Torsors, Reductive Group Schemes and Extended Affine Lie Algebras
aa r X i v : . [ m a t h . R A ] F e b Torsors, Reductive Group Schemes and ExtendedAffine Lie Algebras
Philippe Gille and Arturo Pianzola , UMR 8553 du CNRS, Ecole Normale Sup´erieure, 45 rue d’Ulm, 75005 Paris,France. Department of Mathematical Sciences, University of Alberta, Edmonton, AlbertaT6G 2G1, Canada. Centro de Altos Estudios en Ciencia Exactas, Avenida de Mayo 866, (1084)Buenos Aires, Argentina.
Abstract
We give a detailed description of the torsors that correspond to multiloop al-gebras. These algebras are twisted forms of simple Lie algebras extended overLaurent polynomial rings. They play a crucial role in the construction of Ex-tended Affine Lie Algebras (which are higher nullity analogues of the affineKac-Moody Lie algebras). The torsor approach that we take draws heavilyfor the theory of reductive group schemes developed by M. Demazure and A.Grothendieck. It also allows us to find a bridge between multiloop algebras andthe work of F. Bruhat and J. Tits on reductive groups over complete local fields.
Keywords:
Reductive group scheme, torsor, multiloop algebra. Extended AffineLie Algebras.
MSC 2000
Contents Loop, finite and toral torsors 16 k –loop adjoint groups . . . . . . . . . . . . . . . 688.4 Action of GL n ( Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 R -groups. . . . . . . . . . . . . . . 819.2.2 Applications to the classification of EALAs in nullity 2. . . . . . . . 879.2.3 Rigidity in nullity 2 apart from type A. . . . . . . . . . . . . . . . . 909.2.4 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 G F , E and simply connected E in nullity d A . . . . . . . . . . . . . . . . . . . . . . . . . . 110
14 Invariants attached to EALAs and multiloop algebras 11115 Appendix 1: Pseudo-parabolic subgroup schemes 112 GL n, Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11315.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
16 Appendix 2: Global automorphisms of G–torsors over the projective line117
To our good friend Benedictus Margaux
Many interesting infinite dimensional Lie algebras can be thought as being “finitedimensional” when viewed, not as algebras over the given base field, but rather asalgebras over their centroids. From this point of view, the algebras in question looklike “twisted forms” of simpler objects with which one is familiar. The quintessentialexample of this type of behaviour is given by the affine Kac-Moody Lie algebras.Indeed the algebras that we are most interested in, Extended Affine Lie Algebras (orEALAs for short), can roughly be thought of as higher nullity analogues of the affineKac-Moody Lie algebras. Once the twisted form point of view is taken the theoryof reductive group schemes developed by Demazure and Grothendieck [SGA3] arisesnaturally.Two key concepts which are common to [GP2] and the present work are thoseof a twisted form of an algebra, and of a multiloop algebra . At this point we brieflyrecall what these objects are, not only for future reference, but also to help us redacta more comprehensive Introduction. ***3nless specific mention to the contrary throughout this paper k will denote a fieldof characteristic 0 , and k a fixed algebraic closure of k. We denote k – alg the categoryof associative unital commutative k –algebras, and R object of k – alg. Let n ≥ m > R = R n = k [ t ± , . . . , t ± n ] and R ′ = R n,m = k [ t ± m , . . . , t ± m n ] . Forconvenience we also consider the direct limit R ′∞ = lim −→ R n,m taken over m whichin practice will allow us to “see” all the R n,m at the same time. The natural map R → R ′ is not only faithfully flat but also ´etale. If k is algebraically closed thisextension is Galois and plays a crucial role in the study of multiloop algebras. Theexplicit description of Gal( R ′ /R ) is given below.Let A be a k –algebra. We are in general interested in understanding forms (forthe f ppf -topology) of the algebra A ⊗ k R, namely algebras L over R such that(1.1) L ⊗ R S ≃ A ⊗ k S ≃ ( A ⊗ k R ) ⊗ R S for some faithfully flat and finitely presented extension S/R.
The case which is of mostinterest to us is when S can be taken to be a Galois extension R ′ of R of Laurentpolynomial algebras described above. Given a form L as above for which (1.1) holds, we say that L is trivialized by S. The R –isomorphism classes of such algebras can be computed by means of cocycles,just as one does in Galois cohomology:(1.2) Isomorphism classes of
S/R –forms of A ⊗ k R ←→ H fppf (cid:0) S/R,
Aut ( A ) (cid:1) . The right hand side is the part “trivialized by S ” of the pointed set of non-abeliancohomology on the flat site of Spec( R ) with coefficients in the sheaf of groups Aut ( A ) . In the case when S is Galois over R we can indeed identify H fppf (cid:0) S/R,
Aut ( A ) (cid:1) withthe “usual” Galois cohomology set H (cid:0) Gal(
S/R ) , Aut ( A )( S ) (cid:1) as in [Se].Assume now that k is algebraically closed and fix a compatible set of primitive m –th roots of unity ξ m , namely such that ξ eme = ξ m for all e > . We can then identifyGal( R ′ /R ) with ( Z /m Z ) n where for each e = ( e , . . . , e n ) ∈ Z n the correspondingelement e = ( e , · · · , e n ) ∈ Gal( R ′ /R ) acts on R ′ via e t m i = ξ e i m t m i . The primary example of forms L of A ⊗ k R which are trivialized by a Galoisextension R ′ /R as above are the multiloop algebras based on A. These are definedas follows. Consider an n –tuple σσσ = ( σ , . . . , σ n ) of commuting elements of Aut k ( A )satisfying σ mi = 1 . For each n –tuple ( i , . . . , i n ) ∈ Z n we consider the simultaneouseigenspace A i ...i n = { x ∈ A : σ j ( x ) = ξ i j m x for all 1 ≤ j ≤ n } . Then A = P A i ...i n , The Isotriviality Theorem of [GP1] and [GP3] shows that this assumption is superflous if
Aut ( A )is a an algebraic k –group whose connected component is reductive, for example if A is a finitedimensional simple Lie algebra. A = L A i ...i n if we restrict the sum to those n –tuples ( i , . . . , i n ) for which0 ≤ i j < m j . The multiloop algebra corresponding to σσσ , commonly denoted by L ( A, σσσ ) , is de-fined by L ( A, σσσ ) = ⊕ ( i ,...,i n ) ∈ Z n A i ...i n ⊗ t i m . . . t inm n ⊂ A ⊗ k R ′ ⊂ A ⊗ k R ′∞ Note that L ( A, σσσ ) , which does not depend on the choice of common period m, is notonly a k –algebra (in general infinite-dimensional), but also naturally an R –algebra.It is when L ( A, σσσ ) is viewed as an R –algebra that Galois cohomology and the theoryof torsors enter into the picture. Indeed a rather simple calculation shows that L ( A, σσσ ) ⊗ R R ′ ≃ A ⊗ k R ′ ≃ ( A ⊗ k R ) ⊗ R R ′ . Thus L ( A, σσσ ) corresponds to a torsor over Spec( R ) under Aut ( A ) . When n = 1 multiloop algebras are called simply loop algebras. To illustrateour methods, let us look at the case of (twisted) loop algebras as they appear inthe theory of affine Kac-Moody Lie algebras. Here n = 1 , k = C and A = g is afinite-dimensional simple Lie algebra. Any such L is naturally a Lie algebra over R := C [ t ± ] and L ⊗ R S ≃ g ⊗ C S ≃ ( g ⊗ C R ) ⊗ R S for some (unique) g , and somefinite ´etale extension S/R.
In particular, L is an S/R –form of the R –algebra g ⊗ C R ,with respect to the ´etale topology of Spec( R ). Thus L corresponds to a torsor overSpec( R ) under Aut ( g ) whose isomorphism class is an element of the pointed set H (cid:0) R, Aut ( g ) (cid:1) . We may in fact take S to be R ′ = C [ t ± m ] . Assume that A is a finite-dimensional. The crucial point in the classification offorms of A ⊗ k R by cohomological methods is the exact sequence of pointed sets(1.3) H et (cid:0) R, Aut ( A ) (cid:1) → H et (cid:0) R, Aut ( A ) (cid:1) ψ −→ H et (cid:0) R, Out ( A ) (cid:1) , where Out ( A ) is the (finite constant) group of connected components of the algebraic k –group Aut ( A ) . Grothendieck’s theory of the algebraic fundamental group allows us to identify H et (cid:0) R, Out ( A ) (cid:1) with the set of conjugacy classes of n –tuples of commuting elementsof the corresponding finite (abstract) group Out( A ) (again under the assumption that k is algebraically closed). This is an important cohomological invariant attached toany twisted form of A ⊗ k R. We point out that the cohomological information is alwaysabout the twisted forms viewed as algebras over R (and not k ). In practice, as the Strictly speaking we should be using the affine R –group scheme Aut ( A ⊗ k R ) instead of thealgebraic k –group Aut ( A ) . This harmless and useful abuse of notation will be used throughout thepaper. k (and not R ). A technical tool (the centroid trick) developed andused in [ABP2] and [GP2] allows us to compare k vs R information.We begin by looking at the nullity n = 1 case. The map ψ of (1.3) is injective[P1]. This fundamental fact follows from a general result about the vanishing of H for reductive group schemes over certain Dedekind rings which includes k [ t ± ] . This result can be thought of as an analogue of “Serre Conjecture I” for some veryspecial rings of dimension 1. It follows from what has been said that we can attach aconjugacy class of the finite group Out( A ) that characterizes L up to R –isomorphism.In particular, if Aut ( A ) is connected, then all forms (and consequently, all twistedloop algebras) of A are trivial , i.e. isomorphic to A ⊗ k R as R –algebras. This yields theclassification of the affine Kac-Moody Lie algebras by purely cohomological methods.One can in fact define the affine algebras by such methods (which is a completelydifferent approach than the classical definition by generators and relations).Surprisingly enough the analogue of “Serre Conjecture II” for k [ t ± , t ± ] fails, asexplained in [GP2]. The single family of counterexamples known are the the so-called Margaux algebras. The classification of forms in nullity 2 case is in fact quiteinteresting and challenging. Unlike the nullity one case there are forms which are notmultiloop algebras (the Margaux algebra is one such example). The classification innullity 2 by cohomological methods, both over R and over k, will be given in § k but not over R ) can also be attained entirely by EALA methods [ABP3]. Thetwo approaches complement each other and are the culmination of a project starteda decade ago. We also provide classification results for loop Azumaya algebras in § A = g is a finite dimensionalsimple Lie algebra over k. The twisted forms relevant to EALA theory are alwaysmultiloop algebras based on g [ABFP]. It is therefore desirable to try to characterizeand understand the part of H et (cid:0) R, Aut ( g ) (cid:1) corresponding to multiloop algebras. Weaddress this problem by introducing the concept of loop and toral torsors (with k notnecessarily algebraically closed). These concepts are key ideas within our work. It iseasy to show using a theorem of Borel and Mostow that a multiloop algebra based on g , viewed as a Lie algebra over R n , always admits a Cartan subalgebra (in the senseof [SGA3]). We establish that the converse also holds.Central to our work is the study of the canonical map(1.4) H et (cid:0) R n , Aut ( g ) (cid:1) → H et (cid:0) F n , Aut ( g ) (cid:1) where F n stands for the iterated Laurent series field k (( t )) . . . (( t n )) . The AcyclicityTheorem proved in § H loop (cid:0) R n , Aut ( g ) (cid:1) ⊂ H (cid:0) F n , Aut ( g ) (cid:1) of classes of loop torsors is bijective.This has strong applications to the classification of EALAs. Indeed H et (cid:0) F n , Aut ( g ) (cid:1) can be studied using Tits’ methods for algebraic groups over complete local fields.In particular EALAs can be naturally attached Tits indices and diagrams, combi-natorial root data and relative and absolute types. These are important invariantswhich are extremely useful for classification purposes. Setting any applications aside,and perhaps more importantly, we believe that the theory and methods that we areputting forward display an intrinsic beauty, and show just how powerful the methodsdeveloped in [SGA3] really are. Acknowledgement
The authors would like to thank M. Brion, V. Chernousov andthe referee for their valuable comments.
Throughout this section X will denote a scheme, and G a group scheme over X . Assume that X is connected and locally noetherian. Fix a geometric point a of X i.e. a morphism a : Spec(Ω) → X where Ω is an algebraically closed field.Let X f´et be the category of finite ´etale covers of X , and F the covariant functorfrom X f´et to the category of finite sets given by F ( X ′ ) = { geometric points of X ′ above a } . That is, F ( X ′ ) consists of all morphisms a ′ : Spec (Ω) → X ′ for which the diagram X ′ a ′ ր ↓ Spec (Ω) → a X commutes. The group of automorphism of the functor F is called the algebraic fun-damental group of X at a, and is denoted by π ( X , a ) . If X = Spec( R ) is affine, then a corresponds to a ring homomorphism R → Ω and we will denote the fundamentalgroup by π ( R, a ) . The functor F is pro-representable: There exists a directed set I, objects ( X i ) i ∈ I of X f´et , surjective morphisms ϕ ij ∈ Hom X ( X j , X i ) for i ≤ j and geometric points7 i ∈ F ( X i ) such that(2.1) a i = ϕ ij ◦ a j (2.2) The canonical map f : lim −→ Hom X ( X i , X ′ ) → F ( X ′ ) is bijective, where the map f of (2.2) is as follows: Given ϕ : X i → X ′ then f ( ϕ ) = F ( ϕ )( a i ) . Theelements of lim −→ Hom X ( X i , X ′ ) appearing in (2.2) are by definition the morphisms inthe category of pro-objects over X (see [EGA IV] § −→ Hom( X i , − ) pro-represents F. Since the X i are finite and ´etale over X the morphisms ϕ ij are affine. Thus theinverse limit X sc = lim ←− X i exist in the category of schemes over X [EGA IV] § X ′ over X wethus have a canonical map(2.3) Hom P ro − X ( X sc , X ′ ) def = lim −→ Hom X ( X i , X ′ ) ≃ F ( X ′ ) → Hom X ( X sc , X ′ )obtained by considering the canonical morphisms ϕ i : X sc → X i . Proposition 2.1.
Assume X is noetherian. Then F is represented by X sc ; that is,there exists a bijection F ( X ′ ) ≃ Hom X ( X sc , X ′ ) which is functorial on the objects X ′ of X f´et . Proof.
Because the X i are affine over X and X is noetherian, each X i is noetherian; inparticular, quasicompact and quasiseparated. Thus, for X ′ / X locally of finite presen-tation, in particular for X ′ in X f´et , the map (2.3) is bijective [EGA IV, prop 8.13.1].The Proposition now follows from (2.2). Remark 2.2.
The bijection of Proposition 2.1 could be thought along the same linesas those of (2.2) by considering the “geometric point” a sc ∈ lim ←− F ( X i ) satisfying a sc a i for all i ∈ I. In computing X sc = lim ←− X i we may replace ( X i ) i ∈ I by any cofinal family. Thisallows us to assume that the X i are (connected) Galois, i.e. the X i are connected andthe (left) action of Aut X ( X i ) on F ( X i ) is transitive. We then have F ( X i ) ≃ Hom
P ro − X ( X sc , X i ) ≃ Hom X ( X i , X i ) = Aut X ( X i ) . π ( X , a ) can be identified with the group lim ←− Aut X ( X i ) opp . Each Aut X ( X i ) isfinite, and this endows π ( X , a ) with the structure of a profinite topological group.The group π ( X , a ) acts on the right on X sc as the inverse limit of the finitegroups Aut X ( X i ) . Thus, the group π ( X , a ) acts on the left on each set F ( X ′ ) =Hom P ro − X ( X sc , X ′ ) for all X ′ ∈ X f´et . This action is continuous since the structuremorphism X ′ → X “factors at the finite level”, i.e there exists a morphism X i → X ′ of X –schemes for some i ∈ I. If u : X ′ → X ′′ is a morphism of X f´et , then the map F ( u ) : F ( X ′ ) → F ( X ′′ ) clearly commutes with the action of π ( X , a ) . This constructionprovides an equivalence between X f´et and the category of finite sets equipped with acontinuous π ( X , a )–action.The right action of π ( X , a ) on X sc induces an action of π ( X , a ) on G ( X sc ) =Mor X ( X sc , G ) , namely γ f ( z ) = f ( z γ ) ∀ γ ∈ π ( X , a ) , f ∈ G ( X sc ) , z ∈ X sc . Proposition 2.3.
Assume X is noetherian and that G is locally of finite presentationover X . Then G ( X sc ) is a discrete π ( X , a ) –module and the canonical map lim −→ H (cid:0) Aut X ( X i ) , G ( X i ) (cid:1) → H (cid:0) π ( X , a ) , G ( X sc ) (cid:1) is bijective. Remark 2.4.
Here and elsewhere when a profinite group A acts discretely on amodule M the corresponding cohomology H ( A, M ) is the continuous cohomology asdefined in [Se1]. Similarly, if a group H acts in both A and M, then Hom H ( A, M )stands for the continuous group homomorphism of A into M that commute with theaction of H. Proof.
To show that G ( X sc ) is discrete amounts to showing that the stabilizer in π ( X , a ) of a point of f ∈ G ( X sc ) is open. But if G is locally of finite presentationthen G ( X sc ) = G (lim ←− X i ) = lim −→ G ( X i ) ([EGA IV] prop. 8.13.1), so we may assumethat f ∈ G ( X i ) for some i. The result is then clear, for the stabilizer we are after isthe inverse image under the continuous map π ( X , a ) → Aut X ( X i ) of the stabilizer of f in Aut X ( X i ) (which is then open since Aut X ( X i ) is given the discrete topology).By definition H (cid:0) π ( X , a ) , G ( X sc ) (cid:1) = lim −→ (cid:0) π ( X , a ) /U, G ( X sc ) U (cid:1) where the limit is taken over all open normal subgroups U of π ( X , a ) . But for eachsuch U we can find U i ⊂ U so that U i = ker (cid:0) π ( X , a ) → Aut X ( X i ) (cid:1) . Thus H (cid:0) π ( X , a ) , G ( X sc (cid:1) = lim −→ H (cid:0) Aut X ( X i ) , G ( X i ) (cid:1) as desired. 9uppose now that our X is a geometrically connected k –scheme, where k is ofarbitrary characteristic. We will denote X × k k by X . Fix a geometric point a :Spec( k ) → X . Let a (resp. b ) be the geometric points of X [resp. Spec( k )] given bythe composite maps a : Spec( k ) a → X → X [resp. b : Spec( k ) a → X → Spec( k )] . Thenby [SGA1, th´eor`eme IX.6.1] π (cid:0) Spec( k ) , b (cid:1) ≃ Gal( k ) = Gal( k s /k ) where k s is theseparable closure of k in k , and the sequence(2.4) 1 → π ( X , a ) → π ( X , a ) p −→ Gal( k ) → Recall that a (right) torsor over X under G (or simply a G –torsor if X is un-derstood) is a scheme E over X equipped with a right action of G for which thereexists a faithfully flat morphism Y → X , locally of finite presentation, such that E × X Y ≃ G × X Y = G Y , where G Y acts on itself by right translation.A G –torsor E is locally trivial (resp. ´etale locally trivial ) if it admits a trivializationby an open Zariski (resp. ´etale) covering of X . If G is affine, flat and locally offinite presentation over X , then G –torsors over X are classified by the pointed set ofcohomology H fppf ( X , G ) defined by means of cocycles `a la ˇCech. If G is smooth, any G –torsor is ´etale locally trivial (cf. [SGA3], Exp. XXIV), and their classes are thenmeasured by H et ( X , G ). In what follows the f ppf -topology will be our default choice,and we will for convenience denote H fppf simply by H . Given a base change Y → X ,we denote by H ( Y / X , G ) the kernel of the base change map H ( X , G ) → H ( Y , G Y ) . As it is customary, and when no confusion is possible, we will denote in what follows H ( Y , G Y ) simply by H ( Y , G )Recall that a torsor E over X under G is called isotrivial if it is trivialized by some finite ´etale extension of X , that is,[ E ] ∈ H ( X ′ / X , G ) ⊂ H ( X , G )for some X ′ in X f´et . We denote by H iso ( X , G ) the subset of H ( X , G ) consisting ofclasses of isotrivial torsors. Proposition 2.5.
Assume that X is noetherian and that G is locally of finite presen-tation over X . Then H iso ( X , G ) = ker (cid:0) H ( X , G ) → H ( X sc , G ) (cid:1) . roof. Assume E is trivialized by X ′ ∈ X f´et . Since the connected components of X ′ are also in X f´et there exists a morphism X i → X ′ for some i ∈ I. But then E × X X i = E × X X ′ × X ′ X i = G X ′ × X ′ X i = G X i so that E is trivialized by X i . The image of [ E ] on H ( X sc , G ) is thus trivial.Conversely assume [ E ] ∈ H ( X , G ) vanishes under the base change X sc → X . Sincethe X i are quasicompact and quasiseparated and G is locally of finite presentation, atheorem of Grothendieck-Margaux [Mg] shows that the canonical maplim −→ H ( X i , G ) → H ( X sc , G )is bijective. Thus E × X X i ≃ G X i for some i ∈ I. We look in detail at an example that is of central importance to this work, namelythe case when X = Spec ( R n ) where R n = k [ t ± , . . . , t ± n ] is the Laurent polynomialring in n –variables with coefficients on a field k of characteristic 0 . Fix once and for all a compatible set ( ξ m ) m ≥ of primitive m –roots of unity in k (i.e. ξ ℓmℓ = ξ m ) . Let { k λ } λ ∈ Λ be the set of finite Galois extensions of k which areincluded in k. Let Γ λ = Gal ( k λ /k ) and Γ = lim ←− Γ λ . Then Γ coincides with thealgebraic fundamental group of Spec ( k ) at the geometric point Spec ( k ) . Let ε : R n → k be the evaluation map at t i = 1 . The composite map R n ε → k ֒ → k defines a geometric point a of X and a geometric point a of X = Spec ( R n ) where R n = k [ t ± , . . . , t ± n ] . Let I be the subset of Λ × Z > consisting of all pairs ( λ, m ) for which k λ contains ξ m . Make I into a directed set by declaring that ( λ, ℓ ) ≤ ( µ, n ) ⇐⇒ k λ ⊂ k µ and ℓ | n. Each R λn,m = k λ [ t ± m , . . . , t ± m n ]is a Galois extension of R n with Galois group Γ m,λ = ( Z /m Z ) n ⋊ Γ λ as follows: For e = ( e , . . . , e n ) ∈ Z n we have e t m j = ξ e j j t m j where − : Z n → ( Z /mZ ) n is the canonicalmap, and the group Γ λ acts naturally on R λn,m through its action on k λ . It is immediate There exists a finite Galois connected covering F → X such that F × X X ′ ∼ = F ⊔ · · · ⊔ F ( r times).If we decompose X ′ = Y ⊔ · · · ⊔ Y m into its connected components we have X ′ × X F = Y × X F ⊔ · · · ⊔ Y m × X F = F ⊔ · · · ⊔ F . It follows that each Y i × X F is a disjoint union of copies of F , hence Y i × X F is finite ´etale over F for i = 1 , .., m . Then each Y i is ´etale over X by proposition 17.7.4.vi of [EGA IV]. By descent, each Y i is finite over X [EGA IV, prop. 2.7.1 xv], so each Y i / X is finite ´etale. γ ∈ Γ λ we have γ e γ − : t m j ( γ ξ j ) e j t m j . Thus if γ ξ j = ξ f j j then γ e γ − = e ′ where e ′ = ( f e , . . . f n e n ) . If ( λ, ℓ ) ≤ ( µ, n ) we have a canonical inclusion R λn,ℓ ⊂ R µn,m . For i = ( λ, ℓ ) ∈ I we let X i = Spec ( R λn,ℓ ) . The above gives morphisms ϕ ij : X j → X i of X –schemeswhenever i ≤ j. We have [GP3] X sc = lim ←− X i = Spec(lim −→ X i )= Spec ( R n, ∞ )where R n, ∞ = lim −→ m R n,m with R n,m = k [ t ± m , . . . , t ± m n ] . Thus(2.5) π ( X , a ) = lim ←− Γ m,λ = b Z (1) n ⋊ Gal ( k ) . where b Z (1) denotes the abstract group b Z = lim ←− m µµµ m ( k ) equipped with the naturalaction of the absolute Galois group Gal( k ) = Gal( k/k ) . Remark 2.6.
Consider the affine k –group scheme ∞ µµµ = lim ←− m µµµ m . It corresponds tothe Hopf algebra k [ ∞ µµµ ] = lim −→ m k [ µµµ m ] = lim −→ m k [ t ] t m − . Then we have ∞ µµµ ( k ) ≃ b Z and ∞ µµµ ( k ) is equipped with a (canonical) structure ofprofinite Gal( k )–module. Remark 2.7.
Let the notation be that of Example 2.3. Since Z n is the charactergroup of the the torus G nm,k , we have an automorphism GL n ( Z ) ≃ Aut gr ( G n,km ) op .This defines a left action GL n ( Z ) on R n and a right action of GL n ( Z ) on the torus G nm,k . Furthermore, by universal nonsense, this action extends uniquely to the simplyconnected covering R n, ∞ at the geometric point a . The extended action on the torusSpec( R n,m ) with character group ( m Z ) n is given by the extension of the action of GL n ( Z ) from Z n to ( m Z ) n inside Q n . The group GL n ( Z ) acts (on the right) on π ( R n ), so we can consider the semidirect product of groups GL n ( Z ) ⋉ π ( R n ) whichacts then on R n, ∞ (see § GL n ( Z ) on R n described in Remark 2.7 we can twistthe R n –module R n by an element g ∈ GL n ( Z ). We denote the resulting twisted R n –algebra by R gn . The multiplication on the R n -algebras R gn and R n coincide. It is the action of R n that isdifferent. See § emma 2.8. Let S be a connected finite ´etale cover of R n . Let L ⊂ S be the integralclosure of k in S . Then there exists g ∈ GL n ( Z ) , a , ..., a n ∈ L × and positive integers d , ..., d n such that d | d · · · | d n and S ⊗ R n R gn ≃ R n -alg ( R n ⊗ k L ) (cid:2) d √ a t , · · · , dn √ a n t n (cid:3) . In particular, S is k –isomorphic to R n ⊗ k L and Pic( S ) = 0 .Proof. Note that R gn and R n have the same units. For convenience in what followswe will for simplicity denote ( R n ) g by R gn , ( R n,m ) g by R gn,m and S ⊗ R n R gn by S g . By Galois theory there exists a finite Galois extension k ′ /k and a positive inte-ger m such that µ m ( k ) ⊂ k ′ and S ≃ R n ( R n,m ⊗ k k ′ ) H where H is a subgroup ofGal( R n,m ⊗ k k ′ /R n ) = µµµ m ( k ′ ) n ⋊ Gal( k ′ /k ). Hence S is geometrically connected and S is a finite ´etale cover of R n ⊗ k L . We can assume without loss of generality that k = L . To say that L = k is to say that the map H → Gal( k ′ /k ) is onto. We considerthe following commutative diagram1 −−−→ µµµ m ( k ′ ) n ∩ H −−−→ H −−−→ Gal( k ′ /k ) −−−→ y y || −−−→ µµµ m ( k ′ ) n −−−→ Gal( R n,m ⊗ k k ′ /R n ) −−−→ Gal( k ′ /k ) −−−→ . Note that the action Gal( k ′ /k ) on µµµ m ( k ′ ) n normalizes µµµ m ( k ′ ) n ∩ H . Hence µµµ nm ( k ′ ) ∩ H is the group of k ′ –points of a split k –group ννν of multiplicative type. By consideringthe corresponding characters groups, we get a surjective homomorphism ( Z /m Z ) n = c µµµ nm → b ννν of finite abelian groups. An element g ∈ GL n ( Z ) ⊂ Aut k ( R n ) transformsthe diagram above to yield1 −−−→ ννν ( k ′ ) −−−→ H −−−→ Gal( k ′ /k ) −−−→ y y || −−−→ µµµ m ( k ′ ) n −−−→ Gal( R n,m ⊗ k k ′ /R n ) −−−→ Gal( k ′ /k ) −−−→ . g ∗ y g ∗ y || −−−→ µµµ m ( k ′ ) n −−−→ Gal( R gn,m ⊗ k k ′ /R gn ) −−−→ Gal( k ′ /k ) −−−→ . x x || −−−→ g ννν ( k ′ ) −−−→ H g −−−→ Gal( k ′ /k ) −−−→ GL n ( Z ) on ( Z /m Z ) n = c µµµ nm is the left action provided by the homo-morphism GL n ( Z ) → GL n ( Z /m Z ). By elementary facts about generators of finite13belian groups there exists g ∈ GL n ( Z ) and positive integers d , ..., d n such that d | d · · · | d n | m for which the following holds( Z /m Z ) n = c µµµ nm −−−→ b ννν g ∗ y ≃ ≃ y ( Z /m Z ) n = c µµµ nm −−−→ Z / ( m/d ) Z ⊕ · · · Z / ( m/d n ) Z . This base change leads to the following commutative diagram1 −−−→ µµµ m/d ( k ′ ) × · · · × µµµ m/d n ( k ′ ) −−−→ H g −−−→ Gal( k ′ /k ) −−−→ y y || −−−→ µµµ m ( k ′ ) n −−−→ Gal( R gn,m ⊗ k k ′ /R gn ) −−−→ Gal( k ′ /k ) −−−→ . We claim that S g is equipped with an R gn –torsor structure under µµµ := µµµ d × · · · µµµ d n .The diagram above provides a bijectionGal( R gn,m ⊗ k k ′ /R gn ) /H g ∼ −→ µµµ ( k ′ ) , hence a map ψ : Gal( R gn,m ⊗ k k ′ /R gn ) −→ µµµ ( k ′ ) which is a cocycle for the standardaction of Gal( R gn,m ⊗ k k ′ /R gn ) on µµµ ( k ′ ) as we now check. We shall use the followingtwo facts(I) ψ is right H g –invariant;(II) the restriction of ψ to µµµ m ( k ′ ) n is a morphism of Gal( k ′ /k )–modules.We are given γ , γ ∈ Gal( R gn,m ⊗ k k ′ /R gn ). Since g H surjects onto Gal( k ′ /k ), we canwrite γ i = α i h i with α i ∈ µµµ m ( k ′ ) n and h i ∈ H g for i = 1 ,
2. We have ψ ( γ γ ) = ψ ( α h α h )= ψ ( α h α h − h h )= ψ ( α h α h − ) [ by (I) ]= ψ ( α ) ψ ( h α h − ) [ by (II) ]= ψ ( α ) h ψ ( α ) h − [ by (I) ]= ψ ( γ ) γ .ψ ( γ ) . Denote by ˜ S the µµµ –torsor over R gn defined by ψ , that is˜ S := n x ∈ R gn,m ⊗ k k ′ | ψ ( γ ) .γ ( x ) = x ∀ γ ∈ Gal( R gn,m ⊗ k k ′ /R gn ) o . ψ is trivial over H g , we have ˜ S ⊂ S g . But ˜ S and S g are both finite ´etalecoverings of R gn of degree | µµµ ( k ′ ) | , hence ˜ S = S g . Since Pic( R gn ) = 0, we can useKummer theory (see [M] III 4.10), namely the isomorphism H ( R gn , µµµ ) = Y j =1 ,...,n H ( R gn , µµµ d j ) ≃ Y j =1 ,...,n R gn × / ( R gn × ) d j . for determining the structure of ˜ S. Since R gn × = k × × Z n , there exist scalars a , ..., a n ∈ k × and monomials x , ..., x n in the t i such that the class of ˜ S/R gn in H ( R n , µµµ ) is givenby ( a x , · · · , a n x n ). In terms of covering, this means that ˜ S = k (cid:2) d √ a x , · · · , dn √ a n x n (cid:3) .Extending scalars to k ′ , we have˜ S ⊗ k k ′ = ( R gn,m ⊗ k k ′ ) µµµ ( k ′ ) = k ′ (cid:2) d √ t , · · · , dn √ t n (cid:3) . From this it follows that x i = t i mod (cid:0) ( R gn ⊗ k k ′ ) × (cid:1) di and x i = t i mod ( R gn × ) di . Weconclude that ˜ S = k (cid:2) d √ a t , · · · , dn √ a n t n (cid:3) . The notation that we are going to use throughout the paper deserves some com-ments. We will tend to use boldface characters, such as G , for algebraic groups over k, as also for group schemes over X that are obtained from groups over k. A quintessen-tial example is G X = G × k X . For arbitrary group schemes, or more generally groupfunctors, over X we shall tend to use german characters, such as G . This duality ofnotation will be particularly useful when dealing with twisted forms over X of groupsthat come from k. The concept of reductive group scheme over X and all related terminology is thatof [SGA3]. For convenience we now recall and introduce some concepts and notation attachedto a reductive group scheme H over X . We denote by rad( H ) (resp. corad( H )) itsradical (resp. coradical) torus, that is its maximal central subtorus (resp. its maximaltoral quotient) of H [XII.1.3].We say that a H is reducible if it admits a proper parabolic subgroup P such that P contains a Levi subgroup L (see XXVI). The opposite notion is irreducible. If X is affine, the notion of reducibility for H is equivalent to the existence of a properparabolic subgroup P (XXVI.2.3), so there is no ambiguity with the terminology of[CGP] and [GP2]. The references to [SGA3] are so prevalent that they will henceforth be given by simply listingthe Expos´e number. Thus XII. 1.3, for example, refers to section 1.3 of Expos´e XII of [SGA3]. The concept of proper parabolic subgroup is not defined in [SGA3]. By proper we mean that P s is a proper subgroup of G x for all geometric points x of X .
15y extension, if an affine group G over X acts on H , we say that the action is reducible if it normalizes a couple ( P , L ) where P is a proper parabolic subgroup of H and L a Levi subgroup of P . The action is otherwise called irreducible. We say that H over is isotropic if H admits a subgroup isomorphic to G m, X . Theopposite notion is anisotropic .If the base scheme X is semi-local and connected (resp. normal), one can show that H is anisotropic if and only if H is irreducible and the torus rad( H ) (or equivalentlycorad( H )) is anisotropic (XXVI.2.3, resp. [Gi4]).Similarly we say that the action of G on H is isotropic if it centralizes a splitsubtorus T of H with the property that all geometric fibers of T are non-trivial.Otherwise the action is anisotropic. One checks that this is the case if and onlyif the action of G on H is irreducible and the action of G on the torus rad( H ) (orequivalently to corad( H )) is anisotropic. Throughout this section k will denote a field of arbitrary characteristic, X a ge-ometrically connected noetherian scheme over k , and X sc = lim ←− X i its simply con-nected cover as described in § G a group scheme over k which is locally offinite presentation. We will maintain the notation of the previous section, and assumethat Ω = k. Consider the fundamental exact sequence (2.4). The geometric point a corresponds to a point of X ( k ) . Because of (2.1), the geometric points a i : Spec ( k ) → X i induce a geometricpoint a sc : Spec ( k ) → lim ←− X i = X sc . We thus have a group homomorphism(3.1) G ( k s ) → G ( k ) G ( a sc ) −→ G ( X sc ) . The group π ( X , a ) acts on k s , hence on G ( k s ) , via the group homomorphism π ( X , a ) → Gal ( k ) of (2.4). This action is continuous, and together with (3.1) yieldsa map H (cid:0) π ( X , a ) , G ( k s ) (cid:1) → H (cid:0) π ( X , a ) , G ( X sc ) (cid:1) , where we remind the reader that these H are defined in the “continuous” sense (seeRemark 2.4). On the other hand, by Proposition 2.3 and basic properties of torsorstrivialized by Galois extensions, we have H (cid:0) π ( X , a ) , G ( X sc ) (cid:1) = lim −→ H (cid:0) Aut X ( X i ) , G ( X i ) (cid:1) = lim −→ H ( X i / X , G ) ⊂ H ( X , G ) .
16y means of the foregoing observations we make the following.
Definition 3.1.
A torsor E over X under G is called a loop torsor if its isomorphismclass [ E ] in H ( X , G ) belongs to the image of the composite map(3.2) H (cid:0) π ( X , a ) , G ( k s ) (cid:1) → H (cid:0) π ( X , a ) , G ( X sc ) (cid:1) ⊂ H ( X , G ) . We will denote by H loop ( X , G ) the subset of H ( X , G ) consisting of classes of looptorsors. They are given by (continuous) cocycles in the image of the natural map Z (cid:0) π ( X , a ) , G ( k s ) (cid:1) → Z ( X , G ) , which we call loop cocycles. Examples 3.2. (a) If X = Spec ( k ) then H loop ( X , G ) is nothing but the usual Galoiscohomology of k with coefficients in G . (b) Assume that k is separably closed. Then the action of π ( X , a ) on G ( k s ) istrivial, so that H (cid:0) π ( X , a ) , G ( k s ) (cid:1) = Hom (cid:0) π ( X , a ) , G ( k s ) (cid:1) / Int G ( k s )where the group Int G ( k s ) of inner automorphisms of G ( k s ) acts naturally onHom (cid:0) π ( X , a ) , G ( k s ) (cid:1) . To be precise, Int( g )( φ ) : x → g − φ ( x ) g for all g ∈ G ( k s ) ,φ ∈ Hom (cid:0) π ( X , a ) , G ( k s ) (cid:1) and x ∈ π ( X , a ) . Two particular cases are important:(b1) G abelian: In this case H (cid:0) π ( X , a ) , G ( k s ) (cid:1) is just the group of continuoushomomorphisms from π ( X , a ) to G ( k s ) . (b2) π ( X , a ) = b Z n : In this case H (cid:0) π ( X , a ) , G ( k s ) (cid:1) is the set of conjugacy classesof n –tuples σ = ( σ , . . . , σ n ) of commuting elements of finite order of G ( k s ) . That theelements are of finite order follows from the continuity assumption.(c) Let X = Spec ( k [ t ± ]) with k algebraically closed of characteristic 0 . If G is a connected linear algebraic group over k then H ( X , G ) = 1 ([P1, prop. 5]). We seefrom (b2) above that the canonical map H (cid:0) π ( X , a ) , G ( k s ) (cid:1) → H ( X , G )of (3.2) need not be injective. It need not be surjective either (take X = P k and G = G m,k ) . (d) If the canonical map G ( k s ) → G ( X sc ) is bijective, e.g. if X is a geometricallyintegral projective variety over k (i.e. a geometrically integral closed subscheme of P nk for some n, ) then H loop ( X , G ) = H (cid:0) π ( X , a ) , G ( k s ) (cid:1) . Remark 3.3.
The notion of loop torsor behaves well under twisting by a Galoiscocycle z ∈ Z (cid:0) Gal( k ) , G ( k s ) (cid:1) . Indeed the torsion map τ − z : H ( X , G ) → H ( X , z G )maps loop classes to loop classes. 17 .2 Loop reductive groups Let H be a reductive group scheme over X . Since X is connected, for all x ∈ X the geometric fibers H x are reductive group schemes of the same “type” (see [SGA3,XXII.2.3]. By Demazure’s theorem there exists a unique split reductive group H over k such that H is a twisted form (in the ´etale topology of X ) of H = H × k X . We will call H the Chevalley k –form of H . The X –group H corresponds to a torsor E over X under the group scheme Aut ( H ) , namely E = Isom group ( H , H ) . We recallthat
Aut ( H ) is representable by a smooth and separated group scheme over X byXXII 2.3. By definition H is then the contracted product E ∧ Aut ( H ) H (see [DG] III § o Definition 3.4.
