationsDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen Fakultat¨derRheinischen Friedrich-Wilhelms-Universitat¨ Bonnvorgelegt vonEugen HellmannausRemscheidBonn 2010Angefertigt mit Genehmigung der Mathematisch-NaturwissenschaftlichenFakultat¨ der Rheinischen Friedrich-Wilhelms-Universitat¨ Bonn1. Gutachter: Prof. Dr. Michael Rapoport2. Gutachter: Prof. Dr. Gerd FaltingsTag der Promotion: 21.04.2011Erscheinungsjahr: 2011ZusammenfassungDievorliegendeArbeitbeschäftigtsichmitderParametrisierungvonStruk-turen, die in der p-adischen Hodge-Theorie auftreten. Sei K eine endliche¯ErweiterungdesKörpersQ derp-adischenZahlenundseiG = Gal(K/K)p Kdessen absolute Galoisgruppe. In p-adischer Hodge-Theorie untersucht mangewisse stetige Darstellungen vonG auf endlich dimensionalenQ -Vektor-K präumen. DieKlassenvonDarstellungen,diehierbeiauftreten,werdenmithilfevon Periodenringen definiert. Im Fall der vorliegenden Arbeit sind dies so-genannte kristalline Darstellungen. Fontaine definiert einen Ring B mitcrisstetiger G -Operation und nennt eine Darstellung von G auf einem d-K KdimensionalenQ -Vektorraum V kristallin, fallspGKD (V)=(V ⊗ B )cris Q crispGKvonDimensiondüberK =B ist. DabeiistK diemaximalunverzweigte0 0crisErweiterungvonQ inK.

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• Mathematics

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• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2011
• Nombre de lectures: 40
• Langue: Deutsch
• Poids de l'ouvrage: 1 Mo

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On arithmetic families of filtered ϕ-modules and crystalline representations

Dissertation

zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-NaturwissenschaftlichenFakulta¨t der RheinischenFriedrich-Wilhelms-Universit¨atBonn

vorgelegt von Eugen Hellmann aus Remscheid

Bonn 2010

Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakulta¨tderRheinischenFriedrich-Wilhelms-Universit¨atBonn

1. Gutachter: Prof. Dr. Michael Rapoport 2. Gutachter: Prof. Dr. Gerd Faltings Tag der Promotion: 21.04.2011

Erscheinungsjahr: 2011

Zusammenfassung Die vorliegende Arbeit beschäftigt sich mit der Parametrisierung von Struk- turen, die in derp Sei-adischen Hodge-Theorie auftreten.Keine endliche ¯ Erweiterung des KörpersQ p derp-adischen Zahlen und seiG K = Gal(K/K) dessen absolute Galoisgruppe. Inp-adischer Hodge-Theorie untersucht man gewisse stetige Darstellungen vonG K auf endlich dimensionalenQ p -Vektor- räumen. Die Klassen von Darstellungen, die hierbei auftreten, werden mithilfe von Periodenringen definiert. Im Fall der vorliegenden Arbeit sind dies so- genanntekristalline definiert einen RingDarstellungen. FontaineB cris mit stetigerG K -Operation und nennt eine Darstellung vonG K auf einemd- dimensionalenQ p Ve-orktumraVkristallin, falls D cris (V) = (V⊗ Q p B cris ) G K von DimensiondüberK 0 =B c G ri K s ist. Dabei istK 0 die maximal unverzweigte Erweiterung vonQ p inK Strukturen auf. ZusätzlicheB cris induzieren zusät- zliche Strukturen aufD=D cris (V) Automorphismus: einenΦ, der semi- linear bezüglich des FrobenuisϕaudK 0 ist und eine absteigende, auss- chöpfende und separierte FiltrierungF • aufD cris (V)⊗ K 0 K. Solch ein Tripel (D,Φ,F • )nennt man einen filtriertenϕ filtrierten-Modul. Dieϕ-Moduln im Bild des FunktorsD cris sind dieschwach zulässigenfiltriertenϕ-Moduln. Hi- erbei ist "schwach zulässig" eine Semi-stabilitätsbedingung, die die "slopes" des FrobeniusΦmit der Filtrierung verknüpft. Ziel dieser Arbeit ist es, Familien dieser Strukturen zu untersuchen. Hi- erbei werden die Familien von geometrischen Objekten über denp-adischen Zahlen, genauer von rigid-analytischen und adischen Räumen, parametrisiert. An Stelle der filtriertenϕ-Moduln treten kohärente GarbenDauf einem RaumX, die (lokal aufX) frei überO X ⊗ Q p K 0 sind, zusammen mit einem id⊗ϕ-linearen AutomorphismusΦund einer FiltrierungF • vonD⊗ K 0 K durchO X ⊗ Q p K-Untermoduln, die lokal aufXdirekte Summanden sind. Für diese Familien definieren wir einen Begriffder schwachen Zulässigkeit und zeigen, dass der OrtU⊂X, auf dem eine Familie von filtriertenϕ- Moduln schwach zuläsig ist, eine offene Teilmenge ist. Hierbei arbeiten wir mit der Kategorie der adischen Räume im Sinne von Huber statt in der Kategorie der rigiden Räume. Der Grund dafür ist, dass es schwierig ist, im Kontext von rigiden Räumen eine offene Teilmenge zu konstruieren: da rigide Räume keine topologischen Räume sind, sondern stattdessen eine Grothendieck-Topologie tragen, reicht es nicht aus für jeden Punkt vonUeine Umgebung diese Punktes inUzu finden, um die Teilmenge Utatsächlich als rigiden Raum zu identifizieren. Anhand von Beispielen stellt sich heraus, dass das Resultat, dass der schwach zulässige Ort in einer Familie filtrierterϕ-Moduln offen ist, in der Kategorie von analytischen Räumen im Sinne von Berkovich falsch ist.

