Profinite R-analytic groups Talde R-analitiko profinituak

PhD Thesis / Doktorego Tesia Profinite R-analytic groups Talde R-analitiko profinituak Andoni Zozaya Ursuegui Supervisors / Zuzendariak Gustavo A. Fernández-Alcober Jon González-Sánchez FEBRUARY 2023 OTSAILA (cc)2023 ANDONI ZOZAYA URSUEGUI (cc by-nc 4.0) © Copyright by Andoni Zozaya Ursuegui, 2023. Abstract While the theory of analytic groups is extensively developed over the p-adic numbers, relatively little is known about groups that are analytic over alternative coefficient rings such as Zp[[t1, . . . , tm]] or Fp[[t1, . . . , tm]]. This thesis is a contribution to the theory of analytic groups over general pro-pdomains by means of a systematic investigation of structural concepts of that theory, including associated Lie algebras, submanifolds and analytic quotients. Moreover, we study several group-theoretical properties in the setting of analytic groups, namely linearity, word problems and fractal dimensions. With regard to the first two properties –linear representations and word-conciseness of some analytic groups– we extend results that are well-established in the p-adic case. In contrast, the study of the Hausdorff dimension –where we mainly focus on groups that are analytic over Fp[[t]]– shows significant differences between groups that are analytic over the p-adic numbers and those that are analytic over other coefficient rings. Laburpena Talde analitikoen teoria aski garatuta dago zenbaki p-adikoen gainean, baina alderatuta ezer gutxi ezagutzen da beste koefiziente eraztun batzuen, Zp[[t1, . . . , tm]] edo Fp[[t1, . . . , tm]] kasu, gainean analitikoak diren taldeei buruz. Tesi hau pro-p domeinu orokorren gainean analitikoak diren taldeen inguruko hainbat ekarpenek osatzen dute. Alde batetik, teoria horretako egiturazko kontzeptu anitz xeheki aztertzen dira, esate baterako, elkarturiko Lieren aljebrak, azpibarietateak eta zatidura analitikoak. Beste alde batetik, talde analitikoen testuinguruan talde teoriako zenbait propietate aztertzen dira, linealtasuna, hitz-problemak eta dimentsio fraktalak hain zuzen ere. Lehenengo bi gaiei dagokienez –talde analitiko batzuen adierazpen linealak eta hitz-laburtasuna–, erdietsiriko emaitzek kasu p-adikoan ezagunak diren teoremak orokortzen dituzte. Alabaina, Hausdorffen dimentsioaren azterketan –eskuarki Fp[[t]]-ren gainean analitikoak diren taldeetara mugatuko gara– alde nabarmena dago zenbaki p-adikoen gainean eta beste koefiziente eraztun batzuen gainean analitikoak diren taldeen artean. iii This thesis has been carried out at the University of the Basque Country (UPV/EHU) under the financial support of the Spanish Ministry of Science, Innovation and Universities’ grant FPU17/04822. In addition, the author was supported by the Basque Government, projects IT974-16 and IT483-22, and the Spanish Government, projects MTM2017-86802-P and PID2020-117281GB-I00, partly with ERDF funds. Tesi hau Euskal Herriko Unibertsitatean (UPV/EHU) idatzi da Espainiako Zientzia, Berrikuntza eta Unibertsitate Ministerioaren FPU17/04822 laguntzarekin. Era berean, autoreak Eusko Jaurlaritzaren, IT974-16 eta IT483-22 proiektuak, eta Espainiako Gobernuaren, MTM2017-86802-P eta PID2020-117281GBI00 proiektuak (partzialki EGEFk finantzatuta) laguntza jaso du. v vi Contents I Profinite R-analytic groups 1 Index of notation 3 Introduction 7 1R-analytic groups 11 1.1 Pro-pdomains............................. 11 1.2 R-analyticgroups........................... 15 1.3 R-standardgroups .......................... 19 1.4 Liealgebras.............................. 26 1.5 Construction of manifolds . . . . . . . . . . . . . . . . . . . . . . 35 1.6 Notes ................................. 45 2 Linearity of compact R-analytic groups 47 2.1 Linearity of compact p-adic analytic groups . . . . . . . . . . . . 49 2.2 Change of pro-pdomains....................... 51 2.3 Discrimination ............................ 57 2.4 Model theory and the linearity of compact R-analytic groups . . . 60 2.5 Notes ................................. 64 3 Hausdorff dimension in compact R-analytic groups 65 3.1 Hausdorff and box dimension . . . . . . . . . . . . . . . . . . . . 67 3.2 Hausdorff dimension of submanifolds . . . . . . . . . . . . . . . . 78 3.3 Abelian compact R-analytic groups . . . . . . . . . . . . . . . . . 82 3.4 Compact Fp[[t]]-analytic groups . . . . . . . . . . . . . . . . . . . 83 3.5 Classical Chevalley groups . . . . . . . . . . . . . . . . . . . . . . 88 3.6 Notes ................................. 94 vii 4 Words in compact R-analytic groups 95 4.1 Conciseness in R-standard groups . . . . . . . . . . . . . . . . . . 100 4.2 Conciseness in compact Fp[[t]]-analytic groups . . . . . . . . . . . 101 4.3 Conciseness in compact R-analytic groups . . . . . . . . . . . . . 105 4.4 Strong conciseness in R-analytic groups . . . . . . . . . . . . . . . 107 4.5 Notes ................................. 108 Appendix A Ado’s Theorem over principal ideal domains 111 A.1 Introduction.............................. 111 A.2 Adjoint and regular representations . . . . . . . . . . . . . . . . . 114 A.3 Ado’sTheorem ............................ 116 A.4 Notes ................................. 127 Bibliography 129 Index 135 II Talde R-analitiko profinituak 139 Notazio indizea 141 Sarrera 145 1 Talde R-analitikoak 149 1.1 Pro-pdomeinuak ........................... 149 1.2 Talde R-analitikoak.......................... 154 1.3 Talde R-estandarrak ......................... 157 1.4 Lierenaljebra............................. 164 1.5 Barietateen eraikuntza . . . . . . . . . . . . . . . . . . . . . . . . 174 1.6 Oharrak................................ 184 2 Linealtasuna talde R-analitiko trinkoetan 187 2.1 Talde p-adiko analitiko trinkoen linealtasuna . . . . . . . . . . . . 189 2.2 Pro-pdomeinualdaketa ....................... 191 2.3 Diskriminazioa ............................ 198 2.4 Eredu teoria eta talde R-analitiko trinkoen linealtasuna . . . . . . 201 2.5 Oharrak................................ 205 3 Hausdorffen dimentsioa talde R-analitiko trinkoetan 207 viii Part I Profinite R-analytic groups 1 Notazio indizea Conventions. We assume that all rings are commutative and with identity. Moreover, throughout all the thesis pstands for a prime number, and qfor a power of p. Notation. Most of the notation is standard, except for A(n),which stands for the nth Cartensian power of the set A(we shall use this notation, since it will be common to write expressions of the form (an)(m)for an ideal a, so it is convenient to distinguish the nth Cartesian power a(n)from the nth power ideal an). Moreover, if f:A→Bis a map, we denote by f(n)the map A(n)→B(n), (a1, . . . , an)7→ (f(a1), . . . , f(an)) . The remaining terminology is listed below: Nthe natural numbers N0the natural numbers together with 0 Zthe integers Zpthe p-adic integers Qthe rational numbers Qpthe p-adic numbers Rthe real numbers R≥0the non-negative real numbers Cthe complex numbers Fqthe finite field of size q logathe logarithm of basis a 3 P(A)the parts of A A×Bthe Cartesian product of Aand B Qi∈IAithe Cartesian product of the directed family {Ai}i∈I A(k)the kth Cartesian power of A A⊕Bthe direct sum of Aand B A⋉Bthe semidirect product of Aand B Li∈IAithe direct sum of the directed family {Ai}i∈I Qi∈IAi/Uthe ultraproduct of the directed family {Ai}i∈I AUthe ultrapower of A A⊆B A is a subset of B A⊆oB A is an open subset of B A⊆cB A is a closed subset of B A≤B A is a subgroup of B A≤oB A is an open subgroup of B A≤cB A is a closed subgroup of B A⊴B A is a normal subgroup of B A⊴oB A is a normal open subgroup of B A⊴cB A is a normal closed subgroup of B Achar B A is a characteristic subgroup of B Let Gbe a group and let g, x, y ∈G: xyy−1xy [x, y]x−1y−1xy [x1, . . . , xn] [[x1, . . . , xn−1], xn] Z(G)the centre of G CG(g)the centraliser of g∈G G′= [G, G]the derived subgroup of G:h[x, y]|x, y ∈Gi. Gnthe nth power subgroup: hgn|g∈Gi [H1, . . . , Hn]h[h1, . . . , hn]|hi∈Hii cythe conjugation isomorphism G→G, x 7→ xy Lythe left multiplication map G→G, x 7→ yx Rythe right multiplication map G→G, x 7→ xy 4 ker fthe kernel of the group (resp. ring) homomorphism f im fthe image of the group (resp. ring) homomorphism f Hom(A, B)group (resp. ring) homomorphisms f:A→B. Let Qbe a ring: U(Q)the units of Q char Qthe characteristic of Q dimKrull Qthe Krull dimension of Q Frac(Q)the fraction field of the integral domain Q Q[[t1, . . . , tm]] the ring of formal power series in mvariables and coefficients in Q Mn×m(Q)n×mmatrices with coefficients in Q Mn(Q)n×nmatrices with coefficients in Q GLn(Q)general linear group with coefficients in Q SLn(Q)special linear group with coefficients in Q SOn(Q)special orthogonal group with coefficients in Q Spn(Q)symplectic group with coefficients in Q Un(Q)upper triangular matrices with coefficients in Q Let Kbe a field: Kalg the algebraic closure of K dimKK-vector space dimension Let Mbe an R-analytic manifold: dimxMthe analytic dimension of the manifold Mat x dim Ganalytic dimension of the analytic group G Let Gbe a linear algebraic group: Ru(G)unipotent radical rk rank of a matrix res.rk residual rank of a matrix det determinant of a matrix tr trace of a matrix 5 DUdifferential of a tuple of power series JxFthe Jacobian matrix of Fat x rk Mthe rank of a free module M IsoM(N)isolator of Nin M EndR(M)endomorphisms of the R-module M hdim Hausdorff dimension hspec Hausdorff spectrum hdimst standard Hausdorff dimension hspecst standard Hausdorff spectrum bdim box dimension bdimst standard box dimension lbdim lower box dimension lbdimst standard lower box dimension ubdim upper box dimension ubdimst standard upper box dimension w{G}the set of w-values of G w(G)the verbal subgroup of w w∗(G)the marginal subgroup of w X∗ℓthe set of products of `elements of X∪X−1∪{1} deg Ldegree of the Lie algebra L deg φdegree of the representation φ Z(L)centre of the Lie algebra L Rn(L)nilpotent radical of L Rs(L)soluble radical of L TR(L)torsion algebra of L UR(L)universal enveloping algebra of L DerR(L)derivations of L Cent(L)centroid of L 6 “I tend to think too much. My greatest successes came from decisions I made when I stopped thinking and simply did what felt right. Even if there was no good explanation for what I did. [...] Even if there were very good reasons for me not to do what I did.” Kvothe, (Patrick Rothfuss, The Name of the Wind) Introduction This dissertation is a monograph on analytic groups. These comprise an abstract group together with an analytic manifold structure over a convenient topological ring in such a way that both structures are compatible, in the sense that the multiplication map and the inversion map are analytic functions. The theory of analytic Lie groups over topological fields is a source of examples of profinite groups. Of course, over the classical fields Rand C,analytic Lie groups cannot be profinite unless they are finite, as they should be both compact and locally homeomorphic to a totally disconnected subset of C(n). However, profinite analytic groups might arise if the underlying group of the base ring is a profinite group in its own right. For instance, in the treatise Groupes analytiques p-adiques [50], Lazard extensively studied the p-adic analytic groups, that is, analytic Lie groups over the field of p-adic numbers Qp– equivalently, over the valuation ring of p-adic integers Zp–; and he showed that compact p-adic analytic groups are actually profinite groups. In addition to the original purely analytic point of view, there are several alternative characterisations of p-adic analytic groups (we refer to [24, Interlude A] for a comprehensive list). Among these characterisations, in order to prove what could be regarded as Hilbert’s 5th problem for p-adic analytic groups, Lazard himself proved that compact p-adic analytic groups are precisely the profinite groups that are virtually pro-pgroups of finite rank– these are the profinite groups which contain a pro-pgroup of finite index such that all the subgroups of that pro-p group are finitely generated, and such that the necessary number of generators is bounded. The theory of p-adic analytic groups has since evolved into a rich area, 7 and a number of exciting properties have been established: all of them, with the exception of the group Zp, satisfy Golod-Shafarevich inequality (Lubotzky [52]), they have polynomial subgroup growth (Lubotzky and Mann [53]), they are verbally elliptic (Jaikin-Zapirain [44]), etc. Furthermore, if one starts with a general topological ring Rand define analytic groups by analogy, the concept of p-adic analytic group is generalised to that of analytic group over R, which hereinafter will be referred to as R-analytic group. Thereby, Bourbaki (or better said, the bourbaquists) [11] and Serre [68] studied analytic groups over the local field Fp((t)),i.e., the positive characteristic counterpart of p-adic analytic groups. Moreover, the second edition of the celebrated book Analytic pro-p groups [24] was provided with a further chapter concerning analytic groups over general pro-pdomains and thus took the first steps of this broader theory. We recall that a pro-pdomain is a local Noetherian integral domain Rwhich is complete with respect to the metric defined by the maximal ideal and whose residue field is finite of characteristic p(Section 1.1 is devoted to exploring these rings and the concepts required in their definition). These more general groups possess interesting algebraic properties (see [14], [42], [43], [45] and [54]), albeit not as those enjoyed by the p-adic analytic groups. Worse, there is no characterisation, even at a conjectural level, of R-analytic groups purely in group-theoretic terms. This thesis aims to develop further the theory mentioned above. Its objectives are twofold: on the one hand, to advance in the systematic study of analytic groups over general pro-pdomains, which constitutes a somewhat belated sequel to [24, Chapter 13]; and, on the other hand, to provide this theory with new research results. Those mainly generalise known properties for p-adic to the broader context of R-analytic groups. We point out that, moreover, the p-adic case is usually a fundamental ingredient of our proofs. We now outline the contents in greater detail: Chapter 1 is an introduction to R-analytic groups taking [24, Chapter 13] as a starting point. We will pay special attention to standard groups, which perhaps constitute the main example of Ranalytic groups, as well as to the Lie algebra associated with them. In view of the fact that many elementary concepts concerning analytic groups had still to be developed, we shall establish the machinery we will use throughout. For instance, in [24, p. 349], the authors highlighted that “for more general analytic groups of the present chapter, such concepts [of submanifold and quotient manifold] would 8 need to be developed”, which is precisely what we try to do in Section 1.5. Chapter 2 is about linearity. Specifically we show that when Ris a pro-p domain of characteristic zero, every compact R-analytic group is linear, i.e., it can be embedded in the eneral linear group GLn(K)for a suitable field K. This partially answers a question from Lubotzky and Shalev (see Question 2 in page 311 of [54]). The proof we shall give is neat and based on the linearity of p-adic analytic groups. Besides, it has a model-theoretic flavour, as we embed the group in question in a convenient ultrapower of GLn(Zp).Since model theory is not a central topic of this thesis, the proof is written, as far as possible, in such a way that prior knowledge is not required. Chapter 3 is devoted to the Hausdorff dimension in compact R-analytic groups. We shall show that in a compact R-analytic group G, there exists a metric that encodes its analytic structure, and we will recall how to define the Hausdorff dimension corresponding to that metric, namely a fractal dimension hdim: P(G)→ [0,1]. The chapter consists of two main parts. Firstly, we shall study the relationship between the analytic and the Hausdorff dimensions of a closed submanifold. This study is based on the article [27] by Fernández-Alcober, Giannelli and González-Sánchez. Secondly, we will focus mainly on the case R=Fp[[t]],and describe the Hausdorff spectrum of compact Fp[[t]]-analytic groups, namely the set hspec(G) = {hdim(H)|H≤Gis closed}. Chapter 4 is concerned with words. A word is nothing but an element w= w(x1, . . . , xk)of the free group F(x1, . . . , xk)in k-generators; and given a group G, it naturally defines a map w:G(k)→G, which sends (g1, . . . , gk)to the element of Gobtained by substituting gifor xiin w. In the setting of analytic groups, we will study some problems regarding words that were originally proposed by P. Hall [33]. In keeping with his terminology, we will prove that in a compact R-analytic group every word is concise, i.e., whenever im w, the image of the map win G, is finite, the verbal subgroup w(G) = him wiis also finite. Finally, Appendix A contains a proof of Ado’s Theorem for Lie algebras over principal ideal domains that are additionally free modules, since we will need this version of the theorem in Chapter 2. At the end of each chapter, there is a section of Notes, where we detail the author’s original contributions, or we make various comments. This manuscript intends to be stand-alone, and accordingly, most concepts are thoroughly introduced. However, familiarity with profinite and pro-pgroups is assumed, and if necessary the reader is referred to [24, Chapter 1]. 9 10 (U2, φ2, n2)are two R-charts such that U1∩U26=∅,then n1=n2.Therefore, the dimension of Mat xis well-defined as the common dimension of the R-charts of x, and it will be denoted by dimxM. It is worth remarking that over pro-pdomains, unlike in real or complex manifolds, the dimension is not a topological property, but an analytic property determined by the chart. For example, Zpand Z(2) pare isomorphic to each other as topological groups. (iv) An R-analytic manifold is said to be pure when dimxMis constant for all x∈M. Mimicking Definition 1.9, we can define R-analytic maps between R-analytic manifolds: Definition 1.12. Let Mand Nbe R-analytic manifolds. A function F:N→M is R-analytic at x∈Nif there exists an R-chart (U, φ, n)of xin Nand an R-chart (V, ψ, m)of F(x)in Msuch that F−1(V)is open in Nand ψ◦F◦φ−1|ϕ(U∩F−1(V)) :φU∩F−1(V)→R(m)(1.2) is an R-analytic map (according to Definition 1.9 (i)). Similarly, Fis an R-analytic map when it is R-analytic at all the points of N. Moreover, it is habitual to call the map (1.2) by Fin coordinates, and on occasions we will use this term informally without specifying the R-charts we work with. Furthermore, we duly refer to Fas strictly analytic at S⊆N, if in Definition 1.12 we can take the R-charts such that S⊆U∩F−1(V)and there exists H∈R[[X1, . . . , Xn]](m)such that ψ◦F◦φ−1(φ(x)) = H(x)∀x∈S. Adopting a term used by Serre [68] we will also define the following: Definition 1.13. Let Mbe an R-analytic manifold, U⊆oMand x∈U. A family of R-analytic functions F={fi:U→R}n i=1 is a coordinate system of Mat x, if there exits U′⊆oUsuch that x∈U′and (U′, F|U′),where F= (f1, . . . , fn),is an R-chart. Observe from the definition that whenever Fis a coordinate system at x, then it is so locally around x. 17 Examples 1.14. (i) The canonical example of an R-analytic manifold is M= mN(n)where N, n ∈N.The canonical coordinate system is {πi}n i=1 where πi is the ith projection map πi:mN(n)→mN.Besides, for every x∈M, the components of the translation map tx:M→M, y 7→ y+x, i.e. {πi◦tx}n i=1 , are also a coordinate system. (ii) Let K= Frac(R),and endow K(n)with the topology given by the neighbourhood basis nk+mN(n)oN∈Nof k∈K(n).Then K(n)is an R-analytic manifold with respect to the atlas {(Uk, ϕk, n)}k∈K(n)where Uk=k+R(n) and ϕk:Uk→R(n), x 7→ x−k(recall that the topology in K(n)we are imposing does not in general coincide with the natural topology on the fraction field; in truth it never does unless Ris a PID). (iii) The set of matrices Mn×m(m), which can be naturally identified with m(nm), is clearly an R-analytic manifold, and so is the general linear group GLn(R) with respect to the atlas {(UA, φA, n2)}A∈GLn(R),where UA=A+ Mn(m) and φA:UA→Mn(m), A +M7→ M. (iv) Let Mand Nbe R-analytic manifolds, with atlases {(Ui, φi, ni)}i∈Iand {(Vj, ψj, mj)}j∈J.The direct product M×Nis an R-analytic manifold with respect to the atlas {(Ui×Vj, φi×ψj, ni+mj)}i∈I, j∈J. The following elementary properties can be easily deduced from Lemmata 1.5 and 1.7. Lemma 1.15. (i) Every R-analytic map is continuous. (ii) (cf. [24, Lemma 13.4]) The composition of two R-analytic maps is an Ranalytic map. (iii) The composition of two strictly R-analytic maps is a strictly R-analytic map. We finish with the main definition: Definition 1.16. An R-analytic group is a topological group Gthat is an Ranalytic manifold such that (i) the multiplication map m:G×G→G, (g, h)7→ g·hand (ii) the inversion map ι:G→G, g 7→ g−1 are R-analytic maps. 18 Particularly, Zp-analytic groups are the foremost example of these groups, as well as the germ of the above definition. In the literature these groups have been referred to as p-adic analytic groups. 