Torsors, Reductive Group Schemes and Extended Affine Lie Algebras

Philippe Gille^{1} and Arturo Pianzola^{2,3}

1UMR 8553 du CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France.

2Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada.

3Centro 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 algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that we take draws heavily for the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows us to find a bridge between multiloop algebras and the work of J. Tits on reductive groups over complete local fields.

Keywords: Reductive group scheme, torsor, multiloop algebra. Extended Affine Lie Algebras.

MSC 2000 17B67, 11E72, 14L30, 14E20.

Contents

1 Introduction 1

2 Generalities on the algebraic fundamental group, torsors, and reductive group schemes 5 2.1 The fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 An example: Laurent polynomials in characteristic 0 . . . . . . . . . . . . . 9 2.4 Reductive group schemes: Irreducibility and isotropy . . . . . . . . . . . . . 13 3 Loop, finite and toral torsors 14

3.1 Loop torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Loop reductive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Loop torsors at a rational base point . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Finite torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Toral torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Semilinear considerations 23 4.1 Semilinear morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Semilinear morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Case of affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.4 Group functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.5 Semilinear version of a theorem of Borel-Mostow . . . . . . . . . . . . . . . 30 4.6 Existence of maximal tori in loop groups . . . . . . . . . . . . . . . . . . . . 33 4.7 Variations of a result of Sansuc. . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Maximal tori of group schemes over the punctured line 38 5.1 Twin buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 Proof of Theorem ?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Internal characterization of loop torsors and applications 44 6.1 Internal characterization of loop torsors . . . . . . . . . . . . . . . . . . . . 44 6.2 Applications to (algebraic) Laurent series. . . . . . . . . . . . . . . . . . . . 49

7 Isotropy of loop torsors 51 7.1 Fixed point statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.2 Case of flag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.3 Anisotropic loop torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8 Acyclicity 63 8.1 The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 8.2 Application: Witt-Tits decomposition . . . . . . . . . . . . . . . . . . . . . 65 8.3 Classification of semisimple k–loop adjoint groups . . . . . . . . . . . . . . . 66 8.4 Action of GL_{n}(ℤ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

9 Small dimensions 79 9.1 The one dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.2 The two dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9.2.1 Classification of semisimple loop R_{2}-groups. . . . . . . . . . . . . . . 80 9.2.2 Applications to the classification of EALAs in nullity 2. . . . . . . . 85 9.2.3 Rigidity in nullity 2 outside type A. . . . . . . . . . . . . . . . . . . 88 9.2.4 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 10 The case of orthogonal groups 93

11 Groups of type G_{2} 95

12 Case of groups of type F_{4}, E_{8} and simply connected E_{7} in nullity 3 96

13 The case of PGL_{d} 99 13.1 Loop Azumaya algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 13.2 The one dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 13.3 The geometric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 13.4 Loop algebras of inner type A . . . . . . . . . . . . . . . . . . . . . . . . . . 109

14 Invariants attached to EALAs and multiloop algebras 110

15 Appendix 1: Pseudo-parabolic subgroup schemes 111 15.1 The case of GL_{n,ℤ}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 15.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

16 Appendix 2: Global automorphisms of G–torsors over the projective line116

1 Introduction

To our good friend Benedictus Margaux

Many interesting infinite dimensional Lie algebras can be thought as being “finite dimensional” when viewed, not as algebras over the given base field, but rather as algebras over their centroids. From this point of view, the algebras in question look like “twisted forms” of simpler objects with which one is familiar. The quintessential example 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 (or EALAs for short), can roughly be thought of as higher nullity analogues of the affine Kac-Moody Lie algebras. Once the twisted form point of view is taken the theory of reductive group schemes developed by Demazure and Grothendieck [SGA3] arises naturally.

Two key concepts which are common to [GP2] and the present work are those of a twisted form of an algebra, and of a multiloop algebra. At this point we briefly recall what these objects are, not only for future reference, but also to help us redact a more comprehensive Introduction.

***

Unless specific mention to the contrary throughout this paper k will denote a field of characteristic 0, and \overline{k} a fixed algebraic closure of k. We denote k–alg the category of associative unital commutative k–algebras, and R and object of k–alg. Let n ≥ 0 and m > 0 be integers that we assume are fixed in our discussion. Consider the Laurent polynomial rings R = R_{n} = k[t_{1}^{±1}, …, t_{n}^{±1}] and R′ = R_{n,m} = k[t_{1}^{± \frac{1}{m}} , …, t_{n}^{± \frac{1}{m}}]. For convenience we also consider the direct limit R′_{∞} = \underrightarrow{lim}R_{n,m} taken over m which in 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 étale. If k is algebraically closed this extension is Galois and plays a crucial role in the study of multiloop algebras. The explicit description of Gal(R′/R) is given below.

Let A be a k–algebra. We are in general interested in understanding forms (for the fppf-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 most interest to us is when S can be taken to be a Galois extension R′ of R of Laurent polynomial algebras described above.^{1}

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}^{1}(S/R, Aut(A)).

The right hand side is the part “trivialized by S” of the pointed set of non-abelian cohomology 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}^{1}(S/R, Aut(A)) with the “usual” Galois cohomology set H^{1}( Gal(S/R), Aut(A)(S)) 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_{me}^{e} = ξ_{m} for all e > 0. We can then identify Gal(R′/R) with (ℤ/mℤ)^{n} where for each e = (e_{1}, …, e_{n}) ∈ ℤ^{n} the corresponding

element \overline{e} = (\overline{e}_{1}, ⋯, \overline{e}_{n}) ∈ Gal(R′/R) acts on R′ via \overline{e}ti^{\frac{1}{m}} = ξ_{m}^{e_{i}}ti^{\frac{1}{m}} .

The primary example of forms L of A ⊗_{k} R which are trivialized by a Galois extension R′/R as above are the multiloop algebras based on A. These are defined as follows. Consider an n–tuple σ σ σ = (σ_{1}, …, σ_{n}) of commuting elements of Aut_{k}(A) satisfying σ_{i}^{m} = 1. For each n–tuple (i_{1}, …, i_{n}) ∈ ℤ^{n} we consider the simultaneous eigenspace A_{i_{1}...i_{n}} = {x ∈ A : σ_{j}(x) = ξ_{m}^{i_{j}}x for all 1 ≤ j ≤ n}. Then A = ∑ A_{i_{1}...i_{n}}, and A = ⨁ A_{i_{1}...i_{n}} if we restrict the sum to those n–tuples (i_{1}, …, i_{n}) for which 0 ≤ i_{j} < m_{j}.

1The 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 finite dimensional simple Lie algebra.

The multiloop algebra corresponding to σ σ σ, commonly denoted by L(A,σ σ σ), is defined by

L(A,σ σ σ) = ⊕_{(i_{1},...,i_{n})∈ℤ^{n}} A_{i_{1}...i_{n}} ⊗ t^{\frac{i_{1}}{m}} … tn^{\frac{in}{m}} ⊂ A ⊗_{k} R′ ⊂ A ⊗_{k} R′_{∞}

Note that L(A,σ σ σ), which does not depend on the choice of common period m, is not only a k–algebra (in general infinite dimensional), but also naturally an R–algebra. It is when L(A,σ σ σ) is viewed as an R–algebra that Galois cohomology and the theory of torsors enter into the picture. Indeed a rather simple calculation shows that

L(A,σ σ σ) ⊗_{R} R′ ≃ A ⊗_{k} R′ ≃ (A ⊗_{k} R) ⊗_{R} R′.

Thus L(A,σ σ σ) corresponds to a torsor over Spec(R) under Aut(A).

When n = 1 multiloop algebras are called simply loop algebras. To illustrate our methods, let us look at the case of (twisted) loop algebras as they appear in the theory of affine Kac-Moody Lie algebras. Here n = 1, k = ℂ and A = 𝔤 is a finite-dimensional simple Lie algebra. Any such L is naturally a Lie algebra over R := ℂ[t^{±1}] and L ⊗_{R} S ≃ 𝔤 ⊗_{ℂ} S ≃ (𝔤 ⊗_{ℂ} R) ⊗_{R} S for some (unique) 𝔤, and some finite étale extension S/R. In particular, L is an S/R–form of the R–algebra 𝔤 ⊗_{ℂ} R, with respect to the étale topology of Spec(R). Thus L corresponds to a torsor over Spec(R) under Aut(𝔤) whose isomorphism class is an element of the pointed set H_{ét}^{1}(R, Aut(𝔤)). We may in fact take S to be R′ = ℂ[t^{± \frac{1}{m}}].

Assume that A is a finite-dimensional. The crucial point in the classification of forms of A ⊗_{k} R by cohomological methods is the exact sequence of pointed sets

(1.3)  H_{ét}^{1}(R, Aut^{0}(A)) → H_{ét}^{1}(R, Aut(A)) ⟶^{ψ} H_{ét}^{1}(R, Out(A)),

where Out(A) is the (finite constant) group of connected components of the algebraic k–group Aut(A).^{2}

Grothendieck’s theory of the algebraic fundamental group allows us to identify H_{ét}^{1}(R, Out(A)) with the set of conjugacy classes of n–tuples of commuting elements of the corresponding finite (abstract) group Out(A) (again under the assumption that k is algebraically closed). This is an important cohomological invariant attached to any twisted form of A⊗_{k}R. We point out that the cohomological information is always about the twisted forms viewed as algebras over R (andnot k). In practice, as the affine Kac-Moody case illustrates, one is interested in understanding these algebras as objects over k (and not R). A technical tool (the centroid trick) developed and used in [ABP2] and [GP2] allows us to compare k vs R information.

2Strictly speaking we should be using the affine R–group scheme Aut(A ⊗_{k} R) instead of the algebraic k–group Aut(A). This harmless and useful abuse of notation will be used throughout the paper.

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^{1} for reductive group schemes over certain Dedekind rings which includes k[t^{±1}]. This result can be thought of as an analogue of “Serre Conjecture I” for some very special rings of dimension 1. It follows from what has been said that we can attach a conjugacy 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 twisted loop algebras) of A are trivial, i.e. isomorphic to A⊗_{k}R as R–algebras. This yields the classification 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 completely different approach than the classical definition by generators and relations).

Surprisingly enough the analogue of “Serre Conjecture II” for k[t_{1}^{±1}, t_{2}^{±1}] fails, as explained in [GP2]. The single family of counterexamples known are the the socalled Margaux algebras. The classification of forms in nullity 2 case is in fact quite interesting and challenging. Unlike the nullity one case there are forms which are not multiloop algebras (the Margaux algebra is one such example). The classification in nullity 2 by cohomological methods, both over R and over k, will be given in §9 as an application of one of our main results (the Acyclicity Theorem). This classification (over k but not over R) can also be attained entirely by EALA methods [ABP3]. The two approaches complement each other and are the culmination of a project started a decade ago. We also provide classification results for loop Azumaya algebras in §13.

Questions related to the classification and characterization of EALAs in arbitrary nullity are at the heart of our work. In this situation A = 𝔤 is a finite dimensional simple Lie algebra over k. The twisted forms relevant to EALA theory are always multiloop algebras based on 𝔤 [ABFP]. It is therefore desirable to try to characterize and understand the part of H_{ét}^{1}(R, Aut(𝔤)) corresponding to multiloop algebras. We address this problem by introducing the concept of loop and toral torsors (with k not necessarily algebraically closed). These concepts are key ideas within our work. It is easy 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 sense of [SGA3]). We establish that the converse also holds.

Central to our work is the study of the canonical map

(1.4)  H_{ét}^{1}(R_{n}, Aut(𝔤)) → H_{ét}^{1}(F_{n}, Aut(𝔤))

where F_{n} stands for the iterated Laurent series field k((t_{1})) … ((t_{n})). The Acyclicity Theorem proved in §8 shows that the restriction of the canonical map (1.4) to the subset H_{loopl}^{1}(R_{n}, Aut(𝔤)) ⊂ H_{ét}^{1}(F_{n}, Aut(𝔤)) of classes of loop torsors is bijective. This has strong applications to the classification of EALAs. Indeed H_{ét}^{1}(F_{n}, Aut(𝔤)) can be studied using Tits’ methods for algebraic groups over complete local fields. In particular EALAs can be naturally attached Tits indices and diagrams, combinatorial root data and relative and absolute types. These are important invariants which are extremely useful for classification purposes. Setting any applications aside, and perhaps more importantly, we believe that the theory and methods that we are putting forward display an intrinsic beauty, and show just how powerful the methods developed in [SGA3] really are.

Acknowledgement The authors would like to thank M. Brion and V. Chernousov for their valuable comments.

2 Generalities on the algebraic fundamental group, torsors, and reductive group schemes

Throughout this section 𝔛 will denote a scheme, and 𝔊 a group scheme over 𝔛.

2.1 The fundamental group

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_{fét} be the category of finite étale covers of 𝔛, and F the covariant functor from 𝔛_{fét} 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′} ↓
Spec (Ω) →_{a} 𝔛

commutes. The group of automorphism of the functor F is called the algebraic fundamental group of 𝔛 at a, and is denoted by π_{1}(𝔛, a). If 𝔛 = Spec(R) is affine, then a corresponds to a ring homomorphism R → Ω and we will denote the fundamental group by π_{1}(R, a).

The functor F is pro-representable: There exists a directed set I, objects (𝔛_{i})_{i∈I} of 𝔛_{fét}, surjective morphisms φ_{ij} ∈ Hom_{𝔛}(𝔛_{j}, 𝔛_{i}) for i ≤ j and geometric points a_{i} ∈ F (𝔛_{i}) such that

(2.1)  a_{i} = φ_{ij} ∘ a_{j}

(2.2)  The canonical map f : \underrightarrow{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}). The elements of \underrightarrow{lim} Hom_{𝔛}(𝔛_{i}, 𝔛′) appearing in (2.2) are by definition the morphisms in the category of pro-objects over 𝔛 (see [EGA IV] §8.13 for details). It is in this sense that \underrightarrow{lim} Hom(𝔛_{i}, −) pro-represents F.

Since the 𝔛_{i} are finite and étale over 𝔛 the morphisms φ_{ij} are affine. Thus the inverse limit

𝔛^{sc} = \underleftarrow{lim} 𝔛_{i}

exist in the category of schemes over 𝔛 [EGA IV] §8.2. For any scheme 𝔛′ over 𝔛 we thus have a canonical map

(2.3)  Hom_{P ro−𝔛}(𝔛^{sc}, 𝔛′) =^{def} \underrightarrow{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_{fét}.

Proof. Because the 𝔛_{i} are affine over 𝔛 and 𝔛 is noetherian, each 𝔛_{i} is noetherian; in particular, quasicompact and quasiseparated. Thus, for 𝔛′/𝔛 locally of finite presentation, in particular for 𝔛′ in 𝔛_{fét}, 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 lines as those of (2.2) by considering the “geometric point” a^{sc} ∈ \underleftarrow{lim}F (𝔛_{i}) satisfying a^{sc} ↦ a_{i} for all i ∈ I.

In computing 𝔛^{sc} = \underleftarrow{lim}𝔛_{i} we may replace (𝔛_{i})_{i∈I} by any cofinal family. This allows us to assume that the 𝔛_{i} are (connected) Galois, i.e. the 𝔛_{i} are connected and the (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}).

Thus π_{1}(𝔛, a) can be identified with the group \underleftarrow{lim} Aut_{𝔛}(𝔛_{i})^{opp}. Each Aut_{𝔛}(𝔛_{i}) is finite, and this endows π_{1}(𝔛, a) with the structure of a profinite topological group.

The group π_{1}(𝔛, a) acts on the right on 𝔛^{sc} as the inverse limit of the finite groups Aut_{𝔛}(𝔛_{i}). Thus, the group π_{1}(𝔛, a) acts on the left on each set F (𝔛′) = Hom_{P ro−𝔛}(𝔛^{sc}, 𝔛′) for all 𝔛′ ∈ 𝔛_{fét}. This action is continuous since the structure morphism 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_{fét}, then the map F (u) : F (𝔛′) → F (𝔛′′) clearly commutes with the action of π_{1}(𝔛, a). This construction provides an equivalence between 𝔛_{fét} and the category of finite sets equipped with a continuous π_{1}(𝔛, a)–action.

The right action of π_{1}(𝔛, a) on 𝔛^{sc} induces an action of π_{1}(𝔛, a) on 𝔊(𝔛^{sc}) = Mor_{𝔛}(𝔛^{sc}, 𝔊), namely

γf(z) = f(z^{γ}) ∀γ ∈ π_{1}(𝔛, a), f ∈ 𝔊(𝔛^{sc}), z ∈ 𝔛^{sc}.

Proposition 2.3. Assume 𝔛 is noetherian and that 𝔊 is locally of finite presentation over 𝔛. Then 𝔊(𝔛^{sc}) is a discrete π_{1}(𝔛, a)–module and the canonical map

lim_{⟶} H^{1}(Aut_{𝔛}(𝔛_{i}), 𝔊(𝔛_{i})) → H^{1}(π_{1}(𝔛, a), 𝔊(𝔛^{sc}))

is bijective.

Remark 2.4. Here and elsewhere when a profinite group A acts discretely on a module M the corresponding cohomology H^{1}(A, M) is the continuous cohomology as defined in [Se]. Similarly, if a group H acts in both A and M, then Hom_{H}(A, M) stands for the continuos group homomorphism of A into M that commute with the action of H.

Proof. To show that 𝔊(𝔛^{sc}) is discrete amounts to showing that the stabilizer in π_{1}(𝔛, a) of a point of f ∈ 𝔊(𝔛^{sc}) is open. But if 𝔊 is locally of finite presentation then 𝔊(𝔛^{sc}) = 𝔊(\underleftarrow{lim} 𝔛_{i}) = \underrightarrow{lim} 𝔊(𝔛_{i}) ([EGA IV] prop. 8.13.1), so we may assume that f ∈ 𝔊(𝔛_{i}) for some i. The result is then clear, for the stabilizer we are after is the inverse image under the continuous map π_{1}(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^{1}(π_{1}(𝔛, a), 𝔊(𝔛^{sc})) = \underrightarrow{lim}(π_{1}(𝔛, a)/U, 𝔊(𝔛^{sc})^{U})

where the limit is taken over all open normal subgroups U of π_{1}(𝔛, a). But for each such U we can find U_{i} ⊂ U so that U_{i} = ker(π_{1}(𝔛, a) → Aut_{𝔛}(𝔛_{i})). Thus

H^{1}(π_{1}(𝔛, a), 𝔊(𝔛^{sc}) = \underrightarrow{lim} H^{1}(Aut_{𝔛}(𝔛_{i}), 𝔊(𝔛_{i}))

as desired.

Suppose now that our 𝔛 is a geometrically connected k–scheme, where k is of arbitrary characteristic. We will denote 𝔛 ×_{k} \overline{k} by \overline{𝔛}. Fix a geometric point \overline{a} : Spec(\overline{k}) → \overline{𝔛}. Let a (resp. b) be the geometric points of 𝔛 [resp. Spec(k)] given by the composite maps a : Spec(\overline{k}) →^{\overline{a}} \overline{𝔛} → 𝔛 [resp. b : Spec(\overline{k}) →^{\overline{a}} 𝔛 → Spec(k)]. Then by [SGA1, théorème IX.6.1] π_{1}( Spec(k), b) ≃ Gal(k) = Gal(k_{s}/k) where k_{s} is the separable closure of k in \overline{k}, and the sequence

(2.4)  1 → π_{1}(\overline{𝔛}, \overline{a}) → π_{1}(𝔛, a) ⟶^{p} Gal(k) → 1

is exact.

2.2 Torsors

Recall that a (right) torsor over 𝔛 under 𝔊 (or simply a 𝔊–torsor if 𝔛 is understood) is a scheme 𝔈 over 𝔛 equipped with a right action of 𝔊 for which there exists 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. étale locally trivial) if it admits a trivialization by an open Zariski (resp. étale) covering of 𝔛. If 𝔊 is affine, flat and locally of finite presentation over 𝔛, then 𝔊–torsors over 𝔛 are classified by the pointed set of cohomology H_{fppf}^{1}(𝔛, 𝔊) defined by means of cocycles à la Čech. If 𝔊 is smooth, any 𝔊–torsor is étale locally trivial (cf. [SGA3], Exp. XXIV), and their classes are then measured by H_{ét}^{1}(𝔛, 𝔊). In what follows the fppf-topology will be our default choice, and we will for convenience denote H_{fppf}^{1} simply by H^{1}. Given a base change 𝔜 → 𝔛, we denote by H^{1}(𝔜/𝔛, 𝔊) the kernel of the base change map H^{1}(𝔛, 𝔊) → H^{1}(𝔜, 𝔊_{𝔜}). As it is customary, and when no confusion is possible, we will denote in what follows H^{1}(𝔜, 𝔊_{𝔜}) simply by H^{1}(𝔜, 𝔊)

Recall that a torsor 𝔈 over 𝔛 under 𝔊 is called isotrivial if it is trivialized by some finite étale extension of 𝔛, that is,

[𝔈] ∈ H^{1}(𝔛′/𝔛, 𝔊) ⊂ H^{1}(𝔛, 𝔊)

for some 𝔛′ in 𝔛_{fét}. We denote by H_{iso}^{1}(𝔛, 𝔊) the subset of H^{1}(𝔛, 𝔊) consisting of classes of isotrivial torsors.

Proposition 2.5. Assume that 𝔛 is noetherian and that 𝔊 is locally of finite presentation over 𝔛. Then

H_{iso}^{1}(𝔛, 𝔊) = ker (H^{1}(𝔛, 𝔊) → H^{1}(𝔛^{sc}, 𝔊)).

Proof. Assume 𝔈 is trivialized by 𝔛′ ∈ 𝔛_{fét}. Since the connected components of 𝔛′ are also in 𝔛_{fét}^{3} 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^{1}(𝔛^{sc}, 𝔊) is thus trivial.

Conversely assume [𝔈] ∈ H^{1}(𝔛, 𝔊) vanishes under the base change 𝔛^{sc} → 𝔛. Since the 𝔛_{i} are quasicompact and quasiseparated and 𝔊 is locally of finite presentation, a theorem of Grothendieck-Margaux [Mg] shows that the canonical map

\underrightarrow{lim} H^{1}(𝔛_{i}, 𝔊) → H^{1}(𝔛^{sc}, 𝔊)

is bijective. Thus 𝔈 ×_{𝔛} 𝔛_{i} ≃ 𝔊_{𝔛_{i}} for some i ∈ I.

2.3 An example: Laurent polynomials in characteristic 0

We look in detail at an example that is of central importance to this work, namely the case when 𝔛 = Spec (R_{n}) where R_{n} = k[t_{1}^{±1}, …, t_{n}^{±1}] is the Laurent polynomial ring in n–variables with coefficients on a field k of characteristic 0.

Fix once and for all a compatible set (ξ_{m})_{m≥0} of primitive m–roots of unity in \overline{k} (i.e. ξ_{mℓ}^{ℓ} = ξ_{m}). Let \{k_{λ}\}_{λ∈Λ} be the set of finite Galois extensions of k which are included in \overline{k}. Let Γ_{λ} = Gal (k_{λ}/k) and Γ = \underleftarrow{lim} Γ_{λ}. Then Γ coincided with the algebraic fundamental group of Spec (k) at the geometric point Spec (\overline{k}).

Let ε : R_{n} → k be the evaluation map at t_{i} = 1. The composite map R_{n} →^{ε} k ↪ \overline{k} defines a geometric point a of 𝔛 and a geometric point \overline{a} of \overline{𝔛} = Spec (\overline{R}_{n}) where \overline{R}_{n} = \overline{k}[t_{1}^{±1}, …, t_{n}^{±1}].

Let I be the subset of Λ × ℤ_{>0} 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_{n,m}^{λ} = k_{λ}[t_{1}^{± \frac{1}{m}} , …, t_{n}^{± \frac{1}{m}}]

is a Galois extension of R_{n} with Galois group Γ_{m,λ} = (ℤ/mℤ)^{n} ⋊ Γ_{λ} as follows: For e = (e_{1}, …, e_{n}) ∈ ℤ^{n} we have \overline{e}tj^{\frac{1}{m}} = ξ_{j}^{e_{j}}tj^{\frac{1}{m}} where ^{−} : ℤ^{n} → (ℤ/mZ)^{n} is the canonical map, and the group Γ_{λ} acts naturally on R_{n,m}^{λ} through its action on k_{λ}. It is immediate from the definition that for γ ∈ Γ_{λ} we have γeγ^{−1} : tj^{\frac{1}{m}} ↦ (^{γ}ξ_{j})^{e_{j}}tj^{\frac{1}{m}} . Thus if ^{γ}ξ_{j} = ξ_{j}^{f_{j}} then γeγ^{−1} = e′ where e′ = (f_{1}e_{1}, … f_{n}e_{n}).

3There exists a finite Galois connected covering 𝔉 → 𝔛 such that 𝔉 ×_{𝔛} 𝔛′ ≅ 𝔉 ⊔ ⋯ ⊔ 𝔉 (r times).

If we decompose 𝔛′ = 𝔜_{1} ⊔ ⋯ ⊔ 𝔜_{m} into its connected components we have

𝔛′ ×_{𝔛} 𝔉 = 𝔜_{1} ×_{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 étale over 𝔉 for i = 1, .., m. Then each 𝔜_{i} is étale 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 étale.

If (λ, ℓ) ≤ (μ, n) we have a canonical inclusion R_{n,ℓ}^{λ} ⊂ R_{n,m}^{μ}. For i = (λ, ℓ) ∈ I we let 𝔛_{i} = Spec (R_{n,ℓ}^{λ}). The above gives morphism φ_{ij} : 𝔛_{j} → 𝔛_{i} of 𝔛–schemes whenever i ≤ j.

We have [GP3]

𝔛^{sc} = \underleftarrow{lim} 𝔛_{i} = Spec(\underrightarrow{lim} 𝔛_{i})
= Spec (\overline{R}_{n,∞})

where \overline{R}_{n,∞} = \underrightarrow{lim}_{m} \overline{R}_{n,m} with \overline{R}_{n,m} = \overline{k}[t_{1}^{± \frac{1}{m}} , …, t_{n}^{± \frac{1}{m}}]. Thus

(2.5)  π_{1}(𝔛, a) = \underleftarrow{lim}Γ_{m,λ} = ℤ̂(1)^{n} ⋊ Gal (k).

where ℤ̂(1) denotes the abstract group ℤ̂ = \underleftarrow{lim}_{m} μ μ μ_{m}(\overline{k}) equipped with the natural action of the absolute Galois group Gal(k) = Gal(\overline{k}/k).

Remark 2.6. Consider the affine k–group scheme _{∞}μ μ μ = \underleftarrow{lim}_{m} μ μ μ_{m}. It corresponds to the Hopf algebra

k[_{∞}μ μ μ] = \underrightarrow{lim}_{m} k[μ μ μ_{m}] = \underrightarrow{lim}_{m} \frac{k[t]}{t^{m} − 1}.

Then we have _{∞}μ μ μ(\overline{k}) ≃ ℤ̂ and _{∞}μ μ μ(\overline{k}) is equipped with a (canonical) structure of profinite Gal(k)–module.

Remark 2.7. Let the notation be that of Example 2.3. Since ℤ^{n} is the character group of the the torus G_{m,k}^{n}, we have an automorphism GL_{n}(ℤ) ≃ Aut_{gr}(G_{m}^{n,k})^{op}. This defines a left action GL_{n}(ℤ) on R_{n} and a right action of GL_{n}(ℤ) on the torus G_{m,k}^{n}. Furthermore, by universal nonsense, this action extends uniquely to the simply connected covering \overline{R}_{n,∞} at the geometric point \overline{a}. The extended action on the torus Spec(\overline{R}_{n,m}) with character group (\frac{1}{m}ℤ)^{n} is given by the extension of the action of GL_{n}(ℤ) from ℤ^{n} to (\frac{1}{m}ℤ)^{n} inside ℚ^{n}. The group GL_{n}(ℤ) acts (on the right) on π_{1}(R_{n}), so we can consider the semidirect product of groups GL_{n}(ℤ) ⋉ π_{1}(R_{n}) which acts then on \overline{R}_{n,∞} (see §8.4 for details).

Lemma 2.8. Let S be a connected finite étale cover of R_{n}. Let L ⊂ S be the integral closure of k in S. Then there exists g ∈ GL_{n}(ℤ), a_{1}, ..., a_{n} ∈ L^{×} and positive integers d_{1}, ..., d_{n} such that d_{1} | d_{2} ⋯ | d_{n} and

S ⊗_{R_{n}} R_{n}^{g} ≃_{R_{n}-alg} (R_{n} ⊗_{k} L)[ \sqrt[d_{1}]{a_{1}t_{1}}, ⋯, \sqrt[dn]{a_{n}t_{n}}].

In particular, S is k–isomorphic to R_{n} ⊗_{k} L and Pic(S) = 0.

Proof. By taking the action of GL_{n}(ℤ) on R_{n} described in Remark 2.7 we can twist the R_{n}–module R_{n} by an element g ∈ GL_{n}(ℤ). We denote the resulting twisted R_{n}– algebra by R_{n}^{g}.^{4} Note that R_{n}^{g} and R_{n} have the same units. For convenience in what follows we will for simplicity denote (R_{n})^{g} by R_{n}^{g}, (R_{n,m})^{g} by R_{n,m}^{g} and S ⊗_{R_{n}} R_{n}^{g} by S^{g}.

By Galois theory there exists a finite Galois extension k′/k and a positive integer m such that μ_{m}(\overline{k}) ⊂ k′ and S ≃_{R_{n}} (R_{n,m} ⊗_{k} k′)^{H} where H is a subgroup of Gal(R_{n,m} ⊗_{k} k′/R_{n}) = μ μ μ_{m}(k′)^{n} ⋊ Gal(k′/k). Hence S is geometrically connected and S is a finite étale 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 consider the following commutative diagram

1 ⟶ μ μ μ_{m}(k′)^{n} ∩ H ⟶  H  ⟶ Gal(k′/k) ⟶ 1
↓  ↓  ||
1 ⟶ μ μ μ_{m}(k′)^{n} ⟶ Gal(R_{n,m} ⊗_{k} k′/R_{n}) ⟶ Gal(k′/k) ⟶ 1.

Note that the action Gal(k′/k) on μ μ μ_{m}(k′)^{n} normalizes μ μ μ_{m}(k′)^{n} ∩H. Hence μ μ μ_{m}^{n}(k′)∩H is the group of k′–points of a split k–group ννν of multiplicative type. By considering the corresponding characters groups, we get a surjective homomorphism (ℤ/mℤ)^{n} = μ̂ μ μ_{m}^{n} → ν̂νν of finite abelian groups. An element g ∈ GL_{n}(ℤ) ⊂ Aut_{k}(R_{n}) transforms the diagram above to yield

1 ⟶ ννν(k′) ⟶  H  ⟶ Gal(k′/k) ⟶ 1
↓  ↓  ||
1 ⟶ μ μ μ_{m}(k′)^{n} ⟶ Gal(R_{n,m} ⊗_{k} k′/R_{n}) ⟶ Gal(k′/k) ⟶ 1.
g^{∗}↓  ^{g^{∗}}↓  ||
1 ⟶ μ μ μ_{m}(k′)^{n} ⟶ Gal(R_{n,m}^{g} ⊗_{k} k′/R_{n}^{g}) ⟶ Gal(k′/k) ⟶ 1.
↑  ↑  ||
1 ⟶ ^{g}ννν(k′) ⟶  H^{g}  ⟶ Gal(k′/k) ⟶ 1

The action of GL_{n}(ℤ) on (ℤ/mℤ)^{n} = μ̂ μ μ_{m}^{n} is the left action provided by the homomorphism GL_{n}(ℤ) → GL_{n}(ℤ/mℤ). By elementary facts about generators of finite abelian groups there exists g ∈ GL_{n}(ℤ) and positive integers d_{1}, ..., d_{n} such that d_{1} | d_{2} ⋯ | d_{n} | m for which the following holds

4The multiplication on the R_{n}-algebras R_{n}^{g} and R_{n} coincide. It is the action of R_{n} that is different. See §4.1 for details.

(ℤ/mℤ)^{n} = μ̂ μ μ_{m}^{n} ⟶  ν̂νν
g^{∗}↓^{≃}  ^{≃}↓
(ℤ/mℤ)^{n} = μ̂ μ μ_{m}^{n} ⟶ ℤ/(m/d_{1})ℤ ⊕ ⋯ ℤ/(m/d_{n})ℤ.

This base change leads to the following commutative diagram

1 ⟶ μ μ μ_{m/d_{1}}(k′) × ⋯ × μ μ μ_{m/d_{n}}(k′) ⟶  H^{g}  ⟶ Gal(k′/k) ⟶ 1
↓  ↓  ||
1 ⟶  μ μ μ_{m}(k′)^{n}  ⟶ Gal(R_{n,m}^{g} ⊗_{k} k′/R_{n}^{g}) ⟶ Gal(k′/k) ⟶ 1.

We claim that S^{g} is equipped with a structure of an R_{n}^{g}–torsor structure under μ μ μ := μ μ μ_{d_{1}} × ⋯μ μ μ_{d_{n}}. The diagram above provides a bijection

Gal(R_{n,m}^{g} ⊗_{k} k′/R_{n}^{g})/H^{g} ⟶^{∼} μ μ μ(k′),

hence a map ψ : Gal(R_{n,m}^{g} ⊗_{k} k′/R_{n}^{g}) ⟶ μ μ μ(k′) which is a cocycle for the standard action of Gal(R_{n,m}^{g} ⊗_{k} k′/R_{n}^{g}) on μ μ μ(k′) as we now check. We shall use the following two facts

(I) ψ is right H^{g}–invariant;

(II) the restriction of ψ to μ μ μ_{m}(k′)^{n} is a morphism of Gal(k′/k)–modules.

We are given γ_{1}, γ_{2} ∈ Gal(R_{n,m}^{g} ⊗_{k} k′/R_{n}^{g}). Since ^{g}H surjects onto Gal(k′/k), we can write γ_{i} = α_{i}h_{i} with α_{i} ∈ μ μ μ_{m}(k′)^{n} and h_{i} ∈ H^{g} for i = 1, 2. We have

ψ(γ_{1}γ_{2}) = ψ(α_{1} h_{1} α_{2} h_{2})
= ψ(α_{1} h_{1} α_{2} h_{1}^{−1}h_{1}h_{2})
= ψ(α_{1} h_{1} α_{2} h_{1}^{−1}) [by (I) ]
= ψ(α_{1}) ψ(h_{1} α_{2} h_{1}^{−1}) [by (II) ]
= ψ(α_{1}) h_{1} ψ(α_{2}) h_{1}^{−1} [by (I) ]
= ψ(γ_{1}) γ_{1}.ψ(γ_{2}).

Denote by S̃ the μ μ μ–torsor over R_{n}^{g} defined by ψ, that is

S̃ := { x ∈ R_{n,m}^{g} ⊗_{k} k′ | ψ(γ) .γ(x) = x ∀γ ∈ Gal(R_{n,m}^{g} ⊗_{k} k′/R_{n}^{g}) }.

Since ψ is trivial over H^{g}, we have S̃ ⊂ S^{g}. But S̃ and S^{g} are both finite étale coverings of R_{n}^{g} of degree | μ μ μ(k′) |, hence S̃ = S^{g}. Since Pic(R_{n}^{g}) = 0, we can use Kummer theory, namely the isomorphism

H^{1}(R_{n}^{g},μ μ μ) = ∏_{j=1,...,n} H^{1}(R_{n}^{g},μ μ μ_{d_{j}}) ≃ ∏_{j=1,...,n} R_{n}^{g}^{×}/(R_{n}^{g}^{×})^{d_{j}}.

for determining the structure of S̃. Since R_{n}^{g}^{×} = k^{×}×ℤ^{n}, there exist scalars a_{1}, ..., a_{n} ∈ k^{×} and monomials x_{1}, ..., x_{n} in the t_{i} such that the class of S̃/R_{n}^{g} in H^{1}(R_{n},μ μ μ) is given by (a_{1} x_{1}, ⋯, a_{n}x_{n}). In terms of covering, this means that S̃ = k[ \sqrt[d_{1}]{a_{1}x_{1}}, ⋯, \sqrt[dn]{a_{n}x_{n}}]. Extending scalars to k′, we have

S̃ ⊗_{k} k′ = (R_{n,m}^{g} ⊗_{k} k′)^{μ μ μ(k′)} = k′[ \sqrt[d_{1}]{t_{1}}, ⋯, \sqrt[dn]{t_{n}}].

¿From this it follows that x_{i} = t_{i} mod ((R_{n}^{g} ⊗_{k} k′)^{×})^{d_{i}} and x_{i} = t_{i} mod (R_{n}^{g}^{×})^{d_{i}}. We conclude that S̃ = k[ \sqrt[d_{1}]{a_{1}t_{1}}, ⋯, \sqrt[dn]{a_{n}t_{n}}].

2.4 Reductive group schemes: Irreducibility and isotropy

The notation that we are going to use throughout the paper deserves some comments. 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 quintessential example is G_{𝔛} = G ×_{k} 𝔛. For arbitrary group schemes, or more generally group functors, over 𝔛 we shall tend to use german characters, such as 𝔊. This duality of notation will be particularly useful when dealing with twisted forms over 𝔛 of groups that come from k.

The concept of reductive group scheme over 𝔛 and all related terminology is that of [SGA3].^{5}

For convenience we now recall and introduce some concepts and notation attached to a reductive group scheme ℌ over 𝔛. We denote by rad(ℌ) [resp. corad(ℌ)] its radical (resp. coradical) torus, that is its maximal central subtorus (resp. its maximal toral 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).^{6} The opposite notion is irreducible. If 𝔛 is affine, the notion of reducibility for ℌ is equivalent to the existence of a proper parabolic subgroup 𝔓 (XXVI.2.3), so there is no ambiguity with the terminology of [CGP] and [GP2].

5The references to [SGA3] are so prevalent that they will henceforth be given by simply listing the Exposé number. Thus XII. 1.3, for example, refers to section 1.3 of Exposé XII of [SGA3].

6The concept of proper parabolic subgroup is not defined in [SGA3]. By proper we mean that 𝔓\overline{s} is a proper subgroup of 𝔊\overline{x} for all geometric points \overline{x} of 𝔛.

By 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}. The opposite 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 equivalently corad(ℌ)] is anisotropic (XXVI.2.3, resp. [Gi4]).

Similarly we say that the action of 𝔊 on ℌ is isotropic if it centralizes a split subtorus 𝔗 of H with the property that al geometric fibers of 𝔗 are non-trivial. Otherwise the action is anisotropic. One checks that this is the case if and only if the action is it is irreducible and the action of 𝔊 on the torus rad(ℌ) [or equivalently to corad(ℌ)] is anisotropic.

3 Loop, finite and toral torsors

Throughout this section k will denote a field of arbitrary characteristic, 𝔛 a geometrically connected noetherian scheme over k, and 𝔛^{sc} = \underleftarrow{lim} 𝔛_{i} its simply connected cover as described in §2.1. Let G a group scheme over k which is locally of finite presentation. We will maintain the notation of the previous section, and assume that Ω = \overline{k}. Consider the fundamental exact sequence (2.4). The geometric point a corresponds to a point of 𝔛(\overline{k}).

