A COMPARISON OF LOCALLY ANALYTIC GROUP COHOMOLOGY AND LIE ALGEBRA COHOMOLOGY FOR p-ADIC LIE GROUPS SABINELECHNER 2 Abstract. Themainresultofthisworkisanewproofandgeneraliza- 1 tionof Lazard’scomparison theoremof locally analyticgroup cohomol- 0 ogy with Lie algebra cohomology for K-Lie groups, where K is a finite 2 extension of Q . p n a J 2 2 Contents ] A 1. Introduction 1 R 2. Preliminaries and Notation 3 2.1. Formal group law, Lie algebra and Universal enveloping algebra 3 . h 2.2. Hopf algebra structures 5 t a 2.3. Cohomology complexes 8 m 3. Isomorphism of complete Hopf algebras 10 [ 4. Quasi-isomorphism of complexes 13 1 5. Locally analytic group cohomology 15 v 5.1. K-Lie groups and standard groups 15 0 5.2. Limit morphism 18 5 5 5.3. Proof of the Comparison Theorem for standard groups 22 4 References 24 . 1 0 2 1 : 1. Introduction v i X Lazard’s comparison theorem was one of the main results in his work on p-adic groups [Laz65]. It relates locally analytic group cohomology with Lie r a algebra cohomology for Q -Lie groups in two steps. FirstLazard worked out p an isomorphism between locally analytic group cohomology and continuous group cohomology and secondly between continuous group cohomology and Lie algebra cohomology. The latter is obtained from a difficult isomorphism between the saturated group ring and the saturated universal enveloping algebra. HuberandKingsshowedin[HK11]thatonecandirectlydefineamapfrom locally analytic group cohomology to Lie algebra cohomology by differenting TheseresultsarepartofmyPh.DthesisandIwouldliketothankmyadviserAnnette Huber-Klawitter for her supervision and her inspiring support. The thesis was made possible by grants of DFG Forschergruppe "Algebraische Zykel und L-Funktionen" in Freiburg. 1 2 SABINELECHNER cochains and that in the case of smooth algebraic group schemes H over Z p with formal group H ⊂ H(Z ) the resulting map p Φ : Hn(H,Q )→ Hn(h,Q ) la p p coincides, after identifying continuous group cohomology with locally an- alytic group cohomology, with Lazard’s comparison isomorphism ([HK11, Theorem 4.7.1]). Serre mentioned to the aforementioned authors that this was clear to him at the time Lazard’s paper was written, however it was not included in the published results. In their joint work with N. Naumann in [HKN11] they extended the comparison isomorphism for K-Lie groups attached to smooth group schemes with connected generic fibre over the in- tegers of K ([HKN11, Theorem 4.3.1]). The aim of this thesis is to use this simpler map to obtain an independent proof of Lazard’s result. The context in which Huber and Kings worked out the new description of the Lazard isomorphism is the construction of a p-adic regulator map in complete analogy to Borel’s regulator for the infinite prime. The van Est isomorphism between relative Lie algebra cohomology and continuous group cohomology is replaced by the Lazard isomorphism. Their aim is to use this construction of a p-adic regulator for attacking the Bloch-Kato conjecture for special values of Dedekind Zeta functions. Our strategy to prove the comparison isomorphism between locally ana- lyticgroup cohomology and Liealgebra cohomology isto traceitback tothe case of a formal group law G. Hence the first step is to obtain an isomor- phism of formal group cohomology with Lie algebra cohomology (Corollary 