We say that a group scheme H over X is loop reductive if it isreductive and if E is a loop torsor.We look more closely to the affine case X = Spec( R ). Concretely, let H =Spec( k [ H ]) be a split reductive k –group and consider the corresponding R –group H = H × k R, whose Hopf algebra is R [ H ] = k [ H ] ⊗ k R. Let H be an R –group which is a twisted form of H trivialized by the universalcovering R sc . Then to a trivialization H × R R sc ∼ = H × R R sc , we can attach a cocycle u ∈ Z (cid:0) π ( R, a ) , Aut ( H )( R sc ) (cid:1) from which H can be recovered by Galois descent aswe now explain in the form of a Remark for future reference. Remark 3.5.
There are two possible conventions as to the meaning of the cocycles u of Z (cid:0) π ( R, a ) , Aut ( H )( R sc ) (cid:1) . On the one hand H can be thought of as the affinescheme Spec( R [ H ]) , Aut ( H )( R sc ) as the (abstract) group of automorphisms of the R sc –group Spec( R sc [ H ]) where R sc [ H ] = R [ H ] ⊗ R R sc , and π ( X, a ) as the oppositegroup of automorphisms of Spec( R sc ) / Spec( R ) acting naturally on Aut ( H )( R sc ).We will adopt an (anti) equivalent second point of view that is much more con-venient for our calculations. We view Aut ( H )( R sc ) as the group of automorphismsof the R sc –Hopf algebra R sc [ H ] = R [ H ] ⊗ R R sc ≃ k [ H ] ⊗ k R sc on which the Galoisgroup π ( R, a ) acts naturally. Then the R –Hopf algebra R [ H ] corresponding to H isgiven by R [ H ] = { x ∈ R sc [ H ] : u γγ x = a for all γ ∈ π ( R, a ) } . To say then that H is k –loop reductive is to say that u can be chosen so that u γ ∈ Aut ( H )( k ) ⊂ Aut ( H )( R sc ) = Aut ( H )( R sc ) for all γ ∈ π ( R, a ) . If our geometric point a lies above a k –rational point x of X , then x correspondsto a section of the structure morphism X → Spec( k ) which maps b to a. This yields18 group homomorphism x ∗ : Gal( k ) → π ( X , a ) that splits the sequence (2.4) above.This splitting defines an action of Gal( k ) on the profinite group π ( X , a ), hence asemidirect product identification(3.3) π ( X , a ) ≃ π ( X , a ) ⋊ Gal( k ) . We have seen an example of (3.3) in Example 2.3.
Remark 3.6.
By the structure of extensions of profinite groups [RZ, § π ( X , a ) is the projective limit of a system (cid:0) H α ⋊ Gal( k ) (cid:1) where the H α ’s arefinite groups. The Galois action on each H α defines a twisted finite constant k –group ννν α . We define ννν = lim ←− α ννν α . The ννν α are affine k –groups such that ννν ( k ) = lim ←− α ννν α ( k ) = π ( X , a ) . Note that ννν ( k s ) = ννν ( k ). In the case when X = Spec( R n ) , where as before R n = k [ t ± , . . . , t ± n ] with k of characteristic zero and a is the geometric point described inExample 2.3, the above construction yields the affine k –group ∞ µµµ defined in Remark2.6.By means of the decomposition (3.3) we can think of loop torsors as being com-prised of a geometric and an arithmetic part, as we now explain.Let η ∈ Z (cid:0) π ( X , a ) , G ( k s ) (cid:1) . The restriction η | Gal( k ) is called the arithmetic part of η and its denoted by η ar . It is easily seen that η ar is in fact a cocycle in Z (cid:0) Gal( k ) , G ( k s ) (cid:1) .If η is fixed in our discussion, we will at times denote the cocycle η ar by the moretraditional notation z. In particular, for s ∈ Gal( k ) we write z s instead of η ars . Next we consider the restriction of η to π ( X , a ) that we denote by η geo and calledthe geometric part of η. We thus have a mapΘ : Z (cid:0) π ( X , a ) , G ( k s ) (cid:1) −−−→ Z (cid:0) Gal( k ) , G ( k s ) (cid:1) × Hom (cid:0) π ( X , a ) , G ( k s ) (cid:1) η (cid:0) η ar , η geo (cid:1) The group Gal( k ) acts on π ( X , a ) by conjugation. On G ( k s ) , the Galois groupGal( k ) acts on two different ways. There is the natural action arising for the actionof Gal( k ) on k s that as customary we will denote by s g, and there is also the twistedaction given by the cocycle η ar = z . Following Serre we denote this last by s ′ g . Thus19 ′ g = z ss gz s − . Following standard practice to view the abstract group G ( k s ) as aGal( k )–module with the twisted action by z we write z G ( k s ) . For s ∈ Gal( k ) and h ∈ π ( X , a ), we have η geoshs − = η shs − = η ss ( η hs − ) [ η is a cocycle]= z s s ( η hs − ) [ η s = η ar s = z s ]= z s s ( η geoh z s − ) [ η is a cocycle and h acts trivially on G ( k s )]= z s s η geo h z − s [1 = z s s z s − ].This shows that η geo : π ( X , a ) → z G ( k s ) commutes with the action of Gal( k ) . Inother words, η geo ∈ Hom
Gal( k ) (cid:0) π ( X , a ) , z G ( k s ) (cid:1) . Lemma 3.7.
The map Θ defines a bijection between Z (cid:0) π ( X , a ) , G ( k s ) (cid:1) and couples ( z, η geo ) with z ∈ Z (cid:0) π ( X , a ) , G ( k s ) (cid:1) and η geo ∈ Hom
Gal( k ) (cid:0) π ( X , a ) , z G ( k s ) (cid:1) .Proof. Since a 1–cocycle is determined by its image on generators, the map Θ isinjective. For the surjectivity, assume we are given z ∈ Z ( π ( X , a ) , G ( k s )) and η geo ∈ Hom
Gal( k ) (cid:0) π ( X , a ) , z G ( k s ) (cid:1) . We define then η : π ( X , a ) → G ( k s ) by η hs := η geoh z s This map is continuous, its restriction to π ( X , a ) (resp. Gal( k ) ) is η geo (resp. z ). Finally, since η is a section of the projection map G ( k s ) ⋊ π ( X , a ) → π ( X , a ), itis a cocycle.We finish this section by recalling some basic properties of the twisting bijection.Let η ∈ Z (cid:0) π ( X , a ) , G ( k s ) (cid:1) and consider its corresponding pair Θ( η ) = ( z, η geo ) . Wecan apply the same construction to the twisted k –group z G . This leads to a map Θ z that attaches to a cocycle η ′ ∈ Z (cid:0) π ( X , a ) , z G ( k s ) (cid:1) a pair ( z ′ , η ′ geo ) along the linesexplained above. Note that by Lemma 3.7 the pair (1 , η geo ) is in the image of Θ z . More precisely.
Lemma 3.8.
Let η ∈ Z (cid:0) π ( X , a ) , G ( k s ) (cid:1) . With the above notation, the inverse ofthe twisting map [Se1] τ − z : Z (cid:0) π ( X , a ) , G ( k s ) (cid:1) ∼ −→ Z (cid:0) π ( X , a ) , z G ( k s ) (cid:1) satisfies Θ z ◦ τ − z ( η ) = (1 , η geo ) . Remark 3.9.
Consider the special case when the semi-direct product is direct, i.e. π ( X , a ) = π ( X , a ) × Gal( k ). In other words, the affine k –group ννν defined above isconstant so that η geoh = z s s η geoh z − s h ∈ π ( X , a ) and s ∈ Gal( k ). The torsion map τ − z : Z (cid:0) π ( X , a ) , G ( k s ) (cid:1) → Z (cid:0) π ( X , a ) , z G ( k s ) (cid:1) maps the cocycle η to the homomorphism η geo : π ( X , a ) → z G ( k s ).We give now one more reason to call η geo the geometric part of η . Lemma 3.10.
Let ννν be the affine k –group scheme defined in Remark 3.6. Then foreach linear algebraic k –group H , there is a natural bijection Hom k − gp ( ννν, H ) ∼ −→ Hom
Gal( k ) (cid:0) π ( X , a ) , H ( k s ) (cid:1) . Proof.
First we recall that Hom
Gal( k ) (cid:0) π ( X , a ) , H ( k s ) (cid:1) stands for the continuous ho-momorphisms from π ( X , a ) to H ( k s ) that commute with the action of Gal( k ).Write ννν = lim ←− ννν α as an inverse limit of twisted constant finite k –groups. Since H and the ννν α are of finite presentation we have by applying [SGA3] VI B k − gp ( ννν, H ) = lim −→ α Hom k − gp ( ννν α , H )= lim −→ α Hom
Gal( k ) (cid:0) ννν α ( k s ) , H ( k s ) (cid:1) = Hom Gal( k ) (cid:0) π ( X , a ) , H ( k s ) (cid:1) . This permits to see purely geometric k –loop torsors in terms of homomorphismsof affine k –group schemes. Throughout this section we assume that G is a smooth affine k –group, and X ascheme over k. Let G X = G × k X be the X –group obtained from G by base change.Following our convention a torsor over X under G means under G X , and we write H ( X , G ) instead of H ( X , G X ) . Definition 3.11.
A torsor E over X under G is said to be finite if it admits areduction to a finite k –subgroup S of G ; this is to say, the class of E belongs to theimage of the natural map H ( X , S ) → H ( X , G ) for some finite subgroup S of G .We denote by H finite ( X , G ) the subset of H ( X , G ) consisting of classes of finitetorsors, that is H finite ( X , G ) := [ S ⊂ G Im (cid:0) H ( X , S ) → H ( X , G ) (cid:1) . where S runs over all finite k –subgroups of G .The case when k is of characteristic 0 is well known.21 emma 3.12. Assume that k is of characteristic . Then H finite ( X , G ) ⊂ H loop ( X , G ) . If in addition k is algebraically closed, then H finite ( X , G ) = H loop ( X , G ) . Proof.
Let S be a finite subgroup of the k –group G . Since k is of characteristic 0 thegroup S is ´etale. Thus S corresponds to a finite abstract group S together with acontinuous action of Gal( k ) by group automorphisms. More precisely (see [SGA1] or[K] pg.184) S = S ( k ) with the natural action of Gal( k ) . Similarly the ´etale X -group S X corresponds to S with the action of π ( X , a ) induced from the homomorphism π ( X , a ) → Gal( k ) . By Exp. XI of [SGA1] we have(3.4) H ( X , S ) def = H ( X , S X ) = H (cid:0) π ( X , a ) , S (cid:1) = H ( π ( X , a ) , S ( k ))which shows that H ( X , S ) ⊂ H loop ( X , G ) . If k is algebraically closed any k -loop torsor E is given by a continuous grouphomomorphism f E : π ( X , a ) → G ( k ) , as explained in Example 3.2(b). Then theimage of f E is a finite subgroup of G ( k ) which gives rise to a finite (constant) algebraicsubgroup S of G . By construction [ E ] comes from H ( X , S ) . Let k , G and X be as in the previous section. Given a torsor E over X under G X we can consider the twisted X –group E G X = E ∧ G X G X . Since no confusion will arisewe will denote E G X simply by E G . We say that our torsor E is toral if the twisted X –group E G admits a maximal torus (XII.1.3). We denote by H toral ( X , G ) ⊂ H ( X , G )the set of classes of toral torsors.We recall the following useful result. Lemma 3.13.
1. Let T be a maximal torus of G . Then H toral ( X , G ) = Im (cid:16) H (cid:0) X , N G ( T ) (cid:1) → H ( X , G ) (cid:17) .
2. Let → S → G ′ p → G → be a central extension of G by a k –group S ofmultiplicative type. Then the diagram H toral ( X , G ′ ) ⊂ H ( X , G ′ ) p ∗ y p ∗ y H toral ( X , G ) ⊂ H ( X , G ) We remind the reader that we are abiding by [SGA3] conventions and terminology. In theexpression “maximal torus of G ” we view G as a k –group, namely a group scheme over Spec( k ) . In particular T is a k -group... s cartesian.Proof. (1) is established in [CGR2, 3.1].(2) Consider first the case when S is the reductive center of G ′ , We are given an X –torsor E ′ under G ′ and consider the surjective morphism of X –group schemes E ′ G ′ → E ′ G whose kernel is S × k X . By XII 4.7 there is a natural one-to-one correspondencebetween maximal tori of the X –groups E ′ G ′ and E ′ G . Hence E ′ is a toral G ′ –torsorif and only if E ′ ∧ G ′ X G X is a toral G –torsor. The general case follows form the factthat G ′ / Z ′ ≃ G / Z where Z ′ (resp. Z ) is the reductive center of G .In an important case the property of a torsor being toral is of infinitesimal nature. Lemma 3.14.
Assume that G is semisimple of adjoint type. For a X –torsor E under G the following conditions are equivalent:1. E is toral.2. The Lie algebra L ie ( E G ) admits a Cartan subalgebra.Proof. By XIV th´eor`emes 3.9 and 3.18 there exists a natural one-to-one correspon-dence between the maximal tori of E G and Cartan subalgebras of L ie ( E G ) . Recall the following result [CGR2].
Theorem 3.15.
Let R be a commutative ring and G a smooth affine group schemeover R whose connected component of the identity G is reductive. Assume furtherthat one of the following holds:(a) R is an algebraically closed field, or(b) R = Z , G is a Chevalley group, and the order of the Weyl group of thegeometric fiber G s is independent of s ∈ Spec( Z ) , or(c) R is a semilocal ring, G is connected, and the radical torus rad( G ) is isotrivial.Then there exist a maximal torus T of G , and a finite R –subgroup S ⊂ N G ( T ) , suchthat1. S is an extension of a finite twisted constant R –group by a finite R –group ofmultiplicative type,2. the natural map H fppf ( X , S ) −→ H fppf (cid:0) X , N G ( T ) (cid:1) is surjective for any R –scheme X satisfying the condition: (3.5) Pic( X ′ ) = 0 for every generalized Galois cover X ′ / X ,where by a generalized Galois cover X ′ → X we understand a Γ –torsor for sometwisted finite constant X –group scheme Γ . orollary 3.16. Let G be a linear algebraic k –group whose connected component ofthe identity G is reductive. Assume that one of the following holds:(i) k is algebraically closed;(ii) G is obtained by base change from a smooth affine Z –group satisfying thehypothesis of Theorem 3.15(b);(iii) G is reductive.If the k –scheme X satisfies condition (3.5), then1. H toral ( X , G ) ⊂ H finite ( X , G ) .2. If furthermore char( k ) = 0 , we have H toral ( X , G ) ⊂ H finite ( X , G ) ⊂ H loop ( X , G ) . The first statement is immediate. The second one follows from Lemma 3.12.
Throughout this section ˜ k will denote an object of k – alg. We will denote by Γa subgroup of the group Aut k − alg (˜ k ) . The elements of Γ are thus k –linear automor-phisms of the ring ˜ k. For convenience we will denote the action of an element γ ∈ Γon an element λ ∈ ˜ k by γ λ. Given an object R of ˜ k –alg (the category of associative unital commutative ˜ k –algebras), we will denote the action of and element λ ∈ ˜ k on an element r ∈ R by λ R · r, or simply λ R r or λr if no confusion is possible.Given an element γ ∈ Γ , we denote by R γ the object of ˜ k –alg which coincideswith R as a ring, but where the ˜ k –module structure is now obtained by “twisting ”by γ : λ R γ · r = ( γ λ ) R · r One verifies immediately that(4.1) ( R γ ) τ = R γτ for all γ, τ ∈ Γ . It is important to emphasize that (4.1) is an equality and not acanonical identification.Given a morphism ψ : A → R of ˜ k –algebras and an element γ ∈ Γ we can view ψ as a map ψ γ : A γ → R γ (recall that A = A γ and R = R γ as rings, hence also as sets).24t is immediate to verify that ψ γ is also a morphism of ˜ k –algebras. By (4.1) we have( ψ γ ) τ = ψ γτ for all γ, τ ∈ Γ . The map ψ → ψ γ gives a natural correspondence(4.2) Hom ˜ k − alg ( A, R ) → Hom ˜ k − alg ( A γ , R γ ) . In view of (4.1) we have also a natural (and equivalent) correspondence(4.3) Hom ˜ k − alg ( A, R γ ) → Hom ˜ k − alg ( A γ − , R ) . that we record for future use. Remark 4.1. (i) Let γ, σ ∈ Γ . It is clear from the definitions that the k -algebraisomorphism γ : ˜ k → ˜ k induces a ˜ k -algebra isomorphism γ σ : ˜ k σ → ˜ k γσ . If noconfusion is possible we will denote γ σ simply by γ. One checks that the ˜ k –algebras ( R ⊗ k ˜ k ) γ and R ⊗ k ˜ k γ are equal (recall that bothalgebras have R ⊗ k ˜ k as underlying sets). We thus have a ˜ k –algebra isomorphism1 ⊗ γ : R ⊗ k ˜ k → R ⊗ k ˜ k γ = ( R ⊗ ˜ k ) γ , or more generally 1 ⊗ γ σ : R ⊗ k ˜ k σ → R ⊗ k ˜ k γσ = ( R ⊗ ˜ k ) γσ . (ii) If A is an object ˜ k –alg, and γ ∈ Γ, then the ˜ k –algebras A and A γ have the sameideals.Given a ˜ k –functor X , that is a functor from the category ˜ k –alg to the category ofsets (see [DG] for details), and an element γ ∈ Γ we can define a new the ˜ k –functor γ X by setting(4.4) γ X ( R ) = X ( R γ )and γ X ( ψ ) = X ( ψ γ ) where ψ : R → S is as above. The diagram X ′ ( R γ ) = −−−→ X ( R γ ) γ X ( ψ ) y X ( ψ γ ) y γ X ′ ( R ) = −−−→ γ X ( R )then commutes by definition, and one can indeed easily verify that γ X is a ˜ k –functor.We call γ X the twist of X by γ. k –algebras described in (4.1) we have the equality offunctors(4.5) γ ( τ X ) = γτ X for all γ, τ ∈ Γ . A morphism f : X ′ → X induces a morphism γ f : γ X ′ → γ X by setting γ f ( R ) = f ( R γ ) . We thus have the commutative diagram X ′ ( R γ ) f ( R γ ) −−−→ X ( R γ ) = y = y γ X ′ ( R ) γ f ( R ) −−−→ γ X ( R )This gives a natural bijection(4.6) Hom ˜ k − fun ( X ′ , X ) → Hom ˜ k − fun ( γ X ′ , γ X )given by f γ f . This correspondence is compatible with the action of Γ , this is γ ( τ f ) = γτ f . As before we will for future use explicitly write down an equivalentversion of this last bijection, namely(4.7) Hom ˜ k − fun ( γ − X ′ , X ) → Hom ˜ k − fun ( X ′ , γ X ) A ˜ k –functor morphism f : γ X → Z is called a semilinear morphism of type γ from X to Z . We denoted the set of such morphisms by Hom γ ( X , Z ) , and set Hom Γ ( X , Z ) = ∪ γ ∈ Γ Hom γ ( X , Z ) . These are the Γ– semilinear morphisms from X to Z . If f : γ X → Y and g : τ Y → Z are semilinear of type γ and τ respectively, thenthe map gf : τγ X → Z defined by ( gf )( R ) = g ( R ) ◦ f ( R τ ) according to the sequence(4.8) τγ X ( R ) = γ X ( R τ ) f ( R τ ) → Y ( R τ ) = τ Y ( R ) g ( R ) → Z ( R )is semilinear of type τ γ. The above considerations give the set Aut Γ ( X ) of invertible elements of Hom Γ ( X , X )a group structure whose elements are called Γ– semilinear automorphisms of X . Thereis a group homomorphism t : Aut Γ ( X ) → Γ that assigns to a semilinear automorphismof X its type. The alert reader may question whether the “type” is well defined. Indeed it may happen that γ X and X are the same ˜ k -functor even though γ = 1 . This ambiguity can be formally resolved bydefining semilinear morphism of type γ as pairs ( f : γ X → Z , γ ) . We will omit this complication ofnotation in what follows since no confusion will be possible within our context. Note that the unionof sets ∪ γ ∈ Γ Hom γ ( X , Z ) is thus disjoint by definition. emark 4.2. Fix a ˜ k –functor Y . Recall that the category of ˜ k –functors over Y consists of ˜ k –functors X equipped with a structure morphism X → Y . This categoryadmits fiber products: Given f : X → Y and f : X → Y then X × Y X is givenby ( X × Y X )( R ) = { ( x , x ) ∈ X ( R ) × X ( R ) : f ( R )( x ) = f ( R )( x ) } . Semilinearity extends to fiber products under the right conditions. Suppose f : X → Y and f : X → Y are as above, and that the action of Γ in X i and Y is compatible in the obvious way. Then for each γ ∈ Γ the “structure morphisms” γ f i : γ X i → γ Y defined above can be seen to verify(4.9) γ ( X × Y X ) = γ X × γ Y γ X for all γ ∈ Γ . Assume that X is affine, that is X = Sp ˜ k A = Hom ˜ k − alg ( A, − ) . If γ ∈ Γ then(4.10) γ X = Sp ˜ k A γ − as can be seen from (4.3). In particular γ X is also affine. Our next step is to showthat semilinear twists of schemes are also schemes.Assume that Y is an open subfunctor of X . We claim that γ Y is an open subfunctorof γ X . We must show that for all affine functor Sp ˜ k A and all morphism f : Sp ˜ k A → γ X there exists an ideal I of A such that f − ( γ Y ) = D ( I ) where D ( I )( R ) = { α ∈ Hom(
A, R ) : Rf ( I ) = R } . Let us for convenience denote Sp ˜ k A by X ′ , and γ − by γ ′ . Our morphism f inducesa morphism γ ′ f : γ ′ X ′ → X by the considerations described above. Because Y isopen in X and γ ′ X ′ = Sp ˜ k − alg A γ is affine there exists and ideal I of A γ such that( γ ′ f ) − ( Y ) = D ( I ) . Applying this to the ˜ k –algebra R γ we obtain(4.11) γ ′ f ( R γ ) − ( Y ( R γ ) = { α ∈ Hom ˜ k − alg ( A γ ′ , R γ ) : R γ α ( I ) = R γ } . On the other hand γ ′ f ( R γ ) − = f ( R ) − and Y ( R γ ) = γ Y ( R ) . Finally in the righthand side of (4.11) we have Hom ˜ k − alg ( A γ ′ , R γ ) = Hom ˜ k − alg ( A, R ) and R γ α ( I ) = R γ if and only if Rα ( I ) = R. Since I is also an ideal of the ˜ k –algebra A this completesthe proof that γ Y is an open subfunctor of γ X . If X is local then so is γ X . Indeed, given a ˜ k -algebra R and and element f ∈ R then f can naturally be viewed as an element of R γ (since R and R γ coincide as27ings), and it is immediate to verify that ( R f ) γ = ( R γ ) f . Using that it is then clearthat the sequence(4.12) γ X ( R ) → γ X ( R f i ) ⇒ γ X ( R f i f j )is exact whenever 1 = f + · · · + f n . Since R is a field if and only if R γ is a field it is clear that if X is covered by afamily of open subfunctors ( Y i ) i ∈ I , then γ X is covered by the open subfunctors γ Y i . From this it follows that if X is a scheme then so is γ X . Remark 4.3.
Let X is a ˜ k –scheme defined along traditional lines (and not as aspecial type of ˜ k –functor), and let X also denote the restriction to the category ofaffine ˜ k –schemes of the functor of points of X . If we define (again along traditionallines) γ X = X × Spec(˜ k ) Spec(˜ k γ − ) , then it can be shown that the functor of points of γ X (restricted to the category of affine ˜ k –schemes) coincides with the twist by γ of X that we have defined. Remark 4.4.
We look in detail at the case when our ˜ k –scheme is an affine groupscheme G . Thus G = Sp ˜ k ˜ k [ G ] for some ˜ k –Hopf algebra ˜ k [ G ] (see [DG] II § ǫ G : ˜ k [ G ] → ˜ k denote the counit map. As ˜ k -modules we have ˜ k [ G ] = ˜ k ⊕ I G where I G is the kernel of ǫ G . Let γ ∈ Γ . As explained in (4.10) we have γ G =Sp ˜ k ˜ k [ G ] γ − = Hom ˜ k − alg (˜ k [ G ] γ − , − ) . We leave it to the reader to verify that ǫ γ G : γ ◦ ǫ G . As an abelian group I G = I γ G , but in this last the action of ˜ k is obtained through theaction of ˜ k in ˜ k [ G ] γ . Next we make some relevant observations about Lie algebras from a functorialpoint of view ([DG] II § Lie ( G ) attaches to anobject R in ˜ k -alg the kernel of the group homomorphism G ( R [ ǫ ]) → G ( R ) where R [ ǫ ] is the ˜ k –algebra of dual numbers of R, and the group homomorphism comesfrom the functorial nature of G applied to the morphism R [ ǫ ] → R in ˜ k –alg thatmaps ǫ . By definition L ie ( G ) = Lie ( G )(˜ k ) . In particular L ie ( G ) ⊂ G (˜ k [ ǫ ]) =Hom ˜ k -alg (˜ k [ G ] , ˜ k [ ǫ ]) . Every element x ∈ L ie ( G ) is given by(4.13) x : a ǫ G ( a ) + δ x ( a ) ǫ with δ x ∈ Der ˜ k (˜ k [ G ] , ˜ k ) where ˜ k is viewed as a ˜ k [ G ]–module via the counit map of G . In what follows we write x = ǫ G + δ x ǫ. The map x δ x is in fact a ˜ k -moduleisomorphism L ie ( G ) ≃ Der ˜ k (˜ k [ G ] , ˜ k ) . In particular if λ ∈ ˜ k then λx ∈ L ie ( G ) is suchthat δ λx = λδ x . Similar considerations apply to the affine ˜ k -group γ G . We have L ie ( γ G ) =Der ˜ k (˜ k [ G ] γ − , ˜ k ) . Note that if y ∈ L ie ( γ G ) corresponds to δ y ∈ Der ˜ k (˜ k [ G ] γ − , ˜ k ) , k on L ie ( γ G ) the element λy corresponds to the derivation λδ y and not to ( γ λ ) δ y : The “ γ part” is taken into consideration already by the factthat y ∈ L ie ( γ G ) and that δ y ∈ Der ˜ k (˜ k [ G ] γ − , ˜ k ) . Let from now on G denote a ˜ k -group functor. If H is a subgroup functor of G welet Aut γ ( G , H ) = n f ∈ Aut γ ( G ) | γ H = f − ( H ) o . It is easy to verify then that Aut Γ ( G , H ) = ∪ γ ∈ Γ Aut γ ( G , H ) is a subgroup of Aut Γ ( G ) . Proposition 4.5.
Let Π ˜ k/k G be the Weil restriction of G to k (which we view as a k –group functor). There exists a canonical group homomorphism ˜ : Aut Γ ( G ) → Aut(Π ˜ k/k G ) . Proof.
As observed in Remark 4.1 the map γ : ˜ k → ˜ k γ is an isomorphism of ˜ k –alg,and for R in k –alg ( R ⊗ k ˜ k ) γ = R ⊗ k ˜ k γ . We thus have a ˜ k –algebra isomorphism1 ⊗ γ : R ⊗ k ˜ k → ( R ⊗ ˜ k ) γ . For a given f ∈ Aut Γ ( G ) , the composite map˜ f ( R ) : (Π ˜ k/k G )( R ) = G ( R ⊗ k ˜ k ) G (1 ⊗ γ ) −→ G (cid:0) ( R ⊗ k ˜ k ) γ (cid:1) == γ G ( R ⊗ k ˜ k ) f ( R ⊗ k ˜ k ) → G ( R ⊗ k ˜ k ) = (Π ˜ k/k G )( R )is an automorphism of the group (Π ˜ k/k G )( R ) . One readily verifies that the family˜ f = ˜ f ( R ) R ∈ k − alg is functorial on R, hence an automorphism of Π ˜ k/k G . To check that ˜ is a group homomorphism we consider two elements f , f ∈ Aut Γ ( G ) of type γ and γ respectively. Recall that γ induces a ˜ k -algebra homomor-phism 1 ⊗ γ σ : R ⊗ ˜ k γ → R ⊗ ˜ k γ γ for all σ ∈ Γ [see Remark 4.1(i)]. Since γ will beunderstood from the context we will denote this homomorphism simply by 1 ⊗ γ . Byfunctoriality we get the following commutative diagram [see Remark 4.1(1)] G ( R ⊗ ˜ k γ ) G (1 ⊗ γ ) −−−−−→ G ( R ⊗ ˜ k γ γ ) y = y = γ G ( R ⊗ ˜ k ) γ G (1 ⊗ γ ) −−−−−→ γ G ( R ⊗ ˜ k γ ) y f ( R ⊗ ˜ k ) y f ( R ⊗ ˜ k γ ) G ( R ⊗ ˜ k ) G (1 ⊗ γ ) −−−−−→ G ( R ⊗ ˜ k γ )29ince f ◦ f is of type γ γ , by definition we have ^ f ◦ f ( R ⊗ ˜ k ) = ( f ◦ f )( R ⊗ ˜ k ) ◦ G (1 ⊗ γ γ ) . Thus ^ f ◦ f ( R ⊗ ˜ k ) = ( f ◦ f )( R ⊗ ˜ k ) ◦ G (1 ⊗ γ ◦ ⊗ γ )= ( f ◦ f )( R ⊗ ˜ k ) ◦ G (1 ⊗ γ ) ◦ G (1 ⊗ γ )= f ( R ⊗ ˜ k ) ◦ f ( R ⊗ ˜ k γ ) ◦ G (1 ⊗ γ ) ◦ G (1 ⊗ γ )= f ( R ⊗ ˜ k ) ◦ G (1 ⊗ γ ) ◦ f ( R ⊗ ˜ k ) ◦ G (1 ⊗ γ )= ˜ f ( R ⊗ ˜ k ) ◦ ˜ f ( R ⊗ ˜ k ) . Example 4.6. (a) Consider the case of the trivial ˜ k –group e ˜ k . Each set Aut γ ( e ˜ k ) =Isom( γ e ˜ k , e ˜ k ) consists of one element which we denote by γ ∗ . Then Aut Γ ( e ˜ k ) ≃ Γ . We have Π ˜ k/k e ˜ k = e k . In particular Aut(Π ˜ k/k e ˜ k ) = 1 and the homomorphism˜ : Aut Γ ( G ) → Aut(Π ˜ k/k G ) is in this case necessarily trivial. In affine terms e ˜ k isrepresented by ˜ k and γ e ˜ k by ˜ k γ − . Then the ˜ k –group isomorphism γ ∗ : γ e ˜ k → e ˜ k corresponds to the ˜ k –Hopf algebra isomorphism γ − : ˜ k → ˜ k γ − . (b) Consider the case when Γ is the Galois group of the extension C / R , and G is theadditive C –group. Then Aut Γ ( G ) can be identified with the group of automorphismsof ( C , +) which are of the form z λz or z λz for some λ ∈ C × . The Weilrestriction of G to R is the two-dimensional additive R –group. Thus Aut(Π ˜ k/k G ) =GL ( R ) . The above examples show that, even if ˜ k/k is a finite Galois extension of fieldsand G is a connected linear algebraic group over ˜ k , the homomorphism f ˜ f needbe neither injective nor surjective Corollary 4.7.
Assume that G = Sp ˜ k ˜ k [ G ] is an affine ˜ k -group. The group Aut Γ ( G ) acts naturally on the groups G (˜ k ) and G (˜ k [ ǫ ]) . Furthermore the action of an element f ∈ Aut γ ( G ) on G (˜ k [ ǫ ]) stabilizes L ie ( G ) ⊂ G (˜ k [ ǫ ]) . The induced map L ie ( f ) : L ie ( G ) → L ie ( G ) is an automorphism of L ie ( G ) viewed as a Lie algebra over k. This automorphism is ˜ k –semilinear, i.e., L ie ( f )( λx ) = ( γ λ ) L ie ( f )( x ) for all λ ∈ ˜ k and x ∈ L ie ( G ) . Proof.
We maintain the notation and use the facts presented in Remark 4.4. Let x ∈ L ie ( G ) and write x = ǫ G + δ x ǫ. If λ ∈ ˜ k then λx ∈ L ie ( G ) is such that δ λx = λδ x . By definition (Π ˜ k/k G )( k ) = G (˜ k ) and (Π ˜ k/k G )( k [ ǫ ]) = G (˜ k [ ǫ ]) . The action of anelement f ∈ Aut Γ ( G ) on these two groups is then given by the automorphisms ˜ f k and ˜ f k [ ǫ ] of the previous Proposition. Thus if we let γ ǫ : ˜ k [ ǫ ] → ˜ k [ ǫ ] γ denote the30somorphism of ˜ k –alg induced by γ the map ˜ f k [ ǫ ] is then obtained by restricting to L ie ( G ) the composite map G (˜ k [ ǫ ]) = Hom ˜ k − alg (˜ k [ G ] , ˜ k [ ǫ ]) G ( γ ǫ ) → Hom ˜ k − alg (˜ k [ G ] , ˜ k [ ǫ ] γ ) == Hom ˜ k − alg (˜ k [ G ] γ − , ˜ k [ ǫ ]) = γ G (˜ k [ ǫ ]) f (˜ k [ ǫ ]) → Hom ˜ k − alg (˜ k [ G ] , ˜ k [ ǫ ]) = G (˜ k [ ǫ ]) . Using the fact that γ ǫ ◦ ǫ G = γ ◦ ǫ G = ǫ γ G it easily follows that(4.14) L ie ( f )( x ) = ˜ f k [ ǫ ] ( x ) = f (˜ k [ ǫ ]) ◦ (cid:0) ǫ γ G + ( γ ◦ δ x ) ǫ (cid:1) Let y = ǫ γ G + ( γ ◦ δ x ) ǫ ∈ L ie ( γ G ) . If λ ∈ ˜ k then we have L ie ( f )( λx ) = f (˜ k [ ǫ ]) ◦ (cid:0) ǫ γ G + ( γ ◦ δ λx ) ǫ (cid:1) = f (˜ k [ ǫ ]) ◦ (cid:0) ǫ γ G + ( γ ( λδ x ) ǫ (cid:1) = f (˜ k [ ǫ ]) ◦ (cid:0) ǫ γ G + γ λ ( γ ◦ δ x ) ǫ (cid:1) = f (˜ k [ ǫ ]) (cid:0) ( γ λ ) y ) (cid:1) where ( γ λ ) y is the action of the element γ λ ∈ ˜ k on the element y ∈ L ie ( γ G ), asexplained in the last paragraph of Remark 4.4. Since the restriction of f (˜ k [ ǫ ]) to L ie ( γ G ) induces an isomorphism L ie ( γ G ) → L ie ( G ) of ˜ k -Lie algebras, this restrictionis in particular ˜ k -linear. It follows that L ie ( f )( λx ) = f (˜ k [ ǫ ]) (cid:0) ( γ λ ) y (cid:1) = ( γ λ ) f (˜ k [ ǫ ])( y ) = ( γ λ ) L ie ( f )( x ) . This shows that L ie ( f ) is semilinear. We leave it to the reader to verify that L ie ( f )is an automorphism of L ie ( G ) as a Lie algebra over k. Remark 4.8.
There is no natural action of Aut Γ ( G ) on G . Throughout this section k denotes a field of characteristic 0 . Theorem 4.9. (Semilinear Borel-Mostow) Let ˜ k/k be a finite Galois extension offields with Galois group Γ . Suppose we are given a quintuple (cid:0) g , H, ψ, φ, ( H i ) ≤ i ≤ s (cid:1) where g is a (finite dimensional) reductive Lie algebra over ˜ k,H is a group, ψ is a group homomorphism from H into the Galois group Γ , is a group homomorphism from H into the group Aut k ( g ) of automorphisms of g viewed as a Lie algebra over k, ( H i ) ≤ i ≤ s is a finite family of subgroups of H for which the following two conditionshold:(i) If we let the group H act on g via φ and on Γ via ψ , namely h x = φ ( h ) x and h λ = ψ ( h ) λ for all h ∈ H, x ∈ g , and λ ∈ ˜ k, then the action of H in g is semilinear,i.e., h ( λx ) h = h λ h x. (ii) ker( ψ ) = H s ⊃ H s − ⊃ ... ⊃ H ⊃ H = 0 . Furthermore, each H i is normalin H, the elements of φ ( H i ) are semisimple, and the quotients H i /H i − are cyclic.Then there exists a Cartan subalgebra of g which is stable under the action of H .Proof. We will reason by induction on s. If s = 0 we can identify by assumption (ii) H with a subgroup Γ of Γ via ψ. Let ˜ k = ˜ k Γ . This yields a semilinear action ofΓ on g . By Galois descent the fixed point g Γ is a Lie algebra over ˜ k for which thecanonical map ρ : g Γ ⊗ ˜ k ˜ k ≃ g is a ˜ k –Lie algebra isomorphism. If h is a Cartansubalgebra of g Γ then ρ ( h ⊗ ˜ k ˜ k ) is a Cartan subalgebra of g which is H –stable asone can easily verify with the aid of assumption (i).Assume s ≥ θ of the cyclic group H . As we havealready observed the action of θ on g is ˜ k –linear. If V is a ˜ k –subspace of g stableunder θ we will denote by V θ the subspace of fixed points. Before continuing with weestablish the following crucial fact: Claim 4.10. g θ is a reductive Lie algebra over ˜ k. If h is a Cartan subalgebra of g θ , then z g ( h ) is a Cartan subalgebra of g . Since φ ( θ ) is an automorphism of the ˜ k –Lie algebra g we see that g θ is indeeda Lie subalgebra of g . Let g ′ and z denote the derived algebra and the centre of g respectively. Because g is reductive g ′ is semisimple and g = g ′ × z . Clearly θ inducesby restriction automorphisms (also denoted by θ ) of g ′ and of z . By [Bbk] Ch. 8 § g ′ ) θ is reductive, and therefore g θ = ( g ′ ) θ × z θ is also reductive.Every Cartan subalgebra h of g θ is of the form h = h ′ × z θ for some Cartansubalgebra h ′ of ( g ′ ) θ . Clearly z g ( h ) = z g ′ ( h ′ ) × z . By [P3] theorem 9 the centralizer z g ′ ( h ′ ) is a Cartan subalgebra of g ′ , so the claim follows.We now return to the proof of the Theorem. Since H is normal in H we havean induced action (via φ ) of H ′ = H/H on the reductive ˜ k –Lie algebra g θ . Wehave induced group homomorphisms φ ′ : H ′ → Aut k ( g θ ) and ψ ′ : H ′ → Γ (this lastsince H ⊂ ker( ψ )). For 0 ≤ i < s define H ′ i = H i +1 /H . We apply the inductionassumption to the quintuple (cid:0) g θ , H ′ , ψ ′ , φ ′ , ( H ′ i ) ≤ i ≤ s − (cid:1) . This yields the existence of Because H i ⊂ ker( ψ ) the action of the elements of H i on g is ˜ k –linear. The assumption is that φ ( θ ) be semisimple as a ˜ k –linear endomorphisms of g for all θ ∈ H i .