Im Gegensatz zu den klassischen Periodenbereichen im Sinne von Rapoport und Zink ist der schwach zulässige Ort im vorliegenden Fall eine Zariski- offene Teilmenge, sobald der FrobeniusΦ bedeutet, dass Dasfixiert wird. die Teilmenge in einer Flaggenvarietät, über der die universelle Filtrierung schwach zulässig bezüglich eines gegebenen AutomorphismusΦist, die An- alytifizierung eines offenen Unterschemas ist. In einer Arbeit von Kisin werden filtrierteϕ-Moduln mit Vektorbündeln auf der offenen Kreisscheibe überK 0 zusammen mit einem gewissen semi- linearen Endomorphismus in Verbindung gebracht. Diese Beziehung wird (im Fall, dass die Filtrierung nur Sprünge in Graden0und1besitzt) auf Familien filtrierterϕ-Moduln verallgemeinert. In Kisin’s Arbeit wird gezeigt, dass schwache Zulässigkeit für filtrierteϕ-Moduln sich in die Existenz einer ganzzahligen Struktur für das Vektorbündel auf der offenen Kreisscheibe übersetzt. In dieser Arbeit wird gezeigt, dass eine solche ganzzahlige Struk- tur auf einer offenen Teilmenge des schwach zulässigen Ortes existiert, die alle rigiden Punkte enthält. Dieses Resultat verifiziert eine Vermutung von Pappas und Rapoport über das Bild einer Periodenabbildung von einem Stack, der ganzzahlige Strukturen parametrisiert, in den Stack der filtrierten ϕ-Moduln. Hierbei besteht der wichtigste Schritt darin, ein Resultat von Kedlaya und Liu über den Ort, über dem eine Familie vonϕ-Moduln über dem Robba-Ring étale ist, auf den Fall von adischen Räumen zu verallge- meinern. Im letzten Teil der Arbeit kehren wir zu Darstellungen der absoluten GaloisgruppeG K  zeigen, dass es eine ozurück. Wirffene Teilmenge des Bildes der Periodenabbildung gibt, über der die Familie von filtriertenϕ- Moduln von einer Familie von Galois-Darstellungen induziert wird, d.h. von einem Vektorbündel mit einer stetigenG K -Operation. Anhand eins ein- fachen Beispiels lässt sich zeigen, dass dieserzulässigeOrt nicht mit dem schwach zulässigen übereinstimmt, sondern im Allgemeinen eine echte Teil- menge ist.

ON ARITHMETIC FAMILIES OF FILTEREDϕ-MODULES AND CRYSTALLINE REPRESENTATIONS

EUGEN HELLMANN

Abstract.We consider stacks of filteredϕ-modules over rigid analytic spaces and adic spaces. We show that these modules parametrizep-adic Galois representations of the absolute Galois group of ap-adic field with varying coefficients over an open substack containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrising integral data and determine the image of this morphism.