3.1 Loop torsors

Because of (2.1), the geometric points a_{i} : Spec (\overline{k}) → 𝔛_{i} induce a geometric point a^{sc} : Spec (\overline{k}) → \underleftarrow{lim} 𝔛_{i} = 𝔛^{sc}. We thus have a group homomorphism

(3.1)  G(k_{s}) → G(\overline{k}) ⟶^{G(a^{sc})} G(𝔛^{sc}).

The group π_{1}(𝔛, a) acts on k_{s}, hence on G(k_{s}), via the group homomorphism π_{1}(𝔛, a) → Gal (k) of (2.4). This action is continuous, and together with (3.1) yields a map

H^{1}(π_{1}(𝔛, a), G(k_{s})) → H^{1}(π_{1}(𝔛, a), G(𝔛^{sc})),

where we remind the reader that these H^{1} are defined in the “continuous” sense (see Remark 2.4). On the other hand, by Proposition 2.3 and basic properties of torsors trivialized by Galois extensions, we have

H^{1}(π_{1}(𝔛, a), G(𝔛^{sc})) = \underrightarrow{lim} H^{1}(Aut_{𝔛}(𝔛_{i}), G(𝔛_{i}))
= \underrightarrow{lim} H^{1}(𝔛_{i}/𝔛, G) ⊂ H^{1}(𝔛, G).

By 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 isomorphism class [𝔈] in H^{1}(𝔛, G) belongs to the image of the composite map

(3.2)  H^{1}(π_{1}(𝔛, a), G(k_{s})) → H^{1}(π_{1}(𝔛, a), G(𝔛^{sc})) ⊂ H^{1}(𝔛, G).

We will denote by H_{loop}^{1}(𝔛, G) the subset of H^{1}(𝔛, G) consisting of classes of loop torsors. They are given by (continuous) cocycles in the image of the natural map Z^{1}(π_{1}(𝔛, a), G(k_{s})) → Z^{1}(𝔛, G), which we call loop cocycles.

Examples 3.2. (a) If 𝔛 = Spec (k) then H_{loop}^{1}(𝔛, G) is nothing but the usual Galois cohomology of k with coefficients in G.

(b) Assume that k is separably closed. Then the action of π_{1}(𝔛, a) on G(k_{s}) is trivial, so that

H^{1}(π_{1}(𝔛, a), G(k_{s})) = Hom (π_{1}(𝔛, a), G(k_{s}))/Int G(k_{s})

where the group Int G(k_{s}) of inner automorphisms of G(k_{s}) acts naturally on Hom (π_{1}(𝔛, a), G(k_{s})). To be precise, Int(g)(ϕ) : x → g^{−1}ϕ(x)g for all g ∈ G(k_{s}), ϕ ∈ Hom (π_{1}(𝔛, a), G(k_{s})) and x ∈ π_{1}(𝔛, a). Two particular cases are important:

(b1) G abelian: In this case H^{1}(π_{1}(𝔛, a), G(k_{s})) is just the group of continuous homomorphisms from π_{1}(𝔛, a) to G(k_{s}).

(b2) π_{1}(𝔛, a) = ℤ̂^{n} : In this case H^{1}(π_{1}(𝔛, a), G(k_{s})) is the set of conjugacy classes of n–tuples σ = (σ_{1}, …, σ_{n}) of commuting elements of finite order of G(k_{s}). That the elements are of finite order follows from the continuity assumption.

(c) Let 𝔛 = Spec (k[t^{±1}]) with k algebraically closed of characteristic 0. If G is a connected linear algebraic group over k then H^{1}(𝔛, G) = 1 ([P1, prop. 5]). We see from (b2) above that the canonical map

H^{1}(π_{1}(𝔛, a), G(k_{s})) → H^{1}(𝔛, G)

of (3.2) need not be injective. It need not be surjective either (take 𝔛 = ℙ_{k}^{1} and G = G_{m,k}).

(d) If the canonical map G(k_{s}) → G(𝔛^{sc}) is bijective, e.g. if 𝔛 is a geometrically integral projective variety over k (i.e. a geometrically integral closed subscheme of ℙ_{k}^{n} for some n,) then H_{loop}^{1}(𝔛, G) = H^{1}(π_{1}(𝔛, a), G(k_{s})).

Remark 3.3. The notion of loop torsor behaves well under twisting by a Galois cocycle z ∈ Z^{1}( Gal(k), G(k_{s})). Indeed the torsion map τ_{z}^{−1} : H^{1}(𝔛, G) → H^{1}(𝔛,_{z}G) maps loop classes to loop classes.

3.2 Loop reductive groups

Let ℌ be a reductive group scheme over 𝔛. Since 𝔛 is connected, for all x ∈ 𝔛 the geometric fibers ℌ\overline{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_{0} over k such that ℌ is a twisted form (in the étale topology of 𝔛) of ℌ_{0} = H_{0} ×_{k} 𝔛. We will call H_{0} the Chevalley k–form of ℌ. The 𝔛–group ℌ corresponds to a torsor 𝔈 over 𝔛 under the group scheme Aut(ℌ_{0}), namely 𝔈 = Isom_{gr}(ℌ_{0}, ℌ). We recall that Aut(ℌ_{0}) is representable by a smooth and separated group scheme over 𝔛 by XXII 2.3. By definition ℌ is then the contracted product 𝔈 ∧^{Aut(ℌ_{0})} ℌ_{0} (see [DG] III §4 n^{o}3 for details.)

We now define one of the central concepts of our work.

Definition 3.4. We say that a group scheme ℌ over 𝔛 is loop reductive if it is reductive and if 𝔈 is a loop torsor.

We look more closely to the affine case 𝔛 = Spec(R). Concretely, let H_{0} = Spec(k[H_{0}]) be a split reductive k–group and consider the corresponding R–group ℌ_{0} = H_{0} ×_{k} R, whose Hopf algebra is R[ℌ_{0}] = k[H_{0}] ⊗_{k} R.

Let ℌ be an R–group which is a twisted form of ℌ_{0} trivialized by the universal covering R^{sc}. Then to a trivialization ℌ_{0} ×_{R} R^{sc} ≅ ℌ ×_{R} R^{sc}, we can attach a cocycle u ∈ Z^{1}(π_{1}(R, a), Aut(ℌ_{0})(R^{sc})) from which ℌ can be recovered by Galois descent as we 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 Z^{1}(π_{1}(R, a), Aut(ℌ_{0})(R^{sc})). On the one hand ℌ_{0} can be thought as the affine scheme Spec(R[ℌ_{0}]), Aut(ℌ_{0})(R^{sc}) as the (abstract) group of automorphisms of the R^{sc}– group Spec(R^{sc}[ℌ_{0}]) where R^{sc}[ℌ_{0}] = R[ℌ_{0}] ⊗_{R} R^{sc}, and π_{1}(X, a) as the opposite group of automorphisms of Spec(R^{sc})/ Spec(R) acting naturally on Aut(ℌ_{0})(R^{sc}).

We will adopt an (anti) equivalent second point of view that is much more convenient for our calculations. We view Aut(ℌ_{0})(R^{sc}) as the group of automorphisms of the R^{sc}–Hopf algebra R^{sc}[H_{0}] = R[ℌ_{0}] ⊗_{R} R^{sc} ≃ k[H_{0}] ⊗_{k} R^{sc} on which the Galois group π_{1}(R, a) acts naturally. Then the R–Hopf algebra R[ℌ] corresponding to ℌ is given by

R[ℌ] = {x ∈ R^{sc}[H_{0}] : u_{γ}^{γ}x = a for all γ ∈ π_{1}(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_{0})(\overline{k}) ⊂ Aut(H_{0})(R^{sc}) = Aut(ℌ_{0})(R^{sc}) for all γ ∈ π_{1}(R, a).

3.3 Loop torsors at a rational base point

If our geometric point a lies above a k–rational point x of 𝔛, then x corresponds to a section of the structure morphism 𝔛 → Spec(k) which maps b to a. This yields a group homomorphism x^{∗} : Gal(k) → π_{1}(𝔛, a) that splits the sequence (2.4) above. This splitting defines an action of Gal(k) on the profinite group π_{1}(\overline{𝔛}, \overline{a}), hence a semidirect product identification

(3.3)  π_{1}(𝔛, a) ≃ π_{1}(\overline{𝔛}, \overline{a}) ⋊ Gal(k).

We have seem an example of (3.3) in Example 2.3.

Remark 3.6. By the structure of extensions of profinite groups [RZ, §6.8], it follows that π_{1}(\overline{𝔛}, \overline{a}) is the projective limit of a system (H_{α} ⋊ Gal(k)) where the H_{α}’s are finite groups. The Galois action on each H_{α} defines a twisted finite constant k–group ννν_{α}. We define

ννν = \underleftarrow{lim}_{α}ννν_{α}.

The ννν_{α} are affine k–groups such that

ννν(\overline{k}) = \underleftarrow{lim}_{α} ννν_{α}(\overline{k}) = π_{1}(\overline{𝔛}, \overline{a}).

Note that ννν(k_{s}) = ννν(\overline{k}). In the case when 𝔛 = Spec(R_{n}), where as before R_{n} = k[t_{1}^{±1}, …, t_{n}^{±1}] with k of characteristic zero and a is the geometric point described in Example 2.3, the above construction yields the affine k–group _{∞}μ μ μ defined in Remark 2.6.

By means of the decomposition (3.3) we can think of loop torsors as being comprised of a geometric and an arithmetic part, as we now explain.

Let η ∈ Z^{1}(π_{1}(𝔛, a), G(k_{s})) 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^{1}( Gal(k), G(k_{s})). If η is fixed in our discussion, we will at times denote the cocycle η^{ar} by the more traditional notation z. In particular, for s ∈ Gal(k) we write z_{s} instead of η_{s}^{ar}.

Next we consider the restriction of η to π_{1}(\overline{𝔛}, \overline{a}) that we denote by η^{geo} and called the geometric part of η.

We thus have a map

Θ : Z^{1}(π_{1}(𝔛, a), G(k_{s})) ⟶ Z^{1}( Gal(k), G(k_{s})) × Hom (π_{1}(\overline{𝔛}, \overline{a}), G(k_{s}))
η  ↦  ( η^{ar} , η^{geo} )

The group Gal(k) acts on π_{1}(\overline{𝔛}, \overline{a}) by conjugation. On G(k_{s}), the Galois group Gal(k) acts on two different ways. There is the natural action arising for the action of Gal(k) on k_{s} that as customary we will denote by ^{s}g, and there is also the twisted action given by the cocycle η^{ar} = z. Following Serre we denote this last by ^{s′}g. Thus s′g = z_{s}^{s}gz_{s}^{−1}. Following standard practice to view the abstract group G(k_{s}) as a Gal(k)–module with the twisted action by z we write _{z}G(k_{s}).

For s ∈ Gal(k) and h ∈ π_{1}(\overline{𝔛}, \overline{a}), we have

η_{shs^{−1}}^{geo} = η_{shs^{−1}} = η_{s}^{s}(η_{hs^{−1}})  [η is a cocycle]
= z_{s} ^{s}(η_{hs^{−1})}  [η_{s} = η_{s}^{ar} = z_{s}]
= z_{s} ^{s}(η_{h}^{geo} z_{s^{−1}})  [η is a cocycle and h acts trivially on G(k_{s})]
= z_{s} ^{s}η_{h}^{geo} z_{s}^{−1}  [1z_{s} ^{s}z_{s^{−1}}].

This shows that η^{geo} : π_{1}(\overline{𝔛}, \overline{a}) → _{z}G(k_{s}) commutes with the action of Gal(k). In other words, η^{geo} ∈ Hom_{Gal(k)} (π_{1}(\overline{𝔛}, \overline{a}),_{z}G(k_{s})).

Lemma 3.7. The map Θ defines a bijection between Z^{1}(π_{1}(𝔛, a), G(k_{s})) and couples (z, η^{geo}) with z ∈ Z^{1}(π_{1}(𝔛, a), G(k_{s})) and η^{geo} ∈ Hom_{Gal(k)} (π_{1}(\overline{𝔛}, \overline{a}),_{z}G(k_{s})).

Proof. Since a 1–cocycle is determined by its image on generators, the map Θ is injective. For the surjectivity, assume we are given z ∈ Z^{1}(π_{1}(𝔛, a), G(k_{s})) and η^{geo} ∈ Hom_{Gal(k)} (π_{1}(\overline{𝔛}, \overline{a}),_{z}G(k_{s})). We define then η : π_{1}(𝔛, a) → G(k_{s}) by η_{hs} := η_{h}^{geo} z_{s} This map is continuous, its restriction to π_{1}(\overline{𝔛}, \overline{a}) (resp. Gal(k) ) is η^{geo} (resp. z). Finally, since η is a section of the projection map G(k_{s}) ⋊ π_{1}(𝔛, a) → π_{1}(𝔛, a), it is a cocycle.

We finish this section by recalling some basic properties of the twisting bijection. Let η ∈ Z^{1}(π_{1}(𝔛, a), G(k_{s})) and consider its corresponding pair Θ(η) = (z, η^{geo}). We can apply the same construction to the twisted k–group _{z}G. This would lead to a map Θ_{z} that will attach to a cocycle η′ ∈ Z^{1}(π_{1}(𝔛, a),_{z}G(k_{s})) a pair (z′, η′^{geo}) along the lines explained 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^{1}(π_{1}(𝔛, a), G(k_{s})). With the above notation, the inverse of the twisting map [Se]

τ_{z}^{−1} : Z^{1}(π_{1}(𝔛, a), G(k_{s})) ⟶^{∼} Z^{1}(π_{1}(𝔛, a),_{z}G(k_{s}))

satisfies Θ_{z} ∘ τ_{z}^{−1}(η) = (1, η^{geo}).

Remark 3.9. Consider the special case when the semi-direct product is direct, i.e. π_{1}(𝔛, a) = π_{1}(\overline{𝔛}, \overline{a}) × Gal(k). In other words, the affine k–group ννν defined above is constant so that

η_{h}^{geo} = z_{s} ^{s}η_{h}^{geo} z_{s}^{−1}

for all h ∈ π_{1}(\overline{𝔛}, \overline{a}) and s ∈ Gal(k). The torsion map

τ_{z}^{−1} : Z^{1}(π_{1}(\overline{𝔛}, \overline{a}), G(k_{s})) → Z^{1}(π_{1}(\overline{𝔛}, \overline{a}),_{z}G(k_{s}))

maps the cocycle η to the homomorphism η^{geo} : π_{1}(\overline{𝔛}, \overline{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 for each linear algebraic k–group H, there is a natural bijection

Hom_{k−gp}(ννν, H) ⟶^{∼} Hom_{Gal(k)}(π_{1}(\overline{𝔛}, \overline{a}), H(k_{s})).

Proof. First we recall that Hom_{Gal(k)}(π_{1}(\overline{𝔛}, \overline{a}), H(k_{s})) stands for the continuous homomorphisms from π_{1}(\overline{𝔛}, \overline{a}) to H(k_{s}) that commute with the action of Gal(k).

Write ννν = \underleftarrow{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} 10.4 that

Hom_{k−gp}(ννν, H) = \underrightarrow{lim}_{α}Hom_{k−gp}(ννν_{α}, H)

= \underrightarrow{lim}_{α}Hom_{Gal(k)} (ννν_{α}(k_{s}), H(k_{s})) = Hom_{Gal(k)} (π_{1}(\overline{𝔛}, \overline{a}), H(k_{s})).

This permits to see purely geometric k–loop torsors in terms of homomorphisms of affine k–group schemes.

3.4 Finite torsors

Throughout this section we assume that G is a smooth affine k–group, and 𝔛 a scheme 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^{1}(𝔛, G) instead of H^{1}(𝔛, G_{𝔛}).

Definition 3.11. A torsor 𝔈 over 𝔛 under G is said to be finite if it admits a reduction to a finite k–subgroup S of G; this is to say, the class of 𝔈 belongs to the image of the natural map H^{1}(𝔛, S) → H^{1}(𝔛, G) for some finite subgroup S of G.

We denote by H_{finite}^{1}(𝔛, G) the subset of H^{1}(𝔛, G) consisting of classes of finite torsors, that is

H_{finite}^{1}(𝔛, G) := ⋃_{S⊂G} Im(H^{1}(𝔛, S) → H^{1}(𝔛, G)).

where S runs over all finite k–subgroups of G.

The case when k is of characteristic 0 is well known.

Lemma 3.12. Assume that k is of characteristic 0. Then H_{finite}^{1}(𝔛, G) ⊂ H_{loop}^{1}(𝔛, G). If in addition k is algebraically closed, then H_{finite}^{1}(𝔛, G) = H_{loop}^{1}(𝔛, G).

Proof. Let S be a finite subgroup of the k–group G. Since k is of characteristic 0 the group S is étale. Thus S corresponds to a finite abstract group S together with a continuous action of Gal(k) by group automorphisms. More precisely (see [SGA1] or [K2] pg.184) S = S(\overline{k}) with the natural action of Gal(k). Similarly the étale 𝔛-group S_{𝔛} corresponds to S with the action of π_{1}(𝔛, a) induced from the homomorphism π_{1}(𝔛, a) → Gal(k).

By Exp. XI of [SGA1] we have

(3.4)  H^{1}(𝔛, S) =^{def} H^{1}(𝔛, S_{𝔛}) = H^{1}(π_{1}(𝔛, a), S) = H^{1}(π_{1}(𝔛, a), S(\overline{k}))

which shows that H^{1}(𝔛, S) ⊂ H_{loop}^{1}(𝔛, G).

If k is algebraically closed any k-loop torsor 𝔈 is given by a continuous group homomorphism f_{𝔈} : π_{1}(𝔛, a) → G(k), as explained in Example 3.2(b). Then the image of f_{𝔈} is a finite subgroup of G(k) which gives rise to a finite (constant) algebraic subgroup S of G. By construction [𝔈] comes from H^{1}(𝔛, S).

3.5 Toral torsors

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 arise we 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}^{1}(𝔛, G) ⊂ H^{1}(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.^{7} Then

H_{toral}^{1}(𝔛, G) = Im(H^{1}(𝔛, N_{G}(T)) → H^{1}(𝔛, G)).

2. Let 1 → S → G′ →^{p} G → 1 be a central extension of G by a k–group S of multiplicative type. Then the diagram

H_{toral}^{1}(𝔛, G′) ⊂ H^{1}(𝔛, G′)
p_{∗}↓  ^{p_{∗}}↓
H_{toral}^{1}(𝔛, G) ⊂ H^{1}(𝔛, G)

7We remind the reader that we are abiding by [SGA3] conventions and terminology. In the expression “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...

is 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 correspondence between maximal tori of the 𝔛–groups _{𝔈′}G′ and _{𝔈′}G. Hence 𝔈′ is a toral G′–torsor if and only if 𝔈′ ∧^{G′^{𝔛}} G_{𝔛} is a toral G–torsor. The general case follows form the fact that 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 Lie(_{𝔈}G) admits a Cartan subalgebra.

Proof. By XIV théorèmes 3.9 and 3.18 there exists a natural one-to-one correspondence between the maximal tori of _{𝔈}G and Cartan subalgebras of Lie(_{𝔈}G).

Recall the following result [CGR2].

Theorem 3.15. Let R be a commutative ring and G a smooth affine group scheme over R whose connected component of the identity G^{0} is reductive. Assume further that one of the following holds:

(a) R is an algebraically closed field, or

(b) R = ℤ, G^{0} is a Chevalley group, and the order of the Weyl group of the geometric fiber G\overline{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), such that

1. S is an extension of a finite twisted constant R–group by a finite R–group of multiplicative type,

2. the natural map H_{fppf}^{1}(𝔛, S) ⟶ H_{fppf}^{1}(𝔛, N_{G}(T)) is surjective for any scheme 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 some twisted finite constant 𝔛–group scheme Γ.

Corollary 3.16. Let G be a linear algebraic k–group whose connected component of the identity G^{0} 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 the hypothesis of Theorem 3.15(b);

(iii) G is reductive.

If the k–scheme 𝔛 satisfies condition (3.5), then

1. H_{toral}^{1}(𝔛, G) ⊂ H_{finite}^{1}(𝔛, G).

2. If furthermore char(k) = 0, we have H_{toral}^{1}(𝔛, G) ⊂ H_{finite}^{1}(𝔛, G) ⊂ H_{loop}^{1}(𝔛, G).

The first statement is immediate. The second one follows from Lemma 3.12.

4 Semilinear considerations

Throughout this section k̃ will denote an object of k–alg. We will denote by Γ a subgroup of the group Aut_{k−alg}(k̃). The elements of Γ are thus k–linear automorphisms of the ring k̃. For convenience we will denote the action of an element γ ∈ Γ on an element λ ∈ k̃ by ^{γ}λ.

4.1 Semilinear morphisms

Given an object R of k̃–alg (the category of associative unital commutative k̃– algebras), we will denote the action of and element λ ∈ k̃ 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 k̃–alg which coincides with R as a ring, but where the k̃–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 a canonical identification.

Given a morphism ψ : A → R of k̃–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).

It is immediate to verify that ψ_{γ} is also a morphism of k̃–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_{k̃−alg}(A, R) → Hom_{k̃−alg}(A^{γ}, R^{γ}).

In view of (4.1) we have also a natural (and equivalent) correspondence

(4.3)  Hom_{k̃−alg}(A, R^{γ}) → Hom_{k̃−alg}(A^{γ}^{−1}, R).

that we record for future use.

Remark 4.1. (i) Let γ, σ ∈ Γ. It is clear from the definitions that the k-algebra isomorphism γ : k̃ → k̃ induces a k̃-algebra isomorphism γ_{σ} : k̃^{σ} → k̃^{γσ}. If no confusion is possible we will denote γ_{σ} simply by γ.

One checks that the k̃–algebras (R ⊗_{k} k̃)^{γ} and R ⊗_{k} k̃^{γ} are equal (recall that both algebras have R ⊗_{k} k̃ as underlying sets). We thus have a k̃–algebra isomorphism

1 ⊗ γ : R ⊗_{k} k̃ → R ⊗_{k} k̃^{γ} = (R ⊗ k̃)^{γ},

or more generally

1 ⊗ γ_{σ} : R ⊗_{k} k̃^{σ} → R ⊗_{k} k̃^{γσ} = (R ⊗ k̃)^{γσ}.

(ii) If A is an object k̃–alg, and γ ∈ Γ, then the k̃–algebras A and A^{γ} have the same ideals.

Given a k̃–functor 𝔛, that is a functor from the category k̃–alg to the category of sets (see [DG] for details), and an element γ ∈ Γ we can define a new the k̃–functor γ𝔛 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})
γ𝔛(ψ)↓  ^{𝔛(ψ_{γ})}↓
γ𝔛′(R) ⟶^{=} ^{γ}𝔛(R)

then commutes by definition, and one can indeed easily verify that ^{γ}𝔛 is a k̃–functor. We call ^{γ}𝔛 the twist of 𝔛 by γ.

Similarly to the case of k̃–algebras described in (4.1) we have the equality of functors

(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})
=↓  ^{=}↓
γ𝔛′(R) ⟶^{^{γ}f(R)} ^{γ}𝔛(R)

This gives a natural bijection

(4.6)  Hom_{k̃−fun}(𝔛′, 𝔛) → Hom_{k̃−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 equivalent version of this last bijection, namely

(4.7)  Hom_{k̃−fun}(^{γ}^{−1}𝔛′, 𝔛) → Hom_{k̃−fun}(𝔛′,^{γ}𝔛)

4.2 Semilinear morphisms

A k̃–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).^{8} 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, then the 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 τγ.

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. There is a group homomorphism t : Aut_{Γ}(𝔛) → Γ that assigns to a semilinear automorphism of 𝔛 its type.

8The alert reader may question whether the “type” is well defined. Indeed it may happen that γ𝔛 and 𝔛 are the same k̃-functor even though γ ≠ 1. This ambiguity can be formally resolved by defining semilinear morphism of type γ as pairs (f : ^{γ}𝔛 → ℨ, γ). We will omit this complication of notation in what follows since no confusion will be possible within our context. Note that the union of sets ∪_{γ∈Γ} Hom_{γ}(𝔛, ℨ) is thus disjoint by definition.

Remark 4.2. Fix a k̃–functor 𝔜. Recall that the category of k̃–functors over 𝔜 consists of k̃–functors 𝔛 equipped with a structure morphism 𝔛 → 𝔜. This category admits fiber products: Given f_{1} : 𝔛_{1} → 𝔜 and f_{2} : 𝔛_{2} → 𝔜 then 𝔛_{1} ×_{𝔜} 𝔛_{2} is given by

(𝔛_{1} ×_{𝔜} 𝔛_{2})(R) = {(x_{1}, x_{2}) ∈ 𝔛_{1}(R) × 𝔛_{2}(R) : f_{1}(R)(x_{1}) = f_{2}(R)(x_{2})}.

Semilinearity extend to fiber products under the right conditions. Suppose f_{1} : 𝔛_{1} → 𝔜 and f_{2} : 𝔛_{2} → 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” γf_{i} : ^{γ}𝔛_{i} → ^{γ}𝔜 defined above can be seen to verify

(4.9)  ^{γ}(𝔛_{1} ×_{𝔜} 𝔛_{2}) = ^{γ}𝔛_{1} ×_{^{γ}𝔜} ^{γ}𝔛_{2}

for all γ ∈ Γ.

4.3 Case of affine schemes

Assume that 𝔛 is affine, that is 𝔛 = Spk̃A = Hom_{k̃−alg}(A, −). If γ ∈ Γ then

(4.10)  ^{γ}𝔛 = Spk̃A^{γ}^{−1}

as can be seen from (4.3). In particular ^{γ}𝔛 is also affine. Our next step is to show that semilinear twists of schemes are also schemes.

Assume that 𝔜 is an open subfunctor of 𝔛. We claim that^{γ}𝔜 is an open subfunctor of^{γ}𝔛. We must show that for all affine functor Spk̃A and all morphism f : Spk̃A → ^{γ}𝔛 there exists an ideal I of A such that f^{−1}(^{γ}𝔜) = D(I) where

D(I)(R) = {α ∈ Hom(A, R) : Rf(I) = R}.

Let us for convenience denote Spk̃A by 𝔛′, and γ^{−1} by γ′. Our morphism f induces a morphism ^{γ′}f : ^{γ′}𝔛′ → 𝔛 by the considerations described above. Because 𝔜 is open in 𝔛 and ^{γ}′𝔛′ = Sp_{k̃−alg}A^{γ} is affine there exists and ideal I of A^{γ} such that (^{γ′}f)^{−1}(𝔜) = D(I). Applying this to the k̃–algebra R^{γ} we obtain

(4.11)  ^{γ′}f(R^{γ})^{−1}(𝔜(R^{γ}) = {α ∈ Hom_{k̃−alg}(A^{γ}′, R^{γ}) : R^{γ}α(I) = R^{γ}}.

On the other hand ^{γ′}f(R^{γ})^{−1} = f(R)^{−1} and 𝔜(R^{γ}) = ^{γ}𝔜(R). Finally in the right hand side of (4.11) we have Hom_{k̃−alg}(A^{γ}′, R^{γ}) = Hom_{k̃−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 k̃–algebra A this completes the proof that ^{γ}𝔜 is an open subfunctor of ^{γ}𝔛.

If 𝔛 is local then so is ^{γ}𝔛. Indeed, given a k̃-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 as rings), and it is immediate to verify that (R_{f})^{γ} = (R^{γ})_{f}. Using that it is then clear that the sequence

(4.12)  ^{γ}𝔛(R) → ^{γ}𝔛(R_{f_{i}}) ⇉ ^{γ}𝔛(R_{f_{i}f_{j}})

is exact whenever 1 = f_{1} + ⋯ + 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 a family 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 k̃–scheme defined along traditional lines (and not as a special type of k̃–functor), and let 𝔛 also denote the restriction to the category of affine k̃–schemes of the functor of points of 𝔛. If we define (again along traditional lines) ^{γ}𝔛 = 𝔛 ×_{Spec(k̃)} Spec(k̃^{γ}^{−1}), then it can be shown that the functor of points of γ𝔛 (restricted to the category of affine k̃–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 k̃–scheme is an affine group scheme 𝔊. Thus 𝔊 = Spk̃k̃[𝔊] for some k̃–Hopf algebra k̃[𝔊] (see [DG] II §1 for details).

Let ϵ_{𝔊} : k̃[𝔊] → k̃ denote the counit map. As k̃-modules we have k̃[𝔊] = k̃ ⊕ I_{𝔊} where I_{𝔊} is the kernel of ϵ_{𝔊}. Let γ ∈ Γ. As explained in (4.10) we have ^{γ}𝔊 = Spk̃k̃[𝔊]^{γ}^{−1} = Hom_{k̃−alg}(k̃[𝔊]^{γ}^{−1}, −). We leave it to the reader to verify that ϵ_{^{γ} 𝔊} : γ∘ϵ_{𝔊}. As an abelian group I_{𝔊} = I_{^{γ} 𝔊}, but in this last the action of k̃ is obtained through the action of k̃ in k̃[𝔊]^{γ}.

Next we make some relevant observations about Lie algebras from a functorial point of view ([DG] II §4). Recall that the group functor 𝔏𝔦𝔢(𝔊) attaches to an object R in k̃-alg the kernel of the group homomorphism 𝔊(R[ϵ]) → 𝔊(R) where R[ϵ] is the k̃–algebra of dual numbers of R, and the group homomorphism comes from the functorial nature of 𝔊 applied to the morphism R[ϵ] → R in k̃–alg that maps ϵ ↦ 0. By definition Lie(𝔊) = 𝔏𝔦𝔢(𝔊)(k̃). In particular Lie(𝔊) ⊂ 𝔊(k̃[ϵ]) = Hom_{k̃-alg}(k̃[𝔊], k̃[ϵ]). Every element x ∈ Lie(𝔊) is given by

(4.13)  x : a ↦ ϵ_{𝔊}(a) + δ_{x}(a)ϵ

with δ_{x} ∈ Derk̃(k̃[𝔊], k̃) where k̃ is viewed as a k̃[𝔊]–module via the counit map of 𝔊. In what follows we write x = ϵ_{𝔊} + δ_{x}ϵ. The map x ↦ δ_{x} is in fact a k̃-module isomorphism Lie(𝔊) ≃ Derk̃(k̃[𝔊], k̃). In particular if λ ∈ k̃ then λx ∈ Lie(𝔊) is such that δ_{λx} = λδ_{x}.

Similar considerations apply to the affine k̃-group ^{γ}𝔊. We have Lie(^{γ}𝔊) = Derk̃(k̃[𝔊]^{γ}^{−1}, k̃). Note that if y ∈ Lie(^{γ}𝔊) corresponds to δ_{y} ∈ Derk̃(k̃[𝔊]^{γ}^{−1}, k̃), then under the action of k̃ on Lie(^{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 fact that y ∈ Lie(^{γ}𝔊) and that δ_{y} ∈ Derk̃(k̃[𝔊]^{γ}^{−1}, k̃).

4.4 Group functors

Let from now on 𝔊 denote a k̃-group functor. If ℌ is a subgroup functor of 𝔊 we let

Aut_{γ}(𝔊, ℌ) = {f ∈ Aut_{γ}(𝔊) | ^{γ}ℌ = f^{−1}(ℌ)}.

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_{k̃/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_{k̃/k}𝔊).

Proof. As observed in Remark 4.1 the map γ : k̃ → k̃^{γ} is an isomorphism of k̃–alg, and for R in k–alg (R ⊗_{k} k̃)^{γ} = R ⊗_{k} k̃^{γ}. We thus have a k̃–algebra isomorphism 1 ⊗ γ : R ⊗_{k} k̃ → (R ⊗ k̃)^{γ}. For a given f ∈ Aut_{Γ}(𝔊), the composite map

f̃(R) : (Π_{k̃/k}𝔊)(R) = 𝔊(R ⊗_{k} k̃) ⟶^{𝔊(1⊗γ)} 𝔊((R ⊗_{k} k̃)^{γ}) =

= ^{γ}𝔊(R ⊗_{k} k̃) →^{f(R⊗_{k}k̃)} 𝔊(R ⊗_{k} k̃) = (Π_{k̃/k}𝔊)(R)

is an automorphism of the group (Π_{k̃/k}𝔊)(R). One readily verifies that the family f̃ = f̃(R)_{R∈k−alg} is functorial on R, hence an automorphism of Π_{k̃/k}𝔊.

To check that ˜ is a group homomorphism we consider two elements f_{1}, f_{2} ∈ Aut_{Γ}(𝔊) of type γ_{1} and γ_{2} respectively. Recall that γ_{2} induces a k̃-algebra homomorphism 1 ⊗ γ_{2}_{σ} : R ⊗ k̃^{γ} → R ⊗ k̃^{γ_{2}γ} for all σ ∈ Γ [see Remark 4.1(i)]. Since γ will be understood from the context we will denote this homomorphism simply by 1 ⊗ γ_{2}. By functoriality we get the following commutative diagram [see Remark 4.1(1)]

𝔊(R ⊗ k̃^{γ_{1}}) ⟶^{𝔊(1⊗γ_{2})} 𝔊(R ⊗ k̃^{γ_{2}γ_{1}})
↓^{=}  ↓^{=}
γ_{1}𝔊(R ⊗ k̃) ⟶^{^{γ}𝔊(1⊗γ_{2})} ^{γ_{1}}𝔊(R ⊗ k̃^{γ_{2}})
↓^{f_{1}(R⊗k̃)}  ↓^{f_{1}(R⊗k̃^{γ_{2}})}
𝔊(R ⊗ k̃) ⟶^{𝔊(1⊗γ_{2})} 𝔊(R ⊗ k̃^{γ_{2}})

Since f_{2} ∘ f_{1} is of type γ_{2}γ_{1}, by definition we have

{ \atop f_{2} ∘ f_{1}}(R ⊗ k̃) = (f_{2} ∘ f_{1})(R ⊗ k̃) ∘ 𝔊(1 ⊗ γ_{2}γ_{1}).

Thus

{ \atop f_{2} ∘ f_{1}}(R ⊗ k̃) = (f_{2} ∘ f_{1})(R ⊗ k̃) ∘ 𝔊(1 ⊗ γ_{2} ∘ 1 ⊗ γ_{1})

= (f_{2} ∘ f_{1})(R ⊗ k̃) ∘ 𝔊(1 ⊗ γ_{2}) ∘ 𝔊(1 ⊗ γ_{1})

= f_{2}(R ⊗ k̃) ∘ f_{1}(R ⊗ k̃^{γ_{2}}) ∘ 𝔊(1 ⊗ γ_{2}) ∘ 𝔊(1 ⊗ γ_{1})

= f_{2}(R ⊗ k̃) ∘ 𝔊(1 ⊗ γ_{2}) ∘ f_{1}(R ⊗ k̃) ∘ 𝔊(1 ⊗ γ_{1})

= f̃_{2}(R ⊗ k̃) ∘ f̃_{1}(R ⊗ k̃).

Example 4.6. (a) Consider the case of the trivial k̃–group ek̃. Each set Aut_{γ}(ek̃) = Isom(^{γ}ek̃, ek̃) consists of one element which we denote by γ_{∗}. Then Aut_{Γ}(ek̃) ≃ Γ. We have Π_{k̃/k}ek̃ = e_{k}. In particular Aut(Π_{k̃/k}ek̃) = 1 and the homomorphism ˜ : Aut_{Γ}(𝔊) → Aut(Π_{k̃/k}𝔊) is in this case necessarily trivial. In affine terms ek̃ is represented by k̃ and ^{γ}ek̃ by k̃^{γ^{−1}}. Then the k̃–group isomorphism γ_{∗} : ^{γ}ek̃ → ek̃ corresponds to the k̃–Hopf algebra isomorphism γ^{−1} : k̃ → k̃^{γ^{−1}}.

(b) Consider the case when Γ is the Galois group of the extension ℂ/ℝ, and 𝔊 is the additive ℂ–group. Then Aut_{Γ}(𝔊) can be identified with the group of automorphisms of (ℂ, +) which are of the form z ↦ λz or z ↦ λ\overline{z} for some λ ∈ ℂ^{×}. The Weil restriction of 𝔊 to ℝ is the two-dimensional additive ℝ–group. Thus Aut(Π_{k̃/k}𝔊) = GL_{2}(ℝ).

The above examples show that, even if k̃/k is a finite Galois extension of fields and 𝔊 is a connected linear algebraic group over k̃, the homomorphism f ↦ f̃ need be neither injective nor surjective

Corollary 4.7. Assume that 𝔊 = Spk̃k̃[𝔊] be an affine k̃-group. The group Aut_{Γ}(𝔊) acts naturally on the groups 𝔊(k̃) and 𝔊(k̃[ϵ]). Furthermore the action of an element f ∈ Aut_{γ}(𝔊) on 𝔊(k̃[ϵ]) stabilizes Lie(𝔊) ⊂ 𝔊(k̃[ϵ]). The induced map Lie(f) : Lie(𝔊) → Lie(𝔊) is an automorphism of Lie(𝔊) viewed as a Lie algebra over k. This automorphism is k̃–semilinear, i.e., Lie(f)(λx) = (^{γ}λ)Lie(f)(x) for all λ ∈ k̃ and x ∈ Lie(𝔊).

Proof. We maintain the notation and use the facts presented in Remark 4.4. Let x ∈ Lie(𝔊) and write x = ϵ_{𝔊} + δ_{x}ϵ. If λ ∈ k̃ then λx ∈ Lie(𝔊) is such that δ_{λx} = λδ_{x}.

By definition (Π_{k̃/k}𝔊)(k) = 𝔊(k̃) and (Π_{k̃/k}𝔊)(k[ϵ]) = 𝔊(k̃[ϵ]). The action of an element f ∈ Aut_{Γ}(𝔊) on these two groups is then given by the automorphisms f̃_{k} and f̃_{k[ϵ]} of the previous Proposition. Thus if we let γ_{ϵ} : k̃[ϵ] → k̃[ϵ]^{γ} denote the isomorphism of k̃–alg induced by γ the map f̃_{k[ϵ]} is then obtained by restricting to Lie(𝔊) the composite map

𝔊(k̃[ϵ]) = Hom_{k̃−alg}(k̃[𝔊], k̃[ϵ]) →^{𝔊(γ_{ϵ})} Hom_{k̃−alg}(k̃[𝔊], k̃[ϵ]^{γ}) =

= Hom_{k̃−alg}(k̃[𝔊]^{γ}^{−1}, k̃[ϵ]) = ^{γ}𝔊(k̃[ϵ]) →^{f(k̃[ϵ])} Hom_{k̃−alg}(k̃[𝔊], k̃[ϵ]) = 𝔊(k̃[ϵ]).