1.3 R-standard groups The R-standard groups are a noteworthy family of R-analytic groups. Definition 1.17. An R-standard group of level Nand dimension dis an Ranalytic group Swith a global chart {(S, φ, d)}such that (i) φ(S) = mN(d), (ii) φ(1) = 0and (iii) for all j∈ {1, . . . , d}there exists a formal power series Fj∈R[[X1, . . . , X2d]] such that φ(xy) = (F1(φ(x), φ(y)), . . . , Fd(φ(x), φ(y))) ∀x, y ∈S. Any tuple of power series F= (F1, . . . , Fd)satisfying condition (iii) of the above definition must certainly also satisfy (F1) F(X,0) = Xand F(0,Y) = Y(in particular, each Fihas constant term equal to zero), and (F2) F(X,F(Y,Z)) = F(F(X,Y),Z), (here X,Yand Zare d-tuples of variables) as they are straightforward consequences of the fact that φ(1) = 0and the associativity of the group law. Conversely, any tuple F∈R[[X1, . . . , X2d]](d)that satisfies the preceding two conditions endows mN(d),for any N∈N,with an R-standard group structure. Accordingly, a tuple of power series satisfying (F1) and (F2) is said to be a ddimensional formal group law, and there exists a formal inverse of it, namely a tuple of power series I= (I1, . . . , Id)∈R[[X1, . . . , Xd]](d)such that F(I(X),X) = F(X,I(X)) = 0 (see [24, Proposition 13.16 (ii)]). Sometimes we will denote an R-standard group by (S, φ)or (S, F)to emphasise the rôle of the homeomorphism or the formal 19 group law. From (F1) we can further conclude that F(X,Y) = X+Y+B(X,Y) + G(X,Y),(1.3) where Bis bilinear and where all the monomials involved in Ghave total degree at least 3.Moreover, every monomial involved in Band Gcontains a non-zero power of Xiand Yjfor some i, j ∈ {1, . . . , d}. Remark 1.18 (cf. [68, Part II, Chapter IV, §7]).Starting from (1.3) we can obtain similar expressions for the formal inverse and the conjugation maps. In effect, it is a routine exercise to verify that if Iis the formal inverse of Fin (1.3), then I(X) = −X+B(X,X) + ˜ G(X),(1.4) where every monomial involved in ˜ G(X)has total degree at least 3; and consequently, the conjugation map has the next form in coordinates: F(I(Y),F(X,Y)) = X+B(X,Y)−B(Y,X) + ˆ G(X,Y),(1.5) where every monomial involved in ˆ Ghas total degree at least 3and it contains a non-zero power of Xiand Yjfor some i, j ∈ {1, . . . , d}. Examples 1.19. (i) The additive group mN(d)is an R-standard group with the additive formal group law F(X,Y) = X+Y. (ii) The multiplicative R-standard group G= 1 + mN,which is so with respect to the global chart φ:G→mN,1 + m7→ mand the multiplicative formal group law F(X, Y ) = X+Y+XY. (iii) We can generalise the previous two formal group laws: it is easy to verify that any 1-dimensional polynomial formal group law has the form Fc(X, Y ) = X+Y+cXY for some c∈R(cf. [9, Corollary 2.2.4]). (iv) Let GL1 n(R)be the kernel of the modulo mreduction map GLn(R)→ GLn(R/m),that is, GL1 n(R) = In+ Mn(m).This group is R-standard with the n2-dimensional R-chart given by the assignation In+A7→ Aand the formal group law F(X,Y) = X+Y+X·Y,where Xand Ystand for n-by-nmatrices of indeterminates. 20 (v) In positive characteristic p, we have the 2-dimensional formal group law F(X1, X2, Y1, Y2) = (X1+Y1, X2+Y2+Xp 1Y2), which is attributed to Chevalley (cf. [18, Chapter II, §10, Example V]). Given an R-standard group (S, φ)we can define the R-standard filtration series by Sn:= φ−1mN+n(d)∀n∈N0,(1.6) where Nand dstand for the level and the dimension of S. It readily follows from (1.5) that Snis an open normal subgroup of S, for every n∈N.Moreover, since Ris compact, Sis a compact topological group, so Snhas finite index in S. We can specify better: Lemma 1.20 (cf. [27, Lemma 2.3]).Let (S, φ, d)be an R-standard group of level N. Then, |S:Sn|=mN(d):mN+n(d), where the latter stands for the index as additive groups. Proof. From (1.3), φ(x) = φxy−1y=Fφxy−1, φ(y)=φ(xy−1) + φ(y) + Hφxy−1, φ(y), where all the monomials involved in H(X,Y)have total degree at least 2and contain a non-zero power of Xiand Yjfor some i, j ∈ {1, . . . , d}. Thus, if φ(x)−φ(y)∈mK(d)\mK+1(d),then φxy−1+Hφxy−1, φ(y)=φ(x)−φ(y)∈mK(d)\mK+1(d), and so φ(xy−1)∈mK(d)\mK+1(d).Conversely, if φ(xy−1)∈mK(d)\ mK+1(d),then φ(x)−φ(y)≡φxy−1mod mK+1(d), and so φ(x)−φ(y)∈mK(d)\mK+1(d). In other words, xy−1∈Sn⇐⇒ φ(x)−φ(y)∈mN+n(d). 21 Therefore, since log|R/m||mi:mi+1|= dimR/m(mi/mi+1),we have that |S:Sn|=pcd Pn−1 i=0 dimR/m(mN+i/mN+i+1),(1.7) where pcis the size of the residue field R/m,and particularly, S/Snis a finite p-group. Forasmuch as Sis a compact topological group with an open neighbourhood system of the identity {Sn}n∈Nsuch that S/Snis a finite p-group for all n∈N, we conclude that any R-standard group is actually a countably based pro-pgroup. By the next result the study of R-analytic groups can be reduced, to some extent, to R-standard groups. Lemma 1.21 (cf. [24, Theorem 13.20]).Let Gbe an R-analytic group and (U, φ, d) an R-chart of the identity. Then, Ucontains an open R-standard subgroup of dimension dim1G. In particular, every R-analytic group contains an open Rstandard subgroup. Proof. We can assume that φ(1) = 0by composing with a convenient translation. Since the multiplication map mand the inversion map ιare R-analytic at 1,there exists N∈Nsuch that mN(d)⊆φ(U)and some power series Fj∈Λ0[[X1, . . . , X2d]] and Ij∈Λ0[[X1, . . . , Xd]], j ∈ {1, . . . , d},such that φ◦m◦(φ, φ)−1(x, y) = (F1(x, y), . . . , Fd(x, y)) ∀x, y ∈mN(d) and φ◦ι◦φ−1(x) = (I1(x), . . . , Id(x)) ∀x∈mN(d) (actually, it is sufficient to consider the multiplication, since in mN(d)the tuple of power series Iis nothing but the formal inverse of F). In particular, if F= (F1, . . . , Fd)and I= (I1, . . . , Id),then 0=φ(1) = F(φ(1), φ(1)) = F(0,0), and 0=φ(1) = I(φ(1)) = I(0), so each power series Fjand Ijhas constant term equal to zero. Therefore, mN(d) is closed with respect to both Fand I.Thus, H:= φ−1mN(d)⊆U 22 is an open subgroup of Gsuch that φ(xy) = (F1(φ(x), φ(y)), . . . , Fd(φ(x), φ(y))) ∀x, y ∈H. When Ris not a PID, Λ0[[X]] = R[[X]] so (H, φ|H)is an R-standard group of level N, dimension dand formal group law F.Suppose now that Ris a PID with uniformiser πand fraction field K, according to (1.3), Fj(X,Y) = X+Y+X α,β∈N(d) 0\{0} |α|+|β|≥2 aj,α,βXαYβ∈K[[X,Y]]. Since Fis convergent in mN(d),there exists L∈N0such that aj,α,βπL(|α|+|β|)∈R. Define the power series Fj(X,Y) := π−LFjπLX, πLY=X α,β∈N(d) 0\{0} π−Laj,α,βπL(|α|+|β|)XαYβ, which is a power series with coefficients in R. Consequently, ¯ H:= φ−1mN+L(d) is an open subgroup of G, which is an R-standard group with the R-chart ψ:¯ H→ mN(d), h 7→ π−Lφ(h)and the formal group law F= (F1, . . . , Fd).Indeed, ψj(xy) = π−Lφj(xy) = π−LFj(φ(x), φ(y)) (1.8) =Fjπ−Lφ(x), π−Lφ(y)=Fj(ψ(x), ψ(y)),∀x, y ∈¯ H. An open R-standard group (S, φ, d)can be used to obtain a natural atlas of G. Indeed, consider {(xS, φx, d)}x∈G,where φx:xS →mN(d)is defined by φx(y) = φ(x−1y).Those R-charts are compatible, as φx◦φ−1 y=φ◦Lx−1◦Ly◦φ−1=φ◦Lx−1y◦φ−1 and Lx−1yis R-analytic. Moreover, this atlas is compatible with the initial Ranalytic structure of G. As a by-product we observe that Corollary 1.22. Let Gbe an R-analytic group. Then dimxGis constant for all x∈G. 23 Proof. We have indicated in Remark 1.11, although it will be proved in Corollary 1.30, that dimxGis independent of the R-charts. By Lemma 1.21, there exists an open R-standard subgroup S≤Gof dimension d:= dim1G, and the R-atlas {(xS, φx, d)}x∈Gshows that dimx(G) = dfor all x∈G. This common value is referred to as the (analytic) dimension of an R-analytic group, and on account of it, we will write the R-charts simply as the pair (U, φ). 1.3.1 R-standard groups and group operations By virtue of Lemma 1.21 every R-analytic group contains an open pro-psubgroup, so compact R-analytic groups are profinite groups. Furthermore, assuming compactness Lemma 1.21 can be strengthened: Lemma 1.23. Let Rbe a pro-pdomain that is not a PID. A compact R-analytic group Gcontains an open normal R-standard subgroup S such that for all g∈G the conjugation map cg:S→Sis strictly R-analytic. Proof. Let Gbe a compact R-analytic group of dimension d. By Lemma 1.21, there exists a finite index R-standard subgroup (H, φ)of dimension d, level N, formal group law Fand formal inverse I.Let Tbe a left transversal for Hin G. Since the conjugation maps are R-analytic at 1, for each t∈Tthere exists an integer Nt≥Nand some power series Ct j∈R[[X1, . . . , Xd]], j ∈ {1, . . . , d}, such that φxt=Ct 1(φ(x)), . . . , Ct d(φ(x))∀x∈φ−1mNt(d). Let L= maxt∈TNt, since mL(d)is closed with respect to the tuples of power series Fand I,then S:= φ−1mL(d)is an open R-standard subgroup. Moreover, Ct j(0) = Ct j(φ(1)) = 0, so each Ct jhas constant term equal to zero, and thus, Sis closed with respect to the conjugation by every t∈Tand ct:S→Sis strictly analytic in Sfor every t∈T. In addition, whenever h∈Hand x∈Sthen φxh=F(I(φ(h)),F(φ(x), φ(h))), so Sis closed with respect to conjugating by hand ch:S→Sis strictly analytic in S. Since every element g∈Gcan be written as th where t∈Tand h∈H, then Sis closed with respect to the conjugation with g–i.e. Sis normal in G– and cg=ch◦ctis strictly analytic in S. 24 For the remainder of the section, we keep the notation used throughout the previous proof and we recover the atlas associated to (S, φ),accordingly cgis given in coordinates by the tuple of power series Cg= (Cg 1, . . . , Cg d)of the proof, and we can give an explicit description of the group operations in G: Lemma 1.24. Let Gbe an R-analytic group and let Sbe a open normal Rstandard subgroup such that for all g∈Gthe conjugation map cg:S→Sis strictly analytic. Suppose that with respect to the R-atlas induced by Sthe map cg is given in coordinates by the tuple of power series Cg.Let t, r ∈G. (i) The inverse in tS is given in coordinates by the tuple of power series Ct−1◦I. That is, φt−1x−1= (Ct−1◦I) (φt(x)) ∀x∈tS. (ii) The multiplication in tS ×rS is given in coordinates by the tuple of power series F(Cr(X),Y).That is, φtr(xy) = F(Cr(φt(x)), φr(y)) ∀x∈tS, y ∈rS. Proof. (i) Take x=tx ∈tS, then Ct−1(I(φt(x))) = Ct−1φx−1=φx−1t−1=φt−1x−1. (ii) Take x=tx ∈tS and y=ry ∈rS, then φtr(xy) = φ(xry) = F(Cr(φ(x)), φ(y)) = F(Cr(φt(x)), φr(y)). Finally, if we fix a left transversal Tfor Sin G, we can work solely with the atlas {(tS, φt)}t∈T.In fact, for x, y ∈Gsuch that xS =yS, let Ay x:mN(d)→mN(d) be the R-analytic homeomorphism φy◦φ−1 x.Since φy=Ay x◦φx,Lemma 1.24(i) is restated as φrx−1= (Ar t−1◦Ct−1◦I) (φt(x)) ∀x∈tS, whenever rS =t−1S. Ditto multiplication, i.e. φp(xy) = Ap tr (F(Cr(φt(x)), φr(y))) ∀x∈tS, y ∈rS, whenever trS =pS. 25 1.4 Lie algebras Any R-standard group is associated with a so-called Lie algebra. The objective of this section is to describe this construction by following [24, Section 13.3], as well as to reproduce for general pro-pdomains the results in [68, Part II, Chapter III, § 10]. Let (S, F)be an R-standard group of level Nand dimension dand formal inverse I.In accordance with the notation in (1.3) we can associate to Fthe Lie bracket [X,Y]F:= B(X,Y)−B(Y,X), which will be simply denoted by [·,·]when there is no risk of confusion. Let us verify that [·,·]Fis an actual formal Lie bracket (see Appendix A for the precise definition of Lie bracket). Obviously, it is bilinear and [X,X]=0.Further it satisfies the Jacobi identity: Lemma 1.25 (cf. [24, Lemma 13.24] and [68, Part II, Chapter IV, § 7.6]).Let X,Yand Zbe d-tuples of variables. Then, [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0. Proof. The result follows from the so-called Hall-Witt identity (cf. [68, Part I, Proposition 1.1]), the non-commutative version of Jacobi’s identity. Accordingly, every group Gsatisfies the identity: [xy,[y, z]] [yz,[z, x]] [zx,[x, y]] = 1.(1.9) Hereinafter, O(n)stands for formal power series in two d-tuples of variables Xand Y,all whose monomials have degree at least nand contain a non-zero power of Xiand Yjfor some i, j ∈ {1, . . . , d}.Moreover, the formal conjugation F(I(Y),F(X,Y)) will be abbreviated by XY,and if C(X,Y) := F(I(X),F(I(Y),F(X,Y))) is the formal commutator, according to (1.3) - (1.5), we have that XY=X+O(2) and [X,Y] = C(X,Y) + O(4).Therefore, [X,[Y,Z]] = CXY,C(Y,Z)+O(4). Thus, 0=FCXY,C(Y,Z),FCYZ,C(Z,X),CZX,C(X,Y) = [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] + O(4), 26 assume, by working in coordinates, that x=0∈U=mL(n)⊆oN, that F(x) = 0∈V=mL(m)⊆oMand that F∈R[[X1, . . . , Xn]](m). Secondly, for simplicity, we can assume that the residue rank of the first n columns of J0Fis n, that is, if ˜ F= (F1, . . . , Fn)then res.rk J0˜ F=n. Let W=mL(m−n),and define the map Φ: N×W→M, (x, w)7→ F(x) + (0, w).Then, J0Φ = J0F0 Im−n∈Mm(R), and so res.rk J0Φ = m. By Theorem 1.33, there exists a local inverse of Φ,that is, there exist some open subsets U′, V ′and W′of, respectively, R(n), R(m)and R(m−n)such that Ψ: V′→U′×W′is the inverse of Φ|U′×W′.That is, the following diagram is commutative U′V′ U′×{0}(m−n)U′×W′, F ιΨ Ψ◦F◦ι−1 where ι(x) = (x, 0),and Ψ◦F:U′→U′×W′,(x1, . . . , xn)7→ x1, . . . , xn,0,(m−n) . . . , 0, as we were required. In particular, since an R-bianalytic map is both an immersion and submersion, it looks in coordinates as the identity. Definition 1.39. An R-analytic map F:N→Mis a subimmersion at xwhen there exist x∈U⊆oN, F(x)∈V⊆oMand an R-analytic manifold Wsuch that F|Uis the composition Uπ →Wι →V, where πis a submersion and ιis an immersion. Lemma 1.40. Let F:N→Mbe an R-analytic map. The following are equivalent: (i) Fis subimmersion at x. 33 (ii) Flooks in coordinates around xlike the R-linear homomorphism F:R(n)→R(m),(x1, . . . , xn)7→ (x1, . . . , xr,0, . . . , 0) for some r≤min{n, m},where n= dimxNand m= dimF(x)M. Proof. The proof follows from Lemma 1.38. (i) ⇒(ii).By definition Fhas locally the form Uπ →Wι →V. Let r= dimπ(x)W. According to Lemma 1.38, πand ιlook in coordinates as π:R(n)→R(r),(x1, . . . , xn)7→ (x1, . . . , xr) and ι:R(r)→R(m),(x1, . . . , xr)7→ (x1, . . . , xr,0,(m−r) . . . , 0), so Flooks like their composition. (ii) ⇒(i).There exist R-charts (U, φ, n)at xand (V, ψ, m)at F(x)such that ψ◦F◦φ−1=F=ι◦πat φ(U),where πand ιare defined as in the preceding implication. Hence W=π(φ(U)) is an open subset of R(r)and the following diagram is commutative: U V φ(U)W ψ(V) ϕ F π ι ψ−1 The result follows since π◦φis a submersion and ψ−1◦ιis an immersion. The following lemma is proved reproducing the arguments of [68, Theorem in pg. 86]. We will rely on the next basic result: Lemma 1.41. Let Qbe a ring of characteristic zero and F∈Q[[X1, . . . , Xn]]. Suppose that the formal derivatives on the last mvariables are 0,that is, ∂jF= 0 for all j∈ {n−m+ 1, . . . , n}.Then F∈Q[[X1, . . . , Xn−m]]. Proposition 1.42. Let Rbe a pro-pdomain of characteristic zero and let F:N→ Mbe an R-analytic map. Suppose that there exists r∈N0such that rk JyF= res.rk JyF=rfor all yin an open neighbourhood Uof x. Then Fis a subimmersion. 34 Proof. Let n= dimxNand m= dimF(x)M. Since the question is local, working in coordinates, we can assume that x=0∈U=mL(n)⊆oN, F(x) = 0∈ V=mL(m)⊆oMand that F∈R[[X1, . . . , Xn]](m).Assume for simplicity that if ˜ F= (F1, . . . , Fr)then res.rk J0˜ F=r. Thus, {F1, . . . , Fr, πr+1, . . . , πn} is a coordinate system of Nat U. Hence, if we consider Uas mL(r)×mL(n−r), after a change of coordinates we can assume that ˜ F(x1, x2) = x1,that is, then F(x1, x2) = (x1, ψ(x1, x2)) , for some ψ∈R[[X1, . . . , Xn]](m−r)such that res.rk J0ψ= 0.We shall check up on ψ, whether it is independent of the second variable in a neighbourhood of 0. First of all, ∂2ψ=0in U, where ∂2ψstands for the matrix of formal derivatives on the last n−rvariables. Otherwise, according to (1.10), there would be y∈U such that rk JyF > r, which is a contradiction. Therefore, since char R= 0,from Lemma 1.41, ψis independent of the last n−rcoordinates. That is, F(x1, x2) = (x1, ψ(x1)). Hence, if π:mL(n)→mL(r)is the projection onto the first rcoordinates, then F= (Id ×ψ)◦π, and thus, Fis locally the composition of a submersion and an immersion. 1.5 Construction of manifolds This section is devoted to constructing new analytic structures starting from an initial R-analytic manifold, both by changing the coefficient ring to a convenient subring of Ror by developing concepts such as submanifolds or quotient manifolds. 1.5.1 Restriction of scalars In this subsection we will illustrate how to induce a manifold structure over a suitable subring. For that we shall follow the procedure of [24, Example 13.6]. Let (R, m)be a pro-pdomain and Qa subring that is itself a pro-pdomain with maximal ideal n:= m∩Q. Suppose further that Ris a finitely generated free Q-module. For instance, in view of Cohen’s Structure Theorem whenever 35 dimKrull(R) = 1,then Ris a finitely generated free Zp-module if char R= 0,or a finitely generated free Fp[[t]]-module if char R=pis positive. Let Mbe an R-analytic manifold and σ:R→Q(e)aQ-module isomorphism, that is, if we fix a basis {v1, . . . , ve}for Ras a Q-module, then σis σ e X i=1 qivi!= (q1, . . . , qe). For each R-chart (U, φ, n)of M, we can define the triple U, σ(n)◦φ, ne,which is actually a Q-chart. Indeed, we only have to check that σ(n)◦φ(U)is open in Q(ne).Firstly, note that if τis the inverse of σ, then τnN(e)⊆mNfor all N∈N.Secondly, let x∈φ(U),since φ(U)is open in R(n),there exists N∈N such that x+mN(n)⊆φ(U), and therefore, σ(n)(x) + nN(ne)⊆σ(n)x+mN(n)⊆σ(n)◦φ(U). Moreover, the Q-charts U, σ(n)◦φ, neand V, σ(n)◦ψ, neare compatible (note that the compatibility is a requirement only when U∩V6=∅,so in view of Corollary 1.30, the charts must be of equal dimension). In fact, σ(n)◦φ◦σ(n)◦ψ−1=σ(n)◦φ◦ψ−1◦σ−1(n), and since φ◦ψ−1is R-analytic, the result follows by the next lemma: Lemma 1.43 (cf. [24, Exercise 13.4]).Let F∈Λ0(R)[[X1, . . . , Xn]],then σ◦F◦σ−1(n):n(en)→Q(e) is strictly Q-analytic. Proof. As customary Λ(R)stands for the fraction field Frac(R)if Ris PID and for Rotherwise, and the same for Λ(Q). Observe that when Ris PID, since Ris an integral ring extension of Q, then dimKrull(Q) = dimKrull(R) = 1,so the pro-pdomain Qis also a discrete valuation ring. Therefore, if πand ρare uniformisers of respectively Rand Q, then ρ=πN for some N∈N. 36 Let {v1, . . . , ve}be the basis for Ras free Q-module that corresponds to σand let us denote σ−1by τ. Moreover, suppose that F(X1, . . . , Xn) = X α∈N(n) 0 aαXα1 1. . . Xαn n∈Λ0(R)[[X1, . . . , Xn]]. We will show that there exist some power series F∗ l∈Λ0(Q)[[X1, . . . , Xen]], l∈ {1, . . . , e},such that F◦τ(n)(y1, . . . , yen) = e X l=1 vlF∗ l(y1, . . . , yen)∀yj∈n. That is, whenever xj=Pe i=1 viyij for some yij ∈n, j ∈ {1, . . . , n},then F(x1, . . . , xn) = e X l=1 vlF∗(y11, . . . , yen). First of all, there exist some elements there exist some elements aα(k, l)∈Λ(Q) such that aαvk= e X l=1 aα(k, l)vl. In order to prove the preceding when Ris a PID, we should have taken into account that since F∈Λ0(R)[[X]] and since ρ=πN,there exists a big enough integer L∈Nsuch that aαρ|α|L∈Rfor all α. In particular, aα(k, l)ρ|α|L∈Q∀k∈ {1, . . . , e}.(1.11) Furthermore, by an application of the multinomial theorem, for any tuple α= (α1, . . . , αn)∈N(n) 0there exist some elements γk(β)∈Qsuch that n Y j=1 e X i=1 viYij!αj =X |β|=|α| e X k=1 γk(β)vk e Y i=1 n Y j=1 Yβij ij , where the Yij’s are indeterminates. Finally, the desired power series are defined as F∗ l(Y11, . . . , Yen) = X α∈N(d) 0X |β|=|α| e X k=1 aα(k, l)γk(β) e Y i=1 n Y j=1 Yβij ij . When Ris not a PID these power series are clearly in Q[[X]].In contrast, when Ris a PID, F∗ l∈Λ0(Q)[[X]] by virtue of (1.11). 37 Furthermore, the preceding Q-manifold structure is independent of the isomorphism σchosen, i.e. of the Q-basis of Rchosen. Actually, it suffices to prove that for the R-analytic manifold mNany two Q-module isomorphisms σand ˜σgive rise to equivalent Q-charts. For that note that ˜σ◦σ−1:σmN→˜σmN is nothing but the linear map defined by the change of basis matrix A∈GLe(Q), and so, it is a Q-bianalytic map. Thus, we will refer to this procedure of generating new manifolds, simply as restriction of scalars, with no need of specifying the isomorphism. To finish, let us observe the following particular case: Corollary 1.44. Let Rbe a pro-pdomain of Krull dimension one. Then any Ranalytic group is by restriction of scalars either a p-adic analytic group if char(R) = 0,or an Fp[[t]]-analytic group if char(R) = pis positive. Under certain conditions the converse of this result is also true. Theorem 1.45. Let Gbe a non-discrete R-analytic group. (i) (cf. [24, Theorem 13.23]) If Gadmits a p-adic analytic group structure, then Ris a finitely generated Zp-module. (ii) (cf. [45, Theorem 1.1]) Suppose that Gis finitely generated (as topological group). If Gadmits an Fp[[t]]-analytic group structure, then Ris a finitely generated Fp[[t]]-module. In the latter article, the authors already observed that the result does not hold for non finitely generated groups, on account of the additive topological groups Fp[[t1]] and Fp[[t1, t2]] being isomorphic to one another. Nevertheless, whether for finitely generated groups the analytic structure determines the base ring is a sensible question, in this direction it is tempting to speculate that: Conjecture 1.46. Let Gbe a non-discrete finitely generated topological group that admits both an R-analytic and a Q-analytic group structure. Then, Rand Qshare the Krull dimension and the characteristic. 1.5.2 Submanifolds The concept of submanifold has to be developed when treating manifolds over general pro-pdomains. The theory of analytic manifolds over local fields supplies various equivalent definitions (cf. [68, pg. 89]). Here we reproduce some of those, underscoring that any definition should reflect the fact that the set has R-analytic manifold structure by itself. 38 Definition 1.47. Let F:N→Mbe an injective weak immersion. Then F(N) is an immersed subset of M. For example, let mbe the maximal ideal of Zp[[t]] with the natural manifold structure. If we endow the set tmwith the global chart ψ:tm→m, tx 7→ x, the inclusion ι:tm→mis a weak immersion, as Jx(Id ◦ι◦ψ−1) = (t)for all x∈tm. However, in this example, the topology of tm,which is chosen purposely to make ψa homeomorphism, does not coincide with the subspace topology. Since it is natural to ask for compatibility between topological structures, we define: Definition 1.48. Let Mbe an R-analytic manifold. Then S⊆Mis an R-analytic submanifold when for each s∈Sthere exist ks∈N0,an open neighbourhood Us of sin Sand an R-chart (Vs, φs, ds)of sin Msuch that •Us⊆Vsand •φs(Us) = φs(Vs)∩R(ks)×{0}(ds−ks). The integer ksis the dimension of Sat s, denoted by dimsS. With the subspace topology Sis an R-analytic manifold, as each point s∈S can be endowed with the R-chart (Us,(φs,1, . . . , φs,ks), ks). An immediate application of Lemma 1.38 yields: Proposition 1.49. Let F:N→Man injective R-analytic map. Then F(N)is an R-analytic submanifold of Mif and only if Fis an immersion. Remark 1.50. Note that R-analytic submanifolds are locally closed. Indeed, let Mbe an R-analytic manifold and S⊆Ma submanifold. For each s∈Sthere exists an two subsets Us⊆oSand Vs⊆oMcontaining ssuch that φs(Us)is defined by some linear equations in φ(Vs).In particular, Usis closed in Vs,that is, Vs\Usis open in Vs.Hence, if V=∪s∈SVs,then V\S=Ss∈SVs\Usis open in V, so Sis closed in the open set V. Another straightforward observation yields: Lemma 1.51. Let Mbe an R-analytic manifold and Sa submanifold. Suppose that dimsM= dimsSfor all s∈S. Then Sis open in M. Proof. In keeping with the notation of Definition 1.48: since ks=dsand φis a homeomorphism, then Us=Vsand so S=Ss∈SVsis open in M. 39 Definition 1.48 strengthens the definition of R-analytic submanifold used in [27]: Definition 1.52. Let Mbe an R-analytic manifold. Then S⊆Mis a weak R-analytic submanifold if for each s∈Sthere exist an open neighbourhood Us of sin S, an R-chart (Vs, φs, ds)of sin Mand a K-vector space Es≤K(ds) (K= Frac R) such that •Us⊆Vsand •φs(Us) = φs(Vs)∩Es. Over fields Definitions 1.48 and 1.52 are equivalent. Certainly, by a convenient change of basis we can assume that Es=K(k)× {0},where k= dimK(Es). Nonetheless, as we shall illustrate with an example, in general they are not equal. Let R=Z2[[t]] with maximal ideal m= (2, t)R, K =Q2((t)),the R-analytic manifold M=m(2) and the K-vector space: E=(x, y)∈K(2) |2x−ty = 0. Then, M∩E={(ta, 2a)|a∈R}is clearly a weak R-analytic submanifold, and it is endowed with the global R-chart ψ:M∩E→R, (ta, 2a)7→ a. However, Js(Id ◦ι◦ψ−1) = t 2for all s∈M∩E, and therefore ιis not an immersion but a weak immersion. In fact, the change of basis of K(2) from the canonical basis to β={(t, 2),(0,1)}is the linear map LA:K(2) →K(2) defined by the change of basis matrix A=1/t0 −2/t1, and it maps Eto K(1) × {0}.However, LAis not an R-bianalytic map. This illustrates forby the difficulty to determine a valid coordinate system for a weak submanifold. Over fields, the natural way of doing so is as before, that is, by fixing a basis β={v1, . . . , vk}for Es,and considering the map ψ:φs(Vs)∩Es→K(k), k X i=1 αivi7→ (α1, . . . , αk)∈K(k). 