4.3) Φ˜ :Hn(G˜,R) → Hn(g,R), where R is an integral domain of characteristic zero. The tilde over Φ and over the formal group law G indicate that one has to modify the formal group cohomology while working with coefficients in R. This means that if the ring of functions to G, called O(G), is given by a formal power series ring over R one has to allow certain denominators. However, we will prove in Lemma 5.23 that functions of this modified ring of functions O˜(G) still converge, with the same region of convergence as the exponential function. Let g be the Lie algebra associated to the formal group law G and U(g) its universal enveloping algebra. Then an essential ingredient to the first step is a morphism of complete Hopf algebras (Proposition 3.1) β⋆ :O(G) → U(g) from the ring of functions to the universal enveloping algebra. We show in Proposition 3.8 that this morphism is an isomorphism if we consider the modified ring of functions O˜(G). Furthermore, we prove in Theorem 4.2 that this isomorphism extends to a quasi-isomorphism of the corresponding complexes and hence to the above isomorphism Φ˜. The second step in the proof of the Main Theorem is a Comparison The- orem for standard K-Lie groups. These standard groups are K-Lie groups associated to a formal group law G, see Definition 5.2. Comparison Theorem for standard groups [Thm.5.7]. Let G be a formal group law over R and let G(h) be the m-standard group of level h to LAZARD ISOMORPHISM 3 G with Lie algebra g⊗ K. Then the map R Φ :Hn(G(h),K) → Hn(g,K) s la given by the continuous extension of f ⊗···⊗f 7→ df ∧···∧df , 1 n 1 n for n ≥ 1 and by the identity for n = 0 is an isomorphism for all h > h = 1 . 0 p−1 The Main Theorem can then be deduced from the Comparison Theorem for standard K-Lie groups, since every K-Lie group contains an open sub- group, which is standard, see Lemma 5.10. Our approach to the proof of the Comparion Theorem for standard K-Lie groups is as follows. In a first step, we will show using the isomorphism Φ˜ that the limit morphism Φ : Hn(Ola(G(0)•) ,K) → Hn(g,K), ∞ e associated to the ring of germs of locally analytic functions in e, denoted by Ola(G(0)) , is an isomorphism. Then injectivity of Φ follows from a e s spectral sequence argument. The proof of this injectivity part will be analo- gous to the proof of Theorem 4.3.1 in [HKN11], however independent of the work of Lazard [Laz65]. For surjectivity we will again use the isomorphism statement for formal group cohomology of Corollary 4.3 in addition to the aforementioned fact that functions of O˜(G) still converge. The article is organized as follows: In Section 2 we give a number of def- initions and well-known facts concerning formal groups, Lie algebras, Hopf algebra structures and cohomology complexes. Section 3 deals, firstly, with the existence of a morphism of complete Hopf algebras between the ring of functionsofaformalgrouplawandthedualoftheuniversalenveloping alge- bra. Secondly we consider the cases where this morphism is an isomorphism and thirdly, considering the modified ring of functions, we can prove that the morphism is in this modified case actually an isomorphism of complete Hopf algebras. In Section 4 we show that the isomorphism of Section 3 can be extended to a quasi-isomorphism of the corresponding complexes and we will give an explicit description, which will be identical to the description of the comparison map in [HKN11] and hence to Lazard’s map. The last Section 5 gives the proof of the Main Theorem. We willbegin by fixingsome notation in Subsection 5.1, in order to formulate the Comparison Theorem forstandardgroups. Theproofofthistheoremwillbegivenintheremaining two subsections. 