Using the fact that γ_{ϵ} ∘ ϵ_{𝔊} = γ ∘ ϵ_{𝔊} = ϵ_{^{γ} 𝔊} it easily follows that

(4.14)  Lie(f)(x) = f̃_{k[ϵ]}(x) = f(k̃[ϵ]) ∘ (ϵ_{^{γ} 𝔊} + (γ ∘ δ_{x})ϵ)

Let y = ϵ_{^{γ} 𝔊} + (γ ∘ δ_{x})ϵ ∈ Lie(^{γ}𝔊). If λ ∈ k̃ then we have

Lie(f)(λx) = f(k̃[ϵ]) ∘ (ϵ_{^{γ} 𝔊} + (γ ∘ δ_{λx})ϵ)
= f(k̃[ϵ]) ∘ (ϵ_{^{γ} 𝔊} + (^{γ}(λδ_{x})ϵ)
= f(k̃[ϵ]) ∘ (ϵ_{^{γ} 𝔊} +^{γ}λ(γ ∘ δ_{x})ϵ)
= f(k̃[ϵ])((^{γ}λ)y))

where (^{γ}λ)y is the action of the element ^{γ}λ ∈ k̃ on the element y ∈ Lie(^{γ}𝔊), as explained in the last paragraph of Remark 4.4. Since the restriction of f(k̃[ϵ]) to Lie(^{γ}𝔊) induces an isomorphism Lie(^{γ}𝔊) → Lie(𝔊) of k̃-Lie algebras, this restriction is in particular k̃-linear. It follows that

Lie(f)(λx) = f(k̃[ϵ])((^{γ}λ)y) = (^{γ}λ)f(k̃[ϵ])(y) = (^{γ}λ)Lie(f)(x).

This shows that Lie(f) is semilinear. We leave it to the reader to verify that Lie(f) is an automorphism of Lie(𝔊) as a Lie algebra over k.

Remark 4.8. There is no natural action of Aut_{Γ}(𝔊) on 𝔊.

4.5 Semilinear version of a theorem of Borel-Mostow

Throughout this section k denotes a field of characteristic 0.

Theorem 4.9. (Semilinear Borel-Mostow) Let k̃/k be a finite Galois extension of fields with Galois group Γ. Suppose we are given a quintuple (𝔤, H, ψ, ϕ, (H_{i})_{0≤i≤s}) where

𝔤 is a (finite dimensional) reductive Lie algebra over k̃, H is a group, ψ is a group homomorphism from H into the Galois group Γ,

ϕ 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})_{1≤i≤s} is a finite family of subgroups of H for which the following two conditions hold:

(i) If we let the group H act on 𝔤 via ϕ and on Γ via ψ, namely ^{h}x = ^{ϕ(h)}x and hλ = ^{ψ(h)}λ for all h ∈ H, x ∈ 𝔤, and λ ∈ k̃, 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−1} ⊃ ... ⊃ H_{1} ⊃ H_{0} = 0. Furthermore, each H_{i} is normal in H, the elements of ϕ(H_{i}) are semisimple,^{9} and the quotients H_{i}/H_{i−1} 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_{0} of Γ via ψ. Let k̃_{0} = k̃^{Γ_{0}}. This yields a semilinear action of Γ_{0} on 𝔤. By Galois descent the fixed point 𝔤^{Γ_{0}} is a Lie algebra over k̃_{0} for which the canonical map ρ : 𝔤^{Γ_{0}} ⊗k̃_{0} k̃ ≃ 𝔤 is a k̃–Lie algebra isomorphism. If 𝔥_{0} is a Cartan subalgebra of 𝔤^{Γ_{0}} then ρ(𝔥_{0} ⊗k̃_{0} k̃) is a Cartan subalgebra of 𝔤 which is H–stable as one can easily verify with the aid of assumption (i).

Assume s ≥ 1 and consider a generator θ of the cyclic group H_{1}. As we have already observed the action of θ on 𝔤 is k̃–linear. If V is a k̃–subspace of 𝔤 stable under θ we will denote by V^{θ} the subspace of fixed points. Before continuing with we establish the following crucial fact:

Claim 4.10. 𝔤^{θ} is a reductive Lie algebra over k̃. 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 k̃–Lie algebra 𝔤 we see that 𝔤^{θ} is indeed a 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 induces by restriction automorphisms (also denoted by θ) of 𝔤′ and of 𝔷. By [Bbk] Ch. 8 §1 cor. to prop. 12. (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 Cartan subalgebra 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_{1} is normal in H we have an induced action (via ϕ) of H′ = H/H_{1} on the reductive k̃–Lie algebra 𝔤^{θ}. We have induced group homomorphisms ϕ′ : H′ → Aut_{k}(𝔤^{θ}) and ψ′ : H′ → Γ [this last since H_{1} ⊂ ker(ψ)]. For 0 ≤ i < s define H′_{i} = H_{i+1}/H_{1}. We apply the induction assumption to the quintuple (𝔤^{θ}, H′, ψ′, ϕ′, (H′_{i})_{0≤i≤s−1}). This yields the existence of a Cartan subalgebra 𝔥 of 𝔤^{θ} which is stable under the action of H′ given by ϕ′. This means 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 now conclude by (4.10) .

9Because H_{i} ⊂ ker(ψ) the action of the elements of H_{i} on 𝔤 is k̃–linear. The assumption is that ϕ(θ) be semisimple as a k̃–linear endomorphisms of 𝔤 for all θ ∈ H_{i}.

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 direct proof of this result.

We shall use the above semilinear version of Borel-Mostow’s theorem 4.12 to establish the following result which will play a crucial role in the the proof of the existence of maximal tori on twisted groups corresponding to loop torsors.

Corollary 4.12. Let k̃/k be a finite Galois extension with Galois group Γ. Let G be a reductive group over k̃. Let H be a group, and assume we are given a group homomorphism ρ : H → Aut_{Γ}(G) for which we can find a family of subgroups (H_{i})_{0≤i≤s} of H as in the Theorem, that is ker(t ∘ ρ) = H_{s} ⊃ H_{s−1} ⊃ ... ⊃ H_{1} ⊃ H_{0} = 0 where t : Aut_{Γ}(G) → Γ is the type morphism, each H_{i} is normal in H, the elements of ρ(H_{i}) act semisimply on the k̃–Lie algebra Lie(G), and the quotients H_{i}/H_{i−1} 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 have the following commutative diagram.

G(k̃) ⟶^{G(γ)} G(k̃^{γ}) = ^{γ}G(k̃)  ⟶^{ρ(h)(k̃)} G(k̃)
↓  ↓  ↓
G(k̃[ϵ]) ⟶^{G(γ_{ϵ})} G(k̃[ϵ]^{γ}) = ^{γ}G(k̃[ϵ]) ⟶^{ρ(h)(k̃[ϵ])} G(k̃[ϵ])
↑  ↑  ↑
Lie(G) ⟶^{𝔏𝔦𝔢(𝔊)(γ)}  Lie(G^{γ})  ⟶^{𝔏𝔦𝔢(𝔊)(ρ(h))} Lie(G).

where we have denoted by γ_{ϵ} : k̃[ϵ] → k̃[ϵ]^{γ} the k̃-algebra isomorphism induced by γ. For convenience in what follows we will denote Lie(G) by 𝔤. By Corollary 4.7 we obtain by composing ρ with the map ˜ defined in Proposition 4.5 a group homomorphism ϕ : H → Aut_{k}(𝔤), namely ϕ(h) = ρ̃(h), which together with the group homomorphism ψ = t ∘ ρ : H → Γ and the H_{i} satisfy the assumptions of Theorem 4.9. It follows that there exists a Cartan subalgebra 𝔱 of 𝔤 which is stable under the action of H defined by ϕ.

Note that by definition

(4.15)  ρ̃(h) = ρ(h)(k̃) ∘ G(γ)

which is nothing but the top row of our diagram above. Similarly with the notation of Corollary 4.7 we have

(4.16)  Lie(ρ̃(h)) = { \atop ρ_{ϵ}(h)}|_{𝔤} = 𝔏𝔦𝔢(G)(ρ(h)) ∘ 𝔏𝔦𝔢(G)(γ)

where { \atop ρ_{ϵ}(h)} stands for the middle row of our diagram, namely ρ(h)(k̃[ϵ]) ∘ G(γ_{ϵ}).

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.^{10}

Given an element g ∈ G(k̃) we will denote its natural image in G(k̃[ϵ]) by g_{ϵ}. Since we are working over a base field the k̃-points of T = Z_{G}(𝔱) can be computed in the naive way, namely

(4.17)  T(k̃) = {g ∈ G(k̃) : g_{ϵ}xg_{ϵ}^{−1} = x for all x ∈ 𝔱 ⊂ G(k̃[ϵ])}

Since ρ̃_{ϵ}(h) is an automorphism of the (abstract) group G(k̃[ϵ]) we obtain

(4.18) T(k̃) = {g ∈ G(k̃) : ρ̃_{ϵ}(h)(g_{ϵ})ρ̃_{ϵ}(h)(x)(ρ̃_{ϵ}(h)(g_{ϵ}))^{−1} = ρ̃_{ϵ}(h)(x) for all x ∈ 𝔱}

But since ρ̃_{ϵ}(h) stabilizes 𝔱 this last reads

(4.19)  T(k̃) = {g ∈ G(k̃) : ρ̃_{ϵ}(h)(g_{ϵ})x(ρ̃_{ϵ}(h)(g_{ϵ}))^{−1} = x for all x ∈ 𝔱}

Note that by the commutativity of the top square of our diagram we have (ρ̃(h)(g))_{ϵ} = ρ̃_{ϵ}(h)(g_{ϵ}). Thus from (4.19) we obtain that ρ̃_{ϵ}(h)(T(k̃)) = T(k̃). On the other hand by

(4.15) we have ρ̃_{ϵ}(h)(T(k̃)) = ρ(h)(k̃)(G(γ)(T(k̃))). But by definition (G(γ)(T(k̃))) =

γT(k̃). Thus our k̃-group homomorphism ρ(h) : ^{γ}G :→ G is such that the two tori ρ(h)(^{γ}T) and T of G have the same k̃-points. This forces ρ(h)(^{γ}T) = T.

Next we give a crucial application of the semilinear considerations developed thus far to the existence of maximal tori for certain loop groups.

10We 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).

4.6 Existence of maximal tori in loop groups

We come back to the case of R = R_{n} = k[t_{1}^{±1}, …, t_{n}^{±1}] where k is a field of characteristic zero. This is the ring that plays a central role in all applications to infinite dimensional Lie theory. It is not true in general that a reductive R_{n}–group admits a maximal torus; however.

Proposition 4.13. Let 𝔊 be a loop reductive group scheme over R_{n} (see definition 3.4). 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 R̃ = k̃[t_{1}^{±1/m}, …, t_{n}^{±1/m}] where m is a positive integers and k̃/k is a finite Galois extension of fields containing all primitive m-th roots of unity of \overline{k}. Recall from Example 2.3 that R̃ is a Galois extension of R with Galois group Γ̃ = (ℤ/mℤ)^{n} ⋊ Γ as follows: For e = (e_{1}, …, e_{n}) ∈ ℤ^{n} we have \overline{e}(λtj^{\frac{1}{m}}) = λξ_{m}^{e_{j}}tj^{\frac{1}{m}} for all λ ∈ k̃, where − : ℤ^{n} → (ℤ/mZ)^{n} is the canonical map, while the Galois group Γ = Gal(k̃/k) acts naturally on R̃ through its action on k̃.

Let G_{0} be the Chevalley k–form of 𝔊 (see §3.2). By assumption, we can assume that 𝔊 is the twist of 𝔊_{0} = G_{0} ×_{k} R by a loop cocycle

u : Γ̃ → Aut(G_{0})(k̃).

The homomorphism ψ : Γ̃ = (ℤ/mℤ)^{n}⋊Γ → Γ is defined to be the natural projection. For convenience we will adopt the following notational convention. The elements of Γ̃ will be denoted by γ̃, and the image under ϕ of such an element (which belongs to Γ), by the corresponding greek character: that is ψ(γ̃) = γ.

Consider the reductive k̃–group G = Spec(k̃[G_{0}]) where, as usual, k̃[G_{0}] denotes the k̃–Hopf algebra k̃ ⊗_{k} k[G_{0}]. Consider for each γ̃ the map f(γ̃) : k̃[G] → k̃[G] defined by

(4.20)  f(γ̃) = uγ̃ ∘ γ.

Since each uγ̃ is an automorphism of the k̃–Hopf algebra k̃[G], it follows that f(γ̃) is in fact a k̃-Hopf algebra isomorphism k̃[G] → k̃[G]^{γ}. As such it can be thought, by Yoneda considerations and (4.10), as an element of Aut_{γ}(G) of type γ^{−1} which we will denote by ρ(γ̃).

Since the restriction of the action of γ̃ on R̃[G] to k̃[G] is given by γ, the cocycle condition on u shows that for all α̃, β̃ ∈ Γ̃ we have

(4.21)  ρ(α̃β̃) = ρ(α̃)ρ(β̃)

where this last product takes place in Aut_{Γ}(G). Thus ρ is a group homomorphism and f(γ̃) can be viewed as a k̃-Hopf algebra morphism from k̃[G] to k̃[G]^{γ}

¿From (4.21), the various definitions and the “anti equivalent” nature of Yoneda’s correspondence it follows that the map γ̃ → ρ(γ̃) can be viewed as a group homomorphism ρ : Γ̃^{opp} → Aut_{Γ}(G), where Γ̃^{opp} is the opposite group of Γ̃. Since ρ(γ̃) is of type γ^{−1} we can complete the necessary semilinear picture by defining ϕ : Γ̃^{opp} → Γ to be the map γ̃ → γ^{−1}. The kernel of the composite map t ∘ ρ is precisely (ℤ/mℤ)^{n}, and the elements of this kernel act trivially on k̃[G], in particular their corresponding action on the Lie algebra of G is trivial, hence semisimple. We can thus apply Corollary 4.12; the role of H now being played by Γ̃^{opp}.

Let T be a torus G such that ρ(γ̃)(^{γ}^{−1}T) = T for all γ̃ ∈ Γ̃. The torus T corresponds to a Hopf ideal I of the Hopf k̃–algebra k̃[G] representing G. Each ρ(γ̃), which corresponds to the k̃-Hopf algebra isomorphism f(γ̃) described in (4.20), induces a k̃–Hopf algebra isomorphism \overline{f}(γ̃) from k̃[T] to k̃[T]^{γ} where k̃[G]/I = k̃[T] is the Hopf algebra representing T. For future use we observe that the resulting action of Γ̃ on k̃[T] is Γ–semilinear in the sense that if λ ∈ k̃ and a ∈ k̃[T] then

(4.22)  \overline{f}(γ̃)(λa) = \overline{f}(γ̃)(λ_{k̃[T]}.a) = λ_{k̃[T]^{γ}}.(\overline{f}(γ̃)(a)) = (^{γ}λ)\overline{f}(γ̃)(a)

This follows immediately from the definition of f(γ̃).

Consider the reductive R̃-group 𝔊̃ = G ×k̃ R̃ and its maximal torus 𝔗̃ = T ×k̃ R̃. We want to define an action of Γ̃ as automorphisms of the R–Hopf algebra R̃[𝔗] = k̃[T] ⊗k̃ R̃ so that the action is Γ̃–semilinear, this is

(4.23)  γ̃(xs) = ^{γ̃}x^{γ̃}s

for all γ̃ ∈ Γ̃, s ∈ R̃ and x ∈ R̃[𝔗]. By Galois descent this will show that the maximal torus 𝔗̃ of 𝔊̃ descends to a torus (necessarily maximal) 𝔗 of 𝔊.

To give the desired semilinear action consider, for a given fixed γ̃ ∈ Γ̃, the map

k̃[T] × R̃ → k̃[T] ⊗k̃ R̃ = R̃[𝔗]

defined by

(4.24)  (a, s) ↦ \overline{f}(γ̃)(a) ⊗ γ̃s

for all a ∈ k̃[T] and s ∈ R̃. From (4.22) and the fact that ^{γ̃}s = ^{γ}s if s ∈ k̃ ⊂ R̃ it follows that the above map is k̃-balanced, hence that induces a morphism of k̃-spaces

(4.25)  f̂(γ̃) : k̃[T] ⊗k̃ R̃ = R̃[𝔗] → R̃[𝔗]

satisfying

(4.26)  f̂(γ̃) : a ⊗ s ↦ \overline{ρ}(γ̃)(a) ⊗^{γ̃}s

for all a ∈ k̃[T] and s ∈ R̃. From (4.22) and (4.26) we then obtain an action of the group Γ̃ on the Hopf algebra R̃[𝔗] as prescribed by (4.23).

4.7 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 algebraic group 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^{1}(Γ, H(k′)) → H^{1}(Γ, (H/U)(k′))

is bijective.

2. Let R be an object in k–alg. Then the map

H^{1}(R, H) → H^{1}(R, H/U)

is bijective.

Proof. The k–group U admits a non-trivial characteristic central split unipotent subgroup U_{0} ≃ G_{a}^{n} [DG, IV.4.3.13]. We can then form the following commutative diagram of exact sequence of algebraic k–groups

1

↓

1  U/U_{0}
↓  ↓
1 ⟶ U_{0} ⟶ H ⟶ H/U_{0} ⟶ 1
↓  ≅↓  ↓
1 ⟶ U ⟶ H ⟶ H/U ⟶ 1
↓  ↓
U/U_{0}  1

↓

1

If the Lemma holds for the morphisms H → H/U_{0} and H/U_{0} → H/U, it holds for H → H/U. Without loss of generality, we can therefore assume by devissage that U = G_{a}^{n}.

(1) Since H^{1}(k′, U) = 0, we have an exact sequence of Γ–groups

1 → U(k′) → H(k′) → (H/U)(k′) → 1.

For each c ∈ Z^{1}(Γ, (H/U)(k′)), the group _{c}(U(k′)) is a uniquely divisible abelian group, so H^{i}(Γ,_{c}(U(k′))) = 0 for all i > 0. By applying a basic result on non-abelian cohomology [Se, §I.5, corollary to prop. 41], the vanishing of these H^{2} implies that the map H^{1}(Γ, H(k′)) → H^{1}(Γ, (H/U)(k′)) is surjective. Similarly, for each z ∈ Z^{1}(Γ, H(k′)), the group 0 = H^{1}(Γ,_{z}(U(k′))) maps onto the subset of H^{1}(Γ, H(k′)) consisting of classes of cocycles whose image in H^{1}(Γ, (H/U)(k′)) coincides with that of [z]. We conclude that the map H^{1}(Γ, H(k′)) → H^{1}(Γ, (H/U)(k′)) is bijective.

(2) Let us first prove the injectivity by using the classical torsion trick. We are given a H/U–torsor 𝔈 over Spec(R). We can twist the exact sequence of R–group schemes 1 → U_{R} → H_{R} → H_{R}/U_{R} → 1 by 𝔈 and get the twisted sequence 1 → _{𝔈}U → _{𝔈}H → 𝔈H/_{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^{1}(R, H) ⟶ H^{1}(R, H/U)
torsion↑^{≃}  ^{torsion}↑^{≃}
H^{1}(R,_{𝔈}U) ⟶ H^{1}(R,_{𝔈}H) ⟶ H^{1}(R,_{𝔈}H/_{𝔈}U)

where the bottom map is an exact sequence of pointed sets. Indeed GL_{n} is the group of automorphisms of the group scheme G_{a}^{n} (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 Ȟ^{i}(R,_{𝔈}U) = 0 for all i > 0.^{11} So the the map H^{1}(R,_{𝔈}H) → H^{1}(R,_{𝔈}H/_{𝔈}U) has trivial kernel and the fiber of H^{1}(R, H) → H^{1}(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

Δ([𝔈]) ∈ Ȟ^{2}(R,_{𝔈}U)

which is the obstruction to the existence of a lift of [𝔈] to H^{1}(R, H). Here _{𝔈}U is the R–group scheme obtained by twisting G_{a}^{n} by the R–torsor 𝔈. Since _{𝔈}U corresponds to a locally free sheaf, the same reasoning used above shows that the obstruction Δ([𝔈]) vanishes as desired.

Lemma 4.15. Let 𝔛 be a scheme of characteristic 0. Let E be a locally free 𝔛–sheaf of finite rank and let V(E) be the associated “additive” 𝔛–group scheme. Then the natural homomorphism of fpqc sheaves

α : GL(E) → Aut_{𝔛−gr}(V(E))

11All the Ȟ^{i} that we consider coincide with the corresponding H^{i} defined in terms of derived functors.

is an 𝔛–group sheaf isomorphism. In particular, Aut_{𝔛−gr}(V(E)) is an 𝔛–group scheme.

Our convention is that of [DG, §2], namely V(E)(𝔛′) = H^{0}(𝔛′, E ⊗_{O_{𝔛}} O_{X′}) for every scheme 𝔛′ over 𝔛.

Proof. It is clear that α is an morphism of 𝔛–groups. For showing that α is an isomorphism 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 suffice to establish the result for R = \overline{ℚ} and E = (\overline{ℚ})^{n}. Now on \overline{ℚ}-schemes the functor S ↦ Aut_{gr}(V(E))(S) is representable by a linear algebraic \overline{ℚ}-group H according to Hochschild-Mostow’s criterion [HM, th. 3.2]. Therefore we can check the fact that α : GL_{n} → H is an isomorphism on \overline{ℚ}–points. But this readily follows from the equivalence of categories between nilpotent Lie algebras and algebraic unipotent groups [DG, §IV §2 cor.4.5]. Since GL(E) is an 𝔛–group scheme, α is an isomorphism of 𝔛–group schemes.

5 Maximal tori of group schemes over the punctured line

Let G be a linear algebraic k–group. One of the central results of [CGP] is the existence of maximal tori for twisted groups of the form_{𝔈}G where [𝔈] ∈ H^{1}(k[t^{±1}], G).^{12} This result is used to describe the nature of torsors over k[t^{±1}] under G. In our present work we are ultimately interested in the classification of reductive groups over Laurent polynomial rings when k is of characteristic 0, and applications to infinite dimensional Lie 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] result mentioned above, but for arbitrary twisted groups, not just inner forms.^{13} This is one of the crucial theorems of our paper.

Theorem 5.1. Let R = k[t^{±1}] where k is a field of characteristic 0. Every reductive group 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 scheme over R whose connected component of the identity 𝔊^{0} is reductive. Then 1. H_{toral}^{1}(R, 𝔊) = H^{1}(R, 𝔊).

12If the characteristic of k is sufficiently large.

13Aut(G) need not be an algebraic group. Even if it is, the fact that it need not be connected leads to considerable technical complications (stemming from the fact that, unlike the affine line, the punctured line has non-trivial geometric étale coverings). As already mentioned, these difficulties have to be dealt with if one is interested in the study of twisted forms of G_{R} or its Lie algebra.

2. If 𝔊 is constant, i.e. 𝔊 = G ×_{k} R for some linear algebraic k–group G, then

H_{toral}^{1}(R, 𝔊) = H_{loop}^{1}(R, 𝔊) = H^{1}(R, 𝔊).

The first assertion is an immediate corollary of the Theorem while the second then follows from Corollary 3.16.2 and Lemma 2.8.

The proof of the Theorem relies on Bruhat-Tits twin buildings and Galois descent considerations. We begin by establishing the following useful reduction.

Lemma 5.3. It suffices to establish Theorem 5.1 under the assumption that 𝔊 is a twisted form of a simple simply connected Chevalley R–group.^{14}

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 𝔊̃_{ad} of 𝔊_{ad}. We can thus assume without lost of generality that 𝔊 is simply connected. By [XXIV.5.10] we have

𝔊 = ∏_{i=1,...,l} ∏_{S_{i}/R} 𝔊_{i}

where each S_{i} is a connected finite étale covering of R and each 𝔊_{i} a simple simply connected S_{i}–group scheme. By Demazure’s main theorem, the S_{i}–groups 𝔊_{i} are twisted forms of simple simply connected Chevalley groups. Since by Lemma 2.8 S_{i} is a Laurent polynomial ring, our hypothesis implies that each of the S_{i}–groups 𝔊_{i} admits a maximal torus 𝔗_{i} Then our R–group 𝔊 admits the maximal torus ∏_{i=1,...,l} ∏_{S_{i}/R} 𝔗_{i}.

5.1 Twin buildings

Throughout this section k denotes a field of characteristic 0. We set R = k[t^{±1}], K = k(t) and K̂ = K((t)). For a “survival kit” on euclidean buildings, we recommend Landvogt’s paper [L].

Let R̃ be a finite Galois extension of R of the form R̃ = k̃[t^{± \frac{1}{n}}] where k̃/k is a finite Galois extension of k containing all n-roots of unity in \overline{k}. Then as we have already seen Γ̃ := Gal(R̃/R) = μ μ μ_{n}(k̃) ⋊ Γ where Γ = Gal(k̃/k).

Set t̃ = t^{\frac{1}{n}} We let L = k̃( t̃) = k̃(t̃^{−1}), and consider the two completions L̂_{+} = k̃(( t̃)) and L̂_{−} = k̃((t̃^{−1})) of L at 0 and ∞ respectively, as well as their corresponding valuation rings Â_{+} = k̃[[ t̃]] and Â_{−} = k̃[[t̃^{−1}]].

14The usual algebraic group literature would use the term “almost simple” in this situation. We adhere throughout to the terminology of [SGA3].

Let G be a split simple simply connected group over k. Let T be a maximal split torus of G, B^{+} a Borel subgroup of G which contains T, and B^{−} the corresponding opposite Borel subgroup (which also contains T). We denote by W = N_{G}(T) the corresponding Weyl group and by Δ_{±} the Dynkin diagram attached to (G, B^{±}, T).

Following Tits [T3], we consider the twin building B = B_{+} × B_{−} of G_{0} ×_{k} L with respect to the two completions L̂_{+} and L̂_{−}. Recall that B comes equipped with an action of the group G(L), hence also of G(R̃). The split torus T_{0} ×_{k} L gives rise to a twin apartment A = A_{+} × A_{−} of B. The Borel subgroups B^{±} define the fundamental chambers C_{±} of A_{±}, each of which is an open simplex whose vertices are given by the extended Dynkin diagram Δ̃_{±} of Δ_{±}.

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.^{15} This leads to an action of 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 account the natural action of Γ̃ ≃ Gal(L̂_{+}/K̂) on B we conclude that the twin building B is equipped with an action of the semi-direct product Aut(G)(R̃) ⋊ Γ which is compatible (via the adjoint action) with the action of G(R̃).

The hyperspecial group G(Â_{±}) fixes a unique point ϕ_{±} of A_{±} [BT1, §9.1.19.c]. Recall that the hyperspecial points of B_{±} are G(L̂_{±})-conjugate to ϕ_{±} of B_{±}, and can therefore be identified with the set of left cosets

G(L̂_{±})/G(Â_{±}) ≃ G(L̂_{±}). ϕ_{±} ⊂ B_{±}.

More generally each facet of the building B_{±} has a type [BT1, §2.1.1 ] which is a subset of Δ̃_{±} and the type of a point x ∈ B_{±} is the type of its underlying facet F_{x}.^{16} The type of the chamber C_{±} is ∅ and the type of an hyperspecial point is Δ̃_{±} \ Δ_{±}, namely the extra vertice of the affine Dynkin diagram.

5.2 Proof of Theorem 5.1

By Lemma 5.3, we can assume that 𝔊 is simple simply connected. By the Isotriviality Theorem [GP1, cor. 2.16], we know that our R–group 𝔊 is isotrivial. 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 our terminology, G is the Chevalley k–form of 𝔊.

Because of the structure of the algebraic fundamental group of R we may assume without loss of generality that S = R̃ is as in §5.1, and we keep all the notation therein. What is so special about this situation is that R and R̃ “look the same”, namely they are both Laurent polynomial rings in one variable with coefficients in a field.

15The group in question acts on the set of maximal split tori, hence permutes the apartments around.

16Namely the smallest facet containing x in its closure.

We have Spec(R̃) = ℙ1 ; k̃ \ {0, ∞} and the action of Γ̃ on R̃ extends to ℙ1 ; k̃ since R̃ is regular of dimension 1.

For γ̃ ∈ Γ̃ consider the map zγ̃ = f^{−1} ∘ γ̃f : Γ̃ → Aut(G)(R̃), where Aut(G) stands for the group scheme of automorphisms of the ℤ-group G. Then z = (zγ̃)_{γ̃∈Γ̃} is a cocycle in Z^{1}(Γ̃, Aut(G)(R̃)) where the Galois group Γ̃ acts naturally on Aut(G)(R̃) via its action on R̃. Descent theory tells us that G is isomorphic to the twisted Rgroup _{z}G.^{17}

The action of Γ̃ on Aut(G)(R̃) allows us to consider the semidirect product group Aut(G)(R̃) ⋊ Γ̃. We then have a group homomorphism

(5.1)  ψ_{z} : Γ̃ → Aut(G)(R̃) ⋊ Γ̃

given by ψ_{z}(γ) = z_{γ} γ which is a section of the projection map Aut(G)(R̃) ⋊ Γ̃ → Γ̃.

Let T be a maximal split of G. Set L = k̃(t̃), and let A_{+} (resp. A_{−}) be the local ring of ℙ1 ; k̃ at 0 (resp. ∞). The composite map (see §5.1)

Γ̃ ⟶^{ψ_{z}} Aut(G)(R̃) ⋊ Γ̃ → Aut(B)

is a group homomorphism. The corresponding action of Γ̃ on B will be referred to as the twisted action of Γ̃ on the building. We now appeal to the Bruhat-Tits fixed point theorem [BT1, §3.2] to obtain a point p = (p_{+}, p_{−}) ∈ B which is fixed under the twisted action, i.e. ψ_{z}(γ̃).p = zγ̃ γ̃(p) = p for all γ̃ ∈ Γ̃. Abramenko’s result [A, Proposition 5] states that G(R̃).A = B. There thus exists g ∈ G(R̃) such that p belongs to the apartment g.A. Up to replacing zγ̃ by Int(g)^{−1} zγ̃ Int(γ̃(g)), we can therefore assume that p belongs to A.

We shall use several times that Γ̃ acts trivially on A. Here since γ̃(p) = p, it follows that zγ̃ . p = p for all γ̃ ∈ Γ̃.

Let F_{p_{±}} be the facet associated to p_{±} and choose a vertex q_{±} of \overline{F}_{p_{±}}. The transforms of q_{±} by zγ̃ and γ̃ are vertices of \overline{F}_{p_{±}}, so ψ_{z}(γ̃).q_{±} belongs to A. We define

x_{±} = Barycentre(ψ_{z}(γ̃).q_{±}, γ̃ ∈ Γ̃) ∈ A.

Let d be the integer attached to G in [Gi1, §2], and set m = d | Γ̃ |. Let s ∈ \overline{L} (a fixed algebraic closure of L) be such that s^{m} = t̃. We have accordingly s^{mn} = t. Set

R′ = k′[ s^{±1} ]

17Recall that for convenience _{z}G is shorthand notation for _{z}(G ×_{k} R) = _{z}(G_{R}).

where k′ is a Galois extension of k which contains k̃ and all mn-roots of unity in \overline{k}. Then R′ is Galois over R of Galois group

Γ′ = μ μ μ_{mn}(k′) ⋊ Gal(k′/k).

By Galois theory the map υ : Γ′ → Γ̃ given by

(5.2)  υ : (ξ, θ) ↦ (ξ^{m}, θ_{|k̃})

is a surjective group homomorphism.

We consider the twin building B′ = B′_{+} × B′_{−} of G which is constructed in the manner described above after replacing, mutatis mutandis, the relevant objects attached to R̃ 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′) = γ̃ then the following diagram commutes

(∗) \begin{matrix}B & ⟶^{ρ} & B′ \\ γ̃↓ & & ^{γ′}↓ \\ B & ⟶^{ρ} & B′ \end{matrix}

where the actions of Γ̃ and Γ′ are the twisted actions. If we define z′ : Γ′ → Aut(G)(R′) by

z′ : γ′ ↦ z′_{γ′} = zγ̃ ∈ Aut(G)(R̃) ⊂ Aut(G)(R′)

then z′ is a cocycle and the classes [z] and [z′] in H^{1}(R, Aut(G)) are the same.

Lemma 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

L̂_{+} = k̃(( t̃)) ⊂ k′(( t̃)) ⊂ k′(( s^{d})) ⊂ k′(( s)).

The map ρ_{+} is the composite of the corresponding maps for the intermediate fields, namely

B_{+} = B_{+}(G_{0, k̃(( t̃))}) ⟶^{ρ_{1,+}} B(G_{0, k′(( t̃))}) ⟶^{ρ_{d,+}} B(G_{0, k′(( s^{d}))}) ⟶^{ρ_{\frac{m}{d} ,+}} B(G_{0, k′(( s))})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_{+}[Barycenter(ψ_{z}(γ̃).q_{±}, γ ∈ Γ̃)]
= ρ_{\frac{m}{d} ,+}[Barycenter(ρ_{d,+} ∘ ρ_{1,+}(ψ_{z}(γ̃).q_{±}), γ ∈ Γ̃)].

By [Gi1, Lemma 2.3’ in the errata], we know that the image under ρ_{\frac{m}{d} ,+} of the barycentre of \frac{m}{d} hyperspecial points of a common apartment (namely the one attached to the torus T) is a hyperspecial point, so we conclude that ρ_{+}(x_{+}) is a hyperspecial point.

In view of diagram (*) above it follows that by replacing R̃ by R′ we may assume without loss of generality that the points p_{±} ∈ A_{±} are hyperspecial. Note that by construction, the points ρ_{±}(x_{±}) of B′_{±} are fixed by both actions (standard and twisted) 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 Γ̃.

Since T(R̃) acts transitively on the sets ϕ_{±} + (T̂)^{0} ⊂ A_{±} of hyperspecial points of A_{±}, there exists g ∈ T(R̃) and a cocharacter λ ∈ (T̂)^{0} such that

g.(ψ_{+}, x_{−}^{λ}) = (x_{+}, x_{−}) = x.

where x_{−}^{λ} := ϕ_{−} + λ (recall that we have a map (T̂)^{0} → A_{−} = (T̂)^{0} ⊗_{ℤ} ℝ defined by θ ↦ ϕ_{−} + θ). Up to replacing the cocycle z by z′ where z′ ; γ̃ = Int(g)^{−1} zγ̃ ^{γ̃}Int(g), we may assume that ψ_{z}(γ̃).(ϕ_{+}, x_{−}^{λ}) = z_{γ}.(ϕ_{+}, x_{−}^{λ}) = (ϕ_{+}, x_{−}^{λ}) for every γ̃ ∈ Γ̃. In particular,

zγ̃ ∈ Stab_{Aut(G)(L̂_{+})}(ϕ_{+}) = Aut(G)(Â_{+})

hence

zγ̃ ∈ Aut(G)(R̃) ∩ Aut(G)(Â_{+})Aut(G)(k̃[ t̃])

for each γ̃ ∈ Γ̃. Let g_{λ} := λ( t̃) ∈ T(R̃) ⊂ G(R̃). We have g_{λ}.ϕ_{−} = x_{−}^{λ} and therefore

zγ̃ ∈ Stab_{Aut(G)(L̂_{−})}(x_{−}^{λ})
Int(g_{λ}) Stab_{Aut(G)(L̂_{−})}(ϕ_{−}) Int(g_{λ})^{−1} = Int(g_{λ})Aut(G)(Â_{−})Int(g_{λ}^{−1}).

It follows that for each γ̃ ∈ Γ̃

zγ̃ ∈ J_{λ} := Aut(G)(k̃[ t̃]) ∩ Int(g_{λ}) Aut(G)(Â_{−}) Int(g_{λ}^{−1})
= Aut(G)(k[ t̃]) ∩ Int(g_{λ}) Aut(G)(k̃[ t̃^{−1} ]) Int(g_{λ}^{−1}).

Note that for γ̃ ∈ Γ̃ we have

γ̃g_{λ} = ^{γ̃}λ(t′) = λ(t′) (λ(t′)^{−1} ^{γ̃}λ(t′)) ∈ λ(t′)T(k̃) ⊂ λ(t′)G(Â_{−}) = g_{λ} G(Â_{−})

From this it follows not only that the subgroup J_{λ} of Aut(G)(R̃) is stable under the (standard) action of Γ̃, but also that

[z] ∈ Im(H^{1}(Γ̃, J_{λ}) → H^{1}(Γ̃, Aut(G)(R̃))).

It turns out that the structure of the group J_{λ} is known by a computation of Ramanathan, as we shall see in Proposition 16.2 below. We have

J_{λ} = U_{λ}(k̃) ⋊ Z_{Aut(G)}(λ)(k̃) ⊂ Aut(G)(k̃[ t̃]),

where U_{λ} is a unipotent k–group. Lemma 4.14 shows that the map

H^{1}(Γ̃, Z_{Aut(G)}(λ)( k̃)) → H^{1}(Γ̃, J_{λ})

is bijective. Summarizing, we have the commutative diagram

H^{1}(Γ̃, Z_{Aut(G)}(λ)( k̃)) ⟶ H^{1}(Γ̃, Aut(G)( k̃))
↓≅  ↓
H^{1}(Γ̃, J_{λ})  ⟶ H^{1}(Γ̃, Aut(G)(R̃))

which shows that

[z] ∈ Im(H^{1}(Γ̃, Aut(G)( k̃)) → H^{1}(Γ̃, Aut(G)(R̃))).

This means that [z] is cohomologous to a loop cocycle, and we can now conclude by Proposition 4.13

6 Internal characterization of loop torsors and applications

We continue to assume that our base field k of characteristic zero. Let R_{n} = k[t_{1}^{±1}, ..., t_{n}^{±1}] and 𝔛 = Spec(R_{n}). As explained in Example 2.5 we have π_{1}(R_{n}, a) ≃ ℤ̂^{n}⋊Gal(k), where the action of Gal(k) on ℤ̂^{n} is given by our fixed choice of compatible roots of unity in \overline{k}. For convenience in what follows we will denote π_{1}(R_{n}, a) simply by π_{1}(R_{n}).

Throughout this section G denotes a linear algebraic k–group.

6.1 Internal characterization of loop torsors

We first observe that loop torsors make sense over R_{0} = k, namely H_{loop}^{1}(R_{0}, G) is the usual Galois cohomology H^{1}(k, G).

Section 3.3 shows that Z^{1}(π_{1}(R_{n}), G(\overline{k})) is given by couples (z, η^{geo}) where z ∈ Z^{1}(Gal(k), G(\overline{k})) and η^{geo} ∈ Hom_{Gal(k)}(π_{1}(\overline{X}, \overline{a}),_{z}G(\overline{k})) = Hom_{Gal(k)}(ℤ̂^{n},_{z}G(\overline{k})) ≃ Hom_{Gal(k)}(_{∞}μ μ μ,_{z}G). We are now ready to state and establish the internal characterization of k-loop torsors as toral torsors.

Theorem 6.1. Assume that G^{0} is reductive. Then H_{toral}^{1}(R_{n}, G) = H_{loop}^{1}(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 ≥ 1 the natural abstract group homomorphisms

H^{i}(π_{1}(R_{n}), ℌ(\overline{R}_{n,∞})) → H^{i}(R_{n}, ℌ) → H^{i}(F_{n}, ℌ).

are all isomorphisms.

2) Let T be a k–torus. Let c ∈ Z^{1}(π_{1}(R_{n}), Aut(T)(\overline{k})) ⊂ Z^{1}(π_{1}(R_{n}), Aut(T)(\overline{R}_{n,∞})) be a cocycle, and consider the twisted R_{n}–torus _{c}T = c(T ×_{k} R_{n}). Consider the natural maps

H^{i}(π_{1}(R_{n}), _{c}(T(\overline{k})) → H^{i}(π_{1}(R_{n}),_{c}T(\overline{R}_{n,∞})) → 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 an isomorphism.

(ii) If i > 1 then all the maps are group isomorphisms.