40 However, over general pro-pdomains depending on the choice of β, it might occur that im ψis not contained in R(k),and so ψmight not be an R-chart. In classical Lie theory, authors already distinguish between immersed (the equivalent of Definition 1.47) and embedded (the equivalent of Definition 1.48) submanifolds (see [51, Chapter 5]), the latter being a stronger condition. Nevertheless, when the manifold is compact both concepts coincide, and so do they when working with analytic manifolds over principal ideal pro-pdomains (cf. [68, Part II, Section III.11.2]). Over general pro-pdomains, however, it is apparent now that the notion of submanifold we work with must be categorically specified. In [45, Section 4], Fq[[t]]-analytic submanifolds are characterized as fibers of analytic maps. More precisely, they prove the following: Proposition 1.53 (see [45, Corollary 4.2]).Let Mbe an Fq[[t]]-analytic manifold and a subset S⊆M. Suppose that (i) Sis homogeneous, i.e. it is contained in a single orbit of the action of the group of R-bianalytic automorphisms of the manifold M, and (ii) Sis an analytic subset, i.e. for each s∈Sthere exist an open neighbourhood Uand some Fq[[t]]-analytic maps {fi:U→Fq[[t]]}i∈Isuch that S∩U={x∈U|fi(x) = 0 ∀i∈I}. Then Sis an Fq[[t]]-analytic submanifold.† Observe that when Mis an R-analytic group, the action of the left multiplication maps is transitive, and thus in this situation, every subset is homogeneous. In order to replicate a version of this for general pro-pdomains, given an Ranalytic manifold Mand S⊆M, we say that Sis an R-analytic subset of Mwhen for every s∈Sthere exist an open neighbourhood Uof sand some R-analytic maps {fi:U→R|i= 1, . . . , rs}such that S∩U={y∈U|fi(y) = 0 ∀i= 1, . . . , rs}. In other words, an analytic subset is locally the nullset of some analytic functions. This definition obviously extends that of submanifold. Moreover, note that since R[[X1, . . . , Xn]] is Noetherian (see [49, Theorem IV.9.4]), the definition can be in principle relaxed allowing rsto be infinite. †This result is valid regardless of the definition of submanifold, as the base ring is a PID. 41 Definition 1.54. Let Sbe an R-analytic subset of an R-analytic manifold M. A point s∈Sis a regular point when there exists an open neighbourhood Uof s and some R-analytic maps {fi:U→R|i= 1, . . . , rs}such that (i) S∩U={y∈U|fi(y) = 0 ∀i= 1, . . . , rs}and (ii) if F:= (f1, . . . , frs),then res.rk (JsF) = rs. The integer rsis termed the corank of Sat s, and by Lemma 1.38, we know that rs≤dimsS. Lemma 1.55. Let Mbe an R-analytic manifold and S⊆M. Then, Sis a submanifold if and only if all the points of Sare regular points. Proof. The only if is immediate from the definition and Lemma 1.38. For the if, let s∈Sand d= dimsS, suppose that sis a regular point of corank r, then there exists an open neighbourhood Uof ssuch that S∩U={y∈U|fi(y) = 0 ∀i= 1, . . . , r}. If F= (f1, . . . , fr),then res.rk JsF=r, so we can extend Fto a coordinate system {fi}d i=1 of sin M, and with respect to that coordinate system for a possibly smaller open set U′⊆Uwe have S∩U′={y∈U′|f1(y) = ··· =fr(y) = 0}, and thus Mis a submanifold of dimension d−rat s. Additionally, in R-analytic groups, we can blend the notions of analytic and group substructure. Accordingly, Definition 1.56. Let Gbe an R-analytic group. An analytic subgroup is a subgroup H≤Gthat is besides an R-analytic submanifold. Lemma 1.57. An R-analytic subgroup is closed. Proof. It is a general fact that in a topological group a locally closed subgroup is actually closed, so the result is straightforward from Remark 1.50. To make it explicit, Hbeing locally closed means that His open in its topological closure H. Suppose by contradiction that H6=H, so that there exists a non-trivial left coset gH ⊆H. Since Hand gH are open dense subsets of H, then gH ∩H6=∅, contradicting the disjointness of the cosets. 42 Let Ibe the ideal of R[[X]] generated by the formal power series nπi,α |α∈N(d) 0, i ∈ {1, . . . , d}o.(2.1) Since R[[X]] is Noetherian (see [49, Theorem 9.4]), Iis generated by a finite subset of (2.1), say F,and so consider m= max {|α| | πi,α ∈ F} ∈ N.Denote by Mthe ideal (X1, . . . , Xd)R[[X]],then W=M/Mm+1 is a free R-module of finite rank, and Sacts on W. Clearly Z(S)acts trivially on W. Conversely, suppose that gacts trivially on W. Then, π(g)=0,for all π∈ F.Hence, πi,α(g)=0for all i∈ {1, . . . , d}and α∈N(d) 0,so g·πi(X) = Xi,that is, g∈Z(S).Thus, S/Z(S)acts faithfully on W. On other hand, Lazard’s proof – it will be briefly explained in Section 2.1–, relies on Ado’s Theorem in conjunction with the Baker-Campbell-Hausdorff formula in order to link the group operation with the corresponding Lie algebra operation. Based on that procedure, Camina and Du Sautoy [14] proved that perfect Zp[[t]]- standard groups, namely Zp[[t]]-standard groups Sof level Nsuch that S′=S2N (see (1.6)), are linear. Furthermore, Jaikin-Zapirain [43] exploited similar ideas to prove that over a pro-pdomain Rof characteristic zero, every finitely generated compact R-analytic group is linear. The main result of this chapter is to extend this, and prove that whenever Rhas characteristic zero, any compact R-analytic group is linear. 2.1 Linearity of compact p-adic analytic groups In this section, we will succinctly describe the construction of the aforementioned faithful linear representation for compact p-adic analytic groups, with a view towards controlling its degree. We start by recalling Ado’s Theorem, more precisely a generalised version of it, due by Churkin [19] and Weigel [74] (see Appendix A for a detailed proof as well as for pertinent definitions on the topic). Theorem 2.4. Let Lbe a Zp-Lie algebra which is a free Zp-module of rank r. There exists a Zp-Lie algebra monomorphism φ:L,→Mn(Zp),where ndepends only on r. Fix a prime number p, and for simplicity of notation set p=pif pis odd and p= 4 if p= 2.Any Zp-standard group (S, φ)is a so-called uniformly powerful group, that is, a finitely generated torsion-free group such that S′≤Sp(cf. [24, 49 Thms 4.5 and 8.31]). There is a categorical isomorphism between the category UGroup of d-dimensional uniformly powerful pro-pgroups and the category pLie of d-dimensional powerful Zp-Lie lattices, namely Zp-Lie algebras Lthat are free Zpmodules of rank dsuch that [L,L]≤pL.More concretely, there exists a categorical isomorphism L:UGroup →pLie,with inverse E,and to each uniformly powerful group Sof dimension d, it assigns a Zp-Lie algebra L(S)that is a free Zp-module of rank d, where the underlying set is Sitself, and the module operations are given in terms of the group operations as follows: let r∈Zpand x, y ∈Sthen r·x=xr x+y= lim n→∞ xpnypnp−n [x, y] = lim n→∞ xpn, ypnp−2n, we refer to [24, Section 4.3] for the precise definitions of the right-hand side formulas. For instance, pMn(Zp)is a powerful Lie lattice with Lie bracket [A, B] = AB −BA, and E(pMn(Zp)) ⊆GLn(Zp)(this can be viewed as the usual p-adic matrix exponentiation).∗ Conversely, the Baker-Hausdorff-Campbell formula can be regarded as a formal power series H(x, y)in two non-commuting variables satisfying the identity eH(x,y)=exey, and given a powerful Zp-Lie lattice S, the set Sis a uniformly powerful group with the group operation xy =H(x, y). Consequently, if φ:L(S),→Mn(Zp)is the injective Zp-Lie algebra homomorphism provided by Theorem 2.4, then φ|pL(S):pL(S)→pMn(Zp) is an injective Lie algebra homomorphism between powerful Zp-Lie lattices, and hence, since Eis a functor, we obtain a group monomorphism E(φ): E(pL(S)) ,→ E(pMn(Zp)) ≤GLn(Zp). Finally, the additive cosets of the Zp-Lie algebra are the same as the multiplicative cosets of the uniformly powerful group (cf. [24, Proposition 4.31(iii)]), so |S:E(pL(S))|=|L(S) : L◦E(pL(S))|=|L(S) : pL(S)|=Z(d) p:pZ(d) p=pd, ∗Actually the image of pMn(Zp)by Eis the first congruence subgroup GL1 n(Zp). 50 and thus by taking the induced representation we obtain a group monomorphism m:S ,→GLpdn(Zp). We gather all this in the following result: Theorem 2.5. Let Sbe a Zp-standard group of dimension d. There exists a faithful linear representation m:S ,→GLn(Zp),whose degree ndepends only on pand d. Finally, since every compact p-adic analytic group has a Zp-standard subgroup of finite index, the induced linear representation leads to: Corollary 2.6 (Lazard).Every compact p-adic analytic group is linear over Zp. 2.2 Change of pro-pdomains The idea behind various results in this thesis is to reduce the problem to analytic groups over pro-pdomains of Krull dimension one and use the already known structural results there. For that purpose a homomorphism ϕ:R→Qbetween pro-pdomains can be used to construct Q-analytic groups that are natural images of R-analytic groups. Indeed, given a power series F(X) = Pα∈N(d) 0aαXα∈R[[X]] by applying ϕto the coefficients we obtain the power series Fφ=X α∈N(d) 0 ϕ(aα)Xα∈Q[[X]]. This is no more than a specific instance of the wider universal property of topological power series rings (see Section 1.1). To transform coefficients, we restrict to natural ring homomorphisms between local rings, namely local ring homomorphisms. Those are ring homomorphisms ϕ: (R, m)→(Q, n)between local rings such that ϕ(m)⊆n.Observe that, every local ring homomorphism is continuous, considering that ϕ(mn)⊆nnfor any n∈N. The following lemma shows that the foregoing change of rings commutes with the composition of power series. Lemma 2.7. Let ϕ: (R, m)→(Q, n)be a local ring homomorphism. Let F∈ R[[X1, . . . , Xn]](m)and G∈R[[X1, . . . , Xm]](l)be formal power series, and assume that F(0)∈m(m).Then (G◦F)φ(X1, . . . , Xn) = Gφ◦Fφ(X1, . . . , Xn). 51 Proof. Using the universal property of power series rings, there exists a unique continuous ring homomorphism Φφ:R[[X1, . . . , Xn]] →Q[[X1, . . . , Xn]] such that Φφ(H(X)) = Hφ(X)for all H∈R[[X]],where X= (X1, . . . , Xn). Let F(X)=(F1(X), . . . , Fm(X)). Since F(0)∈m(m)and ϕis local, then Φφ(F1(X)) , . . . , Φφ(Fm(X)) are in (n, X1, . . . , Xn)Q[[X]],the maximal ideal of Q[[X]], therefore using the universal property of power series rings we can define the continuous map Φ1:R[[Y1, . . . , Ym]] →Q[[X1, . . . , Xn]] such that Φ1(r) = ϕ(r)for all r∈R, and Φ1(Yi) = (Fi)φ(X)for all i∈ {1, . . . , m}. Similarly, since F1(X), . . . , Fm(X)are in the maximal ideal of R[[X]], define the map Φ2:R[[Y1, . . . , Ym]] →R[[X1, . . . , Xn]] such that Φ2|R= IdR,and Φ2(Yi) = Fi(X)for all i∈ {1, . . . , m}. Notice that Φ1(r)=Φφ◦Φ2(r) = ϕ(r)for all r∈R, and that Φ1(Yi) = Φφ◦Φ2(Yi) = (Fi)φ(X)for all i∈ {1, . . . , m}.Therefore, by the uniqueness of the universal property we have that Φ1= Φφ◦Φ2,and so, in particular, (G◦F)φ(X) = Φφ◦Φ2(G(Y)) = Φ1(G(Y)) = Gφ◦Fφ(X). Hence, the change of rings preserves formal power series identities, so in particular: Corollary 2.8. Let Rand Qbe pro-pdomains, let F∈R[[X1, . . . , X2d]](d)be a formal group law with formal inverse Iand let ϕ:R→Qbe a local ring homomorphism. Then Fφis a formal group law with formal inverse Iφ. Proof. Let X,Yand Zbe d-tuples of variables. Since F(0) = 0and using Lemma 2.7 we have: (i) since F(F(X,Y),Z) = F(X,F(Y,Z)),then Fφ(Fφ(X,Y),Z) = Fφ(X,Fφ(Y,Z)), (ii) and since F(X,0) = F(0,X) = X,then Fφ(X,0) = Fφ(0,X) = X. 52 Therefore Fφ∈Q[[X1, . . . , X2d]](d)is a formal group law. Finally, since I(0) = 0 and F(I(X),X) = F(X,I(X)) = 0,by Lemma 2.7: Fφ(Iφ(X),X) = Fφ(X,Iφ(X)) = 0, and so Iφis the formal inverse corresponding to Fφ. Let now ϕ: (R, m)→(Q, n)be a local ring homomorphism between pro-p domains and let Sbe an R-standard group, which can be identified for simplicity with mN(d),whose formal group law is F.Using the above-constructed formal group law Fφ, L := nN(d)can be endowed with a group operation making it into a Q-standard group. Indeed, the group operation is simply x∗y=Fφ(x, y),(2.2) for all x, y ∈L. In the following lemma we keep all the preceding notation. Lemma 2.9. The map ϕ(d):mN(d),F→nN(d),Fφ,(r1, . . . , rd)7→ (ϕ(r1), . . . , ϕ(rd)) is a group homomorphism. Proof. If we write F= (F1, . . . , Fd)and Fi(X,Y) = X α,β∈N(d) 0 aα,βXαYβ∈R[[X,Y]], then ϕ(Fi(x, y)) = ϕ  X α,β∈N(d) 0 aα,βxα1 1. . . xαd dyβ1 1. . . yβd d   =X α,β∈N(d) 0 ϕ(aα,β)ϕ(x1)α1. . . ϕ(xd)αdϕ(y1)β1. . . ϕ(yd)βd = (Fi)φϕ(d)(x), ϕ(d)(y)∀x, y ∈mN(d), using the continuity of ϕin the second equality; and consequently ϕ(d)is a group homomorphism. 53 Suppose now that Ris not a PID. Let Gbe a compact R-analytic group and let (S, F)be an open normal R-standard group such that the conjugation maps are strictly analytic (we can assure its existence by virtue of Lemma 1.23). We shall construct a compact Q-analytic group, whose open normal Q-standard subgroup will be the Q-standard group L=nN(d),Fφbuilt upon Sas in (2.2). More precisely, let Tbe a left transversal for Sin G, and assume that 1∈T. For notational convenience, we will use the following: whenever g∈Gthen ˜gis the representative of gS in T. By (1.3.1), and using the notation therein, if x∈tS and y∈rS, their product is given by φe tr (xy) = Ae tr tr (F(Cr(φt(x)), φr(y))) . Define H:= T×Land the homeomorphisms ψt: (t, L)→L, ψt(t, l)7→ l. If x∈(t, L)and y∈(r, L), imitating the previous formula define the operation: x∗y=e tr, Ae tr trφFφ(Cr)φ(ψt(x)), ψr(y).(2.3) Remark. We can identify (1, L)with Lfor plainness, and then (2.3) extends (2.2). Lemma 2.10. With the notation above, (H, ∗)is a compact Q-analytic group with open normal Q-standard subgroup L. Proof. Firstly, His a compact Q-analytic manifold with atlas {(tL, ψt)}t∈T– abusing the notation we will use tL to denote (t, L)–. Moreover, His a group. Indeed, (i) let t, r, p ∈T, from Lemma 1.24 and the associativity of Gwe know that as formal power series: Af trp t·erp FCerp(X), A erp rp(F(Cp(Y),Z))= Af trp e tr·pFCpAe tr tr (F(Cr(X),Y)),Z. Thus, let x∈tL, y ∈rL and z∈pL, then by Lemma 2.7 and (2.3): ψf trp(x∗(y∗z)) =Af trp t·erpφFφ(Cerp)φ(ψt(x)),Aerp rpφFφ(Cp)φ(ψr(y)), ψp(z) =Af trp e tr·pφFφ(Cp)φAe tr trφFφ(Cr)φ(ψt(x)), ψr(y), ψp(z) =ψf trp((x∗y)∗z). 54 (ii) The neutral element is (1,0)∈L. (iii) The inverse of x∈tL is given by y=f t−1,Ag t−1 t−1φ◦(Ct−1)φ◦Iφ(ψt(x)). Indeed, clearly x∗y∈Land by Lemma 1.24, we know that A1 t·g t−1FCg t−1(X), Ag t−1 t−1(Ct−1(I(X)))=0. Hence, by Lemma 2.7 and (2.3), 0=A1 t·g t−1φFφCg t−1φ(ψt(x)),Ag t−1 t−1φ(Ct−1)φ(Iφ(ψt(x))) =A1 t·g t−1φFφCg t−1φ(ψt(x)), ψg t−1(y) =ψ1(x∗y). In a similar fashion, y∗x= (1,0). Finally, Lis normal in H, as 1t= 1 for all t∈T. We will finish with a trivial (but helpful in upcoming chapters) observation: Remark 2.11. By definition, if we have that φe tr(x·y) = M(φt(x), φr(y)) ∀(x, y)∈tS ×rS. for a suitable tuple of power series M∈R[[X1, . . . , X2d]](d).Then, ψe tr(¯x∗¯y) = Mφ(ψt(¯x), ψr(¯y)) ∀(¯x, ¯y)∈tL ×rL. Moreover, the same holds for the inversion map. 2.2.1 Evaluation epimorphisms In practise, for the aforesaid change of pro-pdomains we will chiefly use evaluation epimorphisms. More precisely, let (R, m)be a pro-pdomain and a∈m(m), the evaluation epimorphism at ais the continuous local ring homomorphism sa:R[[t1, . . . , tm]] →R, F 7→ F(a).Those epimorphisms can be extended to any integral extension of R[[t1, . . . , tm]],by virtue of the following classical result: 55 Theorem 2.12 (Going Up Theorem (cf. [75, Theorem V.2.3])).Let A⊆Bbe an integral ring extension. For every prime ideal p⊆Athere exists a prime ideal q⊆Bsuch that q∩A=p. Corollary 2.13. Let A⊆Bbe an integral ring extension, let Pbe an integral domain and let ϕ:A→Pbe a ring epimorphism. There exists an integral domain Qsuch that ϕextends to a ring epimorphism ˜ϕ:B→Q. Proof. Let p= ker ϕ, by the Going Up Theorem, there exists a prime ideal q⊆B such that q∩A=p.Thus, the following diagram is commutative: B B/q A A/p ˜φ φ ψ where ψ(x+p) = x+qis injective. Identifying A/pwith P, then ˜ϕextends ϕ. Remark 2.14. The proof above gives more information about Q. Actually, since Bis an integral extension of A,Qis also an integral extension of P. Indeed, any A-integral dependence in B, remains so modulo q: it is now an A/p-integral dependence in B/q. Furthermore, if Bis a pro-pdomain, so is Qas a quotient of Bby a prime ideal. Finally, if A⊆Bis besides a finitely generated ring extension, then B/qis a finitely generated A/p-module. Indeed, reducing the generators of Bas A-module modulo q,we obtain a generating set for B/qas A/p-module. These extended evaluation epimorphisms appear in any pro-pdomain R. Recall that by virtue of Cohen’s Structure Theorem, Ris a finitely generated, and so integral, extension of P[[t1, . . . , tm]] where m= dimKrull(R)−1and (P, n)is a pro-pdomain of Krull dimension one – actually we can specify even more by recollecting that Pis either Zpor Fp[[t]] depending on the characteristic of R–, and therefore for each a∈n(m)we obtain a continuous epimorphism ˜sa:R→Qa. Lastly, we shall present a property that resembles Lemma 1.8. In keeping with prior notation: Corollary 2.15. Let U⊆on(m)and Da dense subset of U, then ∩a∈Dker ˜sa={0}. Proof. We write A=P[[t1, . . . , tm]],pa= ker saand qa= ker ˜safor all a∈n(m). First of all, evaluating a power series F∈P[[t1, . . . , tm]] is continuous, so if 56 F(a)=0for all a∈D, then F(a)=0for all a∈U. Moreover, by construction, pa=qa∩A, so Lemma 1.8 yields that (∩a∈Dqa)∩A=∩a∈Dpa=∩a∈Upa={0}. Suppose by contradiction that there exists r∈ ∩a∈Dqa\ {0}.Since the ring extension A⊆Ris integral, there exists a monic polynomial fr(X)∈A[X],such that fr(r) = rn+ n−1 X i=0 airi= 0. We can additionally assume frto be of minimal degree. But since ∩a∈Dqais an ideal of R, we have that a0∈ ∩a∈Dqa∩A={0},which contradicts the minimality of n, since we could consider fr(X) = Xn−1+Pn−1 i=1 aiXi−1. 2.3 Discrimination The purpose of this section is to study the following concept: Definition 2.16. Let Aand Bbe two instances of the same algebraic structure, Ais said to be fully residually Bor Ais discriminated by B– equivalently, B discriminates A– if for any finite subset S⊆A, there exists a homomorphism h:A→Bin the corresponding category such that the restriction h|Sis injective. More generally, a family B={Bi}i∈Idiscriminates Aor Ais fully residually-B, if for each finite subset S⊆Athere exists a morphism h:A→Bi,for some i∈I, such that h|Sis injective. Example 2.17. Let us exemplify this with rings: let Aand B={Bi}i∈Ibe rings and suppose that Ais an integral domain. Then Abeing fully residually-Bis equivalent to the existence of a collection of homomorphisms F ⊆ ∪i∈IHom(A, Bi) such that \ f∈F ker f={0}. The necessity is clear, since otherwise there would be a non-zero element x∈A such that f(x) = 0 for any ring homomorphism f:A→Bi, and so no homomorphism would be injective when restricted to {0, x}. For the sufficiency, let S={ri}i∈I⊆Abe a finite set, and define r=Qi=j(ri− rj)∈R. Notice that r6= 0,as the ri’s are distinct and Ais an integral domain. 57 Therefore, there exists f∈Hom(A, Bi)such that 06=f(r) = Y i=j (f(ri)−f(rj)) , and thus f(ri)6=f(rj)for all i6=j∈I. For instance, Lemma 1.8 yields that the pro-pdomain Rdiscriminates R[[X]] for any finite number of variables in X. Lemma 2.18. Let Rbe a pro-pdomain. For each finite S⊆Rthere exists a pro-pdomain Qof Krull dimension one and characteristic char R, and a local ring epimorphism ϕ:R→Qthat is injective when restricted to S. In particular, a pro-pdomain Ris discriminated by the set of pro-pdomains of Krull dimension one and characteristic char R. Proof. Let m= dimKrull(R)−1and let (P, n)be the pro-pdomain Zpif char(R) = 0or Fp[[t]] if char(R) = pis positive. According to Subsection 2.2.1, for each a∈n(m),there exists a pro-pdomain Qaand a local ring epimorphism ˜sa:R→Qa that extends the evaluation homomorphism sa:P[[t1, . . . , tm]] →P. Moreover, by Remark 2.14, Qais an integral extension of P, and thus dimKrull Qa= dimKrull P= 1 and char Qa= char P= char R. Finally, by Corollary 2.15, ∩a∈n(m)ker ˜sa={0},and Example 2.17 yields the result. As mentioned in the preceding proof, if Rhas characteristic zero, it is a finitely generated extension of Zp[[t1, . . . , tm]].Let us denote by µ(R)the minimum number of elements that is necessary to generate Ras a Zp[[t1, . . . , tm]]-module. We recall from Remark 2.14 that Qais a free Zp-module of rank at most µ(R). Proposition 2.19. Let (R, m)be a pro-pdomain of characteristic zero, and let Gbe an R-standard group. There exists an integer n∈N,depending on R, the dimension of Gand the level of G, such that Gis discriminated by GLn(Zp). Proof. We identify Gwith mN(d),where the group operation is given by the formal group law F(X,Y) = X α,β∈N(d) 0 aα,βXαYβ∈R[[X,Y]]. 