32 Cartan subalgebra h of g θ which is stable under the action of H ′ given by φ ′ . Thismeans that, back in g , the algebra h is stable under our original action of H given by φ. But then the centralizer of h in g is also stable under this action, and we can nowconclude by (4.10) . Remark 4.11. If ψ is the trivial map the Theorem reduces to the “Main result (B)”of Borel and Mostow [BM] for g . The use of (4.10) allows for a slightly more directproof of this result.We shall use the above semilinear version of Borel-Mostow’s theorem 4.12 to estab-lish the following result which will play a crucial role in the the proof of the existenceof maximal tori on twisted groups corresponding to loop torsors.
Corollary 4.12.
Let ˜ k/k be a finite Galois extension with Galois group Γ . Let G bea reductive group over ˜ k. Let H be a group, and assume we are given a group homo-morphism ρ : H → Aut Γ ( G ) for which we can find a family of subgroups ( H i ) ≤ i ≤ s of H as in the Theorem, that is ker( t ◦ ρ ) = H s ⊃ H s − ⊃ ... ⊃ H ⊃ H = 0 where t : Aut Γ ( G ) → Γ is the type morphism, each H i is normal in H, the elementsof ρ ( H i ) act semisimply on the ˜ k –Lie algebra L ie ( G ) , and the quotients H i /H i − are cyclic. Then there exists a maximal torus T of G such that ρ has values in Aut Γ ( G , T ) ⊂ Aut Γ ( G ) . Namely if h ∈ H and ( t ◦ ρ )( h ) = γ ∈ Γ , then ρ ( h ) : γ G → G induces by restriction an isomorphism γ T → T . Proof.
Let h ∈ H. If ( t ◦ ρ )( h ) = γ then according to the various definitions we havethe following commutative diagram. G (˜ k ) G ( γ ) −−−→ G (˜ k γ ) = γ G (˜ k ) ρ ( h )(˜ k ) −−−−→ G (˜ k ) y y y G (˜ k [ ǫ ]) G ( γ ǫ ) −−−→ G (˜ k [ ǫ ] γ ) = γ G (˜ k [ ǫ ]) ρ ( h )(˜ k [ ǫ ]) −−−−−→ G (˜ k [ ǫ ]) x x x L ie ( G ) Lie ( G )( γ ) −−−−−→ L ie ( G γ ) Lie ( G )( ρ ( h )) −−−−−−−→ L ie ( G ) . where we have denoted by γ ǫ : ˜ k [ ǫ ] → ˜ k [ ǫ ] γ the ˜ k -algebra isomorphism inducedby γ. For convenience in what follows we will denote L ie ( G ) by g . By Corollary4.7 we obtain by composing ρ with the map ˜ defined in Proposition 4.5 a grouphomomorphism φ : H → Aut k ( g ) , namely φ ( h ) = g ρ ( h ), which together with the grouphomomorphism ψ = t ◦ ρ : H → Γ and the H i satisfy the assumptions of Theorem4.9. It follows that there exists a Cartan subalgebra t of g which is stable under theaction of H defined by φ. g ρ ( h ) = ρ ( h )(˜ k ) ◦ G ( γ )which is nothing but the top row of our diagram above. Similarly with the notationof Corollary 4.7 we have(4.16) L ie ( g ρ ( h )) = ] ρ ǫ ( h ) | g = Lie ( G )( ρ ( h )) ◦ Lie ( G )( γ )where ] ρ ǫ ( h ) stands for the middle row of our diagram, namely ρ ( h )(˜ k [ ǫ ]) ◦ G ( γ ǫ ) . Let T be the maximal torus of G whose Lie algebra is t [XIV.6.6.c]. We have T = Z G ( t ) where the centralizer is taken respect to the adjoint action of G on g . Given an element g ∈ G (˜ k ) we will denote its natural image in G (˜ k [ ǫ ]) by g ǫ . Since we are working over a base field the ˜ k -points of T = Z G ( t ) can be computed inthe naive way, namely(4.17) T (˜ k ) = { g ∈ G (˜ k ) : g ǫ xg ǫ − = x for all x ∈ t ⊂ G (˜ k [ ǫ ]) } Since ˜ ρ ǫ ( h ) is an automorphism of the (abstract) group G (˜ k [ ǫ ]) we obtain(4.18) T (˜ k ) = { g ∈ G (˜ k ) : ˜ ρ ǫ ( h )( g ǫ ) ˜ ρ ǫ ( h )( x ) (cid:0) ˜ ρ ǫ ( h )( g ǫ ) (cid:1) − = ˜ ρ ǫ ( h )( x ) for all x ∈ t } But since ˜ ρ ǫ ( h ) stabilizes t this last reads(4.19) T (˜ k ) = { g ∈ G (˜ k ) : ˜ ρ ǫ ( h )( g ǫ ) x (cid:0) ˜ ρ ǫ ( h )( g ǫ ) (cid:1) − = x for all x ∈ t } Note that by the commutativity of the top square of our diagram we have (cid:0) ˜ ρ ( h )( g ) (cid:1) ǫ =˜ ρ ǫ ( h )( g ǫ ) . Thus from (4.19) we obtain that ˜ ρ ǫ ( h ) (cid:0) T (˜ k ) (cid:1) = T (˜ k ) . On the other hand by(4.15) we have ˜ ρ ǫ ( h ) (cid:0) T (˜ k ) (cid:1) = ρ ( h )(˜ k ) (cid:16) G ( γ ) (cid:0) T (˜ k ) (cid:1)(cid:17) . But by definition (cid:16) G ( γ ) (cid:0) T (˜ k ) (cid:1)(cid:17) = γ T (˜ k ) . Thus our ˜ k -group homomorphism ρ ( h ) : γ G : → G is such that the two tori ρ ( h )( γ T ) and T of G have the same ˜ k -points. This forces ρ ( h )( γ T ) = T . Next we give a crucial application of the semilinear considerations developed thusfar to the existence of maximal tori for certain loop groups. We could not find a reference for this basic fact in the literature. By [XIII 5.3] we have N G ( T ) = N G ( t ) . Since the natural homomorphism N G ( T ) / T → Aut( t ) is injective we obtain T = Z G ( t ) . .6 Existence of maximal tori in loop groups We come back to the case of R = R n = k [ t ± , . . . , t ± n ] where k is a field ofcharacteristic zero. This is the ring that plays a central role in all applications toinfinite-dimensional Lie theory. It is not true in general that a reductive R n –groupadmits a maximal torus; however. Proposition 4.13.
Let G be a loop reductive group scheme over R n (see definition3.4). Then G admits a maximal torus.Proof. We try to recreate the situation of the semilinear Borel-Mostow theorem.We can assume that G is split after base extension to the Galois covering ˜ R =˜ k [ t ± /m , . . . , t ± /mn ] where m is a positive integers and ˜ k/k is a finite Galois exten-sion of fields containing all primitive m -th roots of unity of k. Recall from Example2.3 that ˜ R is a Galois extension of R with Galois group ˜Γ = ( Z /m Z ) n ⋊ Γ as fol-lows: For e = ( e , . . . , e n ) ∈ Z n we have e ( λt m j ) = λξ e j m t m j for all λ ∈ ˜ k , where − : Z n → ( Z /mZ ) n is the canonical map, while the Galois group Γ = Gal(˜ k/k ) actsnaturally on ˜ R through its action on ˜ k. Let G be the Chevalley k –form of G (see § G is the twist of G = G × k R by a loop cocycle u : ˜Γ → Aut ( G )(˜ k ) . The homomorphism ψ : ˜Γ = ( Z /m Z ) n ⋊ Γ → Γ is defined to be the natural projection.For convenience we will adopt the following notational convention. The elements of˜Γ will be denoted by ˜ γ , and the image under φ of such an element (which belongs toΓ), by the corresponding greek character: that is ψ (˜ γ ) = γ. Consider the reductive ˜ k –group G = Spec(˜ k [ G ]) where, as usual, ˜ k [ G ] denotesthe ˜ k –Hopf algebra ˜ k ⊗ k k [ G ] . Consider for each ˜ γ the map f (˜ γ ) : ˜ k [ G ] → ˜ k [ G ]defined by(4.20) f (˜ γ ) = u ˜ γ ◦ γ. Since each u ˜ γ is an automorphism of the ˜ k –Hopf algebra ˜ k [ G ] , it follows that f (˜ γ ) isin fact a ˜ k -Hopf algebra isomorphism ˜ k [ G ] → ˜ k [ G ] γ . As such it can be thought of,by Yoneda considerations and (4.10), as an element of Aut γ ( G ) of type γ − which wewill denote by ρ (˜ γ ) . Since the restriction of the action of ˜ γ on ˜ R [ G ] to ˜ k [ G ] is given by γ, the cocyclecondition on u shows that for all ˜ α, ˜ β ∈ ˜Γ we have(4.21) ρ ( ˜ α ˜ β ) = ρ ( ˜ α ) ρ ( ˜ β )35here this last product takes place in Aut Γ ( G ) . Thus ρ is a group homomorphismand f (˜ γ ) can be viewed as a ˜ k -Hopf algebra morphism from ˜ k [ G ] to ˜ k [ G ] γ From (4.21), the various definitions and the “anti equivalent” nature of Yoneda’scorrespondence it follows that the map ˜ γ → ρ (˜ γ ) can be viewed as a group homomor-phism ρ : ˜Γ opp → Aut Γ ( G ) , where ˜Γ opp is the opposite group of ˜Γ . Since ρ (˜ γ ) is oftype γ − we can complete the necessary semilinear picture by defining φ : ˜Γ opp → Γto be the map ˜ γ → γ − . The kernel of the composite map t ◦ ρ is precisely ( Z /m Z ) n ,and the elements of this kernel act trivially on ˜ k [ G ], in particular their correspond-ing action on the Lie algebra of G is trivial, hence semisimple. We can thus applyCorollary 4.12; the role of H now being played by ˜Γ opp . Let T be a torus G such that ρ (˜ γ )( γ − T ) = T for all ˜ γ ∈ ˜Γ . The torus T corresponds to a Hopf ideal I of the Hopf ˜ k –algebra ˜ k [ G ] representing G . Each ρ (˜ γ ) , which corresponds to the ˜ k -Hopf algebra isomorphism f (˜ γ ) described in (4.20),induces a ˜ k –Hopf algebra isomorphism f (˜ γ ) from ˜ k [ T ] to ˜ k [ T ] γ where ˜ k [ G ] /I = ˜ k [ T ]is the Hopf algebra representing T . For future use we observe that the resulting actionof ˜Γ on ˜ k [ T ] is Γ– semilinear in the sense that if λ ∈ ˜ k and a ∈ ˜ k [ T ] then(4.22) f (˜ γ )( λa ) = f (˜ γ )( λ ˜ k [ T ] .a ) = λ ˜ k [ T ] γ . (cid:0) f (˜ γ )( a ) (cid:1) = ( γ λ ) f (˜ γ )( a )This follows immediately from the definition of f (˜ γ ) . Consider the reductive ˜ R -group ˜ G = G × ˜ k ˜ R and its maximal torus ˜ T = T × ˜ k ˜ R. We want to define an action of ˜Γ as automorphisms of the R –Hopf algebra ˜ R [ T ] =˜ k [ T ] ⊗ ˜ k ˜ R so that the action is ˜Γ–semilinear, this is(4.23) ˜ γ ( xs ) = ˜ γ x ˜ γ s for all ˜ γ ∈ ˜Γ, s ∈ ˜ R and x ∈ ˜ R [ T ] . By Galois descent this will show that the maximaltorus ˜ T of ˜ G descends to a torus (necessarily maximal) T of G . To give the desired semilinear action consider, for a given fixed ˜ γ ∈ ˜Γ , the map˜ k [ T ] × ˜ R → ˜ k [ T ] ⊗ ˜ k ˜ R = ˜ R [ T ]defined by(4.24) ( a, s ) f (˜ γ )( a ) ⊗ ˜ γ s for all a ∈ ˜ k [ T ] and s ∈ ˜ R. From (4.22) and the fact that ˜ γ s = γ s if s ∈ ˜ k ⊂ ˜ R itfollows that the above map is ˜ k -balanced, hence that induces a morphism of ˜ k -spaces(4.25) b f (˜ γ ) : ˜ k [ T ] ⊗ ˜ k ˜ R = ˜ R [ T ] → ˜ R [ T ]satisfying(4.26) b f (˜ γ ) : a ⊗ s ρ (˜ γ )( a ) ⊗ ˜ γ s for all a ∈ ˜ k [ T ] and s ∈ ˜ R. From (4.22) and (4.26) we then obtain an action of thegroup ˜Γ on the Hopf algebra ˜ R [ T ] as prescribed by (4.23).36 .7 Variations of a result of Sansuc. We shall need the following variantion of a well-known and useful result [Sa, 1.13].
Lemma 4.14.
Assume that k is of characteristic zero. Let H be a linear algebraicgroup over k and let U be a normal unipotent subgroup of H .1. Let k ′ /k be a finite Galois extension of fields. Let Γ be a finite group acting on k ′ /k . Then the map H (cid:0) Γ , H ( k ′ ) (cid:1) → H (cid:0) Γ , ( H / U )( k ′ ) (cid:1) is bijective.2. Let R be an object in k – alg. Then the map H ( R, H ) → H ( R, H / U ) is bijective.Proof. The k –group U admits a non-trivial characteristic central split unipotent sub-group U ≃ G na [DG, IV.4.3.13]. We can then form the following commutativediagram of exact sequence of algebraic k –groups 1 y U / U y y −−−→ U −−−→ H −−−→ H / U −−−→ y ∼ = y y −−−→ U −−−→ H −−−→ H / U −−−→ y y U / U y H → H / U and H / U → H / U , it holds for H → H / U . Without loss of generality, we can therefore assume by devissage that U = G na . 371) Since by Hilbert’s Theorem 90 (additive form) and devissage H ( k ′ , U ) = 0, wehave an exact sequence of Γ–groups1 → U ( k ′ ) → H ( k ′ ) → ( H / U )( k ′ ) → . For each c ∈ Z (cid:0) Γ , ( H / U )( k ′ ) (cid:1) , the group c (cid:0) U ( k ′ ) (cid:1) is a uniquely divisible abeliangroup, so H i (cid:0) Γ , c ( U ( k ′ )) (cid:1) = 0 for all i >
0. By applying a basic result on non-abeliancohomology [Se1, § I.5, corollary to prop. 41], the vanishing of these H implies thatthe map H (cid:0) Γ , H ( k ′ ) (cid:1) → H (cid:0) Γ , ( H / U )( k ′ ) (cid:1) is surjective. Similarly, for each z ∈ Z (cid:0) Γ , H ( k ′ ) (cid:1) , the group 0 = H (cid:0) Γ , z ( U ( k ′ )) (cid:1) maps onto the subset of H (cid:0) Γ , H ( k ′ ) (cid:1) consisting of classes of cocycles whose image in H (cid:0) Γ , ( H / U )( k ′ ) (cid:1) coincides with thatof [ z ] . We conclude that the map H (cid:0) Γ , H ( k ′ ) (cid:1) → H (cid:0) Γ , ( H / U )( k ′ ) (cid:1) is bijective.(2) Let us first prove the injectivity by using the classical torsion trick. We are givena H / U –torsor E over Spec( R ). We can twist the exact sequence of R –group schemes1 → U R → H R → H R / U R → E and get the twisted sequence 1 → E U → E H → E H / E U →
1, where as usual we write E U instead of E U R and E H instead of E H R . We consider the following commutative diagram of sets [Gi, III.3.3.4] H ( R, H ) −−−→ H ( R, H / U ) torsion x ≃ torsion x ≃ H ( R, E U ) −−−→ H ( R, E H ) −−−→ H ( R, E H / E U )where the bottom map is an exact sequence of pointed sets. Indeed GL n is the groupof automorphisms of the group scheme G na (see Lemma 4.15 below). It follows that E U corresponds to a locally free sheaf over Spec( R ). By [Gr1, pp 16-17] (or [M, III.3.7]),we have ˇH i ( R, E U ) = 0 for all i > So the the map H ( R, E H ) → H ( R, E H / E U )has trivial kernel and the fiber of H ( R, H ) → H ( R, H / U ) is only [ E ].For surjectivity, if we are given a H / U –torsor E over Spec( R ) then by [Gi, IV.3.6.1]there is a class ∆([ E ]) ∈ ˇH ( R, E U )which is the obstruction to the existence of a lift of [ E ] to H ( R, H ). Here E U is the R –group scheme obtained by twisting G na by the R –torsor E . Since E U correspondsto a locally free sheaf, the same reasoning used above shows that the obstruction∆([ E ]) vanishes as desired. See [GMB] Lemme 7.3 for a more general result. All the ˇH i that we consider coincide with the corresponding H i defined in terms of derivedfunctors. emma 4.15. Let X be a scheme of characteristic . Let E be a locally free X –sheafof finite rank and let V ( E ) be the associated “additive” X –group scheme. Then thenatural homomorphism of fpqc sheaves α : GL ( E ) → Aut X − gr ( V ( E )) is an X –group sheaf isomorphism. In particular, Aut X − gr (cid:0) V ( E ) (cid:1) is an X –groupscheme. Our convention is that of [DG, § V ( E )( X ′ ) = H ( X ′ , E ⊗ O X O X ′ ) forevery scheme X ′ over X . Proof.
It is clear that α is a morphism of X –groups. For showing that α is an iso-morphism of sheaves, we may assume that X = Spec( R ) is affine and that E = R n .This in turn reduces to the case of R = Q and E = Q n . By descent, it will sufficeto establish the result for R = Q and E = ( Q ) n . Now on Q -schemes the functor S Aut gr ( V ( E ))( S ) is representable by a linear algebraic Q -group H accordingto Hochschild-Mostow’s criterion [HM, th. 3.2]. Therefore we can check the factthat α : GL n → H is an isomorphism on Q –points. But this readily follows fromthe equivalence of categories between nilpotent Lie algebras and algebraic unipotentgroups [DG, § IV § GL ( E ) is an X –group scheme, α is an isomorphismof X –group schemes. Let G be a linear algebraic k –group. One of the central results of [CGP] is the ex-istence of maximal tori for twisted groups of the form E G where [ E ] ∈ H ( k [ t ± ] , G ) . This result is used to describe the nature of torsors over k [ t ± ] under G . In our presentwork we are ultimately interested in the classification of reductive groups over Laurentpolynomial rings when k is of characteristic 0 , and applications to infinite dimensionalLie theory. In understanding twisted forms of G the relevant objects are torsors under Aut ( G ), and not G . It is therefore essential to have an analogue of the [CGP] resultmentioned above, but for arbitrary twisted groups, not just inner forms. This isone of the crucial theorems of our paper. If the characteristic of k is sufficiently large. Aut ( G ) need not be an algebraic group. Even if it is, the fact that it need not be connectedleads to considerable technical complications (stemming from the fact that, unlike the affine line, thepunctured line has non-trivial geometric ´etale coverings). As already mentioned, these difficultieshave to be dealt with if one is interested in the study of twisted forms of G R or its Lie algebra. heorem 5.1. Let R = k [ t ± ] where k is a field of characteristic . Every reductivegroup scheme G over R admits a maximal torus. Corollary 5.2.
Let k and R be as above. Let G be a smooth affine group schemeover R whose connected component of the identity G is reductive. Then1. H toral ( R, G ) = H ( R, G ) .
2. If G is constant, i.e. G = G × k R for some linear algebraic k –group G , then H toral ( R, G ) = H loop ( R, G ) = H ( R, G ) . The first assertion is an immediate corollary of the Theorem while the second thenfollows from Corollary 3.16.2 and Lemma 2.8.The proof of the Theorem relies on Bruhat-Tits twin buildings and Galois descentconsiderations. We begin by establishing the following useful reduction.
Lemma 5.3.
It suffices to establish Theorem 5.1 under the assumption that G is atwisted form of a simple simply connected Chevalley R –group. Proof.
Assume that Theorem 5.1 holds in the simple simply connected case. By[XII.4.7.c], there is a natural one-to-one correspondence between the maximal tori of G , its adjoint group G ad and those of the simply connected covering ˜ G ad of G ad . Wecan thus assume without lost of generality that G is simply connected. By [XXIV.5.10]we have G = Y i =1 ,...,l Y S i /R G i where each S i is a connected finite ´etale covering of R and each G i a simple simply con-nected S i –group scheme. By Demazure’s main theorem, the S i –groups G i are twistedforms of simple simply connected Chevalley groups. Since by Lemma 2.8 S i is a Lau-rent polynomial ring, our hypothesis implies that each of the S i –groups G i admits amaximal torus T i . Then our R –group G admits the maximal torus Q i =1 ,...,l Q S i /R T i . Throughout this section k denotes a field of characteristic 0 . We set R = k [ t ± ] ,K = k ( t ) and b K = K (( t )) . For a “survival kit” on euclidean buildings, we recommendLandvogt’s paper [L]. The usual algebraic group literature would use the term “almost simple” in this situation. Weadhere throughout to the terminology of [SGA3]. R be a finite Galois extension of R of the form ˜ R = ˜ k [ t ± n ] where ˜ k/k is a finiteGalois extension of k containing all n -roots of unity in k. Then as we have alreadyseen ˜Γ := Gal( ˜
R/R ) = µµµ n (˜ k ) ⋊ Γ where Γ = Gal(˜ k/k ).Set ˜ t = t n We let L = ˜ k ( ˜ t ) = ˜ k (˜ t − ), and consider the two completions b L + =˜ k (( ˜ t )) and b L − = ˜ k ((˜ t − )) of L at 0 and ∞ respectively, as well as their correspondingvaluation rings b A + = ˜ k [[ ˜ t ]] and b A − = ˜ k [[˜ t − ]].Let G be a split simple simply connected group over k. Let T be a maximal splittorus of G , B + a Borel subgroup of G which contains T , and B − the correspondingopposite Borel subgroup (which also contains T ). We denote by W = N G ( T ) thecorresponding Weyl group and by ∆ ± the Dynkin diagram attached to ( G , B ± , T ).Following Tits [T3], we consider the twin building B = B + × B − of G × k L withrespect to the two completions b L + and b L − . Recall that B comes equipped with anaction of the group G ( L ), hence also of G ( ˜ R ) . The split torus T × k L gives rise to atwin apartment A = A + × A − of B . The Borel subgroups B ± define the fundamentalchambers C ± of A ± , each of which is an open simplex whose vertices are given by theextended Dynkin diagram ˜∆ ± of ∆ ± .Recall that the group functor Aut ( G ) is an affine group scheme. The group Aut ( G )( L ) acts on B by “transport of structure” [L] 1.3.4. This leads to an actionof G ( L ) on B via Int : G → Aut ( G ) . This action coincides with the “standard”action of G ( L ) mentioned before because G is semisimple. By taking into accountthe natural action of ˜Γ ≃ Gal( b L + / b K ) on B we conclude that the twin building B is equipped with an action of the semi-direct product Aut( G )( ˜ R ) ⋊ Γ which iscompatible (via the adjoint action) with the action of G ( ˜ R ) . The hyperspecial group G ( b A ± ) fixes a unique point φ ± of A ± [BT1, § B ± are G ( b L ± )-conjugate to φ ± of B ± , and cantherefore be identified with the set of left cosets G ( b L ± ) / G ( b A ± ) ≃ G ( b L ± ) . φ ± ⊂ B ± . More generally each facet of the building B ± has a type [BT1, § ± and the type of a point x ∈ B ± is the type of its underlying facet F x . The type of the chamber C ± is ∅ and the type of an hyperspecial point is ˜∆ ± \ ∆ ± , namely the extra vertex of the affine Dynkin diagram. The group in question acts on the set of maximal split tori, hence permutes the apartmentsaround. Namely the smallest facet containing x in its closure. .2 Proof of Theorem 5.1 By Lemma 5.3, we can assume that G is simple simply connected. By theIsotriviality Theorem [GP1, cor. 2.16], we know that our R –group G is isotriv-ial. This means that there exists a finite Galois covering S/R and a “trivialization” f : G × k S ≃ G × R S where G is a split simple simply connected k –group. In ourterminology, G is the Chevalley k –form of G . Because of the structure of the algebraic fundamental group of R we may assumewithout loss of generality that S = ˜ R is as in § R and ˜ R “look the same”,namely they are both Laurent polynomial rings in one variable with coefficients in afield.We have Spec( ˜ R ) = P k \ { , ∞} and the action of ˜Γ on ˜ R extends to P k since ˜ R is regular of dimension 1.For ˜ γ ∈ ˜Γ consider the map z ˜ γ = f − ◦ ˜ γ f : ˜Γ → Aut ( G )( ˜ R ), where Aut ( G )stands for the group scheme of automorphisms of the Z -group G . Then z = ( z ˜ γ ) ˜ γ ∈ ˜Γ isa cocycle in Z (cid:0) ˜Γ , Aut ( G )( ˜ R ) (cid:1) where the Galois group ˜Γ acts naturally on Aut ( G )( ˜ R )via its action on ˜ R . Descent theory tells us that G is isomorphic to the twisted R -group z G . The action of ˜Γ on
Aut ( G )( ˜ R ) allows us to consider the semidirect product group Aut ( G )( ˜ R ) ⋊ ˜Γ . We then have a group homomorphism(5.1) ψ z : ˜Γ → Aut ( G )( ˜ R ) ⋊ ˜Γgiven by ψ z ( γ ) = z γ γ which is a section of the projection map Aut ( G )( ˜ R ) ⋊ ˜Γ → ˜Γ.Let T be a maximal split of G . Set L = ˜ k (˜ t ) , and let A + (resp. A − ) be the localring of P k at 0 (resp. ∞ ). The composite map (see § ψ z −→ Aut ( G )( ˜ R ) ⋊ ˜Γ → Aut( B )is a group homomorphism. The corresponding action of ˜Γ on B will be referred toas the twisted action of ˜Γ on the building. We now appeal to the Bruhat-Tits fixedpoint theorem [BT1, § p = ( p + , p − ) ∈ B which is fixed underthe twisted action, i.e. ψ z (˜ γ ) .p = z ˜ γ ˜ γ ( p ) = p for all ˜ γ ∈ ˜Γ. Abramenko’s result[A, Proposition 5] states that G ( ˜ R ) . A = B . There thus exists g ∈ G ( ˜ R ) such that p belongs to the apartment g. A . Up to replacing z ˜ γ by Int( g ) − z ˜ γ Int(˜ γ ( g )), we cantherefore assume that p belongs to A .We shall use several times that ˜Γ acts trivially on A under the standard action.To see this one reduces to the action of e Γ on A + = φ + + b T ⊗ Z R . Firstly e Γ stabilizes Recall that for convenience z G is shorthand notation for z ( G × k R ) = z ( G R ). G ( b A + ) so it fixes φ + . Secondly it acts trivially on b T so acts trivially on A + .Observe that since ˜ γ ( p ) = p , we have that z ˜ γ . p = p for all ˜ γ ∈ ˜Γ.Let F p ± be the facet associated to p ± and choose a vertex q ± of F p ± . The trans-forms of q ± by z ˜ γ and ˜ γ are vertices of F p ± , so ψ z (˜ γ ) .q ± belongs to A . We define x ± = Barycentre (cid:16) ψ z (˜ γ ) .q ± , ˜ γ ∈ ˜Γ (cid:17) ∈ A , where the barycentre stands for the riemannian’s one as defined by Pansu [P, § d be the integer attached to G in [Gi1, § m = d | ˜Γ | . Let s ∈ L (afixed algebraic closure of L ) be such that s m = ˜ t. We have accordingly s mn = t. Set R ′ = k ′ [ s ± ]where k ′ is a Galois extension of k which contains ˜ k and all mn -roots of unity in k. Then R ′ is Galois over R of Galois groupΓ ′ = µµµ mn ( k ′ ) ⋊ Gal( k ′ /k ) . By Galois theory the map υ : Γ ′ → ˜Γ given by(5.2) υ : ( ξ, θ ) ( ξ m , θ | ˜ k )is a surjective group homomorphism.We consider the twin building B ′ = B ′ + × B ′− of G which is constructed inthe manner described above after replacing, mutatis mutandis, the relevant objectsattached to ˜ R by those of R ′ . We have a restriction map [Ro, § II.4] ρ ± : B ± → B ′± which gives rise to ρ = ( ρ + , ρ − ) : B → B ′ . Furthermore, if γ ′ ∈ Γ ′ and we set υ ( γ ′ ) = ˜ γ then the following diagram commutes (cid:0) ∗ ) B ρ −→ B ′ ˜ γ y γ ′ y B ρ −→ B ′ where the actions of ˜Γ and Γ ′ are the twisted actions. If we define z ′ : Γ ′ → Aut ( G )( R ′ ) by z ′ : γ ′ z ′ γ ′ = z ˜ γ ∈ Aut ( G )( ˜ R ) ⊂ Aut ( G )( R ′ )then z ′ is a cocycle and the classes [ z ] and [ z ′ ] in H (cid:0) R, Aut ( G ) (cid:1) are the same.43 emma 5.4. ρ ± ( x ± ) is a hyperspecial point of B ′± .Proof. We look at the case ρ + ( x + ). We need to consider the intermediate extensions b L + = ˜ k (( ˜ t )) ⊂ k ′ (( ˜ t )) ⊂ k ′ (( s d )) ⊂ k ′ (( s )) . The map ρ + is the composite of the corresponding maps for the intermediate fields,namely B + = B + ( G , ˜ k (( ˜ t )) ) ρ , + −→ B (cid:0) G , k ′ (( ˜ t )) (cid:1) ρ d, + −→ B (cid:0) G , k ′ (( s d )) (cid:1) ρ md , + −→ B (cid:0) G , k ′ (( s )) (cid:1) B ′ + . The first map does not change the type. By [Gi1, lemma 2.2.a], the image under ρ d, + of any vertex is a hyperspecial point. We have ρ + ( x + ) = ρ + h Barycenter (cid:16) ψ z (˜ γ ) .q ± , γ ∈ ˜Γ (cid:17)i = ρ md , + h Barycenter (cid:16) ρ d, + ◦ ρ , + ( ψ z (˜ γ ) .q ± ) , γ ∈ ˜Γ (cid:17)i . By [Gi1, Lemma 2.3’ in the errata], we know that the image under ρ md , + of thebarycentre of md hyperspecial points of a common apartment (namely the one attachedto the torus T ) is a hyperspecial point, so we conclude that ρ + ( x + ) is a hyperspecialpoint.In view of diagram (*) above it follows that by replacing ˜ R by R ′ we may assumewithout loss of generality that the points p ± ∈ A ± are hyperspecial. Note thatby construction, the points ρ ± ( x ± ) of B ′± are fixed by both actions (standard andtwisted) of Γ ′ , so that after our further extension of base ring we may assume that ρ ± ( x ± ) of B ± are fixed by both actions of ˜Γ . Since T ( ˜ R ) acts transitively on the sets φ ± + ( b T ) ⊂ A ± of hyperspecial points of A ± , there exists g ∈ T ( ˜ R ) and a cocharacter λ ∈ ( b T ) such that g. ( ψ + , x λ − ) = ( x + , x − ) = x. where x λ − := φ − + λ (recall that we have a map ( b T ) → A − = ( b T ) ⊗ Z R definedby θ φ − + θ ). Up to replacing the cocycle z by z ′ where z ′ ˜ γ = Int( g ) − z ˜ γ ˜ γ Int( g ),we may assume that ψ z (˜ γ ) . ( φ + , x λ − ) = z γ . ( φ + , x λ − ) = ( φ + , x λ − ) for every ˜ γ ∈ ˜Γ. Inparticular, z ˜ γ ∈ Stab
Aut ( G )( b L + ) ( φ + ) = Aut ( G )( b A + )hence z ˜ γ ∈ Aut ( G )( ˜ R ) ∩ Aut ( G )( b A + ) Aut ( G )(˜ k [ ˜ t ])44or each ˜ γ ∈ ˜Γ. Let g λ := λ ( ˜ t ) ∈ T ( ˜ R ) ⊂ G ( ˜ R ). We have g λ .φ − = x λ − and therefore z ˜ γ ∈ Stab
Aut ( G )( b L − ) ( x λ − )Int( g λ ) Stab Aut ( G )( b L − ) ( φ − ) Int( g λ ) − = Int( g λ ) Aut ( G )( b A − )Int( g − λ ) . It follows that for each ˜ γ ∈ ˜Γ z ˜ γ ∈ J λ := Aut ( G )(˜ k [ ˜ t ]) ∩ Int( g λ ) Aut ( G )( b A − ) Int( g − λ )= Aut ( G )( k [ ˜ t ]) ∩ Int( g λ ) Aut ( G )(˜ k [ ˜ t − ]) Int( g − λ ) . Note that for ˜ γ ∈ ˜Γ we have ˜ γ g λ = ˜ γ λ ( t ′ ) = λ ( t ′ ) (cid:0) λ ( t ′ ) − γ λ ( t ′ ) (cid:1) ∈ λ ( t ′ ) T (˜ k ) ⊂ λ ( t ′ ) G ( b A − ) = g λ G ( b A − )From this it follows not only that the subgroup J λ of Aut ( G )( ˜ R ) is stable under the(standard) action of ˜Γ , but also that[ z ] ∈ Im (cid:16) H (cid:0) ˜Γ , J λ (cid:1) → H (cid:0) ˜Γ , Aut ( G )( ˜ R ) (cid:1)(cid:17) . It turns out that the structure of the group J λ is known by a computation of Ra-manathan, as we shall see in Proposition 16.2 below. We have J λ = U λ (˜ k ) ⋊ Z Aut ( G ) ( λ )(˜ k ) ⊂ Aut ( G )(˜ k [ ˜ t ]) , where U λ is a unipotent k –group. Lemma 4.14 shows that the map H (cid:0) ˜Γ , Z Aut ( G ) ( λ )( ˜ k ) (cid:1) → H (cid:0) ˜Γ , J λ (cid:1) is bijective. Summarizing, we have the commutative diagram H (cid:0) ˜Γ , Z Aut ( G ) ( λ )( ˜ k ) (cid:1) −−−→ H (cid:0) ˜Γ , Aut ( G )( ˜ k ) (cid:1)y ∼ = y H (cid:0) ˜Γ , J λ (cid:1) −−−→ H (cid:0) ˜Γ , Aut ( G )( ˜ R ) (cid:1) which shows that[ z ] ∈ Im (cid:16) H (cid:0) ˜Γ , Aut ( G )( ˜ k ) (cid:1) → H (cid:0) ˜Γ , Aut ( G )( ˜ R ) (cid:1)(cid:17) . This means that [ z ] is cohomologous to a loop cocycle, and we can now conclude byProposition 4.13 45 Internal characterization of loop torsors and ap-plications
We continue to assume that our base field k of characteristic zero. Let R n = k [ t ± , ..., t ± n ] and X = Spec( R n ) . As explained in Example 2.5 we have π ( R n , a ) ≃ b Z n ⋊ Gal( k ) , where the action of Gal( k ) on b Z n is given by our fixed choice of compatibleroots of unity in k . For convenience in what follows we will denote π ( R n , a ) simplyby π ( R n ) . Throughout this section G denotes a linear algebraic k –group. We first observe that loop torsors make sense over R = k , namely H loop ( R , G )is the usual Galois cohomology H ( k, G ) . Section 3.3 shows that Z (cid:0) π ( R n ) , G ( k ) (cid:1) is given by couples ( z, η geo ) where z ∈ Z (Gal( k ) , G ( k )) and η geo ∈ Hom
Gal( k ) ( π ( X, a ) , z G ( k )) = Hom Gal( k ) ( b Z n , z G ( k )) ≃ Hom
Gal( k ) ( ∞ µµµ, z G ). We are now ready to state and establish the internal characteri-zation of k -loop torsors as toral torsors. Theorem 6.1.
Assume that G is reductive. Then H toral ( R n , G ) = H loop ( R n , G ) . First we establish an auxiliary useful result.
Lemma 6.2.
1) Let H be an R n group of multiplicative type. Then for all i ≥ thenatural abstract group homomorphisms H i (cid:0) π ( R n ) , H ( R n, ∞ ) (cid:1) → H i ( R n , H ) → H i ( F n , H ) . are all isomorphisms.2) Let T be a k –torus. Let c ∈ Z (cid:0) π ( R n ) , Aut ( T )( k ) (cid:1) ⊂ Z (cid:0) π ( R n ) , Aut ( T )( R n, ∞ ) (cid:1) be a cocycle, and consider the twisted R n –torus c T = c ( T × k R n ) . Consider the natural maps H i (cid:0) π ( R n ) , c ( T ( k ) (cid:1) → H i (cid:0) π ( R n ) , c T ( R n, ∞ ) (cid:1) → H i ( R n , c T ) → H i ( F n , c T ) Then.(i) If i = 1 then the first group homomorphism is surjective and the last one is anisomorphism.(ii) If i > then all the maps are group isomorphisms. roof.
1) The second isomorphism is proposition 3.4.3 of [GP2]. As for the first iso-morphism we consider the Hochschild-Serre spectral sequence H p (cid:0) π ( R n ) , H q ( R n, ∞ , H ) (cid:1) = ⇒ H p + q ( R n , H ) . From the fact that the group H p ( R n, ∞ , H )is torsion for p ≥ −→ m H p ( R n, ∞ , m H ) → H p ( R n, ∞ , H ), where m H stands for the kernel of the “multiplication by m ” map, is surjective. By loc. cit. cor. 3.3 H p ( R n, ∞ , m H ) vanishes for all m ≥ . Hence H p ( R n, ∞ , H ) = 0 . The spectralsequence degenerates and yields the isomorphisms H i (cid:0) π ( R n ) , H ( R n, ∞ ) (cid:1) ≃ H i ( R n , H )for all i ≤ .
2) We begin with an observation about the notation used in the statement ofthe Lemma. The subgroup T ( k ) of T ( R n, ∞ ) is stable under the (twisted) action of π ( R n ) on c T ( R n, ∞ ) . To view T ( k ) as a π ( R n )–module with this twisted action wewrite c ( T ( k )) . The fact that the last two maps are isomorphism for all i ≥ π ( R n )–module A = T ( R n, ∞ ) / T ( k ). We have A = lim −→ m T ( R n, ∞ ) / T ( k )= lim −→ m ( b T ) ⊗ Z R × n,m / ( b T ) ⊗ Z k × = ( b T ) ⊗ Z lim −→ m R × n,m / k × = ( b T ) ⊗ Z lim −→ m ( Z n ) m (where ( Z n ) m = Z n )= ( b T ) ⊗ Z Q n given that the transition map ( Z n ) m → ( Z n ) md is multiplication by d . It follows that A, hence also c A, is uniquely divisible.We consider the sequence of continuous π ( R n )–modules(6.1) 1 → c (cid:0) T ( k ) (cid:1) → c T ( R n, ∞ ) → c A → . From the fact that c A is uniquely divisible it follows that the group homomorphisms(6.2) H i (cid:0) π ( R n ) , c ( T ( k )) (cid:1) → H i (cid:0) π ( R n ) , c T ( R n, ∞ ) (cid:1) are surjective for all i ≥ i > . We can proceed now with the proof of Theorem 6.1.47 roof.
Let us show first show that H loop ( R n , G ) ⊂ H toral ( R n , G ). Case 1: G = Aut ( H ) where H is a semisimple Chevalley k –group : Let φ : π ( R n ) → Aut ( H )( k ) be a loop cocycle. Consider the twisted R –group φ G . Propo-sition 4.13 shows that the connected component of the identity ( φ G ) = φ ( G ) of φ G admits a maximal R -torus. Therefore φ G admits a maximal R -torus, hence φ definesa toral R -torsor. Case 2: G = Aut ( H ) where H is a semisimple k –group : Denote by H the Cheval-ley k –form of H . There exists a cocycle z : Gal( k ) → G ( k ) such that H is isomorphicto the twisted k –group z H . We can assume then that H = z H and G = z Aut ( H ).The torsion bijection τ z : H ( R, G ) = H ( R, z Aut ( H )) ∼ −→ H ( R, Aut ( H )) ex-changes loop classes (resp. toral classes) according to Remark 3.3. Case 1 then yields H loop ( R n , G ) ⊂ H toral ( R n , G ). General case.