Proof. 1) The second isomorphism is proposition 3.4.3 of [GP2]. As for the first isomorphism we consider the Hochschild-Serre spectral sequence H^{p}(π_{1}(R_{n}), H^{q}(\overline{R}_{n,∞}, ℌ)) ⟹ H^{p+q}(R_{n}, ℌ). From the fact tha the group H^{p}(\overline{R}_{n,∞}, ℌ) is torsion for p ≥ 1 it follows that the map \underrightarrow{lim}_{m} H^{p}(\overline{R}_{n,∞},_{m}ℌ) → H^{p}(\overline{R}_{n,∞}, ℌ), where mℌ stands for the kernel of the “multiplication by m” map, is surjective. By loc. cit. cor. 3.3 H^{p}(\overline{R}_{n,∞},_{m}ℌ) vanishes for all m ≥ 1. Hence H^{p}(\overline{R}_{n,∞}, ℌ) = 0. The spectral sequence degenerates and yields the isomorphisms H^{i}(π_{1}(R_{n}), ℌ(\overline{R}_{n,∞})) ≃ H^{i}(R_{n}, ℌ) for all i ≤ 1.

2) We begin with an observation about the notation used in the statement of the Lemma. The subgroup T(\overline{k}) of T(\overline{R}_{n,∞}) is stable under the (twisted) action of π_{1}(R_{n}) on _{c}T(\overline{R}_{n,∞}). To view T(\overline{k}) as a π_{1}(R_{n})–module with this twisted action we write _{c}(T(\overline{k})).

The fact that the last two maps are isomorphism for all i ≥ 1 is a special case (1). For the first map, we first analyse the π_{1}(R_{n})–module A = T(\overline{R}_{n,∞})/T(\overline{k}). We have

A = \underrightarrow{lim}_{m} T(\overline{R}_{n,∞}) / T(\overline{k})
= \underrightarrow{lim}_{m} (T̂)^{0} ⊗_{ℤ} \overline{R}_{n,m}^{×} /(T̂)^{0} ⊗_{ℤ} \overline{k}^{×}
= (T̂)^{0} ⊗_{ℤ} \underrightarrow{lim}_{m} \overline{R}_{n,m}^{×} / \overline{k}^{×}
= (T̂)^{0} ⊗_{ℤ} \underrightarrow{lim}_{m} (ℤ^{n})_{m} (where (ℤ^{n})_{m} = ℤ^{n})
= (T̂)^{0} ⊗_{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_{1}(R_{n})–modules

(6.1)  1 → _{c}(T(\overline{k})) → _{c}T(\overline{R}_{n,∞}) → _{c}A → 1.

¿From the fact that _{c}A is uniquely divisible it follows that the group homomorphisms

(6.2)  H^{i}(π_{1}(R_{n}), _{c}(T(\overline{k}))) → H^{i}(π_{1}(R_{n}), _{c}T(\overline{R}_{n,∞}))

are surjective for all i ≥ 1 and bijective if i > 1.

We can proceed now with the proof of Theorem 6.1.

Proof. Let us show first show that H_{loop}^{1}(R_{n}, G) ⊂ H_{toral}^{1}(R_{n}, G). Case 1: G = Aut(H_{0}) where H_{0} is a semisimple Chevalley k–group : Let ϕ : π_{1}(R_{n}) → Aut(H_{0})(\overline{k}) be a loop cocycle. Consider the twisted R–group_{ϕ}G.^{18} Proposition 4.13 shows that the connected component of the identity (_{ϕ}G)^{0} = _{ϕ}(G^{0}) of _{ϕ}G admits a maximal R-torus. Therefore _{ϕ}G admits a maximal R-torus, hence ϕ defines a toral R-torsor. Case 2: G = Aut(H) where H is a semisimple k–group : Denote by H_{0} the Chevalley k–form of H. There exists a cocycle z : Gal(k) → G(\overline{k}) such that H is isomorphic to the twisted k–group _{z}H_{0}. We can assume then that H = _{z}H_{0} and G = _{z}Aut(H_{0}). The torsion bijection τ_{z} : H^{1}(R, G) = H^{1}(R,_{z}Aut(H_{0})) ⟶^{∼} H^{1}(R, Aut(H_{0})) exchanges loop classes (resp. toral classes) according to Remark 3.3. Case 1 then yields H_{loop}^{1}(R_{n}, G) ⊂ H_{toral}^{1}(R_{n}, G).

18We recall, for the last time, that _{ϕ}G is short hand notation for _{ϕ}(G_{R}).

General case. The k–group acts by conjugacy on G^{0}, its center Z(G^{0}) and then on its adjoint quotient G_{ad}^{0}. Denote by f : G → Aut(G_{ad}^{0}) this action. Let ϕ : π_{1}(R_{n}) → G(\overline{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)^{0} =_{ϕ} (G^{0}) admits a maximal torus which is in turn equivalent to the fact that (_{f_{∗}ϕ}G)_{ad}^{0} = _{f_{∗}ϕ}(G_{ad}^{0}) admits a maximal torus [SGA3, XII.4.7]. But f_{∗}ϕ is a loop cocycle for Aut(G_{ad}^{0}), so defines a toral R–torsor under Aut(G^{0}) according to Case 2. Thus _{f_{∗}ϕ}(G_{ad}^{0}) admits a maximal torus as desired.

To establish the reverse inclusion we consider the quotient group ννν = G/G^{0}, which is finite and étale k–group. In particular ννν ×_{k} \overline{k} is constant and finite, and one can easily see as a consequence that the natural map ννν(\overline{k}) → ννν(\overline{R}_{n,∞}) is an isomorphism. We first establish the result for tori and then the general case. G^{0} is a torus T: We again appeal to the isotriviality theorem of [GP1] to see that H^{1}(π_{1}(R_{n}), G(\overline{R}_{n,∞})) ⟶^{∼} H^{1}(R_{n}, G). We consider the following commutative diagram of continuous π_{1}(R_{n})–groups

1 ⟶ T(\overline{k}) ⟶ G(\overline{k}) ⟶ ννν(\overline{k}) ⟶ 1
↓  ↓  ||
1 ⟶ T(\overline{R}_{n,∞}) ⟶ G(\overline{R}_{n,∞}) ⟶ ννν(\overline{R}_{n,∞}) ⟶ 1.

This gives rise to an exact sequence of pointed sets

1 ⟶ H^{1}(π_{1}(R_{n}), T(\overline{k})) ⟶ H^{1}(π_{1}(R_{n}), G(\overline{k})) ⟶ H^{1}(π_{1}(R_{n}),ννν(\overline{k}))
↓  ↓  ||
1 ⟶ H^{1}(π_{1}(R_{n}), T(\overline{R}_{n,∞})) ⟶ H^{1}(π_{1}(R_{n}), G(\overline{R}_{n,∞})) ⟶ H^{1}(π_{1}(R_{n}),ννν(\overline{R}_{n,∞})) .

We are given a cocycle z ∈ Z^{1}(π_{1}(R_{n}), G(\overline{R}_{n,∞})). Denote by c the image of z in Z^{1}(π_{1}(R_{n}),ννν(\overline{k})) = Z^{1}(π_{1}(R_{n}),ννν(\overline{R}_{n,∞})) under the bottom map. Under the top map, the obstruction to lifting [c] to H^{1}(π_{1}(R_{n}), G(\overline{k})) is given by a class Δ([c]) ∈ H^{2}(π_{1}(R_{n}), _{c}(T(\overline{k}))) [Se, §I.5.6]. This class vanishes in H^{2}(π_{1}(R_{n}), _{c}T(\overline{R}_{n,∞}))), so Lemma 6.2 shows that Δ([c]) = 0. Hence c lifts to a loop cocycle a ∈ Z^{1}(π_{1}(R_{n}), G(\overline{k})).

By twisting by a we obtain the following commutative diagram of pointed sets

1 ⟶ H^{1}(π_{1}(R_{n}), _{a}(T(\overline{k})) ) ⟶ H^{1}(π_{1}(R_{n}), _{a}(G(\overline{k})) ) ⟶ H^{1}(π_{1}(R_{n}), _{c}(ννν(\overline{k})) )
↓  ↓  ||
1 ⟶ H^{1}(π_{1}(R_{n}), _{a}(T(\overline{R}_{n,∞})) ) ⟶ H^{1}(π_{1}(R_{n}),_{a}(G(\overline{R}_{n,∞}))) ⟶ H^{1}(π_{1}(R_{n}), _{c}(ννν(\overline{R}_{n,∞})) )

↓

1

where the surjectivity of the left map comes from Lemma 6.2, _{a}(G(\overline{k})) denotes G(\overline{k}) as a π_{1}(R_{n})–submodule of _{a}G(\overline{R}_{n,∞}), and similarly for _{c}(ννν)(\overline{k})) = _{c}(ννν(\overline{R}_{n,∞})). We consider the torsion map

τ_{a} : H^{1}(π_{1}(R_{n}),_{a}G(\overline{R}_{n,∞})) ⟶^{∼} H^{1}(π_{1}(R_{n}), G(\overline{R}_{n,∞})).

Then

τ_{a}^{−1}([z]) ∈ ker(H^{1}(π_{1}(R_{n}), _{a}G(\overline{R}_{n,∞}) ) → H^{1}(π_{1}(R_{n}), _{c}(ννν(\overline{R}_{n,∞}))).

The diagram above shows that τ_{a}^{−1}([z]) comes from H^{1}(π_{1}(R_{n}), _{a}(G(\overline{k})) ), hence [z] comes from H^{1}(π_{1}(R_{n}), G(\overline{k})) as desired. We conclude that H^{1}(R_{n}, G) is covered by loop torsors. General case : Let T be a maximal torus of G. Consider the commutative diagram

H_{loop}^{1}(R_{n}, N_{G}(T)) ⟶ H^{1}(R_{n}, N_{G}(T))
↓  ↓
H_{loop}^{1}(R_{n}, G) ⟶ H_{toral}^{1}(R_{n}, G).

The right vertical map is surjective according to Lemma 3.13.1. The top horizontal map is surjective by the previous case. We conclude that the bottom horizontal map is surjective as desired.

Corollary 6.3. Let 𝔊 be a reductive R_{n}–group. Then 𝔊 is loop reductive if and only if 𝔊 admits a maximal torus.

Proof. Let G be the Chevalley k–form of 𝔊. To 𝔊 corresponds a class [𝔈] ∈ H^{1}(R_{n}, Aut(G)). When G is semisimple, Aut(G) is an affine algebraic k–group and the Corollary follows from H_{toral}^{1}(R_{n}, Aut(G)) = H_{loop}^{1}(R_{n}, Aut(G)).

We deal now with the general case. We already know by 4.13 that every loop reductive group is toral. Conversely assume that 𝔊 admits a maximal torus. Consider the exact sequence of smooth k–groups [XXIV.1.3.(iii)]

1 → G_{ad} → Aut(G) ⟶^{p} Out(G) → 1,

where G_{ad} is the adjoint group of G and Out(G) is a constant k–group. Since R_{n} is a noetherian normal domain, we know that R_{n}–torsors under Out(G) are isotrivial [X.6]. Furthermore by [SGA1, XI §5]

Hom_{ct}(π_{1}(R_{n}), Out(G)(k))/ ∼ ⟶^{∼} H^{1}(R_{n}, Out(G)).

So p_{∗}[𝔈] is given by a continous homomorphism π_{1}(R_{n}) → Out(G)(k) whose image we denote by Out(G)^{♯}. This is a finite group so that Aut(G)^{♯} := p^{−1}(Out(G)^{♯}) is an affine algebraic k–group. We consider the square of pointed sets

H^{1}(R_{n}, Aut(G)^{♯}) ⟶^{p_{∗}^{♯}} H^{1}(R_{n}, Out(G)^{♯})
↓  ↓
H^{1}(R_{n}, Aut(G)) ⟶^{p_{∗}} H^{1}(R_{n}, Out(G)).

Since Aut(G)/Aut(G)^{♯} = Out(G)/Out(G)^{♯}, this square is cartesian as can be seen by using the criterion of reduction of a torsor to a subgroup [Gi, III.3.2.1]. By construction, [p_{∗}𝔈] comes from H^{1}(R_{n}, Out(G)^{♯}), hence [𝔈] comes from a class [𝔉] ∈ H^{1}(R_{n}, Aut(G)^{♯}). Our assumption is that the R_{n}–group 𝔊 = _{𝔈}G = _{𝔉}G contains a maximal torus, so 𝔊_{ad} = _{𝔉}G_{ad} contains a maximal torus and _{𝔉}(Aut(G)^{♯}) contains a maximal torus. In other words, 𝔉 is a toral R_{n}–torsor under Aut(G)^{♯}. ¿From the equality H_{toral}^{1}(R_{n}, Aut(G)^{♯}) = H_{loop}^{1}(R_{n}, Aut(G)^{♯}), it follows that that 𝔉 is a loop torsor under Aut(G)^{♯}. By applying the change of groups Aut(G)^{♯} → Aut(G), we conclude 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 1 → S → G′ →^{p} G → 1 be a central extension of G by a k–group S of multiplicative type. Then the diagram

H_{loop}^{1}(R_{n}, G′) ⊂ H^{1}(R_{n}, G′)
p_{∗}↓  ^{p_{∗}}↓
H_{loop}^{1}(R_{n}, G) ⊂ H^{1}(R_{n}, G)

is cartesian.

Remark 6.5. For a k–group G satisfying the condition of Corollary 3.16, one can prove in a simpler way that toral G–torsors over R_{n} are loop torsors by reducing to the case of a finite étale group.

Remark 6.6. Given an integer d ≥ 2, the Margaux algebra (both the Azumaya and Lie versions) [GP2, 3.22 and example 5.7 ] provides an example of a PGL_{d}-torsor over ℂ[t_{1}^{±1}, t_{2}^{±1}] which is not a loop torsor. The underlying PGL_{d}-torsor is therefore not toral. This means that the Margaux Azumaya algebra does not contain any (commutative) étale ℂ[t_{1}^{±1}, t_{2}^{±1}]-subalgebra of rank d, and that the Margaux Lie algebra, viewed as a Lie algebra over ℂ[t_{1}^{±1}, t_{2}^{±1}], does not contain any Cartan subalgebras (in the sense of [SGA3]).

Remark 6.7. More generally, for each each positive integer d, we have H_{toral}^{1}(R_{n}[x_{1}, ..., x_{d}], G) = H_{loop}^{1}(R_{n}[x_{1}, ..., x_{d}], G). Since π_{1}(R_{n}[x_{1}, ..., x_{d}]) ≃ π_{1}(R_{n}) and N_{G}(T)(S) = N_{G}(T)(S[x_{1}, ..., x_{d}]) for every finite étale covering S of R_{n}, the proof we have given works just the same in this case.

6.2 Applications to (algebraic) Laurent series.

Let F_{n} = k((t_{1}))((t_{2}))...((t_{n})). In an analogous fashion to what we did in the case

of R_{n} we define F_{n,m} = k((t1^{\frac{1}{m}}))((t2^{\frac{1}{m}}))...((tn^{\frac{1}{m}})) and F_{n,∞} = \underrightarrow{lim}F_{n,m}.

Remark 6.8. (a) If k̃ is a field extension of k the natural map k̃ ⊗_{k} F_{n,m} → k̃((t1^{\frac{1}{m}}))((t2^{\frac{1}{m}}))...((tn^{\frac{1}{m}})) is injective. If the extension is finite, then this map is an isomorphism. We will find it convenient (assuming that the field k̃ is fixed in our discussion) to denote k̃((t1^{\frac{1}{m}}))((t2^{\frac{1}{m}}))...((tn^{\frac{1}{m}})) simply by F̃_{n,m}.

(b) The field \underrightarrow{lim}\overline{k}((t1^{\frac{1}{m}}))((t2^{\frac{1}{m}}))...((tn^{\frac{1}{m}})) is algebraically closed. We will denote the algebraic closure of F_{n} (resp. F_{n,m}, F_{n,∞}) in this field by \overline{F_{n}} (resp. \overline{F_{n,m}}, \overline{F_{n,∞}}). As mentioned in (a) we have a natural injective ring homomorphsm \overline{k} ⊗_{k} F_{n,∞} → \overline{F_{n}}.

(c) There is a natural group morphism π_{1}(R_{n}) → Gal(F_{n}) given by considering the Galois extensions R̃_{n,m} = k̃ ⊗_{k} R_{n,m} of R_{n} and F̃_{n,m} of F_{n} respectively, where k̃ ⊂ \overline{k} is a finite Galois extension of k containing all m-roots of unity. These homomorphisms are in fact isomorphisms.^{19} For by applying successively the structure theorem for local fields [GMS] §7.1 p. 17, we have Gal(F_{n}) = _{∞}μ μ μ^{n}(\overline{k}) ⋊ Gal(k). This means that

Gal(F_{n}) = \underleftarrow{lim}Gal(k̃((t1^{\frac{1}{m}}))((t2^{\frac{1}{m}}))...((tn^{\frac{1}{m}}))/F_{n})

19If k is algebraically closed this was proved in [GP3] cor 2.14.

for m running over all integers and k̃ running over all finite Galois extensions of k inside \overline{k} containing a primitive m–root of unity. Since at each step we have an isomorphism

Gal(k̃ ⊗ R_{n,m}/R_{n}) ≅ Gal(k̃((t1^{\frac{1}{m}}))((t2^{\frac{1}{m}}))...((tn^{\frac{1}{m}}))/F_{n}) ≅ μ μ μ_{m}^{n}(k̃) ⋊ Gal(k̃/k),

we conclude that π_{1}(R_{n}) ≅ Gal(F_{n}).

(d) It follows from (c) that the base change R_{n} → F_{n} induces an equivalence of categories between finite étale coverings of R_{n} and finite étale coverings of F_{n}. Furthermore, if 𝔈/R_{n} is a finite étale covering of R_{n}, we have 𝔈(R_{n}) = 𝔈(F_{n}). Indeed, 𝔈 is split by some Galois covering R̃_{n,m} = k̃ ⊗_{k} R_{n,m} and 𝔈(R_{n}) = 𝔈(R̃_{n,m})^{Gal(R̃_{n,m}/R_{n})} = 𝔈(F̃_{n,m})^{Gal(F̃_{n,m}/F_{n})} = 𝔈(F_{n}).

Proposition 6.9. The canonical map

H_{loop}^{1}(R_{n}, G) → H^{1}(F_{n}, G)

is surjective.

Proof. We henceforth identify π_{1}(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 notation therein. Again we proceed in two steps. First case: G^{0} is a torus T: We consider the following commutative diagram of continuous π_{1}(R_{n})–groups

1 ⟶ T(\overline{k}) ⟶ G(\overline{k}) ⟶ ννν(\overline{k}) ⟶ 1
↓  ↓  ||
1 ⟶ T(\overline{F_{n}}) ⟶ G(\overline{F_{n}}) ⟶ ννν(\overline{F_{n}}) ⟶ 1.

This gives rise to an exact sequence of pointed sets

1 ⟶ H^{1}(π_{1}(R_{n}), T(\overline{k})) ⟶ H^{1}(π_{1}(R_{n}), G(\overline{k})) ⟶ H^{1}(π_{1}(R_{n}),ννν(\overline{k}))
↓  ↓  ||
1 ⟶  H^{1}(F_{n}, T)  ⟶  H^{1}(F_{n}, G)  ⟶ H^{1}(F_{n},ννν) .

We are given a cocycle z ∈ Z^{1}(Gal(F_{n}), G(\overline{F_{n}})) = Z^{1}(π_{1}(R_{n}), G(\overline{F_{n}})), and consider its image c ∈ Z^{1}(π_{1}(R_{n}),ννν(\overline{F_{n}})). By reasoning as in Theorem 6.1 we see that [z] comes from H^{1}(π_{1}(R_{n}), G(\overline{k})) as desired. We conclude that H^{1}(F_{n}, G) is covered by k-loop torsors.

General case: Let T be a maximal torus of G.

H_{loop}^{1}(R_{n}, N_{G}(T)) ⟶ H^{1}(F_{n}, N_{G}(T))
↓  ↓
H_{loop}^{1}(R_{n}, G) ⟶ H^{1}(F_{n}, G).

The reasoning is again identical to the one used in Theorem 6.1.

7 Isotropy of loop torsors

As before G denotes a linear algebraic group over a field k of characteristic zero. R_{n} and π_{1}(R_{n}) are as in the previous section.

7.1 Fixed point statements

Let η : π_{1}(R_{n}) → G(\overline{k}) be a continuous cocycle. Consider as before a Galois extension R̃_{n,m} = k̃ ⊗_{k} R_{n,m} of R_{n} where k̃ ⊂ \overline{k} is a finite Galois extension of k containing all m–roots of unity in \overline{k}, chosen so that our cocycle η factors through the Galois group

(7.1)  Γ̃_{n,m} = Gal(R̃_{n,m}/R_{n}) ≅ μ μ μ_{m}^{n}(k̃) ⋊ Gal(k̃/k)

We assume henceforth that G acts on a k–scheme Y. The Galois group Γ̃_{n,m} acts naturally on Y(R̃_{n,m}), and we denote this action by γ : y ↦ ^{γ}y. By means of η we get a twisted action of Γ̃_{n,m} on Y(R̃_{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 knows that this twisted form is up to isomorphism independent of the Galois extension R̃_{n,m} chosen through which η factors. We will denote this twisted form by _{η}Y_{R_{n}}, or simply by _{η}Y following the conventions that have been previously mentioned regarding this matter.

Let (z, η^{geo}) be the couple associated to η according to Lemma 3.7. Thus z ∈ Z^{1}( Gal(k), G(\overline{k})) and η^{geo} ∈ Hom_{k−gp}(_{∞}μ μ μ^{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 comes equipped with an action of _{z}G. Via η^{geo}, this defines an algebraic action of the affine k-group _{∞}μ μ μ^{n} on _{z}Y. At the level of \overline{k}–points of _{∞}μ μ μ^{n}, the action is given by

(7.3)  n̂.y = η^{geo}(n̂).y

for all n̂ ∈ _{∞}μ μ μ^{n}(\overline{k}) = \underleftarrow{lim}_{m} μ μ μ_{m}^{n}(\overline{k}) and y ∈ _{z}Y(\overline{k}). We denote by (_{z}Y)^{η^{geo}} the closed fixed point subscheme for the action of _{∞}μ μ μ^{n} (see [DG] II §1 prop. 3.6.d). We have

(_{z}Y)^{η^{geo}}(\overline{k}) = \{y ∈ _{z}Y(\overline{k}) = Y(\overline{k}) | y = η^{geo}(n̂).y ∀ n̂ ∈ _{∞}μ μ μ^{n}(\overline{k})\}

and in terms of rational points (7.4) (_{z}Y)^{η^{geo}}(k) = _{z}Y(k) ∩ (_{z}Y)^{η^{geo}}(\overline{k}) = \{y ∈ _{z}Y(k) | y = η^{geo}(n̂).y ∀ n̂ ∈ _{∞}μ μ μ^{n}(\overline{k})\}.

where we recall that

zY(k) = \{y ∈ Y(\overline{k}) | y = z_{γ}.^{γ}y ∀ γ ∈ Gal(k)\}.

Theorem 7.1. 1. Let G act on Y as above, and assume that Y is projective ( i.e. a closed subscheme in ℙ_{k}^{n}). Let η : π_{1}(R_{n}) → G(\overline{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 compactification of G/S (i.e., Y is projective k–variety that contains G/S as an open dense G-subvariety). Then the following are equivalent:

(a) [η]_{K_{n}} ∈ Im(H^{1}(K_{n}, S) → H^{1}(K_{n}, G)),

(b) [η]_{F_{n}} ∈ Im(H^{1}(F_{n}, S) → H^{1}(F_{n}, G)),

(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 → zY. Lemma 3.8 enables us to assume without loss of generality that z is the trivial cocycle. We are thus left to deal with a k–homomorphism η^{geo} : _{∞}μ μ μ^{n} → G which factors at the finite level through μ μ μ_{m}^{n} → G for m large enough. This allows us to reason by means of a suitable covering R̃_{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 ∈ Γ̃_{n,m} induces an automorphism of R̃_{n,m} ⊗_{R_{n}} F_{n} ≃ F_{n,m} ⊗_{k} k̃ = F̃_{n,m} which we also denote by γ (even though the notation γ ⊗ 1 would be more accurate.) Since R̃_{n,m} trivializes _{η}Y, the Galois extension F̃_{n,m} of F_{n} (whose Galois group we identify with Γ̃_{n,m}) splits _{η}Y_{F_{n}}. By Galois descent

ηY(F_{n}) = {y ∈ Y(F̃_{n,m}) | η_{γ} .^{γ}y = y ∀γ ∈ Γ̃_{n,m}}.

Since z is trivial, this last equality reads

ηY(F_{n}) = \{y ∈ Y(F̃_{n,m}) | η^{geo}(\overline{γ}) .^{γ}y = y ∀γ ∈ Γ̃_{n,m}\}.

where \overline{γ} is the image of γ under the map Γ̃_{n,m} → μ μ μ_{m}^{n}(k̃) given by (7.1). Hence we have _{η}Y(F_{n}) ⊂ Y(F_{n,m}) and

ηY(F_{n}) = \{y ∈ Y(F_{n,m}) | η^{geo}(γ) .^{γ}y = y ∀γ ∈ μ μ μ_{m}^{n}(\overline{k})\}.

Since Y is proper over k, we have

ηY(F_{n}) = \{y ∈ Y(F_{n−1,m}[[tn^{\frac{1}{m}}]]) | η^{geo}(γ).^{γ}y = y γ ∈ μ μ μ_{m}^{n}(\overline{k})\}.

Our hypothesis is that this last set is not empty. By specializing at t_{n} = 0, we get that

(7.5)  \{y ∈ Y(F_{n−1,m}) | η^{geo}(γ).^{γ}y = y ∀γ ∈ μ μ μ_{m}^{n}(\overline{k})\} ≠ ∅.

We write now μ μ μ_{m}^{n}(\overline{k}) = μ μ μ_{m}^{n−1}(\overline{k})×μ μ μ_{m}(\overline{k}) which provides a decomposing of η^{geo} into two k-homomorphisms η′^{geo} : μ μ μ_{m}^{n−1} → G and η_{n}^{geo} : μ μ μ_{m} → G. We define η′ = (1, η′^{geo}), η_{n} = (1, η_{n}^{geo}) and

Y′ := Y^{η_{n}^{geo}}.

By [DG] II §1 prop. 3.6.(d) we know that Y′ is a closed subscheme of Y, hence a projective k–variety. Observe that μ μ μ_{m}^{n−1} acts on Y′.

Claim 7.2. The set (7.5) is included in _{η′}Y′(F_{n−1}).^{20}

To look at the invariants under the action of μ μ μ_{m}^{n}(\overline{k}), we first look at the invariants under the last factor μ μ μ_{m}(\overline{k}), and then the first (n − 1)-factor μ μ μ_{m}^{n−1}(\overline{k}) By restricting the condition to elements of the form (1, γ_{n}) for γ_{n} ∈ μ μ μ_{m}(\overline{k}), we see that our set is included in

\{y ∈ Y(F_{n−1,m}) | η_{n}^{geo}(γ_{n}).y = y ∀γ_{n} ∈ μ μ μ_{m}(\overline{k})\}

20This inclusion is in fact an equality, but this stronger statement is not needed.

because μ μ μ_{m}(\overline{k}) acts trivially on F_{n−1,m}. By identity (7.4) applied to the base change of η_{n}^{geo} to the field F_{n−1}, this is nothing but Y^{η_{n}^{geo}}(F_{n−1,m}). Looking now at the invariant condition for the elements of the form (γ′, 1) for γ′ ∈ μ μ μ_{m}^{n−1}(\overline{k}), it follows that

\{y ∈ Y(F_{n−1,m}) | η^{geo}(γ).^{γ}y = y ∀γ ∈ μ μ μ_{m}^{n}(\overline{k})\}

⊂ \{y ∈ Y^{η_{n}^{geo}}(F_{n−1,m}) | η′^{geo}(γ′).^{γ′}y = y ∀γ′ ∈ μ μ μ_{m}^{n−1}(\overline{k})\} = _{η′}Y′(F_{n−1}).

By induction on n, we get that inside (_{η′}Y′)(F_{n−1}) we have Y′^{η′_{geo}}(k) ≠ ∅. Thus Y(k)^{η^{geo}} ≠ ∅ as desired.

(d) ⟹ (a): Since

(_{η}Y)(R_{n}) = \{y ∈ Y(R̃_{n,m}) | η^{geo}(\overline{γ}) .^{γ}y = y ∀γ ∈ Γ̃_{n,m}\},

the inclusion Y(k) ⊂ Y(R̃_{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]. The only non trivial implication is (c) ⟹ (a). Let X = (G/S) ×_{k} R_{n}. By (1), we have ηY(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).

7.2 Case of flag varieties

The k–group G^{0}/R_{u}(G) is reductive. Let T be a maximal k–torus of G^{0}/R_{u}(G). This data permits to choose a basis Δ the root system Φ(G^{0}/R_{u}(G) ×_{k} \overline{k}, T ×_{k} \overline{k}) or in other words to choose a Borel subgroup B of the \overline{k}–group G^{0}/R_{u}(G) ×_{k} \overline{k}. It is well known that there is a one-to-one correspondence between the subsets of Δ and the parabolic subgroups of G^{0} ×_{k} \overline{k} containing B, which is provided by the standard parabolic subgroups (P_{I})_{I⊂Δ} of (G^{0}/R_{u}(G)) ×_{k} \overline{k} [Bo, §21.11]. We have P_{Δ} = (G^{0}/R_{u}(G)) ×_{k} \overline{k} and P_{∅} = B. Furthermore we know that each parabolic subgroup of (G^{0}/R_{u}(G)) ×_{k} \overline{k} is (G^{0}/R_{u}(G))(\overline{k})–conjugate to a unique standard parabolic subgroup. This allows us to define the type of an arbitrary parabolic subgroup of G^{0}/R_{u}(G). It can happen that two different standard parabolic subgroups of the \overline{k}–group (G^{0}/R_{u}(G)) ×_{k} \overline{k} are conjugate under G(\overline{k}): There are in general fewer conjugacy classes of parabolic subgroups. If P is a parabolic subgroup of the k–group G^{0}/R_{u}(G), we denote by N_{G}(P) its normalizer for the conjugacy action of G on G^{0}/R_{u}(G).

Lemma 7.3. The quotient G/N_{G}(P) is a projective k–variety.

Proof. We can assume that G^{0} is reductive. Since G^{0}/P is projective and is a connected 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 finite, étale 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_{1}(R_{n}) → G(\overline{k}) with coordinates (z, η^{geo}) as before, we focus on the special case of flag varieties of parabolic subgroups of G^{0}/R_{u}(G).

Corollary 7.4. 1. Let I ⊂ Δ. The following are equivalent:

(a) The R_{n}–group _{η}(G^{0}/R_{u}(G)) admits a parabolic subgroup scheme of type I;

(b) The R_{n}–group _{η}(G^{0}/R_{u}(G))_{R_{n}} admits a parabolic subgroup of type I;

(c) The F_{n}–group _{η}(G^{0}/R_{u}(G))_{F_{n}} admits a parabolic subgroup of type I;

(d) There exists a parabolic subgroup P of the k–group_{z}(G^{0}/R_{u}(G)) which is of 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}(G^{0}/R_{u}(G))_{R_{n}} is irreducible (i.e has no proper parabolic subgroups);

(b) _{η}(G^{0}/R_{u}(G))_{K_{n}} is irreducible;

(c) _{η}(G^{0}/R_{u}(G)oftype)_{F_{n}} is irreducible;

(d) The k–group homomorphism η^{geo} : _{∞}μ μ μ^{n} → _{z}G → Aut(_{z}G^{0}) is irreducible (see §2.4).

3. The following are equivalent:

(a) _{η}(G^{0}/R_{u}(G))_{R_{n}} is anisotropic;

(b) _{η}(G^{0}/R_{u}(G))_{K_{n}} is anisotropic;

(c) _{η}(G^{0}/R_{u}(G))_{F_{n}} is anisotropic;

(d) The k–group homomorphism η^{geo} : _{∞}μ μ μ^{n} → _{z}G → Aut(_{z}G^{0}) is anisotropic (see §2.4).

Proof. Without loss of generality, we can factor out by R_{u}(G) and assume that G^{0} is reductive. As in the proof of Theorem 7.1, we can assume by twisting that z is trivial and reason “at the finite level”:

Claim 7.5. There exists a positive integer m such that [η] ∈ H^{1}(R_{n}, G) is trivialized by the base change R_{n,m}/R_{n}.

Indeed by continuity η^{geo} : _{∞}μ μ μ^{n} → G factors through a morphism f : μ μ μ_{m}^{n} → G and [η] = f_{∗}[𝔈_{n,m}] where 𝔈_{n,m} = Spec(R_{n,m})/ Spec(R_{n}) stands for the standard μ μ μ_{m}^{n}- torsor. In particular, the class [η] ∈ H^{1}(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^{0}_{F_{n}} admits a F_{n}-parabolic Q of type I. Hence ηG^{0}_{F_{n}} ×_{F_{n}} F_{n,m} = G_{F_{n,m}}^{0} admits a F_{n,m}–parabolic of type I. Since F_{n,m} is an iterated Laurent serie field over k, it implies that G^{0} admits a parabolic P of type I (see the proof of [CGP, lemma 5.24]). We consider the R_{n}–scheme X := _{η}(G/N_{G}(P)) ×_{k} R_{n} which by descent considerations [EGA IV, 2.7.1.vii] is proper since G/N_{G}(P) is.

Claim 7.6. X(F_{n}) ≠ ∅.

The F_{n}–group _{η}G^{0}/F_{n} admits a subgroup Q such that Q ×_{k} \overline{F}_{n} is G^{0}(\overline{F}_{n})- conjugate to P ×_{k} \overline{F}_{n} ⊂ G^{0} ×_{k} \overline{F}_{n}. Let g ∈ G(\overline{F}_{n}) be such that Q ×_{F_{n}} \overline{F}_{n} = g (P ×_{k} \overline{F}_{n}) g^{−1}. As in [Se, III.2, lemme 1], we check that the cocycle γ ↦ g^{−1} η_{γ} ^{γ}g is cohomologous to η and has value in N_{G}(P)(\overline{F}_{n}). In other words, the F_{n}–torsor corresponding to η admits a reduction to the subgroup N_{G}(P), i.e.

[η] ∈ Im(H^{1}(F_{n}, N_{G}(P)) → H^{1}(F_{n}, G)).

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^{1}(k, N_{G}(P)) injects in H^{1}(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_{0} where x_{0} stands for the image of 1 in X(k). We have

η^{geo}(μ μ μ_{m}^{n}(\overline{k})) ⊂ Stab_{G(\overline{k})}(x)g Stab_{G(\overline{k})}(x_{0})g^{−1} = gN_{G}(P)(\overline{k})g^{−1} = N_{G}(gPg^{−1})(\overline{k}).

Thus η^{geo} normalizes a parabolic subgroup of type I of the k–group of G^{0}.

(d) ⟹ (a): We may assume that η has values in N_{G}(P)(\overline{k}). In that case, the twisted R_{n}–group scheme _{η}G^{0} admits the parabolic subgroup _{η}P/R_{n}.

(2) Follows of (1).

(3) Recall that a k–group H with reductive connected component of the identity H^{0} is anisotropic if and only if it is irreducible and its connected center Z(H^{0})^{0} is an anisotropic torus. Statement (3) reduces then to the case where G^{0} is a k–torus T. We are then given a continuous action of π_{1}(R_{n}) on the cocharacter group T̂^{0}(\overline{k}). It is convenient to work with the opposite assertions to (a), (b) (c)and (d), which we denote 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 have

Hom_{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} : _{∞}μ μ μ^{n} → zG fixes a cocharacter of T = (G)^{0}, hence (c′). (d′) ⟹ (a′): We assume that the morphism η^{geo} : _{∞}μ μ μ^{n} → G fixes a cocharacter λ : G_{m} → T. Since

Hom_{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 the following consequence.

Corollary 7.7. The minimal elements (with respect to inclusion) of the set of parabolic subgroups of the k–group _{z}G^{0} which are normalized by η^{geo} are all conjugate under zG^{0}(k). The type I(η) of this conjugacy class is the Witt-Tits index of the F_{n}–group η(G^{0}/R_{u}(G)) ×_{R_{n}} F_{n}.

7.3 Anisotropic loop torsors

For anisotropic loop classes, we have the following beautiful picture.

Theorem 7.8. Assume that G^{0} is reductive. Let η, η′ : π_{1}(R_{n}) → G(\overline{k}) be two loop cocycles. Assume that (_{η}G)_{F_{n}} is anisotropic. Then the following are equivalent:

1. [η] = [η′] ∈ H^{1}(R_{n}, G),

2. [η]_{K_{n}} = [η′]_{K_{n}} ∈ H^{1}(K_{n}, G), 3. [η]_{F_{n}} = [η′]_{F_{n}} ∈ H^{1}(F_{n}, G).

We consider first the case of purely geometric loop cocycles. Note that this is the set of all loop cocycles if k is algebraically closed.

Theorem 7.9. Let η, η′ : π_{1}(R_{n}) → G(\overline{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^{1}(R_{n}, G),

3. [η]_{K_{n}} = [η′]_{K_{n}} ∈ H^{1}(K_{n}, G),

4. [η]_{F_{n}} = [η′]_{F_{n}} ∈ H^{1}(F_{n}, G).

Proof. Recall that η^{geo}, η′^{geo} : _{∞}μ μ μ → G are affine k–group homomorphisms that factor through the algebraic group μ μ μ_{m}^{n} for m large enough. The meaning of (1) is that there exists g ∈ G(k) such that η′^{geo} = Int(g) ∘ η^{geo}.

The implications 1) ⟹ 2) ⟹ 3) ⟹ 4) are obvious. We shall prove the implication 4) ⟹ 1). Assume, therefore, that [η]_{F_{n}} = [η′]_{F_{n}} ∈ H^{1}(F_{n}, G).

Let T be a maximal torus of G^{0} and let N = N_{G}(T) and W = N/T. Since the maximal tori of G^{0} ×_{k} \overline{k} are all conjugate under G^{0}(\overline{k}), the map N_{G}(T) → G/G^{0} is surjective. Let k̃ be a finite Galois extension which contains μ μ μ_{m}(\overline{k}), splits T and such that the natural map N(k̃) → (G/G^{0})(\overline{k}) is surjective. We furthermore assume without loss of generality that our choice of m and k̃ trivialize η and η′.

Set Γ̃_{n,m} = μ μ μ_{m}^{n}(\overline{k}) ⋊ Gal(k̃/k). In terms of cocycles, our hypothesis means that there exists h_{n} ∈ G(F̃_{n,m}) such that

(7.6)  h_{n}^{−1} η(γ)^{γ}h_{n} = η′(γ) ∀γ ∈ Γ̃_{n}.

Our goal is to show that we can actually find such an element inside G(k). We reason by means of a building argument, and appeal to Remark 6.8 to view F̃_{n,m} as a complete local field with residue field F̃_{n−1,m}. Note that F_{n} = (F̃_{n,m})Γ̃_{n,m}, and that F_{n} can be viewed as complete local field with residue field F_{n−1}.