1.Introduction Inp-adic Hodge theory one considers filteredϕ-modules as a category of linear algebra data describingcrystalline MoreGalois representations. precisely, there is an equivalence of categories between crystalline represen- tations and weakly admissible filteredϕ-modules (cf. [CF, Theorem A]), where weak admissibility is a certain condition relating the slopes of the FrobeniusΦwith the filtration. In their book [RZ], Rapoport and Zink consider geometric families of these ¯ modules. More precisely they fix an isocrystal overW(F p )[1/p]and consider ¯ for a given coweightνthe flag varietyF ν overW(F p )[1/p], whereνprescribes the jumps of the filtration. They show that there is an admissible open subspaceF ν wa ⊂F rig parametrizing those filteredϕ-modules that are weakly ν admissible with respect to the given isocrystal. In this paper we considerarithmetic familiesof filteredϕ-modules. That is, we fix a local fieldKof characteristic0, and study a stack on the category of rigid analytic spaces parametrisingK-filteredϕ-modules with coefficients defined by Pappas and Rapoport in [PR]. The difference with the geometric situation is that we do not fix an isocrystal (i.e. the Frobenius may vary) and the filtration also varies. We define a notion of weak admissibility in this context, and show that the weakly admissible locus in this stack is an open substack. Here we work in the category of adic spaces introduced by Huber (see [Hu3] for example), instead of the category of rigid analytic spaces. The work of Kedlaya and Liu [KL] shows that the category of Berkovich spaces (see [Be] for example) is not the right category in which to address these questions, since the locus where certainϕ-modules over the Robba ring 1

2 E. HELLMANN are étale is not an open subspace. This phenomenon appears here again, since the weakly admissible locus will not be a Berkovich space. Kedlaya and Liu formulate the local étaleness ofϕ-modules over the Robba ring in terms of rigid analytic spaces. However, we cannot apply their result, since a covering by admissible open subsets is not necessarily an admissible covering. This is the reason why we are forced to work in the category of adic spaces. In fact, we will generalize the theorem of Kedlaya and Liu to the setting of adic spaces. In the geometric setting of [RZ], there is a period morphism from the moduli space ofp-divisible groups to a certain period domainF ν wa . The image of this morphism was described by Hartl in [Ha1], [Ha2] and Faltings, [Fa]. As an analogue to the moduli space ofp-divisible groups, Pappas and Rapoport define in [PR] a formal stack parametrising modules that appear in integralp-adic Hodge theory developed by Breuil and Kisin (see [Br2] for example). They also define an analogue of the period map of [RZ]. In this paper we will determine the image of this period map, confirming a conjecture of Pappas and Rapoport. Finally we construct an open substack of the stack of weakly admissible filteredϕ-modules and a family of crystalline representations on this stack, i.e.avectorbundlewithacontinuousGaloisactionwhichiscrystalline.We will show that this subspace is in fact universal for families of crystalline representations. Our main results are as follows. LetKbe a finite extension ofQ p and denote byK 0 the maximal unramified extension ofQ p insideK fix an. We integerd >0and a dominant coweightνof the algebraic groupRes K/Q p GL d with associated reflex fieldE denote by. WeD ν the fpqc-stack on the cat- egory of rigid analytic spaces overE(or, slightly more generally, on the category of adic spaces locally of finite type) whoseX-valued points are triples(D,Φ,F • )with a locally freeO X ⊗ Q p K 0 -moduleD, a semi-linear automorphismΦand a filtration ofD⊗ Q p Kwhich is of typeν. It is easy to see that this stack is anArtin stack, where we define an Artin stack on the category of rigid spaces (or adic spaces) exactly as on the category of schemes. We will define a notion ofweak admissibilityfor all points of the adic space generalizing the usual notion (cf. [CF, 3.4]) at the rigid analytic points. Theorem 1.1.The weakly admissible locus is an open substackD ν wa ⊂D ν on the category of adic spaces locally of finite type overE. It turns out that in some sense the weakly admissible locus behaves like an algebraic variety overQ p  precisely, denote by. MoreA⊂GL d the diagonal torus and byWthe Weyl group ofGL d . We define a morphism α:D ν −→(A/W) ad