The k –group acts by conjugacy on G , its center Z ( G ) and then on itsadjoint quotient G ad . Denote by f : G → Aut ( G ad ) this action. Let φ : π ( R n ) → G ( k ) be a loop cocycle. We have to show that the twisted R –group scheme φ G admits a maximal torus. Equivalently, we need to show that ( φ G ) = φ ( G ) admitsa maximal torus which is in turn equivalent to the fact that ( f ∗ φ G ) ad = f ∗ φ ( G ad )admits a maximal torus [SGA3, XII.4.7]. But f ∗ φ is a loop cocycle for Aut ( G ad ), sodefines a toral R –torsor under Aut ( G ) according to Case 2. Thus f ∗ φ ( G ad ) admitsa maximal torus as desired.To establish the reverse inclusion we consider the quotient group ννν = G / G , whichis a finite and ´etale k –group. In particular ννν × k k is constant and finite, and one caneasily see as a consequence that the natural map ννν ( k ) → ννν ( R n, ∞ ) is an isomorphism.We first establish the result for tori and then the general case. G is a torus T : We again appeal to the isotriviality theorem of [GP1] to see that H ( π ( R n ) , G ( R n, ∞ )) ∼ −→ H ( R n , G ). We consider the following commutative dia-gram of continuous π ( R n )–groups1 −−−→ T ( k ) −−−→ G ( k ) −−−→ ννν ( k ) −−−→ y y || −−−→ T ( R n, ∞ ) −−−→ G ( R n, ∞ ) −−−→ ννν ( R n, ∞ ) −−−→ . This gives rise to an exact sequence of pointed sets1 −−−→ H (cid:0) π ( R n ) , T ( k ) (cid:1) −−−→ H (cid:0) π ( R n ) , G ( k ) (cid:1) −−−→ H (cid:0) π ( R n ) , ννν ( k ) (cid:1)y y || −−−→ H (cid:0) π ( R n ) , T ( R n, ∞ ) (cid:1) −−−→ H (cid:0) π ( R n ) , G ( R n, ∞ ) (cid:1) −−−→ H (cid:0) π ( R n ) , ννν ( R n, ∞ ) (cid:1) . We recall, for the last time, that φ G is short hand notation for φ ( G R ) .
48e are given a cocycle z ∈ Z (cid:0) π ( R n ) , G ( R n, ∞ ) (cid:1) . Denote by c the image of z in Z (cid:0) π ( R n ) , ννν ( k ) (cid:1) = Z (cid:0) π ( R n ) , ννν ( R n, ∞ ) (cid:1) under the bottom map. Under the topmap, the obstruction to lifting [ c ] to H (cid:0) π ( R n ) , G ( k ) (cid:1) is given by a class ∆([ c ]) ∈ H (cid:0) π ( R n ) , c ( T ( k )) (cid:1) [Se1, § I.5.6]. This class vanishes in H (cid:0) π ( R n ) , c T ( R n, ∞ )) (cid:1) , soLemma 6.2 shows that ∆([ c ]) = 0. Hence c lifts to a loop cocycle a ∈ Z (cid:0) π ( R n ) , G ( k ) (cid:1) .By twisting by a we obtain the following commutative diagram of pointed sets1 −−−→ H (cid:16) π ( R n ) , a (cid:0) T ( k ) (cid:1) (cid:17) −−−→ H (cid:16) π ( R n ) , a (cid:0) G ( k ) (cid:1) (cid:17) −−−→ H (cid:16) π ( R n ) , c (cid:0) ννν ( k ) (cid:1) (cid:17)y y || −−−→ H (cid:16) π ( R n ) , a (cid:0) T ( R n, ∞ ) (cid:1) (cid:17) −−−→ H (cid:16) π ( R n ) , a (cid:0) G ( R n, ∞ ) (cid:1)(cid:17) −−−→ H (cid:16) π ( R n ) , c (cid:0) ννν ( R n, ∞ ) (cid:1) (cid:17)y a ( G ( k )) denotes G ( k )as a π ( R n )–submodule of a G ( R n, ∞ ) , and similarly for c (cid:0) ννν )( k ) (cid:1) = c (cid:0) ννν ( R n, ∞ ) (cid:1) . Weconsider the torsion map τ a : H (cid:0) π ( R n ) , a G ( R n, ∞ ) (cid:1) ∼ −→ H (cid:0) π ( R n ) , G ( R n, ∞ ) (cid:1) . Then τ − a ([ z ]) ∈ ker (cid:16) H (cid:0) π ( R n ) , a G ( R n, ∞ ) (cid:1) → H (cid:0) π ( R n ) , c (cid:0) ννν ( R n, ∞ ) (cid:1)(cid:17) . The diagram above shows that τ − a ([ z ]) comes from H (cid:0) π ( R n ) , a ( G ( k )) (cid:1) , hence [ z ]comes from H (cid:0) π ( R n ) , G ( k ) (cid:1) as desired. We conclude that H ( R n , G ) is covered byloop torsors. General case :
Let T be a maximal torus of G . Consider the commutative diagram H loop (cid:0) R n , N G ( T ) (cid:1) −−−→ H (cid:0) R n , N G ( T ) (cid:1)y y H loop ( R n , G ) −−−→ H toral ( R n , G ) . The right vertical map is surjective according to Lemma 3.13.1. The top horizontalmap is surjective by the previous case. We conclude that the bottom horizontal mapis surjective as desired.
Corollary 6.3.
Let G be a reductive R –group. Then G is loop reductive if and onlyif G admits a maximal torus. roof. Let G be the Chevalley k –form of G . To G corresponds a class [ E ] ∈ H (cid:0) R n , Aut ( G ) (cid:1) . When G is semisimple, Aut ( G ) is an affine algebraic k –group and the Corollary fol-lows from H toral (cid:0) R n , Aut ( G ) (cid:1) = H loop (cid:0) R n , Aut ( G ) (cid:1) .We deal now with the general case. We already know tby 4.13 that every loopreductive group is toral. Conversely assume that G admits a maximal torus. Considerthe exact sequence of smooth k –groups [XXIV.1.3.(iii)]1 → G ad → Aut ( G ) p −→ Out ( G ) → , where G ad is the adjoint group of G and Out ( G ) is a constant k –group. Since R n isa noetherian normal domain, we know that R n –torsors under Out ( G ) are isotrivial[X.6]. Furthermore by [SGA1, XI § ct (cid:0) π ( R n ) , Out ( G )( k ) (cid:1) / ∼ ∼ −→ H (cid:0) R n , Out ( G ) (cid:1) . So p ∗ [ E ] is given by a continous homomorphism π ( R n ) → Out ( G )( k ) whose imagewe denote by Out ( G ) ♯ . This is a finite group so that Aut ( G ) ♯ := p − ( Out ( G ) ♯ ) isan affine algebraic k –group. We consider the square of pointed sets H (cid:0) R n , Aut ( G ) ♯ (cid:1) p ♯ ∗ −−−→ H (cid:0) R n , Out ( G ) ♯ (cid:1)y y H (cid:0) R n , Aut ( G ) (cid:1) p ∗ −−−→ H (cid:0) R n , Out ( G ) (cid:1) . Since
Aut ( G ) / Aut ( G ) ♯ = Out ( G ) / Out ( G ) ♯ , this square is cartesian as can beseen by using the criterion of reduction of a torsor to a subgroup [Gi, III.3.2.1]. Byconstruction, [ p ∗ E ] comes from H (cid:0) R n , Out ( G ) ♯ (cid:1) , hence [ E ] comes from a class [ F ] ∈ H (cid:0) R n , Aut ( G ) ♯ (cid:1) . Our assumption is that the R n –group G = E G = F G contains amaximal torus, so G ad = F G ad contains a maximal torus and F (cid:0) Aut ( G ) ♯ (cid:1) containsa maximal torus. In other words, F is a toral R n –torsor under Aut ( G ) ♯ . From theequality H toral (cid:0) R n , Aut ( G ) ♯ (cid:1) = H loop (cid:0) R n , Aut ( G ) ♯ (cid:1) , it follows that that F is a looptorsor under Aut ( G ) ♯ . By applying the change of groups Aut ( G ) ♯ → Aut ( G ), weconclude that E is a loop torsor under Aut ( G ) , hence that G is loop reductive.Lemma 3.13.2 yields the following fact. Corollary 6.4.
Let → S → G ′ p → G → be a central extension of G by a k –group S of multiplicative type. Then the diagram H loop ( R n , G ′ ) ⊂ H ( R n , G ′ ) p ∗ y p ∗ y H loop ( R n , G ) ⊂ H ( R n , G ) is cartesian. emark 6.5. For a k –group G satisfying the condition of Corollary 3.16, one canprove in a simpler way that toral G –torsors over R n are loop torsors by reducing tothe case of a finite ´etale group. Remark 6.6.
Given an integer d ≥
2, the Margaux algebra (both the Azumaya andLie versions) [GP2, 3.22 and example 5.7 ] provides an example of a
PGL d -torsor over C [ t ± , t ± ] which is not a loop torsor. The underlying PGL d -torsor is therefore nottoral. This means that the Margaux Azumaya algebra does not contain any (com-mutative) ´etale C [ t ± , t ± ]-subalgebra of rank d, and that the Margaux Lie algebra,viewed as a Lie algebra over C [ t ± , t ± ], does not contain any Cartan subalgebras (inthe sense of [SGA3]). Remark 6.7.
More generally, for each each positive integer d , we have H toral ( R n [ x , ..., x d ] , G ) = H loop ( R n [ x , ..., x d ] , G ). Since π ( R n [ x , ..., x d ]) ≃ π ( R n ) and N G ( T )( S ) = N G ( T )( S [ x , ..., x d ])for every finite ´etale covering S of R n , the proof we have given works just the samein this case. Let F n = k (( t ))(( t )) ... (( t n )) . In an analogous fashion to what we did in the caseof R n we define F n,m = k (( t m ))(( t m )) ... (( t m n )) and F n, ∞ = lim −→ F n,m . Remark 6.8. (a) If ˜ k is a field extension of k the natural map ˜ k ⊗ k F n,m → ˜ k (( t m ))(( t m )) ... (( t m n )) is injective. If the extension is finite , then this map is anisomorphism. We will find it convenient (assuming that the field ˜ k is fixed in ourdiscussion) to denote ˜ k (( t m ))(( t m )) ... (( t m n )) simply by ˜ F n,m . (b) The field lim −→ k (( t m ))(( t m )) ... (( t m n )) is algebraically closed. We will denote thealgebraic closure of F n (resp. F n,m , F n, ∞ ) in this field by F n (resp. F n,m , F n, ∞ ). Asmentioned in (a) we have a natural injective ring homomorphsm k ⊗ k F n, ∞ → F n . (c) There is a natural group morphism π ( R n ) → Gal( F n ) given by considering theGalois extensions ˜ R n,m = ˜ k ⊗ k R n,m of R n and ˜ F n,m of F n respectively, where ˜ k ⊂ k isa finite Galois extension of k containing all m -roots of unity. These homomorphismsare in fact isomorphisms. For by applying successively the structure theorem forlocal fields [GMS] § F n ) = ∞ µµµ n ( k ) ⋊ Gal( k ). This means thatGal( F n ) = lim ←− Gal (cid:0) ˜ k (( t m ))(( t m )) ... (( t m n )) /F n (cid:1) If k is algebraically closed this was proved in [GP3] cor 2.14. m running over all integers and ˜ k running over all finite Galois extensions of k inside k containing a primitive m –root of unity. Since at each step we have anisomorphismGal (cid:0) ˜ k ⊗ R n,m /R n (cid:1) ∼ = Gal (cid:0) ˜ k (( t m ))(( t m )) ... (( t m n )) /F n (cid:1) ∼ = µµµ nm (˜ k ) ⋊ Gal(˜ k/k ) , we conclude that π ( R n ) ∼ = Gal( F n ).(d) It follows from (c) that the base change R n → F n induces an equivalence ofcategories between finite ´etale coverings of R n and finite ´etale coverings of F n . Fur-thermore, if E /R n is a finite ´etale covering of R n , we have E ( R n ) = E ( F n ). Indeed, E is split by some Galois covering ˜ R n,m = ˜ k ⊗ k R n,m and E ( R n ) = E ( ˜ R n,m ) Gal( ˜ R n,m /R n ) = E ( ˜ F n,m ) Gal( ˜ F n,m /F n ) = E ( F n ). Proposition 6.9.
The canonical map H loop ( R n , G ) → H ( F n , G ) is surjective.Proof. We henceforth identify π ( R n ) with Gal( F n ) as described in Remark 6.8(c).The proof is very similar to that of Theorem 6.1, and we maintain the notationtherein. Again we proceed in two steps. First case: G is a torus T : We consider the following commutative diagram ofcontinuous π ( R n )–groups1 −−−→ T ( k ) −−−→ G ( k ) −−−→ ννν ( k ) −−−→ y y || −−−→ T ( F n ) −−−→ G ( F n ) −−−→ ννν ( F n ) −−−→ . This gives rise to an exact sequence of pointed sets1 −−−→ H (cid:0) π ( R n ) , T ( k ) (cid:1) −−−→ H (cid:0) π ( R n ) , G ( k ) (cid:1) −−−→ H (cid:0) π ( R n ) , ννν ( k ) (cid:1)y y || −−−→ H (cid:0) F n , T (cid:1) −−−→ H (cid:0) F n , G (cid:1) −−−→ H (cid:0) F n , ννν (cid:1) . We are given a cocycle z ∈ Z (cid:0) Gal( F n ) , G ( F n ) (cid:1) = Z (cid:0) π ( R n ) , G ( F n ) (cid:1) , and considerits image c ∈ Z (cid:0) π ( R n ) , ννν ( F n ) (cid:1) . By reasoning as in Theorem 6.1 we see that [ z ]comes from H (cid:0) π ( R n ) , G ( k ) (cid:1) as desired. We conclude that H ( F n , G ) is covered by k -loop torsors. 52 eneral case: Let T be a maximal torus of G . H loop (cid:0) R n , N G ( T ) (cid:1) −−−→ H (cid:0) F n , N G ( T ) (cid:1)y y H loop ( R n , G ) −−−→ H ( F n , G ) . The reasoning is again identical to the one used in Theorem 6.1.
As before G denotes a linear algebraic group over a field k of characteristic zero. R n and π ( R n ) are as in the previous section. Let η : π ( R n ) → G ( k ) be a continuous cocycle. Consider as before a Galoisextension ˜ R n,m = ˜ k ⊗ k R n,m of R n where ˜ k ⊂ k is a finite Galois extension of k containing all m –roots of unity in k , chosen so that our cocycle η factors through theGalois group(7.1) ˜Γ n,m = Gal( ˜ R n,m /R n ) ∼ = µµµ nm (˜ k ) ⋊ Gal(˜ k/k )We assume henceforth that G acts on a k –scheme Y . The Galois group ˜Γ n,m actsnaturally on Y ( ˜ R n,m ) , and we denote this action by γ : y γ y. By means of η weget a twisted action of ˜Γ n,m on Y ( ˜ R n,m ) which we denote by γ : y γ ′ y where(7.2) γ ′ y = η γ . γ y By Galois descent (7.2) leads to a twisted form of the R n –scheme Y R n . One knowsthat this twisted form is up to isomorphism independent of the Galois extension ˜ R n,m chosen through which η factors. We will denote this twisted form by η Y R n , or simplyby η Y following the conventions that have been previously mentioned regarding thismatter.Let ( z, η geo ) be the couple associated to η according to Lemma 3.7. Thus z ∈ Z (cid:0) Gal( k ) , G ( k ) (cid:1) and η geo ∈ Hom k − gp ( ∞ µµµ n , z G ) by taking into account Lemma 3.10.By means of z we construct a twisted form z Y of the k –scheme Y which comesequipped with an action of z G . Via η geo , this defines an algebraic action of the affine k -group ∞ µµµ n on z Y . At the level of k –points of ∞ µµµ n , the action is given by(7.3) b n.y = η geo ( b n ) .y b n ∈ ∞ µµµ n ( k ) = lim ←− m µµµ nm ( k ) and y ∈ z Y ( k ). We denote by ( z Y ) η geo the closedfixed point subscheme for the action of ∞ µµµ n (see [DG] II § z Y ) η geo ( k ) = n y ∈ z Y ( k ) = Y ( k ) | y = η geo ( b n ) .y ∀ b n ∈ ∞ µµµ n ( k ) o and in terms of rational points(7.4)( z Y ) η geo ( k ) = z Y ( k ) ∩ ( z Y ) η geo ( k ) = n y ∈ z Y ( k ) | y = η geo ( b n ) .y ∀ b n ∈ ∞ µµµ n ( k ) o . where we recall that z Y ( k ) = n y ∈ Y ( k ) | y = z γ . γ y ∀ γ ∈ Gal( k ) o . Theorem 7.1.
1. Let G act on Y as above, and assume that Y is projective ( i.e.a closed subscheme in P nk ). Let η : π ( R n ) → G ( k ) be a (continuous) cocycle,and η Y be the corresponding twisted form of Y R n . The following are equivalent:(a) ( η Y )( R n ) = ∅ ,(b) ( η Y )( K n ) = ∅ ,(c) ( η Y )( F n ) = ∅ ,(d) ( z Y ) η geo ( k ) = ∅ .2. Let S be a closed k –subgroup of G . Let Y be a smooth G –equivariant compact-ification of G / S (i.e., Y is projective k –variety that contains G / S as an opendense G -subvariety). Then the following are equivalent:(a) [ η ] K n ∈ Im (cid:0) H ( K n , S ) → H ( K n , G ) (cid:1) ,(b) [ η ] F n ∈ Im (cid:0) H ( F n , S ) → H ( F n , G ) (cid:1) ,(c) ( z Y ) η geo ( k ) = ∅ .Proof. (1) Again we twist the action G × Y → Y by z to obtain an action z G × z Y → z Y . Lemma 3.8 enables us to assume without loss of generality that z is the trivialcocycle. We are thus left to deal with a k –homomorphism η geo : ∞ µµµ n → G whichfactors at the finite level through µµµ nm → G for m large enough. This allows us toreason by means of a suitable covering ˜ R n,m as in (7.1).( a ) = ⇒ ( b ) = ⇒ ( c ) are obtained by applying the base change R n ⊂ K n ⊂ F n . ( c ) = ⇒ ( d ): Each γ ∈ ˜Γ n,m induces an automorphism of ˜ R n,m ⊗ R n F n ≃ F n,m ⊗ k ˜ k =˜ F n,m which we also denote by γ (even though the notation γ ⊗ R n,m trivializes η Y , the Galois extension ˜ F n,m of F n (whose Galoisgroup we identify with ˜Γ n,m ) splits η Y F n . By Galois descent η Y ( F n ) = n y ∈ Y ( ˜ F n,m ) | η γ . γ y = y ∀ γ ∈ ˜Γ n,m o . Since z is trivial, this last equality reads η Y ( F n ) = n y ∈ Y ( ˜ F n,m ) | η geo ( γ ) . γ y = y ∀ γ ∈ ˜Γ n,m o . where γ is the image of γ under the map ˜Γ n,m → µµµ nm (˜ k ) given by (7.1). Hence wehave η Y ( F n ) ⊂ Y ( F n,m ) and η Y ( F n ) = n y ∈ Y ( F n,m ) | η geo ( γ ) . γ y = y ∀ γ ∈ µµµ nm ( k ) o . Since Y is proper over k , we have η Y ( F n ) = n y ∈ Y ( F n − ,m [[ t m n ]]) | η geo ( γ ) . γ y = y γ ∈ µµµ nm ( k ) o . Our hypothesis is that this last set is not empty. By specializing at t n = 0, we getthat(7.5) n y ∈ Y ( F n − ,m ) | η geo ( γ ) . γ y = y ∀ γ ∈ µµµ nm ( k ) o = ∅ . We write now µµµ nm ( k ) = µµµ n − m ( k ) × µµµ m ( k ) which provides a decomposition of η geo into two k -homomorphisms η ′ geo : µµµ n − m → G and η ngeo : µµµ m → G . We define η ′ = (1 , η ′ geo ), η n = (1 , η ngeo ) and Y ′ := Y η geon . By [DG] II § Y ′ is a closed subscheme of Y , hence aprojective k –variety. Observe that µµµ n − m acts on Y ′ . Claim 7.2.
The set (7.5) is included in η ′ Y ′ ( F n − ) . To look at the invariants under the action of µµµ nm ( k ), we first look at the invariantsunder the last factor µµµ m ( k ) , and then the first ( n − µµµ n − m ( k ) By restrictingthe condition to elements of the form (1 , γ n ) for γ n ∈ µµµ m ( k ), we see that our set isincluded in n y ∈ Y ( F n − ,m ) | η geon ( γ n ) .y = y ∀ γ n ∈ µµµ m ( k ) o This inclusion is in fact an equality, but this stronger statement is not needed. µµµ m ( k ) acts trivially on F n − ,m . By identity (7.4) applied to the base change of η geon to the field F n − , this is nothing but Y η geon ( F n − ,m ). Looking now at the invariantcondition for the elements of the form ( γ ′ ,
1) for γ ′ ∈ µµµ n − m ( k ), it follows that n y ∈ Y ( F n − ,m ) | η geo ( γ ) . γ y = y ∀ γ ∈ µµµ nm ( k ) o ⊂ n y ∈ Y η geon ( F n − ,m ) | η ′ geo ( γ ′ ) . γ ′ y = y ∀ γ ′ ∈ µµµ n − m ( k ) o = η ′ Y ′ ( F n − ) . By induction on n , we get that inside ( η ′ Y ′ )( F n − ) we have Y ′ η ′ geo ( k ) = ∅ . Thus Y ( k ) η geo = ∅ as desired.( d ) = ⇒ ( a ): Since( η Y )( R n ) = n y ∈ Y ( ˜ R n,m ) | η geo ( γ ) . γ y = y ∀ γ ∈ ˜Γ n,m o , the inclusion Y ( k ) ⊂ Y ( ˜ R n,m ) yields the inclusion( Y η geo )( k ) ⊂ ( η Y )( R n ) . Thus if ( Y η geo )( k ) = ∅ , then ( η Y )( R n ) = ∅ .(2) The quotient G / S is representable by Chevalley’s theorem [DG, § III.3.5]. Theonly non trivial implication is ( c ) = ⇒ ( a ) . Let X = ( G / S ) × k R n . By (1), we have η Y ( K n ) = ∅ . In other words, the K n -homogeneous space η X under η G has a K n -rational point on the compactification η Y . By Florence’s theorem [F], η X ( K n ) = ∅ ,hence ( a ). The k –group G /R u ( G ) is reductive. Let T be a maximal k –torus of G /R u ( G ).This data permits to choose a basis ∆ of the root system Φ( G /R u ( G ) × k k, T × k k )or in other words to choose a Borel subgroup B of the k –group G /R u ( G ) × k k . It iswell known that there is a one-to-one correspondence between the subsets of ∆ andthe parabolic subgroups of G × k k containing B , which is provided by the standardparabolic subgroups ( P I ) I ⊂ ∆ of (cid:0) G /R u ( G ) (cid:1) × k k [Bo, § P ∆ = (cid:0) G /R u ( G ) (cid:1) × k k and P ∅ = B . Furthermore we know that each parabolic subgroupof (cid:0) G /R u ( G ) (cid:1) × k k is (cid:0) G /R u ( G ) (cid:1) ( k )–conjugate to a unique standard parabolicsubgroup. This allows us to define the type of an arbitrary parabolic subgroup of G /R u ( G ). It can happen that two different standard parabolic subgroups of the k –group (cid:0) G /R u ( G ) (cid:1) × k k are conjugate under G ( k ): There are in general fewerconjugacy classes of parabolic subgroups. If P is a parabolic subgroup of the k –group G /R u ( G ), we denote by N G ( P ) its normalizer for the conjugacy action of G on G /R u ( G ). 56 emma 7.3. The quotient G / N G ( P ) is a projective k –variety.Proof. We can assume that G is reductive. Since G / P is projective and is a con-nected component of G / P , G / P is projective as well. The point is that the morphism G / P → G / N G ( P ) is a N G ( P ) / P –torsor. Since the affine k –group N G ( P ) / P is fi-nite, ´etale descent tells us that G / N G ( P ) is proper [EGA IV, prop. 2.7.1]. But G / N G ( P ) is quasiprojective, hence projective.Given a loop cocycle η : π ( R n ) → G ( k ) with coordinates ( z, η geo ) as before, wefocus on the special case of flag varieties of parabolic subgroups of G /R u ( G ). Corollary 7.4.
1. Let I ⊂ ∆ . The following are equivalent:(a) The R n –group η (cid:0) G /R u ( G ) (cid:1) admits a parabolic subgroup scheme of type I ;(b) The R n –group η (cid:0) G /R u ( G ) (cid:1) R n admits a parabolic subgroup of type I ;(c) The F n –group η (cid:0) G /R u ( G ) (cid:1) F n admits a parabolic subgroup of type I ;(d) There exists a parabolic subgroup P of the k –group z ( G /R u ( G )) which isof type I and which is normalized by η geo , i.e., η geo factorizes through N z G ( P ) .2. The following are equivalent:(a) η (cid:0) G /R u ( G ) (cid:1) R n is irreducible (i.e has no proper parabolic subgroups);(b) η (cid:0) G /R u ( G ) (cid:1) K n is irreducible;(c) η (cid:0) G /R u ( G ) of type (cid:1) F n is irreducible;(d) The k –group homomorphism η geo : ∞ µµµ n → z G → Aut ( z G ) is irreducible(see § η (cid:0) G /R u ( G ) (cid:1) R n is anisotropic;(b) η (cid:0) G /R u ( G ) (cid:1) K n is anisotropic;(c) η (cid:0) G /R u ( G ) (cid:1) F n is anisotropic;(d) The k –group homomorphism η geo : ∞ µµµ n → z G → Aut ( z G ) is anisotropic(see § roof. Without loss of generality, we can factor out by R u ( G ) and assume that G is reductive. As in the proof of Theorem 7.1, we can assume by twisting that z istrivial and reason “at the finite level”: Claim 7.5.
There exists a positive integer m such that [ η ] ∈ H ( R n , G ) is trivializedby the base change R n,m /R n . Indeed by continuity η geo : ∞ µµµ n → G factors through a morphism f : µµµ nm → G and [ η ] = f ∗ [ E n,m ] where E n,m = Spec( R n,m ) / Spec( R n ) stands for the standard µµµ nm -torsor. In particular, the class [ η ] ∈ H ( R n , G ) is trivialized by the covering R n,m /R n as above.(1) ( a ) = ⇒ ( b ) = ⇒ ( c ): obvious.( c ) = ⇒ ( d ): We assume that η G F n admits a F n -parabolic subgroup Q of type I .Hence η G F n × F n F n,m = G F n,m admits a F n,m –parabolic subgroup of type I . Since F n,m is an iterated Laurent serie field over k , it implies that G admits a parabolic sub-group P of type I (see the proof of [CGP, lemma 5.24]). We consider the R n –scheme X := η (cid:0) G / N G ( P ) (cid:1) × k R n which by descent considerations [EGA IV, 2.7.1.vii] isproper since G / N G ( P ) is. Claim 7.6. X ( F n ) = ∅ . The F n –group η G /F n admits a subgroup Q such that Q × k F n is G ( F n )-conjugate to P × k F n ⊂ G × k F n . Let g ∈ G ( F n ) be such that Q × F n F n = g ( P × k F n ) g − . As in [Se1, III.2, lemme 1], we check that the cocycle γ g − η γ γ g is cohomologous to η and has value in N G ( P )( F n ). In other words, the F n –torsorcorresponding to η admits a reduction to the subgroup N G ( P ) , i.e.[ η ] ∈ Im (cid:16) H (cid:0) F n , N G ( P ) (cid:1) → H ( F n , G ) (cid:17) . This implies that X ( F n ) = ∅ ( ibid , I.5, prop. 37) and the Claim is proven.By Theorem 7.1.1, we have ( X η geo )( k ) = ∅ , so that there exists an element x ∈ ( X η geo )( k ). Since H (cid:0) k, N G ( P ) (cid:1) injects in H ( k, G ) (see [Gi4, cor. 2.7.2]), we have X ( k ) = G ( k ) / N G ( P )( k ), i.e. X ( k ) is homogeneous under G ( k ).Hence there exists g ∈ G ( k ) such that x = g.x where x stands for the image of1 in X ( k ). We have η geo ( µµµ nm ( k )) ⊂ Stab G ( k ) ( x ) g Stab G ( k ) ( x ) g − = g N G ( P )( k ) g − = N G ( g P g − )( k ) . Thus η geo normalizes a parabolic subgroup of type I of the k –group of G . ( d ) = ⇒ ( a ): We may assume that η has values in N G ( P )( k ). In that case, the twisted R n –group scheme η G admits the parabolic subgroup η P /R n .582) Follows of (1).(3) Recall that a k –group H with reductive connected component of the identity H is anisotropic if and only if it is irreducible and its connected center Z ( H ) is ananisotropic torus. Statement (3) reduces then to the case where G is a k –torus T .We are then given a continuous action of π ( R n ) on the cocharacter group b T ( k ). Itis convenient to work with the opposite assertions to ( a ), ( b ) ( c )and ( d ) , which wedenote by ( a ′ ), ( b ′ ) ( c ′ ) and ( c ′ ) respectively.( a ′ ) = ⇒ ( b ′ ): If the R n –torus η T := η T R n is isotropic, so is the K n –torus η T × R n K n .( b ′ ) = ⇒ ( c ′ ): If the K n –torus η T × R n K n is isotropic, so is the F n –torus η T × R n F n .( c ′ ) = ⇒ ( d ′ ): By Lemma 3.8 we haveHom F n − gr ( G m , η T F n ) = Hom F n − gr ( G m , T F n ) η geo . If η T is isotropic, then this group is not zero and the k –group morphism η geo : ∞ µµµ n → z G fixes a cocharacter of T = ( G ) , hence ( c ′ ).( d ′ ) = ⇒ ( a ′ ): We assume that the morphism η geo : ∞ µµµ n → G fixes a cocharacter λ : G m → T . SinceHom K n − gr ( G m , η T K n ) ≃ Hom K n − gr ( G m , T K n ) η geo . it follows that λ provides a non-zero cocharacter of η G K n , hence ( a ′ ).As in the case of loop torsors [GP2, cor. 3.3], the Borel-Tits theorem has thefollowing consequence. Corollary 7.7.
The minimal elements (with respect to inclusion) of the set of parabolicsubgroups of the k –group z G which are normalized by η geo are all conjugate under z G ( k ) . The type I ( η ) of this conjugacy class is the Witt-Tits index of the F n –group η (cid:0) G /R u ( G ) (cid:1) × R n F n . For anisotropic loop classes, we have the following beautiful picture.
Theorem 7.8.
Assume that G is reductive. Let η, η ′ : π ( R n ) → G ( k ) be two loopcocycles. Assume that ( η G ) F n is anisotropic. Then the following are equivalent:1. [ η ] = [ η ′ ] ∈ H ( R n , G ) , . [ η ] K n = [ η ′ ] K n ∈ H ( K n , G ) ,3. [ η ] F n = [ η ′ ] F n ∈ H ( F n , G ) . We consider first the case of purely geometric loop cocycles. Note that this is theset of all loop cocycles if k is algebraically closed. Theorem 7.9.
Let η, η ′ : π ( R n ) → G ( k ) be two loop cocycles of the form η = (1 , η geo ) and η ′ = (1 , η ′ geo ) . Assume that η is anisotropic. Then the following are equivalent:1. η geo and η ′ geo are conjugate under G ( k ) ,2. [ η ] = [ η ′ ] ∈ H ( R n , G ) ,3. [ η ] K n = [ η ′ ] K n ∈ H ( K n , G ) ,4. [ η ] F n = [ η ′ ] F n ∈ H ( F n , G ) .Proof. Recall that η geo , η ′ geo : ∞ µµµ → G are affine k –group homomorphisms that factorthrough the algebraic group µµµ nm for m large enough. The meaning of (1) is that thereexists g ∈ G ( k ) such that η ′ geo = Int( g ) ◦ η geo . The implications 1) = ⇒
2) = ⇒
3) = ⇒
4) are obvious. We shall prove the impli-cation 4) = ⇒ η ] F n = [ η ′ ] F n ∈ H ( F n , G ).Let T be a maximal torus of G and let N = N G ( T ) and W = N / T . Since themaximal tori of G × k k are all conjugate under G ( k ), the map N G ( T ) → G / G is surjective. Let ˜ k be a finite Galois extension which contains µµµ m ( k ), splits T andsuch that the natural map N (˜ k ) → ( G / G )( k ) is surjective. We furthermore assumewithout loss of generality that our choice of m and ˜ k trivialize η and η ′ . Set ˜Γ n,m = µµµ nm ( k ) ⋊ Gal(˜ k/k ). In terms of cocycles, our hypothesis means thatthere exists h n ∈ G ( ˜ F n,m ) such that(7.6) h − n η ( γ ) γ h n = η ′ ( γ ) ∀ γ ∈ ˜Γ n . Our goal is to show that we can actually find such an element inside G ( k ). Wereason by means of a building argument, and appeal to Remark 6.8 to view ˜ F n,m asa complete local field with residue field ˜ F n − ,m . Note that F n = ( ˜ F n,m ) ˜Γ n,m , and that F n can be viewed as complete local field with residue field F n − .Let C = G / DG be the coradical of G . This is a k –torus which is split by ˜ k .Recall that the (enlarged) Bruhat-Tits building B n of the ˜ F n,m –group G × k ˜ F n,m [T2, § B n = B × V V = b C ⊗ Z R , and B is the building of the semisimple ˜ F n,m –group DG × k ˜ F n,m . The building B n is equipped with a natural action of G ( ˜ F n,m ) ⋊ ˜Γ n,m . By[BT1, 9.1.19.c] the group DG ( ˜ F n − ,m [[ t m n ]]) fixes a unique (hyperspecial) point φ d ∈B ( DG × k ˜ F n,m ) and Stab DG ( ˜ F n,m ) ( φ d ) = DG (cid:0) ˜ F n − ,m [[ t m n ]] (cid:1) .Since the bounded group G ( ˜ F n − ,m [[ t m n ]]) ⋊ ˜Γ m,n fixes at least one point of thebuilding B ( DG × k ˜ F n,m ); such a point is necessarily φ d which is then fixed under G ( ˜ F n − ,m [[ t m n ]]) ⋊ ˜Γ m,n . Claim 7.10.
There exists a point φ = ( φ d , v ) ∈ B n such that1. ˜Γ m,n fixes φ ;2. Stab G ( ˜ F n,m ) ( φ ) = G (cid:0) ˜ F n − ,m [[ t m n ]] (cid:1) . We use the fact that G ( ˜ F n,m ) acts on V by translations under the map G ( ˜ F n,m ) q −→ C ( ˜ F n,m ) = b C ⊗ Z ˜ F × n,m − ord tn −→ b C . It follows that for each v ∈ V , we have( ∗ ) Stab G ( ˜ F n,m ) ( v ) = Stab G ( ˜ F n,m ) ( V ) = q − (cid:0) C ( ˜ F n − ,m [[ t m n ]]) (cid:1) . Since q maps G ( ˜ F n − ,m [[ t m n ]]) into C ( ˜ F n − ,m [[ t m n ]]), it follows that G ( ˜ F n − ,m [[ t m n ]])fixes pointwise φ d × V .Let us choose now the vector v by considering the action of the group N (˜ k ) ⋊ ˜Γ m,n on V . Since this action is trivial on T (˜ k ), it provides an action of the finite group W (˜ k ) ⋊ ˜Γ m,n on V . But this action is affine, so there is at least one v ∈ V which isfixed under N (˜ k ) ⋊ ˜Γ m,n . The point φ = ( φ d , v ) is ˜Γ m,n -invariant, hence (1). We nowuse that N (˜ k ) surjects onto ( G / G )(˜ k ) = ( G / G )( ˜ F n − ,m [[ t m n ]]) = ( G / G )( ˜ F n,m ),hence that G ( ˜ F n − ,m [[ t m n ]]) = G ( ˜ F n − ,m [[ t m n ]]) . N (cid:0) ˜ k (cid:1) , G ( ˜ F n,m ) = G ( ˜ F n,m ) . N (cid:0) ˜ k (cid:1) . Since N (cid:0) ˜ k (cid:1) fixes φ , we haveStab G ( ˜ F n,m ) ( φ ) = Stab G ( ˜ F n,m ) ( φ ) . N (cid:0) ˜ k (cid:1) , and it remains to show that G ( ˜ F n − ,m [[ t m n ]]) = Stab G ( ˜ F n,m ) ( φ ). Since T × k ˜ k is split,the map T × k ˜ k → C × k ˜ k is split and we have the decompositions G ( ˜ F n − ,m [[ t m n ]]) = DG ( ˜ F n − ,m [[ t m n ]]) . T ( ˜ F n − ,m [[ t m n ]])61nd G ( ˜ F n,m ) = DG ( ˜ F n,m ) . T ( ˜ F n,m ) . The first equality shows that G ( ˜ F n − ,m [[ t m n ]]) fixes φ hence that G ( ˜ F n − ,m [[ t m n ]]) ⊂ Stab G ( ˜ F n,m ) ( φ ).As for the reversed inclusion consider an element g ∈ Stab G ( ˜ F n,m ) ( φ ). Then q ( g ) ∈ C ( ˜ F n − ,m [[ t m n ]]). The map G ( ˜ F n − ,m [[ t m n ]]) q −→ C ( ˜ F n − ,m [[ t m n ]]) is surjective, hencewe can assume that g ∈ DG ( ˜ F n,m ). Since g.φ d = φ d , g belongs to DG (cid:0) ˜ F n − ,m [[ t m n ]] (cid:1) as desired. This finishes the proof of our claim.We consider the twisted action of ˜Γ n,m on B n defined by γ ∗ x = η ( γ ) . γ x. The extension of local fields (with respect to t n ) ˜ F n,m /F n is tamely ramified. TheBruhat-Tits-Rousseau theorem states that the Bruhat-Tits building of ( η G ) F n can beidentified with B ˜Γ n,m n , i.e. the fixed points of the building B n under the twisted action([Ro] and [Pr]). But by Corollary 7.4.3, the F n − (( t n ))–group η G × R n F n − (( t n )) isanisotropic, so its building consists of a single point, which is in fact φ. Indeed sinceour loop cocycle has value in G (˜ k ) ⋊ Γ n,m , φ is fixed under the twisted action of ˜Γ n,m .This shows that B Γ n,m n = { φ } . We claim that h n .φ = φ . We have γ ∗ ( h n . φ ) = η ( γ ) γ ( h n ) . γ φ = η ( γ ) γ ( h n ) . φ [ φ is invariant under ˜Γ m,n ]= h n . η ′ ( γ ) φ [relation 7.6]= h n . φ [ η ′ ( γ ) ∈ G (˜ k )] and claim 7.10]for every γ ∈ ˜Γ n,m . Hence h n . φ ∈ B Γ n,m n and therefore h n . φ = φ as desired.It then follows that h n ∈ G ( ˜ F n − ,m [[ t m n ]]). By specializing (7.6) at t n = 0, weobtain an element h n − ∈ G ( ˜ F n − ,m ) such that(7.7) h − n − η ( γ ) γ h n − = η ′ ( γ ) ∀ γ ∈ ˜Γ n,m . Since η and η ′ have trivial arithmetic part, it follows that h n − is invariant underGal(˜ k/k ). We apply now the relation (7.7) to the generator τ n of Gal (cid:0) ˜ F n,m / ˜ F n − ,m (( t n )) (cid:1) . This yields(7.8) h − n − η ( τ n ) h n − = η ′ ( τ n ) , η ( τ n ) , η ′ ( τ n ) ∈ G (˜ k ) and h n − ∈ G ( F n − ,m ) = G ( ˜ F n − ,m ) Gal(˜ k/k ) . If we denoteby µµµ ( n ) m the last factor of µµµ nm then (7.8) implies that η geo | µµµ ( n ) m and η ′ geo | µµµ ( n ) m are conjugateunder G ( F n − ,m ). Claim 7.11. η geo | µµµ ( n ) m and η ′ geo | µµµ ( n ) m are conjugate under G ( k ) . The transporter X := { h ∈ G | Int( h ) ◦ η geo | µµµ ( n ) n = η ′ geo | µµµ ( n ) n } is a non-empty k –variety which is a homogeneous space under the group Z G ( η geo | µµµ ( n ) n ). Since X ( F n − ,m ) = ∅ and F n − ,m is an iterated Laurent series field over k , Florence’s theorem [F, §
1] showsthat X ( k ) = ∅ . The claim is thus proven.Without loss of generality we may therefore assume that η geo | µµµ ( n ) m = η ′ geo | µµµ ( n ) m . Thefinite multiplicative k –group µµµ ( n ) m acts on G via η geo , and we let G n − denote the k –group which is the centralizer of this action [DG, II 1.3.7]. The connected componentof the identity of G n − is reductive ([Ri], proposition 10.1.5). Since the action of µµµ ( n − m on G given by η geo commutes with that of µµµ ( n ) m , the k –group morphism η geo : µµµ nm → G factors through G n − . Similarly for η ′ geo . Denote by η geon − (resp. η ′ geon − ) therestriction of η geo (resp. η ′ geo ) to the k –subgroup µµµ n − m = µµµ (1) n × · · · × µµµ ( n − n of µµµ nm . Set˜Γ n − ,m := µµµ n − m ( k ) ⋊ Gal(˜ k/k ) and consider the loop cocycle η n − : ˜Γ n − ,m → G n − (˜ k )attached to (1 , η geon − ) , and similarly for η ′ geon − .The crucial point for the induction argument we want to use is the fact that η geon − : µµµ n − m → G n − is anisotropic. For otherwise the k –group G n − admits a non-trivial split subtorus S which is normalized by µµµ n − m . But then S is a non-trivialsplit subtorus of G which is normalized by µµµ nm , and this contradicts the anisotropicassumption on η geo .Inside G n − ( ˜ F n − ,m ), relation (7.7) yields that h − n − η n − ( γ ) γ h n − = η ′ ( γ ) ∀ γ ∈ ˜Γ n − . which is similar to (7.6). By induction on n , we may assume that η geon − is G n − ( k )–conjugate to η ′ geon − . Thus η geo is G ( k )–conjugate to η ′ geo as desired.Before establishing Theorem 7.8, we need the following preliminary step. Lemma 7.12.