58 3 Hausdorff dimension in compact R-analytic groups Fractal dimensions arose as a generalisation of the notion of topological dimension and there are several alternative definitions for that purpose. However, the bulk of them depends on some sort of measurement. Amongst all these fractal dimension, the most prominent ones are the Hausdorff dimension and the Minkowski-Bouligand dimension (also known as box dimension). These dimensions can be defined in any metric space, and in the specific group theoretical context, the study of the Hausdorff dimension in the setting of profinite groups has attracted considerable attention. Actually, if Gis a countably based profinite infinite group, there exists a filtration series of G, that is, a family {Gn}n∈Nof descending open subgroups which is a neighbourhood system of the identity, i.e. Tn∈NGn={1}.Such filtration defines a metric on Gby letting d(x, y) = inf |G:Gn|−1|xy−1∈Gn. This notion of distance makes Ginto a metric space. Thus, one can define the Hausdorff and the Minkowski-Bouligand dimensions of a subset X⊆G(see Section 3.1 for the precise definitions), which will be denoted, respectively as hdim(X)and lbdim(X).There is a unique measure on a profinite group, namely 65 the Haar measure, whereas there might be several non-equivalent metrics; and usually fractal dimensions depend on the metric –or equivalently on the filtration series employed to define it–. Furthermore, for a fixed filtration series {Gn}n∈Nwe can consider the collection of values hdim{Gn}(H)where Hranges over the closed subgroups of G, that is hspec{Gn}(G) := hdim{Gn}(H)|H≤cG, which is called the Hausdorff spectrum of Gwith respect to the filtration series {Gn}n∈N.Although we could define the Minkowski-Bouligand spectrum similarly, for natural filtration series we have that hdim(H) = lbdim(H)for every closed subgroup H≤cG(see upcoming Theorem 3.7). Consequently, in keeping with classical terminology, we will merely use the name ”Hausdorff”. It turns out that these spectra might have little or no resemblance as one changes the filtration. For instance, consider the additive pro-pgroup Zp⊕Zp. For finitely generated pro-pgroups of this kind, there exists a natural filtration series, namely the p-power filtration series, given by Gn=Gpn.With respect to this series hspec{Gn}(G) = {0,1/2,1},so it is finite; whilst, by [48, Theorem 1.3] there exists a filtration series {Hn}n∈Nsuch that hspec{Hn}(G)contains the real interval h1 p+1,p−1 p+1i,so, whenever p > 2, it is uncountable. The article [6] of Barnea and Shalev is one of the earliest works concerning Hausdorff dimension in profinite groups, and amid other results, there is shown that hspec{Gpn}(G)is finite for any p-adic analytic pro-pgroup G. Nonetheless the converse remains open: Question 3.1 (cf. [6, Problem 1]).Let Gbe a finitely generated pro-pgroup such that hspec{Gpn}(G)is finite. Is G p-adic analytic? It is worthwhile mentioning that the question has positive answer when Gis besides a soluble group (see [48, Theorem 1.7]). However, the p-power filtration series typically can not be used in the setting of profinite R-analytic groups, as Gpnis not normally an open subgroup of a compact R-analytic group G. Nevertheless, those groups possess a canonical filtration series, which depends on the group’s analytic structure. By a way of example, in Section 3.2 we study the connection between the analytic dimension of a closed submanifold Mand the aforesaid fractal dimensions computed with respect to this natural filtration series, and we obtain the following identity: hdim(M) = lbdim(M) = max{dimxM|x∈M} dim H.(3.1) 66 The Hausdorff dimension relative to this filtration series, which is introduced insightfully in Subsection 3.1.3, is called R-standard Hausdorff dimension. The corresponding Hausdorff spectrum, the R-standard Hausdorff spectrum, is denoted as hspecst .For p-adic analytic pro-pgroups, the finiteness of the Hausdorff spectrum with respect to the p-power filtration can be stated differently: Theorem 3.2 (cf. [6, Corollary 1.2] and [27, Corollary 3.4]).Let Gbe a compact p-adic analytic group. Then hspecst(G)is finite and rational. Nevertheless, the situation is radically dissimilar when the base ring is distinct from a finitely generated extension of Zp(recall that an R-analytic group is p-adic analytic if and only if Ris a finitely generated ring extension of Zp). In this chapter, we shall mostly restrict to the case R=Fp[[t]],and the main findings of our investigation can be summarised as follows: Theorem 3.3. Let Gbe a compact Fp[[t]]-analytic group. (i) The standard Hausdorff spectrum of Gcontains the real interval [0,1/dim G]. (ii) If Gis soluble, then hspecst(G) = [0,1]. These suggest that the standard spectrum of an R-analytic group might be sufficient to isolate p-adic analytic groups, as the prior results are consonant with the next conjecture: Conjecture 3.4. Let Gbe a compact R-analytic group such that hspecst(G)is finite. Then Gis p-adic analytic. 3.1 Hausdorff and box dimension In this section, we briefly describe the above-mentioned fractal dimensions. Furthermore, we collect their basic properties and some preliminary results, focusing chiefly on the setting of profinite groups. 3.1.1 Basic definitions and properties Let us shortly present the fractal dimensions alluded to throughout the previous introductory section. Let (M, d)be a metric space, let X⊆Mand let δand zbe positive numbers. We define Hz δ(X) := inf ∞ X n=1 diam(Un)z, 67 where {Un}n∈Nis a δ-covering, namely a covering of Xconsisting of sets of diameter at most δ, and the infimum is taken over all those coverings. Observe that the limit Hz(X) := lim δ→0Hz δ(X) exists, since Hz δ(X)is non-decreasing as δtends to zero. Besides, Hzis an outer measure (see [28, Proposition 11.17]), called the z-Hausdorff measure in M. The following result holds: Lemma 3.5 (cf. [26, Section 3.2]).Suppose that Hs(X)<∞and t≥s. Then Ht(X) = 0. Accordingly, we can define the Hausdorff dimension of Xwith respect to the metric das hdimd(X) := inf {s| Hs(X) = 0}= sup {s| Hs(X) = ∞} –over profinite groups, the metric depends on a filtration series {Gn}n∈N,and consequently, we will use the notation hdim{Gn}–. It is straightforward to verify that the Hausdorff dimension is •monotone, that is, hdimd(X)≤hdimd(Y)whenever X⊆Yand •countably stable, that is, hdimd(∪n∈NXn) = supn∈Nhdimd(Xn) (compare with [26, pp. 48-49]). Furthermore, we highlight the following property: Proposition 3.6 (cf. [26, Proposition 3.3]).Let f: (M1, d1)→(M2, d2)be a bi-Lipschitz map between metric spaces, i.e. there exist two positive constants C, c ∈R≥0such that c·d1(x, y)≤d2(f(x), f(y)) ≤C·d1(x, y)∀x, y ∈M1. Then hdimd2(f(X)) = hdimd1(X)for all X⊆M1.In particular, isometries preserve Hausdorff dimension. The other fractal dimension we will treat with is the Minkowski-Bouligand dimension or the (lower) box dimension. Let Nδ(X)be the minimal number of sets of diameter at most δthat are required to cover X, and define respectively the lower box dimension and the upper box dimension (albeit we will mainly focus on the former) as follows: lbdimd(X) := lim inf δ→0+ log Nδ(X) −log δand ubdimd(X) := lim sup δ→0+ log Nδ(X) −log δ. 68 Observe that these expressions are independent of the base to which we take the logarithm. When these two values coincide, and therefore the underlying sequence of real numbers is convergent, that common value is referred to as bdimd(X),the box dimension of X. In the context of countably based profinite groups, we can obtain a purely group theoretical formula for the above expressions. Indeed, in a profinite group G, the sole values that take the metric defined by using the filtration series {Gn}n∈Nare |G:Gn|−1,and the ball of center xand radius δ=|G:Gn|−1is simply the coclass xGn.Thus, for every X⊆Gwe have Nδ(X) = |XGn:Gn|(this expression stands for the number of cosets of the form xGnfor some x∈X), and consequently the preceding definitions can be rewritten as lbdim{Gn}(X) = lim inf n→∞ log |XGn:Gn| log |G:Gn| and ubdim{Gn}(X) = lim sup n→∞ log |XGn:Gn| log |G:Gn| –we will substitute the subscript with {Gn},as the metric is completely determined by the filtration series–. Moreover, it easily follows from these definitions that both box dimensions are monotone and bi-Lipschitz invariant (see [26, Proposition 2.5]). In addition, the upper box dimension is finitely stable, that is, ubdim(X∪Y) = max{ubdim(X),ubdim(Y)}.However, the lower box dimension might not have this property. In his pioneering work [1], Abercrombrie proved that for closed subgroups –and some filtration series– the Hausdorff and the lower box dimension coincide. In other words, Theorem 3.7 (cf. [6, Theorem 2.4]).Let Gbe a countably based profinite group with normal filtration series {Gn}n∈N,that is, Gn⊴Gfor all n∈N.For every closed subgroup H≤cGwe have hdim{Gn}(H) = lbdim{Gn}(H) = lim inf n→∞ log |HGn:Gn| log |G:Gn|.(3.2) In fact, in the referred work the author proved that lbdim(H)≤hdim(H),as the other inequality is true in any metric space. In accordance with mathematical literature, we will use the name Hausdorff dimension inasmuch as we will mainly be concerned about the dimension of closed subgroups. 69 3.1.2 Formulae: subgroups and quotients Throughout this chapter, relating the Hausdorff dimension of a countably based profinite group to that of its subgroups and quotients will be of vital importance. Therefore, it is sometimes convenient to use the notation hdimG {Gn}to emphasize that the dimension, with respect to the filtration series {Gn}n∈N, is calculated within the group G. Lemma 3.8 (cf. [48, Lemma 5.3]).Let Gbe a countably based profinite group, {Gn}n∈Na normal filtration series of Gand H≤cGa closed subgroup whose Hausdorff dimension is given by a proper limit. Then hdimG {Gn}(K) = hdimG {Gn}(H) hdimH {H∩Gn}(K) for all K≤cH. Remark 3.9. The Hausdorff dimension of Habove being a proper limit means that hdim{Gn}(H) = lim n→∞ log |HGn:Gn| log |G:Gn|. Proof. A simple computation shows that hdimG {Gn}(K) = lim inf n→∞ log |K:K∩Gn| log |G:Gn| = lim n→∞ log |H:H∩Gn| log |G:Gn|lim inf n→∞ log |K:K∩H∩Gn| log |H:H∩Gn| = hdimG {Gn}(H) hdimH {H∩Gn}(K). Moreover, for quotients of countably based profinite groups we have the following result: Lemma 3.10 (cf. [47, Lemma 2.2]).Let Gbe a countably based profinite group, {Gn}n∈Na normal filtration series of Gand N⊴cGa closed normal subgroup. Assume that the Hausdorff dimension of Nis given by a proper limit. Then for every subgroup H≤cGcontaining None has hdimG {Gn}(H) = 1−hdimG {Gn}(N)hdimG/N {GnN/N}(H/N) + hdimG {Gn}(N). 70 Proof. We observe that log |HGn:NGn| log |G:Gn|=log |G:NGn| log |G:Gn|·log |HGn:NGn| log |G:NGn| =log |G:Gn|−log |NGn:Gn| log |G:Gn|·log |HGn:NGn| log |G:NGn| =1−log |NGn:Gn| log |G:Gn|log |HGn:NGn| log |G:NGn|. Therefore, since hdimG {Gn}(N) = ηis given by a proper limit hdimG {Gn}(H) = lim inf n→∞ log |HGn:Gn| log |G:Gn| = lim inf n→∞ log |HGn:NGn| log |G:Gn|+log |NGn:Gn| log |G:Gn| = lim inf n→∞ 1−log |NGn:Gn| log |G:Gn|log |HGn:NGn| log |G:NGn|+η = (1 −η) lim inf n→∞ log |HGn/N :NGn/N| log |G/N :NGn/N|+η = (1 −η) hdimG/N {NGn/N}(H/N) + η, as required. Corollary 3.11. Let Gbe a countably based profinite group with normal filtration series {Gn}n∈Nand let N⊴Gbe a finite normal subgroup. Then hspec{Gn}(G) = hspec{GnN/N}(G/N). Proof. Since hdimG {Gn}(N) = 0 is given by a proper limit, the inclusion hspec{GnN/N}(G/N)⊆hspec{Gn}(G) is a direct consequence of the Correspondence Theorem and Lemma 3.10. For the converse, consider η∈hspec{Gn}(G); then there exists H≤cGsuch that hdimG {Gn}(H) = η. Since Nis finite and the right multiplication is an isometry by Lemma 3.10 one has hdimG {Gn}(H) = hdimG {Gn} [ n∈N Hn! = hdimG {Gn}(HN) = hdimG {GnN/N}(HN/N), as required. 71 Finally, the combination of the above results yields the following corollary. Corollary 3.12. Let Gbe a countably based profinite group, {Gn}n∈Na normal filtration series and let N⊴K≤Gbe closed subgroups such that hdimG {Gn}(N) = η and hdimG {Gn}(K) = κare given by proper limits. If hspec{(K∩Gn)N N}(K/N) = [0,1] then [η, κ]⊆hspec{Gn}(G). Proof. Firstly, by Lemma 3.8 it follows that hdimK {K∩Gn}(N) = η/κ, and using the Correspondence Theorem and Lemma 3.10 we obtain [η/κ, 1] = (1 −η/κ)α+η/κα∈hspecn(K∩Gn)N No(K/N)⊆hspec{K∩Gn}(K). By another application of Lemma 3.8, one concludes [η, κ]⊆hspec{Gn}(G). We conclude this subsection by stating the following result, due to Klopsch, Thillaisundaram and Zugadi-Reizabal in [48], that will be of utility to find profinite groups with full Hausdorff spectrum: Theorem 3.13 (cf. [48, Theorem 5.4]).Let Gbe a countably based pro-pgroup and let {Gn}n∈Nbe a normal filtration series. Suppose that every finitely generated closed subgroup H≤cGsatisfies hdim{Gn}(H) = 0.Then hspec{Gn}(G) = [0,1]. 3.1.3 R-standard Hausdorff dimension In the context of compact R-analytic groups a natural filtration is available. Indeed, let Gbe a compact R-analytic group of dimension dand let (S, φ)be an open R-standard subgroup of level N. As already presented in (1.6), the R-standard filtration series induced by Sis the filtration series {Sn}n∈Ndefined as Sn:= φ−1mN+n(d),∀n∈N0. These are obviously open subgroups and, by virtue of the Krull Intersection Theorem (loc. cit.) an R-standard filtration series is indeed a filtration series. Furthermore, from (1.5) one has that Sn⊴Sfor every n∈N,and thus formula (3.2) holds for R-standard groups with the above filtration. Because of the dependence of hdim on the chosen filtration we should not assume a priori that the Hausdorff dimension (resp. lower box dimension) of a subgroup of a compact R-analytic group is the same when computed with respect to two different R-standard filtrations. To prove this actual independence, we start by recalling this consequence of (1.7): 72 Remark 3.14. The Hilbert function of (R, m)is defined as H:N0→N, n 7→ dimR/m(mn/mn+1),and for large enough nit coincides with a polynomial p(n) of degree dimKrull(R)−1,called the Hilbert polynomial of R(cf. [25, Chapter 6, Theorem C]). Hence, according to the Euler-Maclaurin formula the sum Pn−1 i=1 p(i) is asymptotically equivalent to a polynomial f(n)of degree dimKrull(R),i.e. their ratio tends to 1as ntends to infinity. Let qbe the size of the residue field R/mand let (S, φ)be an R-standard group of dimension dand level N. In view of (1.7), logq|S:Sn|=d N+n−1 X i=N H(i) is asymptotically equivalent to df(n). The following result shows that the lower box dimension is independent of the standard filtration chosen. Lemma 3.15 (cf. [27, Theorem 3.1]).Let Gbe a compact R-analytic group and let (S, φ)and (T, ψ)be two open R-standard subgroups of G. For every X⊆Gwe have that lbdim{Sn}(X) = lbdim{Tn}(X). Proof. Let us denote by N(S)and N(T)respectively the levels of Sand T. Firstly, we shall prove the existence of two integers a, b ∈Nsuch that for every integer n satisfying n−b∈N Sn+a≤Tn≤Sn−b.(3.3) Indeed, since the R-analytic map ψ◦φ−1is convergent in φ(S∩T)⊆mN(S)(d) and since ψ◦φ−1(0) = 0, according to (1.1) there exists L≥N(S)such that ψ◦φ−1mL+n(d)⊆(mn)(d) for any n∈N.Hence, by setting a=L−N(S)+N(T),we obtain that Sn+a≤Tn. Arguing similarly with φ◦ψ−1we obtain (3.3). Consequently, lbdim{Tn}(X) = lim inf n→∞ log |XTn:Tn| log |G:Tn| ≤lim inf n→∞ log |XSn+a:Sn+a| log |G:Sn+a|·log |G:Sn+a| log |G:Tn| = lim inf n→∞ log |XSn+a:Sn+a| log |G:Sn+a|·log |G:Sn+a| log |G:Sn+a|−log |Tn:Sn+a| = lbdim{Sn}(X), 73 using in the ultimate equality that lim n→∞ log |G:Sn+a| log |G:Sn+a|−log |Tn:Sn+a|= 1. Indeed, according to the previous remark log |Tn:Sn+a| ≤ log |Sn−b:Sn+a|= N+n+a−1 X i=N+n−b H(i). Hence, for large enough nthe right hand side term is the sum of a+bpolynomials of degree dimKrull(R)−1,while log |G:Sn+a|= log |G:S|+ log |S:Sn+a| is asymptotically equivalent to a polynomial of degree dimKrull(R).We finish the proof by swapping Sand T. This allows us to define the standard or R-standard lower box dimension, denoted lbdimst,and the R-standard upper box dimension, denoted ubdimst,disregarding the chosen standard filtration. Besides, it is worth noting as an aside that when Ris a PID the analogue of Lemma 3.15 can be proved for the Hausdorff dimension. Lemma 3.16. Let Rbe pro-pdomain that is a PID, let Gbe a compact R-analytic group and let (S, φ)and (T, ψ)be two open R-standard subgroups of G. For every X⊆G, we have hdim{Sn}(X) = hdim{Tn}(X). Proof. Let qbe the size of the residue field R/mand d= dim G. Since Ris a PID, |Sn:Sn+1|=|Tn:Tn+1|=qdfor every n∈N.From (3.3) there exist two integers a, b ∈Nsuch that Sn+a≤Tn≤Sn−bfor every integer nsuch that n−b∈N. Thus, |G:Tn|−1≥ |G:Sn+a|−1=q−d(a+b)|G:Sn−b|−1. Hence, if we denote by δSand δTthe distances in Ginduced respectively by the filtration series {Sn}n∈Nand {Tn}n∈N,then δT(x, y)≥q−d(a+b)δS(x, y). By swapping Sand Twe obtain the existence of a constant C > 0such that C·δS(x, y)≥δT(x, y).Hence, the identity map between the metric spaces (G, δS) and (G, δT)is bi-Lipschitz, and the result follows by Proposition 3.6. 74 where k= dimxM. Reproducing the arguments of (3.6) verbatim, if δSis the distance in Ginduced by S, we obtain that hdimδS(Ux) = dimxM dim G. Let (T, ψ)be another open R-standard subgroup of Gand let δTbe the distance defined in Gby using T. On the one hand, from [26, Proposition 3.4] and (3.6), hdimδT(Ux)≤ubdimst(Ux) = dimxM dim G. Let us define ˜ Ux=x−1Ux∩T, so in particular ψ(˜ Ux)⊆ψ(S∩T).Then φ(˜ Ux) = φ(x−1Ux∩T) = mN(k)×{0}(d−k)∩φ(S∩T), and since φ(S∩T)is open in m(d),there exists K∈Nsuch that mK(k)×{0}(d−k)⊆φ(˜ Ux). Let us denote by dthe distance defined on m(d)using the standard filtration series n(mn)(d)on∈N.Since φand ψare isometries, then φ◦ψ−1is also an isometry from (ψ(S∩T), d)to (φ(S∩T), d).Hence, in view of Lemma 3.17, Proposition 3.6 and the monotonicity, hdimG δT(Ux)≥hdimT δT˜ Ux= hdimm(d) δψ(˜ Ux)= hdimm(d) δφ◦ψ−1◦ψ(˜ Ux) = hdimm(d) δ(φ(˜ Ux)) ≥hdimm(d) {(mn)(d)}mK(k)×{0}(d−k)=k/d, using Lemma 3.23 in the last equality. Therefore, hdim(Ux) = dimxM/dim G,unregarding the standard filtration chosen. We finish as in (3.7) and (3.8), but replacing the box dimension with the Hausdorff dimension. Considering that R-analytic subgroups are closed pure R-analytic submanifolds, we recover the principal result in [27]: Corollary 3.27 (cf. [27, Main Theorem]).Let Gbe a compact R-analytic group and let Hbe an R-analytic subgroup. Then bdimst(H) = hdimst(H) = dim H dim G. In particular, both dimensions are a proper limit. 81 When R=Zpevery closed subgroup is p-adic analytic (see [24, Theorem 9.6]), so we obtain an alternative proof of Theorem 3.2, and in passing we get a clear-cut expression for standard spectra in this setting. In fact, if Gis a d-dimensional compact p-adic analytic group, then hspecst(G)⊆0,1 d, . . . , d−1 d,1. 3.3 Abelian compact R-analytic groups Henceforth, we will carry on describing the standard spectra of profinite R-analytic groups that are not p-adic analytic, i.e. Ris not a finite extension of Zp.In the first place, we will deal with the abelian case. Remark. From now on we will work with closed subgroups, and Theorem 3.7 will be implicitly used. Proposition 3.28. Let Rbe a pro-pdomain of characteristic por Krull dimension at least 2,and let (S, φ)be an abelian R-standard group. Then hspecst(S) = [0,1]. Proof. By Theorem 3.13 it suffices to prove that every finitely generated closed subgroup H≤cSsatisfies hdimst(H) = 0.Let dbe the dimension of Sand let H≤Sbe a topologically r-generated closed subgroup. If Rhas characteristic p, since the group operation in Sis given by a formal group law, by (1.3) whenever x∈Snwe have φ(xp)≡pφ(x) = 0mod m2n(d), and thus xp≡1 (mod S2n).Therefore Sn/S2nis an elementary abelian p-group. Since Sis abelian, H/(H∩Sn)is an abelian p-group of exponent pewhere e≤ dlog2(n)e.Moreover, His topologically r-generated, so H/(H∩Sn)is rgenerated, and thus |H:H∩Sn| ≤ per ≤p⌈log2(n)⌉r. According to Remark 3.14, if |R/m|=q=pc,then |S:Sn|is asymptotically equivalent to qdf(n)where f(n)is a polynomial of degree dimKrull(R).Consequently, hdimst(H) = lim inf n→∞ logp|H:H∩Sn| logp|S:Sn|≤lim inf n→∞ rdlog2(n)e cdf(n)= 0, as desired. 82 Similarly, if Rhas Krull dimension of at least 2,by (1.3), whenever x∈Snwe have φ(xp)≡pφ(x)≡0mod mn+1(d), so Sn/Sn+1 is an elementary abelian p-group. Consequently, H/(H∩Sn)is an r-generated abelian group of exponent pe,where e≤n−1.Therefore |H:H∩Sn| ≤ pre ≤pr(n−1), so, according to Remark 3.14, hdimst(H) = lim inf n→∞ logp|H:H∩Sn| logp|S:Sn|≤lim inf n→∞ r(n−1) cdf(n)= 0, as f(n)is a polynomial of degree dimKrull(R)≥2. Clearly, in view of of Corollary 3.20, this result can be generalised to abelian compact R-analytic groups. Corollary 3.29. Let Rbe a pro-pdomain of characteristic por Krull dimension at least 2.If Gis an abelian compact R-analytic group, then hspecst(G) = [0,1]. Furthermore, it is known that any R-standard group of dimension one is abelian (see [35, Theorem 1.6.7]), and we thus have the following: Corollary 3.30. Let Rbe a pro-pdomain of characteristic por Krull dimension at least 2and let Gbe a compact R-analytic group of dimension one. Then hspecst(G) = [0,1]. 3.4 Compact Fp[[t]]-analytic groups Section 3.3 invites to surmise that when Gis a soluble compact R-analytic group that is not p-adic analytic, its R-standard spectrum is the whole real interval [0,1]. The main strategy to prove that would lie in adding successive intervals to the spectrum, using the consecutive abelian quotients of a subnormal series. In fact, we have the following result: Lemma 3.31. Let Gbe a compact R-analytic group and let N⊴K≤Gbe R-analytic subgroups such that hspecst(K/N) = [0,1].Then dim N dim G,dim K dim G= [hdimst(N),hdimst(K)] ⊆hspecst(G). 