FAMILIES OF FILTEREDϕ 3-MODULES AND CRYSTALLINE REPRESENTATIONS to the adjoint quotient of the groupGL d , where(−) ad means the adic spaces associated with a scheme, and prove the following theorem. Theorem 1.2.Letx∈(A/W) ad and form the2-fiber product α −1 (x) wa ￿￿ D ν wa

 x ￿￿ (A/W  ) ad . Then there is a finite extensionFofQ p insidek(x)and an Artin stack in schemesAoverFsuch that α −1 (x) wa =A ad ⊗ F k(x). The stackAis the stack quotient of a quasi-projective variety overF. ￿ We will also consider the following stackC K on the category ofZ p -schemes ￿ on whichp Theis nilpotent.S-valued points ofC K are tuples(M,Φ), where Mis a (fpqc-locally onS) freeO S ⊗ ￿ Z p W[[u]]-module together with a semi- linear injectionΦ:M→Mwhose cokernel is killed by the minimal polyno- mial overK 0 of some fixed uniformizer ofK category of those modules. The was introduced by Breuil and studied by Kisin in order to describe finite flat group schemes ofp-power order overSpecO K and hence to describep- divisible groups in the limit. Given a miniscule cocharacterν, Pappas and ￿ ￿ Rapoport define a closed substackC K,ν ofC K by posing an extra condition on the cokernel of the semi-linear injectionΦ a formal scheme. ForXover Z p they also define aperiod map Π(X) :C ￿ K,ν (X)−→D ν (X rig ), that maps theZ p -point associated with ap-divisible group overO K to the filtered isocrystal of thep-divisible group. We will show that the map indeed factors through the weakly admissible locus and we describe the image of the period map as follows. In [Ki2], Kisin shows that there is an equivalence between the category of filteredϕ-modules and a certain category of vector bundles on the open unit disc together with aϕ will show-linear map. We that the equivalence of categories generalizes to families. As the notion of weak admissibility (for filtered isocrystals overQ p ) translates to the property of being étale over the Robba ring, we will define a substackD i ν nt ofD ν consisting of those filteredϕ-modules whose associated vector bundle is étale. Theorem 1.3.The stackD i ν nt is an open substack ofD ν wa (on the cate- gory of adic spaces locally of finite type over the reflex field ofν). It is the image of the period morphism in the sense that a morphismX→D ν wa , defining(D,Φ,F • )∈D ν (X), factors throughD i ν nt if and only if there ex- ists a covering(U i ) i∈I ofXandp-adic formal schemesU i together with ￿ (M i ,Φ i )∈C K,ν (U i )such thatU i ad =U i and Π(U i )(M i ,Φ i ) = (D,Φ,F • )| U i .

4 E. HELLMANN Finally we go back to crystalline representations. We consider vector bun- dlesEon an adic spaceXendowed with an action of the absolute Galois groupG K ofK the structure sheaf. AsO X is a sheaf of topological rings and Eis a vector bundle, all spaces of sectionsΓ(U,E)ofEover some open subset Uhave a natural topology and hence the notion of a continuousG K -action onE vector bundle Amakes sense.Ewith a continuousG K -action such that D cris (E) = (E ￿ s ) G ⊗ Q p B cri K is locally onXfree of rankdoverO X ⊗ Q p K 0 is called afamily of crystalline representations onX. Theorem 1.4.Letνbe a miniscule cocharacter ofRes K/Q p GL d  the. Then groupoid of families of crystalline representations of Hodge-Tate weightsνis an open substackD ν adm ofD i ν nt . The following diagram summarizes the stacks that appear in this paper. D a ν dm ￿ 