Let H be a linear algebraic k –group. It two loop classes [ η ] , [ η ′ ] of H (cid:0) π ( R n ) , H ( k ) (cid:1) have same image in H ( F n , H ) , then [ η ar ] = [ η ′ ar ] in H ( k, H ) .Proof. Up to twisting H by η ar , the standard torsion argument allows us to assumewith no loss of generality that η ar is trivial, i.e. that η is purely geometrical. Weare thus left to deal with the case of a k –group homomorphism η geo : ∞ µµµ → H thatfactors through some µµµ nm → H for m > η ] is trivialized by63he extension ˜ R n,m /R n and its image in H ( F n , H ) by the extension ˜ F n,m /F n , where˜ R n,m and ˜ F n,m are as above.By further increasing m , the same reasoning allows us to assume that the imageof η ′ in H (cid:0) R n,m , H ( k ) (cid:1) is purely arithmetic. More precisely, that the map Z (cid:0) π ( R n ) , H ( k ) (cid:1) → Z (cid:0) π ( R n,m ) , H ( k ) (cid:1) maps ( η ′ geo , η ′ ar ) to (1 , η ′ ar ) where the coordinates are as in Section 3.3. But ourhypothesis implies that the image of [ η ′ ] in H ( F n,m , H ) is trivial, hence[ η ′ ar ] ∈ ker (cid:0) H ( k, H ) → H ( F n , H ) (cid:1) . Since F n is an iterated Laurent series field over k , this kernel is trivial (see [F, § η ′ ar ] = 1 ∈ H ( k, H ).We are now ready to proceed with the proof of Theorem 7.8. Proof.
The implications 1) = ⇒
2) = ⇒
3) are obvious. We shall prove the implication3) = ⇒
1) by using the previous result. By assumption [ η ] F n = [ η ′ ] F n ∈ H ( F n , G ). Itis convenient to work at finite level as we have done previously, namely with cocycles η, η ′ : ˜Γ n,m → G (˜ k )with ˜Γ n,m := µµµ nm (˜ k ) ⋊ Gal(˜ k/k ) where m > k/k is a finite Ga-lois extension extension containing all m –roots of unity in k. We associate to η itsarithmetic-geometric coordinate pair ( z, η geo ) where z ∈ Z (Gal(˜ k/k ) , G (˜ k )) and η geo : µµµ nm → z G is a k –group homomorphism. Similar considerations apply to η ′ , and its corresponding pair ( z ′ , η ′ geo ) . By Lemma 7.12, we have [ z ] = [ z ′ ] ∈ H ( k, G ).Without lost of generality we may assume that z = z ′ . Consider the commutativediagram H (cid:0) ˜Γ n,m , z G (˜ k ) (cid:1) −−−→ H (cid:0) ˜Γ n,m , z G ( ˜ F n,m ) (cid:1) τ z y ≀ τ z y ≀ H (cid:0) ˜Γ n,m , G (˜ k ) (cid:1) −−−→ H (cid:0) ˜Γ n,m , G ( ˜ F n,m ) (cid:1) . where the vertical arrows are the torsion bijections. Thus τ − z [ η ] = τ − z [ η ′ ] ∈ H (cid:0) ˜Γ n,m , z G ( ˜ F n,m ) (cid:1) .By Corollary 7.4.3, η geo : µµµ nm → z G is an anisotropic k –group homomorphism. Wecan thus apply Theorem 7.9 to conclude that η geo and η ′ geo are conjugate under z G ( k )hence τ − z [ η ] = τ − z [ η ′ ] in H (cid:0) ˜Γ n,m , z G (˜ k ) (cid:1) , and therefore [ η ] = [ η ′ ] in H (cid:0) ˜Γ n,m , z G (˜ k ) (cid:1) as desired. Corollary 7.13.
Let the assumptions be as in the Theorem, and let H (cid:0) π ( R n ) , G ( k ) (cid:1) an denote the preimage of H ( F n , G ) an under the composite map H (cid:0) π ( R n ) , G ( k ) (cid:1) → H ( R n , G ) → H ( F n , G ) . Then H (cid:0) π ( R n ) , G ( k ) (cid:1) an injects into H ( F n , G ) . Acyclicity
We have now arrived to one of the main results of our work
Theorem 8.1.
Let G be a linear algebraic group over a field k of characteristic .Then the natural restriction map H ( R n , G ) → H ( F n , G ) induces a bijection H loop ( R n , G ) ∼ −→ H ( F n , G ) . In particular, the inclusion map H loop ( R n , G ) → H ( R n , G ) admits a canonical sec-tion. For any k –scheme X , we denote by H ( X , G ) irr ⊂ H ( X , G ) the subset consistingof classes of G –torsor E over X for which the twisted reductive X –group scheme E (cid:0) G /R u ( G ) (cid:1) X does not contain a proper parabolic subgroup which admits a Levisubgroup. Set H loop ( X , G ) irr = H loop ( X , G ) ∩ H ( X , G ) irr . We begin with thefollowing special case. Lemma 8.2. H loop ( R n , G ) irr injects into H ( F n , G ) .Proof. By Lemma 4.14, we can assume without loss of generality that G is reductive.We have an exact sequence 1 → G i −→ G p −→ ννν → ννν is a finite ´etale k –group. We are given two loop cocycles η , η ′ in Z ( R n , G ) which have the same imagein H ( F n , G ), and for which the twisted F n –groups η G , η ′ G are irreducible. Since H (cid:0) π ( R n ) , ννν ( k ) (cid:1) ∼ −→ H ( F n , ννν ), it follows that p ∗ [ η ] = p ∗ [ η ′ ] in H (cid:0) π ( R n ) , ννν ( k ) (cid:1) .We can thus assume without loss of generality that p ∗ η = p ∗ η ′ in Z (cid:0) π ( R n ) , ννν ( k ) (cid:1) .Furthermore, as in the proof of Theorem 7.8 the standard twisting argument reducesthe problem to the case of purely geometric loop torsors. In particular, the groupactions of η geo and η ′ geo are irreducible according to Corollary 7.4.3.Let C be the connected center of G . Then C is a k –torus equipped with an actionof ννν . We consider its k –subtorus C ♯ := ( C p ◦ η geo ) and denote by C its maximal k –split subtorus which is defined by c C = c C ♯ ( k ). By construction, η geo : ∞ µµµ n → G centralizes C ♯ and C . Similarly for η ′ geo . The k –torus C is a split subtorus of C centralized by η geo and maximal for this property. We consider the exact sequence of k –groups 1 → C → G q −→ G / C → We remind the reader that H ( X , G ) stands for H ( X , G X ) . Recall that the assumption on the existence of the Levi subgroup is superfluous whenever X isaffine. laim 8.3. The composite q ◦ η geo : ∞ µµµ n → G / C is anisotropic. Let us establish the claim. We are given a split subtorus S of the k –group G which is centralized by q ◦ η geo . Since q ◦ η geo is irreducible, S is central in G / C .We consider M = q − ( S ), this is an extension of S by C , so it is a split k –torus. Bythe semisimplicity of the category of representations of ∞ µµµ n , we see that ∞ µµµ n actstrivially on M . Then M = C and S = 1 , and the claim thus holds.Next we twist the sequence of R n –groups1 → C → G q −→ G / C → η to obtain 1 → C → η G → η ( G / C ) → . This leads to the commutative exact diagram of pointed sets0 = H ( R n , C ) −−−→ H ( R n , G ) q ∗ −−−→ H ( R n , G / C ) τ η x ≃ τ η x ≃ H ( R n , C ) −−−→ H ( R n , η G ) −−−→ H (cid:0) R n , η ( G / C ) (cid:1) where the vertical maps are the torsion bijections. Note that H ( R n , C ) vanishessince Pic( R n ) = 0. By diagram chasing we have [ η ] = [ η ′ ] in H ( R n , G ) if and onlyif q ∗ [ η ] = q ∗ [ η ′ ] in H ( R n , G / C ). Since q ∗ [ η ] F n = q ∗ [ η ′ ] F n in H ( F n , G / C ) it willsuffice to establish the Lemma for G / C . The claim states that q ∗ η geo is anisotropic,therefore q ∗ [ η ] = q ∗ [ η ′ ] in H ( R n , G / C ) by Theorem 7.9.We can now proceed to prove Theorem 8.1. Proof.
The surjectivity of the map H loop ( R n , G ) → H ( F n , G ) is a special case ofProposition 6.9. Let us establish injectivity. We are given two loop cocycles η, η ′ ∈ Z (cid:0) π ( R n ) , G ( k ) (cid:1) having the same image in H ( F n , G ). Lemma 7.12 shows that[ η ar ] = [ η ′ ar ] in H ( k, G ). Up to twisting G by η ar , we may assume that η and η ′ are purely geometrical loop torsors. The proof now proceeds by reduction to theirreducible case, i.e. to the case when η G × R n F n is irreducible.Let Q be a minimal F n –parabolic subgroup of η G × R n F n . Corollary 7.4 showsthat the k –group G admits a parabolic subgroup P of the same type as Q which isnormalized by η. The same statement shows that η ′ normalizes a parabolic subgroup,say P ′ , of the same type than P . Since by Borel-Tits theory P ′ is G ( k )–conjugateto P we may assume that η ′ normalizes P as well. Furthermore, P is minimal for66 (and η ′ ) with respect to this property. We can view then η , η ′ as elements of Z loop (cid:0) R n , N G ( P ) (cid:1) irr . We look at the following commutative diagram H loop (cid:0) R n , N G ( P ) (cid:1) irr −−−→ H ( R n , G ) y y H (cid:0) F n , N G ( P ) (cid:1) irr ∼ −−−→ H ( F n , G )Since the bottom map is injective (see [Gi4, th. 2.15]), it will suffice to show that H loop (cid:0) R n , N G ( P ) (cid:1) irr injects in H (cid:0) F n , N G ( P I ) (cid:1) . Since the unipotent radical U of P is a split unipotent group, we have H (cid:0) R n , N G ( P ) (cid:1) ≃ H (cid:0) R n , N G ( P ) / U (cid:1) , and similarly for F n by Lemma 4.14. So we are reduced to showing that H loop (cid:0) R n , N G ( P ) / U (cid:1) irr injects in H (cid:0) F n , N G ( P ) / U (cid:1) , which is covered by Lemma 8.2. This completes theproof of injectivity. By using the Witt-Tits decomposition over F n [Gi4, th. 2.15], we get the following. Corollary 8.4.
Assume that G is a split reductive k –group. Let P I , ... , P I l berepresentatives of the G ( k ) -conjugacy classes of parabolic subgroups of G . Let L I j bea Levi subgroup of P I j for j = 1 , ...l . Then G j =1 ,...,l H loop (cid:0) R n , N G ( P I j , L I j ) (cid:1) irr ≃ H loop ( R n , G ) ≃ H ( F n , G ) . Remark 8.5.
It follows from the Corollary that we have a “Witt-Tits decomposi-tion” for loop torsors. Furthermore, if we are interested in purely geometrical irre-ducible loop torsors, then we have a nice description in terms of k –group homomor-phisms ∞ µµµ → G as described in Theorem 7.9. This corresponds to the embedding ofHom k − gp ( ∞ µµµ nirr , G ) / G ( k ) in H ( R n , G ).For future use we record the connected case. Corollary 8.6.
Assume that G is a split reductive group.1. Let P I , ... , P I l be the standard k –parabolic subgroups containing a given Borelsubgroup of G /k . Then G j =1 ,...,l H loop ( R n , P I j ) irr ≃ H loop ( R n , G ) ≃ H ( F n , G ) . . If k is algebraically closed Hom k − gp,irr ( ∞ µµµ n , G ) / G ( k ) ≃ H loop ( R n , G ) irr ≃ H ( F n , G ) irr . Using our choice of roots of unity (2.3), we have ∞ µµµ ≃ b Z . So the left handsideis Hom gp (cid:0)b Z n , G ( k ) (cid:1) irr / G ( k ), namely the G ( k )–conjugacy classes of finite order ir-reducible pairwise commuting elements ( g , ..., g n ) (irreducible in the sense that theelements do not belong to a proper parabolic subgroup). k –loop adjoint groups Next we discuss in detail the important case where our algebraic group is the group
Aut ( G ) of automorphisms of a split semisimple group G of adjoint type. This is thesituation needed to classify forms of the R n –group G × k R n and of the corresponding R n –Lie algebra g ⊗ k R n where g is the Lie algebra of G . Indeed it is this particularcase, and its applications to infinite- dimensional Lie theory as described in [P2] and[GP2] for example, that have motivated our present work.We fix a “Killing couple” T ⊂ B of G , as well as a base ∆ of the correspondingroot system. For each subset I ⊂ ∆ we define as usual T I = (cid:0) \ α ∈ I ker( α ) (cid:1) . Since G is adjoint, we know that the roots define an isomorphism T ≃ ( G m ) | ∆ | , hence T I ≃ ( G m ) | ∆ \ I | . The centralizer L I := C G ( T I ) , is the standard Levi subgroup of theparabolic subgroup P I = U I ⋊ L I attached to I . Since G is of adjoint type, we knowthat L I / T I is a semisimple k –group of adjoint type.We have a split exact sequence of k –groups1 → G → Aut ( G ) → Out ( G ) → Out ( G ) is the finite constant k –group corresponding to the finite (abstract)group Out( G ) of symmetries of the Dynkin diagram of G . [XXIV § For I ⊂ ∆,we need to describe the normalizer N Aut( G ) ( L I ) of L I . Following [Sp, 16.3.9.(4)], wedefine the subgroup of I -automorphisms of G by Aut I ( G ) = Aut ( G , P I , L I )where the latter group is the subgroup of Aut ( G ) that stabilizes both P I and L I . There is then an exact sequence1 → L I → Aut I ( G ) → Out I ( G ) → , In [SGA3] the group
Out ( G ) is denoted by Aut (Dyn) . Out I ( G ) is the finite constant group corresponding to the subgroup of Out( G )consisting of elements that stabilize I ⊂ ∆ . Then the preceding Corollary reads G [ I ] ⊂ ∆ / Out( G ) H loop (cid:0) R n , Aut I ( G ) (cid:1) irr ≃ H loop (cid:0) R n , Aut ( G ) (cid:1) ≃ H (cid:0) F n , Aut ( G ) (cid:1) . By [Gi4, cor. 3.5], H loop (cid:0) R n , Aut I ( G ) (cid:1) irr ≃ H (cid:0) F n , Aut I ( G ) (cid:1) irr can be seen as asubset of H loop (cid:0) R n , Aut I ( G ) / T I (cid:1) an ≃ H (cid:0) F n , Aut I ( G ) / T I (cid:1) an . We come now toanother of the central results of the paper. Theorem 8.7.
Assume that k is algebraically closed and of characteristic . Let G be a simple k –group of adjoint type. Let T ⊂ B , I , and ∆ be as above. On the set Hom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) (cid:1) define the equivalence relation φ ∼ I φ ′ if there exists g ∈ Aut I ( G )( k ) such that φ and gφ ′ g − have same image in Hom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) / T I (cid:1) .Then we have a decomposition G [ I ] ⊂ ∆ / Out( G ) Hom k − gp ( ∞ µµµ n , Aut I ( G )) an / ∼ I ∼ −→ H loop ( R n , G ) ∼ −→ H ( F n , G ) . where Hom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) / T I (cid:1) an stands for the set of anisotropic group homo-morphisms ∞ µµµ n → Aut I ( G ) / T I . Remark 8.8.
As an application of Margaux’s rigidity theorem [Mg2], the right hand-side does not change by extension of algebraically closed fields. Hence H loop ( R n , G )does not change by extension of algebraically closed fields. This allows us in practicewhenever useful to work over Q or C . Proof.
The group (
Aut I ( G ) / T I )( k ) acts naturally on the set Hom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) / T I (cid:1) an by conjugation, and we denote the resulting quotient set by Hom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) / T I (cid:1) an . The commutative squareHom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) (cid:1) irr −−−→ Hom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) / T I (cid:1) an y y H loop (cid:0) R n , Aut I ( G ) (cid:1) irr −−−→ H loop (cid:0) R n , Aut I ( G ) / T I (cid:1) an is well defined as one can see by taking into account Corollary 7.4. Since k is al-gebraically closed, loop torsors are purely geometric, hence the two vertical mapsare onto. As we have seen, the bottom horizontal map is injective; this defines anequivalence relation ∼ ′ I on the set Hom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) (cid:1) such thatHom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) (cid:1) irr / ∼ ′ I ∼ −→ H loop ( R n , Aut I ( G )) .
69t remains to establish that the equivalence relations ∼ ′ I and ∼ I coincide, and we dothis by using that the right vertical map in the above diagram is injective. (Theorem7.9). We are given φ , φ ∈ Hom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) (cid:1) irr . Then φ ∼ ′ I φ if and onlyif the image of φ and φ in Hom k − gp (cid:0) ∞ µµµ n , Aut I ( G ) / T I (cid:1) an are conjugate by anelement of (cid:0) Aut I ( G ) / L I (cid:1) ( k ). Since the map Aut I ( G )( k ) → (cid:0) Aut I ( G ) / L I (cid:1) ( k ) isonto, it follows that φ ∼ ′ I φ if and only if φ ∼ I φ as desired. Corollary 8.9.
Under the hypothesis of Theorem 8.7, the classification of loop torsorson R n “is the same” as the classification, for each subset I ⊂ ∆ , of irreduciblecommuting n –uples of elements of finite order of Aut I ( G )( k ) up to the equivalencerelation ∼ I . n ( Z ) The assumptions are as in the previous section. We fix a pinning (´epinglage)( G , B , T ) [XXIV § s : Out ( G ) → Aut ( G ) . The group GL n ( Z ) acts on the left as automorphisms of the k –algebra R n via(8.1) g = ( a ij ) ∈ GL n ( Z ) : t i t a i t a i . . . t a ni n We denote the resulting k –automorphism of R n corresponding to g also by g since noconfusion will arise. By Yoneda considerations g (anti)corresponds to an automor-phism g ∗ of the k –scheme Spec( R n ) . Applying (8.1) where we now replace t i by t /m and k by k gives a left actionof GL n ( Z ) as automorphisms of R n,m = k [ t ± /m , . . . , t ± /mn ] . If we denote by g m theautomorphism corresponding to g then the diagram R n g −−−→ R n y y R n,m g m −−−→ R n,m commutes. Passing to the direct limit on (8.4) the element g induces an automorphism g ∞ of R n, ∞ = lim −→ k [ t ± /m , . . . , t ± /mn ] . If no confusion is possible, we will denote g ∞ and g m simply by g. Recall that π ( R n ) = b Z (1) n ⋊ Gal( k ) . Our fixed choice of compatible roots of unity( ξ m ) allows us to identify π ( R n ) with b Z n ⋊ Gal( k ) where the left action of Gal( k ) oneach component b Z = lim ←− Z /m Z is as follows: If a ∈ Gal( k ) and m ≥ ≤ a ( m ) ≤ m − a ( ξ m ) = ξ a ( m ) m . This defines an automorphism a m of the additive group Z /m Z . Passing to the limit on each component yields thedesired group automorphism b a of b Z n . (cid:0) Z /m Z (cid:1) n as row vectors. Then GL n ( Z ) acts on the right on this group byright multiplication g : e m e gm = e m g where (cid:0) Z /m Z (cid:1) n is viewed as a Z –module in the natural way. By passing to theinverse limit we get a right action of GL n ( Z ) as automorphisms of b Z n that we denoteby e e g . We extend this to a right action on π ( R n ) = b Z n ⋊ Gal( k ) by letting GL n ( Z ) act trivially on Gal( k ) . Thus if γ = (e , a ) ∈ b Z n ⋊ Gal( k ) and g ∈ GL n ( Z ) , then γ g = (e g , a ) . By taking the foregoing discussion into consideration we can define the (right)semidirect product group GL n ( Z ) ⋉ π ( R n ) with multiplication(8.2) (cid:0) h, (e , a ) (cid:1)(cid:0) g, (f , b ) (cid:1) = (cid:0) hg, (e g , a )(f , b ) (cid:1) = (cid:0) hg, (e g b a (f) , ab ) (cid:1) for all h, g ∈ GL n ( Z ), e , f ∈ b Z n and a, b ∈ π ( R n ) . For future use we point out thatunder that under the natural identification of GL n ( Z ) and π ( R n ) with subgroups of GL n ( Z ) ⋉ π ( R n ) we have(8.3) γg = gγ g for all g ∈ GL n ( Z ) and γ ∈ π ( R n ) . By definition π ( R n ) acts naturally on R n, ∞ . Under our identification π ( R n ) = b Z n ⋊ Gal( k ) the action is given by(8.4) (e , a ) : λt /mi a ( λ ) ξ e m,i m t /mi where e = (e , . . . , e n ) ∈ b Z n , e i = (e m,i ) m ≥ with 0 ≤ e m,i < m, and λ ∈ k. Using (8.2)and (8.4) it is tedious but straightforward to check that the group GL n ( Z ) ⋉ π ( R n )defined above acts on the left as automorphisms of the k –algebra R n, ∞ in a way whichis compatible with the left actions of each of the groups, i.e.(8.5) ( gγ ) .x = g. ( γ.x )for all g ∈ GL n ( Z ) , γ ∈ π ( R n ) and x ∈ R n, ∞ . In the reminder of this section we let H denote a linear algebraic group over k .Each element g ∈ GL n ( Z ) viewed as an automorphism g ∗ of the k –scheme Spec( R n )induces by functoriality a bijection, also denoted by g ∗ , of the pointed set H ( R n , H )onto itself. This leads to a left action of GL n ( Z ) on this pointed set which we After our identifications, this is nothing but the natural action of g ∗ on π (cid:0) Spec( R n ) (cid:1) . base change. Our objective is to have a precise description of this action. The isotriviality theorem [GP3, th. 2.9] shows tha it will suffice to trace the basechange action at the level of 1-cocycles in Z (cid:0) π ( R n ) , H ( R n, ∞ ) (cid:1) . Following standardconventions for cocycles we denote the action of an element γ ∈ π ( R n ) on an element h ∈ H ( R n, ∞ ) by γ h. Then (8.5) implies that(8.6) γ.h = γ h. Lemma 8.10.
The base change action of GL n ( Z ) on H ( R n , H ) is induced by theaction η g η of GL n ( Z ) on Z (cid:0) π ( R n ) , H ( R n, ∞ ) (cid:1) given by ( g η )( γ ) = g.η ( γ g ) for all γ ∈ π ( R n ) and g ∈ GL n ( Z ) . Proof.
For all α, β ∈ π ( R n ) we have g η ( αβ ) = g.η (cid:0) ( αβ ) g ) (cid:1) [definition]= g.η ( α g β g )= g. (cid:0) η ( α g ) α g η ( β g ) (cid:1) [ η a cocycle]= g. (cid:16) η ( α g ) (cid:0) α g .η ( β g ) (cid:1)(cid:17) [by (8.6)]= (cid:0) g. ( η ( α g ) (cid:1)(cid:0) g.α g .η ( β g ) (cid:1) [by action axiom]= (cid:0) g. ( η ( α g ) (cid:1)(cid:0) α.g.η ( β g ) (cid:1) [by action axiom and (8.3)]= g η ( α ) (cid:0) α. g η ( β ) (cid:1) [by definition ]= g η ( α ) α (cid:0) g η ( β ) (cid:1) [by (8.6) ].This shows that g η is a cocycle (which is clearly continuous since η is). That thisdefines a left action of GL n ( Z ) on Z (cid:0) π ( R n ) , H ( R n, ∞ ) (cid:1) is easy to verify using thedefinitions.Next we verify that the action factors through H . Assume µ is a cocycle coho-mologous to η, and let h ∈ H ( R n, ∞ ) be such that µ ( γ ) = h − ηγ γ h for all γ ∈ π ( R n ) . Then Our main interest is the case when H = Aut ( G ) with G simple. The reason behind theimportance of this case lies in the applications to infinite-dimensional Lie theory. µ ( γ ) = g.µ ( g γ ) [definition]= g. (cid:0) h − η ( γ g ) γ h (cid:1) = g.h − g.η ( γ g ) g. γ h [action axiom]= ( g.h ) − g η ( γ ) γ g .γ h [action axiom, definition, and g = γ g ]= ( g.h ) − g η ( γ ) γ ( g.h ) . Thus g µ and g η are cohomologous.It remains to verify that the action we have defined on H (cid:0) π ( R n ) , H ( R n, ∞ ) (cid:1) = H (cid:0) R n , H (cid:1) coincides with the base change action. To see this we consider a faithfulrepresentation H → GL d and the corresponding quotient variety Y = GL d / H . Since H ( R n , GL n ) = 1 by a variation of a theorem of Quillen and Suslin ([Lam] V.4), wehave a short exact sequence of pointed sets1 → H ( R n ) → GL d ( R n ) → Y ( R n ) ϕ → H ( R n , H ) → . Therefore it is enough to verify our assertion for the image of the characteristic map ϕ. Given y ∈ Y ( R n ), by definition ϕ ( y ) is the class of the cocycle γ → η ( γ ) = Y − γ Y = Y − γ.Y where Y ∈ GL d ( R n, ∞ ) is a lift of y [the last equality holds by (8.6)]. If g ∈ GL d ( Z )we have g ∗ ( ϕ ( y )) = ϕ ( g.y )by the equivariance of the characteristic map relative to k –schemes. Since g.Y is alift of g.y , we conclude that ϕ ( g.y ) is the class of the cocycle ( g.Y ) − γ ( g.Y ) . Usingidentities and compatibility of actions that have already been mentioned, we have( g.Y ) − γ ( g.Y ) = ( g.Y ) − γ. ( g.Y ) = g.Y − gγ g .Y == g.Y − g.γ g .Y = g. (cid:0) Y − γ g .Y (cid:1) = g.η ( γ g ) = g η ( γ )as desired. Remark 8.11.
The action of GL n ( Z ) on Z (cid:0) π ( R n ) , H ( R n, ∞ ) (cid:1) stabilizes Z (cid:0) π ( R n ) , H ( k ) (cid:1) .In particular, it preserves loop cocycles.We pause to observe that the decomposition 8.4 is equivariant under the actionof GL n ( Z ) on R n . Thus 73 orollary 8.12. With the assumptions and notation as above G j =1 ,...,l GL n ( Z ) \ H loop (cid:0) R n , Aut ( G , P I j ) (cid:1) irr ≃ GL n ( Z ) \ H loop (cid:0) R n , Aut ( G ) (cid:1) In particular, if E is a loop R n –torsor under Aut ( G ) , the Witt-Tits index of the loopgroup scheme E G /R n depends only of the class of E in GL n ( Z ) \ H loop (cid:0) R n , Aut ( G ) (cid:1) . Remark 8.13.
Assume η is a loop cocycle. Since GL n ( Z ) acts trivially on H ( k ) wehave ( g η )( γ ) = η ( γ g ) for all γ ∈ π ( R n ) and g ∈ GL n ( Z ). Lemma 8.14.