83 Proof. The equality is a direct consequence of Corollary 3.27, namely hdimst(H) = dim H/dim Gfor every analytic subgroup H≤Gand such dimension is given by a proper limit, and the inclusion is straightforward from Corollary 3.12, Lemma 3.21 and Lemma 3.22. Thus, we shall establish a useful criterion for finding R-analytic subgroups of a compact R-analytic group. The main obstacle compared with classical Lie theory arises here: it is well known that any closed subgroup of a real (p-adic) Lie group is a real (p-adic) Lie subgroup; nevertheless for R-analytic groups, closeness is a necessary condition (see Lemma 1.57), but not sufficient. For example, the additive group Fp[[t]] is an Fp[[t]]-analytic group and Fp[[t2]] is a closed subgroup with its own Fp[[t]]-analytic group structure. However, those manifold structures are not compatible, so Fp[[t2]] is not an Fp[[t]]-analytic subgroup of Fp[[t]]. Now we will turn to the case when R=Fp[[t]].The task of finding Fp[[t]]- analytic subgroups can be carried out by using Proposition 1.53, which shows that analytic subsets have a manifold structure over Fp[[t]].According to the definition therein, a set X⊆Mis an analytic subset if for each x∈Xthere exist an open neighbourhood Uof xand some Fp[[t]]-analytic functions f1, . . . , frdefined on U (for some r=rx) such that X∩U={y∈U|fi(y) = 0 ∀i= 1, . . . , r}. We then have: Theorem 3.32 (cf. [45, Corollary 4.2]).Let Gbe an Fp[[t]]-analytic group and let Hbe both a subgroup of Gand an analytic subset of G. Then His an Fp[[t]]- analytic subgroup of G. Let us see some examples of applications of the preceding theorem: Corollary 3.33. Let Sbe an Fp[[t]]-standard group and ain S. Then Z(S)and CS(a)are Fp[[t]]-analytic subgroups. Proof. By the previous theorem it is enough to show that Z(S)and CS(a)are analytic subsets. The former is proved in [45, Corollary 4.3], while the latter follows the same spirit. Indeed, since Sis Fp[[t]]-standard of level say Nand dimension say d, then it can be identified with tN(d).Since the group operation is given by a formal group law, by (1.5) there exist some gi,α ∈Fp[[t]][[X1, . . . , Xd]] such that πiy−1ay=ai+X |α|≥1 gi,α(a)yα1 1. . . yαd d=ai+hi(y) 84 for all yin S, where the map πi:tN(d)→tNis the projection to the ith coordinate. Moreover, the maps hi(y) = P|α|≥1gi,α(a)yα1 1. . . yαd dare clearly Fp[[t]]- analytic. Therefore CS(a) = y∈S|πiy−1ay=ai∀i= 1, . . . , d ={y∈S|hi(y) = 0 ∀i= 1, . . . , d}, and CS(a)is an analytic subset. The second application involves the general linear group GLn(R).In addition to the usual topology in GLn(R),namely the one induced by the ring topology of R, we also have the so-called Zariski topology, in which the closed subsets are the affine sets, i.e. subsets of the form {A∈GLn(R)|f(A) = 0 ∀f∈ F}, where F ⊆ R[X]is a subset of polynomials in n2variables. Note as well that any subgroup H≤GLn(R)can be likewise endowed with both the usual subspace topology or the weaker (polynomial maps are continuous with respect to the madic topology) Zariski topology. Let us present some general facts concerning the Zariski topology: Proposition 3.34 (cf. [73, Lemma 5.9 and Theorem 5.11]).Let H≤GLn(R) and let Hbe its Zariski closure. (i) Then H ≤ GLn(R). (ii) If His normal in GLn(R),so is H. (iii) Suppose that His nilpotent of class c. Then, Hhas a central series of length cconsisting of Zariski closed subgroups. In particular, His nilpotent of class c. (iv Suppose that His soluble of length c. Then Hhas a subnormal series of length cconsisting of Zariski closed subgroups whose quotient groups are abelian. In particular, His soluble of length c. (iv) Let Kbe a subgroup of Hsuch that |H:K|is finite and let Kbe the Zariski closure of Kin GLn(R).Then Khas finite index in H. Corollary 3.35. Let G⊆GLn(Fp[[t]]) be a linear Fp[[t]]-analytic group and let Hbe a Zariski closed subgroup of GLn(Fp[[t]]).Then H∩Gis an Fp[[t]]-analytic subgroup of G. 85 Proof. Since His closed in the Zariski topology, it is an affine set, that is, there exists a subset Fof Fp[[t]][X],where Xis a tuple of n2variables, such that H={A∈GLn(Fp[[t]]) |f(A) = 0 ∀f∈ F}. But since Fp[[t]][X]is Noetherian we can assume Fto be finite, and thus H∩G={A∈G|f(A) = 0 ∀f∈ F} is an analytic subset, so it is an Fp[[t]]-analytic subgroup by Theorem 3.32. We are now in a position to prove part of Theorem 3.3 by using the previous results: Theorem 3.36. Let Gbe a soluble compact Fp[[t]]-analytic group. Then, hspecst(G) = [0,1]. Proof. By Corollary 3.20, we can assume without loss of generality that Gis Fp[[t]]-standard. We first prove the theorem for the case when Gis linear over Fp[[t]],that is, G⊆GLn(Fp[[t]]).Let Gbe the Zariski closure of Gin GLn(Fp[[t]]). According to Proposition 3.34, Gis a soluble group, and there exists a subnormal series G=H1⊵H2⊵···⊵Hℓ−1⊵Hℓ={1} consisting of Zariski closed subgroups whose quotient groups are all abelian. Then G=H1∩G⊵H2∩G⊵···⊵Hℓ−1∩G⊵Hℓ∩G={1} is a soluble series of Ggiven by Fp[[t]]-analytic subgroups by Corollary 3.35. Denote Hi=Hi∩G. Since each Hiis an Fp[[t]]-analytic subgroup of Gthen Hi−1/Hiis a compact abelian Fp[[t]]-analytic group for all i∈ {2, . . . , `}, so by Corollary 3.29 it follows that hspecst(Hi/Hi−1) = [0,1].Hence by Lemma 3.31 one has that [hdimst(Hi),hdimst(Hi−1)] ⊆hspecst(G)for all i∈ {2, . . . , `},and thus hspecst(G) = [0,1]. Let us finally turn to the general case. By Corollary 3.33, Z(G)is an abelian Fp[[t]]-analytic subgroup of Gand thus by Corollary 3.29 and Lemma 3.31 [0,hdimst Z(G)] ⊆hspecst(G). 86 Moreover, by Propositions 1.59 and 2.3 one has that G/Z(G)is a compact soluble Fp[[t]]-analytic group that is linear over Fp[[t]]. Hence, according to Lemmata 3.21 and 3.22, hspec{SnZ(G)/Z(G)}(G/Z(G)) = hspecst (G/Z(G)) = [0,1], and so by Corollary 3.12 [hdimst Z(G),1] ⊆hspecst(G), thus obtaining the whole interval in the spectrum. More generally, a suitable way to find an interval in the Fp[[t]]-standard Hausdorff spectrum of a compact Fp[[t]]-analytic group Gis looking for a soluble Fp[[t]]- analytic subgroup. This search will rely heavily on the topological analogue of the Tits alternative. But we first observe the following: Lemma 3.37. Let Gbe an Fp[[t]]-standard group. Suppose that either (i) Z(G)is infinite or (ii) Gcontains an element xof infinite order. Then [0,1/dim G]⊆hspec(G). Proof. Under the first hypothesis, by Corollary 3.33, Z(G)is an abelian infinite Fp[[t]]-analytic subgroup. Similarly, under the second hypothesis Z(CG(x)) is an abelian Fp[[t]]-analytic subgroup which is infinite, because hxi ≤ Z(CG(x)).In both cases, there exists an infinite abelian Fp[[t]]-analytic subgroup H≤G. Since Gis compact, Hhas strictly positive analytic dimension, and according to Corollary 3.29 the Fp[[t]]-standard spectrum of His the whole interval [0,1].Finally, by Lemma 3.31, [0,dim H/dim G]is contained in the Fp[[t]]-standard spectrum. Theorem 3.38. Let Gbe a compact Fp[[t]]-analytic group. Then, [0,1/dim G]⊆hspecst(G). Proof. We can assume, in view of Corollary 3.20, that Gis R-standard. Firstly, observe that when Z(G)is infinite the result follows by Lemma 3.37(i), so we shall deal with the case when Z(G)is finite. But then G/Z(G)is an Fp[[t]]-analytic group of dimension dim Gand according to Corollary 3.11 it follows that hspecst(G) = hspecst(G/Z(G)). 87 Furthermore, by Proposition 2.3, G/Z(G)is an Fp[[t]]-analytic group that is linear over Fp[[t]].Hence by the topological Tits alternative (cf. [12, Theorem 1.3]) it follows that G/Z(G)contains either an open soluble subgroup, say H, or contains a dense free subgroup. In the former case, His a soluble Fp[[t]]-analytic group of dimension dim G/Z(G) = dim Gand thus hspecst(G/Z(G)) = [0,1] . In the latter case G/Z(G)contains an element of infinite order and the statement follows by Lemma 3.37(ii). 3.5 Classical Chevalley groups The spectrum of a compact Fp[[t]]-analytic group need not be the whole interval [0,1].For instance, consider the special linear group SLn(Fp[[t]]).It is well-known that SLn(Fp[[t]]) is a compact Fp[[t]]-analytic group of dimension n2−1,containing as an open subgroup the Fp[[t]]-standard group SL1 n(Fp[[t]]) := ker {SLn(Fp[[t]] →SLn(Fp[[t]]/tFp[[t]])}. In [6, Corollary 1.5], the standard spectrum of SL2(Fp[[t]]) is completely established when p > 2,to wit hspecst (SL2(Fp[[t]])) = [0,2/3]∪{1}. Moreover, in [6, Theorem 1.4] it is proved that when p > 2, hspecst (SLn(Fp[[t]])) ∩1−1 n+ 1,1=∅, and 1is an isolated point of the spectrum thereof. We will provide further examples of compact Fp[[t]]-analytic groups whose spectrum is not the whole interval, by proving an analogous result for the other classical Chevalley groups. For that purpose, we will follow the same techniques already used in [6] and work in the corresponding graded Lie algebra. We start with a brief summary of those matrix groups. For basic definitions regarding root systems and a comprehensive analysis on the topic we refer to [15]. Let Rbe a general pro-pdomain. • The Chevalley group over Rassociated to a root system of type An(n≥1) is SLn+1(R). 88 • A root system of type Bn(n≥2) defines the odd special orthogonal group SO2n+1(R) := A∈M2n+1(R)AtK2n+1A=K2n+1, where Kn=     0. . . 0 1 0. . . 1 0 . . . ... . . .. . . 1. . . 0 0     ∈Mn(R),which is an R-analytic group of dimension n(2n+ 1). • A root system of type Cn(n≥3) defines the symplectic group Sp2n(R) := A∈M2n(R)|AtJ2nA=J2n, where J2n=0Kn −Kn0,which is an R-analytic group of dimension n(2n+ 1). • A root system of type Dn(n≥4) defines the even special orthogonal group SO2n(R) := A∈M2n(R)|AtK2nA=K2n, which is an R-analytic group of dimension n(2n−1). All those groups are compact, being closed subsets of the compact space Mn(R) ∼ =R(n2).That is, classical Chevalley groups over Rare actually compact R-analytic groups. Furthermore, the following result describes their associated Lie algebras. Theorem 3.39 (cf. [24, Exercise 13.11(iii)]).Let Xnbe a root system of type An (n≥1), Bn(n≥2), Cn(n≥3) or Dn(n≥4). Let G(R)be the Chevalley group associated to Xnover a pro-pdomain R. (i) If Xn=An,there exists an open R-standard group Ssuch that L(S)∼ =sln+1(R) = {A∈Mn+1(R)|tr(A) = 0}. (ii) If Xn=Bn,there exists an open R-standard subgroup Ssuch that L(S)∼ =so2n+1(R) = A∈M2n+1(R)|At=−A. 89 (iii) If Xn=Cn,there exists an open R-standard subgroup Ssuch that L(S)∼ =sp2n(R) = A∈M2n(R)|J2nA+AtJ2n= 0, where J2nis defined as before. (iv) If Xn=Dn,there exists an open R-standard subgroup Ssuch that L(S)∼ =so2n(R) = A∈M2n(R)|At=−A. Here L(S)is an abbreviation for the Lie algebra associated to S(compare with Section 1.4). We start by presenting the construction in [54, Definition 2.9]. Given an Rstandard group (S, φ)and the corresponding R-standard filtration {Sn}n∈Nwe define the graded Lie R/m-algebra grL(S) = Ln≥0Sn/Sn+1,which is so with the Lie bracket obtained extending by bilinearity the rule [xSn+1, ySm+1]grL(S):= [x, y]Sn+m+1 (the right-hand side brackets stand for the group commutator in S). On the one hand, [,]grL(S)is a Lie bracket. Indeed, from (1.5), φ([x, y]) = B(φ(x), φ(y)) −B(φ(y), φ(x)) mod φ(Sn+m+1), and therefore [·,·]grL(S)satisfies Jacobi’s identity by virtue of Lemma 1.25. On the other hand, whenever x∈Snand y∈Sm,then [x, y]∈Sn+m,so the above-defined algebra is graded over the natural numbers. Finally, let qbe the cardinality of R/m.Since each Sn/Sn+1 is an R/m-vector space, grL(S)is a graded Fq-Lie algebra. Any closed subgroup H≤cSdefines a graded subalgebra of grL(S), which by abuse of notation we will denote by grL(H)and is given by grL(H) := M n≥0 (H∩Sn)Sn+1 Sn+1 . Although every closed subgroup defines a graded subalgebra, there might be graded subalgebras that do not arise in this way. 90 Examples 4.5. Let Gbe a group. (i) Both Gitself and the trivial subgroup {1}are verbal subgroups corresponding, respectively, to the word w(x) = xand the empty word. (ii) One of the most common words is the commutator γ2(x, y)=[x, y] = x−1y−1xy. Its verbal subgroup is the derived subgroup γ2(G) = G′and its marginal subgroup is the centre γ∗ 2(G) = Z(G). (iii) Lower central words are defined recursively as γn(x1, . . . , xn) := [γn−1(x1, . . . , xn−1), xn]∀n≥3, and derived words are defined recursively as δ1(x1, x2) := γ(x1, x2)and δn(x1, . . . , x2n) := [δn−1(x1, . . . , x2n−1), δn−1(x2n−1+1, . . . , x2n)] ∀n≥2. (iv) The Burnside word wm(x) = xmdefines the verbal subgroup Gm,the subgroup generated by the mth powers of elements of G, and for instance, the marginal subgroup w∗ 2(G)consists on all central elements of order dividing 2, that is, w∗ 2(G) = {g∈Z(G)|g2= 1}. P. Hall [33] posed several questions regarding the relation between the set of w-values and the corresponding verbal and marginal subgroups. The following definitions will serve to summarise some of those: Definition 4.6. Let wbe a word and let Cbe a class of groups. (i) A word wis concise in Cif for every G∈ C we have that w(G)is finite whenever w{G}is finite. (ii) A word wis robust in Cif for every G∈ C we have that w(G)is finite whenever |G:w∗(G)|is finite. Consonantly, wis concise (resp. robust) if it is concise (resp. robust) in the class of all groups. Likewise, if every word is concise in a group G, we will say that G is verbally concise. In general, since |w{G}| ≤ |G:w∗(G)|k(being kthe number of variables of the word w), robustness is stronger than conciseness: if wis concise in C,then w is robust in C.However, over residually finite groups, and all the groups we are concerned about are so, both concepts are equivalent: 97 Lemma 4.7 (cf. [67, Lemma 1.4.1]).Let Gbe a group and let wbe a word. (i) If |G:w∗(G)|is finite, then w{G}is finite. (ii) If Gis residually finite and w{G}is finite, then |G:w∗(G)|is finite. One of the original predictions of P. Hall was that all groups were verbally concise. However, this conjecture was refuted almost three decades later by Ivanov [39], by finding a group Gand a word wsuch that w{G}has two elements, but w(G)is infinite cyclic. Nonetheless, this counterexample, or the comparable counterexample constructed by Ol’shanskiĭ (see [62, Theorem 39.7]), is not residually finite. This leads us to the following conjecture, proposed by Jaikin-Zapirain [44] and Segal [67]: Conjecture 4.8 (Conciseness conjecture for residually finite groups).Every word is concise in the class of residually finite groups. There are few known examples of classes of verbally concise groups. Apart from the obvious examples of abelian (see Lemma 4.2) and periodic (see upcoming Lemma 4.15) groups; in the decade of 1960s, Merzjalkov [57] and Turner-Smith [72] proved, respectively, that linear groups and groups all of whose quotients are residually finite (e.g. virtually nilpotent groups) are verbally concise. When the set of w-values is infinite, the comparable notion to that of conciseness is verbal ellipticity. In order to define it, we will use the following notation: for a subset X⊆G, we denote by X∗ℓthe set of all products of at most `elements of Xand their inverses. Definition 4.9. Let wbe a word and let Gbe a group. We say that wis elliptic in G, if there exists `∈Nsuch that w(G) = w{G}∗ℓ. The smallest of such integers `is the verbal width of win G. In consonance, a group Gis verbally elliptic when every word is elliptic in G. Moreover, ellipticity is stronger than conciseness: whenever wis elliptic in all the groups of a class C, then wis also concise in C. It follows from Lemma 4.2 that abelian groups are verbally elliptic and in them all words have verbal width equal to 1. Aside from them, it is known that linear algebraic groups∗(cf. [56]), finitely generated virtually abelian-by-nilpotent ∗by a linear algebraic group we mean a Zariski closed subgroup of GLn(K)for some n∈N and an algebraically closed field K. 98 groups (cf. [29] and [71]) or, directly related to the topic of this thesis, compact p-adic analytic groups (cf. [44]) are verbally elliptic. Nonetheless, there are natural examples of non-verbally elliptic groups. For instance, Roman’kov [66] presented a finitely generated soluble pro-pgroup where the second derived word δ2(x1, . . . , x4) = [[x1, x2],[x3, x4]] has infinite width. Regarding profinite groups, verbal width is related to whether the corresponding verbal subgroup is closed. Proposition 4.10. Let Gbe a compact Hausdorff topological group and let wbe a word. Then wis elliptic in Gif and only if w(G)is closed. Proof. For the only if implication, note that for every integer nthe set w{G}∗n is closed, as it is the continuous image of a compact set. Thus, if `is the verbal width of w, then w(G) = w{G}∗ℓis closed. For the if, note that w(G) = [ n∈N w{G}∗n where each w{G}∗nis closed. Therefore, since w(G)is a compact Hausdorff topological space, by the Baire Category Theorem (cf. [59, Theorem 48.2]), there exists an integer msuch that w{G}∗mhas non-empty interior in w(G), i.e. it contains a non-empty open subset U⊆ow(G).Hence, w(G) = [ g∈w(G) gU, and by the compactness of w(G) w(G) = r [ i=1 giU, for some elements g1, . . . , gr∈w(G).Take k∈Nsuch that gi∈w{G}∗kfor all i∈ {1, . . . , r},then w(G) = r [ i=1 giU⊆w{G}∗(k+m), as desired. 99 Generally, knowing that a (verbal) subgroup is closed can be helpful when working with profinite groups. That is why we should finish this introduction by pointing out the following critical result due to Jaikin-Zapirain. Theorem 4.11 (cf. [44, Theorem 1.1]).Let w∈Fkbe a word in kvariables. Then whas finite width in all finitely generated pro-pgroups if and only if w /∈ δ2(Fk) (F′ k)p. This chapter is devoted to proving that compact R-analytic groups are verbally concise. It is worth mentioning that when char R= 0,this result is a direct consequence of Theorem 2.27 –every compact R-analytic group is linear– together with Merzjalkov’s Theorem –linear groups are verbally concise–. But in spite of that, it is interesting to provide an independent proof, which is valid for any pro-p domain regardless of the characteristic. Furthermore, this general result provides yet another evidence for a positive answer to the question of whether compact analytic groups are linear in positive characteristic. 4.1 Conciseness in R-standard groups In the class of R-standard groups, conciseness is straightforward: Proposition 4.12. Let Sbe an R-standard group and let wbe a word such that w{S}is finite. Then w(S) = {1}. Proof. Firstly, Scan be identified with mN(d),where Nis the level and dthe dimension of S, such that the multiplication and the inversion are defined by two tuples of power series and the identity is 0.Consequently, the word map wis a single power series W∈R[[X1, . . . , Xdk]](d),where kis the number of variables of the word. Since w{S}is finite and the word map is continuous, Wis locally constant, so by Lemma 1.8 Wis constant, i.e W(X1, . . . , Xdk) = W(0, . . . , 0) = 0, and thus w{S}={0}. We should take notice of a couple of consequences of the previous result. On the one hand, any R-analytic group Gsatisfies a weaker version of the conciseness conjecture: Corollary 4.13. Let Gbe an R-analytic group and let wbe a word such that w{G}is finite. There exists an open R-standard subgroup Swhere wis a law, that is, w(S) = {1}. 100 Proof. According Lemma 1.21, there exists an open R-standard subgroup Sof G. Since |w{S}| ≤ |w{G}|, from Proposition 4.12 it follows that w(S) = {1}. On the other hand, if Gis a compact R-analytic group such that w{G}is finite, we can obtain the set of w-values just by looking at a transversal of a convenient R-standard subgroup. Indeed, let Gbe a compact R-analytic group, S⊴oGan open normal R-standard subgroup whose conjugation maps are strictly analytic, which exists by Lemma 1.23 (provided that Ris not a PID), and let Tbe a left transversal for Sin G. We should bring the atlas induced by Sto mind, namely the atlas {(tS, φt)}t∈Twhere φt(x) = φ(t−1x).As a consequence of Lemma 1.24, the R-analytic word map w:G(k)→G, which is nothing but an adequate composition of multiplication and inversion maps, is given by a single tuple of power series on the open subset t1S×···×tkS(ti∈T), i.e. there exists a tuple of power series Wt1,...,tk∈R[[X1, . . . , Xdk]](d)such that φp(w(x1, . . . , xk)) = Wt1,...,tk(φt1(x1), . . . , φtk(xk)) ∀xj∈tjS, (4.2) where pis the element of Tsuch that w(t1, . . . , tk)p−1∈S. If we further assume that w{G}is finite, the continuous map wis locally constant, so by Lemma 1.8, Wt1,...,tkis constant, i.e. Wt1,...,tk(X1, . . . , Xdk) = c∈R(d). That is, φp(w(x1, . . . , xk)) = cfor all xj∈tjS. In other words, Proposition 4.14. Let Gbe a compact R-analytic group, let wbe a word and let Sbe an open normal R-standard subgroup whose conjugation maps are strictly analytic. If w{G}is finite, then Sis marginal for w. 4.2 Conciseness in compact Fp[[t]]-analytic groups The demonstration technique will consist in reducing the problem to a group that is analytic over a pro-pdomain of Krull dimension one, by using the change of rings described in Section 2.2. Hence, firstly we shall deal with the 1-dimensional case. Compact p-adic analytic groups are verbally concise, as they are linear by virtue of Corollary 2.6. Hence, in view of Corollary 1.44 we can restrict to the case R=Fp[[t]]. We shall proceed via several technical results. Some of them will be proved, whilst others will be simply stated. We start with the following ”folklore” result: 101 Lemma 4.15. Let Gbe a group and let wbe a word such that w{G}is finite. Then w(G)′is finite, and w(G)is finite if and only every w-value in Ghas finite order. Proof. Let g∈G. By Lemma 4.3, w{G}g⊆w{G},i.e. for all x∈w{G}the conjugacy class xGis contained in w{G}.Hence |G:CG(x)|=wG≤ |x{G}|, so CG(x)has finite index in G. Therefore, CG(w(G)) = ∩x∈w{G}CG(x)has finite index in G, and so |w(G) : Z(w(G))|is also finite. Thus, by Schur’s Theorem (cf. [65, Theorem 10.1.4]), w(G)′is finite. Finally, if w(G)is finite, it must have finite exponent. Conversely, suppose that the elements of w{G}have finite order, then w(G)/w(G)′is an abelian group finitely generated by elements of finite order, in particular, it is finite; and the finiteness of w(G)′yields the result. Moreover, we will need the following Schur-type result: Lemma 4.16 (cf. [45, Proposition 5.1]).Let Gbe a group with a nilpotent normal subgroup N. Suppose that NZ(G) Z(G)has finite exponent. Then [N, G]has finite exponent. Proof. The Hall-Petrescu formula (see [38, III.9.4]) states that for all m∈N, xmym= (xy)mc2(x, y)(m 2). . . cm(x, y)(m m),(4.3) where cr(x, y)∈γr(hx, yi). Let mbe the exponent of NZ(G) Z(G),according to (4.3), for all n∈Nand g∈G: [n, g]m≡n−m(n[n, g])m=n−m(ng)m= [nm, g] = 1 mod γ2(K),(4.4) where K:= hn, [n, g]i ≤ N. Moreover, by (4.3) and (4.4), for every l∈Nwe have ([n1, g1]. . . [nl, gl])m≡[n1, g1]m. . . [nl, gl]m≡1 mod γ2(N). Let η(m)be†the order of the largest 2-generated nilpotent group of exponent m. It suffices to prove that γ2(N)η(m)={1}.For that, let x, y ∈Nand H= hx, yi ≤ N. Since H/Z(H)is a 2-generated nilpotent group of exponent dividing †This integer exists because, as Baer [4] proved, nilpotent groups satisfy the Burnside problem (see [20, Theorem 2.23]). 