Let C = G^{0}/DG^{0} be the coradical of G^{0}. This is a k–torus which is split by k̃. Recall that the (enlarged) Bruhat-Tits building B_{n} of the F̃_{n,m}–group G ×_{k} F̃_{n,m} [T2, §2.1] is defined by

B_{n} = B × V

where V = Ĉ^{0} ⊗_{ℤ} ℝ, and B is the building of the semisimple F̃_{n,m}–group DG^{0} ×_{k} F̃_{n,m}. The building B_{n} is equipped with a natural action of G(F̃_{n,m}) ⋊ Γ̃_{n,m}. By [BT1, 9.1.19.c] the group DG^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) fixes a unique (hyperspecial) point ϕ_{d} ∈

B(DG^{0} ×_{k} F̃_{n,m}) and Stab_{DG^{0}(F̃_{n,m})}(ϕ_{d}) = DG^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]).

Since the bounded group G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) ⋊ Γ̃_{m,n} fixes at least one point of the building B(DG^{0} ×_{k} F̃_{n,m}); such a point is necessarily ϕ_{d} which is then fixed under

G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) ⋊ Γ̃_{m,n}.

Claim 7.10. There exists a point ϕ = (ϕ_{d}, v) ∈ B_{n} such that

1. Γ̃_{m,n} fixes ϕ;

2. Stab_{G(F̃_{n,m})}(ϕ) = G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]).

We use the fact that G^{0}(F̃_{n,m}) acts on V by translations under the map

G^{0}(F̃_{n,m}) ⟶^{q} C(F̃_{n,m}) = Ĉ^{0} ⊗_{ℤ} F̃_{n,m}^{×} ⟶^{−ord_{tn}} Ĉ^{0}.

It follows that for each v ∈ V, we have

(∗)  Stab_{G^{0}(F̃_{n,m})}(v) = Stab_{G^{0}(F̃_{n,m})}(V) = q^{−1}(C(F̃_{n−1,m}[[tn^{\frac{1}{m}}]])).

Since q maps G^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) into C(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]), it follows that G^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) fixes pointwise ϕ_{d} × V.

Let us choose now the vector v by considering the action of the group N(k̃)⋊Γ̃_{m,n} on V. Since this action is trivial on T(k̃), it provides an action of the finite group W(k̃) ⋊ Γ̃_{m,n} on V. But this action is affine, so there is at least one v ∈ V which is fixed under N(k̃) ⋊ Γ̃_{m,n}. The point ϕ = (ϕ_{d}, v) is Γ̃_{m,n}-invariant, hence (1). We now use that N(k̃) surjects onto (G/G^{0})(k̃) = (G/G^{0})(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) = (G/G^{0})(F̃_{n,m}), hence that

G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) = G^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) . N(k̃), G(F̃_{n,m}) = G^{0}(F̃_{n,m}) . N(k̃).

Since N(k̃) fixes ϕ, we have

Stab_{G(F̃_{n,m})}(ϕ) = Stab_{G^{0}(F̃_{n,m})}(ϕ) . N(k̃),

and it remains to show that G^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) = Stab_{G^{0}(F̃_{n,m})}(ϕ). Since T×_{k} k̃ is split, the map T ×_{k} k̃ → C ×_{k} k̃ is split and we have the decompositions

G^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) = DG^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) . T(F̃_{n−1,m}[[tn^{\frac{1}{m}}]])

and

G^{0}(F̃_{n,m}) = DG^{0}(F̃_{n,m}) . T(F̃_{n,m}).

The first equality shows that G^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) fixes ϕ hence that G^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) ⊂ Stab_{G^{0}(F̃_{n,m})}(ϕ).

As for the reversed inclusion consider an element g ∈ Stab_{G(F̃_{n,m})}(ϕ). Then q(g) ∈

C(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]). The map G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) ⟶^{q} C(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) is surjective, hence we can assume that g ∈ DG^{0}(F̃_{n,m}). Since g.ϕ_{d} = ϕ_{d}, g belongs to DG^{0}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) as desired. This finishes the proof of our claim.

We consider the twisted action of Γ̃_{n,m} on B_{n} defined by

γ ∗ x = η(γ) .^{γ}x.

The extension of local fields (with respect to t_{n}) F̃_{n,m}/F_{n} is tamely ramified. The Bruhat-Tits-Rousseau theorem states that the Bruhat-Tits building of (_{η}G^{0})_{F_{n}} can be identified with BΓ̃_{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−1}((t_{n}))–group _{η}G^{0} ×_{R_{n}} F_{n−1}((t_{n})) is anisotropic, so its building consists of a single point, which is in fact ϕ. Indeed since our loop cocycle has value in G(k̃) ⋊ Γ_{n,m}, ϕ is fixed under the twisted action of Γ̃_{n,m}. This shows that

B_{n}^{Γ_{n,m}} = {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 Γ̃_{m,n}]
= h_{n} . η′(γ) ϕ [relation 7.6]
= h_{n} . ϕ [η′(γ) ∈ G(k̃)] and claim 7.10]

for every γ ∈ Γ̃_{n,m}. Hence h_{n} . ϕ ∈ B_{n}^{Γ_{n,m}} and therefore h_{n} . ϕ = ϕ as desired.

It then follows that h_{n} ∈ G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]). By specializing (7.6) at t_{n} = 0, we obtain an element h_{n−1} ∈ G(F̃_{n−1,m}) such that

(7.7)  h_{n−1}^{−1} η(γ) ^{γ}h_{n−1} = η′(γ) ∀γ ∈ Γ̃_{n,m}.

Since η and η′ have trivial arithmetic part, it follows that h_{n−1} is invariant under Gal(k̃/k). We apply now the relation (7.7) to the generator τ_{n} of Gal(F̃_{n,m}/F̃_{n−1,m}((t_{n}))). This yields

(7.8)  h_{n−1}^{−1} η(τ_{n}) h_{n−1} = η′(τ_{n}),

where η(τ_{n}), η′(τ_{n}) ∈ G(k̃) and h_{n−1} ∈ G(F_{n−1,m}) = G(F̃_{n−1,m})^{Gal(k̃/k)}. If we denote by μ μ μ_{m}^{(n)} the last factor of μ μ μ_{m}^{n} then (7.8) implies that η^{geo}_{|μ μ μ_{m}^{(n)}} and η^{′geo}_{|μ μ μ_{m}^{(n)}} are conjugate under G(F_{n−1,m}).

Claim 7.11. η^{geo}_{|μ μ μ_{m}^{(n)}} and η^{′geo}_{|μ μ μ_{m}^{(n)}} are conjugate under G(k).

The transporter X := {h ∈ G | Int(h) ∘ η^{geo}_{|μ μ μ_{n}^{(n)}} = η^{′geo}_{|μ μ μ_{n}^{(n)}}} is a non-empty k– variety which is a homogeneous space under the group Z_{G}(η^{geo}_{|μ μ μ_{n}^{(n)}}). Since X(F_{n−1,m}) ≠ ∅ and F_{n−1,m} is an iterated Laurent series field over k, Florence’s theorem [F, §1] shows that X(k) ≠ ∅. The claim is thus proven.

Without loss of generality we may therefore assume that η^{geo}_{|μ μ μ_{m}^{(n)}} = η^{′geo}_{|μ μ μ_{m}^{(n)}}. The

finite multiplicative k–group μ μ μ_{m}^{(n)} acts on G via η^{geo}, and we let G_{n−1} denote the k– group which is the centralizer of this action [DG, II 1.3.7]. The connected component of the identity of G_{n−1} is reductive ([Ri], proposition 10.1.5). Since the action of μ μ μ_{m}^{(n−1)} on G given by η^{geo} commutes with that of μ μ μ_{m}^{(n)}, the k–group morphism η^{geo} : μ μ μ_{m}^{n} → G factors through G_{n−1}. Similarly for η^{′geo}. Denote by η_{n−1}^{geo} (resp. η′_{n−1}^{geo}) the restriction of η^{geo} (resp. η′^{geo}) to the k–subgroup μ μ μ_{m}^{n−1} = μ μ μ_{n}^{(1)} ×⋯×μ μ μ_{n}^{(n−1)} of μ μ μ_{m}^{n}. Set Γ̃_{n−1,m} := μ μ μ_{m}^{n−1}(\overline{k})⋊Gal(k̃/k) and consider the loop cocycle η_{n−1} : Γ̃_{n−1,m} → G_{n−1}(k̃) attached to (1, η_{n−1}^{geo}), and similarly for η′_{n−1}^{geo}.

The crucial point for the induction argument we want to use is the fact that η_{n−1}^{geo} : μ μ μ_{m}^{n−1} → G_{n−1} is anisotropic. For otherwise the k–group G_{n−1}^{0} admits a nontrivial split subtorus S which is normalized by μ μ μ_{m}^{n−1}. But then S is a non-trivial split subtorus of G^{0} which is normalized by μ μ μ_{m}^{n}, and this contradicts the anisotropic assumption on η^{geo}.

Inside G_{n−1}(F̃_{n−1,m}), relation (7.7) yields that

h_{n−1}^{−1} η_{n−1}(γ)^{γ}h_{n−1} = η′(γ) ∀γ ∈ Γ̃_{n−1}.

which is similar to (7.6). By induction on n, we may assume that η_{n−1}^{geo} is G_{n−1}(k)– conjugate to η′_{n−1}^{geo}. 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^{1}(π_{1}(R_{n}), H(\overline{k})) have same image in H^{1}(F_{n}, H), then [η^{ar}] = [η′^{ar}] in H^{1}(k, H).

Proof. Up to twisting H by η^{ar}, the standard torsion argument allows us to assume with no loss of generality that η^{ar} is trivial, i.e. that η is purely geometrical. We are thus left to deal with the case of a k–group homomorphism η^{geo} : _{∞}μ μ μ → H that factors through some μ μ μ_{m}^{n} → H for m large enough. Hence [η] is trivialized by the extension R̃_{n,m}/R_{n} and its image in H^{1}(F_{n}, H) by the extension F̃_{n,m}/F_{n}, where R̃_{n,m} and F̃_{n,m} are as above.

By further increasing m, the same reasoning allows us to assume that the image of η′ in H^{1}big(R_{n,m}, H(\overline{k})big) is purely arithmetic. More precisely, that the map

Z^{1}(π_{1}(R_{n}), H(\overline{k})) → Z^{1}(π_{1}(R_{n,m}), H(\overline{k}))

maps (η′^{geo}, η′^{ar}) to (1, η′^{ar}) where the coordinates are as in Section 3.3. But our hypothesis implies that the image of [η′] in H^{1}(F_{n,m}, H) is trivial, hence

[η′^{ar}] ∈ ker(H^{1}(k, H) → H^{1}(F_{n}, H)).

Since F_{n} is an iterated Laurent series field over k, this kernel is trivial (see [F, §5.4]), and we conclude that [η′^{ar}] = 1 ∈ H^{1}(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 implication

3) ⟹ 1) by using the previous result. By assumption [η]_{F_{n}} = [η′]_{F_{n}} ∈ H^{1}(F_{n}, G). It is convenient to work at finite level as we have done previously, namely with cocycles

η, η′ : Γ̃_{n,m} → G(k̃)

with Γ̃_{n,m} := μ μ μ_{m}^{n}(k̃) ⋊ Gal(k̃/k) where m > 0 large enough and k̃/k is a finite Galois extension extension containing all m–roots of unity in \overline{k}. We associate to η its arithmetic-geometric coordinate pair (z, η^{geo}) where z ∈ Z^{1}(Gal(k̃/k), G(k̃)) and η^{geo} : μ μ μ_{m}^{n} → _{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^{1}(k, G). Without lost of generality we may assume that z = z′. Consider the commutative diagram

H^{1}(Γ̃_{n,m},_{z}G(k̃)) ⟶ H^{1}(Γ̃_{n,m},_{z}G(F̃_{n,m}))
τ_{z}↓^{≀}  ^{τ_{z}}↓^{≀}
H^{1}(Γ̃_{n,m}, G(k̃)) ⟶ H^{1}(Γ̃_{n,m}, G(F̃_{n,m})).

where the vertical arrows are the torsion bijections. Thus τ_{z}^{−1}[η] = τ_{z}^{−1}[η′] ∈ H^{1}(Γ̃_{n,m},_{z}G(F̃_{n,m})). By Corollary 7.4.3, η^{geo} : μ μ μ_{m}^{n} → _{z}G is an anisotropic k–group homomorphism. We can thus apply Theorem 7.9 to conclude that η^{geo} and η′^{geo} are conjugate under_{z}G(k) hence τ_{z}^{−1}[η] = τ_{z}^{−1}[η′] in H^{1}(Γ̃_{n,m},_{z}G(k̃)), and therefore [η] = [η′] in H^{1}(Γ̃_{n,m},_{z}G(k̃)) as desired. Corollary 7.13. Let the assumptions be as in the Theorem, and let H^{1}(π_{1}(R_{n}), G(\overline{k}))_{an} denote the preimage of H^{1}(F_{n}, G)_{an} under the composite map H^{1}(π_{1}(R_{n}), G(\overline{k})) → H^{1}(R_{n}, G) → H^{1}(F_{n}, G). Then H^{1}(π_{1}(R_{n}), G(\overline{k}))_{an} injects into H^{1}(F_{n}, G).

8 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 0. Then the natural restriction map H^{1}(R_{n}, G) → H^{1}(F_{n}, G) induces a bijection

H_{loop}^{1}(R_{n}, G) ⟶^{∼} H^{1}(F_{n}, G).

In particular, the inclusion map H_{loop}^{1}(R_{n}, G) → H^{1}(R_{n}, G) admits a canonical section.

8.1 The proof

For any k–scheme 𝔛, we denote by H^{1}(𝔛, G)_{irr} ⊂ H^{1}(𝔛, G)^{21} the subset consisting of classes of G–torsor 𝔈 over 𝔛 for which the twisted reductive 𝔛–group scheme 𝔈(G^{0}/R_{u}(G^{0}))_{𝔛} does not contain a proper parabolic subgroup which admits a Levi subgroup.^{22} Set H_{loop}^{1}(𝔛, G)_{irr} = H_{loop}^{1}(𝔛, G) ∩ H^{1}(𝔛, G)_{irr}. We begin with the following special case.

Lemma 8.2. H_{loop}^{1}(R_{n}, G)_{irr} injects into H^{1}(F_{n}, G).

Proof. By Lemma 4.14, we can assume without loss of generality that G^{0} is reductive. We have an exact sequence 1 → G^{0} ⟶^{i} G ⟶^{p} ννν → 1 where ννν is a finite étale k– group. We are given two loop cocycles η, η′ in Z^{1}(R_{n}, G) which have the same image in H^{1}(F_{n}, G), and for which the twisted F_{n}–groups _{η}G^{0}, _{η′}G^{0} are irreducible. Since

H^{1}(π_{1}(R_{n}),ννν(\overline{k})) ⟶^{∼} H^{1}(F_{n},ννν), it follows that p_{∗}[η] = p_{∗}[η′] in H^{1}(π_{1}(R_{n}),ννν(\overline{k})).

We can thus assume without loss of generality that p_{∗}η = p_{∗}η′ in Z^{1}(π_{1}(R_{n}),ννν(\overline{k})). Furthermore, as in the proof of Theorem 7.8 the standard twisting argument reduces the problem to the case of purely geometric loop torsors. In particular, the group actions of η^{geo} and η′^{geo} are irreducible according to Corollary 7.4.3.

Let C be the connected center of G^{0}. Then C is a k–torus equipped with an action of ννν. We consider its k–subtorus C^{♯} = (C^{p∘η^{geo}})^{0} and denote by C_{0} its maximal k–

split subtorus which is defined by Ĉ_{0}^{0} = Ĉ^{♯}^{0}(k). By construction, η^{geo} : _{∞}μ μ μ^{n} → G centralizes C^{♯} and C_{0}. Similarly for η′^{geo}. The k–torus C_{0} 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_{0} → G ⟶^{q} G/C_{0} → 1

21We remind the reader that H^{1}(𝔛, G) stands for H^{1}(𝔛, G_{𝔛}).

22Recall that the assumption on the existence of the Levi subgroup is superfluous whenever 𝔛 is affine.

Claim 8.3. The composite q ∘ η^{geo} : _{∞}μ μ μ^{n} → G/C_{0} is anisotropic.

Let us establish the claim. We are given a split subtorus S of the k–group G^{0} which is centralized by q ∘ η^{geo}. Since q ∘ η^{geo} is irreducible, S is central in G^{0}/C_{0}. We consider M = q^{−1}(S), this is an extension of S by C_{0}, so it is a split k–torus. By the semisimplicity of the category of representations of _{∞}μ μ μ^{n}, we see that _{∞}μ μ μ^{n} acts trivially on M. Then M = C_{0} and S = 1, and the claim thus holds.

Next we twist the sequence of R_{n}–groups

1 → C_{0} → G ⟶^{q} G/C_{0} → 1

by η to obtain

1 → C_{0} → _{η}G → _{η}(G/C_{0}) → 1.

This leads to the commutative exact diagram of pointed sets

0 = H^{1}(R_{n}, C_{0}) ⟶ H^{1}(R_{n}, G) ⟶^{q_{∗}} H^{1}(R_{n}, G/C_{0})
τ_{η}↑^{≃}  ^{τ_{η}}↑^{≃}
0 = H^{1}(R_{n}, C_{0}) ⟶ H^{1}(R_{n},_{η}G) ⟶ H^{1}(R_{n},_{η}(G/C_{0}))

where the vertical maps are the torsion bijections. Note that H^{1}(R_{n}, C_{0}) vanishes since Pic(R_{n}) = 0. By diagram chasing we have [η] = [η′] in H^{1}(R_{n}, G) if and only if q_{∗}[η] = q_{∗}[η′] in H^{1}(R_{n}, G/C_{0}). Since q_{∗}[η]_{F_{n}} = q_{∗}[η′]_{F_{n}} in H^{1}(F_{n}, G/C_{0}) it will suffice to establish the Lemma for G/C_{0}. The claim states that q_{∗}η^{geo} is anisotropic, therefore q_{∗}[η] = q_{∗}[η′] in H^{1}(R_{n}, G/C_{0}) by Theorem 7.9.

We can now proceed to prove Theorem 8.1.

Proof. The surjectivity of the map H_{loop}^{1}(R_{n}, G) → H^{1}(F_{n}, G) is a special case of Proposition 6.9. Let us establish injectivity. We are given two loop cocycles η, η′ ∈ Z^{1}(π_{1}(R_{n}), G(\overline{k})) having the same image in H^{1}(F_{n}, G). Lemma 7.12 shows that [η^{ar}] = [η′^{ar}] in H^{1}(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 the irreducible case, i.e. to the case when _{η}G^{0} ×_{R_{n}} F_{n} is irreducible.

Let Q be a minimal F_{n}–parabolic subgroup of _{η}G^{0} ×_{R_{n}} F_{n}. Corollary 7.4 shows that the k–group G^{0} admits a parabolic subgroup P of the same type than Q which is normalized 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^{0}(k)–conjugate to P we may assume that η′ normalizes P as well. Furthermore, P is minimal for η (and η′) with respect to this property. We can view then η, η′ as elements of Z_{loop}^{1}(R_{n}, N_{G}(P))_{irr}. We look at the following commutative diagram

H_{loop}^{1}(R_{n}, N_{G}(P))_{irr} ⟶ H^{1}(R_{n}, G)
↓  ↓
H^{1}(F_{n}, N_{G}(P))_{irr} ⟶^{∼} H^{1}(F_{n}, G)

Since the bottom map is injective (see [Gi4, th. 2.15]), it will suffice to show that H_{loop}^{1}(R_{n}, N_{G}(P))_{irr} injects in H^{1}(F_{n}, N_{G}(P_{I})). Since the unipotent radical U of P is a split unipotent group, we have

H^{1}(R_{n}, N_{G}(P)) ≃ H^{1}(R_{n}, N_{G}(P)/U),

and similarly for F_{n} by Lemma 4.14. So we are reduced to showing that H_{loop}^{1}(R_{n}, N_{G}(P)/U)_{irr} injects in H^{1}(F_{n}, N_{G}(P)/U), which is covered by Lemma 8.2. This completes the proof of injectivity.

8.2 Application: Witt-Tits decomposition

By using the Witt-Tits decomposition over F_{n} [Gi4, th. 2.15], we get the following.

Corollary 8.4. Assume that G^{0} is a split reductive k–group. Let P_{I_{1}}, ... , P_{I_{l}} be representatives of the G(\overline{k})-conjugacy classes of parabolic subgroups of G^{0}. Let L_{I_{j}} be a Levi subgroup of P_{I_{j}} for j = 1, ...l. Then

⨆_{j=1,...,l} H_{loop}^{1}(R_{n}, N_{G}(P_{I_{j}}, L_{I_{j}}))_{irr} ≃ H_{loop}^{1}(R_{n}, G) ≃ H^{1}(F_{n}, G).

Remark 8.5. It follows from the Corollary that we have a “Witt-Tits decomposition” for loop torsors. Furthermore, if we are interested in purely geometrical irreducible loop torsors, then we have a nice description in terms of k–group homomorphisms _{∞}μ μ μ → G as described in Theorem 7.9. This corresponds to the embedding of Hom_{k−gp}(_{∞}μ μ μ_{irr}^{n}, G)/G(k) in H^{1}(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_{1}}, ... , P_{I_{l}} be the standard k–parabolic subgroups containing a given Borel subgroup of G/k. Then

⨆_{j=1,...,l} H_{loop}^{1}(R_{n}, P_{I_{j}})_{irr} ≃ H_{loop}^{1}(R_{n}, G) ≃ H^{1}(F_{n}, G).

2. If k is algebraically closed

Hom_{k−gp,irr}(_{∞}μ μ μ^{n}, G)/G(k) ≃ H_{loop}^{1}(R_{n}, G)_{irr} ≃ H^{1}(F_{n}, G)_{irr}.

Using our choice of roots of unity (2.3), we have _{∞}μ μ μ ≃ ℤ̂. So the left handside is Hom_{gp} (ℤ̂^{n}, G(k))_{irr}/G(k), namely the G(k)–conjugacy classes of finite order irreducible pairwise commuting elements (g_{1}, ..., g_{n}) (irreducible in the sense that the elements do not belong to a proper parabolic subgroup).

8.3 Classification of semisimple 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 the situation 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 particular case, 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 Δ of the corresponding root system. For each subset I ⊂ Δ we define as usual

T_{I} = (⋂_{α∈I} ker(α))^{0}.

Since G is adjoint, we know that the roots define an isomorphism T ≃ (G_{m})^{|Δ|}, hence T_{I} ≃ (G_{m})^{|Δ\I|}. The centralizer L_{I} := C_{G}(T_{I}), is the standard Levi subgroup of the parabolic subgroup P_{I} = U_{I} ⋊ L_{I} attached to I. Since G is of adjoint type, we know that L_{I}/T_{I} is a semisimple k–group of adjoint type.

We have a split exact sequence of k–groups

1 → G → Aut(G) → Out(G) → 1

where 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 §3].^{23} For I ⊂ Δ, we need to describe the normalizer N_{Aut(G)}(L_{I}) of L_{I}. Following [Sp, 16.3.9.(4)], we define the subgroup of I-automorphisms of G by

Aut_{I}(G) = Aut(G, P_{I}, L_{I}).

where this last is the subgroup of Aut(G) that stabilizes both P_{I} and L_{I}. There is then an exact sequence

1 → L_{I} → Aut_{I}(G) → Out_{I}(G) → 1,

23In [SGA3] the group Out(G) is denoted by Aut(Dyn).

where Out_{I}(G) is the finite constant group corresponding to the subgroup of Out(G) consisting of elements that stabilize I ⊂ Δ. Then the preceding Corollary reads

⨆_{[I]⊂Δ/Out(G)} H_{loop}^{1}(R_{n}, Aut_{I}(G))_{irr} ≃ H_{loop}^{1}(R_{n}, Aut(G)) ≃ H^{1}(F_{n}, Aut(G)).

By [Gi4, cor. 3.5], H_{loop}^{1}(R_{n}, Aut_{I}(G))_{irr} ≃ H^{1}(F_{n}, Aut_{I}(G))_{irr} can be seen as a subset of H_{loop}^{1}(R_{n}, Aut_{I}(G)/T_{I})_{an} ≃ H^{1}(F_{n}, Aut_{I}(G)/T_{I})_{an}. We come now to another of the central results of the paper.

Theorem 8.7. Assume that k is algebraically closed and of characteristic 0. Let G be a simple k–group of adjoint type. Let T ⊂ B, I, and Δ be as above. On the set Hom_{k−gp} (_{∞}μ μ μ^{n}, Aut_{I}(G)) define the equivalence relation ϕ ∼_{I} ϕ′ if there exists g ∈ Aut_{I}(G)(k) such that ϕ and gϕ′g^{−1} have same image in Hom_{k−gp} (_{∞}μ μ μ^{n}, Aut_{I}(G)/T_{I}). Then we have a decomposition

⨆_{[I]⊂Δ/Out(G)} Hom_{k−gp}(_{∞}μ μ μ^{n}, Aut_{I}(G))_{an}/ ∼_{I} ⟶^{∼} H_{loop}^{1}(R_{n}, G) ⟶^{∼} H^{1}(F_{n}, G).

where Hom_{k−gp} (_{∞}μ μ μ^{n}, Aut_{I}(G)/T_{I})_{an} stands for the set of anisotropic group homomorphisms _{∞}μ μ μ^{n} → Aut_{I}(G)/T_{I}.

Remark 8.8. As an application of Margaux’s rigidity theorem [Mg2], the right handside does not change by extension of algebraically closed fields. Hence H_{loop}^{1}(R_{n}, G) does not change by extension of algebraically closed fields. This allows us in practice whenever useful to work over \overline{ℚ} or ℂ.

Proof. The group (Aut_{I}(G)/T_{I})(k) acts naturally on the set Hom_{k−gp} (_{∞}μ μ μ^{n}, Aut_{I}(G)/T_{I})_{an} by conjugation, and we denote the resulting quotient set by \overline{Hom}_{k−gp}(_{∞}μ μ μ^{n}, Aut_{I}(G)/T_{I})_{an}. The commutative square

Hom_{k−gp} (_{∞}μ μ μ^{n}, Aut_{I}(G))_{irr} ⟶ \overline{Hom}_{k−gp}(_{∞}μ μ μ^{n}, Aut_{I}(G)/T_{I})_{an}
↓  ↓
H_{loop}^{1}(R_{n}, Aut_{I}(G))_{irr} ⟶ H_{loop}^{1}(R_{n}, Aut_{I}(G)/T_{I})_{an}

is well defined as one can see by taking into account Corollary 7.4. Since k is algebraically closed, loop torsors are purely geometric, hence the two vertical maps are onto. As we have seen, the bottom horizontal map is injective; this defines an equivalence relation ∼′_{I} on the set Hom_{k−gp} (_{∞}μ μ μ^{n}, Aut_{I}(G)) such that

Hom_{k−gp} (_{∞}μ μ μ^{n}, Aut_{I}(G))_{irr}/ ∼′_{I} ⟶^{∼} H_{loop}^{1}(R_{n}, Aut_{I}(G)).

It remains to establish that the equivalence relations ∼′_{I} and ∼_{I} coincide, and we do this by using that the right vertical map in the above diagram is injective. (Theorem 7.9). We are given ϕ_{1}, ϕ_{2} ∈ Hom_{k−gp} (_{∞}μ μ μ^{n}, Aut_{I}(G))_{irr}. Then ϕ_{1} ∼′_{I} ϕ_{2} if and only if the image of ϕ_{1} and ϕ_{2} in Hom_{k−gp} (_{∞}μ μ μ^{n}, Aut_{I}(G)/T_{I})_{an} are conjugate by an element of (Aut_{I}(G)/L_{I})(k). Since the map Aut_{I}(G)(k) → (Aut_{I}(G)/L_{I})(k) is onto, it follows that ϕ_{1} ∼′_{I} ϕ_{2} if and only if ϕ_{1} ∼_{I} ϕ_{2} as desired.

Corollary 8.9. Under the hypothesis of Theorem 8.7, the classification of loop torsors on R_{n} “is the same” as the classification, for each subset I ⊂ Δ, of irreducible commuting n–uples of elements of finite order of Aut_{I}(G)(k) up to the equivalence relation ∼_{I}.

8.4 Action of GL_{n}(ℤ)

The assumptions are as in the previous section. We fix a pinning (epinglage) (G, B, T) [XXIV §1]. This determines a section We fix a section 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_{1}^{a_{1i}}t_{2}^{a_{2i}} … t_{n}^{a_{ni}}

We denote the resulting k–automorphism of R_{n} corresponding to g also by g since no confusion will arise. By Yoneda considerations g (anti)corresponds to an automorphism g^{∗} of the k–scheme Spec(R_{n}).

Applying (8.1) where we now replace t_{i} by t_{1}^{1/m} and k by \overline{k} gives a left action of GL_{n}(ℤ) as automorphisms of \overline{R}_{n,m} = \overline{k}[t^{±1/m}, …, t_{n}^{±1/m}]. If we denote by g_{m} the automorphism corresponding to g then the diagram

R_{n} ⟶^{g} R_{n}
↓  ↓
\overline{R}_{n,m} ⟶^{g_{m}} \overline{R}_{n,m}

commutes. Passing to the direct limit on (8.4) the element g induces an automorphism g_{∞} of \overline{R}_{n,∞} = \underrightarrow{lim}\overline{k}[t^{±1/m}, …, t_{n}^{±1/m}]. If no confusion is possible, we will denote g_{∞} and g_{m} simply by g.

Recall that π_{1}(R_{n}) = ℤ̂(1)^{n} ⋊ Gal(k). Our fix choice of compatible roots of unity (ξ_{m}) allows us to identify π_{1}(R_{n}) with ℤ̂^{n} ⋊ Gal(k) where the left action of Gal(k) on each component ℤ̂ = \underleftarrow{lim}ℤ/mℤ is as follows: If a ∈ Gal(k) and m ≥ 1 there exists a unique 1 ≤ a(m) ≤ m − 1 such that a(ξ_{m}) = ξ_{m}^{a(m)}. This defines an automorphism a_{m} of the additive group ℤ/mℤ. Passing to the limit on each component yields the desired group automorphism â of ℤ̂^{n}.

View (ℤ/mℤ)^{n} as row vectors. Then GL_{n}(ℤ) acts on the right on this group by right multiplication

g : e_{m} ↦ e_{m}^{g} = e_{m}g

where (ℤ/mℤ)^{n} is viewed as a ℤ–module in the natural way. By passing to the inverse limit we get a right action of GL_{n}(ℤ) as automorphisms of ℤ̂^{n} that we denote by e ↦ e^{g}. We extend this to a right action on π_{1}(R_{n}) = ℤ̂^{n} ⋊ Gal(k) by letting GL_{n}(ℤ) act trivially on Gal(k). Thus if γ = (e, a) ∈ ℤ̂^{n} ⋊ Gal(k) and g ∈ GL_{n}(ℤ), then γ^{g} = (e^{g}, a).^{24}

By taking the foregoing discussion into consideration we can define the (right) semidirect product group GL_{n}(ℤ) ⋉ π_{1}(R_{n}) with multiplication

(8.2)  (h, (e, a))(g, (f, b)) = (hg, (e^{g}, a)(f, b)) = (hg, (e^{g}â(f), ab))

for all h, g ∈ GL_{n}(ℤ), e, f ∈ ℤ̂^{n} and a, b ∈ π_{1}(R_{n}). For future use we point out that under that under the natural identification of GL_{n}(ℤ) and π_{1}(R_{n}) with subgroups of GL_{n}(ℤ) ⋉ π_{1}(R_{n}) we have

(8.3)  γg = gγ^{g}

for all g ∈ GL_{n}(ℤ) and γ ∈ π_{1}(R_{n}).

By definition π_{1}(R_{n}) acts naturally on \overline{R}_{n,∞}. Under our identification π_{1}(R_{n}) = ℤ̂^{n} ⋊ Gal(k) the action is given by

(8.4)  (e, a) : λt_{i}^{1/m} ↦ a(λ)ξ_{m}^{e_{m,i}}t_{i}^{1/m}

where e = (e_{1}, …, e_{n}) ∈ ℤ̂^{n}, e_{i} = (e_{m,i})_{m≥1} 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_{1}(R_{n}) defined above acts on the left as automorphisms of the k–algebra \overline{R}_{n,∞} in a way which is 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_{1}(R_{n}) and x ∈ \overline{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^{1}(R_{n}, H) onto itself. This leads to a left action of GL_{n}(ℤ) on this pointed set which we called base change. Our objective is to have a precise description of this action.^{25} The isotriviality theorem [GP3, th. 2.9] shows tha it will suffice to trace the base change action at the level of 1-cocycles in Z^{1}(π_{1}(R_{n}), H(\overline{R}_{n,∞})). Following standard conventions for cocycles we denote the action of an element γ ∈ π_{1}(R_{n}) on an element h ∈ H(\overline{R}_{n,∞}) by ^{γ}h. Then (8.5) implies that

24After our identifications, this is nothing but the natural action of g^{∗} on π_{1}( Spec(R_{n})).

(8.6)  γ.h = ^{γ}h.

Lemma 8.10. The base change action of GL_{n}(ℤ) on H^{1}(R_{n}, H) is induced by the action η ↦ ^{g}η of GL_{n}(ℤ) on Z^{1}(π_{1}(R_{n}), H(\overline{R}_{n,∞})) given by

(^{g}η)(γ) = g.η(γ^{g})

for all γ ∈ π_{1}(R_{n}) and g ∈ GL_{n}(ℤ).

Proof. For all α, β ∈ π_{1}(R_{n}) we have

(^{g}η(αβ) = g.η((αβ)^{g})) [definition]
= g.η(α^{g}β^{g})
= g.(η(α^{g})^{α^{g}}η(β^{g})) [η a cocycle]
= g.(η(α^{g}) (α^{g}.η(β^{g}))) [by (8.6)]
= (g.(η(α^{g}))(g.α^{g}.η(β^{g})) [by action axiom]
= (g.(η(α^{g}))(α.g.η(β^{g})) [by action axiom and (8.3)]
= ^{g}η(α)(α.^{g}η(β)) [by definition ]
= ^{g}η(α)^{α}(^{g}η(β)) [by (8.6) ].

This shows that ^{g}η is a cocycle (which is clearly continuous since η is). That this defines a left action of GL_{n}(ℤ) on Z^{1}(π_{1}(R_{n}), H(\overline{R}_{n,∞})) is easy to verify using the definitions.

Next we verify that the action factors through H^{1}. Assume μ is a cocycle cohomologous to η, and let h ∈ H(\overline{R}_{n,∞}) be such that μ(γ) = h^{−1}ηγ^{γ}h for all γ ∈ π_{1}(R_{n}). Then

25Our main interest is the case when H = Aut(G) with G simple. The reason behind the importance of this case lies in the applications to infinite dimensional Lie theory.

gμ(γ) = g.μ(^{g}γ) [definition]
= g.(h^{−1}η(γ^{g})^{γ}h)
= g.h^{−1} g.η(γ^{g}) g.^{γ}h [action axiom]
= (g.h)^{−1} ^{g}η(γ)^{γ}g^{.}^{γ}h [action axiom, definition, and g = ^{γ}g]
= (g.h)^{−1} ^{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^{1}(π_{1}(R_{n}), H(\overline{R}_{n,∞})) = H^{1}(R_{n}, H) coincides with the base change action. To see this we consider a faithfull representation H → GL_{d} and the corresponding quotient variety Y = GL_{d}/H. Since H^{1}(R_{n}, GL_{n}) = 1 by a theorem of Quillen and Suslin, we have a short exact sequence of pointed sets

1 → H(R_{n}) → GL_{d}(R_{n}) → Y(R_{n}) →^{φ} H^{1}(R_{n}, H) → 1.

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^{−1} ^{γ}Y = Y^{−1} γ.Y

where Y ∈ GL_{d}(\overline{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 a lift of g.y, we conclude that φ(g.y) is the class of the cocycle (g.Y)^{−1}^{γ}(g.Y). Using identities and compatibility of actions that have already been mentioned, we have

(g.Y)^{−1}^{γ}(g.Y) = (g.Y)^{−1}γ.(g.Y) = g.Y^{−1} gγ^{g}.Y =
= g.Y^{−1} g.γ^{g}.Y = g.(Y^{−1}γ^{g}.Y ) = g.η(γ^{g}) = ^{g}η(γ)

as desired.

Remark 8.11. The action of GL_{n}(ℤ) on Z^{1}(π_{1}(R_{n}), H(\overline{R}_{n,∞})) stabilizes Z^{1}(π_{1}(R_{n}), H(\overline{k})). In particular, it preserves loop cocycles.

We pause to observe that the decomposition 8.4 is equivariant under the action of GL_{n}(ℤ) on R_{n}. Thus Corollary 8.12. With the assumptions and notation as above

⨆_{j=1,...,l} GL_{n}(ℤ)\H_{loop}^{1}(R_{n}, Aut(G, P_{I_{j}}))_{irr} ≃ GL_{n}(ℤ)\H_{loop}^{1}(R_{n}, Aut(G))

In particular, if 𝔈 is a loop R_{n}–torsor under Aut(G), the Witt-Tits index of the loop group scheme _{𝔈}G/R_{n} depends only of the class of 𝔈 in GL_{n}(ℤ)\H_{loop}^{1}(R_{n}, Aut(G)).

Remark 8.13. Assume η is a loop cocycle. Since GL_{n}(ℤ) acts trivially on H(k) we have (^{g}η)(γ) = η(γ^{g}) for all γ ∈ π_{1}(R_{n}) and g ∈ GL_{n}(ℤ).

Lemma 8.14. Assume that H acts on a quasi-projective k–variety M. Let η ∈ Z^{1}(π_{1}(R_{n}), H(\overline{k})) 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^{1}(π_{1}(R_{n}), H(\overline{k})).

(1) Λ_{η} = { g ∈ GL_{n}(ℤ) | η(γ^{g}) = η(γ) ∀γ ∈ π_{1}(R_{n}) }.

(2) The map

(GL_{n}(ℤ) ⋉ π_{1}(R_{n})) × M(\overline{R}_{n,∞}) → M(\overline{R}_{n,∞}), ((g, γ), x) ↦ g.η(γ).γ.x

defines an action of Λ_{η} ⋉ π_{1}(R_{n}) on (_{η}X)(\overline{R}_{n,∞}).

(3) Assume that M is a linear algebraic k–group on which H acts as group automorphisms. Let g ∈ Λ_{η} and ζ ∈ Z^{1}(π_{1}(R_{n}),_{η}M(\overline{R}_{n,∞})), and set

gζ(γ) = g.ζ(γ^{g}).

This defines a (left) action of Λ_{η} on Z^{1}(π_{1}(R_{n}),_{η}M(\overline{R}_{n,∞})) which induces an action of Λ_{η} on H^{1}(R_{n},_{η}M). The action is functorial in M. If H = M, the diagram

H^{1}(R_{n},_{η}H) ⟶_{∼}^{τ_{η}} H^{1}(R_{n}, H)
g_{∗}↓^{≀}  ^{g_{∗}}↓^{≀}
H^{1}(R_{n},_{η}H) ⟶_{∼}^{τ_{η}} H^{1}(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}(π_{1}(R_{n}),_{η}M) of degree i ≥ 0, set

(^{g}y)(γ_{1}, …, γ_{i}) = ^{g(}y(γ_{1}^{g}, …, γ_{i}^{g})).