Assume that H acts on a quasi-projective k –variety M . Let η ∈ Z (cid:0) π ( R n ) , H ( k ) (cid:1) be a loop cocycle. Let Λ η ⊂ GL n ( Z ) be the stabilizer of η for the(left) action of GL n ( Z ) on Z (cid:0) π ( R n ) , H ( k ) (cid:1) .(1) Λ η = n g ∈ GL n ( Z ) | η ( γ g ) = η ( γ ) ∀ γ ∈ π ( R n ) o . (2) The map (cid:0) GL n ( Z ) ⋉ π ( R n ) (cid:1) × M ( R n, ∞ ) → M ( R n, ∞ ) , (cid:0) ( g, γ ) , x (cid:1) g.η ( γ ) .γ.x defines an action of Λ η ⋉ π ( R n ) on ( η X )( R n, ∞ ) .(3) Assume that M is a linear algebraic k –group on which H acts as group auto-morphisms. Let g ∈ Λ η and ζ ∈ Z (cid:0) π ( R n ) , η M ( R n, ∞ ) (cid:1) , and set g ζ ( γ ) = g.ζ ( γ g ) . This defines a (left) action of Λ η on Z (cid:0) π ( R n ) , η M ( R n, ∞ ) (cid:1) which induces an actionof Λ η on H ( R n , η M ) . The action is functorial in M . If H = M , the diagram H ( R n , η H ) τ η −−−→ ∼ H ( R n , H ) g ∗ y ≀ g ∗ y ≀ H ( R n , η H ) τ η −−−→ ∼ H ( R n , H ) commutes for all g ∈ GL n ( Z ) , where τ η is the twisting bijection.(4) Assume that in (3) H is finite and that M is of multiplicative type. For g ∈ Λ η and an inhomogeneous (continuous) cochain y ∈ C i (cid:0) π ( R n ) , η M (cid:1) of degree i ≥ , set ( g y )( γ , . . . , γ i ) = g (cid:0) y ( γ g , . . . , γ gi ) (cid:1) . his defines a left action of Λ η on the chain complex C ∗ (cid:0) π ( R n ) , η M ( R n, ∞ ) (cid:1) of (cont-nuous) inhomogeneous cochains and on H ∗ (cid:0) π ( R n ) , η M ( R n, ∞ ) (cid:1) = H ∗ ( R n , η M ) whichis functorial with respect to short exact sequences of H –equivariant k –groups of mul-tiplicative type.(5) Assume that H is finite and let → M → M → M → be a an ex-act sequence of linear algebraic k –groups equipped with an equivariant action of H as group automorphism. The action of Λ η commutes with the characteristic map η M ( R n ) → H ( R n , η M ) . If M is central in M , then the action of Λ η commuteswith the boundary map ∆ : H ( R n , η M ) → H ( R n , η M ) .(6) Assume k is algebraically closed and let d be a postive integer. The base changeaction of GL n ( Z ) on H ( R , µµµ ) and on Br( R ) is given by g.α = det( g ) .α .Proof. (1) is obvious by taking into account Remark 8.13.In what follows we take the “Galois” point of view and notation: η M ( R n, ∞ )coincides with M ( R n, ∞ ) as a set, but the action of π ( R n ) is the twisted action,which we denote by ⋆ : γ ⋆ x = η ( γ ) . ( γ.x )(2) The groups GL n ( Z ) and π ( R n ) act on X ( R n, ∞ ) and H ( R n, ∞ ) via their naturalaction on R n, ∞ . We will denote these actions by x g.x , and x γ.x for all g ∈ GL n ( Z ), γ ∈ π ( R n ) and x ∈ X ( R n, ∞ ) . It follows from (8.5) and (8.6) that forall γ ∈ π ( R n ) we have γ.g.x = g.γ g .x One also verifies using the axioms of action that γ. ( h.x ) = ( γ.h ) . ( γ.x )for all h ∈ H ( R n, ∞ ). The content of (2) is that(8.7) ( g, γ ) ⋆ x = g. ( γ ⋆ x )defines an action of Λ η ⋉ π ( R n ) on ( η X )( R n, ∞ ) . Write for convenience g.γ ⋆ x insteadof g. ( γ ⋆ x ) since no confusion is possible. Then75 f, α ) ⋆ ( g, β ) ⋆ x = f.η ( α ) .α.g.η ( β ) .b.x [definition of the twisted action]= f.η ( α ) .g.α g .η ( β ) .β.x = f.g.η ( α ) .α g .η ( β ) .β.x [ η is a loop cocycle]= f.g.η ( α ) . ( α g .η ( β )) .α g . ( β.x )= f.g.η ( α g ) . ( α g .η ( β )) .α g . ( β.x ) [ g ∈ Λ η ]= f g.η ( α g β ) .α g . ( β.x ) [ η a cocycle]= ( f g, α g β ) ⋆ x = (cid:0) ( f, α )( g, β ) (cid:1) ⋆ x. (3) One checks that g η is a cocycle and that two equivalent cocycles remain equivalentunder this action along the same lines as for the proof of Lemma 8.10.The commutativity of the diagram takes place already at the level of cocycles.Indeed. Consider the square Z (cid:0) π ( R n ) , η H ( R n, ∞ ) (cid:1) τ η −−−→ ∼ Z (cid:0) π ( R n ) , H ( R n, ∞ ) (cid:1) g ∗ y ≀ g ∗ y ≀ Z (cid:0) π ( R n ) , η H ( R n, ∞ ) (cid:1) τ η −−−→ ∼ Z (cid:0) π ( R n ) , H ( R n, ∞ ) (cid:1) Given a cocycle φ ∈ Z (cid:0) π ( R n ) , η H ( R n, ∞ ) (cid:1) recall that ( τ η φ )( γ ) = φ ( γ ) η ( γ ), hence g (cid:0) ( τ η )( φ ) (cid:1) ( γ ) = g. (cid:0) τ η ( φ )( γ g ) (cid:1) = g. (cid:0) ( φ )( γ g ) η ( γ g ) (cid:1) , γ ∈ Λ η .g. ( φ )( γ g ) g . η ( γ g ) = g φ ( γ ) g . η ( γ g ) = g φ ( γ ) η ( γ )since g.η ( γ g ) = η ( γ g ) because η is a loop cocycle, and η ( γ g ) = η ( γ ) because g ∈ Λ η . On the other hand by definition of the twisting map τ η ( g φ )( γ ) = g φ ( γ ) η ( γ )so that the diagram above commutes.(4) The continuous profinite cohomology is the direct limit of discrete group coho-mology of finite quotients. Hence it is enough to establish the desired results at the“finite level”, namely for a group Γ = Gal( ˜ R n,m /R n ) where ˜ R n,m = ˜ k ⊗ k R n,m is afinite Galois covering of R n through wich η factors, and such that H (˜ k ) = H ( k ).Recall that the action of GL n ( Z ) on R n, ∞ preserves ˜ k ⊗ k R n,m , so that GL n ( Z ) actson Γ. 76e need to check that the given action of Λ η on C ∗ (Γ , η A ) commutes with thedifferentials. We are given g ∈ Λ η and y ∈ C i (Γ , η A ). Recall that the boundary map ∂ i : C i (Γ , η A ) → C i +1 (Γ , η A ) is given by (cid:0) ∂ i ( y ) (cid:1) ( γ , . . . , γ i +1 ) = γ . η ( γ ) .y ( γ , . . . , γ i +1 ) + i X j =1 ( − j y ( γ , . . . , γ j − , γ j γ j +1 , γ j +2 , . . . γ i +1 ) + ( − i +1 y ( γ , . . . , γ i ) . Thus (cid:16) g (cid:0) ∂ i ( y ) (cid:1)(cid:17) ( γ , . . . , γ i +1 ) = g. (cid:0) ∂ i ( y )( γ g , . . . , γ gi +1 ) (cid:1) = g. (cid:0) γ g η ( γ g ) . y ( γ g , . . . , γ gi +1 ) (cid:1) + g. (cid:0) i X j =1 ( − j y ( γ g , . . . , γ gj − , γ gj γ gj +1 , γ gj +2 , . . . γ gi +1 ) (cid:1) + g. (cid:0) ( − i +1 y ( γ g , . . . , γ gi ) (cid:1) = γ η ( γ ) . g y ( γ , . . . , γ i +1 ) [ g ∈ Λ η ]+ i X j =1 ( − j g y ( γ , . . . , γ j − , γ j γ j +1 , γ j +2 , . . . γ i +1 )+( − i +1 g y ( γ , . . . , γ i )= (cid:0) ∂ i ( g y ) (cid:1) ( γ , . . . , γ i +1 ) . This shows that the action of Λ η on C i (Γ , η A ) commutes with the boundary mapsas desired.(5) We are given an exact sequence of linear algebraic groups 1 → M → M → M → H . We twist it by η to obtain 1 → η M → η M → η M → , and look at the characteristic map ψ : η M ( R n ) → H ( R n , η M ) . Let x ∈ η M ( R n ) ⊂ M ( R n, ∞ ). Lift x to an element x ∈ M ( R n, ∞ ) . Then ψ ( x ) = [ z γ ] with z γ = x − (cid:0) η ( γ ) γ x (cid:1) . Now if g ∈ Λ η the element g x lifts g x , hence ψ ( g x ) is represented by the 1–cocycle( g x ) − (cid:0) η ( γ ) γ ( g x ) (cid:1) = g x − (cid:0) η ( γ ) γg x (cid:1) = g. (cid:0) x − ( η ( γ ) γ g x ) (cid:1) = ( g z ) γ by using again η ( γ g ) = η ( γ ) and the fact that g acts trivially on H ( k ). This showsthat ψ ( g x ) = g ψ ( x ). 77ssuming that M is central and of multiplicative type, we consider the bound-ary map ∆ : H ( R n , η M ) → H ( R n , η M ). By isotriviality, the precise natureof this map can be computed at the “finite level” by means of Galois cocycles.Let ( a γ ) be a cocycle with value η M ( R n, ∞ ) = M ( R n, ∞ ) and choose a lifting( b γ ) in M ( R n, ∞ ) which is trivial on an open subgroup of π ( R n ). Recall that∆([ a γ ]) ∈ H (cid:0) π ( R n ) , η M ( R n, ∞ ) (cid:1) is the class of the 2–cocycle [Se1, I.5.6] c γ,τ = b γ ( η ( γ ) . γ b τ ) b − γτ . Similarly, the ( g b γ g ) lift the ( g a γ g ), so ∆( g. [ a γ ]) is the class of the 2–cocycle g b γ g (cid:0) η ( γ ) . γ ( g b τ g ) (cid:1) g b − γ g τ g = g. (cid:16) b γ ( η ( γ ) . γ g b gτ ) b − γ g τ g (cid:17) = g. ∆([ a γ ])as desired.6) Since H ( R , µµµ d ) injects in Br( R ) = Q / Z [GP2, 2.1], it is enough to check theformula on Br( R ). Since GL ( Z ) is generated by the matrices (cid:18) − − (cid:19) , (cid:18) − (cid:19)(cid:18) (cid:19) and (cid:18) (cid:19) , it is enough to show that the desired compatibility holds when g is one of these four elements. Consider the cyclic Azumaya R -algebra A = A (1 , d )with presentation T d = t , T d = t , T T = ζ d T T . Then for g in the above list wehave g. [ A ] = [ A ] (resp. − [ A ], [ A ], − [ A ]) respectively, so that g. [ A ] = det( g ) . [ A ] . Sincethe classes of these cyclic Azumaya algebras generate Br( R ) the result follows. Remark 8.15. (a) In (4), we have H ∗ (cid:0) π ( R n ) , η M ( R n, ∞ ) (cid:1) ∼ −→ H ∗ ( R n , η M ) [GP3,prop 3.4], hence we have a natural action of Λ η on H ∗ ( R n , η M ). We have used anexplicit description of this action in our proof, but the result can also be establishedin a more abstract setting. For g ∈ Λ η , we claim that the map g ∗ : A → A, a g.a applies H (Γ , η A ) into itself. Indeed for a ∈ H (Γ , η A ) and γ ∈ Γ, we compute thetwisted action just as we did in (2) of the Lemma. γ ⋆ ( g.a ) = ( η ( γ ) γg ) .a = ( η ( γ ) g γ g ) .a [definition of γ g ]= ( g η ( γ ) γ g ) .a [ GL n ( Z ) commutes with H (˜ k )]= ( g η ( γ g ) γ g ) .a [ g ∈ Λ η ]= g.a [ a ∈ H (Γ , η A )] . We get then a morphism of functors g ∗ : F → F which extends uniquely as a mor-phism of δ -functors [W, § g ∗ : H i (Γ , η A ) → H i (Γ , η A ) for each GL n ( Z ) ⋉ (cid:0) H (˜ k ) ⋊ Γ (cid:1) -module.(b) There is an analogous statement to (5) for homogeneous spaces.78or each class [ E ] ∈ H (cid:0) R n , Out ( G ) (cid:1) , we denote by H (cid:0) R n , Aut ( G ) (cid:1) [ E ] the fiberat [ E ] of the map H (cid:0) R n , Aut ( G ) (cid:1) → H (cid:0) R n , Out ( G ) (cid:1) . We then have the decom-position(8.8) H (cid:0) R n , Aut ( G ) (cid:1) = G [ E ] ∈ H ( R n , Out ( G )) H (cid:0) R n , Aut ( G ) (cid:1) [ E ] The group GL n ( Z ) acts on H (cid:0) R n , Out ( G ) (cid:1) and on H (cid:0) R n , Aut ( G ) (cid:1) by base change(see Lemma 8.10). It follows that GL n ( Z ) permutes the subsets of the partition(8.8), and that for each class [ E ] ∈ H (cid:0) R n , Out ( G ) (cid:1) , its stabilizer under the actionof GL n ( Z ) preserves H (cid:0) R n , Aut ( G ) (cid:1) [ E ] .Let Out( G ) = Out ( G )( k ) . The (abstract) group Out( G ) acts naturally on theright on the set of (continuous) homomorphisms Hom (cid:0) π ( R n ) , Out ( G ) (cid:1) . This action,which we denote by int , is given by int( a )( φ )( γ ) = φ a ( γ ) = a − φ ( γ ) a. We have H (cid:0) R n , Out ( G ) (cid:1) = Hom (cid:0) π ( R n ) , Out( G ) (cid:1) / int (cid:0) Out( G ) (cid:1) We consider a system of representatives ([ φ j ]) j ∈ J of the set of double cosets GL n ( Z ) \ Hom (cid:0) π ( R n ) , Out( G ) (cid:1) / int (cid:0) Out( G ) (cid:1) . Consider a fixed element j ∈ J . De-note by Λ j ⊂ GL n ( Z ) the stabilizer of [ φ j ] ∈ H (cid:0) R n , Out ( G ) (cid:1) for the base changeaction of GL n ( Z ) on Spec( R n ). An element g ∈ GL n ( Z ) belongs to Λ j if and only ifthere exists a g ∈ Out( G ) such that the following diagram commutes φ j : π ( R n ) −−−→ Out ( G ) g ∗ x ≀ int( a g ) x ≀ φ j : π ( R n ) −−−→ Out ( G )Note that Λ φ j ⊂ Γ j . We have GL n ( Z ) \ H (cid:0) R n , Aut ( G ) (cid:1) = F j ∈ J Λ j \ H (cid:0) R n , Aut ( G ) (cid:1) [ φ j ] . (8.9)Recall that our section s : Out ( G ) → Aut ( G ) is determined by our choice ofpinning of ( G , B , T ) . This allows us to trace the action of Λ j . Indeed [ s ∗ ( φ j )] ∈ H (cid:0) R n , Aut ( G ) (cid:1) [ φ j ] , so that the classical twisting argument (see [Gi4, lemme 1.2])shows that the map H ( R n , s ∗ ( φ j ) G ) → H (cid:0) R n , s ∗ ( φ j ) Aut ( G ) (cid:1) τ s ∗ ( φj ) → H (cid:0) R n , Aut ( G ) (cid:1) induces a bijection H ( R n , s ∗ ( φ j ) G ) /H (cid:0) R n , φ j Out ( G ) (cid:1) ∼ −→ H (cid:0) R n , Aut ( G ) (cid:1) [ φ j ] . (8.10) 79ote that the action of an element a ∈ H (cid:0) R n , φ j Out ( G ) (cid:1) on H (cid:0) R n , s ∗ ( φ j ) G (cid:1) isgiven by H ( R n , s ∗ ( φ ) G ) ( φj s ∗ )( a ) ∼ −→ H ( R n , s ∗ ( φ j ) G ) . where ( φ j s ∗ ) is the twist of s ∗ by the cocycle φ j . Furthermore the map (8.10) pre-serves toral or, what is equivalent, loop classes. Feeding this information into thedecomposition (8.9), we get(8.11) GL n ( Z ) \ H (cid:0) R n , Aut ( G ) (cid:1) ∼ −→ G j ∈ J Λ j \ (cid:16) H ( R n , s ∗ ( φ j ) G ) /H (cid:0) R n , φj Out ( G ) (cid:1)(cid:17) . At least in certain cases, the action of Λ j on H ( R n , s ∗ ( φ j ) G ) /H (cid:0) R n , φ j Aut ( G ) (cid:1) can be understood quite nicely (see Remark 8.17 below). Lemma 8.16. (1) For each g ∈ Λ φ j , the following diagrams H ( R n , s ∗ ( φ j ) G ) −−−→ H (cid:0) R n , s ∗ ( φ j ) Aut ( G ) (cid:1) τ s ∗ ( φj ) −−−−→ H (cid:0) R n , Aut ( G ) (cid:1) [ φ j ] g ∗ y g ∗ y g ∗ y H ( R n , s ∗ ( φ j ) G ) −−−→ H (cid:0) R n , s ∗ ( φ j ) Aut ( G ) (cid:1) τ s ∗ ( φj ) −−−−→ H (cid:0) R n , Aut ( G ) (cid:1) [ φ j ] ,H ( R n , s ∗ ( φ j ) G ) × H (cid:0) R n , φ j Out ( G ) (cid:1) −−−→ H ( R n , s ∗ ( φ j ) G ) g ∗ y id y g ∗ y H ( R n , s ∗ ( φ j ) G ) × H (cid:0) R n , φ j Out ( G ) (cid:1) −−−→ H ( R n , s ∗ ( φ j ) G ) commute where the maps g ∗ are the base change maps defined in Lemma 8.14.(2) The map (8.10) H ( R n , s ∗ ( φ j ) G ) → H (cid:0) R n , Aut ( G ) (cid:1) [ φ j ] is Λ φ j × H ( R n , φ j Out ( G )) op –equivariant and Λ φ j × H (cid:0) R n , φ j Out ( G ) (cid:1) op \ H (cid:0) R n , s ∗ ( φ j ) G (cid:1) ∼ −→ Λ φ j \ H (cid:0) R n , Aut ( G ) (cid:1) [ φ j ] . Remark 8.17.
Of course (2) is useful provided that Λ φ j = Γ j . This is the case forsimple groups which are not of type D since Out ( G ) = 1 or Z / Z .80 roof. (1) We are given g ∈ Λ φ j . The left square of the first diagram commutes bythe functoriality of the base change map g ∗ . The commutativity of the right squarefollows from Lemma 8.14.(3) applied to the k –group Aut ( G ) and the cocycle s ∗ ( φ j ).The commutativity of the second diagram follows from the action on cocycles givenin Lemma 8.10.(2) By (1), the left action of Λ φ j and the right action of H (cid:0) R n , φ j Out ( G ) (cid:1) on H ( R n , s ∗ ( φ j ) G ) commute. HenceΛ φ j \ (cid:16) H ( R n , s ∗ ( φ j ) G ) /H (cid:0) R n , s ∗ ( φ j ) Aut ( G ) (cid:1)(cid:17) ∼ −→ Λ φ j × H ( R n , s ∗ ( φ j ) Out ( G )) op \ H ( R n , s ∗ ( φ j ) G )and this set maps bijectively onto Λ φ j \ H ( R n , Aut ( G )) [ φ j ] . By combining Theorem 8.1, Corollary 5.2 and Lemma 4.14 we get the followinggeneralization (in characteristic 0) of theorem 2.4 of [CGP].
Corollary 9.1.
Let G be a linear algebraic k –group Then we have bijections H toral ( k [ t ± ] , G ) ∼ −→ H loop ( k [ t ± ] , G ) ∼ −→ H ( k [ t ± ] , G ) ∼ −→ H ( k (( t )) , G ) . In the case when k is algebraically closed, we also recover the original results of[P1] and [P2] that began the “cohomological approach” to classification problems ininfinite-dimensional Lie theory. Throughout this section we assume that k is algebraically closed of characteristic0 and G a semisimple Chevalley k –group of adjoint type. We let G sc → G be itssimply connected covering and denote by µµµ its kernel. R -groups. Serre’s conjecture II holds for the field F by Bruhat–Tits theory [BT3, cor. 3.15],i.e. H ( F , H ) = 1 for every semisimple simply connected group H over F . Fur-thermore, we know explicitly how to compute the Galois cohomology of an arbitrarysemisimple F group [CTGP, th. 2.1] and [GP2, th. 2.5]. We thus have.81 orollary 9.2. We have a decomposition H loop (cid:0) R , Aut ( G ) (cid:1) ∼ −→ G [ E ] ∈ H ( R , Out ( G )) E Out ( G )( R ) \ H ( R , E µµµ ) and the inner R –forms of G are classified by the coset Out ( G )( R ) \ H ( R , µµµ ) . Note that the case when
Out ( G ) is trivial recovers theorem 3.17 of [GP2]. Wecan thus view the last Corollary as an extension of this theorem to the case when theautomorphism group of G is not connected. Proof.
Our choice of splitting s : Out ( G ) → Aut ( G ) of the exact sequence1 → G → Aut ( G ) → Out ( G ) → H (cid:0) F , Aut ( G ) (cid:1) ∼ −→ G [ E ] ∈ H ( F , Out ( G )) H ( F , E G ) / E Out ( G )( F )with respect to the Dynkin-Tits invariant. On the other hand, the boundary map H ( F , E G ) → H ( F , E µµµ ) is bijective by [CTGP, th. 2.1] and [GP2, th. 2.5]. Theright action of E Out ( G )( F ) can then be transferred to H ( F , E µµµ ) , and is the oppositeof the natural left action of E Out ( G )( F ) on H ( F , E µµµ ). Hence H ( F , Aut ( G )) ∼ −→ G [ E ] ∈ H ( F , Out ( G )) E Out ( G )( F ) \ H ( F , E µµµ ) . But E Out ( G ) is finite ´etale over R n hence E Out ( G )( R ) = E Out ( G )( F ) by Remark6.8.(d). On the other hand, we have H ( R , E µµµ ) ∼ −→ H ( F , E µµµ ) since E µµµ is an R –group of multiplicative type [GP3, prop. 3.4]. Taking into account the acyclicitytheorem for Aut ( G ) and Out ( G ), we get the square of bijections H ( F , Aut ( G )) −−−→ ∼ F [ E ] ∈ H ( F , Out ( G )) E Out ( G )( F ) \ H ( F , E µµµ ) . x ≀ x ≀ H loop ( R , Aut ( G )) −−−→ ∼ F [ E ] ∈ H ( R , Out ( G )) E Out ( G )( R ) \ H ( R , E µµµ ) , and this establishes the Corollary. 82ext we give a complete list of the isomorphism classes of loop R –forms of G inthe case when G is simple of adjoint type. We have Out ( G ) = 1 in type A B , C , E , E , F and G , Out ( G ) = Z / Z in type A n ( n ≥ D n ( n ≥
5) and E , and Out ( G ) = S in type D . In the case
Out ( G ) = 1, then by theorem 3.17 of [GP2] we have H loop ( R , Aut ( G )) ∼ −→ H ( R , µµµ ). But µµµ = µµµ n for n = 1 or 2. We have H ( R , µµµ ) ∼ = Z / Z [GP2, § Corollary 9.3. (1) If G has type A , then H loop (cid:0) R , Aut ( G ) (cid:1) ≃ Z / Z .(2) If G has type B , C or E , then H loop (cid:0) R , Aut ( G ) (cid:1) ≃ Z / Z .(3) If G has type E , F or G , then H loop (cid:0) R , Aut ( G ) (cid:1) = 1 . Remark 9.4.
In Case (1) and Case (2) the non-trivial twisted groups are not qua-sisplit (because their “Brauer invariant” in H ( R , µµµ ) is not trivial.) In Case (1) thenon-trivial twisted group is in fact anisotropic (see [GP2] for details).In the case Out ( G ) = Z / Z , we have H ( R , Z / Z ) ∼ = Z / Z ⊕ Z / Z . The Z / Z -Galois extensions of R under consideration are R × R , R [ √ t ] , R [ √ t ]and R [ √ t t ] which correspond to the elements (0 , , (1 , , (0 ,
1) and (1 ,
1) respec-tively. These can also be thought as Z / Z -torsors over R that we will denote by E , , E , , E , and E , respectively. In the first case the generator of the Galois groupacts by permuting the two factors, while in the other three is of the form √ x
7→ −√ x. Since E Out ( G ) ∼ = Out ( G ) = Z / Z , for any of our four torsors we have H loop (cid:0) R , Aut ( G ) (cid:1) ∼ −→ Z / Z \ H ( R , µµµ ) G E = E , , E , , E , Z / Z \ H ( R , E µµµ ) . This leads to a case by case discussion.
Corollary 9.5. (1) For G of type A n ( n ≥ H loop (cid:0) R , Aut ( G ) (cid:1) ≃ {± } \ (cid:16) Z / (2 n + 1) Z (cid:17) G E = E , , E , , E , { E G } . There are n + 1 inner and three outer loop R –forms of G . All outer forms arequasisplit.(2) For G of type A n − ( n ≥ ) H loop (cid:0) R , Aut ( G ) (cid:1) ≃ {± } \ (cid:16) Z / n Z (cid:17) G E = E , , E , , E , { E G ± } . Of course here 1 , Z / Z and S are here viewed as constant R groups or finite (abstract) groupsas the situation requires. here are n + 1 inner and six outer loop R –forms of G . The outer forms come inthree pairs. Each pair has one form which is quasisplit and one which is not.(3) For G of type D n − ( n ≥ ) H loop (cid:0) R , Aut ( G ) (cid:1) ≃ {± } \ (cid:16) Z / Z (cid:17) G E = E , , E , , E , { E G ± } . There are three inner and six outer loop R –forms of G . The outer forms come inthree pairs. Each pair has one form which is quasisplit and one which is not.(4) For G of type D n ( n ≥ ) H loop (cid:0) R , Aut ( G ) (cid:1) ≃ switch \ (cid:16) Z / Z ⊕ Z / Z (cid:17) G E = E , , E , , E , { E G ± } . There are three inner and six outer loop R –forms of G . The outer forms come inthree pairs. Each pair has one form which is quasisplit and one which is not.(5) For G of type E H loop ( R , Aut ( G )) ≃ {± } \ (cid:16) Z / Z (cid:17) G E = E , , E , , E , { E G } . There are two inner and three outer loop R –forms of G . All outer forms are quasis-plit.Proof. (1) We have µµµ = µµµ n +1 = ker (cid:0) µµµ n +1 Q → µµµ n +1 (cid:1) and the action of Z / Z switchesthe two factors. We have H ( R , µµµ ) ≃ Z / (2 n + 1) Z and the outer action of Z / Z isby signs. Let E = E (1 , . It follows that E µµµ = ker (cid:0) Q R [ √ t ] /R µµµ n +1 norm → µµµ n +1 (cid:1) . Since 2 n + 1is odd, the norm is split and Shapiro lemma yields H ( R , E µµµ ) = ker (cid:0) H ( R [ √ t ] , µµµ n +1 ) Cores → H ( R , µµµ n +1 ) (cid:1) . This reads ker (cid:0) Z / (2 n + 1) Z id → Z / (2 n + 1) Z (cid:1) = 0 by taking into account proposition2.1 of [GP3]. The same calculation holds for E (0 , and E (1 , and we obtain the desireddecomposition. In particular there are n + 1 inner forms and three outer forms. Theouter forms are all quasiplit. Strictly speaking we are looking, here and in what follows, at the action of Out( G ) on R –groups or cohomology of R –groups which are of multiplicative type. Since we have an equivalenceof categories between R and F groups of multiplicative type [GP3]. By Remark 6.8(d)) we cancarry all relevant calculations at the level of fields, in which case the situation is well understood.See for example the table in page 332 of [PR]. µµµ = µµµ n = ker (cid:0) µµµ n Q → µµµ n (cid:1) and the action of Z / Z switches the twofactors. The coset Z / Z \ H ( R, µµµ n ) is as before {± } \ ( Z / n Z ). However, thecomputation of H ( R , E µµµ ) is different. The exact sequence1 → E µµµ n → Y R [ √ t ] /R µµµ n norm → µµµ n → · · · → H ( R [ √ t ] , µµµ n ) norm → H ( R , µµµ n ) δ → H ( R , E µµµ ) → H ( R [ √ t ] , µµµ n ) norm → H ( R , µµµ n ) . The norm map appearing on the righthand side is the identity map id : Z / n Z → Z / n Z , so δ is onto. By the choices of coordinates √ t and t on R [ √ t ] and t , t on R , the beginning of the exact sequence decomposes as Z / n Z ⊕ Z / n Z ( id, × −→ Z / n Z ⊕ Z / n Z . So H ( R , E µµµ ) ≃ Z / Z and the action of Z / Z on H ( R , E µµµ ) is therefore necessarilytrivial. Thus E leads to two distinct twisted forms E G ± . More precisely E G + = E G (which is quasiplit), while E G − is not quasiplit (since its “Brauer invariant”in H ( R , E µµµ ) is not trivial). Similarly for E (0 , and E (1 , . This gives the desireddecomposition. There are n + 1 inner forms and six outer forms (three of which arequasisplit).(3) In this case µµµ = µµµ . The computation of the H are exactly as in case (2) for n = 2. There are three inner forms and six outer forms (three of which are quasisplit).(4) This case is rather different since µµµ = µµµ × µµµ where Z / Z switches the twosummands. We have H ( R , µµµ ) ≃ Z / Z ⊕ Z / Z where again Z / Z acts by switchingthe two summandsGiven that E µµµ = Q R [ √ t ] /R µµµ , we have H ( R , E µµµ ) ∼ −→ H ( R [ √ t ] , µµµ ) = Z / Z . Similarly for E (0 , and E (1 , , whence our decomposition. Again we have three innerforms and six outer forms (three of which are quasisplit).(5) This is exactly as in case (1) for n = 1. There are two inner forms and three outerforms (all three of them quasisplit).It remains to look at the case when G is of type D . The set H ( R , S ) classifies alldegree 3 ´etale extensions S of R . Then S is a direct product of connected extensions.There are tree cases: S = R × R × R (the split case), S = S ′ × R with S ′ /R ofdegree 2 , and the connected case.The case of S ′ × R is already understood: They correspond to a 1-cocycle φ : π ( R n ) → Z / Z ⊂ S , where we view Z / Z as a subgroup of S generated by a85ermutation. Note that up to conjugation by S , there are exactly three such maps φ. These are three non-isomorphic quadratic extensions which were denoted by E ( i,j ) above for ( i, j ) = (0 , . We shall denote them by E ( i,j )2 in the present situation toavoid confusion.In the connected case there are four cubic extensions of R . They correspond toadjoining to R a cubic root in R , ∞ of t , t , t t and t t respectively. We willdenote the corresponding four S –torsors by E ( i,j )3 with the obvious values for ( i, j ) . The cubic case, which a priori appears as the most complicated, ends up being quitesimple due to cohomological vanishing reasons, as we shall momentarily see.According to Corollary 9.2 we have the decomposition H loop ( R , Aut ( G )) ≃ S \ H ( R , µµµ )(9.1) G E ( i,j )2 ( E ( i,j )2 S )( R ) \ H ( R , E ( i,j )2 µµµ ) G E ( i,j )3 ( E ( i,j )3 S )( R ) \ H ( R , E ( i,j )3 µµµ ) . The centre is µµµ = µµµ × µµµ = ker (cid:0) µµµ Q → µµµ (cid:1) and S acts by permutation on µµµ .Hence H ( R , µµµ ) = ker (cid:0) H ( R , µµµ ) → H ( R , µµµ ) (cid:1) ⊂ H ( R , µµµ ) ≃ ( Z / Z ) . Thereare two orbits for the action of S on H ( R , µµµ ), namely (0 , ,
0) and (1 , , E (1 , by E and E (1 , by E . Since the group GL ( Z )acts transitively on the set of quadratic and cubic extensions of R we may consideronly the case of E := E , [resp. E := E , ] for the purpose of determining the coset( E ( i,j )2 S )( R ) \ H ( R , E ( i,j )2 µµµ ) [resp. ( E ( i,j )3 S )( R ) \ H ( R , E ( i,j )3 µµµ )]. that all the twists of µµµ and S by quadratic or cubic torsors are of the form E i µµµ and E i S for i = 2 (resp. i = 3) in the quadratic (resp. cubic) case.We have E µµµ = ker (cid:0) Q R [ √ t ] /R µµµ × µµµ norm × id → µµµ (cid:1) , hence H ( R , E µµµ ) = ker (cid:0) H ( R [ √ t ] , µµµ ) ⊕ H ( R , µµµ ) → H ( R , µµµ ) (cid:1) ∼ = H ( R [ √ t ] , µµµ ) = Z / Z . Since Z / Z has trivial automorphism group, we get three copies of Z / Z in the secondsummand of the decomposition (9.1).In the cubic case we have E µµµ = ker (cid:0) Q R [ √ t ] /R µµµ norm → µµµ (cid:1) . Since 2 is prime to 3,the norm is split and H ( R , E µµµ ) = ker (cid:0) H ( R [ √ t ] , µµµ ) Cores −→ H ( R , µµµ ) (cid:1) = ker (cid:0) Z / Z id → Z / Z ) = 0 . E S )( R ) ∼ = Z / Z and( E S )( R ) ∼ = Z / Z .Looking at (9.1) we obtain. Corollary 9.6.
For G of type D there are twelve loop R –forms, two inner andten outer. Six of the outer forms are “quadratic”, and come divided into three pairs,where each pair contains exactly one quasiplit group. The remaining four outer formsare “cubic” and are all quasiplit. The Extended Affine Lie Algebras (EALAs), as their name suggests, are a classof Lie algebras which generalize the affine Kac-Moody Lie algebras. To an EALA E one can attached its so called centreless core, which is usually denoted by E cc . Thisis a Lie algebra over k (in general infinite-dimensional) which satisfies the axioms ofa Lie torus. Neher has shown that all Lie torus arise as centreless cores of EALAs,and conversely. He has also given an explicit procedure that constructs all EALAshaving a given Lie torus L as their centreless cores. To some extent this reduces manycentral questions about EALAs (such as their classification) to that of Lie tori.The centroid of a Lie tori L is always of the form R n . This gives a natural R n –Liealgebra structure to L . If L as an R n –module is of finite type, then L is necessarilya multiloop algebra L ( g , σσσ ) as explained in the Introduction. Let G be a Chevalley k –group of adjoint type with Lie algebra g . Since
Aut ( g ) ≃ Aut ( G ) the n –loopalgebras based on g (as R n –Lie algebras) are in bijective correspondence with theloop R n –forms of G . Indeed, they are precisely the Lie algebras of the loop R n –groups. The subtlety comes from the fact that in infinite-dimensional Lie theory oneis interested in these Lie algebras as Lie algebras over k, and not R n . In the presentcontext the “centroid trick” (see [GP2, § GL n ( Z ) action on H (cid:0) R n , Aut ( g ) (cid:1) we have defined. This allows us to describe, in terms of orbits, allthe isomorphism classes of R n –multiloop algebras L ( g , σσσ ) that become isomorphicwhen viewed as Lie algebras over k. In what follows “loop algebras based on g ” will be though as Lie algebras over k. In the case
Out ( G ) = 1 we have seen that H loop (cid:0) R , Aut ( G ) (cid:1) ≃ H ( R , µµµ ) , and this latter H is either trivial or Z / Z . In both cases the action of GL ( Z ) on H loop (cid:0) R , Aut ( G ) (cid:1) is necessarily trivial. In particular. This terminology is due to Neher and Yoshii. It may seem strange to call a Lie algebra a Lietorus (since tori have already a meaning in Lie theory). The terminology was motivated by theconcept of Jordan tori, which are a class of Jordan algebras. orollary 9.7. (1) If g has type A B , C or E , there exists two isomorphism classesof –loop algebras based on g denoted by g (the split case) and g . (2) All 2–loop algebras based on g of type E , F or G are trivial, i.e isomorphicas k –Lie algebras to g = g ⊗ k R . In the case
Out ( G ) = Z / Z , we have H ( R , Z / Z ) ∼ = Z / Z ⊕ Z / Z and theaction of GL ( Z ) on H ( R , Z / Z ) is given by the linear action mod 2. Since SL ( Z / Z ) = GL ( Z / Z ) and SL ( Z / Z ) is generated by elementary matrices, thereduction map GL ( Z ) → GL ( Z / Z ) is onto. Hence there are two orbits for theaction of GL ( Z ) on H ( R , Z / Z ), namely the trivial one and H ( R , Z / Z ) \ { } .The last one is represented by the quadratic Galois extension R [ √ t ] /R , denotedby E (1 , in the previous section, which we will again denote simply by E in whatfollows. The action of GL ( Z ) we have just described shows that in all cases theouter forms, which came in three families (each with one or two classes) in the caseof loop R –groups, collapse into a single family. This single family consists of eithera single class, namely the quasi-split algebra E g = E g + , or two classes E g + and E g − .The algebra E g − is not quasisplit. Corollary 9.8. If Out ( G ) = Z / Z the classification of isomorphism classes of –loop algebras based on g is as follows:(1) In type A n ( n ≥ GL ( Z ) \ H loop ( R , Aut ( g )) ≃ {± } \ (cid:16) Z / (2 n + 1) Z (cid:17) G { E g } . There are n + 1 inner forms, denoted by g q with ≤ q ≤ n , and one outer form(which is quasisplit).(2) In type A n − ( n ≥ ) GL ( Z ) \ H loop ( R , Aut ( g )) ≃ {± } \ (cid:16) Z / n Z (cid:17) G { E g + } G { E g − } . There are n + 1 inner forms, denoted by g q with ≤ q ≤ n, and two outer forms (oneof them quasiplit, the other one not).(3) In type D n − ( n ≥ ) GL ( Z ) \ H loop ( R , Aut ( g )) ≃ {± } \ (cid:16) Z / Z (cid:17) G { E g + } G { E g − } . As pointed out in [GP2], the case of E has an amusing story behind it. The existence of a k –Lie algebra L ( g , σ , σ ) which is not isomorphic to g ⊗ k R was first established by van de Leurwith the aid of a computer. In nullity 1 inner automorphisms always lead to trivial loop algebras.van de Leur’s example shows that this fails already in nullity two. here are inner forms, denoted by g , , , and two outer forms (one of them quasiplit,the other one not).(4) In type D n ( n ≥ , we have GL ( Z ) \ H loop ( R , Aut ( g )) ≃ switch \ (cid:16) Z / Z ⊕ Z / Z (cid:17) G { E g + } G { E g − } . There are inner forms, denoted by g , , , and two outer forms (one of them quasiplit,the other one not).(5) In type E GL ( Z ) \ H loop ( R , Aut ( g )) ≃ {± } \ (cid:16) Z / Z (cid:17) G { E g } . There are inner forms, g and g , and one outer form (which is quasisplit).Proof. The nature of the collapse of outer forms when passing from R to k wasexplained before the statement of the Corollary. It remains to understand the innercases. According to Corollary 9.2 and (8.9), we need to trace the action of GL ( Z )on Z / Z \ H ( R , µµµ ) and use the fact that this action lifts to an action of GL ( Z ) on H ( R , µµµ ) which commutes with that of Z / Z .(1) We have µµµ = µµµ n +1 = ker (cid:0) µµµ n +1 Q → µµµ n +1 (cid:1) and the action of Z / Z switches the twofactors. We have H ( R , µµµ ) ∼ = Z / (2 n + 1) Z and the action of GL ( Z ) on H ( R , µµµ )is given by the determinant (Lemma 8.14.6), that of Z / Z is given by signs. Thus GL ( Z ) acts trivially on Z / Z \ H ( R , µµµ ) and the result follows.(2) We have µµµ = µµµ n = ker (cid:0) µµµ n Q → µµµ n (cid:1) and the action of Z / Z switches the twofactors. The action of GL ( Z ) on H ( R, µµµ ) is given by the determinant, hence the setof cosets ( GL ( Z ) × Z / Z )) \ H ( R, µµµ n ) can still be identified with {± } \ ( Z / n Z ).(3) In this case µµµ = µµµ . The computation of H and reasoning are exactly as in (2)above for n = 2.(4) This case is rather different since µµµ = µµµ × µµµ where Z / Z switches the twosummands. We have H ( R , µµµ ) ∼ = Z / Z ⊕ Z / Z with respect to the switch action.Again GL ( Z ) acts by g.α = det( g ) .α , hence trivially.(5) This is exactly as in case (1) for n = 1.It remains to look at the case when G is of type D . Lemma 9.9.
There are three orbits for the action of GL ( Z ) on H ( R , S ) :- the trivial class;- n E , , E , , E , o ; n E , , E , , E , , E , o where the notations is as in Corollary 9.6 supra.Proof. The three classes above correspond to case of the split ´etale cubic R -algebra,the case S ′ × R where S ′ /R is quadratic and the cubic case. Obviously each ofthe above sets is GL ( Z )-stable, so we need to check that there is a single orbit.The quadratic case was dealt with in Corollary 9.8. In the cubic case, we have E , = R [ √ t ]. By applying the base change corresponding to (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19) we obtain E , , E , and E , respectively. Corollary 9.10.
Up to k -isomorphism there are five 2-loop algebras based on g oftype D : two inner forms, denoted by g and g ; two “quadratic” algebras, E , g + (which is quasisplit) and E , g − (which is not quasiplit); and one “cubic” algebra E , g ( which is quasisplit).Proof. By Lemma 9.9, the quadratic (resp. cubic) classes of Corollary 9.6 are inthe GL ( Z )–orbit of those having Dynkin-Tits invariant E , (resp. E , ). So thecubic case is done. In the quadratic case, there are then one or two non-isomorphic“quadratic” 2-loop algebras. Since one of these R –algebras is quasisplit and theother one is not, Corollary 8.12 shows that they remain non-isomorphic as k –algebras.Finally, there are two orbits for the action of S on H ( R , µµµ ), namely (0 , ,
0) and(1 , , R -forms. The action of GL ( Z ) istrivial in this set, so the algebras remain non-isomorphic over k. A. The following theorem extends results of Steinmetz from classical types [SZ, th.6.4] (which involves certain small rank restrictions) to all types. This establishesConjecture 6.4 of [GP2]. Theorem 9.11.
Let g be a finite dimensional simple Lie algebra over k which isnot of type A . Let L and L ′ be two -loop algebras based on g . The following areequivalent:(1) L and L ′ are isomorphic (as Lie algebras over k );(2) L and L ′ have the same Witt-Tits index. An even stronger version of this Conjecture will be established in the next section. roof. Of course, we use that L (resp. L ′ ) arise as the Lie algebra of R –loop adjointgroups H (resp. H ′ ) which are forms of G = Aut( g ) . Then condition (1) readsthat [ H ] = [ H ′ ] in GL ( Z ) \ H loop (cid:0) R , Aut ( g ) (cid:1) and condition (2) reads that H × R K and H ′ × R K have the same Witt-Tits index.(1) = ⇒ (2) : This is is the simple part of the the equivalence (and it is not necessary toexclude type A ). Let G be the corresponding adjoint group. If [ H ] and [ H ′ ] are equalin GL ( Z ) \ H loop (cid:0) R , Aut ( G ) (cid:1) , it is obvious that their Dynkin-Tits invariant coincidein GL ( Z ) \ H loop (cid:0) R , Out ( G ) (cid:1) , and also that their Tits index over K coincide byCorollary 8.12.(2) = ⇒ (1) : Without loss of generality we can assume that H and H ′ have sameDynkin-Tits invariant in H (cid:0) R , Out ( g ) (cid:1) . The proof is given by a case-by-case dis-cussion. The cases of type E , F and G follow directly from Corollary 9.7.2. Types B , C and E are also straightforward since (over R ) there is only one class of non-split2–loop algebras. For obvious reasons, this non-split Lie algebras necessarily remainnon-isomorphic to the split Lie algebra g ⊗ k R when viewed as Lie algebras over k. Type D n , n ≥ GL ( Z ) \ H loop ( R , Aut ( g )) corresponding to [ E ] has only one non quasi-split class,so [ H ] and [ H ′ ] are equal in GL ( Z ) \ H loop (cid:0) R , Aut ( g ) (cid:1) . Corollary 9.8 states that theinner part of GL ( Z ) \ H loop ( R , Aut( g )) is Z / Z ⊕ Z / Z modulo the switch action,so is represented by (0 , ,
0) and (1 , R -loop group PSO ( q ) with q = h , t , t , t t i ⊥ (2 n − h , − i . Its Witt-Tits K -index is(9.2) . . . . . . ❍❍✟✟ r r ✐r ✐r ✐r ✐r ✐r ✐r✐r✐r α α α n − α n The other one is PSU(
A, h ) where A = A (2 ,
1) is the R -quaternion algebra T = t , T = t , T T + T T = 0 and h is the hyperbolic hermitian form over A n withrespect to the quaternionic involution q q . Indeed PSU( A, h ) is an adjoint innerloop R -group of type D n and its Witt-Tits K -index is(9.3) . . . ❍❍✟✟ r ✐r r r✐ r r ✐r rr✐r α α α n − α n So [ H ] and [ H ′ ] are equal in GL ( Z ) \ H loop ( R , Aut( g )) to the split form or one ofthese two forms. Strictly speaking “...the Lie algebra...” is an R -Lie algebra, but we view this in a natural awayas a k -Lie algebra. ype D n − , n ≥ R -loop groups with distinct K -Witt-Tits index. The first one is the R -loop group PSO ( q ) with q = h , t , t , t t i ⊥ (2 n − h , − i . Its Witt-Tits K -index is(9.4) . . . . . . ❍❍✟✟ r r ✐r ✐r ✐r ✐r ✐r ✐r✐r✐r α α α n − α n − The other one is PSU(
A, h ) where h is the hyperbolic hermitian form over A n − which is the orthogonal sum of h i and the hyperbolic form over A n − . Its Witt-Tits K -index is(9.5) . . . ❍❍✟✟ r ✐r r r✐ r r✐ r rr✐r α α α n − α n − Type D : Follows from Corollary 9.10.
Type E : This case is straightforward because there is only one class of 2–loop algebraswhich is not quasi-split.
Remark 9.12.
There is some redundancy in the statement of the Theorem. It iswell known, by descent considerations, that if L and L ′ are isomorphic as Lie algebrasover k, then their absolute type coincide, i.e. they are both 2–loop algebras basedon the same g (see [ABP2.5] for further details). It will thus suffice to assume in theTheorem that neither L nor L ′ are of absolute type A. The following table summarizes the classification on 2-loop algebras. The tableincludes the Cartan-Killing (absolute) type g , its name, the Witt-Tits index (withTits’ notations) of an R n –representative of the k –Lie algebra in question, and thetype of the relative root system. For example, van de Leur’s algebra has absolutetype E , Tits index E , and relative type F . The way in which the Witt-Tits indexare determined was illustrated in the previous section. The procedure of how toobtain the relative type from the index is described by Tits.In all cases the “trivial” loop algebra g ⊗ k [ t ± , t ± ] is denoted by g . When therelative type is A , the loop algebra in question is anisotropic. For example, inabsolute type A the Lie algebra g is sl ([ t ± , t ± ]) . The Lie algebra g is the derivedalgebra of the Lie algebra that corresponds to the quaternion algebra over k [ t ± , t ± ]with relations T T = − T T and T i = t i . This rank 3 free Lie algebra over k [ t ± , t ± ]is anisotropic, and is a twisted form of sl ⊗ k [ t ± , t ± ] split by the quadratic extension k [ t ± , t ± ]( T ) . g Name Tits index Relative root system A g A (1)1 , A A g A (2)1 , A A n ( n ≥ g q A ( n +1 r )2 n,r − r = gcd( q, n + 1) A r − A n ( n ≥ E g A (1)2 n,n BC n A n − ( n ≥ g q A ( nr )2 n − ,r − r = gcd( q, n ) A r − A n − ( n ≥ E g + 2 A (1)2 n − ,n C n A n − ( n ≥ E g − A (1)2 n − ,n − BC n − B n ( n ≥ g B n,n B n B n ( n ≥ g B n,n − B n − C n ( n ≥ g C (1) n,n C n C n +1 ( n ≥ g C (2)2 n +1 ,n BC n C n ( n ≥ g C (2)2 n,n C n D g D (1)4 , D D g D (1)4 , B D E g + 2 D (1)4 , B D E g − D (2)4 , BC D E g D , G D n − ( n ≥ g D (1)2 n − , n − D n − D n − ( n ≥ g D (1)2 n − , n − B n − D n − ( n ≥ g D (2)2 n − ,n − BC n − D n − ( n ≥ E g + 2 D (1)2 n − , n − B n − D n − ( n ≥ E g − D (2)2 n − ,n − BC n − D n ( n ≥ g D (1)2 n, n D n D n ( n ≥ g D (1)2 n, n − B n − D n ( n ≥ g D (2)2 n,n C n D n ( n ≥ E g + 2 D (1)2 n, n − B n − D n ( n ≥ E g − D (2)2 n,n − BC n − E g E , E E g E , G E E g E , F E g E , E E g E , F E g E , E F g F , F G g G , G
93y taking Remark 9.12 into consideration, an inspection of the Table shows thata stronger version of Theorem 9.11 holds.