2. Preliminaries and Notation The main objects we are dealing with are introduced in this section. We denote by R an integral domain of characteristic zero. 2.1. Formal group law, Lie algebra and Universal enveloping alge- bra. Definition 2.1. [Haz78, 9.1 Definitions] Let X = (X ,...,X ) and Y = 1 m (Y ,...,Y ) betwo sets of m variables. An m-dimensional formal group law 1 m over R is an m-tuple of power series G(X,Y) = (G(1)(X,Y),...,G(m)(X,Y)) 4 SABINELECHNER with G(j)(X,Y) ∈ R[[X,Y]] such that for all j = 1,...,m (1) G(j)(X,Y) = X +Y + m γj X Y +O(d ≥ 3),γj ∈ R j j l,k=1 lk l k lk (2) G(j)(G(X,Y),Z) = G(j)(PX,G(Y,Z)), where the notation O(d ≥ n) stands for a formal power series whose homo- geneous parts vanish in degree strictly less than n. If in addition G(j)(X,Y) = G(j)(Y,X) holdsforallj = {1,...,m},thentheformalgrouplawiscalledcommutative. The ring of functions O(G) to a formal group law G is the ring of formal powerseriesinmvariablest(1),...,t(m) overR,i.e. O(G) = R[t(1),...,t(m)]. Proposition 2.2. [Haz78, Appendix A.4.5] Let G(X,Y) be an m- dimensional formal group law over R. Then there exists a power series s(X) such that G(X,s(X)) = 0. Remark. 2.3. The proof of the existence of the power series s(X) gives an explicit construction of this power series. The first step of the construction yields s(X) = −X mod (degree 2), a fact we will need later in Section 3. Definition 2.4. [Haz78, 9.4 Homomorphisms and isomorphisms] Let G(X,Y) and G′(X,Y) be m-dimensional formal group laws over R. A ho- momorphism G(X,Y)→ G′(X,Y) over R is an m-tuple of power series α(X) in n indeterminantes such that α(X) ≡ 0 mod (degree 1) and α(G(X,Y)) = G′(α(X),α(Y)). Thehomomorphismα(X)isanisomorphism ifthereexistsahomomorphism β(X): G′(X,Y) → G(X,Y) such that α(β(X)) = β(α(X)). Lemma2.5. TheringoffunctionsO(G)toaformalgrouplawGiscomplete with respect to the topology induced by the following descending filtration (1)FiO(G) = {f ∈O(G)|all monomials of f have total degree ≥ i}. Notation. 2.6. • Wedenotebythesign⊗ˆ thecompletedtensorprod- uctwithrespecttotheabovetopology. ThuswecanidentifyO(G)⊗ˆn with the ring of formal power series in nm indeterminates R[[t(1),...t(m),t(1),...,t(m),...,t(1),...,t(m)]]. 1 1 2 2 n n (j) • For the elements t of the ring of functions O(G) we will use equiv- i alently the notation 1⊗...⊗1⊗t(j) ⊗1⊗...⊗1 (with t(j) at the i-th entry) for all j = 1,...,m. • For simplicity we write (1) (m) – t for t ,...,t , 1 1 1 (1) (m) (1) (m) (1) (m) – t for t ,...,t ,t ,...,t ,...,t ,...,t . 1,...,n 1 1 2 2 n n • We use the general multi-index notation j for the tuple (j ,...,j ). 