102 m, it is finite and k=|H:Z(H)|divides η(m).Moreover, θ:H→Z(H),h7→ hk is the transfer of Hinto Z(H)(compare with the proof of [65, Theorem 10.1.3]). In particular, θis a homomorphism, so (xy)k=xkyk.Since xkand ykcommute and kdivides η(m),we have that (xy)η(m)=xη(m)yη(m)∀x, y ∈N. In particular, θ′:N→Z(N), n 7→ nη(m),is a homomorphism, and since im θ′ is abelian, γ2(N)must be contained in ker θ′, that is, γ2(N)η(m)={1}. The upcoming auxiliary results use ideas from the theory of linear algebraic groups. The reader is directed to [37] for a thorough account of the theory behind these results. Let Kbe an algebraically closed field. For our purposes a linear algebraic group will be a closed subgroup Gof GLn(K)endowed with the Zariski topology, and the identity component of Gis the connected component of the identity. Proposition 4.17 (cf. [37, Proposition 7.3]).Let Gbe a linear algebraic group. (i) The identity component of Gis a normal subgroup of finite index. (ii) Let H ≤ G be a closed connected subgroup of finite index, then H=G◦. The identity component of a linear algebraic group Gis the unique normal closed connected subgroup of finite index in G. Furthermore, a matrix is unipotent if its unique eigenvalue is 1,and a subgroup of GLn(K)is a unipotent subgroup if all its elements are unipotent. The key structural result about unipotent groups is that any unipotent subgroup is a conjugate of a subgroup of Un(K),the group of upper triangular matrices with 1’s along the diagonal (see [37, Corollary 17.5]). In particular, every unipotent group Gis nilpotent, and when the base field is of positive characteristic, Ghas finite exponent, compare with [38, Chapter III, Lemma 16.2 and Theorem 16.5] (although in this reference the base field is finite, the arguments are still valid for fields of positive characteristic). Additionally, given an arbitrary linear algebraic group G,the unipotent radical of G, denoted Ru(G),is the subgroup consisting of all the unipotent elements of G,and it is also characterised as the largest connected unipotent subgroup of G. Since Ru(G)is connected and nilpotent, it is contained in the soluble radical of G, namely the identity component of the largest soluble subgroup of G.A non-trivial 103 linear algebraic group Gis said to be reductive, when it is connected and Ru(G) is trivial. Although all those constructions play an important rôle in the theory of algebraic groups, we have just summarised the definitions and the relations between them, considering that we will simply use the following technical result: Proposition 4.18 (cf. [37, Lemma 17.9]).Let Gbe a connected linear algebraic group and let Nbe its soluble radical. Then [N,G]is unipotent. Proof. Let Rube the unipotent radical of G.Then G/Ruis reductive, so according to [37, Lemma 17.9], N/Ru⊆Z(G/Ru),and therefore [N,G]⊆ Ru. Now, we can prove the desired result: Theorem 4.19. Compact Fp[[t]]-analytic groups are verbally concise. Proof. Let Gbe a compact Fp[[t]]-analytic group and let wbe a word such that w{G}is finite. Firstly, by Lemma 4.15, w(G)′is finite, and thus up to a quotient we can assume that w(G)is a finitely generated abelian subgroup. According to Corollary 4.13, there exists an open Fp[[t]]-standard group Swhere wis a law. For abbreviation, Zstands for Z(S)and Kfor the algebraic closure of the local field Fp((t)).By Proposition 2.3, S/Z is a linear group over Fp[[t]],so it is also linear over the fields Fp((t)) and K. According to the topological Tits alternative (loc. cit.) S/Z contains either an open soluble subgroup or a dense free subgroup. But S/Z satisfies an identity, so it must be virtually soluble. Let Sbe the Zariski closure of S/Z in GLn(K),which is also virtually soluble by Proposition 3.34. Let Nbe the largest normal soluble subgroup of Sand N◦its identity component, i.e. the soluble radical of the algebraic group S.In view of Proposition 4.17(i), N◦has finite index in S, so according to Proposition 4.17(ii), N◦=S◦.Let N/Z be the intersection of S/Z with N◦,then, passing to the normal core if necessary, we can assume that Nis a normal subgroup of finite index in G. Besides, according to Proposition 4.18, [N◦,N◦]is unipotent. In particular, [N◦,N◦]is nilpotent and, since Khas characteristic p, it has finite exponent. Therefore, [N, N]Z/Z is nilpotent of finite exponent. Thus [N, N]Zis nilpotent and according to Lemma 4.16, H:= [N, N, S]has finite exponent. On the one hand, H= [N, N, S]≥[N, N, N], and so G/H is virtually nilpotent of class at most 2.Since |w{G/H}| ≤ |w{G}| and G/H is virtually nilpotent, we conclude that w(G/H)is finite by TurnerSmith’s Theorem (cf. [72, Corollary 2]). 104 On the other hand, w(G)∩His a finitely generated abelian group of finite exponent, so it is finite. Finally, the isomorphism wG H=w(G)H H∼ =w(G) w(G)∩H yields the result. Corollary 4.20. Let Rbe a pro-pdomain of Krull dimension one. Every compact R-analytic group is verbally concise. 4.3 Conciseness in compact R-analytic groups Now, we are primed to prove the principal result. Theorem 4.21. Every compact R-analytic group is verbally concise. Proof. Suppose, in view of Corollary 4.20, that Rhas Krull dimension at least 2. Let Gbe a compact R-analytic group and let wbe a word in kvariables such that w{G}is finite. By virtue of Lemma 4.15, it suffices to prove that every w-value is of finite order. By Lemma 1.23, there exists an open normal R-standard subgroup (S, φ)such that for every g∈Gthe conjugation map cg:S→S, x 7→ xgis strictly analytic. If n=|G:S|, then wn{G} ⊆ Sand, by Lemma 4.15, wn(G)is finite if and only if w(G)is finite. Therefore, without loss of generality assume that w{G} ⊆ S. Let (P, m)be the principal ideal pro-pdomain Zpif char R= 0 or Fp[[t]] if char R=p. According to Cohen’s Structure Theorem (see Theorem 1.2), Ris a finitely generated integral extension of P[[t1, . . . , tm]],where m= dimKrull(R)−1. For each a∈m(m),let sabe the evaluation epimorphism sa:P[[t1, . . . , tm]] → P, F(t1, . . . , tm)7→ F(a).By Corollary 2.13, saextends to a continuous ring epimorphism ˜sa:R→Q, where, in view of Remark 2.14, Q= (Q, n)is a pro-p domain and a finitely generated integral extension of P, in particular, Qhas Krull dimension 1. Fix a∈m(m),throughout this proof we use Wato denote W˜safor any tuple of power series W(see Section 2.2). In particular, if Fis the formal group law of S, Fais the formal group law F˜sa(see Corollary 2.8). Let Tbe a left transversal for Sin G, and assume that 1∈T. We will use the atlas induced by S, namely {(tS, φt)}t∈Twhere φt(x) := φ(t−1x)(compare with Section 1.3). 105 Using Lemma 2.10, we define the Q-standard group L:= nN(d),whose group operation is given by Fa,and a compact Q-analytic group H:= T×L, that can be regarded as an overgroup of Land whose group operation, say ∗a,is defined as in (2.3). Recall that the Q-analytic structure of His given by the atlas {(tL, ψt)}t∈T where ψt(t, l) = l. For the rest of the proof fix (t1, . . . , tk)∈T(k)and assume, by (4.2), that for any `∈N,the word map wℓis given in t1S×···×tkSby the tuple of power series Wℓ,that is, recalling that w{G} ⊆ Swe have that φwℓ(x1, . . . , xk)=Wℓ(φt1(x1), . . . , φtk(xk)) ∀xj∈tjS (even though in order to lighten the notation it is not written explicitly, the power series Wℓalso depends on t1, . . . , tk). Let wℓ:H(k)→Hbe the word map wℓwith respect to the operation ∗aof H. By Lemma 2.7 and Remark 2.11, ψ1wℓ(x1, . . . , xk)=Wℓ a(ψt1(x1), . . . , ψtk(xk)) ∀xj∈tjL. Furthermore, according to Proposition 4.14, since w{G}is finite, Sis marginal for w. That is, wis constant in each open subset t1S×···×tkS. Hence, the word map w:H(k)→His constant in each t1L×···×tkL, and thus |w{H}| ≤ |H: L|k=nkis finite. According to Corollary 4.20, His verbally concise, so there exists `a∈Nsuch that wℓa(H) = {(1,0)}.In particular, by Lemma 1.8 Wℓa a(X1, . . . , Xdk) = 0.(4.5) Define the following partition of the space m(m): mℓ=a∈m(m)Wℓ a=0, ` ∈N. Since m(m)=Sℓ∈Nmℓand m(m)is complete, by the Baire Category Theorem there exists `′such that mℓ′contains a non-empty open subset V⊆om(m).Thus, Wℓ′is a constant tuple of power series, i.e. Wℓ′(X1, . . . , Xdk) = (c1, . . . , cd)∈R(d), that whenever a∈mℓ′,satisfies (˜sa(c1), . . . , ˜sa(cd)) = Wℓ′ a(X1, . . . , Xdk) = 0. Consequently, ˜sa(ci)=0for all a∈mℓ′∩V. Besides, mℓ′∩Vis dense in V, so by Corollary 2.15, we have that ci= 0 for all i∈ {1, . . . , d}.Finally, repeating the process for all the tuples in T(k),we obtain an integer `such that φwℓ(x1, . . . , xk)=0for all xi∈G, and thus wℓ(G) = {1}. 106 There are several proofs of Ado’s Theorem (see, for instance, [41, Chapter VI, Section 2]), and from most of them we can conclude that the degree of the representation depends only on the vector space dimension of the Lie algebra. More precisely, let deg L:= min{deg φ|φis a faithful representation of L}. If Ris a field of characteristic zero and Lis an R-Lie algebra of dimension r, then deg L≤f(r)for some non-decreasing function f:N0→N.For instance, from [13] and [60], we know that deg L≤r+α2r √r,(A.1) for some α≈2.763 (we refer to [58, Section 1.1.2] for a detailed proof). There are various other works studying deg Lover fields, and we ought to mention [8], [31] and [63]. For general PIDs, meanwhile, we can not directly make the same deduction from the initially-mentioned two references. In fact, in [74, Proposition 3.4], the finiteness of the degree-to-be follows from the fact that since Ris Noetherian any ascending chain of ideals must be stationary, although we can not determine the number of ideals in the chain. In view of this, we shall present a quantitative way of constructing a faithful representation of an R-Lie lattice. This procedure will be grounded on the ideas that appear in [8] and [63]. Theorem A.4. Let Lbe an R-Lie lattice of rank r. Then deg L≤r+rr+ 1 r4r. Lastly, it is worth mentioning that the positive characteristic counterpart of Ado’s Theorem is also true, as proved by Iwasawa [40]. In fact, the general version of Theorem A.2, without any restriction on the characteristic of the base field, is referred to as the Ado-Iwasawa Theorem. Nevertheless, in positive characteristic the result can be stated with much more generality: Theorem A.5 (cf. [19, Theorem 3]).Let Ra ring of positive characteristic and Lan R-Lie lattice. Then there exist a free R-module Wof finite rank and an injective R-Lie algebra homomorphism φ:L,→EndR(W). 113 In order to prove this theorem it suffices to reproduce the original proof word by word, and thus we obtain for deg Lthe same bound we already knew over fields, namely deg L≤nrk3L, where n= char R(compare with [5, Section 6.2.4]). Remarks. Thoughout the proofs we will use some basic properties of free Rmodules over PIDs. Here is what we should take into account: (i) submodules of a free R-module Mare free, and they have rank at most rk M. Let Mbe an R-module and N≤Ma submodule. The isolator of Nin Mis the submodule IsoM(N) = {x∈M| ∃ r∈R\{0}such that rx ∈N}, and Nis isolated in Mif IsoM(N) = N. (ii) M/ IsoM(N)is a torsion-free R-module. (iii) If Mis a free R-module, N≤Mis an isolated submodule and M/N is finitely generated, then M/N is a free R-module and rk(M) = rk(N) + rk(M/N). The integer rk(M/N)is referred to as the corank of Nin M. (iv) If Mis a free R-module, N≤Mis an isolated submodule and M/N is finitely generated, then Nhas a complementary in M, i.e. there exists a free R-module Lsuch that M=N⊕L. A.2 Adjoint and regular representations We shall introduce a couple of representations that arise naturally in every R-Lie lattice L. Firstly, x∈Ldefines the linear endomorphism adx:L→L, y 7→ [x, y]. This assignation gives the adjoint representation of degree rk L,namely Ad: L→EndR(L), x 7→ adx, 114 which is an R-Lie algebra homomorphism in view of Jacobi’s identity. Nonetheless, the adjoint representation is not faithful in general, as its kernel is the centre of the algebra, Z(L) := {x∈L|[x, y] = 0 ∀y∈L}. Therefore, if Lis a semisimple R-Lie algebra –it has no non-trivial abelian ideal–, the adjoint representation is faithful, and, in this situation, deg L≤rk L. In order to present the second representation, we must define the universal enveloping algebra: Definition A.6 (cf. [41, Chapter V, Section 1]).Let Lbe an R-Lie algebra. The R-tensor algebra of Lis TR(L) = R⊕L1⊕L2⊕···⊕Li⊕. . . , where Li:= L⊗(i) . . . ⊗L.Each Liis an R-module with the natural R-module structure of tensorial modules and the multiplication in TR(L)is defined by the rule (x1⊗···⊗xi)⊗(y1⊗···⊗yj) = x1⊗···⊗xi⊗y1⊗···⊗yj. Let Rbe the ideal of TR(L)generated by the elements [x, y]−(x⊗y−y⊗x), x, y ∈L. The universal enveloping algebra of Lis the associative R-algebra with identity UR(L) := TR(L) R. Identifying Lwith L1,we obtain a homomorphism ι:L→ UR(L),which, since Ris a PID, is injective whenever Lis finitely generated (see [74, Theorem 3.2]). Hence, for simplicity we can assume that L⊆ UR(L).Besides, in order to simplify the notation, the element xi1⊗···⊗xikwill be written as the monomial xi1. . . xik. The universal algebra is described by the Poincaré-Birkhoff-Witt Theorem: Theorem A.7 (cf. [74, Theorem 3.2]).Let Lbe an R-Lie lattice with basis {x1, . . . , xr}.Then UR(L)is a free R-module, and the monomials {xα1 1. . . xαr r|αi∈N0}(A.2) form a basis. 115 Indeed, given two monomials their product can be expressed as a linear combination of monomials of the form (A.2) by successively applying the identity xjxi=xixj−[xi, xj]to reorder the terms until all the involved monomials have the required order. Any R-Lie lattice Lacts by left multiplication on UR(L).Indeed, for each x∈L we get the R-linear endomorphism `x:UR(L)→ UR(L), u 7→ xu. On the one hand, for every x, y ∈L [x, y] = xy −yx in UR(L),and therefore L:L→EndR(UR(L)), x 7→ `xis an R-Lie algebra homomorphism. On the other hand, since the universal enveloping algebra has an identity, whenever x6=ywe have that `x(1) = x6=y=`y(1), and so Lis a faithful representation, called (left) regular representation. However, since UR(L)is of infinite rank, the regular representation is not finite. Nevertheless, there is a property that characterises the universal enveloping algebra, and as a consequence of it any finite representation factors though UR(L). Theorem A.8 (Universal property, cf. [11, Chapter I, § 2.1, Proposition 1]).Let Lbe an R-Lie lattice, Aan associative R-algebra with identity together with the Lie bracket [a, b] = ab −ba (a, b ∈A) and an R-Lie algebra homomorphism ψ:L→ (A, [,]).Then, there exists a unique R-algebra homomorphism ψ∗:UR(L)→A such that ψ=ψ∗◦ι. That is, the following diagram commutes: UR(L) LA. ψ∗ ι ψ Actually, the name of UR(L)comes from this universal property. Further, it can be proved that any R-algebra that satisfies the universal property is isomorphic to UR(L)(compare with [41, Chapter V, Theorem 1.1]). A.3 Ado’s Theorem In order to prove Ado’s Theorem we will usually move from the R-Lie lattice L to the K-vector space LK:= L⊗RK. Note that LKis a K-Lie algebra, whose Lie bracket is nothing but the K-linear application induced by the Lie bracket of L. We should bear in mind the following facts: 116 •LKis K-vector space of dimension rk L, • if L=hx1, . . . , xriRthen LK=hx1, . . . , xriK,and • whenever I⊴LKis an ideal, then I∩L⊴Lis an isolated ideal. We will prove the theorem in three steps: A.3.1 Nilpotent Lie lattices First of all, let us suppose that Lis a nilpotent R-Lie lattice of rank r. Recall that the lower central series of Lis defined as: γ1(L) := L, γi(L) := [γi−1(L),L]∀i≥2, and that Lis nilpotent if there exists an integer c∈Nsuch that γc+1(L) = {0}. The smallest of such integers, when it exists, is the nilpotency class of the R-Lie lattice. Considering that the tensor product is linear, we can easily prove the following: Lemma A.9. Let Lbe an R-Lie lattice and let I,H⊴Lbe ideals. Then, [I⊗RK, H⊗RK] = [I,H]⊗RK. Thus, if Lis a nilpotent R-Lie lattice of nilpotency class c,LKis a nilpotent K-Lie algebra of nilpotency class c. Therefore, LKis nilpotent of nilpotency class say c, i.e LK=γ1(LK)>··· > γi(LK)>··· > γc+1(LK) = {0} is a strictly descending chain of K-vector spaces and thus c≤dimKLK=r. Define now the isolated ideals Li:= γi(LK)∩L⊴L,and choose a basis {x1, . . . , xr}of Lsuch that the first x1, . . . , xr1elements are a basis for Lc,the first x1, . . . , xr2 (r2> r1) elements form a basis for Lc−1and so forth. According to Theorem A.7, the monomials xα:= xα1 1. . . xαr r, α = (α1, . . . , αr)∈N(r) 0 form a basis for the universal enveloping algebra UR(L),and accordingly we can define a weight function ω:UR(L)→N0∪{∞} in the following fashion: 117 ω(xi) = max{m|xi∈Lm}, ω(xα) = Pr i=1 αiω(xi), ω(Pαcαxα) = min {ω(xα)|cα6= 0}and ω(0) = ∞. For each m∈N0,we define Um(L) := {u∈ UR(L)|ω(u)> m} –when the lattice is clear from the context, we will simply write Um–. Let us show that Um(L)⊴UR(L)is an isolated ideal: (i) Note that 0∈Umfor all m∈Nand that ω(rx) = ω(x)and ω(x+y)≥min {ω(x), ω(y)}, for all r∈R\ {0}and x, y ∈ UR(L).Additionally, since ω([xi, xj]) ≥ ω(xi) + ω(xj)for every i, j ∈ {1, . . . , r},we have that ω(xy)≥ω(x) + ω(y).(A.3) Consequently, Umis an ideal. (ii) Let r∈R\{0}, rx ∈Um=⇒ω(x) = ω(rx)> m =⇒x∈Um, i.e. Umis isolated in UR(L). Moreover, UR(L)/Umis a finitely generated R-module, as it is generated by Bm={xα+Um|ω(xα)≤m}. Therefore, Umis an isolated ideal of finite corank, and in view of (A.3), for every x∈Lwe obtain that `x(Um)⊆Um.In consequence, for any mthe regular representation induces the finite representation Lm:L→EndR(UR(L)/Um), x 7→ `x, whose kernel is L∩Um–with an abuse of notation, whenever f∈EndR(UR(L)) satisfies f(X)⊆Xfor some ideal X⊴UR(L),we will still call fto the element in 118 EndR(UR(L)/X)that sends x+Xto f(x) + X–. Besides, L∩Uc(L) = {0}.Indeed, whenever x=Pr i=1 αixi∈L,then ω(x) = ω r X i=1 αixi!≤max i=1,...,r ω(xi) = c. Therefore, Lcis a finite faithful representation of L,and in order to bound its degree it suffices to determine an upper bound for |Bc|.All the monomials in Bc have weight at most c, so they have polynomial degree at most c. Moreover, the number of monomials of polynomial degree at most cin rvariables is exactly the number of monomials of polynomial degree cin r+ 1 variables (by adding an auxiliary variable, homogenise the monomials such that they have polynomial degree c). Lemma A.10. The number of monomials in rvariables and of polynomial degree cis r+c−1 c. This observation gives us a simple bound (compare with [31, Corollary 5.1]): deg L≤rk UR(L) Uc(L)≤r+c c. We conclude this section by giving a not very sharp bound for deg Lin terms of r. In fact, c∈ {1, . . . , r},that is, c=αr where α∈ {1/r, . . . , 1}.Remember that according to the Stirling approximation formula, √2πr (r/e)r≤r!≤√2πr (r/e)re1 12r∀r∈N, and therefore, r+αr αr ≤p2π(r+αr)(r+αr)r+αr √2παr(αr)αr√2πr rre1 12(1+α)r ≤e1/12(1 + α)r √2πr √1 + α √α(1 + α)1+α ααr . In addition, the left-hand side and the right-hand side terms are asymtotically equivalent as rtends to infinity. Further, since α∈[1/r,1] ,then √1 + α √α≤√r+ 1 and (1 + α)1+α αα≤4. 119 That is, e1/12(1 + α)r √2π 1 √r √1 + α √α(1 + α)1+α ααr ≤rr+ 1 r4r. Consequently, r+c c≤rr+ 1 r4r∀c∈ {1, . . . , r}.(A.4) A.3.2 Splittable R-Lie lattices The second step consists on obtaining a suitable R-Lie algebra representation for the so-called splittable R-Lie lattices. Since the sum of nilpotent ideals is again nilpotent (compare with [41, Chapter I, Proposition 7.6]), every R-Lie lattice Lhas a nilpotent ideal that contains any other nilpotent ideal. This is called the nilpotent radical of L,and it will be represented as Rn(L). Additionally, Rn(L)is an isolated ideal, as it is nothing but Rn(LK)∩L. We will say that an R-Lie algebra Lis splittable if there exists an R-Lie subalgebra S≤Lsuch that L=Rn(L)⊕S,that is, Lis the semidirect product of S with the ideal Rn(L).For the categoricaly minded reader we point out that this condition is equivalent to the fact that the short exact sequence 0→Rn(L)→L→L/Rn(L)→0 splits in the category of R-Lie algebras. To deal with the splittable case, we can blend the preceding regular representation and the representations induced from derivations: Definition A.11. Let Abe an R-algebra. A derivation of Ais an R-module endomorphism D∈EndR(A)that satisfies Leibniz identity, i.e. D(ab) = aD(b) + D(a)b∀a, b ∈A. The set of all derivations of Ais denoted by DerR(A). For example, by virtue of Jacobi’s identity, for all x∈Lwe have that adxis a derivation of the R-Lie algebra L.Starting from a derivation Dof Lwe can induce a derivation of UR(L),which will be denoted by D∗.For that, we should 120 extend Dby imposing Leibniz identity, i.e. by taking the linear extension of the rule D∗(x1. . . xt) = t X i=1 x1. . . xi−1D(xi)xi+1 . . . xt, together with D∗(1) = 0 as it must happen for every derivation of an algebra with identity. Actually, this extension is a consequence of the universal property (compare with [41, Chapter V, Theorem 1.1(7)]). Lemma A.12. Let Lbe a nilpotent R-Lie lattice and D∈DerR(L).Then D∗(Um(L)) ⊆Um(L)for every m∈N. Proof. Let {x1, . . . , xr}be the basis of Lwith respect to which the weight function ωis defined (compare with Subsection A.3.1). Since Dis a derivation, D(Li)⊆ Lifor all i∈ {1, . . . , c},so ω(D(xi)) ≥ω(xi)for all i∈ {1, . . . , r}.Hence, if xi1. . . xit∈Um(L),by (A.3) ω(D∗(xi1. . . xit)) ≥min j=1,...,t ω(xi1. . . D(xij). . . xit)≥ω(xi1. . . xit)> m. Proposition A.13 (Zassenhaus extension, cf. [11, Chapter I, § 7.3, Theorem 1] and [41, Chapter VI, Theorem 2.1]).Let Lbe a splittable R-Lie lattice and let c be the nilpotency class of Rn(L). Then, there exists a finite representation Φ: L→EndRUR(Rn(L)) Uc(Rn(L))  of Lthat is injective in Rn(L).In particular, deg Φ depends only on rk Rn(L). Proof. Denote for simplicity Rn(L)as N.Then, L=N⊕Sfor some R-Lie subalgebra S≤L. From Lemma A.12, ad∗ x(Uc(N)) ⊆Uc(N)for all x∈L,so we can define the map Φ: L=N⊕S→EndR(UR(N)/Uc(N)), n +s7→ `n+ ad∗ s. In order to show that it is an R-Lie algebra homomorphism, it suffices to confirm that Φ ([s, n]) = [Φ(s),Φ(n)] = [ad∗ s, `n] for all n∈Nand s∈S.For that, note that for any n∈Nand any D∈ DerR(UR(N)) : [D, `n](u) = D◦`n(u)−`n◦D(u) = D(n)u=`D(n)(u),∀u∈N. 121 Moreover, since Nis an ideal, then [s, n]∈N,so Φ ([s, n]) = `[s,n]=`ad∗ s(n)= [ad∗ s, `n] = [Φ(s),Φ(n)] . Consequently, Φis an R-Lie algebra representation. In addition, its kernel has trivial intersection with the nilpotent radical, as Φ|Nis nothing but the faithful representation Lcof N. Finally, from (A.4) we conclude that deg Φ ≤srk Rn(L)+1 rk Rn(L)·4rk Rn(L).