Theorem 9.13.
Let L and L ′ be two -loop algebras neither of which is of absolutetype A. The following are equivalent:(1) L and L ′ are isomorphic (as Lie algebras over k );(2) L and L ′ have the same absolute and relative type. Remark 9.14.
This result was established, also by inspection, in Cor.13.3.3 of[ABP3]. In this paper the classification of nullity 2 multiloop algebras over k isachieved by considering loop algebras of the affine algebras. More precisely, it isshown that every multiloop algebra of nulllity2 is isomorphic as a Lie algebra over k to a Lie algebra of the form L ( g ⊗ k [ t ± ] , π ) where π is a diagram automorphism ofthe untwisted affine Lie algebra g ⊗ k [ t ± ] . For example, van de Leur’s algebra appearsby taking g of type E and considering the diagram automorphism of order two ofthe corresponding extended Coxeter-Dynkin.Note that in the present work we have outlined a general procedure to classifyloop adjoint groups and algebras over R n , and that the classification of multiloopalgebras over k follows by GL n -considerations from that over R n . This is not the casein [ABP3]. The nullity 2 classification relies on the structure of the affine algebrasand only yields results over k.
10 The case of orthogonal groups
These groups are related to quadratic forms, which allows for a very precise un-derstanding of their nature based on our results.We consider the example of the split orthogonal group O ( d ) for d ≥
1. If d = 2 m (resp. d = 2 m + 1), this is the orthogonal group corresponding to the quadraticform m P i =1 X i X m +1 − i (resp. m P i =1 X i X m +1 − i + X m +1 ). Since R n -projective modules offinite type are free, we know that H (cid:0) R n , O ( d ) (cid:1) classifies regular quadratic forms over R dn [K, § H loop (cid:0) R n , O ( d ) (cid:1) ∼ −→ H (cid:0) F n , O ( d ) (cid:1) . By iterating Springer’stheorem for quadratic forms over k (( t ))) [Sc, § F n -quadraticforms reads as follows: For each subset I ⊂ { , ..., n } , we put t I = Q i ∈ I t i with theconvention 1 = t ∅ ; we denote by H the hyperbolic plane, that is the rank two splitform. The isometry classes of d -dimensional R n –forms are then of the form ⊥ I ⊂{ ,...,n } t I q I ⊥ H v q I ’s are anisotropic quadratic k -forms and v a non negative integer suchthat P I ⊂{ ,...,n } dim k ( q I ) + 2 v = d . Corollary 10.1.
The set H loop (cid:0) R n , O ( d ) (cid:1) is parametrized by the quadratic forms ⊥ I ⊂{ ,...,n } t I q I ⊥ H v where the q I are anisotropic quadratic k -forms and v a non-negative integer such that P I ⊂{ ,...,n } dim k ( q I ) + 2 v = 2 d . We denote by P ( n ) the set of subsets of { , ..., n } and by P even ≤ d ( n ) ⊂ P ( n ) the setof subsets of { , ..., n } of even cardinal ≤ d. In a similar fashion we define P odd ≤ d ( n ) . Corollary 10.2.
Assume that k is quadratically closed.(1) If d = 2 m , then the map P even ≤ d ( n ) −−−→ H loop ( R n , O ( d )) S
7→ ⊥ I ⊂ S h t I i ⊥ H m − | S | is a bijection.(2) If d = 2 m + 1 , then the map P odd ≤ d ( n ) −−−→ H loop ( R n , O ( d )) S
7→ ⊥ I ⊂ S h t I i ⊥ H m + − | S | is a bijection. Corollary 10.3.
Assume that k is quadratically closed. Inside O ′ d = O ( h , . . . , i ) ≃ O d . there is a single O ′ d ( k ) -conjugacy class of maximal anisotropic abelian constantsubgroup of O ′ ( d ) , that of the diagonal subgroup µµµ d . In particular anisotropic abeliansubgroups of O ′ ( d ) are -elementary.Proof. Let A be a finite abelian constant group of O ′ d . There exist an even integer m ≥ φ : ( Z /m Z ) n → A ( k ) . Then the corre-sponding loop torsor [ φ ] ∈ H ( R n , O ′ d ) is anisotropic. Indeed the map H ( R n , µµµ d ) → H ( R n , O ′ d ) is surjective. Hence there exists ψ : ( Z /m Z ) n → µµµ d such that [ φ ] = [ ψ ] ∈ H ( R n , O ′ d ). Theorem 7.9 shows that φ and ψ are O ′ d ( k )-conjugate. By consideringtheir images, we conclude that A ( k ) is O ′ d ( k )-conjugate to a subgroup of µµµ d ( k ). Remark 10.4. (1) All anisotropic abelian constant subgroups of O ′ d are related tocodes, and these are not explicitly enumerated (see [Gs] for details).(2) Under the hypothesis of the Corollary, let f : Spin ′ d → SO ′ d be the universalcovering of O ′ d . Since the image of a finite abelian constant anisotropic subgroup of Spin ′ d in O ′ d is still anisotropic, it follows that an anisotropic finite constant abeliansubgroup of Spin ′ d is of rank ≤ d and has 4-torsion.95 G G the split Chevalley group of type G over k. If F is a fieldof characteristic zero containing k , we know that H ( F n , G ) classifies octonion F -algebras or alternatively 3-Pfister forms [Se2, § r F : H ( F, G ) → H ( F, Z / Z )is injective and sends the class of an octonion algebra to the Arason invariant of itsnorm form.Consider the standard non-toral constant abelian subgroup f : ( Z / Z ) ⊂ G .Then the composite map( F × /F × ) ∼ = H ( F, ( Z / Z ) ) f ∗ −→ H ( F, G ) r F −→ H ( F, Z / Z ) . sends an element (cid:0) ( a ) , ( b ) , ( c ) (cid:1) to the cup product ( a ) . ( b ) . ( c ) ∈ H ( F, Z / Z ) [GiQ, § n ≥
0, we consider the mapping( R × n /R × n ) ≃ H (cid:0) R n , ( Z / Z ) (cid:1) f ∗ −→ H ( R n , G ) . For a class (cid:0) ( x ) , ( y ) , ( z ) (cid:1) ∈ (cid:0) R × / ( R × ) (cid:1) we write only ( x, y, z ). Corollary 11.1.
The map above surjects onto H loop ( R n , G ) .Proof. By the Acyclicity Theorem, it suffices to observe that the analogous statementholds for H loop ( F n , G ).By using the Rost invariant, we get a full classification of the multiloop algebrasbased on the split Lie algebra of type G . Corollary 11.2.
Assume that k is quadratically closed. Assume that n ≥ .1) H loop ( R n , G ) \ { } consists in the images by f ∗ of the (cid:16) t I , t I , t I (cid:17) where I , I , I are non-empty subsets of { , .., n } such that i < i < i for all ( i , i , i ) ∈ I × I × I .2) GL n ( Z ) \ (cid:0) H loop ( R n , G ) \ { } (cid:1) consists of the image by f ∗ of ( t , t , t ) .Proof. (1) Again by aciclicity it suffices to establish the analogous result over F n .Since k is quadratically closed, we have R × n / ( R × n ) × ∼ = F × n / ( F × n ) × ∼ = ( Z / Z ) n . Hence H ( F n , G ) consists of the image of f ∗ ( t I , t I , t I ) for I , I , I running over the sub-sets of { , .., n } . The Rost invariant of such a class is ( t I ) . ( t I ) . ( t I ) ∈ H ( F n , Z / Z ).Since ( t i ) . ( t i ) = 0 and ( t i ) . ( t j ) = ( t j )( t i ) ∈ H ( F n , Z / Z ), it follows that H ( F n , G )96onsists of the trivial class and the images by f ∗ of the (cid:16) t I , t I , t I (cid:17) where I , I , I arenon-empty subsets of { , .., n } such that i < i < i for each ( i , i , i ) ∈ I × I × I .The last classes are non-trivial pairwise distinct elements since the ( t I ) . ( t I ) . ( t I ) ∈ H ( F n , Z / Z ) are distinct pairwise elements by residue considerations (see for exam-ple prop. 3.1.1 of [GP3]).(2) Follows easily from (1).The following corollary refines Griess’ classification in the G -case [Gs]. Corollary 11.3.
Assume that k is algebraically closed. Let A be an anisotropicconstant abelian subgroup of G . Then A is G ( k ) -conjugate to the standard non-toral subgroup ( Z / Z ) .Proof. Let A be a finite abelian constant anisotropic subgroup of G . We reasonas before. There exist an even integer m ≥ φ : ( Z /m Z ) n → A ( k ) so that the corresponding loop torsor [ φ ] ∈ H ( R n , G ) isanisotropic. By part (1) of Corollary 11.1 there exists ψ : ( Z /m Z ) n → ( Z / Z ) suchthat [ φ ] = [ ψ ] ∈ H ( R n , G ). Theorem 7.9 shows that φ and ψ are G ( k )-conjugate.By taking the images, we conclude that A ( k ) is G ( k )–conjugate to the standard( Z / Z ) .
12 Case of groups of type F , E and simply con-nected E in nullity In this section, we assume that k is algebraically closed. We denote by F , and E the split algebraic k –group of type F and E respectively, and by E the splitsimply connected k –group of type E . For either of these three groups we know that G = Aut ( G ) and that H ( R , G ) = 1 [GP2, th. 2.7]. The goal is then to compute H loop ( R , G ) , or at least the anisotropic classes.Since we want to use Borel-Friedman-Morgan’s classification of rank zero (i.e.with finite centralizer) abelian subgroups and triples of the corresponding compactLie group [BFM, § k = C . Note that there is no loss ofgenerality in doing this as explained in Remark 8.8. Denote by G the anisotropic real form of G (viewed as algebraic group over R )and let K = G ( R ) . This is a compact Lie group.In the F and E case K has a single conjugacy class of rank zero abelian subgroupof rank 3. In the E case, K has two conjugacy classes of rank zero abelian subgroup All the results that we need about rank zero abelian groups and triples can also be found in[KS].
97f rank 3, ( Z / Z ) and ( Z / Z ) . To translate this to the complex case we establishthe following fact. Lemma 12.1.
Let H be a complex affine algebraic group whose connected componentof the identity is reductive. Denote by H its anisotropic real form, viewed as algebraicgroup over R (see [OV, § K H = H ( R ) .(1) Let A is a finite abelian subgroup of K H and denote by A the underlyingconstant subgroup of the algebraic R -group H . Then A is a rank zero subgroup of K H if and only if A × R C is an anisotropic subgroup of H .(2) Let A be an anisotropic abelian constant subgroup of H and put A = A ( C ) .Then there exists h ∈ H ( C ) such that h A ⊂ K H and N K H ( h A ) = N H ( h A )( C ) , bothgroups being finite. Furthermore Z K H ( h A ) = Z H ( h A )( C ) . Recall that K is a maximal subgroup of H ( C ) and that maximal compact sub-groups are conjugate under H ( C ). Proof. (1) Let C denote the connected component of the identity of the centralizer Z H ( A ). It is a real reductive group [BMR, 10.1.5]. If A is an anisotropic subgroupof H , then the maximal tori of Z H ( A ) are trivial and C = 1. Hence C ( C ) is finite and Z K H ( A ) is finite, i.e. A is a rank zero subgroup of K H . Conversely, if A is a rank zerosubgroup of K H then C ( R ) is finite. Since C ( R ) is Zariski dense in the connectedgroup C , we see that C = 1, and A × R C is an anisotropic constant abelian subgroupof H .(2) We are given a finite anisotropic constant subgroup A of H . Since 1 = Z H ( A ) = N H ( A ) , N H ( A ) is a finite algebraic group and N H ( A )( C ) is finite. Since N H ( A )( C )is included in a maximal compact group of H ( C ), we know that there exists h ∈ H ( C )such that A ⊂ N H ( A )( C ) ⊂ h − K H . We have then h A ⊂ N H ( h A )( C ) ⊂ K H , hence N K H ( h A ) = N H ( h A )( C ). It follows that Z K H ( h A ) = Z H ( h A )( C ). Lemma 12.2. (1) The group F has a single conjugacy class of anisotropic finiteabelian (constant) subgroups of rank , denoted by f : ( Z / Z ) ⊂ F . Furthermore N F (cid:0) ( Z / Z ) (cid:1) / Z F (cid:0) ( Z / Z ) (cid:1) ≃ SL ( Z / Z ) . (2) The group E has a single conjugacy class of anisotropic finite abelian (con-stant) subgroups of rank , denoted by f : ( Z / Z ) ⊂ E . The finite group f is a sub-group the maximal subgroup SL /µµµ . Furthermore N E (cid:0) ( Z / Z ) (cid:1) / Z E (cid:0) ( Z / Z ) (cid:1) ≃ SL ( Z / Z ) .(3) The group E has two conjugacy classes of anisotropic finite abelian (constant)subgroups of rank , denoted by f and f . We have: a) f : ( Z / Z ) ⊂ E and N E (cid:0) ( Z / Z ) (cid:1) / Z E (cid:0) ( Z / Z ) (cid:1) ≃ SL ( Z / Z ) .(b) f : ( Z / Z ) ⊂ E is a subgroup of the subgroup ( SL × SL × SL ) /µµµ . Furthermore N E (cid:0) Z / Z ) (cid:1) / Z E (cid:0) ( Z / Z ) (cid:1) ≃ SL ( Z / Z ) . Remark 12.3.
The finite subgroups (1) and 3 (a) are described precisely in [GiQ, § f : ( Z / Z ) ⊂ E sits inside the subgroup( SL × SL × SL ) /µµµ , follows from its very construction (see the proof of lemma5.1.1 [BFM] for details). Proof.
As explained above we may assume that k = C . The previous Lemma 12.1shows that any rank 0 finite abelian constant subgroup A of G arises from rank 0abelian subgroup A of K , so the list of Borel-Friedman-Morgan [BFM, § f , f , f and f describedabove. Given two rank 0 finite abelian constant subgroups A and A ′ of G arisingrespectively from rank 0 abelian subgroups A, A ′ of K , it remains to check that A ( C )and A ′ ( C ) are G ( C )–conjugate if and only if A and A ′ are K -conjugate. But this isobvious since the subgroups from the list are distinct as groups. We investigate nowthe normalizers and centralizers. Claim 12.4.
Let A ⊂ K be a rank zero subgroup. Then N K ( A ) = N G ( A )( C ) , Z K ( A ) = Z G ( A )( C ) . Indeed Lemma 12.1.(2) shows the existence of an element g ∈ G ( C ) such that g A ⊂ K and N K ( g A ) = N G ( g A )( C ) , Z K ( A ) = Z G ( g A )( C ) . But A and g A are K –conjugate by Borel-Friedman-Morgan’s theorem, so the samefact holds for g = 1.It is then enough to know the quotient “normalizer/centralizer” in the compactgroup case. For each relevant d , we have an exact sequence of groups1 → Z K (cid:0) ( Z /d Z ) (cid:1) → N K (cid:0) Z /d Z ) (cid:1) θ → GL ( Z /d Z )and we want to determine the image of θ . Denote by S d the set of K -conjugacyclasses of rank zero triples of K of order d . Since such a triple generates a rankzero abelian subgroup of order d of K , the set S d is covered by rank zero triplesinside ( Z /d Z ) , namely GL ( Z /d Z )-conjugates of the standard triple (1 , , GL ( Z /d Z ) / Im( θ ) ∼ = S d . Proposition 5.1.5 of [BFM] states that the K -conjugacy classes of rank zero triples of K of order d consists of the classes f d (1 , , i )for i = 1 , .., d − i prime to d . Hence the image of θ in GL ( Z /d Z ) is exactly SL ( Z /d Z ) as desired. 99iven f d : ( Z /d Z ) → G as above consider the map f d, ∗ : (cid:0) R × / ( R × ) d (cid:1) ≃ H (cid:0) R , ( Z /d Z ) (cid:1) → H ( R , G ) . A class (cid:0) ( x ) , ( y ) , ( z ) (cid:1) ∈ (cid:0) R × / ( R × ) d (cid:1) will for convenience simply be written as( x, y, z ). Corollary 12.5. (1) The set H loop ( R , F ) an consists of the classes of f , ∗ ( t , t , t ) and f , ∗ ( t , t , t ) .(2) The set H loop ( R , E ) an consists of the classes of f , ∗ ( t , t , t ) , f , ∗ ( t , t , t ) .(3) The set H loop ( R , E ) an consists in the classes of f , ∗ ( t , t , t i ) for i = 1 , , , , f , ∗ ( t , t , t ) and f , ∗ ( t , t , t ) .Proof. We do in detail the case of F , the other cases being similar. The set H loop ( R , F ) an is covered by the image of the anisotropic loop cocycles φ : π ( R ) → F ( C ). Theimage of such a φ is an anisotropic finite abelian subgroup of F , so Lemma 12.2.1allows us to assume that its image is the subgroup ( Z / Z ) . Furthermore, we knowthat two such homomorphisms φ and φ ′ have the same image in H loop ( R , F ) an ifand only if there exists g ∈ F ( C ) such that gφg − = φ ′ , or equivalently if there exists g ∈ N F (cid:0) ( Z / Z ) (cid:1) ( C ) such that gφg − = φ ′ . Note the importance of the isomorphism N F (( Z / Z ) ) / Z F (( Z / Z ) ) ≃ SL ( Z / Z ).Rephrasing what has been said in terms of the mapping f , ∗ , we see that H loop ( R , F ) an is the image under f , ∗ of the classes ( x, y, z ) where x, y, z ∈ R × are such that ( x, y, z )generates R × / ( R × ) ; furthermore, two such classes ( x, y, z ) and ( x ′ , y ′ , z ′ ) have thesame image in H loop ( R , F ) an if and only if there exists τ ∈ SL ( Z / Z ) such that( x ′ , y ′ , z ′ ) = τ ∗ (cid:0) ( x, y, z ) (cid:1) . We conclude that H loop ( R , F ) an consists of the classes of f , ∗ ( t , t , t ) and f , ∗ ( t , t , t ). Corollary 12.6. (1) The set GL ( Z ) \ H loop ( R , F ) an consists of the class of f , ∗ ( t , t , t ) .(2) The set GL ( Z ) \ H loop ( R , E ) an consists of the class f , ∗ ( t , t , t ) .(3) The set GL ( Z ) \ H loop ( R , E ) an consists of the classes of f , ∗ ( t , t , t ) and f , ∗ ( t , t , t ) . Remark 12.7.
The above Corollary gives the full classification of nullity 3 anisotropicmultiloop algebras of absolute type F or E .
13 The case of PGL d For any base scheme X , the set H ( X , PGL d ) classifies the isomorphism classesof Azumaya O X -algebras A of degree d , i.e. O X -algebras which are locally isomorphicfor the ´etale topology to the matrix algebra M d ( O X ) [Gr2] and [K, § III].100he exact sequence 1 → G m → GL d p −→ PGL d → X ) → H ( X , GL d ) → H ( X , PGL d ) δ −→ H ( X , G m ) = Br( X ) . We denote again by [ A ] ∈ Br( X ) the class of δ ([ A ]) in the cohomological Brauergroup.By [GP3, 3.1], we have an isomorphism Br( R n ) ∼ = Br( F n ). We look now at thediagram H loop ( R n , PGL d ) δ −−−→ Br( R n ) ∼ = y ∼ = y H ( F n , PGL d ) δ −−−→ Br( F n )where the bottom map is injective [GS, § Corollary 13.1.
The boundary map H loop ( R n , PGL d ) → Br( R n ) is injective. Azumaya R n –algebras whose classes are in H loop ( R n , PGL d ) are called loop Azu-maya algebras. They are isomorphic to twisted form of M d by a loop cocycle. Onecan rephrase the last Corollary by saying that loop Azumaya algebras of degree d areclassified by their “Brauer invariant”.Similarly, Wedderburn’s theorem [GS, 2.1] for F n –central simple algebras has itscounterpart. Corollary 13.2.
Let A be a loop Azumaya R n –algebra of degree d . Then there existsa unique positive integer r dividing d and a loop Azumaya R n –algebra B (unique upto R n –algebras isomorphism) of degree d/r such that A ≃ M r ( B ) and B ⊗ R n F n is adivision algebra. This reduces the classification of loop Azumaya R n -algebras to the “anisotropic”case, namely to the case of loop Azumaya R n –algebras A such that A ⊗ R n F n is adivision algebra.In the same spirit, the Brauer decomposition [GS, 4.5.16] for central F n –divisionalgebras yields the following. Corollary 13.3.
Write d = p m · · · p m l l . Let A be an anisotropic loop Azumaya R n -algebra of degree d . Then there exists a unique decomposition A ≃ A ⊗ R n · · · ⊗ R n A l where A i is an anisotropic k –loop Azumaya R n -algebra of degree p m i i for i = 1 , .., l . R n –algebra reduces the classification of anisotropic loop Azumaya R n -algebras of degree p m . Though the Brauer group of R n and F n are well understood, the understandingof H loop ( R n , PGL d ) an is much more delicate.We are given a loop cocycle φ = ( φ geo , z ) with values in PGL d ( k ). Set A = z ( M d ) . This is a central simple k –algebra such that z PGL d = PGL ( A ). Recall that φ geo is given by a k –group homomorphism φ geo : µµµ nm → PGL ( A ). To say that φ isanisotropic is to say that φ geo : µµµ nm → PGL ( A ) is anisotropic.We discuss in detail the following two special cases : the one-dimensional case,and the geometric case (i.e. k is algebraically closed). If k is algebraically closed H ( R , PGL d ) is trivial. The interesting new case iswhen k is not algebraically closed, e.g. the case of real numbers. Since the map H ( F , PGL d ) → Br( F ) is injective, as a consequence of Corollary 9.1, we have H ( R , PGL d ) ≃ H ( F , PGL d ) and the map(13.1) H ( R , PGL d ) → Br( R ) = Br( k ) ⊕ H ( k, Q / Z )is injective. Theorem 13.4.
The image of the map 13.1 consists of all pairs [ A ] ⊕ χ where A isa central simple algebra of degree d and χ : Gal( k s /k ) → Q / Z a character for whichthat there exists an ´etale algebra K/k of degree d inside A such that χ K = 0 . Remark 13.5.
The indices of such algebras over F are known ([Ti, prop. 2.4] inthe prime exponent case, and [FSS, Lemma 4.6] in the general case). The index of a F –algebra of invariant [ A ] ⊕ χ is deg( χ ) × ind k χ ( A ⊗ k k χ ) where k χ /k stands for thecyclic extension associated to χ .The proof needs some preparatory material from homological algebra based onCartier duality for groups of multiplicative type. More precisely, the dual of anextension of k –groups of multiplicative type1 → G m → E → µµµ m → → Z /m Z → b E → Z → . We have then an isomorphism
Ext k − gr ( µµµ m , G m ) ≃ Ext k ) ( Z , Z /m Z ) = H ( k, Z /m Z )102hich permits to attach to the first extension a character. Up to isomorphism, thereexists a unique extension E χ of µµµ m by G m of class [ χ ]. Lemma 13.6.
Let χ : Gal( k ) → Z /m Z be a character for some m ≥ .1. The boundary map k × / ( k × ) m ∼ −→ H ( k, µµµ m ) → H ( k, G m ) = Br( k ) is given by ( x ) χ ∪ ( x ) .2. Let K/k be an ´etale algebra. The following are equivalent:(a) There exists a morphism of extensions E χ → R K/k ( G m ) rendering thediagram −−−→ G m −−−→ E χ −−−→ µµµ m −−−→ y ∼ = y y −−−→ G m −−−→ R K/k ( G m ) −−−→ R K/k ( G m ) / G m −−−→ commutative.(b) χ K = 0 .Proof. (1) The cocharacter group b E χ is Z /m Z ⊕ Z together with the Galois action γ ( α, β ) = ( α + χ ( γ ) , β ). The Galois action on E χ ( k ) ≃ k × × µµµ m ( k ) is then given by γ ( y, ζ ) = (cid:16) γ ( y ) ζ χ ( γ ) , γ ( ζ ) (cid:17) for every γ ∈ Gal( k ). The class ( x ) ∈ H ( k, µµµ m ) is represented by the cocycle c γ = γ ( m √ x ) / m √ x . The element b γ = (1 , c γ ) ∈ E χ ( k ) lifts c γ . The boundary ∂ (cid:0) ( x ) (cid:1) ∈ H ( k, k × ) is then represented by the 2–cocycle a γ,τ = b γ × γ ( b τ ) b − γτ = c χ ( γ ) τ χ ( γ ) . c τ . (2) We decompose K = k × · · · × k l as a product of field extensions and denote by M j the cocharacter module of R k j /k ( G m ). Then the character module of R K/k ( G m )is M = ⊕ M j . By dualizing we are interested in morphism of extensions0 −−−→ I −−−→ M −−−→ Z −−−→ y y y ≃ −−−→ Z /m Z −−−→ ˆ E χ −−−→ Z −−−→ . Ext ( M j , Z /m Z ) = H ( k j , Z /m Z ) and the map Ext ( Z , Z /m Z ) → Ext ( M j , Z /m Z ) yields the restriction map H ( k, Z /m Z ) → H ( k j , Z /m Z ). It fol-lows that the bottom extension above is killed by the pull-back M j → Z , and thereforethat χ k j = 0 for j = 1 , .., l . This shows that ( a ) = ⇒ ( b ) . ( b ) = ⇒ ( a ): We assume that χ K = 0, namely χ k j = 0 for j = 1 , .., l . Hence b E χ belongs to the kernel of Ext ( Z , Z /m Z ) → Ext ( M j , Z /m Z ) for for j = 1 , .., l so b E χ belongs to the kernel of Ext ( Z , Z /m Z ) → Ext ( M, Z /m Z ). This means that themap M → Z of Galois modules lifts to b E χ → Z as desired.We can proceed now with the proof of Theorem 13.4. Proof.
We show first that the image of ∂ consists of pairs with the desired proper-ties. Again by Corollary 9.1, we have H loop ( R , PGL d ) = H ( R , PGL d ) and weare reduced to twisted algebras given by loop cocycles φ = ( φ geo , z ) with value in PGL d ( k ). Recall that A = z ( M d ) and that we have then a k –group homomorphism φ geo : µµµ m → PGL ( A ). We may assume that φ geo is injective. We pull back thecentral extension 1 → G m → GL ( A ) p −→ PGL ( A ) → φ geo and get a centralextension of algebraic k –groups1 → G m → E p ′ −→ µµµ m → E is a k –subgroup of GL ( A ). By extending the scalars to k , we see that E is commutative, hence is a k –group of multiplicative type. Then E is contained in amaximal torus of the k –group GL ( A ) and is of the form R K/k ( G m ) where K ⊂ A is an ´etale algebra of degree d . We have then the commutative diagram1 −−−→ G m −−−→ E −−−→ µµµ m −−−→ y ≃ y y −−−→ G m −−−→ R K/k ( G m ) −−−→ R K/k ( G m ) / G m −−−→ . Lemma 13.6.2 tells us that χ K = 0. We compute the Brauer class of this loop algebraby taking into account the commutative diagram H ( F , PGL d ) −−−→ Br( F ) τ z x ≀ ≀ x ?+[ A ] H (cid:0) F , PGL ( A ) (cid:1) ∂ −−−→ Br( F ) x x H ( F , µµµ m ) ∂ −−−→ Br( F ) . § I.5.7], whilethat of the bottom square is trivial. The image of ( t ) ∈ F × / ( F × ) × is χ ∪ ( t ) byLemma 13.6.1. The diagram yields the formula ∂ ([ φ ]) = [ A ] ⊕ χ which has therequired properties.Conversely, let K/k be an ´etale algebra of degree d inside A and let χ be acharacter such that χ K = 0. Let m be the order of χ ; by restriction-corestrictionconsiderations m divides d . Lemma 13.6.2 shows that there exists a morphism ofextensions E χ → R K/k ( G m ) . This yields a morphism ψ geo : µµµ m → R K/k ( G m ) / G m → PGL ( A ). The previous computation shows that the loop torsor ( ψ geo , z ) has Brauerinvariant [ A ] ⊕ χ .As an example, we consider the real case. Corollary 13.7.
Assume that k = R . Then the image of the injective map H ( R , PGL d ) → Br( R ) = Br( R ) ⊕ H ( R , Q / Z ) is as follows:1. ⊕ if d is odd;2. ⊕ , ⊕ χ C / R , [( − , − ⊕ and [( − , − ⊕ χ C / R if d is even. Remark 13.8.
In the case d = 2, the four classes under consideration correspondsto the quaternion algebras (1 , , t ), ( − , − − , t ). We assume that k is algebraically closed. According to Corollary 8.6.2, our goalis to extract information from the bijectionsHom gp (cid:0)b Z n , PGL d ( k ) (cid:1) irr / PGL d ( k ) ∼ −→ H loop ( R n , PGL d ) irr ∼ −→ H ( F n , PGL d ) irr . The right hand set is known from the work of Amitsur [Am], Tignol-Wadsworth [TiW]and Brussel [Br], the left hand-side is known by a classification due to Mumford[Mu, Prop. 3] See also [L], [Ne], [RY1, §
8] and [RY2]. These last two references relate to finite abelian constantsubgroups of
PGL d which have been investigated by Reichstein-Youssin in their construction ofdivision algebras with large essential dimension. [Ne] is more interested in the “quantum tori” pointof view and its relation to EALAs of absolute type A. PGL d from the knowledgeof the Brauer group of the field F n . Let us state first Mumford’s classification. If d = s ....s l ˆ s with s | s · · · | s l and s ≥
2, we consider the embedding
PGL s × · · · × PGL s l → PGL d and define the subgroup H ( s , ..., s l ) to be the image of the product of the standardanisotropic subgroups H ( s j ) = ( Z /s j Z ) of PGL s j for j = 1 , .., l defined by thegenerators a j = · · · · · ·
00 1 · · · · · · , b j = · · · ζ s j · · · · · · · · · ζ s j − s j . (13.2) Remark 13.9.
The way of expressing the group H ( s , s ) in the form H ( s ) × H ( s )is not unique when s and s are coprime. There is then a unique way to ar-range such a group H as H ( s ′ , ..., s ′ l ′ ) with s ′ | s ′ · · · | s ′ l and s ′ ≥
2. Note thatrank (cid:0) H ( s ′ , ..., s ′ l )( k ) (cid:1) = 2 l ′ .We can now state and establish the classification of irreducible finite abelian groupsof the projective linear group. Theorem 13.10. [Mu, Prop. 3] (see also [BL, § d = s × ... × s l if and only H ( s , ..., s l ) is irreducible in PGL d .2. If H is an irreducible finite abelian constant subgroup of PGL d , then H is PGL d ( k ) –conjugate to a unique H ( s , ..., s l ) with d = s ...s l , s | s · · · | s l and s ≥ . As mentioned above, our proof makes use of Galois cohomology results over R n for n ≥ F n ) collected from our previous paper [GP3].Our convention on the cyclic algebra ( t i , t j ) qp is that of Tate for the Azumaya R n –algebra with presentation X q = t i , Y q = t pj , Y X = ζ q XY. This is the opposite convention than that of [Br] and [GP3], but consistent with that of [GS]which we use in the proof. R n –algebras. Given sequences of length l ≤ [ n ]2 ≤ s · · · ≤ s l , r , · · · , r l we define A ( r , s , ..., r l , s l ) : ( t , t ) r s ⊗ R n · · · ⊗ R n ( t l − , t l ) r l s l . Lemma 13.11.
With the notation as above, set d = s ...s l and define φ : b Z n → H ( s )( k ) × · · · × H ( s l )( k ) = H ( s , ..., s l )( k ) ⊂ PGL d ( k ) , ( e , e , ..., e l − , e l ) ( − a , − r b , ..., − a l , − r l b l )
1. Then φ ( M d ) ∼ −→ A ( r , s , ..., r l , s l ) as R n –algebras.2. The following are equivalent:(a) A ( r , s , ..., r l , s l ) ⊗ R n F n is division F n –algebra;(b) φ is irreducible;(c) H ( s , ..., s l ) is irreducible in PGL d and ( r j , s j ) = 1 for j = 1 , ..., l .Proof. (1) This is done for R and each H ( s i ) in [GP2, proof of Th. 3.17]. This“extends” in an additive way to yield the general case.(2) The equivalence ( a ) ⇐⇒ ( b ) is a special case of [GP3, 3.1].( b ) = ⇒ ( c ): Since φ is irreducible, its image Im( φ ) is an irreducible subgroup of PGL d .This image is a product of the Im( φ i ) which are then irreducible in PGL s i . Accordingto [GP3, 3.13], we have then ( r j , s j ) = 1 for j = 1 , ..., l . Hence Im( φ ) = H ( s , ..., s l )is irreducible in PGL d .( c ) = ⇒ ( a ): Since H ( s , ..., s l ) is irreducible in PGL d , we have d = s ....s l . Thecondition ( r j , s j ) = 1 for j = 1 , ..., l implies that the algebra A ( r , s , ..., r l , s l ) ⊗ R n F n is division [Am, th. 3].We can now proceed with the proof of Theorem 13.10. Proof. (1) If d = s ...s l then A (1 , s , ..., , s l ) ⊗ R n F n is a division F n –algebra [Am,th. 3], so Lemma 13.11 shows that H ( s , ..., s n ) is irreducible in PGL d . If d = s ...s l ,then this algebra is not division and H ( s , ..., s n ) is reducible.(2) If H ( s , ..., s l ) is PGL d ( k )–conjugate to some H ( s ′ , ..., s ′ l ′ ), then H ( s , ..., s l ) isisomorphic to H ( s ′ , ..., s ′ l ′ ) as finite abelian group. So the divisibility conditions yield l = l ′ and s j = s ′ j for j = 1 , ..., l .The delicate points are existence and conjugacy. Let H be a finite abelian constantirreducible subgroup of PGL d . Denote by n the rank of H ( k ) and by m its exponent.107et us prove first that H is PGL d ( k )–conjugate to some H ( s , ..., s l ). We view H ( k ) as the image of an irreducible group homomorphism ψ : ( Z /m Z ) n → PGL d ( k ).Since k is algebraically closed ψ is a cocycle. The loop construction then definesan Azumaya R n –algebra of degree d such that A ⊗ R n F n is division (i.e. the group PGL ( A ) F n is anisotropic). Up to base change by a suitable element of GL n ( Z ),Theorem 4.7 of [GP3] provides an element g ∈ GL n ( Z ) such that g ∗ ( A ) ∼ = A ( r , s , ..., r l , s l )with ( r j , s j ) = 1 for j = 1 , .., l .By Lemma 6.1.1, A ( r , s , ..., r l , s l ) is the loop Azumaya algebra defined by themorphism φ : ˆ Z n → H ( s )( k ) × · · · × H ( s l )( k ) = H ( s , ..., s l )( k ) ⊂ PGL d ( k ) , ( e , e , ..., e l − , e l ) ( − a , − b , ..., − a l , − r l b l ) . which is then irreducible by the second statement of the same lemma. Theorem 7.9tells us that φ and ψ are PGL d ( k )–conjugate, hence H ( k ) is PGL d ( k )–conjugate to H ( s , ..., s l )( k ) = Im( ψ ). Since n = rank( H ( s , ..., s l )( k )), we have s | s ... | s l .We can now go back to Azumaya algebras. Theorem 13.12.
Let A be an anisotropic loop Azumaya R n –algebra of degree d.