1 m • If m = 1 or n = 1 we skip the upper, respectively lower index and write t for t(1) respectively t(j) for t(j). i i 1 LAZARD ISOMORPHISM 5 Lie algebra and universal enveloping algebra Let R[t ] be the polynomial ring over R in m variables and let ∂ for all 1 ∂t(j) 1 j ∈ {1,...,m} be the j-th partial derivative. Let Der (O(G),O(G)) denote R the set of R-derivations of O(G). Then Der (O(G),O(G)) is a free O(G)- R module on the basis ∂ ,..., ∂ . If d,d ,d ∈ Der (O(G),O(G)) and ∂t(1) ∂t(m) 1 2 R 1 1 r ∈ R, then the mappings rd,d +d and [d ,d ] := d d −d d are also 1 2 1 2 1 2 2 1 derivations, see [Bou98b, Chap.III §10.4]. Thus, the set Der (O(G),O(G)) R is an R-module which is also a Lie algebra. Let e denote the j-th partial derivative ∂ evaluated at 0. We denote j ∂t(j) by g the free R-module on the basis e ,...,e1 . If L = m r e ∈ g and 1 m j=1 i j f ∈ O(G), then we can apply L to f by P m ∂f (2) L(f) = r (0). j Xj=1 ∂t(j) 1 Hence we can identify g with the set of R-derivations of O(G) into R, where R is considered as a O(G)-module via evaluation at zero. We denote this set by Der (O(G),R). 0 j The elements γ of Definition 2.1 of a formal group law G define a Lie lk algebra structure on g, as follows (see [Haz78, Chap.II, p.79]): m j j (3) [e ,e ] = (γ −γ )e . l k j=1 lk kl j X However, g inherits also a Lie algebra structure by the canonical bijection of Der (O(G),O(G)) and Der (O(G),R). One can check that both definitions R 0 of the Lie-bracket coincide. Let U(g) denote the universal enveloping algebra of g. The theorem of Poincaré-Birkhoff-Witt ([Bou98a, Chap.I.2.7, Thm. 1., Cor. 3.]) says that the underlying set of U(g) is the polynomial ring R[e ,...,e ]. Therefore 1 m we denote an arbitrary element of U(g) by c ej with c ∈ R. j j P 2.2. Hopf algebra structures. Thealgebras O(G)andU(g)carrya(com- plete) Hopf algebra structure. In the following we will describe the maps defining these structures. As a reference for the definition of a Hopf algebra onecantakethebookofM.E.SweedleraboutHopfalgebras,[Swe69,Chap.I- VI.], or the book of Ch. Kassel about Quantum groups, [Kas95, Chap.III]. For the definition of a complete Hopf algebra we have to consider complete R-modules, i.e. topologized R-modules which are complete with respect to a given topology. In our cases this topology will come from a descending fil- tration {FnM} on the R-module M. By replacing R-modules by complete R-modules, and also tensor products by complete tensor products, one can define in the same way as for Hopf algebras a complete Hopf algebra over R. Proposition 2.7. [Haz78, Chap. VII.36] Let G be a formal group law. Then the ring of functions on G carries a complete Hopf algebra structure (O(G),▽,η,µ,ǫ,s). The maps are given by ▽: O(G)⊗ˆ O(G) → O(G) η : R → O(G) f ⊗g 7→ f ·g 1 7→ 1 6 SABINELECHNER µ: O(G) → O(G)⊗ˆ O(G) ǫ : O(G) → R t(i) 7→ G(i)(t(1),...,t(m),t(1),...,t(m)) f 7→ f(0) 1 1 2 2 s: O(G) → O(G) t(i) 7→ s(t(i)), where the antipode map s is given by Proposition 2.3 by the condition that G(i)(t(1),...,t(m),s(t(1)),...,s(t(m))) = 0. Definition and Proposition 2.8. Let D be the continuous dual of O(G), i.e. D = O(G)◦ = limHom (O(G)/FnO(G),R), where the filtration was −→ R given in Lemma 2.5. Then the complete Hopf algebra structure on O(G) yields a Hopf algebra structure on its continuous dual (D,▽,η,µ,ǫ,s) given by dualizing the structure morphsims of (O(G),▽,η,µ,ǫ,s). Proof. See [Haz78, Chap.VII.36] and note that D is in our case actually a Hopf algebra, since we didn’t require that the antipode s is a D-module homomorphism. (cid:3) Note that we have associated to a formal group law G the complete Hopf algebra O(G) and the Hopf algebra D. These objects are dual to each other, where one gets from