This defines a left action of Λ_{η} on the chain complex C^{∗}(π_{1}(R_{n}),_{η}M(\overline{R}_{n,∞})) of (contnuous) inhomogeneous cochains and on H^{∗}(π_{1}(R_{n}),_{η}M(\overline{R}_{n,∞})) = H^{∗}(R_{n},_{η}M) which is functorial with respect to short exact sequences of H–equivariant k–groups of multiplicative type.

(5) Assume that H is finite and let 1 → M_{1} → M_{2} → M_{3} → 1 be a an exact 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 ηM_{3}(R_{n}) → H^{1}(R_{n},_{η}M_{1}). If M_{1} is central in M_{2}, then the action of Λ_{η} commutes with the boundary map Δ : H^{1}(R_{n},_{η}M_{3}) → H^{2}(R_{n},_{η}M_{1}).

(6) Assume k is algebraically closed and let d be a postive integer. The base change action of GL_{n}(ℤ) on H^{2}(R_{2},μ μ μ) and on Br(R_{2}) 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(\overline{R}_{n,∞}) coincides with M(\overline{R}_{n,∞}) as a set, but the action of π_{1}(R_{n}) is the twisted action, which we denote by ⋆:

γ ⋆ x = η(γ).(γ.x)

(2) The groups GL_{n}(ℤ) and π_{1}(R_{n}) act on X(\overline{R}_{n,∞}) and H(\overline{R}_{n,∞}) via their natural action on \overline{R}_{n,∞}. For We will denote these actions by x ↦ g.x, and x ↦ γ.x for all g ∈ GL_{n}(ℤ), γ ∈ π_{1}(R_{n}) and x ∈ X(\overline{R}_{n,∞}). It follows from (8.5) and (8.6) that for all γ ∈ π_{1}(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(\overline{R}_{n,∞}). The content of (2) is that

(8.7)  (g, γ) ⋆ x = g.(γ ⋆ x)

defines an action of Λ_{η} ⋉ π_{1}(R_{n}) on (_{η}X)(\overline{R}_{n,∞}). Write for convenience g.γ ⋆ x instead of g.(γ ⋆ x) since no confusion is possible. Then

(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}]
= fg.η(α^{g}β).α^{g}(β.x) [η a cocycle]
= (fg, α^{g}β) ⋆ x
= ((f, α)(g, β)) ⋆ x.

(3) One checks that^{g}η is a cocycle and that two equivalent cocycles remain equivalent under this action along the same lines of Lemma 8.10.

The commutativity of the diagram takes place already at the level of cocycles. Indeed. Consider the square

Z^{1}(π_{1}(R_{n}),_{η}H(\overline{R}_{n,∞})) ⟶_{∼}^{τ_{η}} Z^{1}(π_{1}(R_{n}), H(\overline{R}_{n,∞}))
g_{∗}↓^{≀}  ^{g_{∗}}↓^{≀}
Z^{1}(π_{1}(R_{n}),_{η}H(\overline{R}_{n,∞})) ⟶_{∼}^{τ_{η}} Z^{1}(π_{1}(R_{n}), H(\overline{R}_{n,∞}))

Given a cocycle ϕ ∈ Z^{1}(π_{1}(R_{n}),_{η}H(\overline{R}_{n,∞})) recall that (τ_{η}ϕ)(γ) = ϕ(γ) η(γ), hence

g((τ_{η})(ϕ))(γ) = g.(τ_{η}(ϕ)(γ^{g})) = g.((ϕ)(γ^{g}) η(γ^{g})) , 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 cohomology of finite quotients. Hence it is enough to establish the desired results at the “finite level”, namely for a group Γ = Gal(R̃_{n,m}/R_{n}) where R̃_{n,m} = k̃ ⊗_{k} R_{n,m} is a finite Galois covering of R_{n} through wich η factors, and such that H(k̃) = H(\overline{k}). Recall that the action of GL_{n}(ℤ) on \overline{R}_{n,∞} preserves k̃ ⊗_{k} R_{n,m}, so that GL_{n}(ℤ) acts on Γ.

We need to check that the given action of Λ_{η} on C^{∗}(Γ,_{η}A) commutes with the differentials. 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

(∂_{i}(y))(γ_{1}, …, γ_{i+1}) =

γ_{1} . η(γ_{1}).y(γ_{1}, …, γ_{i+1}) + ∑_{j=1}^{i} (−1)^{j} y(γ_{1}, …, γ_{j−1}, γ_{j}γ_{j+1}, γ_{j+2}, … γ_{i+1}) + (−1)^{i+1}y(γ_{1}, …, γ_{i}).

Thus

(^{g}(∂_{i}(y)))(γ_{1}, …, γ_{i+1}) = g.(∂_{i}(y)(γ_{1}^{g}, …, γ_{i+1}^{g}))
= g.(γ_{1}^{g} η(γ_{1}^{g}) . y(γ_{1}^{g}, …, γ_{i+1}^{g}))

+g.(∑_{j=1}^{i} (−1)^{j} y(γ_{1}^{g}, …, γ_{j−1}^{g}, γ_{j}^{g}γ_{j+1}^{g}, γ_{j+2}^{g}, … γ_{i+1}^{g}))

+g.((−1)^{i+1} y(γ_{1}^{g}, …, γ_{i}^{g}))

= γ_{1} η(γ_{1}) .^{g}y(γ_{1}, …, γ_{i+1}) [g ∈ Λ_{η}]

+ ∑_{j=1}^{i} (−1)^{j} ^{g}y(γ_{1}, …, γ_{j−1}, γ_{j}γ_{j+1}, γ_{j+2}, … γ_{i+1})

+(−1)^{i+1} ^{g}y(γ_{1}, …, γ_{i})

= (∂_{i}(^{g}y))(γ_{1}, …, γ_{i+1}).

This shows that the action of Λ_{η} on C^{i}(Γ,_{η}A) commutes with the boundary maps as desired.

(5) We are given an exact sequence of linear algebraic groups 1 → M_{1} → M_{2} → M_{3} → 1 equipped with an action of H. We twist it by η to obtain 1 → _{η}M_{1} → ηM_{2} → _{η}M_{3} → 1, and look at the characteristic map

ψ : _{η}M_{3}(R_{n}) → H^{1}(R_{n},_{η}M_{1}).

Let x_{3} ∈ _{η}M_{3}(R_{n}) ⊂ M_{3}(\overline{R}_{n,∞}). Lift x_{3} to an element x_{2} ∈ M_{2}(\overline{R}_{n,∞}). Then ψ(x_{3}) = [z_{γ}] with z_{γ} = x_{2}^{−1}(η(γ)^{γ}x_{2}). Now if g ∈ Λ_{η} the element ^{g}x_{2} lifts ^{g}x_{3}, hence ψ(^{g}x_{3}) is represented by the 1–cocycle

(^{g}x_{2})^{−1} (η(γ)^{γ}(^{g}x_{2})) = ^{g}x_{2}^{−1}(η(γ)^{γ}^{g}x_{2}) = g.(x_{2}^{−1}(η(γ)^{γ^{g}}x_{2})) = (^{g}z)_{γ}

by using again η(γ^{g}) = η(γ) and the fact that g acts trivially on H(\overline{k}). This shows that ψ(^{g}x_{2}) = ^{g}ψ(x_{2}).

Assuming that M_{1} is central and of multiplicative type, we consider the boundary map Δ : H^{1}(R_{n},_{η}M_{3}) → H^{2}(R_{n},_{η}M_{1}). By isotriviality, the precise nature of this map can be computed at the “finite level” by means of Galois cocycles. Let (a_{γ}) be a cocycle with value _{η}M_{3}(\overline{R}_{n,∞}) = M_{3}(\overline{R}_{n,∞}) and choose a lifting (b_{γ}) in M_{2}(\overline{R}_{n,∞}) which is trivial on an open subgroup of π_{1}(R_{n}). Recall that Δ([a_{γ}]) ∈ H^{2}(π_{1}(R_{n}),_{η}M_{1}(\overline{R}_{n,∞})) is the class of the 2–cocycle [Se, I.5.6]

c_{γ,τ} = b_{γ} (η(γ).^{γ}b_{τ}) b_{γτ}^{−1}.

Similarly, the (^{g}b_{γ^{g}}) lift the (^{g}a_{γ^{g}}), so Δ(g.[a_{γ}]) is the class of the 2–cocycle

gb_{γ^{g}} (η(γ).^{γ}(^{g}b_{τ^{g}}))^{g}b_{γ^{g}τ^{g}}^{−1} = g.(b_{γ} (η(γ).^{γ^{g}}b_{τ}^{g}) b_{γ^{g}τ^{g}}^{−1}) = g.Δ([a_{γ}])

as desired.

6) Since H^{2}(R_{2},μ μ μ_{d}) injects in Br(R_{2}) = ℚ/ℤ [GP2, 2.1], it is enough to check the

formula on Br(R_{2}). Since GL_{2}(ℤ) is generated by the matrices (\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix}), (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix})

(\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) and (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}), it is enough to show that the desired compatibility holds when

g is one of these four elements. Consider the cyclic Azumaya R_{2}-algebra A = A(1, d) with presentation T_{1}^{d} = t_{1}, T_{2}^{d} = t_{2}, T_{2}T_{1} = ζ_{d}T_{1}T_{2}. Then for g in the above list we have g.[A] = [A] (resp. −[A], [A], −[A]) respectively, so that g.[A] = det(g).[A] Since the classes of these cyclic Azumaya algebras generate Br(R_{2}) the result follows.

Remark 8.15. (a) In (4), we have H^{∗}(π_{1}(R_{n}),_{η}M(\overline{R}_{n,∞})) ⟶^{∼} 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 an explicit description of this action in our proof, but the result can also be established in a more abstract setting. For g ∈ Λ_{η}, we claim that the map g_{∗} : A → A, a ↦ g.a applies H^{0}(Γ,_{η}A) into itself. Indeed for a ∈ H^{0}(Γ,_{η}A) and γ ∈ Γ, we compute the twisted 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(k̃)]
= (g η(γ^{g})γ^{g}) .a [g ∈ Λ_{η}]
= g.a [a ∈ H^{0}(Γ,_{η}A)].

We get then a morphism of functors g_{∗} : F → F which extends uniquely as a morphism of δ-functors [W, §2.5]. This then yields the desired natural transformations g_{∗} : H^{i}(Γ,_{η}A) → H^{i}(Γ,_{η} A) for each GL_{n}(ℤ)⋉(H(k̃) ⋊ Γ)-module.

(b) There is an analogous statement to (5) for homogeneous spaces.

For each class [𝔈] ∈ H^{1}(R_{n}, Out(G)), we denote by H^{1}(R_{n}, Aut(G))_{[𝔈]} the fiber at [𝔈] of the map H^{1}(R_{n}, Aut(G)) → H^{1}(R_{n}, Out(G)). We then have the decomposition

(8.8)  H^{1}(R_{n}, Aut(G)) = ⨆_{[𝔈]∈H^{1}(R_{n},Out(G))} H^{1}(R_{n}, Aut(G))_{[𝔈]}

The group GL_{n}(ℤ) acts on H^{1}(R_{n}, Out(G)) and on H^{1}(R_{n}, Aut(G)) 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^{1}(R_{n}, Out(G)), its stabilizer under the action of GL_{n}(ℤ) preserves H^{1}(R_{n}, Aut(G))_{[𝔈]}.

Let Out(G) = Out(G)(k). The (abstract) group Out(G) acts naturally on the right on the set of (continuous) homomorphisms Hom(π_{1}(R_{n}), Out(G)). This action, which we denote by int, is given by int(a)(ϕ)(γ) = ϕ^{a}(γ) = a^{−1} ϕ(γ) a. We have H^{1}(R_{n}, Out(G)) = Hom (π_{1}(R_{n}), Out(G))/int(Out(G))

We consider a system of representatives ([ϕ_{j}])_{j∈J} of the set of double cosets GL_{n}(ℤ) \ Hom (π_{1}(R_{n}), Out(G))/int(Out(G)). Consider a fixed element j ∈ J. Denote by Λ_{j} ⊂ GL_{n}(ℤ) the stabilizer of [ϕ_{j}] ∈ H^{1}(R_{n}, Out(G)) for the base change action of GL_{n}(ℤ) on Spec(R_{n}). An element g ∈ GL_{n}(ℤ) belongs to Λ_{j} if and only if there exists a_{g} ∈ Out(G) such that the following diagram commutes

ϕ_{j} : π_{1}(R_{n}) ⟶ Out(G)
g^{∗}↑^{≀}  ^{int(a_{g})}↑^{≀}
ϕ_{j} : π_{1}(R_{n}) ⟶ Out(G)

Note that Λ_{ϕ_{j}} ⊂ Γ_{j}. We have

(8.9)  GL_{n}(ℤ)\H^{1}(R_{n}, Aut(G)) = ⨆_{j∈J} Λ_{j}\H^{1}(R_{n}, Aut(G))_{[ϕ_{j}]}.

Recall that our section s : Out(G) → Aut(G) is determined by our choice of pinning of (G, B, T). This allows us to trace the action of Λ_{j}. Indeed [s_{∗}(ϕ_{j})] ∈ H^{1}(R_{n}, Aut(G))_{[ϕ_{j}]}, so that the classical twisting argument (see [Gi4, lemme 1.2]) shows that the map

H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G) → H^{1}(R_{n},_{s_{∗}(ϕ_{j})}Aut(G)) →^{τ_{s∗(ϕ_{j})}} H^{1}(R_{n}, Aut(G))

induces a bijection

(8.10) H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G)/H^{0}(R_{n},_{ϕ_{j}}Out(G)) ⟶^{∼} H^{1}(R_{n}, Aut(G))_{[ϕ_{j}]}.

Note that the action of an element a ∈ H^{0}(R_{n},_{ϕ_{j}}Out(G)) on H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G) is given by

H^{1}(R_{n},_{s_{∗}(ϕ)}G) ⟶^{∼}^{(_{ϕ_{j}} s_{∗})(a)} H^{1}(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) preserves toral or, what is equivalent, loop classes. Feeding this information into the decomposition (8.9), we get

(8.11)

GL_{n}(ℤ)\H^{1}(R_{n}, Aut(G)) ⟶^{∼}  ⨆_{j∈J} Λ_{j}\(H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G)/H^{0}(R_{n},_{ϕj}Out(G))).

At least in certain cases, the action of Λ_{j} on H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G)/H^{0}(R_{n},_{ϕ_{j}}Aut(G)) can be understood quite nicely (see Remark 8.17 below).

Lemma 8.16. (1) For each g ∈ Λ_{ϕ_{j}}, the following diagrams

H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G) ⟶ H^{1}(R_{n},_{s_{∗}(ϕ_{j})}Aut(G)) ⟶^{τ_{s∗(ϕ_{j})}} H^{1}(R_{n}, Aut(G))_{[ϕ_{j}]}
g_{∗}↓  ^{g_{∗}}↓  ^{g_{∗}}↓
H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G) ⟶ H^{1}(R_{n},_{s_{∗}(ϕ_{j})}Aut(G)) ⟶^{τ_{s∗(ϕ_{j})}} H^{1}(R_{n}, Aut(G))_{[ϕ_{j}]},
H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G) × H^{0}(R_{n},_{ϕ_{j}}Out(G)) ⟶ H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G)
g_{∗}↓  ^{id}↓  ^{g_{∗}}↓
H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G) × H^{0}(R_{n},_{ϕ_{j}}Out(G)) ⟶ H^{1}(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^{1}(R_{n},_{s_{∗}(ϕ_{j})}G) → H^{1}(R_{n}, Aut(G))_{[ϕ_{j}]}

is Λ_{ϕ_{j}} × H^{0}(R_{n},_{ϕ_{j}}Out(G))^{op}–equivariant and

Λ_{ϕ_{j}} × H^{0}(R_{n},_{ϕ_{j}}Out(G))^{op} \ H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G) ⟶^{∼} Λ_{ϕ_{j}} \ H^{1}(R_{n}, Aut(G))_{[ϕ_{j}]}.

Remark 8.17. Of course (2) is useful provided that Λ_{ϕ_{j}} = Γ_{j}. This is the case for simple groups which are not of type D_{4} since Out(G) = 1 or ℤ/2ℤ.

Proof. (1) We are given g ∈ Λ_{ϕ_{j}}. The left square of the first diagram commutes by the functoriality of the base change map g_{∗}. The commutativity of the right square follows of 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 given in Lemma 8.10.

(2) By (1), the left action of Λ_{ϕ_{j}} and the right action of H^{0}(R_{n},_{ϕ_{j}}Out(G)) on H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G commute. Hence

Λ_{ϕ_{j}}\(H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G)/H^{0}(R_{n},_{s_{∗}(ϕ_{j})}Aut(G))) ⟶^{∼}

Λ_{ϕ_{j}} × H^{0}(R_{n},_{s_{∗}(ϕ_{j})}Out(G))^{op} \ H^{1}(R_{n},_{s_{∗}(ϕ_{j})}G)

and this set maps bijectively onto Λ_{ϕ_{j}}\H^{1}(R_{n}, Aut(G))_{[ϕ_{j}]}.

9 Small dimensions

9.1 The one dimensional case

By combining Theorem 8.1, Corollary 5.2 and Lemma 4.14 we get the following generalization (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}^{1}(k[t^{±1}], G) ⟶^{∼} H_{loop}^{1}(k[t^{±1}], G) ⟶^{∼} H^{1}(k[t^{±1}], G) ⟶^{∼} H^{1}(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 in infinite dimensional Lie theory.

9.2 The two dimensional case

Throughout this section we assume that k is algebraically closed of characteristic 0 and G a semisimple Chevalley k–group of adjoint type. We let G^{sc} → G be its simply connected covering and denote by μ μ μ its kernel.

9.2.1 Classification of semisimple loop R_{2}-groups.

Serre’s conjecture II holds for the field F_{2} by Bruhat–Tits theory [BT3, cor. 3.15], i.e. H^{1}(F_{2}, H) = 1 for every semisimple simply connected group H over F_{2}. Furthermore, we know explicitly how to compute the Galois cohomology of an arbitrary semisimple F_{2} group [CTGP, th. 2.1] and [GP2, th. 2.5]. We thus have.

Corollary 9.2. We have a decomposition

H_{loop}^{1}(R_{2}, Aut(G)) ⟶^{∼} ⨆_{[𝔈]∈H^{1}(R_{2},Out(G))}_{𝔈}Out(G)(R_{2}) \H^{2}(R_{2},_{𝔈}μ μ μ)

and the inner R_{2}–forms of G are classified by the coset Out(G)(R_{2}) \H^{2}(R_{2},μ μ μ).

Note that the case when Out(G) is trivial recovers theorem 3.17 of [GP2]. We can thus view the last Corollary as an extension of this theorem to the case when the automorphism group of G is not connected.

Proof. Our choice of splitting s : Out(G) → Aut(G) of the exact sequence

1 → G → Aut(G) → Out(G) → 1

easily leads to the decomposition (see [Gi4, lemme 1.2])

H^{1}(F_{2}, Aut(G)) ⟶^{∼} ⨆_{[𝔈]∈H^{1}(F_{2},Out(G))} H^{1}(F_{2},_{𝔈}G)/_{𝔈}Out(G)(F_{2})

with respect to the Dynkin-Tits invariant. On the other hand, the boundary map H^{1}(F_{2},_{𝔈}G) → H^{2}(F_{2},_{𝔈}μ μ μ) is bijective by [CTGP, th. 2.1] and [GP2, th. 2.5]. The right action of_{𝔈}Out(G)(F_{2}) can then be transferred to H^{2}(F_{2},_{𝔈} μ μ μ), and is the opposite of the natural left action of _{𝔈}Out(G)(F_{2}) on H^{2}(F_{2},_{𝔈} μ μ μ). Hence

H^{1}(F_{2}, Aut(G)) ⟶^{∼} ⨆_{[𝔈]∈H^{1}(F_{2},Out(G))}_{𝔈}Out(G)(F_{2}) \H^{2}(F_{2},_{𝔈}μ μ μ).

But _{𝔈}Out(G) is finite étale over R_{n} hence _{𝔈}Out(G)(R_{2}) = _{𝔈}Out(G)(F_{2}) by Remark 6.8.(d). On the other hand, we have H^{2}(R_{2},_{𝔈}μ μ μ) ⟶^{∼} H^{2}(F_{2},_{𝔈}μ μ μ) since _{𝔈}μ μ μ is an R_{2}– group of multiplicative type [GP3, prop. 3.4]. Taking into account the acyclicity theorem for Aut(G) and Out(G), we get the square of bijections

H^{1}(F_{2}, Aut(G)) ⟶_{∼} ⨆_{[𝔈]∈H^{1}(F_{2},Out(G))}_{𝔈}Out(G)(F_{2}) \H^{2}(F_{2},_{𝔈}μ μ μ).
↑^{≀}  ↑^{≀}
H_{loop}^{1}(R_{2}, Aut(G)) ⟶_{∼} ⨆_{[𝔈]∈H^{1}(R_{2},Out(G))}_{𝔈}Out(G)(R_{2}) \H^{2}(R_{2},_{𝔈}μ μ μ),

and this establishes the Corollary.

Next we give a complete list of the isomorphism classes of loop R_{2}–forms of G in the case when G is simple of adjoint type. We have Out(G) = 1 in type A_{1} B, C, E_{7}, E_{8}, F_{4} and G_{2}, Out(G) = ℤ/2ℤ in type A_{n} (n ≥ 2), D_{n} (n ≥ 5) and E_{6}, and Out(G) = S_{3} in type D_{4}. ^{26}

In the case Out(G) = 1, then by theorem 3.17 of [GP2] we have

H_{loop}^{1}(R_{2}, Aut(G)) ⟶^{∼} H^{2}(R_{2},μ μ μ). But μ μ μ = μ μ μ_{n} for n = 1 or 2. We have H^{2}(R_{2},μ μ μ_{2}) ≅

ℤ/2ℤ [GP2, §2.1].Thus

Corollary 9.3. (1) If G has type A_{1}, then H_{loop}^{1}(R_{2}, Aut(G)) ≃ ℤ/2ℤ. (2) If G has type B, C or E_{7}, then H_{loop}^{1}(R_{2}, Aut(G)) ≃ ℤ/2ℤ.

(3) If G has type E_{8}, F_{4} or G_{2}, then H_{loop}^{1}(R_{2}, Aut(G)) = 1.

Remark 9.4. In Case (1) and Case (2) the non-trivial twisted groups are not quasisplit (because their “Brauer invariant” in H^{2}(R_{2},μ μ μ) is not trivial.) In Case (1) the non-trivial twisted group is in fact anisotropic (see [GP2] for details).

In the case Out(G) = ℤ/2ℤ, we have H^{1}(R_{2}, ℤ/2ℤ) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ. The ℤ/2ℤ-Galois extensions of R_{2} under consideration are R_{2} × R_{2}, R_{2}[\sqrt{t_{1}}], R_{2}[\sqrt{t_{2}}] and R_{2}[\sqrt{t_{1}t_{2}}] which correspond to the elements (0, 0), (1, 0), (0, 1) and (1, 1) respectively. These can also be thought as ℤ/2ℤ-torsors over R_{2} that we will denote by 𝔈_{0,0}, 𝔈_{1,0}, 𝔈_{0,1} and 𝔈_{1,1} respectively. In the first case the generator of the Galois group acts by permuting the two factors, while in the other three is of the form \sqrt{x} ↦ −\sqrt{x}.

Since _{𝔈}Out(G) ≅ Out(G) = ℤ/2ℤ, for any of our four torsors we have

H_{loop}^{1}(R_{2}, Aut(G)) ⟶^{∼} ℤ/2ℤ \H^{2}(R_{2},μ μ μ) ⨆_{𝔈=𝔈_{1,0},𝔈_{0,1},𝔈_{1,1}} ℤ/2ℤ \H^{2}(R_{2},_{𝔈}μ μ μ).

This leads to a case by case discussion.

Corollary 9.5. (1) For G of type A_{2n} (n ≥ 1)

H_{loop}^{1}(R_{2}, Aut(G)) ≃ {±1} \(ℤ/(2n + 1)ℤ) ⨆_{𝔈=𝔈_{1,0},𝔈_{0,1},𝔈_{1,1}} {_{𝔈}G}.

There are n + 1 inner and three outer loop R_{2}–forms of G. All outer forms are quasisplit.

(2) For G of type A_{2n−1} (n ≥ 2)

H_{loop}^{1}(R_{2}, Aut(G)) ≃ {±1} \(ℤ/2nℤ) ⨆_{𝔈=𝔈_{1,0},𝔈_{0,1},𝔈_{1,1}} {_{𝔈}G^{±}}.

26Of course here 1, ℤ/2ℤ and S_{3} are here viewed as constant R_{2} groups or finite (abstract) groups as the situation requires.

There are n + 1 inner and six outer loop R_{2}–forms of G. The outer forms come in three pairs. Each pair has one form which is quasisplit and one which is not.

(3) For G of type D_{2n−1} (n ≥ 3)

H_{loop}^{1}(R_{2}, Aut(G)) ≃ {±1} \(ℤ/4ℤ) ⨆_{𝔈=𝔈_{1,0},𝔈_{0,1},𝔈_{1,1}} {_{𝔈}G^{±}}.

There are three inner and six outer loop R_{2}–forms of G. The outer forms come in three pairs. Each pair has one form which is quasisplit and one which is not.

(4) For G of type D_{2n} (n ≥ 3)

H_{loop}^{1}(R_{2}, Aut(G)) ≃ switch \(ℤ/2ℤ ⊕ ℤ/2ℤ) ⨆_{𝔈=𝔈_{1,0},𝔈_{0,1},𝔈_{1,1}} {_{𝔈}G^{±}}.

There are three inner and six outer loop R_{2}–forms of G. The outer forms come in three pairs. Each pair has one form which is quasisplit and one which is not.

(5) For G of type E_{6}

H_{loop}^{1}(R_{2}, Aut(G)) ≃ {±1} \(ℤ/3ℤ) ⨆_{𝔈=𝔈_{1,0},𝔈_{0,1},𝔈_{1,1}} {_{𝔈}G}.

There are two inner and three outer loop R_{2}–forms of G. All outer forms are quasisplit

Proof. (1) We have μ μ μ = μ μ μ_{2n+1} = ker(μ μ μ_{2n+1}^{2} \underrightarrow{∏} μ μ μ_{2n+1}) and the action of ℤ/2ℤ switches the two factors. We have H^{2}(R_{2},μ μ μ) ≃ ℤ/(2n + 1)ℤ and the outer action of ℤ/2ℤ is by signs.^{27}

Let 𝔈 = 𝔈_{(1,0)}. It follows that _{𝔈}μ μ μ = ker( ∏_{R_{2}[\sqrt{t_{1}}]/R_{2}} μ μ μ_{2n+1} →^{norm} μ μ μ_{2n+1}). Since 2n + 1

is odd, the norm is split and Shapiro lemma yields

H^{2}(R_{2}, _{𝔈}μ μ μ) = ker(H^{2}(R_{2}[\sqrt{t_{1}}],μ μ μ_{2n+1}) →^{Cores} H^{2}(R_{2},μ μ μ_{2n+1})).

This reads ker(ℤ/(2n + 1)ℤ →^{id} ℤ/(2n + 1)ℤ) = 0 by taking into account proposition 2.1 of [GP3]. The same calculation holds for E_{(0,1)} and E_{(1,1)} and we obtain the desired decomposition. In particular there are n + 1 inner forms and three outer forms. The outer forms are all quasiplit.

27Strictly speaking we are looking, here and in what follows, at the action of Out(G) on R_{2}– groups or cohomology of R_{2}–groups which are of multiplicative type. Since we have an equivalence of categories between R_{2} and F_{2} groups of multiplicative type ([GP3]. See Remark 6.8(d)) we can carry 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].

(2) We have μ μ μ = μ μ μ_{2n} = ker(μ μ μ_{2n}^{2} \underrightarrow{∏} μ μ μ_{2n}) and the action of ℤ/2ℤ switches the two factors. The coset ℤ/2ℤ \H^{2}(R,μ μ μ_{2n}) is as before {±1} \(ℤ/2nℤ). However, the computation of H^{2}(R_{2}, _{𝔈}μ μ μ) is different. The exact sequence

1 → _{𝔈}μ μ μ_{2n} → ∏_{R_{2}[\sqrt{t_{1}}]/R_{2}} μ μ μ_{2n} →^{norm} μ μ μ_{2n} → 1

gives rise to the long exact sequence of étale cohomology

⋯ → H^{1}(R_{2}[\sqrt{t_{1}}],μ μ μ_{2n}) →^{norm} H^{1}(R_{2},μ μ μ_{2n}) →^{δ} H^{2}(R_{2},_{𝔈}μ μ μ) → H^{1}(R_{2}[\sqrt{t_{1}}],μ μ μ_{2n}) →^{norm} H^{2}(R_{2},μ μ μ_{2n}).

The norm map appearing on the righthand side is the identity map id : ℤ/2nℤ → ℤ/2nℤ, so δ is onto. By the choices of coordinates \sqrt{t_{1}} and t_{2} on R_{2}[\sqrt{t_{1}}] and t_{1}, t_{2} on R_{2}, the beginning of the exact sequence decomposes as

ℤ/2nℤ ⊕ ℤ/2nℤ ⟶^{(id,×2)} ℤ/2nℤ ⊕ ℤ/2nℤ.

So H^{2}(R_{2},_{𝔈}μ μ μ) ≃ ℤ/2ℤ and the action of ℤ/2ℤ on H^{2}(R_{2},_{𝔈}μ μ μ) is therefore necessarily trivial. Thus 𝔈 leads to two distinct twisted forms _{𝔈}G^{±}. More precisely _{𝔈}G^{+} = 𝔈G (which is quasiplit), while _{𝔈}G^{−} is not quasiplit (since its “Brauer invariant” in H^{2}(R_{2},_{𝔈}μ μ μ) is not trivial). Similarly for 𝔈_{(0,1)} and 𝔈_{(1,1)}. This gives the desired decomposition. There are n + 1 inner forms and six outer forms (three of which are quasisplit).

(3) In this case μ μ μ = μ μ μ_{4}. The computation of the H^{2} 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 μ μ μ = μ μ μ_{2} × μ μ μ_{2} where ℤ/2ℤ switches the two summands. We have H^{2}(R_{2},μ μ μ) ≃ ℤ/2ℤ ⊕ ℤ/2ℤ where again ℤ/2ℤ acts by switching the two summands

Given that _{𝔈}μ μ μ = ∏_{R_{2}[\sqrt{t_{1}}]/R_{2}} μ μ μ_{2}, we have H^{2}(R_{2},_{𝔈}μ μ μ) ⟶^{∼} H^{2}(R_{2}[\sqrt{t_{1}}],μ μ μ_{2}) = ℤ/2ℤ.

Similarly for E_{(0,1)} and E_{(1,1)}, whence our decomposition. Again we have three inner forms 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 outer forms (all three of them quasisplit).

It remains to look at the case when G is of type D_{4}. The set H^{1}(R_{2}, S_{3}) classifies all degree 3 étale extensions S of R_{2}. Then S is a direct product of connected extensions. There are tree cases: S = R_{2} × R_{2} × R_{2} (the split case), S = S′ × R_{2} with S′/R_{2} of degree 2, and the connected case.

The case of S′ × R_{2} is already understood: They correspond to a 1-cocycle ϕ : π_{1}(R_{n}) → ℤ/2ℤ ⊂ S_{3}, where we view ℤ/2ℤ as a subgroup of S_{3} generated by a permutation. Note that up to conjugation by S_{3}, 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, 0). We shall denote them by 𝔈_{2}^{(i,j)} in the present situation to avoid confusion.

In the connected case there are four cubic extensions of R_{2}. They correspond to adjoining to R_{2} a cubic root in R_{2,∞} of t_{1}, t_{2}, t_{1}t_{2} and t_{1}^{2}t_{2} respectively. We will denote the corresponding four S_{3}–torsors by 𝔈_{3}^{(i,j)} with the obvious values for (i, j). The cubic case, which a priori appears as the most complicated, ends up being quite simple due to vanishing cohomological reasons, as we shall momentarily see.

According to Corollary 9.2 we have the decomposition

(9.1)  H_{loop}^{1}(R_{2}, Aut(G)) ≃ S_{3}\H^{2}(R_{2},μ μ μ)

⨆_{𝔈_{2}^{(i,j)}} (_{𝔈_{2}^{(i,j)}}S_{3})(R_{2}) \H^{2}(R_{2},_{𝔈_{2}^{(i,j)}} μ μ μ)

⨆_{𝔈_{3}^{(i,j)}} (_{𝔈_{3}^{(i,j)}}S_{3})(R_{2}) \H^{2}(R_{2},_{𝔈_{3}^{(i,j)}} μ μ μ).

The centre is μ μ μ = μ μ μ_{2} × μ μ μ_{2} = ker(μ μ μ_{2}^{3} \underrightarrow{∏} μ μ μ_{2}) and S_{3} acts by permutation on μ μ μ_{2}^{3}. Hence H^{2}(R_{2},μ μ μ) = ker(H^{2}(R_{2},μ μ μ_{2})^{3} → H^{2}(R_{2},μ μ μ_{2})) ⊂ H^{2}(R_{2},μ μ μ_{2})^{3} ≃ (ℤ/2ℤ)^{3}. There are two orbits for the action of S_{3} on H^{2}(R_{2},μ μ μ), namely (0, 0, 0) and (1, 1, 0).

For simplicity we will denote 𝔈_{2}^{(1,0)} by 𝔈_{2} and 𝔈_{3}^{(1,0)} by 𝔈_{3}. Since the group GL_{2}(ℤ) acts transitively on the set of quadratic and cubic extensions of R_{2} we may consider only the case of 𝔈_{2} := 𝔈_{2}^{1,0} [resp. 𝔈_{3} := 𝔈_{3}^{1,0}] for the purpose of determining the coset (_{𝔈_{2}^{(i,j)}}S_{3})(R_{2})\H^{2}(R_{2},_{𝔈_{2}^{(i,j)}}μ μ μ) [resp. (_{𝔈_{3}^{(i,j)}}S_{3})(R_{2})\H^{2}(R_{2},_{𝔈_{3}^{(i,j)}}μ μ μ)]. that all the twists of μ μ μ and S_{3} by quadratic or cubic torsors are of the form _{𝔈_{i}}μ μ μ and _{𝔈_{i}}S_{3} for i = 2 (resp. i = 3) in the quadratic (resp. cubic) case.

We have _{𝔈_{2}}μ μ μ = ker( ∏_{R_{2}[\sqrt{t_{1}}]/R_{2}} μ μ μ_{2} × μ μ μ_{2} →^{norm×id} μ μ μ_{2}), hence

H^{2}(R_{2},_{𝔈_{2}}μ μ μ) = ker(H^{2}(R_{2}[\sqrt{t_{1}}],μ μ μ_{2})⊕H^{2}(R_{2},μ μ μ_{2}) → H^{2}(R_{2},μ μ μ_{2})) ≅ H^{2}(R_{2}[\sqrt{t_{1}}],μ μ μ_{2}) = ℤ/2ℤ.

Since ℤ/2ℤ has trivial automorphism group, we get three copies of ℤ/2ℤ in the second summand of the decomposition (9.1).

In the cubic case we have _{𝔈_{3}}μ μ μ = ker( ∏_{R_{2}[ \sqrt[3]{t}_{1}]/R_{2}} μ μ μ_{2} →^{norm} μ μ μ_{2}). Since 2 is prime to 3,

the norm is split and

H^{2}(R_{2},_{𝔈_{3}}μ μ μ) = ker(H^{2}(R_{2}[\sqrt[3]{t}_{1}],μ μ μ_{2}) ⟶^{Cores} H^{2}(R_{2},μ μ μ_{2})) = ker(ℤ/2ℤ →^{id} ℤ/2ℤ) = 0.

Finally we observe, with the aid of Remark 6.8(d), that (_{𝔈_{2}}S_{3})(R_{2}) ≅ ℤ/2ℤ and (_{𝔈_{3}}S_{3})(R_{2}) ≅ ℤ/3ℤ.

Looking at (9.1) we obtain.

Corollary 9.6. For G of type D_{4} there are twelve loop R_{2}–forms, Two inner and ten 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 forms are “cubic” and are all quasiplit.

9.2.2 Applications to the classification of EALAs in nullity 2.

The Extended Affine Lie Algebras (EALAs), as their name implies, are a class of 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}. This is a Lie algebra over k (in general infinite dimensional) which satisfies the axioms of a Lie torus.^{28} 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 EALAs having a given Lie torus L as their centreless cores. To some extent this reduces many central 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}–Lie algebra structure to L. If L as an R_{n}–module is of finite type, then L is necessarily a multiloop algebra L(𝔤,σ σ σ) 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–loop algebras based on 𝔤 (as R_{n}–Lie algebras) are in bijective correspondence with the loop 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 one is interested in these Lie algebras as Lie algebras over k, and not R_{n}. In the present context the “centroid trick” (see [GP2, §4.1]) translates into the GL_{n}(ℤ) action on H^{1}(R_{n}, Aut(𝔤)) we have defined. This allows us to describe, in terms of orbits, all the isomorphism classes of R_{n}–multiloop algebras L(𝔤,σ σ σ) that become isomorphic when 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}^{1}(R_{2}, Aut(G)) ≃ H^{2}(R_{2},μ μ μ),

and this last H^{2} is either trivial or ℤ/2ℤ. In both cases the action of GL_{2}(ℤ) on H_{loop}^{1}(R_{2}, Aut(G)) is necessarily trivial. In particular.

28This terminology is due to Neher and Yoshii. It may seem strange to call a Lie algebra a Lie torus (since tori have already a meaning in Lie theory). The terminology was motivated by the concept of Jordan tori, which are a class of Jordan algebras.

Corollary 9.7. (1) If 𝔤 has type A_{1} B, C or E_{7}, there exists two isomorphism classes of 2–loop algebras based on 𝔤 denoted by 𝔤_{0} (the split case) and 𝔤_{1}.^{29}

(2) All 2–loop algebras based on 𝔤 of type E_{8}, F_{4} or G_{2} are trivial, i.e isomorphic as k–Lie algebras to 𝔤_{0} = 𝔤 ⊗_{k} R_{2}.

In the case Out(G) = ℤ/2ℤ, we have H^{1}(R_{2}, ℤ/2ℤ) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ and the action of GL_{2}(ℤ) on H^{1}(R_{2}, ℤ/2ℤ) is given by the linear action mod 2. Since SL_{2}(ℤ/2ℤ) = GL_{2}(ℤ/2ℤ) and SL_{2}(ℤ/2ℤ) is generated by elementary matrices, the reduction map GL_{2}(ℤ) → GL_{2}(ℤ/2ℤ) is onto. Hence there are two orbits for the action of GL_{2}(ℤ) on H^{1}(R_{2}, ℤ/2ℤ), namely the trivial one and H^{1}(R_{2}, ℤ/2ℤ) \ {0}. The last one is represented by the quadratic Galois extension R_{2}[\sqrt{t_{1}}]/R_{2}, denoted by 𝔈_{(1,0)} in the previous section, which we will again denote simply by 𝔈 in what follows. The action of GL_{2}(ℤ) we have just described shows that in all cases the outer forms, which came in three families (each with one or two classes) in the case of loop R_{2}–groups, collapse into a single family. This single family consists of either a 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/2ℤ the classification of isomorphism classes of 2– loop algebras based on 𝔤 is as follows:

(1) In type A_{2n} (n ≥ 1)

GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)) ≃ {±1} \(ℤ/(2n + 1)ℤ) ⨆ {_{𝔈}𝔤}.