1. There exists a sequence s , ..., s l and an integer r prime to s satisfying s | · · · | s l , ≥ s , d = s · · · s l , ( r , s ) = 1 and an element g ∈ GL n ( Z ) such that g ∗ ( A ) ∼ = A ( r , s , , s , , s , · · · , , s l ) ∼ = A ( − r , s , , s , , s , · · · , , s l ) . Such a sequence s , ..., s l is unique.2. If n = 2 l , ± r is unique modulo s .3. If n > l , g ∗ ( A ) ∼ = A (1 , s , , s , , s , · · · , , s l ) .Proof. (1) By definition, A is the twist of M d ( k ) by a morphism φ : ( Z /m Z ) n → PGL d ( k ). Since A ⊗ R n F n is division, φ is irreducible [GP2, th. 3.1]. Theorem 13.10shows that there exists a unique sequence s , ..., s l such that s | · · · | s l , ≥ s and Im( φ ) is PGL d ( k )–conjugate to H ( s , ..., s l ) := H ( s , ..., s l )( k ). Without lost ofgenerality, we can assume that Im( φ ) = H ( s , ..., s l ).108ecall that a , b , ..., a l , b l stand for the standard generators of H ( s , ..., s l ). Weshall use that Λ l (cid:0) H ( s , ..., s l ) (cid:1) ≃ Z /s Z generated by a ∧ b · · · a l ∧ b l [RY2, Lemma2.1], as well as the following invariant of φ ( ibid , 2.5) δ ( φ ) = φ ( e ) ∧ φ ( e ) ∧ · · · ∧ φ ( e n ) ∈ Λ n (cid:0) H ( s , ..., s l )( k ) (cid:1) This invariant has the remarkable property that a homomorphism φ ′ : ( Z /m Z ) n → H ( s , ..., s l )( k ) is GL n ( Z )–conjugate to φ if and only if δ ( φ ) = ± δ ( φ ′ ).We shall prove (1) together with (2) [resp. (3)] in the case n = 2 l (resp. n > l ). First case. n = 2 l : The family ( φ ( e ) , ..., φ ( e n )) generates H ( s , ..., s l ), and we con-sider the class [ r ♯ ] := φ ( e ) ∧ φ ( e ) ∧ · · · ∧ φ ( e n ) ∈ ( Z /s Z ) × . Let r be an inverse of r ♯ modulo s . We have φ ( r e ) ∧ φ ( e ) · · · ∧ φ ( e n ) = a ∧ b · · · a l ∧ b l so there exists g ∈ GL n ( Z ) such that ( ibid , 2.5) ψ ( r e ) = a , ψ ( e ) = b , · · · , φ ( e n − ) = a l , φ ( e n ) = b l where ψ = φ ◦ g . In terms of algebras, this means that g ∗ ( A ) ≃ A ( r , s , , s , , s , · · · , , s l ) . Let us first prove the uniqueness assertions The uniqueness of ( s , .., s l ) follows fromTheorem 13.10, hence (1) is proven. For (2), we are given then r ′ ∈ Z coprime to s , and an element h ∈ GL n ( Z ) such that h ∗ (cid:0) A ( r , s , , s , , s , · · · , , s l ) (cid:1) ≃ A ( r ′ , s , , s , , s , · · · , , s l ) . Denote by ψ ′ : ( Z /m Z ) n → H ( s , .., s n ) the group homomorphism defined by ψ ′ ( e ) = r ′ a , ψ ′ ( e ) = b , · · · , ψ ( e n ) = b l . Since the ( F n –anisotropic) loop algebras attachedto h ∗ ψ and ψ ′ are isomorphic, Theorem 7.9 provides an element u ∈ PGL d ( k ) suchthat ψ ◦ h = u ◦ ψ ′ . Since H ( s , ..., s l ) = Im( ψ ) = Im( ψ ′ ), it follows that u ∈ N PGL d ( k ) (cid:0) H ( s , ...s l ) (cid:1) .But the map u : H ( s , ..., s l )( k ) → H ( s , ...s l )( k ) preserves the symplectic pairing H ( s , ..., s l )( k ) × H ( s , ..., s l )( k ) → k × arising by taking the commutator of lifts in GL d ( k ). It follows that Λ n ( u ) = id ( ibid , 2.3.a) hence δ ( ψ ) = ± δ ( ψ ◦ h ) = ± δ ( u ◦ ψ ′ ) = ± δ ( u ◦ ψ ′ ) . r = ± r ′ ∈ Z /s Z as prescribed in (2) . Second case. n > l : For i = 2 l + 1 , ..., n we set c i = 0 ∈ H ( s , .., s l ) . Both fam-ilies (cid:0) φ ( e ) , . . . φ ( e n ) (cid:1) and ( r a , b · · · , a l , b l , c l +1 , · · · , c n ) generate H ( s , ..., s l ) andsatisfy φ ( e ) ∧ φ ( e ) · · · ∧ φ ( e n ) = ( r a ) ∧ b · · · a l ∧ b l ∧ c l +1 ∧ · · · ∧ c n ∈ Λ n (cid:0) H ( s , ..., s l ) (cid:1) = 0 . The same fact [RY2, Lemma 2.5] shows that there exists g ∈ GL n ( Z ) such that( g ∗ φ )( e ) = r a , ( g ∗ φ )( e ) = b , ( g ∗ φ )( e i − ) = a i and ( g ∗ φ )( e i ) = b i for i = 2 , .., l and ( g ∗ φ )( e i ) = c i for i = 2 l + 1 , ..., n . Therefore the preceding case with 2 l variablesyields the existence and the uniqueness of the s i ’s. It remains to prove (3), namelythat we can assume that r = 1. But this follows along the same lines of the argumentgiven above since ( r a ) ∧ b · · · a l ∧ b l ∧ c l +1 · · ·∧ c n = ( a ) ∧ b · · · a l ∧ b l ∧ c l +1 · · ·∧ c n ∈ Λ n (cid:0) H ( s , ..., s l ) (cid:1) . A To the Azumaya R n –algebra A ( r , s , ..., r l , s l ) we can attach (using the commu-tator [ x, y ] = xy − yx ) a Lie algebra over R n . We denote by L ( r , s , ..., r l , s l ) thederived algebra of this Lie algebra. It is a twisted form of sl d ( R n ) where d = s . . . s l . Corollary 13.13.
Let d be a positive integer. Let L be a nullity n loop algebra ofinner (absolute) type A d − .1. If L is not split, it is k –isomorphic to L ( r , s , , s , · · · , , s l ) where s | · · · | s l , ≥ s , d = s · · · s l , ( r , s ) = 1 and l ≤ h n i and such a sequence s , ..., s l is unique.2. If n = 2 l , r is unique modulo s and up to the sign.3. If n > l , L is k –isomorphic to L (1 , s , , s , · · · , , s l ) Proof.
The classification in question is given by considering the image of the naturalmap H loop ( R n , PGL d ) → GL n ( Z ) \ H ( R n , Aut ( PGL d ))The image can be identified with ( Z / Z × GL n ( Z )) \ H loop ( R n , PGL d ) where Z / Z actsby the opposite Azumaya algebra construction. Corollary 8.6 reduces the problem tothe “anisotropic case”. Theorem 13.12 determines the set GL n ( Z ) \ H loop ( R n , PGL d ) , and as it turns out the action of Z / Z is trivial. Therefore the desired classificationis also provided by GL n ( Z ) \ H loop ( R n , PGL d ) and we can now appeal to Theorem13.12 to obtain the Corollary. 110 Both the finite dimensional simple Lie algebras over k (nullity 0) and their affinecounterparts (nullity 1) have Coxeter-Dynkin diagrams attached to them that containa considerable amount of information about the algebras themselves. It has been along dream to find a meaningful way of attaching some kind of diagram to multiloop,or at least EALAs of arbitrary nullity (perhaps with as many nodes as the nullity).Our work shows that this can indeed be done and in a very natural way.Let us recall (see the Introduction for more details) the multiloop algebras basedon a finite dimensional simple Lie algebra g over an algebraically closed field k of char-acteristic 0 . Consider an n –tuple σσσ = ( σ , . . . , σ n ) of commuting elements of Aut( g )satisfying σ mi = 1 . For each n –tuple ( i , . . . , i n ) ∈ Z n we consider the simultaneouseigenspace g i ...i n = { x ∈ g : σ j ( x ) = ξ i j m x for all 1 ≤ j ≤ n } . The multiloop algebra L ( g , σσσ ) corresponding to σσσ is defined by L ( g , σσσ ) = ⊕ ( i ,...,i n ) ∈ Z n g i ...i n ⊗ t i m . . . t inm n ⊂ g ⊗ k R n,m ⊂ g ⊗ k R ∞ Recall that L ( A, σσσ ) , which does not depend on the choice of common period m, isnot only a k –Lie algebra (in general infinite-dimensional), but also naturally an R –algebra. It is when L ( g , σσσ ) is viewed as an R –algebra that Galois cohomology andthe theory of torsors enter into the picture. Indeed a rather simple calculation showsthat L ( g , σσσ ) ⊗ R n R n.m ≃ g ⊗ k R n,m ≃ ( g ⊗ k R n ) ⊗ R n R n,m . Thus L ( g , σσσ ) corresponds to a torsor E σσσ over Spec( R ) under Aut ( g ) . It is, however,the k –Lie algebra structure that is of interest in infinite-dimensional Lie theory andPhysics.Let G be the k –Chevalley group of adjoint type corresponding to g . Since
Aut ( G )and Aut ( g ) coincide we can also consider the twisted R n –group E G R n . By functori-ality and the definition of Lie algebra of a group functor in terms of dual numbers wesee that L ie ( E G R n ) = L ( g , σσσ ) . By the aciclicity Theorem to E G R n we can attach aWitt-Tits index, and this is the “diagram” that we attach to L ( g , σσσ ) as as Lie algebraover k. Note that by Corollary 8.12 this is well defined. The diagram carries theinformation about the absolute and relative type of L ( g , σσσ ) . The relative type as an invariant of L ( g , σσσ ) is defined in in § C is the centroid of L ( g , σσσ ) and e C denotes its field of quotients, then L ( g , σσσ ) ⊗ C e C is afinite dimensional central simple algebra over e C . As such it has an absolute and relative type. Thisconstruction applies to an arbitrary prime perfect Lie algebra which is finitely generated over itscentroid. k (( t )) one obtains precisely the diagrams of the affine algebras.
15 Appendix 1: Pseudo-parabolic subgroup schemes
We extend the definition of pseudo-parabolic subgroups of affine algebraic groups(Borel-Tits [BoT2], see also [Sp, § G which is offinite type and affine over a fixed base scheme X . We begin by establishing somenotation.We will denote by G m, X and G a, X the multiplicative and additive X -groups. Theunderlying schemes of these groups will be denoted by A × X and A X respectively. Afterapplying a base change X → X ′ we obtain corresponding X ′ -groups and schemes thatwe denote by G m, X ′ , G a, X ′ , A × X and A X ′ . The structure morphism of the X ′ –scheme A × X ′ gives by functoriality a group ho-momorphism(15.1) η X ′ : G ( X ′ ) → G ( A × X ′ )Let λ : G m, X → G be a cocharacter. By applying λ X ′ to the identity map id A × X ′ ∈ G m, X ′ ( A × X ′ ) we obtain an element of λ X ′ (id A × X ′ ) ∈ G ( A × X ′ ) . We have a natural group homomorphism G ( A X ′ ) → G ( A × X ′ ) . Given an element x ′ ∈ G ( A × X ′ ) we will write x ′ ∈ G ( A X ′ ) if x ′ is in the image of this map.After these preliminary definitions we are ready to define the three group functorsthat are relevant to the definition of pseudo-parabolic subgroups.Let Z G ( λ ) denote the centralizer of λ . Recall that this is the X -group functor thatto a scheme X ′ over X attaches the group(15.2) Z G ( λ )( X ′ ) = { x ′ ∈ G ( X ′ ) : x ′′ commutes with λ (cid:0) G m, X ( X ′′ ) (cid:1) ⊂ G ( X ′′ ) } where X ′′ is a scheme over X ′ and x ′′ denotes the image of x ′ under the natural grouphomomorphism G ( X ′ ) → G ( X ′′ ) . We consider the two following X –functors P ( λ )( X ′ ) = n g ∈ G ( X ′ ) | λ X ′ (id A × X ′ ) η X ′ ( g ) (cid:0) λ X ′ ( id A × X ′ ) (cid:1) − ∈ G ( A X ′ ) o In [CoGP] the groups that we are about to define are called limit subgroups . We have decided,since we are only dealing with analogues of pseudo-parabolic subgroups over fields, to abide by thisterminology. This material can in part be recovered from their work, but we have decided to includeit in the form that we needed for the sake of completeness. U ( λ )( X ′ ) = n g ∈ G ( X ′ ) | λ X ′ (id A × X ′ ) η X ′ ( g ) (cid:0) λ X ′ ( id A × X ′ ) (cid:1) − ∈ ker (cid:0) G ( A X ′ ) → G ( X ′ ) (cid:1) o for every X -scheme X ′ . The centralizer Z G ( λ ) is an X –subgroup functor of P ( λ ) whichnormalizes U ( λ ).We look at the previous definitions in the case when X = Spec( R ) and X ′ =Spec( R ′ ) are both affine. We have A R ′ = Spec( R ′ [ x ]) and A × R ′ = Spec( R ′ [ x ± ]) . Then x ∈ R ′ [ x ± ] × = G m,R ′ ( R ′ [ x ± ]) = G m,R ′ ( A × R ′ ) , and by applying our cocharacterwe obtain an element λ R ′ ( x ) ∈ G ( R ′ ) . Under Yoneda’s correspondence G m,R ′ ( R ′ [ x ± ]) ≃ Hom R ′ ( R ′ [ x ± ] , R ′ [ x ± ]) our element x corresponds to the identity map, namely tothe element id A × R ′ ∈ G m,R ′ ( A × R ′ ) if we rewrite our ring theoretical objects in terms ofschemes. We thus have P ( λ )( R ′ ) = n g ∈ G ( R ′ ) | λ R ′ ( x ) η R ′ ( g ) (cid:0) λ R ′ ( x ) (cid:1) − ∈ G ( R ′ [ x ]) o and U ( λ )( R ′ ) = n g ∈ G ( R ′ ) | λ R ′ ( x ) η R ′ ( g ) (cid:0) λ R ′ ( x ) (cid:1) − ∈ ker (cid:0) G ( R ′ [ x ]) → G ( R ′ ) (cid:1) o where η R ′ ( g ) is the natural image of g ∈ G ( R ′ ) in G ( R ′ [ x ± ]) , and the group homo-morphism G ( R ′ [ x ]) → G ( R ′ ) comes from the ring homomorphism R ′ [ x ] → R ′ thatmaps x to 0 . n, Z . Assume S = Spec( Z ) and let G denote the general linear group GL n, Z over Z . We let T denote the standard maximal torus of G . Let λ : G m, Z → T ֒ → G bea cocharacter of G that factors through T . We review the structure of the groups Z G ( λ ) , P ( λ ) and U ( λ ) . After replacing λ by int( θ ) ◦ λ for some suitable θ ∈ G ( Z ) we may assume thatthere exists (unique) integers 1 ≤ ℓ < ℓ < · · · < ℓ j ≤ n and e , . . . , e n such that e i = e j if ℓ k ≤ i, j < ℓ k +1 for some ke ℓ k +1 > e ℓ k for all 1 ≤ k ≤ j so that the functor of points of our map λ : G m, Z → T is given by λ R : G m, Z ( R ) −→ T ( R ) As customary we write G m,R instead of G m, X ... ∗ ) r r e
0. . .0 r e n for any (commutative) ring R and for all r ∈ G m, Z ( R ) = R × . At the level of coordinate rings if G m, Z = Spec (cid:0) Z [ x ± ] (cid:1) and T = Spec (cid:0) Z [ t ± , . . . , t ± n ] (cid:1) , then λ corresponds (under Yoneda) to the ring homomorphism λ ∗ : Z [ t ± , . . . , t ± n ] −→ Z [ x ± ]given by λ ∗ : t i x e i . From this it follows that Z G ( λ )( R ) consists of block diagonal matrices inside GL n ( R ) of size ℓ , . . . , ℓ j . Note that one “cannot see” this by looking at the centralizerof λ (cid:0) G m, Z ( R ) (cid:1) inside G ( R ) . This is clear, for example, if n = 2 , R = Z , j = 1 , ℓ = 1and 1 = e < e = 3 . The easiest way to eliminate “naive” contralizers in Z G ( λ )( R )is to look at their image in G (cid:0) R [ x ± ] (cid:1) . In fact
Lemma 15.1.
With the above notation we have Z G ( λ )( R ) = (cid:8) A ∈ G ( R ) ⊂ G ( R [ x ± ]) : A commutes with λ R (cid:0) G m, Z ( R [ x ± ]) (cid:1) (cid:9) . Proof.
The inclusion ⊂ follows from the definition of Z G ( λ ) . Conversely suppose that A ∈ G ( R ) is not an element of Z G ( λ )( R ) . Then there exists a ring homomorphism R → S and an element s ∈ S × such that the image of A in G ( S ) does not commutewith the diagonal matrix λ S ( s ) = s e . . . s e n . But then A, viewed now as an element of G (cid:0) R [ x ± ] (cid:1) cannot commute with λ R [ x ± ] ( x ) = x e . . . x e n . For if it did, we would reach a contradiction by functoriality considerations appliedto the (natural) ring homomorphism R [ x ± ] → S that maps x to s. Z G ( λ ) . If j = 1 and ℓ = n then Z G ( λ ) = G . At the other extreme if j = n then λ is regularand Z G ( λ ) = T . In all cases we see from the diagonal block description that Z G ( λ )is a closed subgroup of G (in particular affine).We now turn our attention to P ( λ ) and U ( λ ) . By using ( ∗ ) one immediately seesthat P ( λ )( R ) = { A = ( a ij ) ∈ G ( R ) : a ji = 0 if e i > e j } . Thus the P ( λ ) are the standard parabolic subgroups of G . Example 15.2.
We illustrate with the case n = 5 with j = 2 and ℓ = 2 , ℓ =5 , e = 1 , e = 3 . Then A = ( a ij ) ∈ GL ( R ) is of the form A = (cid:18) × + − × (cid:19) . We have two blocks, the top left of size 2 and the bottom right of size 3 . Given A = ( a ij ) ∈ GL ( R ) ⊂ GL ( R [ x ± ]) define P by x x x x x A x − x − x − x − x − = P that is λ ( x ) Aλ ( x ) − = P where P = ( p ij ) and p ij = P p ijk x k ∈ R [ x ± ] . To belong to P ( λ ) the element A mustbe such that p ijk = 0 for k < . This forces all entries in the 3 × − to vanish. For the elements in Z G ( λ )( R ) both blocks − and + must vanish.It is easy to determine that if A ∈ P ( λ ) the matrix P is such that the p ij = a ij ∈ R whenever 1 ≤ i, j ≤ ≤ i, j ≤ . If, on the other hand, i ≤ < j then p ij = a ij x . This makes the meaning of U ( λ ) quite clear in general. If λ ( x ) Aλ ( x ) − ∈ G ( R [ x ])is mapped to the identity element of G ( R ) under the map R [ x ] → R which sends x a ii = 1 and a ij = 0 if e i > e j . That is U ( λ )( R ) = { A = ( a ij ) ∈ P ( λ ) : a ii = 1 and a ij = 0 if e i > e j } . In particular U ( λ ) is an unipotent subgroup of P ( λ ) . Lemma 15.3.
Assume that there exists locally for the f pqc -topology a closed embed-ding of G in a linear group scheme. Then1. the X –functor U ( λ ) (resp. P ( λ ) , resp. Z G ( λ ) ) is representable by a closedsubgroup scheme of G which is affine over X .
2. The geometric fibers of U ( λ ) are unipotent.3. P ( λ ) = U ( λ ) ⋊ Z G ( λ ) .4. Z G ( λ ) = P ( λ ) × G P ( − λ ) .5. P ( λ ) = N G (cid:0) P ( λ ) (cid:1) .Proof. The case of GL n,S : The question is local with respect to the fpqc topology,so we can assume then that X is the spectrum of a local ring R. Since all maximalsplit tori of the R –group GL n,R are conjugate under GL n ( R ) [SGA3, XXVI.6.16],we can assume that λ : G m,R → T R < GL n,R where T is the standard maximal torusof GL n, Z . Since Hom Z ( G m , T ) ≃ Hom R ( G m,R T R ), we can reduce our problem to thecase when R = Z , which has been already done in Example 15.1. General case: By f pqc -descent, we can assume that X is the spectrum of a ring R, and that weare given a R –group scheme homomorphism ρ : G → G ′ = GL n,R which is a closedimmersion.(1) Denote by P ′ ( λ ) and U ′ ( λ ) the R –subfunctors of G ′ attached to the cocharacter ρ ◦ λ . The identities P ( λ ) = P ′ ( λ ) × G ′ G and U ( λ ) = U ′ ( λ ) × G ′ G can be establishedby reducing to the case of G ′ = GL n,R . This reduces the representability questionsto the case when G = GL n,R considered above.(2) This follows as well from the GL n,R case.(3) We know that the result holds for G ′ . Let R ′ be a ring extension of R and let g ∈ G ( R ′ ). Then g = uz with u ∈ U ′ ( λ )( R ′ ) and z ∈ P ′ ( λ )( R ′ ). We have λ ( x ) g λ ( x ) − λ ( x ) u λ ( x ) − z ∈ G ( A R ′ ) . By specializing at 0, we get that z ∈ G ( R ′ ). Thus g ∈ Z G ( λ )( R ′ ) and u ∈ U ( λ )( R ′ ).We conclude that P ( λ ) = U ( λ ) ⋊ Z G ( λ ).(4) and (5) follows from the GL n,R case. This condition is satisfied if X is locally noetherian of dimension ≤ § X –group schemes. Trivial, in the terminology of [SGA3]. efinition 15.4. An X –subgroup of G is pseudo-parabolic if it is of the form P ( λ )for some X –group homomorphism G m,X → G . Proposition 15.5.
Let G be a reductive group scheme over X .1. Let λ : G m, X → G be a cocharacter. Then P ( λ ) is a parabolic subgroup schemeof G and Z G ( λ ) is a Levi subgroup of the X –group scheme P ( λ ) .2. Assume that X is semi-local, connected and non-empty. Then the pseudo parabolicsubgroup schemes of G coincide with the parabolic subgroup schemes of G . We shall use that this fact is known for reductive groups over fields [Sp, § Proof.
We can assume that X = Spec( R ) is affine.(1) The geometric fibers of P ( λ ) are parabolic subgroups. By definition [SGA3, § XXVI.1], it remains to show that P ( λ ) is smooth. The question is then local withrespect to the fpqc topology, so that we can assume that R is local and that G issplit. By Demazure’s theorem [SGA3, XXIII.4], we can assume that G arises by basechange from a (unique) split Chevalley group G over Z .We now reason along similar lines than the ones used in studying the GL n, Z caseabove. Let T ⊂ G be a maximal split torus. Since all maximal split tori of G areconjugate under G ( R ), we can assume that our cocharacter λ factors through T R .Since Hom Z ( G m, Z , T ) ∼ = Hom R ( G m,R , T R ), the problem again reduces to the casewhen R = Z and of G = G , and λ : G m, Z → T . By the field case, the morphism P ( λ ) → Spec( Z ) is equidimensional. Since Z is a normal ring and the geometricfibers are smooth, we can conclude by [SGA1, prop. II.2.3] that P ( λ ) is smooth andis a parabolic subgroup scheme of the Z –group G .The geometric fibers of P ( λ ) × S P ( − λ ) are Levi subgroups. By applying [SGA3,th. XXVI.4.3.2], we get that Z G ( λ ) = P ( λ ) × G P ( − λ ) is a Levi S -subgroup schemeof P ( λ ).(2) Using the theory of relative root systems [SGA3, § XXVI.7], the proof is the sameas in the field case.
16 Appendix 2: Global automorphisms of G–torsorsover the pro jective line
In this appendix there is no assumption on the characteristic of the base field k .Let G be a linear algebraic k –group such that G is reductive. One way to stateGrothendieck-Harder’s theorem is to say that the natural mapHom gp ( G m , G ) / G ( k ) → H Zar ( P , G )117hich maps a cocharacter λ : G m → G to the G -torsor E λ := ( − λ ) ∗ (cid:0) O ( − (cid:1) over P k where O ( −
1) stands for the Hopf bundle A k \ { } → P k , is bijective. We fix now a cocharacter λ : G m → G . We are interested in the twisted P k –groupscheme E λ ( G ) = Isom G ( E λ , E λ ) , as well as the abstract group E λ ( G )( P k ). This groupis the group of global automorphisms of the G –torsor E λ over P k . It has a concretedescription. Lemma 16.1. E λ ( G )( P k ) = G ( k [ t ]) ∩ λ ( t ) G ( k [ t − ]) λ ( t − ) .Proof. We recover P k by two affine lines U = Spec( k [ t ]) and U = Spec( k [ t − ]). TheHopf bundle is isomorphic to the twist of G m by the cocycle z ∈ Z ( U ⊔ U / P k , G m )where z , = 1, z , = t − , z , = t , z , = 1. Then λ ( z ) ∈ Z ( U ⊔ U / P k , G m ) is thecocycle of E λ . Hence E λ ( G )( P k ) = n ( g , g ) ∈ G ( U ) × G ( U ) | λ − ( z , ) .g = g o = G ( k [ t ]) ∩ λ ( t ) G ( k [ t − ]) λ ( t − ) . In the split connected case, this group has been computed by Ramanathan [Ra,prop. 5.2] and by the first author in the split case (see proposition II.2.2.2 of [Gi0]).We provide here the general case by computing the Weil restriction H λ = Y P k /k E λ ( G ) , which is known to be a representable by an algebraic affine k –group. Let P ( λ ) = U ( λ ) ⋊ Z G ( λ ) ⊂ G be the parabolic subgroup attached to λ (lemma 15.3).Denote by Z ( λ ) the center of Z G ( λ ). Then λ factors through Z ( λ ) and thisallows us to define the Z ( λ )–torsor S λ := ( − λ ) ∗ (cid:0) O ( − (cid:1) over P k . We can twist themorphism Z G ( λ ) → G by S λ , so we get a morphism Z G ( λ ) × k P k → E λ ( G ) and thena morphism Z G ( λ ) → H λ . Proposition 16.2.
The homomorphisms of k –groups (cid:16) Y P k /k S λ ( U ( λ )) (cid:17) ⋊ Z G ( λ ) → Y P k /k S λ (cid:0) P ( λ ) (cid:1) → H λ . are isomorphisms. Furthermore, Q P k /k S λ ( U ( λ )) is a unipotent k –group. This is not the usual way to state the theorem (see [Gi0, II.2.2.1]), but it is easy to derive theformulation that we are using. roof.
Write P , U for P ( λ ), U ( λ ). Y P k /k S λ ( P ) = Y P k /k S λ ( U ) ⋊ Y P k /k Z G ( λ ) = Y P k /k S λ ( U ) ⋊ Z G ( λ )so it remains to show that Q P k /k S λ ( P ) ∼ −→ H λ . Consider a faithful representation ρ : G → G ′ = GL n . Denote by P ′ the parabolic subgroup of G ′ attached to λ . Wehave P = G × G ′ P ′ , hence S λ ( P ) = E λ ( G ) × E λ ( G ′ ) S λ ( P ′ ). It follows that Y P k /k S λ ( P ) = Y P k /k E λ ( G ) × Q P k/k E λ ( G ′ ) Y P k /k S λ ( P ′ )as can be seen by reducing to the case of GL n already done in Example 15.1. Thiscase also shows that Q P k /k S λ ( U ( λ )) is a unipotent k –group. References [A] P. Abramenko,
Group actions on twin buildings , Bull. Belg. Math. Soc. SimonStevin (1996), 391–406.[Am] S.A. Amitsur, On central division algebras , Israel J. Math. (1972), 408–420.[AABFP] S. Azem, B. Allison, S. Berman, Y. Gao and A. Pianzola, Extended affineLie algebras and their root systems , Mem. Amer. Math. Soc. (1997) 128 pp.[ABP2] B.N. Allison, S. Berman and A. Pianzola,
Covering algebras II: Isomorphismof loop algebras , J. Reine Angew. Math. (2004) 39-71.[ABP2.5] B.N. Allison, S. Berman and A. Pianzola,
Iterated loop algebras , PacificJour. Math. (2006), 1–41.[ABP3] B.N. Allison, S. Berman and A. Pianzola,
Covering algebras III: Multiloopalgebras, Iterated loop algebras and Extended Affine ie Algebras of nullity 2. ,Preprint. arXiv:1002.2674[ABFP] B. Allison, S. Berman, J. Faulkner and A. Pianzola,
Realization of graded-simple algebras as loop algebras , Forum Mathematicum (2008), 395–432.[BMR] M. Bate, B. Martin and G. R¨ohrle, A geometric approach to complete re-ducibility , Invent. math , 177-218 (2005).119BL] C. Birkenhake and H. Lange,
Complex abelian varieties , Gr¨undlehren der math-ematischen Wissenschaften (2004), Springer.[BFM] A. Borel, R. Friedman and J.W. Morgan,
Almost commuting elements incompact Lie groups
Mem. Amer. Math. Soc. , (2002), no. 747.[Bo] A. Borel,
Linear Algebraic Groups (Second enlarged edition), Graduate text inMathematics (1991), Springer.[BM] A. Borel and G.D Mostow,
On semi-simple automorphisms of Lie algebras ,Ann. Math. (1955), 389-405.[BoT1] A. Borel and J. Tits, Groupes r´eductifs , Pub. Math. IHES , (1965), 55–152.[BoT2] A. Borel and J. Tits, Th´eor`emes de structure et de conjugaison pour lesgroupes alg´ebriques lin´eaires , C. R. Acad. Sci. Paris S´er. A-B (1978),55-57.[Bbk] N. Bourbaki,
Groupes et alg`ebres de Lie , Ch. 7 et 8, Masson.[BT1] F. Bruhat and J. Tits,
Groupes alg´ebriques sur un corps local , Pub. IHES (1972), 5–251.[BT2] F. Bruhat and J. Tits, Groupes alg´ebriques sur un corps local II. Sch´emas engroupes. Existence d’une donn´ee radicielle valu´ee , Pub. IHES (1984), 197–376.[BT3] F. Bruhat and J. Tits, Groupes alg´ebriques sur un corps local III. Compl´ementset application `a la cohomologie galoisienne , J. Fac. Sci. Univ. Tokyo (1987),671–698.[Br] E. S. Brussel, The division algebras and Brauer group of a strictly Henselianfield , J. Algebra (2001) 391–411.[CGP] V. Chernousov, P. Gille and A. Pianzola,
Torsors on the punctured line , toappear in American Journal of Math. (2011).[CGR1] V. Chernousov, P. Gille and Z. Reichstein,
Resolution of torsors by abelianextensions , Journal of Algebra (2006), 561–581.[CGR2] V. Chernousov, P. Gille and Z. Reichstein,
Resolution of torsors by abelianextensions II , Manuscripta Mathematica (2008), 465-480.[CTGP] J.–L. Colliot–Th´el`ene, P. Gille and R. Parimala,
Arithmetic of linear alge-braic groups over two-dimensional geometric fields , Duke Math. J. (2004),285-321. 120CTS] J.–L. Colliot–Th´el`ene and J.–J. Sansuc,
Fibr´es quadratiques et composantesconnexes r´eelles , Math. Annalen (1979), 105–134.[CoGP] B. Conrad, O. Gabber and G. Prasad,
Pseudo-reductive groups , CambridgeUniversity Press, 2010.[D] M. Demazure,
Sch´emas en groupes r´eductifs , Bull. Soc. Math. France (1965),369–413.[DG] M. Demazure and P. Gabriel, Groupes alg´ebriques , Masson (1970).[EGA IV] A. Grothendieck (avec la collaboration de J. Dieudonn´e),
El´ements deG´eom´etrie Alg´ebrique IV , Publications math´ematiques de l’I.H. ´E.S. no 20, 24,28 and 32 (1964 - 1967).[F] M. Florence,
Points rationnels sur les espaces homog`enes et leurs compactifica-tions , Transformation Groups (2006), 161-176.[FSS] B. Fein, D. Saltman and M. Schacher, Crossed products over rational functionfields , J. Algebra (1993), 454-493.[GMS] R.S. Garibaldi, A.A. Merkurjev and J.-P. Serre,
Cohomological invariants inGalois cohomology , University Lecture Series, 28 (2003). American MathematicalSociety, Providence.[Gi0] P. Gille,
Torseurs sur la droite affine et R -´equivalence , Th`ese de doctorat(1994), Universit´e Paris-Sud, on author’s URL.[Gi1] P. Gille, Torseurs sur la droite affine , Transform. Groups (2002), 231-245,errata dans Transform. Groups (2005), 267–269.[Gi2] P. Gille, Unipotent subgroups of reductive groups of characteristic p >
0, DukeMath. J. (2002), 307–328.[Gi3] P. Gille,
Serre’s conjecture II: a survey , Proceedings of the Hyderabad con-ference “Quadratic forms, linear algebraic groups, and cohomology”, Springer,41–56.[Gi4] P. Gille,
Sur la classification des sch´emas en groupes semi-simples , see author’sURL.[GM] G. van der Geer and B. Moonen,
Abelian varieties , book in preparation, secondauthor’s URL. 121GMB] P. Gille, L. Moret-Bailly, Actions alg´ebriques de groupes arithm´etiques, toappear in the procedings of the conference “Torsors, theory and applications”,Edinburgh (2011), Proceedings of the London Mathematical Society, edited byV. Batyrev and A. Skorobogatov.[GiQ] P. Gille and A. Qu´eguiner-Mathieu,
Formules pour l’invariant de Rost , to ap-pear in Algebra and Number Theory.[GP1] P. Gille and A. Pianzola,
Isotriviality of torsors over Laurent polynomials rings ,C. R. Acad. Sci. Paris, Ser. I (2005) 725–729.[GP2] P. Gille and A. Pianzola,
Galois cohomology and forms of algebras over Laurentpolynomial rings , Mathematische Annalen (2007), 497-543.[GP3] P. Gille and A. Pianzola,
Isotriviality and ´etale cohomology of Laurent polyno-mial rings , J. Pure Appl. Algebra (2008), 780–800.[GR] P. Gille and Z. Reichstein,
A lower bound on the essential dimension of a con-nected linear group , Commentarii Mathematici Helvetici (2009), 189-212.[GS] P. Gille and T. Szamuely, Lectures on the Merkurjev-Suslin’s theorem , Cam-bridge Studies in Advanced Mathematics (2006), Cambridge UniversityPress.[Gi] J. Giraud,
Cohomologie non ab´elienne , Die Grundlehren der mathematischenWissenschaften (1971), Springer-Verlag.[Gs] R. Griess,
Elementary abelian p-subgroups of algebraic groups , Geom. Ded. (1991), 253–305.[Gr1] A. Grothendieck, Technique de descente et th´eor`emes d’existence en g´eom´etriealg´ebrique. I. G´en´eralit´es. Descente par morphismes fid`element plats , S´eminaireBourbaki (1958-1960), Expos´e No. 190, 29 p.[Gr2] A. Grothendieck,
Le groupe de Brauer. II , Th´eorie cohomologique (1968), DixExpos´es sur la Cohomologie des Sch´emas pp. 67–87, North-Holland.[HM] G. Hochschild and D. Mostow,
Automorphisms of affine algebraic groups ,[KS] V.G. Kac and A.V Smilga,
Vacuum structure in supersymmetric Yang-Millstheories with any gauge group.
The many Faces of the Superworld, pp. 185234.World Sci. Publishing, River Edge (2000) J. Algebra (1969), 535–543.122K] M. A. Knus, Quadratic and hermitian forms over rings , Grundlehren der mat.Wissenschaften (1991), Springer.[L] E. Landvogt,
Some functorial properties of the Bruhat-Tits building , J. reineangew. math. (2000), 213–241.[Lam] T.Y. Lam, Serre’s problem on projective modules, Springer Monographs inMathematics, Springer, Berlin, Heidelberg, New York, 2006.[Mg] B. Margaux,
Passage to the limit in non-abelian ˇCech cohomology , Journal ofLie Theory (2007), 591–596.[Mg2] B. Margaux, Vanishing of Hochschild Cohomology for Affine Group Schemesand Rigidity of Homomorphisms between Algebraic Groups , Doc. Math. (2009), 653-672.[Mt] G. D. Mostow, Fully reducible subgroups of algebraic groups , Amer. J. Math. (1956), 200–221.[M] J.S. Milne, Etale cohomology , Princeton University Press (1980).[Mu] D. Mumford,
On the equations defining abelian varieties. I. , Invent. Math. (1966), 287–354.[NSW] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields,second edition , Grundlehren des math. Wiss. 323 (2008), Springer.[N] Y. A. Nisnevich,
Espaces homog`enes principaux rationnellement triviaux etarithm´etique des sch´emas en groupes r´eductifs sur les anneaux de Dedekind , C.R. Acad. Sci. Paris S´er. I Math. (1984), 5–8.[Ne] K.H. Neeb,
On the classification of rational quantum tori and the structure oftheir automorphism groups , Canad. Math. Bull. (2008), 261–282.[OV] A.L. Onishik and B.E. Vinberg, Lie groups and algebraic groups , Springer(1990).[PR] V. Platonov and A. Rapinchuk,
Algebraic Groups and Number Theory , Aca-demic Press (1993).[P] P. Pansu,
Superrigidit´e g´eom´etrique et applications harmoniques , S´eminaires etcongr`es (2008), Soc. Math. France, 375-422.[P1] A. Pianzola, Affine Kac-Moody Lie algebras as torsors over the punctured line ,Indagationes Mathematicae N.S. (2) (2002) 249-257.123P2] A. Pianzola, Vanishing of H for Dedekind rings and applications to loop alge-bras , C. R. Acad. Sci. Paris, Ser. I (2005), 633-638.[P3] A. Pianzola, On automorphisms of semisimple Lie algebras
Algebras, Groups,and Geometries (1985) 95–116.[Pr] G. Prasad, Galois-fixed points in the Bruhat-Tits building of a reductive group ,Bull. Soc. Math. France , (2001), 169–174.[Ra] A. Ramanathan,
Deformations of principal bundles on the projective line , Invent.Math. (1983), 165–191.[RZ] L. Ribes and P. Zaleski, Profinite Groups , Ergebnisse der Mathematik und ihrerGrenzgebiete vol. 40, Springer (2000).[Ri] R. W. Richardson,
On orbits of algebraic groups and Lie groups , Bull. Austral.Math. Soc. (1982), 1–28.[Ro] G. Rousseau, Immeubles des groupes r´eductifs sur les corps locaux , Th`ese, Uni-versit´e de Paris-Sud (1977).[RY1] Z. Reichstein and B. Youssin
Essential Dimensions of Algebraic Groups anda Resolution Theorem for G-varieties , with an appendix by J. Koll´ar and E.Szab´o , Canada Journal of Math. (2000), 265-304.[RY2] Z. Reichstein and B. Youssin A birational invariant for algebraic group actions ,Pacific J. Math. (2002), 223–246.[Sa] J.-J. Sansuc,
Groupe de Brauer et arithm´etique des groupes alg´ebriques lin´eairessur un corps de nombres , J. reine angew. Math. (Crelle) (1981), 12–80.[Sc] W. Scharlau,
Quadratic and hermitian forms , Grundlehren der math. Wiss. (1985), Springer.[Se1] J.-P. Serre,
Cohomologie Galoisienne , cinqui`eme ´edition r´evis´ee et compl´et´ee,Lecture Notes in Math. , Springer-Verlag.[Se2] J.-P. Serre, Cohomologie galoisienne: progr`es et probl`emes , S´eminaire BourbakiExp. No. 783 (1994), Ast´erisque (1995), 229257.[Sp] T.-A. Springer,
Linear algebraic groups , second edition (1998), Birkha¨user.[SZ] A Steinmetz-Zikesch,
Alg`ebres de Lie de dimension infinie et th´eorie de la de-scente , to appear in M´emoire de la Soci´et´e Math´ematique de France.124SGA1]
S´eminaire de G´eom´etrie alg´ebrique de l’I.H.E.S., Revˆetements ´etales etgroupe fondamental, dirig´e par A. Grothendieck , Lecture Notes in Math. 224.Springer (1971).[SGA3]
S´eminaire de G´eom´etrie alg´ebrique de l’I.H.E.S., 1963-1964, sch´emas engroupes, dirig´e par M. Demazure et A. Grothendieck , Lecture Notes in Math.151-153. Springer (1970).[SGA4]
Th´eorie des topos et cohomologie ´etale des sch´emas. Tome 2 , S´eminairede G´eom´etrie Alg´ebrique du Bois-Marie 1963–1964, dirig´e par M. Artin, A.Grothendieck et J. L. Verdier, Lecture Notes in Mathematics (1972),Springer-Verlag.[Ti] J.-P. Tignol,
Alg`ebres ind´ecomposables d’exposant premier , Adv. in Math. (1987), 205 228.[TiW] J.-P. Tignol and A. R. Wadsworth Totally ramified valuations on finite-dimensional division algebras , Trans. Amer. Math. Soc. (1987), 223–250.[T1] J. Tits,
Classification of algebraic semisimple groups , Algebraic Groups andDiscontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965),33–62 Amer. Math. Soc. (1966).[T2] J. Tits,
Reductive groups over local fields , Proceedings of the Corvallis conferenceon L - functions etc., Proc. Symp. Pure Math. (1979), part 1, 29-69.[T3] J. Tits, Twin buildings and groups of Kac-Moody type , Groups, combinatoricsand geometry (Durham, 1990), 249–286, London Math. Soc. Lecture Note Ser. (1992), Cambridge Univ. Press.[V] A. Vistoli,
Grothendieck topologies, fibered categories and descent theory , Fun-damental algebraic geometry, 1–104, Math. Surveys Monogr. (2005), Amer.Math. Soc.[W] C. Weibel,
An introduction to homological algebra , Cambridge studies in a ad-vanced mathematics38