O(G) to D by taking continuous linear duals and from D to O(G) by taking linear duals. This duality extends to the categories formed by these objects and is known as Cartier duality, see for example [Die73, Chap.I.2]. Proposition 2.9. [Haz78, Chap.II.14.3] Let g be a Lie algebra and U(g) the universal enveloping algebra of g. Then U(g) carries a Hopf algebra structure (U(g),▽,η,µ,ǫ,s). The maps are given by: ▽: U(g)⊗U(g) → U(g) η : R → U(g) x⊗y 7→ x·y 1 7→ 1 µ: g → U(g)⊗U(g) ǫ : g → R L 7→ L⊗1+1⊗L L 7→ 0 s: g → U(g) L 7→ −L. Letgbeanm-dimensionalLiealgebraoverRwhichisfreeasanR-module with basis e ,...,e and let U(g) be the universal enveloping algebra of g. 1 m Then we denote by U⋆(g,R) := Hom (U(g),R) or by U⋆ - if g and R are R clear from the context - the R-linear dual of U(g). Let dj1t(1)···djmt(m) be the dual basis of eji1···ejmm with ji ∈ {1,...n} for all i∈ {1,...m}. Then U⋆ has a ring structure with underlying set (4) R{{dt}} := Rdj1t(1)···djmt(m) j Y andthetwobinaryoperations +asusualadditionandmultiplication •given by the comultiplication of U(g): •: U⋆⊗U⋆ → U⋆ ψ⊗ϕ 7→ [x 7→ ρ(ψ⊗ϕ)(µ(x))], where ρ : U⋆⊗U⋆ → (U(g)⊗U(g))⋆ is the linear injection given by ρ(ψ⊗ϕ)(x⊗y)= ψ(x)·ϕ(y). LAZARD ISOMORPHISM 7 Analogous to O(G) we can define a filtration on U⋆ by (5) FiU⋆ = ϕ ∈ U⋆|for all monomials djt is |j| ≥ i , for all i ∈ N, such t(cid:8)hat U⋆ is a completed ring with respect(cid:9)to the topology induced by this filtration. And we can identify U⋆⊗ˆ U⋆ with (U(g)⊗U(g))⋆. The underlying set of (U⋆)⊗ˆn is given by R{{dt ,...,dt }}, where we use 1 n the multi-index notation drt - or equivalently 1⊗...⊗1⊗drt⊗1⊗...⊗1 i with drt at the i-th entry - for dr1t(1)···drmt(m). i i Proposition 2.10. Let g be a free Lie algebra, U(g) the universal enveloping algebra of g and U⋆ the linear dual of U(g). Then U⋆ carries a complete Hopf algebra structure (U⋆,•,η,µ,ǫ,s). The maps are given by: •: U⋆⊗ˆ U⋆ → U⋆ η : R → U⋆ ψ⊗ϕ 7→ [x 7→ ρ(ψ⊗ϕ)(µ(x))] 1 7→ [x 7→ ǫ(x)] µ: U⋆ → U⋆⊗ˆ U⋆ ǫ : U⋆ → R ϕ 7→ [x⊗y 7→ ϕ(▽(x⊗y))] ϕ 7→ ϕ(1) s: U⋆ → U⋆ ϕ 7→ [x 7→ ϕ(s(x))]. Proof. Notefirstthatagainsince•isacontinuous mapitisenoughtodefine this map on ϕ⊗ψ ∈ U⋆⊗U⋆. After the definition of a Hopf algebra we have to check that (U⋆,•,η,µ,ǫ) is a complete bialgebra. For this see [Swe69, Chap.I-IV] and note that the finiteness condition in Sweedlers book can in our case be replaced by the identification (U(g)⊗U(g))⋆ ∼= U⋆⊗ˆ U⋆. Secondly we have to show that s is actually an antipode, i.e. that •◦(s⊗1)◦µ = η◦ǫ = •◦(1⊗s)◦µ but this can be easily verified from the antipode condition of U(g). (cid:3) In the following lemma, we will provide explicit formulas for the multi- plication and comultiplication in U⋆. Especially the explicit formula for the multiplication will play an essential role in the next section. Lemma 2.11. [Ser06, Chap.V.5-6] Let g be an m-dimensional Lie algebra over R which is free as an R-module with basis e ,...,e and let U⋆ be the 1 m linear dual of the universal enveloping algebra of g. Let dj1t(1)···djmt(m) be the dual basis of ej1···ejm with j ∈ {1,...n} for all i ∈ {1,...m}, so i m i that an element of U⋆ is of the form c dj1t(1)···djmt(m) with c ∈ R. j j j Then multiplication and comultiplicatioPn in U⋆ are given by the continuous R-linear extension of •: U⋆⊗U⋆ → U⋆ and µ: U⋆ → U⋆⊗ˆ U⋆ drt⊗dst 7→ r+s dr+st drt 7→ dlt⊗dkt. r l+k=r (cid:0) (cid:1) P Example 2.12. Let m = 1. Then the explicit formulas for multiplication and comultiplication amount to dt•···•dt = dt•···•dt •2d2t =n!dnt n−times (n−2)−times | {z } | {z } µ(dt) = 1⊗dt+dt⊗1. 