 ￿ ad ￿ nt C K,ν ￿ D i ν  ￿

Π D  ν wa ￿ 

 ∼  C ν = ￿￿ D ν . HereC ν is a stack of vector bundles on the open unit disc that appears as an intermediate step of the period morphism. By the adification of the stack ￿ ￿ C K,ν we mean the stack mapping an adic spaceXto the limit ofC K,ν (X) for all formal modelsXofX map. TheC ￿ a K d ,ν →D i ν nt then is the localization map ofC ￿ a K d ,ν to its isogeny category. vertical arrows on the right are all The open embeddings. We now outline the structure of this paper. In section 2 we will recall some basic facts and concepts fromp-adic Hodge- theory and the theory of adic spaces. In section 3 we introduce filtered isocrystals with coefficients in a valuated field. These objects will appear as the fibers of our families. We introduce the notions of semi-stability and weak admissibility, and establish a Harder- Narasimhan formalism. In section 4 we define the stacks of filtered isocrystals and prove our first main result, Theorem1.1. Section 5 is an aside to the rest of the paper. Here we discuss a morphism from the stack of filtered isocrystals to the adjoint quotient of the groupGL d

FAMILIES OF FILTEREDϕ-MODULES AND CRYSTALLINE REPRESENTATIONS 5 which was already considered by Breuil and Schneider in [BS]. We show that the fibers of this morphism are algebraic in the sense of Theorem1.2. Further we study the image of the weakly admissible locus in the adjoint quotient and show that this image is identified with a closed Newton-stratum in the sense of Kottwitz [Ko]. In section 6 we study the relation of filtered isocrystals and vector bundles on the open unit disc which was introduced by Kisin. In section 7 we discuss the notion of being étale over the Robba ring in fam- ilies, slightly generalizing a result of Kedlaya and Liu about local étaleness of those modules. This allows us to determine the image of the period map in section 8 and prove Theorem1.3. In section 9 we construct a family of crystalline Galois representations on an open substack and show that this family is universal. Acknowledgements thank my advisor M. Rapoport for his interest: I and advice. Further I would like to thank R. Huber for answering my ques- tions about adic spaces. I also acknowledge the hospitality of the Institute Henri Poincare during the Galois trimester in spring 2010 and the hospitality of Harvard University in fall 2010, where part of this work was done. This work was supported by the SFB/TR 45 "Periods, Moduli Spaces and Arith- metic of Algebraic Varieties" of the DFG (German Research Foundation).

2.Preliminaries Throughout the whole paper we fix the following notations: LetKbe a finite extension ofQ p and writeO K for its ring of integers. a uniformizer Fix π∈O K and writek=O K /πO K for the residue field. LetW=W(k)be the Witt ring ofkandK 0 =W[1/p]the maximal unramified extension of Q p insideK we denote by. FurtherE(u)∈W[u]the minimal polynomial of πoverK 0 . ¯ ¯ Fix an algebraic closureKofKand writeG K = Gal(K/K)for the absolute Galois group. Further we choose a compatible systemπ n ofp n - ¯ th roots ofπinKand writeK ∞ for the field obtained fromKby adjoining ¯ theπ n . LetG K ∞ = Gal(K/K ∞ )denote the absolute Galois group ofK ∞ . 2.1.Somep-adic Hodge-theory.For further use we define some rings of p-adic Hodge theory used in Kisin’s papers [Ki1] and [Ki2] 1 . WriteA [0,1) =W[[u]]andAfor thep-adic completion ofW((u)). Further B=A[1/p]. LetRdenote Fontaine’s ring R= lim ←− O C p /pO C p where the transition maps in the limit are given by thep-th power map. Let πdenote the element(π,π 1 ,π 2 , . . .)∈Rand write[π]for the Teichmüller 1 Our notations here differ from the ones in Kisin’s papers.