(A.5) A.3.3 Embedding theorem Like nilpotency in R-Lie algebras, we can define solubility: the derived series of an R-Lie algebra Lis defined recursively as L(1) := L,L(i):= L(i−1),L(i−1)∀i≥2, Lis said to be soluble when L(ℓ)={0}for some `∈N,and the smallest of such integers is the derived length of L.In addition, the sum of soluble ideals is again soluble (compare with [41, Chapter I, Proposition 7.4]), and therefore, if Lis finitely generated, there exists a soluble radical of L,namely a soluble ideal Rs(L) that contains any other soluble ideal. Obviously, any nilpotent R-Lie algebra is soluble and thus Rn(L)≤Rs(L). According to Levi’s Theorem (see [41, Chapter III, Section 9]), if Ris a field of characteristic zero, every R-Lie algebra Lsplits as Rs(L)⊕Sfor some semisimple Lie subalgebra S≤L, called Levi factor of L.This decomposition plays a fundamental rôle in the majority of proofs of Ado’s Theorem for fields, as they firstly obtain a finite faithful representation of Rs(L),and then it is extended to Lusing Zassenhaus extension (compare with Propositon A.13). However, Levi’s Theorem does not hold for general R-Lie algebras: the Z-Lie algebra sl2(2Z)⊕t2(2Z),i.e. the direct sum of 2×2matrices of trace 0and 2×2 upper triangular matrices over the ring 2Z,is not decomposable in the desired way (see [19, Example in pg. 838]). In this third and last step we shall prove the main theorem. For that, we will embed the initial R-Lie lattice Lin a splittable R-Lie lattice and make use of the previous subsection. Actually the problem reduces to fields: 122 Bibliography [1] Abercombrie, A. 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[77] Zozaya, A.: A remark on Ado’s Theorem for principal ideal domains, in preparation. 134 Index δ-covering, 68 m-adic topology, 11 p-adic analytic, 19 Ado Theorem, 112 affine set, 85 algebra central simple ∼, 92 semisimple ∼, 115 splittable ∼, 120 tensor ∼, 115 analytic ∼dimension, 16, 24 ∼group, 18 ∼manifold, 16 ∼map, 15, 17 ∼subgroup, 42 ∼submanifold, 39 ∼subset, 41 strictly ∼map, 17 atlas, 16 compatible ∼, 16 maximal ∼, 16 Baire Category Theory, 99 Baker-Hausdorff formula, 50 bi-Lipschitz, 68 bianalytic map, 30 box dimension, 68, 69 standard lower ∼, 74 upper ∼, 68 lower ∼, 68 standard ∼, 79 centroid, 92 chain rule, 27 chart, 16 adapted ∼, 43 compatible ∼, 16 regular ∼, 16 Chevalley classical group, 88 Cohen Structure Theorem, 12, 35 conciseness strong ∼, 107 verbal ∼, 97 coordinate ∼change map, 16 ∼system, 17 canonical ∼system, 18 change of ∼, 31 corank, 42, 114 derivation, 120 135 derived ∼lenght, 122 differential, 27 dimension ∼submanifold, 39 analytic ∼, 16, 24 box ∼, 68 Hausdorff ∼, 68 local analytic ∼, 17 Minkowski-Bouligand ∼, 68 discrimination, 57 evaluation ∼epimorphism, 55 ∼map, 14 filter, 60 filtration series, 65 p-power ∼, 66 normal ∼, 69 standard ∼, 21, 72 finite intersection property, 61 formal group law, 19 additive ∼, 20 multiplicative ∼, 20 formal morphism, 27 fraction field, 12 full residuality, 57 Going Up Theorem, 56 Hall-Petrescu formula, 102 Hall-Witt identity, 26 Hausdorff ∼density, 91 ∼dimension, 68 ∼measure, 68 ∼spectrum, 66 standard ∼dimension, 67, 76 standard ∼spectrum, 67, 76 Hilbert ∼function, 73 ∼polynomial, 73 identity component, 103 immersed subset, 39 immersion, 32 weak ∼, 30 inverse formal ∼, 19 inversion ∼map, 18 isolated module, 114 isolator, 114 Iwasawa Theorem, 113 Jacobi identity, 112 Jacobian, 27 Krull Intersection Theorem, 11 Leibniz identity, 120 Levi ∼factor, 122 Theorem, 122 Lie algebra, 27, 111 ∼centre, 115 ∼homomorphism, 112 ∼representation, 112 associated ∼, 26 concrete ∼, 112 graded ∼, 90 nilpotent ∼, 117 perfect ∼, 92 136 soluble ∼, 122 Lie bracket, 26, 111 Lie lattice, 112 powerful ∼, 50 linear ∼algebraic group, 98 ∼representation, 47 ∼algebraic group, 98, 103 ∼group, 47 general ∼group, 18, 85 special ∼group, 88 local ∼ring, 11 ∼ring homomorphism, 51 manifold analytic ∼, 16 pure ∼, 17 marginal ∼subgroup, 96 Minkowski-Bouligand ∼dimension, 68 monotonicity, 68 multiplication ∼map, 18 left ∼map, 29 nil-representation, 127 nilpotency ∼class, 117 nilpotent ∼radical, 120 nowhere constant map, 107 Poincaré-Birkhoff-Witt Theorem, 115 power series ∼universal property, 14 ∼ring, 13 convergent ∼, 14 principal ideal domain (PID), 12 pro-pdomain, 12 profinite space, 107 radical nilpotent ∼, 120 soluble ∼, 103, 122 unipotent ∼, 103 reductive group, 104 regular point, 42 representation, 112 ∼degree, 112 adjoint ∼, 114 faithful ∼, 112 finite ∼, 112 induced ∼, 48 linear ∼, 47 matricial ∼, 112 regular ∼, 116 residue field, 11 residue rank, 31 restriction of scalars, 38 Schur Theorem, 102 series derived ∼, 122 lower central ∼, 117 soluble ∼radical, 103, 122 special orthogonal group, 89 stability countable ∼, 68 finite ∼, 69 standard ∼filtration series, 21 ∼group, 19 strong triangle inequality, 12 137 subimmersion, 33 submanifold ∼dimension, 39 analytic ∼, 39 weak ∼, 40 submersion, 32 weak ∼, 30 symplectic group, 89 Tits topological alternative, 88 topologically nilpotent, 14 translation, 18 ultrafilter, 60 ∼theorem, 61 non-principal ∼, 60 ultrametric space, 12 ultrapower, 61 ultraproduct, 61 uniformiser, 13 uniformly powerful group, 49 unipotent ∼matrix, 103 ∼radical, 103 ∼subgroup, 103 unique factorisatin domain (UFD), 45 universal enveloping algebra, 115 universal property, 116 verbal ∼conciseness, 97 ∼ellipticity, 98 ∼robustness, 97 ∼subgroup, 96 ∼width, 98 word, 95 ∼map, 95 ∼value, 95 Burnside ∼, 97 commutator ∼, 97 derived ∼, 97 elliptic ∼, 98 empty ∼, 95 equivalent ∼, 95 lower central ∼, 97 Zassenhaus extension, 121 Zorn Lemma, 92 138 antisimetria itxurazkoa baino ez da. Izan ere, hartu g∈Gelementua eta demagun w(x1, . . . , xig,...,xk) = w(x1, . . . , xk)dela xj∈Gguztietarako, orduan w(x1, . . . , gxi, . . . , xk) = w(x1, . . . , xigxi, . . . , xk) =w(x1, . . . , (xig)xi, . . . , xk) =wxx−1 i 1, . . . , xig,...,xx−1 i kxi =wxx−1 i 1, . . . , xi, . . . , xx−1 i kxi=w(x1, . . . , xi, . . . , xk) da. Adibideak 4.5. Izan bedi Gtaldea. (i) Gtalde totala eta {1}azpitalde tribiala hitzezko azpitaldeak dira, hurrenez huren, w(x) = xeta hitz hutsari dagozkionak. (ii) Hitz arrunt eta ezagunena kommutadore hitza da, hau da, γ2(x, y) = [x, y] = x−1y−1xy. Haren hitzezko azpitaldea γ2(G) = G′azpitalde deribatua da eta dagokion azpitalde marjinala γ∗ 2(G) = Z(G)zentroa. (iii) Hitz behe zentralak errekurtsiboki γn(x1, . . . , xn) := [γn−1(x1, . . . , xn−1), xn]∀n≥3 gisa definitzen dira, eta hitz deribatuak errekutsiboki δ1(x1, x2) := γ2(x1, x2) eta δn(x1, . . . , x2n) := [δn−1(x1, . . . , x2n−1), δn−1(x2n−1+1, . . . , x2n)] ∀n≥2 moduan definitzen dira. (iv) Burnsideren hitzak wm(x) = xmdira. Horiek Gmhitzezko azpitaldeak definitzen dituzte, hau da, G-ko elementuen mgarren berreturek sorturiko taldeak. Eta, adibidez, w∗ 2(G)azpitalde marjinala gehienez 2ordenako elementu zentralek osatzen dute, hau da, w∗ 2(G) = {g∈Z(G)|g2= 1}da. P. Hallek [33] hainbat galdera egin zituen w-balioen multzoaren eta haren hitzezko azpitaldearen eta azpitalde marjinalaren arteko erlazioaren inguruan. Hurrengo definizioak itaun horiek laburtzeko balioko du: Definizioa 4.6. Izan bitez whitza eta Ctalde klasea. 241 (i) whitza laburra da C-n, G∈ C guztietarako w{G}finitua izateak w(G)ere finitua dela inplikatzen badu. (ii) whitza sendoa da C-n, G∈ C guztietarako |G:w∗(G)|finitua izateak w(G) finitua dela inplikatzen badu. Horrela, wlaburra (sendoa) da talde guztien klasean laburra (sendoa) denean. Antzeko moduan, Gtaldean hitz guztiak laburrak badira, Ghitzez laburra dela diogu. Orokorrean, |w{G}| ≤ |G:w∗(G)|kdenez (kzenbaki osoa whitzaren aldagai kopurua da), sendotasuna laburtasuna baino gogorragoa da: wlaburra bada C-n, orduan wsendoa da C-n. Haatik, talde erresidualki finituetan, eta guri ardura zaizkigun taldeak horrelakoak dira, bi kontzeptuak baliokideak dira: Lema 4.7 (cf. [67, Lema 1.4.1]).Izan bitez Gtaldea eta whitza. (i) |G:w∗(G)|finitua bada, orduan w{G}finitua da. (ii) Gerresidualki finitua eta w{G}finitua badira, orduan |G:w∗(G)|finitua da. P. Hallek bat hitz guztiak laburrak zirela iragarri zuen. Aitzitik, ia hiru hamarkadaren ostean, Ivanovek [39] aieru hori errefuxatu zuen, Gtalde bat eta w hitz bat topatu baitzituen non w{G}multzoak bi elementu dituen, baina w(G) talde zikliko infinitua den. Hala ere, kontradibide hori ez da erresidualki finitua, ezta Ol’shanskiĭk (ikusi [62, Teorema 39.7]) eraiki zuen antzeko kontradibidea ere. Horrek Jaikin-Zapirainek [44] eta Segalek [67] proposaturiko aieru honetara garamatza: Aierua 4.8 (Laburtasunaren aierua talde erresidualki finituetan).Hitz guztiak laburrak dira talde erresidualki finituen klasean. Talde hitzez laburren klase gutxi batzuk baino ez dira ezagutzen. Talde abeldarren (ikusi Lema 4.2) eta periodikoen (ikusi datorren Lema 4.15) ageriko adibideez gain; 1960ko hamarkadan, Merzjalkovek [57] eta Turner-Smithek [72] hurrenez hurren frogatu zuten talde linealak eta zatidura guztiak erresidualki finituak dituzten taldeak (e.g. talde birtualki nilpotenteak) hitzez laburrak direla. Hitz balioen multzoa infinitua denean, laburtasunaren pareko kontzeptua hitz eliptikotasuna da. Hori definitzeko notazio hau erabiliko da: X⊆Gazpimultzo baterako, izendatu X∗ℓmoduan X∪X−1∪{1}multzoko `elementuren biderketek osatzen duten multzoa. 242 Definizioa 4.9. Izan bitez Gtaldea eta whitza. Orduan, weliptikoa da G-n existitzen bada `∈Nnon w(G) = w{G}∗ℓden. Aurreko baldintza betetzen duten `zenbaki osoetan txikienari w-ren hitz zabalera deritzo. Horrela, Gtaldea hitzez eliptikoa da hitz guztiak G-n eliptikoak direnean. Eliptikotasuna laburtasuna baino gogorragoa da: weliptikoa bada C talde klaseko talde guztietan, orduan wlaburra da C-n. Lema 4.2ren arabera, talde abeldarrak hitzez eliptikoak dira eta 1hitz zabalera dute. Horiez gain, talde aljebraiko linealak∗(ikusi [56]), finituki sortutako talde abeldar-bider-nilpotenteak (ikusi [29] eta [71]) edo, tesi honen gaiarekin zerikusi zuzena duena, talde p-adiko analitiko trinkoak (ikusi [44]) hitzez eliptikoak dira. Haatik, hitzez eliptikoak ez diren taldeen adibide naturalak daude (kontradibideak ez dira hitzezko laburtasunarenak bezain konplexuak bederen). Esate baterako, Roman’kovek [66] finituki sortutako pro-ptalde ebazgarri bat aurkeztu zuen non δ2(x1, . . . , x4) = [[x1, x2],[x3, x4]] hitz deribatuak zabalera infinitua duen. Talde profinituei dagokionez, hitz zabalera eta hitzezko azpitaldea itxia izatea lotuta daude. Proposizioa 4.10. Izan bitez GHausdorff talde topologiko trinkoa eta whitza. Orduan, weliptikoa da G-n baldin eta soilik baldin w(G)itxia bada. Froga. Soilik baldin norantzan, ohartu edozein n-tarako w{G}∗nitxia dela, multzo trinko baten irudi jarraitua baita. Hortaz, w-ren hitz zabalera `bada, w(G) = w{G}∗ℓitxia da. Bestetik baldina frogatzeko, ohartu w(G) = [ n∈N w{G}∗n dela eta w{G}∗nguztiak itxiak direla. Hori dela eta, w(G)Hausdorff eta trinkoa denez, Baireren Kategoria Teoremaren (ikusi [59, Teorema 48.2]) arabera, existitzen da mzenbaki osoa non w{G}∗m-k barnealde ez-hutsa duen, hau da, U⊆ow(G)azpimultzo ireki ez-huts bat du barruan. Hortaz, w(G) = [ g∈w(G) gU ∗talde aljebraiko lineal diogunean, GLn(K)-ren azpitalde Zariski itxi bat esan nahi dugu, K gorputz aljebraikoki itxia delarik. 243 da, eta w(G)-ren trinkotasunagatik w(G) = r [ i=1 giU da g1, . . . , gr∈w(G)elementu batzuetarako. Hartu k∈Nnon gi∈w{G}∗kden i∈ {1, . . . , r}guztietarako, orduan w(G) = r [ i=1 giU⊆w{G}∗(k+m), da, nahi genuen moduan. Orokorrean, (hitzezko) azpitalde bat itxia den ala ez jakitea erabilgarria da talde profinituekin jarduterakoan. Horregatik, Jaikin-Zapirainen emaitza nabarmen hau enuntziatu behar dugu: Teorema 4.11 (cf. [44, Teorema 1.1]).Izan bedi w∈Fkhitza kaldagaitan. Orduan, whitzak zabalera finitua finituki sortutako pro-ptalde guztietan baldin eta soilik baldin w /∈δ2(Fk) (F′ k)pbada. Kapitulu honen xedea talde R-analitiko trinkoak hitzez laburrak direla frogatzea da. Aipatu beharrekoa da char R= 0 denean, emaitza hori Teorema 2.27ren –talde R-analitiko trinkoak linealak dira– eta Merzjalkoven Teoremaren –talde linealak hitzez laburrak dira– ondorio zuzena dela. Aitzitik, interesgarria da horren froga independentea ematea, zeina pro-pdomeinu guztietarako, karakteristika edozein delarik ere, betetzen den. Are gehiago, emaitza orokor hau talde R-analitiko trinko guztiak linealak izatearen aldeko beste ebidentzia bat da. 4.1 Laburtasuna talde R-estandarretan Talde R-estandarren klasean laburtasuna berehalakoa da: Proposizioa 4.12. Izan bitez Stalde R-estandarra eta whitza. Demagun w{S} finitua dela, orduan w(S) = {1}da. Froga. Lehenik eta behin, Staldea mN(d)-rekin identifikatu daiteke, non N taldearen maila eta ddimentsioa diren. Horrela, biderketa eta alderantzizkoa bi 244 berretura serie formalen tuplak definitzen dituzte eta eta identitatea 0da. Horrenbestez, whitz funtzioa W∈R[[X1, . . . , Xdk]](d)berretura serie tupla bakarra da (khitzeko indeterminatu kopurua da). Horrela, w{S}finitua eta hitz funtzioa jarraitua direnez, Wlokalki konstantea da; eta, beraz, Lema 1.8ren arabera, Wkonstantea da. Hots, W(X1, . . . , Xdk) = W(0, . . . , 0) = 0da, eta, beraz, w{S}={0}. Emaitza horren pare bat ondorio aipatu behar ditugu. Alde batetik, talde Ranalitiko guztiek laburtasunaren aieruaren bertsio ahulago hau betetzen dute: Korolarioa 4.13. Izan bitez Gtalde R-analitikoa eta whitza. Demagun w{G} finitua dela. Orduan, existitzen da Sazpitalde R-estandar irekia non wlegea den, hau da, w(S) = {1}da. Froga. Lema 1.21en arabera, badago Sazpitalde R-estandar ireki bat G-n. Horrela, |w{S}| ≤ |w{G}| denez, Proposizioa 4.12 dela eta, w(S) = {1}da. Beste alde batetik, Gtalde R-analitiko trinkoa bada eta w{G}finitua, wbalioen multzoa soilik azpitalde R-estandar jakin baten ezker koklaseei begira kalkula daiteke. Hots, izan bitez Gtalde R-analitiko trinkoa eta S⊴oGazpitalde R-estandar irekia zeinaren konjokazio funtzioak hertsiki analitikoak diren (bigarrena Lema 1.23 dela eta existitzen da Rez denean IND bat), eta izan bedi Tezker transbertsal bat S-rentzat G-n. Oroitu S-tik eratorritako atlasa, hau da, {(tS, φt)}t∈Tnon φt(x) = φ(t−1x)den. Lema 1.24ren ondorioz, w:G(k)→G funtzio R-analitikoa t1S×···×tkS(ti∈T) multzo irekian berretura serie tupla bakarrak emanda dago –wez da biderketa eta alderantzizko funtzioen konposaketa egokia besterik–. Alegia, existitzen da Wt1,...,tk∈R[[X1, . . . , Xdk]](d)berretura serie formalen tupla non φp(w(x1, . . . , xk)) = Wt1,...,tk(φt1(x1), . . . , φtk(xk)) ∀xj∈tjS(4.2) den, hemen pelementua w(t1, . . . , tk)p−1∈Sbetetzen duen T-ko elementu bakarra da. Gainera, w{G}finitua bada, wfuntzio jarraitua lokalki konstantea da, eta Lema 1.8ren eraginez, Wt1,...,tkkonstantea da, hau da, Wt1,...,tk(X1, . . . , Xdk) = c∈R(d). Hots, φp(w(x1, . . . , xk)) = cda xj∈tjSguztietarako. Beste era batera esanda: Proposizioa 4.14. Izan bitez whitza, Gtalde R-analitiko trinkoa eta Sazpitalde normal R-estandarra zeinaren konjokazio funtzioak hertsiki analitikoak diren. Orduan, w{G}finitua bada, Smarjinala da w-rentzat. 245 4.2 Laburtasuna talde Fp[[t]]-analitiko trinkoetan Frogapen teknika hasierako problema bat Krull dimentsioko pro-pdomeinu baten gainean analitikoa den talde batera murriztean datza, eta horretarako Atala 2.2n deskribaturiko eraztun aldaketa erabiliko da. Hori dela eta, lehenbiziko bat dimentsioko kasua aztertu behar da. Orobat, talde p-adiko analitiko trinkoak hitzez laburrak dira, linealak baitira Korolarioa 2.6ren arabera. Hortaz, Korolarioa 1.44 kontuan hartuta, R=Fp[[t]] kasura murriz gaitezke. Hainbat emaitza tekniko erabiliko ditugu. Horietako batzuk frogatuko dira, baina beste batzuk enuntziatu baino ez ditugu eginen. Lehenik, gogora dezagun emaitza ezagun hau: Lema 4.15. Izan bitez Gtaldea eta whitza. Demagun w{G}finitua dela. Orduan, w(G)′finitua da, eta w(G)finitua da baldin eta soilik baldin w-balio guztiek ordena finitua badute G-n. Froga. Izan bedi g∈G. Lema 4.3ren arabera, w{G}g⊆w{G}da, hau da, x∈w{G}guztietarako xGkonjokazio klasea w{G}-n dago. Hortaz, |G:CG(x)|=xG≤ |w{G}| da, eta CG(x)-k indize finitua du G-n. Horrenbestez, CG(w(G)) = ∩x∈w{G}CG(x) azpitaldeak indize finitua du G-n, eta, ondorioz, |w(G) : Z(w(G))|ere finitua da. Beraz, Schurren Teoremaren arabera (ikusi [65, Teorema 10.1.4]), w(G)′finitua da. Azkenik, w(G)finitua bada, exponente finitua eduki behar du. Alderantziz, demagun w{G}-ko elementuek ordena finitua dutela, orduan w(G)/w(G)′talde abeldarra ordena finituko elementu kopuru finitu batek sortzen du, bereziki, finitua da; eta w(G)′finitua denez emaitza erdiesten dugu. Schur motako emaitza hau ere beharko dugu: Lema 4.16 (cf. [45, Proposizioa 5.1]).Izan bitez Gtaldea eta Nazpitalde normal nilpotentea. Demagun NZ(G) Z(G)zatidurak exponente finitua duela. Orduan, [N, G] azpitaldeak exponente finitua du. Froga. Hall-Petrescuren formularen arabera (ikusi [38, III.9.4]), m∈Nguztietarako xmym= (xy)mc2(x, y)(m 2). . . cm(x, y)(m m)(4.3) 246 da, non cr(x, y)∈γr(hx, yi)den. Izan bedi mzenbakia NZ(G) Z(G)zatidura taldearen exponentea, orduan (4.3) dela eta, n∈Neta g∈Gguztietarako: [n, g]m≡n−m(n[n, g])m=n−m(ng)m= [nm, g] = 1 mod γ2(K)(4.4) da, non K:= hn, [n, g]i ≤ Nden. Are gehiago, (4.3) eta (4.4) direla eta, l∈N guztietarako ([n1, g1]. . . [nl, gl])m≡[n1, g1]m. . . [nl, gl]m≡1 mod γ2(N) da. Izan bedi η(m)zenbakia 2sortzaileko eta mexponenteko talde nilpotente handienaren ordena†. Nahikoa da γ2(N)η(m)={1}dela frogatzea. Horretarako, izan bitez x, y ∈Neta H=hx, yi ≤ N. Horrela, H/Z(H)taldea 2sortzaileko talde nilpotentea denez eta haren exponenteak mzatitzen duenez, finitua da eta k=|H:Z(H)|zenbakiak η(m)zatitzen du. Halaber, θ:H→Z(H),h7→ hk funtzioa H-ren transferra da Z(H)-ra (konparatu [65, Teorema 10.1.3]ren frogarekin). Bereziki, θhomomorfismoa da eta (xy)k=xkykda. Horrela xketa yk elkarrekin trukatzen direnez eta kzenbaki osoak η(m)zatitzen duenez, (xy)η(m)=xη(m)yη(m)∀x, y ∈N da. Bereziki, θ′:N→Z(N), n 7→ nη(m)talde homomorfismoa da. Beraz, im θ′ abeldarra denez, γ2(N)≤ker θ′da, hau da, γ2(N)η(m)={1}. Datozen emaitzek talde aljebraiko linealen teoriako ideiak darabiltzate. Irakurleak [37]ra jo dezake emaitza horien atzeko teorian sakondu nahi badu. Izan bedi Kgorputz aljebraikoki itxia. Testu honetan zehar talde aljebraiko lineal bat GLn(K)-ren Gazpitalde Zariski itxi bat izanen da, eta G-ren identitate osagaia identitatearen osagai konexua da. Proposizioa 4.17 (cf. [37, Proposizioa 7.3]).Izan bedi Gtalde aljebraiko lineal konexua. (i) Orduan, G◦indize finituko azpitalde normala da. (ii) Izan bedi, H ≤ G indize finituko azpitalde itxi konexua, orduan H=G◦da. †Zenbaki hau finitua da, Baerrek [4] frogatu zuenez, talde nilpotenteek Burnsideren problema betetzen dutelako (ikusi [20, Teorema 2.23]). 247 Matrize bat unipotentea da haren autobalio bakarra 1bada, eta GLn(K)-ren azpitalde bat azpitalde unipotentea da elementu guztiak unipotenteak badira. Talde horien inguruko egiturazko emaitza nagusia talde unipotente oro Un(K)- ren –diagonalean 1ak dituzten matrize goi triangeluarren taldea– azpitalde baten konjokatua dela da (ikusi [37, Korolarioa 17.5]). Bereziki, talde unipotente guztiak nilpotenteak dira, eta oinarriko Kgorputza karakteristika positibokoa bada eta G ⊆ GLn(K)talde unipotentea, G-k exponente finitua du, konparatu [38, Kapitulua III, Lema 16.2 eta Teorema 16.5] (nahiz eta erreferentzia gorputz finituetarako izan, argumentuek exponente positiboko gorputzetarako berdin-berdin balio dute). Bestalde, Gtalde aljebraiko lineala emanda, haren erradikal unipotentea,Ru(G) izendatuko duguna, G-ko elementu unipotente guztiek osatutako azpitaldea da, edo baliokideki G-ren azpitalde unipotente konexu handiena. Horrela, Ru(G) konexua eta nilpotentea denez, G-ren erradikal ebazgarrian dago, hau da, G-ren azpitalde ebazgarri handienaren identitate osagaiaren barruan. Halaber, Gtalde aljebraiko lineala erreduktiboa dela diogu konexua bada eta Ru(G)tribiala bada. Eraikuntza horiek guztiak talde aljebraiko linealen teorian garrantzia handikoak dira, haatik soilik definizio nagusiak laburtu eta haien arteko erlazioak enuntziatuko ditugu. Izan ere, honako emaitza teknikoa baino ez dugu behar: Proposizioa 4.18 (cf. [37, Lema 17.9]).Izan bitez Gtalde aljebraiko lineal konexua eta Nharen erradikal ebazgarria. Orduan, [N,G]unipotentea da. Froga. Izan bedi Ru(G)erradikal unipotentea. Orduan, G/Ru(G)erreduktiboa da. Beraz, [37, Lema 17.9]ren arabera, N/Ru⊆Z(G/Ru)da eta, ondorioz, [N,G]⊆ Ru(G)da. Orain atal honetan bila genbiltzan emaitza froga dezakegu: Teorema 4.19. Talde Fp[[t]]-analitiko trinkoak hitzez laburrak dira. Froga. Izan bitez Gtalde Fp[[t]]-analitiko trinkoa eta whitza. Demagun w{G} finitua dela. Lehenik eta behin, Lema 4.15 dela eta, w(G)′finitua da. Ondorioz, behar izanez gero zatidura batera pasata, orokortasunik galdu gabe w(G)finituki sortutako talde abeldarra dela suposa dezakegu. Korolarioa 4.13ren arabera, existitzen da Stalde Fp[[t]]-estandarra non wlegea den. Laburtzearren izenda ditzagun Z=Z(S)eta K=Fp((t))alg,Fp((t)) gorputz lokalaren itxitura aljebraikoa. Proposizioa 2.3gatik, S/Z lineala da Fp[[t]]-ren gainean, eta, ondorioz, lineala da Fp((t)) eta Kgorputzen gainean ere. Gainera, Titsen alternatiba topologikoa (loc. cit.) dela eta, S/Z-k azpitalde ebazgarri 248 ireki bat edo azpitalde aske dentso bat du barruan. Baina S/Z taldeak lege bat betetzen duenez, birtualki ebazgarria izan behar da. Izan bedi Staldea S/Z-ren Zarizki itxitura GLn(K)-n, orudan Sbirtualki ebazgarria den Proposizioa 3.34ren arabera, eta izan bitez N S-ren erradikal ebazgarria, hau da, azpitalde ebazgarri konexu handiena eta N◦bere osagai konexua, hau da, S-ren erradikal ebazgarria. Proposizioa 4.17(i) dela eta, Nindize finituko azpitaldea da S-n, beraz, Proposizioa 4.17(ii)ren eraginez, N◦=S◦da. Izan bedi N/Z taldea S/Z-ren ebakidura N◦-rekin, orduan, behar izanez gero muina normalera pasata, Nindize finituko azpitalde normala da G-n. Proposizioa 4.18ren arabera, [N◦,N◦]unipotentea da. Bereziki, [N◦,N◦]nilpotentea da eta, Kgorputzak karakteristika positiboa duenez, exponente finitua du. Hortaz, [N, N]Z/Z exponente finituko talde nilpotentea da. Horrenbestez, [N, N]Z nilpotentea da eta Lema 4.16ren arabera, H:= [N, N, S]-k exponente finitua du. Alde batetik, H= [N, N, S]≥[N, N, N] da, eta, beraz, G/H birtualki gehienez 2klaseko talde nilpotentea da. Horrela, |w{G/H}| ≤ |w{G}| denez eta G/H birtualki nilpotentea denez, w(G/H)finitua da Turner-Smithen Teoremaren arabera (ikusi [72, Korolarioa 2]). Beste alde batetik, w(G)∩Hfinituki sortutako talde abeldarra da eta exponente finitua du, beraz finitua da. Azkenik, wG H=w(G)H H∼ =w(G) w(G)∩H isomorfismoak ematen du emaitza. Korolarioa 4.20. Izan bedi Rbat Krullen dimentsioko pro-pdomeinua. Orduan, talde R-analitiko trinkoak hitzez laburrak dira. 4.3 Laburtasuna talde R-analitiko trinkoetan Prest gaude emaitza nagusia frogatzeko. Teorema 4.21. Talde R-analitiko trinkoak hitzez laburrak dira. Froga. Korolarioa 4.20 aintzat hartuta, demagun Rpro-pdomeinuak gutxienez 2 Krull dimentsioa duela. Izan bitez Gtalde R-analitiko trinkoa eta whitz bat k aldagaitan non w{G}finitua den. Lema 4.15 dela eta, nahikoa da w-balio guztiek ordena finitua dutela frogatzea. 