8 SABINELECHNER 2.3. Cohomology complexes. Definition and Proposition 2.13. [Kas95, Chap.XVIII.5] Let (H,▽,η,µ,ǫ,s) be a complete Hopf algebra over R. Set Tn(H) = H⊗ˆn if n > 0 and T0(H) = R. We define linear maps ∂0,...,∂n+1 from Tn(H) to n n Tn+1(H) by the continuous extension of ∂0(x ⊗···⊗x ) = 1⊗x ⊗···⊗x , n 1 n 1 n ∂n+1(x ⊗···⊗x ) = x ⊗···⊗x ⊗1, n 1 n 1 n ∂i(x ⊗···⊗x ) = x ⊗···⊗x ⊗µ(x )⊗x ⊗···⊗x , n 1 n 1 i−1 i i+1 n if 1 ≤ i ≤ n. If n = 0, we set ∂0(1) = ∂1(1) = 1. We have ∂j ∂i = 0 0 n+1 n ∂i ∂j−1 for all integers i,j such that 0 ≤ i < j ≤ n+ 2. We define the n+1 n differential ∂ :Tn(H) → Tn+1(H) by n+1 (6) ∂ = (−1)i∂i. n i=0 X Then ∂ ◦ ∂ = 0 and we obtain a cochain complex (T•(H),∂) called cobar complex of the complete Hopf algebra H. In the case of the ring of functions O(G) to a formal group law and in the case of the dual of the universal enveloping algebra of g we will use the following notation. Definition2.14. LetGbeaformalgrouplaw. Aninhomogeneous n-cochain of G with coefficients in R is an element of O(G)⊗ˆn. We will denote the set of inhomogeneous n-cochains also by Kn(G,R). The coboundary homomor- phisms ∂n :Kn(G,R) → Kn+1(G,R) of definition 2.13 transform into: ∂n(f)(t )= f(t ) 1,...,n+1 2,...,n+1 n + (−1)if(t ,...,G(1)1(t ,t ),...,G(m)(t ,t ),...,t ) 1 i i+1 i i+1 n+1 i=1 X +(−1)n+1f(t ). 1,...,n We obtain a cochain complex (K•(G,R),∂) whose cohomology group Hn(G,R) is called n-th group cohomology of G with coefficients in R. Definition and Proposition 2.15. [NSW00, Chap. I.2] Let g be a Lie algebra over R and let U(g) be its universal enveloping alge- bra. An inhomogeneous n-cochain of U(g) with coefficients in R is an element of Hom ((U(g))⊗n,R). The coboundary homomorphisms R ∂n : Hom ((U(g))⊗n,R) → Hom ((U(g))⊗n+1,R) given by u R R n ∂n(u ,...u )= f(u ,...,u ) + (−1)if(u ,...,u u ,...,u ) u 1 n+1 2 n+1 1 i i+1 n+1 i=1 X + (−1)n+1f(u ,...u ) 1 n define a cochain complex (Hom (U•,R),∂ ). R u Remark. 2.16. Sincethesetofinhomogeneous n-cochainscanbeidentified with U⋆⊗ˆn, the definition of the coboundary homomorphisms of Definition 2.15 is equivalent to the definition of the differential defined by (6) of Defi- nition 2.13 for the complete Hopf algebra U⋆. LAZARD ISOMORPHISM 9 Definition and Proposition 2.17. Ahomogeneous n-cochain ofU(g) with coefficients in R is an element of Hom ((U(g))⊗n+1,R), where U(g)⊗n is U(g) considered as an U(g)-module via the following operation: u.(u ,...u ) = (uu ,...u ). 0 n−1 0 n−1 The map ιn : Hom (U(g)⊗n,R) → Hom (U(g)⊗n+1,R) R U(g) ϕ 7→ [(u ,...,u ) 7→ ϕ(u ,...u )] 0 n 1 n [(u ,...,u ) 7→ ϕ(1,u ,...u )] →7 ϕ 1 n 1 n is an isomorphism from the set of inhomogeneous to the set of homogeneous n-cochains. If we consider the following coboundary homomorphisms ∂n : Hom ((U(g))⊗n+1,R) → Hom ((U(g))⊗n+2,R) uh U(g) U(g) defined by ∂n = ιn+1◦∂n ◦(ιn)−1 uh u we obtain a complex (Hom (U•,R),∂ ) of homogeneous n-cochains and U(g) h uh ιn yields an isomorphism ι : (Hom (U•,R),∂ ) → (Hom (U•,R),∂ ) of R u U(g) h uh complexes. Proof. It is enough to prove that ι is an isomorphism, since the remaining statement can be easily deduced from this. We check first that ιn(ϕ) is U(g)-invariant: (u.