6 E. HELLMANN representative ofπin the Witt ringW(R)ofR. We may regardBas a subring ofW(FracR)[1/p]by mapinguto[π]. The liftϕof the absolute Frobenius onRmaps[π]to[π] p and hence this embeddings isϕ-equivariant.  WriteBfor the closure of the maximal unramified extension ofBinside   W(FracR)[1/p]and denote byAits ring of integers. ThenAis a complete discrete valuation ring (for thep-adic topology) with residue field the closure  ofk((u)) sep inFracRand we have an injectionA￿→W(FracR)which is  continuous for the canonical topology ofA. We write  A [0,1) =A  ∩W(R)⊂W(FracR). Then we have A  ϕ=id =Z p ,A  G K ∞ =A,  d B ϕ=i =Q p ,  B G K ∞ =B . The Galois groupG K ∞ is isomorphic to the absolute Galois group of k((u)), see [BrCo, Theorem 11.1.2] for example, and its representations on finite dimensionalQ p -vector spaces are described in terms of étaleϕ-modules overB dimensional finite, i.e.B-vector spacesNtogether with an isomor- phismΦ:ϕ ∗ N→Nsuch that there exists anA-latticeN⊂Nwith Φ(ϕ ∗ N) =N an étale. Givenϕ-module(N,Φ)overB, we write  Φ=id (2.1)V B (N) = (N⊗ B B) for the associatedG K ∞ -representation. Remark2.1. Instead ofIn fact Kisin considers slightly smaller period rings.  Bhe considers thepcompletion of the maximal unramified extension of-adic  BinsideW(FracR)[1/p]. But asBhas the sameϕandG K ∞ -invariants the period ring considered in Kisin’s papers, the theory works with our definitions as well. The advantage of our definition is that is makes it easier to define the sheafified versions of the period rings. LetA cris denote thepcompletion of the divided power envelope of-adic W(R)with respect to the kernel of the surjectionθ:W(R)→O C p induced by [(x, x 1/p , x 1/p 2 , . . .)]→−￿x. LetB + cris =A cris [1/p]andB cris =B iscr + [1/t]denote Fontaine’s ring of crys- talline periods, where t= log[(1, 1 , 2 , . . .)] is the period of the cyclotomic character. Recall that a representation ofG K on adnsional-dimeQ p -vector spaceV is calledcrystallineif D cris (V) = (V⊗ Q p B cris ) G K is of dimensiondoverK 0 =B c G ri K s . TheK 0 -vector spaceD cris (V)is equipped with a semi-linear automorphismΦand a descending, separated and ex- haustive filtrationF • onD cris (V)⊗ K 0 K an object is called a filtered. Such

FAMILIES OF FILTEREDϕ 7-MODULES AND CRYSTALLINE REPRESENTATIONS ϕ-module (or filtered isocrystal) overK. A filtered isocrystal(D,Φ,F • )is calledweakly admissibleif v p (detΦ) =  idim K gr i (D⊗ K 0 K), i v p (detΦ| D ￿ )≥  idim K gr i (D ￿ ⊗ K 0 K)∀D ￿ ⊂D, D ￿ Φ-stable. i Then there is an equivalence of categories between the category of crystalline representations and the category of weakly admissible filteredϕ-modules over K, see [CF, Theorem A].

2.2.Adic spaces.LetAbe a topological ring overQ p . By avaluation(in the sense of [Hu1, 2, Definition]) onAwe mean a mapv:A→Γ v ∪{0}to a totally ordered abelian groupΓ v (written multiplicatively) such that v(0) = 0 v(1) = 1 v(ab) =v(a)v(b) v(a+b)≤max{v(a), v(b)}, where the order onΓ v is extended toΓ v ∪{0}by0<γfor allγ∈Γ v . The valuation is called continuous if{a∈A|v(a)≤γ}is open for all γ∈Γ v . IfA + ⊂Ais an open and integrally closed subring, Huber defines the adic spectrum of(A, A + )as Spa(A, A + ) =  v: i A so → mo Γ r v ph ∪ is { m 0}thac o u c t f h os s essalc v n ( t a in ) u ≤ ou 1o ti rf a l lu al asv a on ∈ s A +  . This space is equipped with a structure sheafO X and a sheaf of integral subringsO + X , see [Hu2, 1]. Given a finite extensionEofQ p we denote byRig E the category of rigid analytic varieties overE Part C]) and by(cf. [BGR,Ad E the category of adic spaces overE have a fully faithful embedding of Wein the sense of Huber. Rig E intoAd E that factors through the full subcategoryAd l E ft of adic spaces locally of finite type overE adic spaces that are locally isomorphic, i.e. toSpa(A, A ◦ )for anE-algebraAthat is topologically of finite type over Eand whereA ◦ ⊂A Inis the subring of power bounded elements. fact, this embedding identifies the category of quasi-separated rigid spaces with the category of quasi-separated adic spaces locally of finite type overE(see [Hu3, 1.1.11]). For an adic spaceXand a pointx∈Xwe will writeO X,x for the local ring atxandm X,x  Wefor its maximal ideal. denote the residue field at xbyk(x)and writev x :k(x)→Γ x ∪{0}for the corresponding valuation.  Further we writek(x)the completion of the residue field (with respectfor