There are n + 1 inner forms, denoted by 𝔤_{q} with 0 ≤ q ≤ n, and one outer form (which is quasisplit).

(2) In type A_{2n−1} (n ≥ 2)

GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)) ≃ {±1} \(ℤ/2nℤ) ⨆ {_{𝔈}𝔤^{+}} ⨆ {_{𝔈}𝔤^{−}}.

There are n + 1 inner forms, denoted by 𝔤_{q} with 0 ≤ q ≤ n, and two outer forms (one of them quasiplit, the other one not).

(3) In type D_{2n−1} (n ≥ 3)

GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)) ≃ {±1} \(ℤ/4ℤ) ⨆ {_{𝔈}𝔤^{+}} ⨆ {_{𝔈}𝔤^{−}}.

29As pointed out in [GP2], the case of E_{7} has an amusing story behind it. The existence of a k–Lie algebra L(𝔤, σ_{1}, σ_{2}) which is not isomorphic to 𝔤 ⊗_{k} R_{2} was first established by van de Leur with 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.

There are 3 inner forms, denoted by 𝔤_{0,1,2}, and two outer forms (one of them quasiplit, the other one not).

(4) In type D_{2n} (n ≥ 3), we have

GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)) ≃ switch \(ℤ/2ℤ ⊕ ℤ/2ℤ) ⨆ {_{𝔈}𝔤^{+}} ⨆ {_{𝔈}𝔤^{−}}.

There are 3 inner forms, denoted by 𝔤_{0,1,2}, and two outer forms (one of them quasiplit, the other one not).

(5) In type E_{6}

GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)) ≃ {±1} \(ℤ/3ℤ) ⨆ {_{𝔈}𝔤}.

There are 2 inner forms, 𝔤_{0} and 𝔤_{1}, and one outer form (which is quasisplit).

Proof. The nature of the collapse of outer forms when passing from R_{2} to k was explained before the statement of the Corollary. It remains to understand the inner cases. According to Corollary 9.2 and (8.9), we need to trace the action of GL_{2}(ℤ) on ℤ/2ℤ\H^{2}(R_{2},μ μ μ) and use the fact that this action lifts to an action of GL_{2}(ℤ) on H^{2}(R_{2},μ μ μ) which commutes with that of ℤ/2ℤ.

(1) We have μ μ μ = μ μ μ_{2n+1} = ker(μ μ μ_{2n+1}^{2} \underrightarrow{∏} μ μ μ_{2n+1}) and the action of ℤ/2ℤ switches the two factors. We have H^{2}(R_{2},μ μ μ) ≅ ℤ/(2n + 1)ℤ and the action of GL_{2}(ℤ) on H^{2}(R_{2},μ μ μ) is given by the determinant (Lemma 8.14.6), that of ℤ/2ℤ is given by signs. Thus GL_{2}(ℤ) acts trivially on ℤ/2ℤ\H^{2}(R_{2},μ μ μ) and the result follows.

(2) We have μ μ μ = μ μ μ_{2n} = ker(μ μ μ_{2n}^{2} \underrightarrow{∏} μ μ μ_{2n}) and the action of ℤ/2ℤ switches the two factors. The action of GL_{2}(ℤ) on H^{2}(R,μ μ μ) is given by the determinant, hence the set of cosets (GL_{2}(ℤ) × ℤ/2ℤ)) \H^{2}(R,μ μ μ_{2n}) can still be identified with {±1} \(ℤ/2nℤ).

(3) In this case μ μ μ = μ μ μ_{4}. The computation of H^{2} and reasoning are exactly as in (2) above for n = 2.

(4) This case is rather different since μ μ μ = μ μ μ_{2} × μ μ μ_{2} where ℤ/2ℤ switches the two summands. We have H^{2}(R_{2},μ μ μ) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ with respect to the switch action. Again GL_{2}(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_{4}.

Lemma 9.9. There are three orbits for the action of GL_{2}(ℤ) on H^{1}(R_{2}, S_{3}) : - the trivial class;

- {𝔈_{2}^{0,1}, 𝔈_{2}^{1,0}, 𝔈_{2}^{1,1}};

- {𝔈_{3}^{1,0}, 𝔈_{3}^{0,1}, 𝔈_{3}^{1,1}, 𝔈_{3}^{2,1}}

where the notations is as in Corollary 9.6 supra.

Proof. The three classes above correspond to case of the split étale cubic R_{2}-algebra, the case S′ × R_{2} where S′/R_{2} is quadratic and the cubic case. Obviously each of the above sets is GL_{2}(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_{3}^{1,0} = R_{2}[\sqrt[3]{t}_{1}]. By applying the base change corresponding to (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}), (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}),

(\begin{matrix}2 & 5 \\ 1 & 3 \end{matrix}) we obtain 𝔈_{3}^{0,1}, 𝔈_{3}^{1,1} and 𝔈_{3}^{2,1} respectively.

Corollary 9.10. Up to k-isomorphism there are five 2-loop algebras based on 𝔤 of type D_{4}: Two inner forms, denoted by 𝔤_{0} and 𝔤_{1}; two “quadratic” algebras, _{𝔈_{2}^{1,0}}𝔤^{+} (which is quasisplit) and _{𝔈_{2}^{1,0}}𝔤^{−} (which is not quasiplit); and one “cubic” algebra _{𝔈_{3}^{1,0}}𝔤 ( which is quasisplit).

Proof. By Lemma 9.9, the quadratic (resp. cubic) classes of Corollary 9.6 are in the GL_{2}(ℤ)–orbit of those having Dynkin-Tits invariant 𝔈_{2}^{1,0} (resp. 𝔈_{3}^{1,0}). So the cubic case is done. In the quadratic case, there are then one or two non-isomorphic “quadratic” 2-loop algebras. Since one of these R_{2}–algebras is quasisplit and the other 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_{3} on H^{2}(R_{2},μ μ μ), namely (0, 0, 0) and (1, 1, 0), and these correspond to the two inner R_{2}-forms. The action of GL_{2}(ℤ) is trivial in this set, so the algebras remain non-isomorphic over k.

9.2.3 Rigidity in nullity 2 outside type 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 establishes Conjecture 6.4 of [GP2].^{30}

Theorem 9.11. Let 𝔤 be a finite dimensional simple Lie algebra over k which is not of type A. Let L and L′ be two 2-loop algebras based on 𝔤. The following are equivalent:

(1) L and L′ are isomorphic (as Lie algebras over k);

(2) L and L′ have the same Witt-Tits index.

30An even stronger version of this Conjecture will be established in the next section.

Proof. Of course, we use that L (resp. L′) arise as the Lie algebra of R_{2}–loop adjoint groups H (resp. H′) which are forms of G = Aut(𝔤)^{0}.^{31} Then condition (1) reads that [H] = [H′] in GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)) and condition (2) reads that H×_{R_{2}} K_{2} and H′ ×_{R_{2}} K_{2} have the same Witt-Tits index.

(1) ⟹ (2) : This is is the simple part of the the equivalence (and it is not necessary to exclude type A). Let G be the corresponding adjoint group. If [H] and [H′] are equal in GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(G)), it is obvious that their Dynkin-Tits invariant coincide in GL_{2}(ℤ)\H_{loop}^{1}(R_{2}, Out(G)), and also that their Tits index over K_{2} coincide by Corollary 8.12.

(2) ⟹ (1) : Without loss of generality we can assume that H and H′ have same Dynkin-Tits invariant in H^{1}(R_{2}, Out(𝔤)). The proof is given by a case-by-case discussion. The cases of type E_{8}, F_{4} and G_{2} follow directly from Corollary 9.7.2. Types B, C and E_{7} are also straightforward since (over R_{2}) there is only one class of non-split 2–loop algebras. For obvious reasons, this non-split Lie algebras necessarily remain non-isomorphic to the split Lie algebra 𝔤 ⊗_{k} R_{2} when viewed as Lie algebras over k. Type D_{2n}, n ≥ 3 : If the Dynkin-Tits invariant is non-trivial, then the summand of GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)) corresponding to [𝔈_{1}] has only one non quasi-split class, so [H] and [H′] are equal in GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)). Corollary 9.8 states that the inner part of GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)) is ℤ/2ℤ ⊕ ℤ/2ℤ modulo the switch action, so is represented by (0, 0), (1, 0) and (1, 1). It has then three elements, the split one and two others. It is then enough to explicitly describe these two other elements and distinguish them by their Witt-Tits index by means of Tits tables [T1]. The first one is the R_{2}-loop group PSO(q) with q = ⟨1, t_{1}, t_{2}, t_{1}t_{2}⟩ ⊥ (2n − 1)⟨1, −1⟩. Its Witt-Tits K_{2}-index is

(9.2)  [figure: …; …; α_{1}; α_{2}; α_{2n−1}; α_{2n}]

The other one is PSU(A, h) where A = A(2, 1) is the R_{2}-quaternion algebra T_{1}^{2} = t_{1}, T_{2}^{2} = t_{2}, T_{1}T_{2} + T_{2}T_{1} = 0 and h is the hyperbolic hermitian form over A^{2n} with respect to the quaternionic involution q ↦ \overline{q}. Indeed PSU(A, h) is an adjoint inner loop R_{2}-group of type D_{2n} and its Witt-Tits K_{2}-index is

(9.3)  [figure: …; α_{1}; α_{2}; α_{2n−1}; α_{2n}]

So [H] and [H′] are equal in GL_{2}(ℤ) \H_{loop}^{1}(R_{2}, Aut(𝔤)) to the split form or one of these two forms.

31Strictly speaking “...the Lie algebra...” is an R_{2}-Lie algebra, but we view this in a natural away as a k-Lie algebra.

Type D_{2n−1}, n ≥ 3 : As in the preceding case, we need to discuss only the inner case and it is enough to provide two non-split R_{2}-loop groups with distinct K_{2}-Witt-Tits index. The first one is the R_{2}-loop group PSO(q) with q = ⟨1, t_{1}, t_{2}, t_{1}t_{2}⟩ ⊥ (2n − 2)⟨1, −1⟩. Its Witt-Tits K_{2}-index is

(9.4)  [figure: …; …; α_{1}; α_{2}; α_{2n−2}; α_{2n−1}]

The other one is PSU(A, h) where h is the hyperbolic hermitian form over A^{2n−1} which is the orthogonal sum of ⟨1⟩ and the hyperbolic form over A^{2n−2}. Its Witt-Tits K_{2}-index is

(9.5)  [figure: …; α_{1}; α_{2}; α_{2n−2}; α_{2n−1}]

Type D_{4}: Follows from Corollary 9.10. Type E_{6}: This case is straightforward because there is only one class of 2–loop algebras which is not quasi-split.

Remark 9.12. There is some redundancy in the statement of the Theorem. It is well known, by descent considerations, that if L and L′ are isomorphic as Lie algebras over k, then their absolute type coincide, i.e. they are both 2–loop algebras based on the same 𝔤 (see [ABP2.5] for further details). It will thus suffice to assume in the Theorem that neither L nor L′ are of absolute type A.

9.2.4 Tables

The following table summarizes the classification on 2-loop algebras that are not of absolute type A. The table includes the Cartan-Killing (absolute) type 𝔤, its name, the Witt-Tits index (with Tits’ notations) of an R_{n}–representative of the k–Lie algebra in question, and the type of the relative root system. For example, van de Leur’s algebra has absolute type E_{7}, Tits index E_{7,4}^{9} and relative index F_{4}. The way in which the Witt-Tits index are determined was illustrated in the previous section. The procedure of how to obtain the relative type from the index is described by Tits.

In all cases the “trivial” loop algebra 𝔤 ⊗ k[t_{1}^{±}, t_{2}^{±}] is denoted by 𝔤_{0}. When the relative type is A_{0}, the loop algebra in question is anisotropic. For example, in absolute type A_{1} the Lie algebra 𝔤_{0} is 𝔰𝔩_{2}([t_{1}^{±}, t_{2}^{±}]). The Lie algebra 𝔤_{1} is the derived algebra of the Lie algebra that corresponds to the quaternion algebra over k[t_{1}^{±}, t_{2}^{±}] with relations T_{1}T_{2} = −T_{2}T_{1} and T_{i}^{2} = t_{i}. This rank 3 free Lie algebra over k[t_{1}^{±}, t_{2}^{±}] is anisotropic, and is a twisted form of 𝔰𝔩_{2} ⊗ k[t_{1}^{±}, t_{2}^{±}] split by the quadratic extension k[t_{1}^{±}, t_{2}^{±}](T_{1}).

Cartan-Killing type 𝔤 Name  Tits index  Relative root system
A_{1}  𝔤_{0}  ^{1}A_{1,1}^{(1)}  A_{1}
A_{1}  𝔤_{1}  ^{1}A_{1,0}^{(2)}  A_{0}
A_{2n} ( n ≥ 1)  𝔤_{q}  ^{1}A_{2n,r−1}^{( \frac{2n+1}{r})} r = gcd(q, 2n + 1)  A_{r−1}
A_{2n} ( n ≥ 1)  _{𝔈}𝔤  ^{2}A_{2n,n}^{(1)}  BC_{n}
A_{2n−1} ( n ≥ 2)  𝔤_{q}  ^{1}A_{2n−1,r−1}^{( \frac{2n}{r})} r = gcd(q, 2n)  A_{r−1}
A_{2n−1} ( n ≥ 2)  _{𝔈}𝔤^{+}  ^{2}A_{2n−1,n}^{(1)}  C_{n}
A_{2n−1} ( n ≥ 2)  _{𝔈}𝔤^{−}  ^{2}A_{2n−1,n−1}^{(1)}  BC_{n−1}
B_{n} (n ≥ 2)  𝔤_{0}  B_{n,n}  B_{n}
B_{n} (n ≥ 2)  𝔤_{1}  B_{n,n−1}  B_{n−1}
C_{n} (n ≥ 3)  𝔤_{0}  C_{n,n}^{(1)}  C_{n}
C_{2n+1} (n ≥ 1)  𝔤_{1}  C_{2n+1,n}^{(2)}  BC_{n}
C_{2n} (n ≥ 2)  𝔤_{1}  C_{2n,n}^{(2)}  C_{n}
D_{4}  𝔤_{0}  ^{1}D_{4,4}^{(1)}  D_{4}
D_{4}  𝔤_{1}  ^{1}D_{4,2}^{(1)}  B_{2}
D_{4}  _{𝔈_{2}}𝔤^{+}  ^{2}D_{4,3}^{(1)}  B_{3}
D_{4}  _{𝔈_{2}}𝔤^{−}  ^{2}D_{4,1}^{(2)}  BC_{1}
D_{4}  _{𝔈_{3}}𝔤  ^{3}D_{4,2}^{2}  G_{2}
D_{2n−1} (n ≥ 3)  𝔤_{0}  ^{1}D_{2n−1,2n−1}^{(1)}  D_{2n−1}
D_{2n−1} (n ≥ 3)  𝔤_{1}  ^{1}D_{2n−1,2n−3}^{(1)}  B_{2n−3}
D_{2n−1} (n ≥ 3)  𝔤_{2}  ^{1}D_{2n−1,n−2}^{(2)}  BC_{n−2}
D_{2n−1} (n ≥ 3)  _{𝔈}𝔤^{+}  ^{2}D_{2n−1,2n−2}^{(1)}  B_{2n−2}
D_{2n−1} (n ≥ 3)  _{𝔈}𝔤^{−}  ^{2}D_{2n−1,n−2}^{(2)}  BC_{n−2}
D_{2n} (n ≥ 3)  𝔤_{0}  ^{1}D_{2n,2n}^{(1)}  D_{2n}
D_{2n} (n ≥ 3)  𝔤_{1}  ^{1}D_{2n,2n−2}^{(1)}  B_{2n−2}
D_{2n} (n ≥ 3)  𝔤_{2}  ^{1}D_{2n,n}^{(2)}  C_{n}
D_{2n} (n ≥ 3)  _{𝔈}𝔤^{+}  ^{2}D_{2n,2n−1}^{(1)}  B_{2n−1}
D_{2n} (n ≥ 3)  _{𝔈}𝔤^{−}  ^{2}D_{2n,n−1}^{(2)}  BC_{n−1}
E_{6}  𝔤_{0}  ^{1}E_{6,6}^{0}  E_{6}
E_{6}  𝔤_{1}  ^{1}E_{6,2}^{16}  G_{2}
E_{6}  _{𝔈}𝔤  ^{2}E_{6,4}^{2}  F_{4}
E_{7}  𝔤_{0}  E_{7,7}^{0}  E_{7}
E_{7}  𝔤_{1}  E_{7,4}^{9}  F_{4}
E_{8}  𝔤_{0}  E_{8,8}^{0}  E_{8}
F_{4}  𝔤_{0}  F_{4,4}^{0}  F_{4}
G_{2}  𝔤_{0}  G_{2,2}^{0}  G_{2}

By taking Remark 9.12 into consideration, an inspection of the Table shows that a stronger version of Theorem 9.11 holds.

Theorem 9.13. Let L and L′ be two 2-loop algebras neither of which is of absolute type 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 is achieved by considering loop algebras of the affine algebras. More precisely, it is shown 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_{1}^{±1}], π) where π is a diagram automorphism of the untwisted affine Lie algebra 𝔤⊗k[t_{1}^{±1}]. For example, van de Leur’s algebra appears by taking 𝔤 of type E_{7} and considering the diagram automorphism of order two of the corresponding extended Coxeter-Dynkin.

Note that in the present work we have outline a general procedure to classify loop adjoint groups and algebras over R_{n}, and that the classification of multiloop algebras over k follows by GL_{n}-considerations from that over R_{n}. This is not the case in [ABP3]. The nullity 2 classification relies on the structure of the affine algebras and only yields results over k.

10 The case of orthogonal groups

These groups are related to quadratic forms, which allows for a very precise understanding of their nature based on our results.

We consider the example of the split orthogonal group O(d) for d ≥ 1. If d = 2m (resp. d = 2m + 1), this is the orthogonal group corresponding to the quadratic

form ∑_{i=1}^{m} X_{i}X_{2m+1−i} (resp. ∑_{i=1}^{m} X_{i}X_{2m+1−i} + X_{2m+1}^{2}). Since R_{n}-projective modules of

finite type are free, we know that H^{1}(R_{n}, O(d)) classifies regular quadratic forms over R_{n}^{d} [K2, §4.6]. We have H_{loop}^{1}(R_{n}, O(d)) ⟶^{∼} H^{1}(F_{n}, O(d)). By iterating Springer’s theorem for quadratic forms over k((t))) [Sc, §6.2], the classification of F_{n}-quadratic forms reads as follows: For each subset I ⊂ {1, ..., n}, we put t_{I} = ∏_{i∈I} t_{i} with the convention 1 = t_{∅}; we denote by H the hyperbolic plane, that is the rank two split form. The isometry classes of d-dimensional R_{n}–forms are then of the form

⊥_{I⊂{1,...,n}} t_{I} q_{I} ⊥ H^{v}

where the q_{I}’s are anisotropic quadratic k-forms and v a non negative integer such

that ∑_{I⊂{1,...,n}} dim_{k}(q_{I}) + 2v = d.

Corollary 10.1. The set H_{loop}^{1}(R_{n}, O(d)) is parametrized by the quadratic forms

⊥_{I⊂{1,...,n}} t_{I} q_{I} ⊥ H^{v}

where the q_{I} are anisotropic quadratic k-forms and v a non-negative integer such that

∑_{I⊂{1,...,n}} dim_{k}(q_{I}) + 2v = 2d.

We denote by P(n) the set of subsets of {1, ..., n} and by P_{≤d}^{even}(n) ⊂ P(n) the set of subsets of {1, ..., n} of even cardinal ≤ d. In a similar fashion we define P_{≤d}^{odd}(n).

Corollary 10.2. Assume that k is quadratically closed.

(1) If d = 2m, then the map

\begin{matrix}P_{≤d}^{even}(n) & ⟶ & H_{loop}^{1}(R_{n}, O(d)) \\ S & ↦ & ⊥_{I⊂S} ⟨t_{I}⟩ ⊥ H^{m− \frac{|S|}{2}} \end{matrix}

is a bijection.

(2) If d = 2m + 1, then the map

\begin{matrix}P_{≤d}^{odd}(n) & ⟶ & H_{loop}^{1}(R_{n}, O(d)) \\ S & ↦ & ⊥_{I⊂S} ⟨t_{I}⟩ ⊥ H^{m+ \frac{1− |S|}{2}} \end{matrix}

is a bijection.

Corollary 10.3. Assume that k is quadratically closed. Inside O′_{d} = O(⟨1, …, 1⟩) ≃ O_{d}. there is a single O′_{d}(k)-conjugacy class of maximal anisotropic abelian constant subgroup of O′(d), that of the diagonal subgroup μ μ μ_{2}^{d}. In particular anisotropic abelian subgroups of O′(d) are 2-elementary.

Proof. Let A be a finite abelian constant group of O′_{d}. There exist an even integer m ≥ 1 and a surjective homomorphism ϕ : (ℤ/mℤ)^{n} → A(k). Then the corresponding loop torsor [ϕ] ∈ H^{1}(R_{n}, O′_{d}) is anisotropic. Indeed the map H^{1}(R_{n},μ μ μ_{2}^{d}) → H^{1}(R_{n}, O′_{d}) is surjective. Hence there exists ψ : (ℤ/mℤ)^{n} → μ μ μ_{2}^{d} such that [ϕ] = [ψ] ∈ H^{1}(R_{n}, O′_{d}). Theorem 7.9 shows that ϕ and ψ are O′_{d}(k)-conjugate. By considering their images, we conclude that A(k) is O′_{d}(k)-conjugate to a subgroup of μ μ μ_{2}^{d}(k).

Remark 10.4. (1) All anisotropic abelian constant subgroups of O′_{d} are related to codes, 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 universal covering 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 abelian subgroup of Spin′_{d} is of rank ≤ d and has 4-torsion.

11 Groups of type G_{2}

We denote by G_{2} the split Chevalley group of type G_{2} over k. If F is a field of characteristic zero containing k, we know that H^{1}(F_{n}, G_{2}) classifies octonion F - algebras or alternatively 3-Pfister forms [Se2, §8.1]. This follows from the fact that the Rost invariant [GMS]

r_{F} : H^{1}(F, G_{2}) → H^{3}(F, ℤ/2ℤ)

is injective and sends the class of an octonion algebra to the Arason invariant of its norm form.

Consider the standard non-toral constant abelian subgroup f : (ℤ/2ℤ)^{3} ⊂ G_{2}. Then the composite map

(F^{×}/F^{×}^{3}) ≅ H^{1}(F, (ℤ/2ℤ)^{3}) ⟶^{f_{∗}} H^{1}(F, G_{2}) ⟶^{r_{F}} H^{3}(F, ℤ/2ℤ).

sends an element ((a), (b), (c)) to the cup product (a).(b).(c) ∈ H^{3}(F, ℤ/2ℤ) [GiQ, §6]. For n ≥ 0, we consider the mapping

(R_{n}^{×}/R_{n}^{×}^{2})^{3} ≃ H^{1}(R_{n}, (ℤ/2ℤ)^{3}) ⟶^{f_{∗}} H^{1}(R_{n}, G_{2}).

For a class ((x), (y), (z)) ∈ (R_{3}^{×}/(R_{3}^{×})^{2})^{3} we write only (x, y, z).

Corollary 11.1. The map above surjects onto H_{loop}^{1}(R_{n}, G_{2}).

Proof. By the Acyclicity Theorem, it suffices to observe that the analogous statement holds for H_{loop}^{1}(F_{n}, G_{2}).

By using the Rost invariant, we get a full classification of the multiloop algebras based on on the split Lie algebra of type G_{2}.

Corollary 11.2. Assume that k is quadratically closed. Assume that n ≥ 3.

1) H_{loop}^{1}(R_{n}, G_{2}) \ {1} consists in the images by f_{∗} of the (t_{I_{1}}, t_{I_{2}}, t_{I_{3}}) where I_{1}, I_{2}, I_{3} are non-empty subsets of {1, .., n} such that i_{1} < i_{2} < i_{3} for all (i_{1}, i_{2}, i_{3}) ∈ I_{1} × I_{2} × I_{3}.

2) GL_{n}(ℤ)\(H_{loop}^{1}(R_{n}, G_{2}) \ {1}) consists of the image by f_{∗} of (t_{1}, t_{2}, t_{3}).

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}^{×})^{×2} ≅ F_{n}^{×}/(F_{n}^{×})^{×2} ≅ (ℤ/2ℤ)^{n}. Hence H^{1}(F_{n}, G_{2}) consists of the image of f_{∗}(t_{I_{1}}, t_{I_{2}}, t_{I_{3}}) for I_{1}, I_{2}, I_{3} running over the subsets of {1, .., n}. The Rost invariant of such a class is (t_{I_{1}}).(t_{I_{2}}).(t_{I_{3}}) ∈ H^{3}(F_{n}, ℤ/2ℤ). Since (t_{i}).(t_{i}) = 0 and (t_{i}).(t_{j}) = (t_{j})(t_{i}) ∈ H^{3}(F_{n}, ℤ/2ℤ), it follows that H^{1}(F_{n}, G_{2}) consists of the trivial class and the images by f_{∗} of the (t_{I_{1}}, t_{I_{2}}, t_{I_{3}}) where I_{1}, I_{2}, I_{3} are non-empty subsets of {1, .., n} such that i_{1} < i_{2} < i_{3} for each (i_{1}, i_{2}, i_{3}) ∈ I_{1} × I_{2} × I_{3}. The last classes are non-trivial pairwise distinct elements since the (t_{I_{1}}).(t_{I_{2}}).(t_{I_{3}}) ∈ H^{3}(F_{n}, ℤ/2ℤ) are distinct pairwise elements by residue considerations (see for example prop. 3.1.1 of [GP3]).

(2) Follows easily from (1).

The following corollary refines Griess’ classification in the G_{2}-case [Gs].

Corollary 11.3. Assume that k is algebraically closed. Let A be an anisotropic constant abelian subgroup of G_{2}. Then A is G_{2}(k)-conjugate to the standard nontoral subgroup (ℤ/2ℤ)^{3}.

Proof. Let A be a finite abelian constant anisotropic subgroup of G_{2}. We reason as before. There exist an even integer m ≥ 1 and a surjective homomorphism ϕ : (ℤ/mℤ)^{n} → A(k) so that the corresponding loop torsor [ϕ] ∈ H^{1}(R_{n}, G_{2}) is anisotropic. By part (1) of Corollary 11.1 there exists ψ : (ℤ/mℤ)^{n} → (ℤ/2ℤ)^{3} such that [ϕ] = [ψ] ∈ H^{1}(R_{n}, G_{2}). Theorem 7.9 shows that ϕ and ψ are G_{2}(k)conjugate. By taking the images, we conclude that A(k) is G_{2}(k)–conjugate to the standard (ℤ/2ℤ)^{3}.

12 Case of groups of type F_{4}, E_{8} and simply connected E_{7} in nullity 3

In this section, we assume that k is algebraically closed. We denote by F_{4}, and E_{8} the split algebraic k–group of type F_{4} and E_{8} respectively, and by E_{7} the split simply connected k–group of type E_{7} For either of these three groups we know that G = Aut(G) and that H^{1}(R_{2}, G) = 1 [GP2, th. 2.7]. The goal is then to compute H_{loop}^{1}(R_{3}, 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 compact Lie group [BFM, §5.2], we will assume that k = ℂ. Note that there is no loss of generality in doing this as explained in Remark 8.8.^{32}

Denote by G_{0} the anisotropic real form of G (viewed as algebraic group over ℝ) and let K = G_{0}(ℝ). This is a compact Lie group.

In the F_{4} and E_{7} case K has a single conjugacy class of rank zero abelian subgroup of rank 3. In the E_{8} case, K has two conjugacy classes of rank zero abelian subgroup of rank 3, (ℤ/5ℤ)^{3} and (ℤ/6ℤ)^{3}. To translate this to the complex case we establish the following fact.

32All the results that we need about rank zero abelian groups and triples can also be found in [KS].

Lemma 12.1. Let H be a complex affine algebraic group whose connected component of the identity is reductive. Denote by H_{0} its anisotropic real form, viewed as algebraic group over ℝ (see [OV, §5.2, th. 12]). Set K_{H} = H_{0}(ℝ).

(1) Let A is a finite abelian subgroup of K_{H} and denote by A the underlying constant subgroup of the algebraic ℝ-group H_{0}. 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), both groups 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 subgroups are conjugate under H^{0}(ℂ).

Proof. (1) Let C denote the connected component of the identity of the centralizer Z_{H_{0}}(A). It is a real reductive group [BMR, 10.1.5]. If A is an anisotropic subgroup of 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 zero subgroup of K_{H} then C(ℝ) is finite. Since C(ℝ) is Zariski dense in the connected group C, we see that C = 1, and A ×_{ℝ} ℂ is an anisotropic constant abelian subgroup of H.

(2) We are given a finite anisotropic constant subgroup A of H. Since 1 = Z_{H}(A)^{0} = N_{H}(A)^{0}, 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^{−1}}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_{4} has a single conjugacy class of anisotropic finite abelian (constant) subgroups of rank 3, denoted by f_{3} : (ℤ/3ℤ)^{3} ⊂ F_{4}. Furthermore

N_{F_{4}}((ℤ/3ℤ)^{3})/Z_{F_{4}}((ℤ/3ℤ)^{3}) ≃ SL_{3}(ℤ/3ℤ).

(2) The group E_{7} has a single conjugacy class of anisotropic finite abelian (constant) subgroups of rank 3, denoted by f_{4} : (ℤ/4ℤ)^{3} ⊂ E_{7}. The finite group f_{4} is a subgroup the maximal subgroup SL_{8} /μ μ μ_{2}. Furthermore N_{E_{7}}((ℤ/4ℤ)^{3})/Z_{E_{7}}((ℤ/4ℤ)^{3}) ≃ SL_{3}(ℤ/4ℤ).

(3) The group E_{8} has two conjugacy classes of anisotropic finite abelian (constant) subgroups of rank 3, denoted by f_{5} and f_{6}. We have.

(a) f_{5} : (ℤ/5ℤ)^{3} ⊂ E_{8} and N_{E_{8}}((ℤ/5ℤ)^{3})/Z_{E_{8}}((ℤ/5ℤ)^{3}) ≃ SL_{3}(ℤ/5ℤ).

(b) f_{6} : (ℤ/6ℤ)^{3} ⊂ E_{8} is a subgroup of the subgroup (SL_{2} × SL_{3} × SL_{6})/μ μ μ_{6}. Furthermore N_{E_{8}}(ℤ/6ℤ)^{3})/Z_{E_{8}}((ℤ/6ℤ)^{3}) ≃ SL_{3}(ℤ/6ℤ).

Remark 12.3. The finite subgroups 1) and 3 (a) are described precisely in [GiQ, §6]. That the third one f_{6} : (ℤ/6ℤ)^{3} ⊂ E_{8} sits inside the subgroup (SL_{2} × SL_{3} × SL_{6})/μ μ μ_{6} is well-known from work of Borel and de Siebenthal.

Proof. As explained above we may assume that k = ℂ. The previous Lemma 12.1 shows that any rank 0 finite abelian constant subgroup A of G arises from rank 0 abelian subgroup A of K, so the list of Borel-Friedman-Morgan [BFM, §5.2] provides all relevant conjugacy classes, and this yields the inclusions f_{3}, f_{4}, f_{5} and f_{6} described above. Given a two rank 0 finite abelian constant subgroups A and A′ of G arising respectively 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 is obvious since the subgroups from the list are distinct as groups. We investigate now the 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 gA ⊂ 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 same fact holds for g = 1.

It is then enough to know the quotient “normalizer/centralizer” in the compact group case. For each relevant d, we have an exact sequence of groups

1 → Z_{K}((ℤ/dℤ)^{3}) → N_{K}(ℤ/dℤ)^{3}) →^{θ} GL_{3}(ℤ/dℤ)

and we want to determine the image of θ. Denote by S_{d} the set of K-conjugacy classes of rank zero triples of K of order d. Since such a triple generates a rank zero abelian subgroup of order d^{3} of K, the set S_{d} is covered by rank zero triples inside (ℤ/dℤ)^{3}, namely GL_{3}(ℤ/dℤ)-conjugates of the standard triple (1, 1, 1). So we have GL_{3}(ℤ/dℤ)/Im(θ) ≅ S_{d}. Proposition 5.1.5 of [BFM] states that the Kconjugacy classes of rank zero triples of K of order d consists of the classes f_{d}(1, 1, i) for i = 1, .., d − 1 with i prime to d. Hence the image of θ in GL_{3}(ℤ/dℤ) is exactly SL_{3}(ℤ/dℤ) as desired.

Given f_{d} : (ℤ/dℤ)^{3} → G as above consider the map

f_{d,∗} : (R_{3}^{×}/(R_{3}^{×})^{d})^{3} ≃ H^{1}(R_{3}, (ℤ/dℤ)^{3}) → H^{1}(R_{3}, G).

A class ((x), (y), (z)) ∈ (R_{3}^{×}/(R_{3}^{×})^{d})^{3} will for convenience simply be written as (x, y, z).

Corollary 12.5. (1) The set H_{loop}^{1}(R_{3}, F_{4})_{an} consists of the classes of f_{3,∗}(t_{1}, t_{2}, t_{3}) and f_{3,∗}(t_{1}, t_{2}, t_{3}^{2}).

(2) The set H_{loop}^{1}(R_{3}, E_{7})_{an} consists of the classes of f_{4,∗}(t_{1}, t_{2}, t_{3}), f_{4,∗}(t_{1}, t_{2}, t_{3}^{3}).

(3) The set H_{loop}^{1}(R_{3}, E_{8})_{an} consists in the classes of f_{5,∗}(t_{1}, t_{2}, t_{3}^{i}) for i = 1, 2, 3, 4, f_{6,∗}(t_{1}, t_{2}, t_{3}) and f_{6,∗}(t_{1}, t_{2}, t_{3}^{5}).

Proof. We do in detail the case of F_{4}, the other cases being similar. The set H_{loop}^{1}(R_{3}, F_{4})_{an} is covered by the image of the anisotropic loop cocycles ϕ : π_{1}(R_{3}) → F_{4}(ℂ). The image of such a ϕ is an anisotropic finite abelian subgroup of F_{4}, so Lemma 12.2.1 allows us to assume that its image is the subgroup (ℤ/3ℤ)^{3}. Furthermore, we know that two such homomorphisms ϕ and ϕ′ have the same image in H_{loop}^{1}(R_{3}, F_{4})_{an} if and only if there exists g ∈ F_{4}(ℂ) such that gϕg^{−1} = ϕ′, or equivalently if there exists g ∈ N_{F_{4}}((ℤ/3ℤ)^{3})(ℂ) such that gϕg^{−1} = ϕ′. Note the importance of the isomorphism N_{F_{4}}((ℤ/3ℤ)^{3})/Z_{F_{4}}((ℤ/3ℤ)^{3}) ≃ SL_{3}(ℤ/3ℤ).

Rephrasing what has been said in terms of the mapping f_{3,∗}, we see that H_{loop}^{1}(R_{3}, F_{4})_{an} is the image under f_{3,∗} of the classes (x, y, z) where x, y, z ∈ R_{3}^{×} are such that (x, y, z) generates R_{3}^{×}/(R_{3}^{×})^{3}; furthermore, two such classes (x, y, z) and (x′, y′, z′) have the same image in H_{loop}^{1}(R_{3}, F_{4})_{an} if and only if there exists τ ∈ SL_{3}(ℤ/3ℤ) such that (x′, y′, z′) = τ_{∗}((x, y, z)). We conclude that H_{loop}^{1}(R_{3}, F_{4})_{an} consists of the classes of f_{3,∗}(t_{1}, t_{2}, t_{3}) and f_{3,∗}(t_{1}, t_{2}, t_{3}^{2}).

Corollary 12.6. (1) The set GL_{3}(ℤ)\H_{loop}^{1}(R_{3}, F_{4})_{an} consists of the class of f_{3,∗}(t_{1}, t_{2}, t_{3}). (2) The set GL_{3}(ℤ)\H_{loop}^{1}(R_{3}, E_{7})_{an} consists of the class f_{4,∗}(t_{1}, t_{2}, t_{3}).

(3) The set GL_{3}(ℤ)\H_{loop}^{1}(R_{3}, E_{8})_{an} consists of the classes of f_{5,∗}(t_{1}, t_{2}, t_{3}) and f_{6,∗}(t_{1}, t_{2}, t_{3}).

Remark 12.7. The above Corollary gives the full classification of nullity 3 anisotropic multiloop algebras of absolute type F_{4} or E_{8}.

13 The case of PGL_{d}

13.1 Loop Azumaya algebras

For any base scheme 𝔛, the set H^{1}(𝔛, PGL_{d}) classifies the isomorphism classes of Azumaya O_{𝔛}-algebras A of degree d, i.e. O_{𝔛}-algebras which are locally isomorphic for the étale topology to the matrix algebra M_{d}(O_{𝔛}) [Gr2] and [K2, §III].

The exact sequence 1 → G_{m} → GL_{d} ⟶^{p} PGL_{d} → 1 induces the sequence of pointed sets

Pic(𝔛) → H^{1}(𝔛, GL_{d}) → H^{1}(𝔛, PGL_{d}) ⟶^{δ} H^{2}(𝔛, G_{m}) = Br(𝔛).

We denote again by [A] ∈ Br(𝔛) the class of δ([A]) in the cohomological Brauer group.

By [GP3, 3.1], we have an isomorphism Br(R_{n}) ≅ Br(F_{n}). We look now at the diagram

H_{loop}^{1}(R_{n}, PGL_{d}) ⟶^{δ} Br(R_{n})
≅↓  ≅↓
H^{1}(F_{n}, PGL_{d}) ⟶^{δ} Br(F_{n})

where the bottom map is injective [GS, §4.4] and the left map is bijective because of Theorem 8.1. We thus have

Corollary 13.1. The boundary map H_{loop}^{1}(R_{n}, PGL_{d}) → Br(R_{n}) is injective.

Azumaya R_{n}–algebras whose classes are in H_{loop}^{1}(R_{n}, PGL_{d}) are called loop Azumaya algebras. They are isomorphic to twisted form of M_{d} by a loop cocycle. One can rephrase the last Corollary by saying that loop Azumaya algebras of degree d are classified by their “Brauer invariant”.

Similarly, Wedderburn’s theorem [GS, 2.1] for F_{n}–central simple algebras has its counterpart.

Corollary 13.2. Let A be a loop Azumaya R_{n}–algebra of degree d. Then there exists a unique positive integer r dividing d and a loop Azumaya R_{n}–algebra B (unique up to R_{n}–algebras isomorphism) of degree d/r such that A ≃ M_{r}(B) and B ⊗_{R_{n}} F_{n} is a division 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 a division algebra.

In the same spirit, the Brauer decomposition [GS, 4.5.16] for central F_{n}–division algebras yields the following.

Corollary 13.3. Write d = p_{1}^{m_{1}} ⋯ p_{l}^{m_{l}}. Let A be an anisotropic loop Azumaya R_{n}- algebra of degree d. Then there exists a unique decomposition

A ≃ A_{1} ⊗_{R_{n}} ⋯ ⊗_{R_{n}} A_{l}

where A_{i} is an anisotropic k–loop Azumaya R_{n}-algebra of degree p_{i}^{m_{i}} for i = 1, .., l.