ιn(ϕ))(u ,...u ) = ιn(ϕ)(uu ,...u )= ϕ(u ,...u ) = ιn(ϕ)(u ,...u ). 0 n 0 n 1 n 0 n Secondly we show that ιn and (ιn)−1 are inverse to each other: (ιn)−1◦ιn(ϕ)(u ,...,u ) = ιn(ϕ)(1,u ,...,u )= ϕ(u ,...,u ) 1 n 1 n 1 n ιn◦(ιn)−1(ϕ)(u ,...,u ) = (ιn)−1(ϕ)(u ,...,u ) =ϕ(1,u ,...,u ) 0 n 1 n 1 n (ϕishomogeneous) = (u0.ϕ)(1,u1,...,un) = ϕ(u0,u1,...,un). (cid:3) For the following definitions let g be an m-dimensional Lie algebra over R which is free as an R-module with basis e ,...,e . Let ng be the n-fold 1 m exteriorproductofgwithbasis{ei1∧...∧ein |i1 < ... < inV},ij ∈{1,...,m}. We endow R with the trivial g-action. Definition and Proposition 2.18. cite[Chap. IV.3]knapp The set Hom ( ng,R) is called the set of n-cochains of g with coefficients in R R and denVoted by Cn(g,R). Note that the rank of Cn(g,R) over R is m . n The boundary operators ∂′n :Cn → Cn+1 are given by the formula (cid:0) (cid:1) ∂′n(ω)(e ∧...∧e ) = (−1)r+sω([e ,e ]∧e ∧...∧e ) , i1 in+1 1≤r<s≤n+1 ir is i1 in+1 r,s X where the notation ([e ,e ]∧e ∧...∧e ) indicates that the elements ir is i1 in+1 r,s e and e are omitted. We thus obtain, after assuring ourself that ∂′2 = 0, ir is a complex (C•(g,R),∂′) whose cohomology group Hn(g,R) is called n-the Lie algebra cohomology of g with coefficients in R. 10 SABINELECHNER Definition 2.19. [Kna88, Chap. IV.3] Let U(g) be the universal enveloping algebra of g. Set i V (g) = U(g)⊗ g i ^ for all i ∈ {0,1,2,...} with the g-module structure induced by the action on the first factor. The differential dn−1 : V (g) → V (g) is given by the n n−1 formula dn−1(u⊗e ∧...∧e ) = (−1)k+l(u⊗[e ,e ]∧e ∧...∧e ) i1 in 1≤k<l≤n ik il i1 in k,l X n + (−1)j+1(ue ⊗e ∧...∧e ) , j=1 ij i1 in j X where the notation (e ∧ ... ∧ e ) , respectively (e ∧ ... ∧ e ) again i1 in k,l i1 in j indicates that the elements e and e , respectively e are omitted. This ik il ij leads, after assuring ourselves that d2 = 0, to a complex, called Koszul complex. Thefollowingtwo propositions relate theKoszulcomplex firsttothestan- dardhomogeneous complex ofU(g)andsecondlytotheLiealgebracomplex. Proposition 2.20. [CE56, Chap. XIII.7, Theorem 7.1] Let (V(g)•,d) be the Koszul complex defined above. Then the map ν : Hom (U•,R) → U(g) h Hom (V(g)•,R) induced by the anti-symmetrisation map U(g) n as : g → U⊗n n ^ given by as (e ∧...∧e ) = sgn(α)e ⊗···⊗e , n i1 in Xα∈Sn iα(1) iα(n) with i ∈ {1,...,m}, is a quasi-isomorphism of complexes. j Proposition 2.21. [Kna88, Chap. IV.3-6] Let (V(g)•,d) be the Koszul com- plex defined above. Then the map κ : Hom (V(g)•,R) → C•(g,R) given U(g) by n κn :Hom (V (g),R) → Hom ( g,R) U(g) n R ^ f 7→ [(e ∧···∧e )7→ f(1⊗e ∧...∧e )] i1 in i1 in is an isomorphism of complexes. 3. Isomorphism of complete Hopf algebras Let R be an integral domain of characteristic zero and let G be an m- dimensional formal group law and O(G) the ring of functions to G (see Definition 2.1). Let g be the associated free m-dimensional Lie algebra over R to G which is free as an R-module with basis e ,...,e (see description 1 m around (3)) and let U⋆ be the linear dual of the universal enveloping algebra of g. This section describes the desired map from O(G) to U⋆ as a compo- sition of maps O(G) → D⋆ → U⋆, where D⋆ is the linear dual of D and the latter map is defined by dualizing the map U(g) → D.