249 Lema 1.23ren arabera, existitzen da (S, φ)azpitalde R-estandar ireki normala non g∈Gguztietarako cg:S→S, x 7→ xgkonjokazio aplikazioak hertsiki analitikoak diren. Horrela, n=|G:S|bada, wn{G} ⊆ Sda eta, Lema 4.15en arabera, wn(G)finitua da baldin eta soilik baldin w(G)is finitua bada. Hortaz, orokortasunik galdu gabe suposa dezagun w{G} ⊆ Sdela. Izan bedi (P, m)ideal nagusietako pro-pdomeinua hau: P=Zp,char R= 0 denean, eta P=Fp[[t]],char R=ppositiboa denean. Cohenen Egitura Teoremaren arabera (ikusi Teorema 1.2), Reraztuna P[[t1, . . . , tm]]-ren finituki sortutako eraztun hedadura integrala da, m= dimKrull(R)−1izanik. Horrela, a∈m(m) bakoitzerako, izan bedi sa:P[[t1, . . . , tm]] →P, F(t1, . . . , tm)7→ F(a)ebaluazio homomorfismoa. Korolarioa 2.13k saepimorfismoa ˜sa:R→Qeraztun epimorfismora hedatzen du, non Oharra 2.14ren arabera, Q= (Q, n)pro-pdomeinua P-ren finituki sortutako eraztun hedadura integrala den, bereziki Q-ren Krullen dimentsioa 1da. Izan bedi a∈m(m),Atala 2.2ko notazioa jarraituz, froga honetan zehar Wa moduan izendatuko dugu W˜sa∈Q[[X]](l)berretura serie formalen tupla, zeinnahi W∈R[[X]](l)berretura serie formalen tuplatarako. Bereziki, Fbada Sren talde eragiketa formala, orduan Fa=F˜satalde eragiketa formala da (ikusi Korolarioa 2.8). Izan bedi Tezker transbertsal bat S-rentzat G-n, eta suposa dezagun 1∈Tdela. Hemendik aurrera S-tik eratorritako atlasaz baliatuko gara, hots, {(tS, φt)}t∈Tnon φt(x) := φ(t−1x)den (konparatu Atala 1.3). Lema 2.10 erabilita, definitu L:= nN(d)talde Q-estandarra, zeinaren talde eragiketa Fatalde eragiketa formalak definitzen duen, eta H:= T×Ltalde Qanalitikoa (2.3)ko eragiketarekin, zeina L-ren gaintaldea balitz bezala ikus daitekeen. Oroitu H-ren egitura Q-analitikoa {(tL, ψt)}t∈T,non ψt(t, l) = l, atlasak ematen duela. Atal hau bukatu arte finka dezagun (t1, . . . , tk)∈T(k)tupla, eta demagun, (4.2) dela eta, edozein l∈Nzenbakitarako wlhitz funtzioa t1S×···×tkSmultzo irekian Wlberretura serie formalen tuplak emanda dagoela, hau da, w{G} ⊆ S dela aintzat hartuta, φwl(x1, . . . , xk)=Wl(φt1(x1), . . . , φtk(xk)) ∀xj∈tjS dugu (notazioa arintzearren explizituki idatziko ez den arren, kontuan eduki Wl berretura seriea t1, . . . , tkbalioen menpekoa ere badela). Izan bedi wl:H(k)→Hhitz funtzioa ∗aeragiketarekin H-n. Lema 2.7 eta Oharra 2.11 direla eta, ψ1wl(x1, . . . , xk)=Wl a(ψt1(x1), . . . , ψtk(xk)) ∀xj∈tjL 250 Hemendik aurrera [74]ko notazioa jarraituz, R-Lie erretikulu bat heina finituko R-modulu askea den R-Lie aljebra bat da. Gorputzen gainean Adoren Teoremaren froga anitz daude (ikusi, esate baterako, [41, Kapitulua VI, 2. Atala]), eta horietako gehienetatik ondorioztatu daiteke eraikitzen den adierazpenaren maila soilik dimKL,aljebraren K-espazio bektorial dimentsioaren, menpekoa dela. Zehatzago esanda, izan bedi deg L:= min{deg φ|φL-ren adierazpen leiala da}, orduan [13] eta [60]ko argudioetan oinarrituta, ikus daiteke Rzero karakteristikako gorputza eta r= dimKLdirenean: deg L≤α2r √r(A.1) dela, α∈Rbatentzat (ikusi [58, 1.1.2 Atala] froga zehatz baterako). Alabaina, badaude gorputzen gainean deg Laztertzen duten beste hainbat lan ere; aipatu beharrekoak dira [8], [31] edota [63] lanak. Haatik, RIND orokor bat denean, aipatutako bi frogetatik ez da zuzenean ondorioztatzen deg Lzenbaki osoa soilik rk L,erretikuluaren heinaren, menpekoa denik. Areago, [74, Proposizioa 3.4]n gerora adierazpenaren maila izanen dena finitua da Reraztun noetherdarra izateagatik ideal segida bat geldikorra delako, baina ezin da zehaztu zenbat idealek osatzen duten segida hori. Apendize honetan, R-Lie erretikulu baten adierazpen leial bat eraikitzeko modu kuantitatibo bat aurkeztuko dugu, [8] eta [63]ko ideietan oinarrituz. Zehatzago: Teorema A.4. Izan bitez Rzero karakteristikako INDa eta Lrheinako R-Lie erretikulua. Orduan, deg L≤r+rr+ 1 r4r da. Orobat, aipatu karakteristika positiboko gorputzetarako Adoren Teoremaren parekoa betetzen dela, hori da Iwasawaren Teorema [40] hain zuzen ere. Are gehiago, Teorema A.2ren bertsio orokorrari, koefizienteen gorputzari beste inolako baldintzarik ezarri gabe, Ado-Iwasawaren Teoerema deitzen zaio. Karakteristika positiboan emaitza orokortasun askoz gehiagorekin eman daiteke: Teorema A.5 (cf. [19, Teorema 3]).Izan bitez Rkarakteristika positiboko eraztun trukakorra eta Lheina finituko R-Lie erretikulua. Orduan, existitzen dira Wheina finituko R-modulu askea eta φ:L→EndR(W)R-Lie aljebra monomorfismoa. 257 Hori frogatzeko nahikoa da froga originala hitzez hitz errepikatzea, eta, horrenbestez, gorputzen gainean lortzen den borne bera lortzen da R-Lie erretikuluen mailarentzat. Hots, deg L≤nrk3L, n= char Rizanik (ikusi [5, Atala 6.2.4]). Oharrak. Frogetan zehar R-modulu askeen (RINDa izanik) inguruko zenbait propietate erabiliko dira. Honatx gogoratu beharrekoak: (i) M R-modulu aske baten azpimoduluak askeak dira, eta gehienez rk(M) heina dute. Izan bitez M R-modulua eta N≤Mazpimodulua. N-ren isolatzailea M-n IsoM(N) = {x∈M| ∃r∈R\{0}non rx ∈N} azpimodulua da, eta N M-n isolatua dela diogu IsoM(N) = Ndenean. (ii) M/ Iso(N)tortsiorik gabeko R-modulua da. (iii) M R-modulu askea, N≤Mazpimodulu isolatua eta M/N R-modulu finituki sortua badira, M/N R-modulu askea da, eta rk(M) = rk(N) + rk(M/N) da. Kasu honetan, rk(M/N)zenbakiari N-ren koheina deritzogu. (iv) Mheina finituko R-modulu askea eta Nazpimodulu isolatua badira, N-k osagarria dauka M-n, hau da, existitzen da L R-modulu askea non M= N⊕Lden. A.2 Adierazpen adjuntua eta erregularra Aurkez ditzagun zein-nahi Lie aljebraren bi adierazpen. Alde batetik, x∈Lelementuak adx:L→L, y 7→ [x, y]aplikazio lineala definitzen du. Jacobiren identitatea dela eta, esleipen horrek L-ren rk Lmailako adierazpen adjuntua definitzen du, hau da, Ad: L→EndR(L), x 7→ adx. Halere, adierazpen hori orohar ez da leila, haren nukleoa L-ren zentroa baita, hau da, Z(L) := {x∈L|[x, y] = 0 ∀y∈L}. 258 Horrenbestez, LR-aljebra semisinplea bada –hots, ez badu ideal abeldar eztribialik–, adierazpen adjuntua leiala da eta deg L≤rk Lda. Bigarren adierazpena aurkezteko inguratze aljebra unibertsala definitu behar da. Definizioa A.6 (cf. [41, Kapitulua V, Teorema 1.1]).Izan bedi LR-Lie aljebra. L-ren R-tentsore aljebra TR(L) = R⊕L1⊕L2⊕···⊕Li⊕. . . da, Li=L⊗(i) . . . ⊗Ltentsio R-modulua delarik. Horren R-modulu egitura biderketa tentsorialarena da, eta biderketa (x1⊗···⊗xi)⊗(y1⊗···⊗yi) = x1⊗···⊗xi⊗y1⊗···⊗yi arauak definitzen du. Izan bedi R [u, v]−(u⊗v−v⊗u), u, v ∈L elementuek sortutako TR(L)-ren ideala. Orduan, L-ren inguratze aljebra unibertsala UR(L) := TR(L) R identitatedun R-aljebra elkarkorra da. Gainera, Leta L1elkarrekin identifikatuz gero, ι:L→ UR(L)homomorfismoa dugu. Ikus daiteke Lfinituki sortua eta RINDa direnean, ιinjektiboa dela (ikusi [74, Teorema 3.2]), eta, beraz, L⊆ UR(L)dela asumi dezakegu. Halaber, notazioa sinplifikatzearren, x1⊗···⊗xkelementua x1. . . xkmonomioa bezala idatziko dugu. Aljebra unibertsala Poincaré-Birkhoff-Witten teoremak deskribatzen du: Teorema A.7 (cf. [74, Teorema 3.2]).Izan bitez Lrheinako R-Lie erretikulua eta {x1, . . . , xr}haren oinarria. Orduan, UR(L)R-modulu askea da, eta {xα1 1. . . xαr r|αi∈N0}(A.2) monomioek oinarri bat osatzen dute. Bi monomio emanda, haien biderketa (A.2) moduko monomioen konbinazio lineal gisa adieraz daiteke, hurrenez hurren xjxi=xixj−[xi, xj]identitatea erabiliz 259 indeterminatuak ordenatzeko. Edozein LR-Lie erretikuluk UR(L)-ren gainean eragiten du ezker biderketaz. Hots, x∈Lbakoitzerako `x:UR(L)→ UR(L), u 7→ xu R-endomorfismoa dugu. Alde batetik, edozein x, y ∈L-tarako [x, y] = xy −yx da UR(L)-n eta, beraz, L:L→EndR(UR(L)), x 7→ `xR-Lie aljebra homomorfismoa da. Halaber, inguratze aljebra unibertsala identitateduna denez x6=y denean `x(1) = x6=y=`y(1) da. Ondorioz, Ladierazpen leiala da, (ezker) adierazpen erregular deituko duguna. Alta, UR(L)heina infinitukoa denez, adierazpen hori ez da finitua. Baina inguratze aljebra unibertsala karakterizatzen duen propietate hau dela eta, Lie aljebra adierazpen guztiek UR(L)-ren zatidura baten gainean ekiten dute. Teorema A.8 (Propietate unibertsala, cf. [11, Kapitulua I, § 2.1, Proposizioa 1]).Izan bitez LR-Lie erretikulua, A R-aljebra elkarkorra, [a, b] = ab −ba Lieren kortxea (a, b ∈Aguztietarako) eta ψ:L→(A, [,]) R-Lie aljebra homomorfismoa. Orduan, existitzen da ψ∗:UR(L)→A R-aljebra homomorfismo bakarra non ψ= ψ∗◦ιden. Alegia, diagrama hau trukakorra da: UR(L) LA. ψ∗ ι ψ Hain zuzen ere, propietate unibertsal hori da UR(L)-ren izenaren arrazoia. Are gehiago, froga daiteke propietate unibertsala betetzen duen edozein R-aljebra UR(L)-ri isomorfoa dela (ikusi [41, Kapitulua V, Teorema 1.1.1]). A.3 Adoren Teorema Adoren Teorema frogatzeko askotan LR-Lie erretikulutik LK:= L⊗RK Kespazio bektorialera pasako gara. Ohartu LKK-aljebra bat dela eta haren Lieren kortxetea L-ren Lieren kortxeteak tentsorizatuz induzituriko K-aplikazio lineala dela. Kontuan hartu berehalako propietate hauek: •LKespazio bektorialak rk Ldimentsioa du, 260 •L=hx1, . . . , xriRbada, orduan LK=hx1, . . . , xriKda eta •I⊴LKideal guztietarako, I∩L⊴Lideal isolatua da. Teorema hiru urratsetan frogatuko dugu. A.3.1 Lieren erretikulu nilpotenteak Lehenik eta behin, demagun Lrheinako R-Lie erretikulu nilpotentea dela. Gogoratu, LLie erretikuluaren serie zentral beherakorra errekurtsiboki γ1(L) = L, γi(L) = [γi−1(L),L]∀i≥2 moduan definitzen dela, eta Lnilpotentea dela existitzen bada γc+1(L) = {0} betetzen duen c∈Nzenbaki osoa. Zenbaki oso horietan txikienari, existituz gero, R-Lie erretikuluaren nilpotentzia klasea deritzo. Biderketa tentsoriala lineala denez, erraz ikus daiteke honako emaitza hau: Lema A.9. Izan bitez LR-Lie erretikulua eta I,H⊴Lidealak. Orduan, [I⊗RK, H⊗RK] = [I,H]⊗RK da. Beraz, Lcklaseko R-Lie erretikulu nilpotentea bada, LKcklaseko K-Lie aljebra nilpotentea da. Orduan, γ0(LK)> γ1(LK)>··· > γi(LK)>··· > γc+1(LK) = {0} K-espazio bektorialen segida hertsiki beherakorra denez, c≤dim LK=rda. Definitu Li=γi(LK)∩L⊴Lideal isolatuak, eta hartu L-ren {x1, . . . , xr}oinarria non lehenengo x1, . . . , xr1elementuak Lc-ren oinarria diren, lehenengo x1, . . . , xr2 (r2> r1) elementuak Lc−1-en oinarria diren, eta horrela hurrenez hurren. Teorema A.7ren arabera, xα:= xα1 1. . . xαr r, α = (α1, . . . , αr)∈N(r) 0 monomioek UR(L)inguratze aljebra unibertsalaren oinarria osatzen dute. Hori kontuan hartuta defini dezagun ω:UR(L)→N0∪ {∞} pisu funtzioa ondoko moduan: 261 ω(xi) = max{m|xi∈Lm}ω(xα) = Pr i=1 αiω(xi) ω(Pr i=1 cαxα) = min {ω(xα)|cα6= 0}eta ω(0) = ∞. Edozein m∈N0-tarako definitu Um(L) := {u∈ UR(L)|ω(u)> m} –Lerretikulua zein den garbi dagoenean, Umbaino ez dugu idatziko–. Ikus dezagun Um⊴UR(L)ideal isolatua dela: (i) Ohartu 0∈Umdela mguztietarako, eta ω(rx) = ω(x)eta ω(x+y)≥min{w(x), w(y)} direla x, y ∈ UR(L)eta r∈R\ {0}guztietarako. Gainera, ω([xi, xj]) ≥ ω(xi) + ω(xj)denez (edozein i, j ∈ {1, . . . , r}-tarako), ω(xy)≥ω(x) + ω(y)(A.3) da. Ondorioz, Umideala da. (ii) Izan bedi r∈R\{0},orduan rx ∈Um=⇒ω(x) = ω(rx)> m =⇒x∈Um, hau da, Umideal isolatua da. Bestalde, UR(L)/Umfinituki sortua da, Bm={xα+Um|ω(xα)≤m} multzoak sortzen baitu. Hortaz, Umkoheina finituko R-modulu aske isolatua da, eta, (A.3)ren arabera, x∈Lguztietarako `x(Um)⊆Umda. Beraz, edozein m-tarako ezker adierazpen erregularrak Lm:L→EndR(UR(L)/Um), x 7→ `x adierazpen finitua ematen du –izan bedi f∈EndR(UR(L)) non f(X)⊆Xden X⊴UR(L)ideal baterako, notazio abusu batekin, berriro ferabiliko da x+X7→ f(x) + Xmoduan definituriko f∈EndR(UR(L)/X)endomorfismoa izendatzeko–. 262 Bestetik, L∩Uc(L) = {0}da. Izan ere, x=Pr i=1 αixibada, ω(x) = ω r X i=1 αixi!≤max i=1,...,r ω(xi) = c da. Hori dela eta, LcL-ren adierazpen leial finitua da, eta horren maila bornatzeko, nahikoa da |Bc|goitik bornatzea. Alde batetik, Bc-n dauden monomio guztiek gehienez cpisua dute, eta, beraz, gehienez cpolinomio maila. Bestetik, raldagaitan gehienez cmailako monomio kopurua r+1 aldagaitan cmailako monomio kopurua da (aldagai laguntzaile bat gehituz homogeneizatu monomio guztiak zehatzmehatz cmaila izan dezaten). Lema A.10. raldagaitan cmailako monomio kopurua r+c−1 cda. Beraz, borne laño hau dugu (konparatu [31, Korolarioa 5.1]): deg L≤rk UR(L) Uc(L)≤r+c c. Atal hau bukatzeko deg L-ren borne ez oso zorrotz bat emanen dugu, baina soilik rk L-ren menpe. Izan ere, c∈ {1, . . . , r}da, hau da, c=αr da α∈ {1/r, . . . , r−1/r,1}izanik. Bestetik, Stirlingen hurbilketa formularen arabera, √2πr (r/e)r≤r!≤√2πr (r/e)re1 12r∀r∈N. da. Ondorioz, r+αr αr ≤p2π(r+αr)(r+αr)r+αr √2παr(αr)αr√2πr rre1 12(1+α)r ≤e1/12(1 + α)r √2πr √1 + α √α(1 + α)1+α ααr . Gainera, bi ezberdintzetan ezkerreko eta eskuineko terminoak asintotikoki baliokideak dira, hau da, haien zatidura 1era doa rinfinitura joan ahala. Halaber, α∈[1/r,1] denez, √1 + α √α≤√r+ 1 eta (1 + α)1+α αα≤4 263 dira. Hots, e1/12(1 + α)r √2π 1 √r √1 + α √α(1 + α)1+α ααr ≤rr+ 1 r4r. Horrenbestez, r+c c≤rr+ 1 r4r∀c∈ {1, . . . , r}.(A.4) A.3.2 Lieren erretikulu banangarriak Bigarren urratsean R-Lie aljebra adierazpen egoki bat emanen dugu R-Lie erretikulu banangarri deituko ditugunetarako. Ideal nilpotenteen batura berriro ideal nilpotentea da (konparatu [41, Kapitulua I, Proposizioa 7.6]), beraz, finituki sortutako LR-Lie aljebra guztiek badute ideal nilpotente oro barruan duen ideal nilpotente bat, L-ren erradikal nilpotente deituko duguna eta Rn(L)adieraziko dena. Bereziki, Z(L)≤Rn(L)da, eta Rn(L) ideal isolatua da, Rn(L) = Rn(LK)∩Lda eta. Horrela, Lheina finituko R-Lie erretikulua banangarria dela diogu, existitzen bada S≤LR-Lie azpialjebra bat non L=Rn(L)⊕Sden, hau da, LLie aljebra S-ren biderketa erdizuzena da Rn(L)idealarekin. Kategoria teoriaren ikuspegitik baliokidea da esatea 0→Rn(L)→L→L/Rn(L)→0 segida zehatz laburra banatu egiten dela R-Lie aljebren kategorian. Kasu banangarrirako, aurreko adierazpen erregularra eta deribazioetatik eratorritako adierazpenak konbina daitezke: Definizioa A.11. Izan bedi A R-aljebra. Orduan, D∈EndR(A)R-modulu endomorfimoa deribazioa da Leibnizen identitatea betetzen badu, hau da, D(ab) = aD(b) + D(a)b∀a, b ∈A. Deribazio guztien multzoa DerR(A)izendatzen da. Esate baterako, Jacobiren identitatearen eraginez, edozein x∈L-tarako adx∈ DerR(L)da. Bestalde, LR-Lie erretikuluaren Dderibaziotik abiatuta UR(L)-ren 264 deribazio bat eraiki daiteke, D∗deituko duguna, Leibnizen identitatea betetzea inposatuz. Hots, D∗(u1. . . ut) = t X i=1 u1. . . ui−1D(ui)ui+1 . . . ut arauaren hedapen lineala hartu eta D∗(1) = 0 definitu (identitatedun aljebra guztietarako bete behar baitu azken horrek). Hedadura hori propietate unibertsalaren kasu partikularra baino ez da (konparatu [41, Kapitulua V, Teorema 1.1(7)]). Lema A.12. Izan bitez LR-Lie erretikulu nilpotentea eta D∈DerR(L).Orduan, D∗(Um(L)) ⊆Um(L)da m∈Nguztietarako. Froga. Demagun ωpisu funtzioa L-ren {x1, . . . , xr}R-oinarriarekiko definitu dela (konparatu Azpiatala A.3.1). Orduan, D R-Lie aljebra homomorfismoa denez, D(Li)⊆Lida i∈ {1, . . . , c}guztietarako, eta, beraz, ω(D(xi)) ≥ω(xi)da i∈ {1, . . . , r}guztietarako, hau da, oinarriko elementu guztietarako. Horrenbestez, xi1. . . xit∈Umbada, (A.3) dela eta, ω(D∗(xi1. . . xit)) = min j=1,...,t {ω(xi1. . . D(xj). . . xit)} ≥ ω(xi1. . . xit)> m. Teorema A.13 (Zassenhaussen hedapena, cf. [11, Kapitulua I, § 7.3, Teorema 1] eta [41, Kapitulua VI, Teorema 2.1]).Izan bedi LR-Lie erretikulu banangarria eta izan bedi czenbaki osoa Rn(L)-ren nilpotentzia klasea. Orduan, existitzen da L-ren Φ: L→EndRUR(Rn(L)) Uc(Rn(L))  adierazpena zeina injektiboa den Rn(L)-n. Bereziki, deg Φ soilik rk Rn(L)-ren menpekoa da. Froga. Izendatu N:= Rn(L).Orduan, L=N⊕Sda, S≤LR-Lie azpialjebra baterako, eta Lema A.12 dela eta, ad∗ x(Uc(N)) ⊆Uc(N)da x∈Lguztietarako. Hortaz, definitu Φ: L=N⊕S→EndR(UR(N)/Uc(N)), n +s7→ `n+ ad∗ s. funtzioa. Ikus dezagun ΦR-Lie aljebra homomorfismoa dela. Horretarako nahikoa da Φ ([s, n]) = [Φ(s),Φ(n)] = [ad∗ s, `n] 265 dela ikustea n∈Neta s∈Sguztietarako. Alde batetik, ohartu n∈Neta D∈DerR(UR(N)) guztietarako [D, `n](u) = D◦`n(u)−`n◦D(u) = D(n)u=`D(n)(u)∀u∈N dela. Bestalde, Nideala denez, [s, n]∈Nda, eta, horrenbestez, Φ ([s, n]) = `[s,n]=`ad∗ s(n)= [ad∗ s, `n] = [Φ(s),Φ(n)] . Beraz, ΦR-Lie aljebra adierazpena da, eta haren nukleoak ebakidura tribiala du erradikal nilpotentearekin, Φ|N=Lceta Lcadierazpen leiala baitira. Bukatzeko, identitatea (A.4)ren arabera, deg Φ ≤srk Rn(L)+1 rk Rn(L)·4rk Rn(L)(A.5) da. A.3.3 Murgilketa teorema Nilpotentzia bezala, R-Lie aljebretan ebazgarritasuna defini daiteke: LR-Lie aljebraren serie deribatua errekurtsiboki L(1) := L,L(i):= L(i−1),L(i−1)∀i≥2 moduan definitzen da, eta Lebazgarria da existitzen bada `∈Nzenbaki osoa non L(ℓ)={0}den. Hori betetzen duten zenbaki osoetan txikiena da, hain zuzen ere, L-ren deribazio luzera. Are gehiago, ideal ebazgarrien batura berriro ere ebazgarria da (konparatu [41, Kapitulua I, Proposizioa 7.4]), eta, horrenbestez, finituki sortutako LLie aljebraren erradikal ebazgarria defini daiteke, hau da, beste ideal ebazgarri oro barruan duen Rs(L)ideal ebazgarria. Noski, aljebra nilpotente guztiak ebazgarriak dira eta, ondorioz, Rn(L)≤Rs(L)da. Leviren Teoremaren arabera (ikusi [41, Kapitulua III, Atala 9]), Rzero karakteristikako gorputza denean, LR-Lie aljebra oro Rs(L)⊕Smoduan deskonposatzen da, S≤LR-Lie azpialjebra semisinple bat delarik, Leviren osagarri deitua. Eskuarki deskonposizio horrek garrantzia handia du gorputzen gainean Adoren Teorema frogatzerakoan. Izan ere, lehenbiziko Rs(L)-ren adierazpen leial finitu bat eraiki ohi da, eta ondotik adierazpen hori L-ra hedatu, Zassenhaussen hedapenaren bidez (konparatu Teorema A.13). 266 Erreferentziak [1] Abercombrie, A. 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[77] Zozaya, A.: A remark on Ado’s Theorem for principal ideal domains, prestatzen. 278 Indizea δ-estalki, 210 p-adiko analitiko, 157 adierazpen ∼adjuntu, 258 ∼eratorri, 188 ∼erregular, 260 ∼finitu, 256 ∼leial, 256 ∼maila, 256 ∼matrizial, 256 lineal, 187 Ado Teorema, 256 alderantzizko ∼formal, 158 ∼funtzio, 157 aljebra ∼bakun zentral, 236 ∼banangarri, 264 ∼semisinple, 259 tentsore ∼, 259 analitiko azpibarietate ∼, 178 azpimultzo ∼, 180 azpitalde ∼, 181 barietate ∼, 155 dimentsio ∼, 162 funtzio ∼, 154, 155 hertsiki ∼, 155 talde ∼, 157 atlas, 154 ∼bateragarri, 154 ∼maximal, 154 azpimurgilketa, 172 azpibarietate, 178 ∼analitiko, 178 ∼ahul, 178 azpimultzo murgildu, 177 azpiraketa, 170 ∼ahul, 168 Baire Kategoria Teorema, 243 Baker-Hausdorff-Campell ∼formula, 190 barietate ∼analitiko, 155 ∼puru, 155 berretura serie ∼propietate unibertsal, 152 ∼eraztun, 151 ∼konbergente, 152 bianalitiko, 169 biderketa ∼funtzio, 157 ezker ∼funtzio, 168 bilipschitziar, 210 279 Chevalley talde klasiko, 232 Cohen Egitura Teorema, 150, 174 deribazio, 264 ∼luzera, 266 diferentzial, 165 dimentsio ∼analitiko, 155, 162 ∼analitiko lokal, 155 azpibarietate ∼, 178 Hausdorffen ∼, 210 Hausdorffen ∼estandar, 218 kutxa ∼, 210 Minkowski-Bouliganden ∼, 210 diskriminazioa, 198 ebakidura finituen propietate, 202 ebaluaketa ∼epimorfismo, 196 ∼funtzio, 152 ebazgarri erradikal ∼, 266 egonkortasun ∼finitu, 211 ∼kontagarri, 210 erradikal ∼ebazgarri, 248, 266 ∼nilpotente, 264 ∼unipotente, 248 erreduktibo talde ∼, 248 eskalare murrizketa, 176 espazio ultrametrikoa, 150 estandar filtrazio serie ∼, 214 talde ∼, 157 ezberdintza triangular gogor, 150 faktorizazio bakarreko domeinu (FBD), 185 filtrazio serie, 207 ∼estandar, 159, 214 ∼normal, 211 p-berretura ∼, 208 filtro, 201 Hall-Petrescu formula, 246 Hall-Witt identitate, 164 Hausdorff ∼dentsitate, 234 ∼dimentsio, 210 ∼dimentsio estandar, 209, 218 ∼espektro, 208 ∼espektro estandar, 209, 218 ∼neurri, 210 Hilbert ∼funtzio, 215 ∼polinomio, 215 hitz, 239 ∼azpitaldea, 240 ∼balio, 239 ∼baliokide, 239 ∼behe zentral, 241 ∼deribatu, 241 ∼eliptikoa, 243 ∼huts, 239 ∼zabalera, 243 ∼funtzio, 239 Burnsideren ∼, 241 kommutadore ∼, 241 hitzezko ∼eliptikotasun, 242 laburtasun ∼, 242 sendotasun ∼, 242 hondar gorputz, 149 280 hondar-heina, 170 ideal nagusietako domeinu (IND), 150 identitate osagai, 247 Igotze Teorema, 196 inguratze aljebra unibertsal, 259 propietate unibertsal, 260 inon konstante funtzio, 252 isolatu modulu, 258 isolatzaile, 258 Iwasawa Teorema, 257 Jacobi identitate, 164, 256 jacobiar, 165 karta, 154 ∼bateragarri, 154 ∼erregular, 155 ∼moldatu, 182 katearen erregela, 165 koheina, 181, 258 koordenatu ∼aldaketa, 169 ∼aldaketa funtzio, 154 ∼sistema, 156 ∼sistema kanoniko, 156 Krull Ebakidura Teorema, 149 kutxa-dimentsio, 210 ∼estandar, 222 behe ∼estandar, 216 behe ∼, 210 goi ∼, 210 labur hitzez ∼, 242 laburtasun, 242 ∼gogorra, 252 Leibniz identitate, 264 Levi ∼osagai, 266 Teorema, 266 Lie algebra ∼nilpotentea, 261 Lie aljebra, 165, 255 ∼adierazpen, 256 ∼ebazgarri, 266 ∼graduatua, 234 ∼homomorfismo, 256 ∼konkretua, 256 ∼perfektu, 235 ∼zentroa, 258 ∼elkartua, 165 Lie erretikulu, 257 ∼berretura-bete, 190 Lieren kortxete, 164, 255 lineal adierazpen ∼, 187 talde ∼, 187 talde ∼berezi, 231 talde ∼orokor, 156, 228 talde aljebraiko ∼, 243 talde aljebraiko ∼, 247 lokal eraztun ∼, 149 eraztun homomorfismo ∼, 191 marjinal azpitalde ∼, 240 Minkowski-Bouligand dimentsio, 210 monotono, 210 morfismo formal, 166 multzo afin, 228 murgilketa, 170 281 ∼ahul, 168 nilpotente erradikal ∼, 264 klase, 261 ortogonal berezi talde ∼, 232 Poincaré-Birkhoff-Witt Teorema, 259 pro-pdomeinu, 150 profinitu espazio, 252 puntu erregular, 181 Schur Teorema, 246 serie ∼deribatu, 266 ∼zentral beherakor, 261 sinplektiko talde ∼, 232 talde eragiketa formal, 158 ∼batukor, 158 ∼biderkakor, 158 Tits alternatiba topologiko, 231 topologia m-adiko, 149 topologikoki nilpotente, 152 translazio, 156 ultraberretura, 203 ultrabiderketa, 202 ultrafiltro, 201 ∼ez-nagusi, 201 ∼teorema, 202 ultrametriko espazio, 150 uniformeki berretura-bete, 189 uniformizatzaile, 151 unipotente azpitalde ∼, 248 erradikal ∼, 248 matrize ∼, 248 Zassenhauss hedapen, 265 zentroide, 236 Zorn Lema, 236 282