The two preceding Corollaries show that the classification of loop Azumaya R_{n}– algebra reduces the 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 understanding of H_{loop}^{1}(R_{n}, PGL_{d})_{an} is much more delicate.

We are given a loop cocycle ϕ = (ϕ^{geo}, z) with values in PGL_{d}(\overline{k}). Set A = _{z}(M_{d}). This is a central simple k–algebra such that _{z} PGL_{d} = PGL_{1}(A). Recall that ϕ^{geo} is given by a k–group homomorphism ϕ^{geo} : μ μ μ_{m}^{n} → PGL_{1}(A). To say that ϕ is anisotropic is to to say that ϕ^{geo} : μ μ μ_{m}^{n} → PGL_{1}(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).

13.2 The one dimensional case

If k is algebraically closed H^{1}(R_{1}, PGL_{d}) is trivial. The interesting new case is when k is not algebraically closed, e.g. the case of real numbers. Since the map H^{1}(F_{1}, PGL_{d}) → Br(F_{1}) is injective, as a consequence of Corollary 9.1, we have H^{1}(R_{1}, PGL_{d}) ≃ H^{1}(F_{1}, PGL_{d}) and the map

(13.1)  H^{1}(R_{1}, PGL_{d}) → Br(R_{1}) = Br(k) ⊕ H^{1}(k, ℚ/ℤ)

is injective.

Theorem 13.4. The image of the map 13.1 consists of all pairs [A_{0}] ⊕ χ where A_{0} is a central simple algebra of degree d and χ : Gal(k_{s}/k) → ℚ/ℤ a character for which that there exists an étale algebra K/k of degree d inside A_{0} such that χ_{K} = 0.

Remark 13.5. The indices of such algebras over F_{1} are known ([Ti, prop. 2.4] in the prime exponent case, and [FSS, Lemma 4.6] in the general case). The index of a F_{1}–algebra of invariant [A_{0}] ⊕ χ is deg(χ) × ind_{k_{χ}}(A ⊗_{k} k_{χ}) where k_{χ}/k stands for the cyclic extension associated to χ.

The proof needs some preparatory material from homological algebra based on Cartier duality for groups of multiplicative type. More precisely, the dual of an extension of k–groups of multiplicative type

1 → G_{m} → E → μ μ μ_{m} → 1

is the exact sequence

0 → ℤ/mℤ → Ê → ℤ → 1.

We have then an isomorphism

Ext_{k−gr}^{1}(μ μ μ_{m}, G_{m}) ≃ Ext_{Gal(k)}^{1}(ℤ, ℤ/mℤ) = H^{1}(k, ℤ/mℤ)

which permits to attach to the first extension a character. Up to isomorphism, there exists a unique extension E_{χ} of μ μ μ_{m} by G_{m} of class [χ].

Lemma 13.6. Let χ : Gal(k) → ℤ/mℤ be a character for some m ≥ 1.

1. The boundary map

k^{×}/(k^{×})^{m} ⟶^{∼} H^{1}(k,μ μ μ_{m}) → H^{2}(k, G_{m}) = Br(k)

is given by (x) ↦ χ ∪ (x).

2. Let K/k be an étale algebra. The following are equivalent:

(a) There exists a morphism of extensions E_{χ} → R_{K/k}(G_{m}) rendering the diagram

1 ⟶ G_{m} ⟶  E_{χ}  ⟶  μ μ μ_{m}  ⟶ 1
↓≅  ↓  ↓
1 ⟶ G_{m} ⟶ R_{K/k}(G_{m}) ⟶ R_{K/k}(G_{m})/G_{m} ⟶ 1;

commutative.

(b) χ_{K} = 0.

Proof. (1) The cocharacter group Ê_{χ} is ℤ/mℤ ⊕ ℤ together with the Galois action γ(α, β) = (α + χ(γ), β). The Galois action on E_{χ}(\overline{k}) ≃ \overline{k}^{×} × μ μ μ_{m}(\overline{k}) is then given by

γ(y, ζ) = (γ(y) ζ^{χ(γ)}, γ(ζ))

for every γ ∈ Gal(k). The class (x) ∈ H^{1}(k,μ μ μ_{m}) is represented by the cocycle c_{γ} = γ( \sqrt[m]{x})/ \sqrt[m]{x}. The element b_{γ} = (1, c_{γ}) ∈ E_{χ}(\overline{k}) lifts c_{γ}. The boundary ∂((x)) ∈ H^{2}(k, \overline{k}^{×}) is then represented by the 2–cocycle

a_{γ,τ} = b_{γ} × γ(b_{τ}) b_{γτ}^{−1} = c_{τ}^{χ(γ)}χ(γ) . c_{τ} .

(2) We decompose K = k_{1} × ⋯ × 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 extensions

0 ⟶ I ⟶ M ⟶ ℤ ⟶ 0
↓  ↓  ↓^{≃}
0 ⟶ ℤ/mℤ ⟶ Ê_{χ} ⟶ ℤ ⟶ 0.

By Shapiro’s lemma Ext^{1}(M_{j}, ℤ/mℤ) = H^{1}(k_{j}, ℤ/mℤ) and the map Ext^{1}(ℤ, ℤ/mℤ) → Ext^{1}(M_{j}, ℤ/mℤ) yields the restriction map H^{1}(k, ℤ/mℤ) → H^{1}(k_{j}, ℤ/mℤ). It follows that the bottom extension above is killed by the pull-back M_{j} → ℤ, and therefore that 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 Ê_{χ} belongs to the kernel of Ext^{1}(ℤ, ℤ/mℤ) → Ext^{1}(M_{j}, ℤ/mℤ) for for j = 1, .., l so Ê_{χ} belongs to the kernel of Ext^{1}(ℤ, ℤ/mℤ) → Ext^{1}(M, ℤ/mℤ). This means that the map M → ℤ of Galois modules lifts to Ê_{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 properties Again by Corollary 9.1, we have H_{loop}^{1}(R_{1}, PGL_{d}) = H^{1}(R_{1}, PGL_{d}) and we are reduced to twisted algebras given by loop cocycles ϕ = (ϕ^{geo}, z) with value in PGL_{d}(\overline{k}). Recall that A_{0} = _{z}(M_{d}) and that we have then a k–group homomorphism ϕ^{geo} : μ μ μ_{m} → PGL_{1}(A_{0}). We may assume that ϕ^{geo} is injective. We pull back the central extension 1 → G_{m} → GL_{1}(A_{0}) ⟶^{p} PGL(A_{0}) → 1 by ϕ^{geo} and get a central extension of algebraic k–groups

1 → G_{m} → E ⟶^{p′} μ μ μ_{m} → 1

such that E is a k–subgroup of GL_{1}(A_{0}). By extending the scalars to \overline{k}, we see that E is commutative, hence a k–group of multiplicative type. Then E is contained in a maximal torus of the k–group GL_{1}(A_{0}) and is of the form R_{K/k}(G_{m}) where K ⊂ A_{0} is an étale algebra of degree d. We have then the commutative diagram

1 ⟶ G_{m} ⟶  E  ⟶  μ μ μ_{m}  ⟶ 1
↓^{≃}  ↓  ↓
1 ⟶ G_{m} ⟶ R_{K/k}(G_{m}) ⟶ R_{K/k}(G_{m})/G_{m} ⟶ 1.

Lemma 13.6.2 tells us that χ_{K} = 0. We compute the Brauer class of this loop algebra by taking into account the commutative diagram

H^{1}(F_{1}, PGL_{d}) ⟶ Br(F_{1})
τ_{z}↑^{≀}  ^{≀}↑^{?+[A_{0}]}
H^{1}(F_{1}, PGL(A_{0})) ⟶^{∂} Br(F_{1})
↑  ↑
H^{1}(F_{1},μ μ μ_{m})  ⟶^{∂} Br(F_{1}).

The commutativity of the upper square is that of the torsion map [Se, §I.5.7], while that of the bottom square is trivial. The image of (t_{1}) ∈ F_{1}^{×}/(F_{1}^{×})^{×} is χ ∪ (t_{1}) by Lemma 13.6.1. The diagram yields the formula ∂([ϕ]) = [A_{0}] ⊕ χ which has the required properties.

Conversely, let K/k be an étale algebra of degree d inside A_{0} and let χ be a character such that χ_{K} = 0. Let m be the order of χ; by restriction-corestriction considerations m divides d. Lemma 13.6.2 shows that there exists a morphism of extensions E_{χ} → R_{K/k}(G_{m}). This yields a morphism ψ^{geo} : μ μ μ_{m} → R_{K/k}(G_{m})/G_{m} → PGL_{1}(A_{0}). The previous computation shows that the loop torsor (ψ^{geo}, z) has Brauer invariant [A_{0}] ⊕ χ.

As an example, we consider the real case.

Corollary 13.7. Assume that k = ℝ. Then the image of the injective map

H^{1}(R_{1}, PGL_{d}) → Br(R_{1}) = Br(ℝ) ⊕ H^{1}(ℝ, ℚ/ℤ)

is as follows:

1. 0 ⊕ 0 if d is odd;

2. 0 ⊕ 0, 0 ⊕ χ_{ℂ/ℝ}, [(−1, −1)] ⊕ 0 and [(−1, −1)] ⊕ χ_{ℂ/ℝ} if d is even.

Remark 13.8. In the case d = 2, the four classes under consideration corresponds to the quaternion algebras (1, 1), (1, t), (−1, −1), (−1, t).

13.3 The geometric case

We assume that k is algebraically closed. According to Corollary 8.6.2, our goal is to extract information from the bijections

Hom_{gp} (ℤ̂^{n}, PGL_{d}(k))_{irr}/ PGL_{d}(k) ⟶^{∼} H_{loop}^{1}(R_{n}, PGL_{d})_{irr} ⟶^{∼} H^{1}(F_{n}, PGL_{d})_{irr}.

The right hand set is known from the work of Amitsur [A], Tignol-Wadsworth [TiW] and Brussel [Br],^{33} the left hand-side is known by a classification due to Mumford [Mu, Prop. 3]

33See also [L], [Ne], [RY1, §8] and [RY2]. These last two references relate to finite abelian constant subgroups of PGL_{d} which have been investigated by Reichstein-Youssin in their construction of division algebras with large essential dimension. [Ne] is more interested in the “quantum tori” point of view and its relation to EALAs of absolute type A.

As a byproduct of our main result, we can provide then a proof of Mumford’s classification of irreducible finite abelian constant subgroups of PGL_{d} from the knowledge of the Brauer group of the field F_{n}. Let us state first Mumford’s classification. If d = s_{1}....s_{l}ŝ with s_{1} | s_{2} ⋯ | s_{l} and s_{1} ≥ 2, we consider the embedding

PGL_{s_{1}} × ⋯ × PGL_{s_{l}} → PGL_{d}

and define the subgroup H(s_{1}, ..., s_{l}) to be the image of the product of the standard anisotropic subgroups H(s_{j}) = (ℤ/s_{j}ℤ)^{2} of PGL_{s_{j}} for j = 1, .., l defined by the generators

(13.2) a_{j} = (\begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 1 & 0 & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & & 0 \\ & ⋯ & & 1 & 0 \end{matrix}) , b_{j} = (\begin{matrix}1 & 0 & ⋯ \\ 0 & ζ_{s_{j}} & 0 & ⋯ & 0 \\ 0 & ⋯ & & & 0 \\ 0 & ⋯ & & 0 & ζ_{s_{j}}^{s_{j}−1} \end{matrix}) .

Remark 13.9. The way of expressing the group H(s_{1}, s_{2}) in the form H(s_{1}) × H(s_{2}) is not unique when s_{1} and s_{2} are coprime. There is then a unique way to arrange such a group H as H(s′_{1}, ..., s′_{l′}) with s′_{1} | s′_{2} ⋯ | s′_{l} and s′_{1} ≥ 2. Note that rank (H(s′_{1}, ..., s′_{l})(k)) = 2l′.

We can now state and establish the classification of irreducible finite abelian groups of the projective linear group.

Theorem 13.10. [Mu, Prop. 3] (see also [BL, §6], [GM, Th. 8.28]).

1. d = s_{1} × ... × s_{l} if and only H(s_{1}, ..., 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_{1}, ..., s_{l}) with d = s_{1}...s_{l}, s_{1} | s_{2} ⋯ | s_{l} and s_{1} ≥ 2.

As mentioned above, our proof makes use of Galois cohomology results over R_{n} for n ≥ 1 (or equivalently F_{n}) collected from our previous paper [GP3].

Our convention on the cyclic algebra (t_{i}, t_{j})_{p}^{q} is that of Tate^{34} for the Azumaya R_{n}–algebra with presentation

X^{q} = t_{i}, Y^{q} = t_{j}^{p}, Y X = ζ_{q} XY.

34This is the opposite convention than that of [Br] and [GP3], but consistent with that of [GS] which we use in the proof.

Let us consider now Brussel R_{n}–algebras. Given sequences of length l ≤ [\frac{n}{2}]

2 ≤ s_{1} ⋯ ≤ s_{l}, r_{1}, ⋯, r_{l}

we define

A(r_{1}, s_{1}, ..., r_{l}, s_{l}) : (t_{1}, t_{2})_{s_{1}}^{r_{1}} ⊗_{R_{n}} ⋯ ⊗_{R_{n}} (t_{2l−1}, t_{2l})_{s_{l}}^{r_{l}}.

Lemma 13.11. With the notation as above, set d = s_{1}...s_{l} and define

ϕ : ℤ̂^{n} → H(s_{1})(k) × ⋯ × H(s_{l})(k) = H(s_{1}, ..., s_{l})(k) ⊂ PGL_{d}(k),

(e_{1}, e_{2}, ..., e_{2l−1}, e_{2l}) ↦ (−a_{1}, −r_{1}b_{1}, ..., −a_{l}, −r_{l}\overline{b}_{l})

1. Then _{ϕ}(M_{d}) ⟶^{∼} A(r_{1}, s_{1}, ..., r_{l}, s_{l}) as R_{n}–algebras.

2. The following are equivalent:

(a) A(r_{1}, s_{1}, ..., r_{l}, s_{l}) ⊗_{R_{n}} F_{n} is division F_{n}–algebra;

(b) ϕ is irreducible;

(c) H(s_{1}, ..., s_{l}) is irreducible in PGL_{d} and (r_{j}, s_{j}) = 1 for j = 1, ..., l.

Proof. (1) This is done for R_{2} 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}}. According to [GP3, 3.13], we have then (r_{j}, s_{j}) = 1 for j = 1, ..., l. Hence Im(ϕ) = H(s_{1}, ..., s_{l}) is irreducible in PGL_{d}.

(c) ⟹ (a): Since H(s_{1}, ..., s_{l}) is irreducible in PGL_{d}, we have d = s_{1}....s_{l}. The condition (r_{j}, s_{j}) = 1 for j = 1, ..., l implies that the algebra A(r_{1}, s_{1}, ..., 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_{1}...s_{l} then A(1, s_{1}, ..., 1, s_{l}) ⊗_{R_{n}} F_{n} is a division F_{n}–algebra [Am, th. 3], so Lemma 13.11 shows that H(s_{1}, ..., s_{n}) is irreducible in PGL_{d}. If d ≠ s_{1}...s_{l}, then this algebra is not division and H(s_{1}, ..., s_{n}) is reducible.

(2) If H(s_{1}, ..., s_{l}) is PGL_{d}(k)–conjugate to some H(s′_{1}, ..., s′_{l′}), then H(s_{1}, ..., s_{l}) is isomorphic to H(s′_{1}, ..., 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 constant irreducible subgroup of PGL_{d}. Denote by n the rank of H(k) and by m its exponent.

Let us prove first that H is PGL_{d}(k)–conjugate to some H(s_{1}, ..., s_{l}). We view H(k) as the image of an irreducible group homomorphism ψ : (ℤ/mℤ)^{n} → PGL_{d}(k). Since k is algebraically closed ψ is a cocycle. The loop construction then defines an Azumaya R_{n}–algebra of degree d such that A ⊗_{R_{n}} F_{n} is division (i.e. the group PGL_{1}(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_{1}, s_{1}, ..., r_{l}, s_{l})

with (r_{j}, s_{j}) = 1 for j = 1, .., l.

By Lemma 6.1.1, A(r_{1}, s_{1}, ..., r_{l}, s_{l}) is the loop Azumaya algebra defined by the morphism

ϕ : ℤ̂^{n} → H(s_{1})(k) × ⋯ × H(s_{l})(k) = H(s_{1}, ..., s_{l})(k) ⊂ PGL_{d}(k),
(e_{1}, e_{2}, ..., e_{2l−1}, e_{2l}) ↦ (−a_{1}, −b_{1}, ..., −a_{l}, −r_{l}b_{l}).

which is then irreducible by the second statement of the same lemma. Theorem 7.9 tells us that ϕ and ψ are PGL_{d}(k)–conjugate, hence H(k) is PGL_{d}(k)–conjugate to H(s_{1}, ..., s_{l})(k) = Im(ψ). Since n = rank(H(s_{1}, ..., s_{l})(k)), we have s_{1} | s_{2}... | 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_{1}, ..., s_{l} and an integer r_{1} prime to s_{1} satisfying

s_{1} | ⋯ | s_{l}, 2 ≥ s_{1}, d = s_{1} ⋯ s_{l}, (r_{1}, s_{1}) = 1

and an element g ∈ GL_{n}(ℤ) such that

g^{∗}(A) ≅ A(r_{1}, s_{1}, 1, s_{2}, 1, s_{3}, ⋯, 1, s_{l}) ≅ A(−r_{1}, s_{1}, 1, s_{2}, 1, s_{3}, ⋯, 1, s_{l}).

Such a sequence s_{1}, ..., s_{l} is unique.

2. If n = 2l, ±r_{1} is unique modulo s_{1}.

3. If n > 2l, g^{∗}(A) ≅ A(1, s_{1}, 1, s_{2}, 1, s_{3}, ⋯, 1, s_{l}).

Proof. (1) By definition, A is the twist of M_{d}(k) by a morphism ϕ : (ℤ/mℤ)^{n} → PGL_{d}(k). Since A ⊗_{R_{n}} F_{n} is division, ϕ is irreducible [GP2, th. 3.1]. Theorem 13.10 shows that there exists a unique sequence s_{1}, ..., s_{l} such that s_{1} | ⋯ | s_{l}, 2 ≥ s_{1} and Im(ϕ) is PGL_{d}(k)–conjugate to H(s_{1}, ..., s_{l}) := H(s_{1}, ..., s_{l})(k). Without lost of generality, we can assume that Im(ϕ) = H(s_{1}, ..., s_{l}).

Recall that a_{1}, b_{1}, ..., a_{l}, b_{l} stand for the standard generators of H(s_{1}, ..., s_{l}). We shall use that Λ^{l}(H(s_{1}, ..., s_{l})) ≃ ℤ/s_{1}ℤ generated by a_{1} ∧ b_{1} ⋯ a_{l} ∧ b_{l} [RY2, Lemma 2.1], as well as the following invariant of ϕ (ibid, 2.5)

δ(ϕ) = ϕ(e_{1}) ∧ ϕ(e_{2}) ∧ ⋯ ∧ ϕ(e_{n}) ∈ Λ^{n}(H(s_{1}, ..., s_{l})(k))

This invariant has the remarkable property that a homomorphism ϕ′ : (ℤ/mℤ)^{n} → H(s_{1}, ..., 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 = 2l (resp. n > 2l). First case. n = 2l: The family (ϕ(e_{1}), ..., ϕ(e_{n})) generates H(s_{1}, ..., s_{l}), and we consider the class

[r_{1}^{♯}] := ϕ(e_{1}) ∧ ϕ(e_{2}) ∧ ⋯ ∧ ϕ(e_{n}) ∈ (ℤ/s_{1}ℤ)^{×}.

Let r_{1} be an inverse of r_{1}^{♯} modulo s_{1}. We have

ϕ(r_{1}e_{1}) ∧ ϕ(e_{2}) ⋯ ∧ ϕ(e_{n}) = a_{1} ∧ b_{1} ⋯ a_{l} ∧ b_{l}

so there exists g ∈ GL_{n}(ℤ) such that (ibid, 2.5)

ψ(r_{1}e_{1}) = a_{1}, ψ(e_{2}) = b_{2}, ⋯, ϕ(e_{n−1}) = a_{l}, ϕ(e_{n}) = b_{l}

where ψ = ϕ ∘ g. In terms of algebras, this means that

g^{∗}(A) ≃ A(r_{1}, s_{1}, 1, s_{2}, 1, s_{3}, ⋯, 1, s_{l}).

Let us first prove the unicity assertions The unicity of (s_{1}, .., s_{l}) follows from Theorem 13.10, hence (1) is proven. For (2), we are given then r′_{1} ∈ ℤ coprime to s_{1}, and an element h ∈ GL_{n}(ℤ) such that

h^{∗}(A(r_{1}, s_{1}, 1, s_{2}, 1, s_{3}, ⋯, 1, s_{l})) ≃ A(r′_{1}, s_{1}, 1, s_{2}, 1, s_{3}, ⋯, 1, s_{l}).

Denote by ψ′ : (ℤ/mℤ)^{n} → H(s_{1}, .., s_{n}) the group homomorphism defined by ψ′(e_{1}) = r′_{1}a_{1}, ψ′(e_{2}) = b_{1}, ⋯, ψ(e_{n}) = b_{2l}. Since the (F_{n}–anisotropic) loop algebras attached to h^{∗}ψ and ψ′ are isomorphic, Theorem 7.9 provides an element u ∈ PGL_{d}(k) such that

ψ ∘ h = u ∘ ψ′.

Since H(s_{1}, ..., s_{l}) = Im(ψ) = Im(ψ′), it follows that u ∈ N_{PGL_{d}(k)}(H(s_{1}, ...s_{l})). But the map u : H(s_{1}, ..., s_{l})(k) → H(s_{1}, ...s_{l})(k) preserves the symplectic pairing H(s_{1}, ..., s_{l})(k) × H(s_{1}, ..., 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′).

Thus r_{1} = ±r′_{1} ∈ ℤ/s_{1}ℤ as prescribed in (2) . Second case. n > 2l: For i = 2l + 1, ..., n we set c_{i} = 0 ∈ H(s_{1}, .., s_{l}). Both families (ϕ(e_{1}), … ϕ(e_{n})) and (r_{1} a_{1}), b_{1} ⋯, a_{l}, b_{l}, c_{2l+1}, ⋯, c_{n}) generate H(s_{1}, ..., s_{l}) and satisfy

ϕ(e_{1})∧ϕ(e_{2}) ⋯∧ϕ(e_{n}) = (r_{1} a_{1})∧b_{1} ⋯ a_{l} ∧b_{l} ∧c_{2l+1} ∧⋯∧c_{n} ∈ Λ^{n}(H(s_{1}, ..., s_{l})) = 0.

The same fact [RY2, Lemma 2.5] shows that there exists g ∈ GL_{n}(ℤ) such that (g^{∗}ϕ)(e_{1}) = r_{1}a_{1}, (g^{∗}ϕ)(e_{2}) = b_{1}, (g^{∗}ϕ)(e_{2i−1}) = a_{i} and (g^{∗}ϕ)(e_{2i}) = b_{i} for i = 2, .., l and (g^{∗}ϕ)(e_{i}) = c_{i} for i = 2l + 1, ..., n. Therefore the preceding case with 2l variables yields the existence and the unicity of the s_{i}’s. It remains to prove (3), namely that we can assume that r_{1} = 1. But this follows along the same lines of the argument given above since (r_{1} a_{1}) ∧ b_{1} ⋯ a_{l} ∧ b_{l} ∧ c_{2l+1} ⋯ ∧ c_{n} = (a_{1}) ∧ b_{1} ⋯ a_{l} ∧ b_{l} ∧ c_{2l+1} ⋯ ∧ c_{n} ∈ Λ^{n}(H(s_{1}, ..., s_{l})).

13.4 Loop algebras of inner type A

To the Azumaya R_{n}–algebra A(r_{1}, s_{1}, ..., r_{l}, s_{l}) we can attach (using the commutator [x, y] = xy − yx) a Lie algebra over R_{n}. We denote by L(r_{1}, s_{1}, ..., r_{l}, s_{l}) the derived algebra of this Lie algebra. It is a twisted form of 𝔰𝔩_{d}(R_{n}) where d = s_{1} … s_{l}.

Corollary 13.13. Let d be a positive integer. Let L be a nullity n loop algebra of inner (absolute) type A_{d−1}.

1. If L is not split, it is k–isomorphic to L(r_{1}, s_{1}, 1, s_{2}, ⋯, 1, s_{l}) where

s_{1} | ⋯ | s_{l}, 2 ≥ s_{1}, d = s_{1} ⋯ s_{l}, (r_{1}, s_{1}) = 1 and l ≤ [\frac{n}{2}]

and such a sequence s_{1}, ..., s_{l} is unique.

2. If n = 2l, r_{1} is unique modulo s_{1} and up to the sign.

3. If n > 2l, L is k–isomorphic to L(1, s_{1}, 1, s_{2}, ⋯, 1, s_{l})

Proof. The classification in question is given by considering the image of the natural map

H_{loop}^{1}(R_{n}, PGL_{d}) → GL_{n}(ℤ)\H^{1}(R_{n}, Aut(PGL_{d}))

The image can be identified with (ℤ/2ℤ×GL_{n}(ℤ))\H_{loop}^{1}(R_{n}, PGL_{d}) where ℤ/2ℤ acts by the opposite Azumaya algebra construction. Corollary 8.6 reduces the problem to the “anisotropic case”. Theorem 13.12 determines the set GL_{n}(ℤ)\H_{loop}^{1}(R_{n}, PGL_{d}), and as it turns out the action of ℤ/2ℤ is trivial. Therefore the desired classification is also provided by GL_{n}(ℤ)\H_{loop}^{1}(R_{n}, PGL_{d}) and we can now appeal to Theorem 13.12 to obtain the Corollary.

14 Invariants attached to EALAs and multiloop algebras

Both the finite dimensional simple Lie algebras over \overline{k} (nullity 0) and their affine counterparts (nullity 1) have Coxeter-Dynkin diagrams attached to them that contain a considerable amount of information about the algebras themselves. It has been a long 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 based on a finite dimensional simple Lie algebra 𝔤 over an algebraically closed field k of characteristic 0. Consider an n–tuple σ σ σ = (σ_{1}, …, σ_{n}) of commuting elements of Aut(𝔤) satisfying σ_{i}^{m} = 1. For each n–tuple (i_{1}, …, i_{n}) ∈ ℤ^{n} we consider the simultaneous eigenspace 𝔤_{i_{1}...i_{n}} = {x ∈ 𝔤 : σ_{j}(x) = ξ_{m}^{i_{j}}x for all 1 ≤ j ≤ n}. The multiloop algebra L(𝔤,σ σ σ) corresponding to σ σ σ is defined by

L(𝔤,σ σ σ) = ⊕_{(i_{1},...,i_{n})∈ℤ^{n}} 𝔤_{i_{1}...i_{n}} ⊗ t^{\frac{i_{1}}{m}} … tn^{\frac{in}{m}} ⊂ 𝔤 ⊗_{k} R_{n,m} ⊂ 𝔤 ⊗_{k} R_{∞}

Recall that L(A,σ σ σ), which does not depend on the choice of common period m, is not only a k–Lie algebra (in general infinite dimensional), but also naturally an R– algebra. It is when L(𝔤,σ σ σ) is viewed as an R–algebra that Galois cohomology and the theory of torsors enter into the picture. Indeed a rather simple calculation shows that

L(𝔤,σ σ σ) ⊗_{R_{n}} R_{n.m} ≃ 𝔤 ⊗_{k} R_{n,m} ≃ (𝔤 ⊗_{k} R_{n}) ⊗_{R_{n}} R_{n,m}.

Thus L(𝔤,σ σ σ) corresponds to a torsor 𝔈_{σ σ σ} over Spec(R) under Aut(𝔤). It is, however, the k–Lie algebra structure that is of interest in infinite dimensional Lie theory and Physics.

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 functoriality and the definition of Lie algebra of a group functor in terms of dual numbers we see that 𝔏ie(_{𝔈}G_{R_{n}}) = L(𝔤,σ σ σ). By the Aciclicity Theorem to _{𝔈}G_{R_{n}} we can attach a Witt-Tits index, and this is the “diagram” that we attach to L(𝔤,σ σ σ) as as Lie algebra over k. Note that by Corollary 8.12 this is well defined. The diagram carries the information about the absolute and relative type of L(𝔤,σ σ σ).^{35}

35The relative type as an invariant of L(𝔤,σ σ σ) is defined in in §3 of [ABP3] by means of the central closure. If C is the centroid of L(𝔤,σ σ σ) and C̃ denotes its field of quotients, then L(𝔤,σ σ σ) ⊗_{C} C̃ is a finite dimensional central simple algebra over C̃. As such it has an absolute and relative type. This construction applies to an arbitrary prime perfect Lie algebra which is finitely generated over its centroid.

To reassure ourselves that this is the correct point of view we can look at the nullity one case. EALAs of nullity one are the same than the affine Kac-Moody Lie algebras. If one uses Tits methods to compute simple adjoint algebraic groups over the field 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^{36} of affine algebraic groups (Borel-Tits [BoT2], see also [Sp, §13.4]) to the case of a group scheme 𝔊 which is of finite type and affine over a fixed base scheme 𝔛. We begin by establishing some notation.

We will denote by G_{m,𝔛} and G_{a,𝔛} the multiplicative and additive 𝔛-groups. The underlying schemes of these groups will be denoted by 𝔸_{𝔛}^{×} and 𝔸_{𝔛} respectively. After applying a base change 𝔛 → 𝔛′ we obtain corresponding 𝔛′-groups and schemes that we 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 homomorphism

(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 functors that 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 that to a scheme 𝔛′ over 𝔛 attaches the group

(15.2)  Z_{𝔊}(λ)(𝔛′) = {x′ ∈ 𝔊(𝔛′) : x′′ commutes with λ(G_{m,𝔛}(𝔛′′)) ⊂ 𝔊(𝔛′′)}

where 𝔛′′ is a scheme over 𝔛′ and x′′ denotes the image of x′ under the natural group homomorphism 𝔊(𝔛′) → 𝔊(𝔛′′).

We consider the two following 𝔛–functors

P(λ)(𝔛′) = {g ∈ 𝔊(𝔛′) | λ_{𝔛′}(id_{𝔸_{𝔛′}^{×}}) η_{𝔛′}(g) (λ_{𝔛′}(id_{𝔸_{𝔛′}^{×}}))^{−1} ∈ 𝔊(𝔸_{𝔛′})}

36In [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 this terminology. This material can in part be recovered from their work, but we have decided to include it in the form that we needed for the sake of completeness.

and

U(λ)(𝔛′) = {g ∈ 𝔊(𝔛′) | λ_{𝔛′}(id_{𝔸_{𝔛′}^{×}}) η_{𝔛′}(g) (λ_{𝔛′}(id_{𝔸_{𝔛′}^{×}}))^{−1} ∈ ker(𝔊(𝔸_{𝔛′}) → 𝔊(𝔛′)) }

for every 𝔛-scheme 𝔛′. The centralizer Z_{𝔊}(λ) is an 𝔛–subgroup functor of P(λ) which normalizes U(λ).

We look at the previous definitions in the case when 𝔛 = Spec(R) and 𝔛′ = Spec(R′) are both affine.^{37} We have 𝔸_{R′} = Spec(R′[x]) and 𝔸_{R′}^{×} = Spec(R′[x^{±1}]). Then x ∈ R′[x^{±1}]^{×} = G_{m,R′}(R′[x^{±1}]) = G_{m,R′}(𝔸_{R′}^{×}), and by applying our cocharacter we obtain an element λ_{R′}(x) ∈ 𝔊(R′). Under Yoneda’s correspondence G_{m,R′}(R′[x^{±1}]) ≃ Hom_{R′}(R′[x^{±1}], R′[x^{±1}]) our element x corresponds to the identity map, namely to the element id_{𝔸_{R′}^{×}} ∈ G_{m,R′}(𝔸_{R′}^{×}) if we rewrite our ring theoretical objects in terms of schemes. We thus have

P(λ)(R′) = {g ∈ 𝔊(R′) | λ_{R′}(x) η_{R′}(g) (λ_{R′}(x))^{−1} ∈ 𝔊(R′[x])}

and

U(λ)(R′) = {g ∈ 𝔊(R′) | λ_{R′}(x) η_{R′}(g) (λ_{R′}(x))^{−1} ∈ ker(𝔊(R′[x]) → 𝔊(R′)) }

where η_{R′}(g) is the natural image of g ∈ 𝔊(R′) in 𝔊(R′[x^{±1}]), and the group homomorphism 𝔊(R′[x]) → 𝔊(R′) comes from the ring homomorphism R′[x] → R′ that maps x ↦ 0.

15.1 The case of GL_{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 be a 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 that there exists (unique) integers 1 ≤ ℓ_{1} < ℓ_{2} < ⋯ < ℓ_{j} ≤ n and e_{1}, …, e_{n} such that

e_{i} = e_{j} if ℓ_{k} ≤ i, j < ℓ_{k+1} for some k
e_{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)

37As customary we write G_{m,R} instead of G_{m,𝔛}...

(∗)  r ↦ (\begin{matrix}r^{e_{1}} & & & & & 0 \\ & & . & . & . \\ & 0 \\ & & & & & r^{e_{n}} \end{matrix} )

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 (ℤ[x^{±1}]) and T = Spec (ℤ[t_{1}^{±1}, …, t_{n}^{±1}]), then λ corresponds (under Yoneda) to the ring homomorphism

λ^{∗} : ℤ[t_{1}^{±1}, …, t_{n}^{±1}] ⟶ ℤ[x^{±1}]

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_{1}, …, ℓ_{j}. Note that one “cannot see” this by looking at the centralizer of λ(G_{m,ℤ}(R)) inside G(R). This is clear, for example, if n = 2, R = ℤ, j = 1, ℓ_{1} = 1 and 1 = e_{1} < e_{2} = 3. The easiest way to eliminate “naive” contralizers in Z_{G}(λ)(R) is to look at their image in G(R[x^{±1}]). In fact

Lemma 15.1. With the above notation we have

Z_{G}(λ)(R) = { A ∈ G(R) ⊂ G(R[x^{±1}]) : A commutes with λ_{R}(G_{m,ℤ}(R[x^{±1}])) }.

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 commute with the diagonal matrix

λ_{S}(s) = (\begin{matrix}s^{e_{1}} \\ & . & . & . \\ & & & & s^{e_{n}} \end{matrix} ) .

But then A, viewed now as an element of 𝔊(R[x^{±1}]) cannot commute with

λ_{R[x^{±1}]}(x) = (\begin{matrix}x^{e_{1}} \\ & . & . & . \\ & & & & x^{e_{n}} \end{matrix} ) .

For if it did, we would reach a contradiction by functoriality considerations applied to the (natural) ring homomorphism R[x^{±1}] → S that takes x ↦ s.

Returning to our example we see that there are two extreme cases for Z_{G}(λ). If j = 1 and ℓ_{1} = n then Z_{G}(λ) = G. At the other extreme if j = n then λ is regular and 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 sees that

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_{1} = 2, ℓ_{2} = 5, e_{1} = 1, e_{2} = 3. Then A = (a_{ij}) ∈ GL_{5}(R) is of the form

A = ( \begin{matrix}× & + \\ − & × \end{matrix} ) .

We have two blocks, the top left of size 2 and the bottom right of size 3. Given A = (a_{ij}) ∈ GL_{5}(R) ⊂ GL_{5}(R[x^{±1}]) define P by

(\begin{matrix}x^{1} & & & 0 \\ & x \\ & & x^{3} \\ & 0 & & x^{3} \\ & & & & x^{3} \end{matrix}) A (\begin{matrix}x^{−1} & & & 0 \\ & x^{−1} \\ & & x^{−3} \\ & 0 & & x^{−3} \\ & & & & x^{−3} \end{matrix}) = P

that is

λ(x)Aλ(x)^{−1} = P

where P = (p_{ij}) and p_{ij} = ∑ p_{ijk}x^{k} ∈ R[x^{±1}]. To belong to P(λ) the element A must be such that p_{ijk} = 0 for k < 0. This forces all entries in the 3 × 2 block marked with a − 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 ≤ 2 or 2 ≤ i, j ≤ 5. If, on the other hand, i ≤ 2 < j then p_{ij} = a_{ij}x^{2}.

This makes the meaning of U(λ) quite clear in general. If λ(x)Aλ(x)^{−1} ∈ G(R[x]) is mapped to the identity element of G(R) under the map R[x] → R which sends x ↦ 0 then 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).

15.2 The general case

Lemma 15.3. Assume that there exists locally for the fpqc-topology a closed embedding of 𝔊 in a linear group scheme.^{38} Then

1. the 𝔛–functor U(λ) (resp. P(λ), resp. Z_{𝔊}(λ)) is representable by a closed subgroup 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}(P(λ)).

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 maximal split^{39} 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 torus of GL_{n,ℤ}. Since Hom_{ℤ}(G_{m}, T) ≃ Hom_{R}(G_{m,R}T_{R}), we can reduce our problem to the case when R = ℤ, which has been already done in Example 15.1. General case:

By fpqc-descent, we can assume that 𝔛 is the spectrum of a ring R, and that we are given a R–group scheme homomorphism ρ : 𝔊 → 𝔊′ = GL_{n,R} which is a closed immersion.

(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 established by reducing to the case of 𝔊′ = GL_{n,R}. This reduces the representability questions to 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)^{−1}λ(x) u λ(x)^{−1} 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.

38This condition is satisfied if 𝔛 is locally noetherian of dimension ≤ 1 [BT2, §1.4], and also for reductive 𝔛–group schemes.

39Trivial, in the terminology of [SGA3].

Definition 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 scheme of 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 parabolic subgroup 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, §15.1].

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 with respect to the fpqc topology, so that we can assume that R is local and that 𝔊 is split. By Demazure’s theorem [SGA3, XXIII.4], we can assume that 𝔊 arises by base change from a (unique) split Chevalley group 𝔊_{0} over ℤ.

We now reason along similar lines than the ones used in studying the GL_{n,ℤ} case above. Let 𝔗 ⊂ 𝔊_{0} be a maximal split torus. Since all maximal split tori of 𝔊 are conjugate 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 case when R = ℤ and of 𝔊 = 𝔊_{0}, 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 geometric fibers are smooth, we can conclude by [SGA1, prop. II.2.3] that P(λ) is smooth and is a parabolic subgroup scheme of the ℤ–group 𝔊_{0}.

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 scheme of P(λ).

(2) Using the theory of relative root systems [SGA3, §XXVI.7], the proof is the same than in the field case.

16 Appendix 2: Global automorphisms of G–torsors over the projective 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^{0} is reductive. One way to state Grothendieck-Harder’s theorem is to say that the natural map

Hom_{gp}(G_{m}, G)/G(k) → H_{Zar}^{1}(ℙ^{1}, G)

which maps a cocharacter λ : G_{m} → G to the G-torsor 𝔈_{λ} := (−λ)_{∗}(O(−1)) over ℙ_{k}^{1} where O(−1) stands for the Hopf bundle A_{k}^{2} \ {0} → ℙ_{k}^{1}, is bijective.^{40}

We fix now a cocharacter λ : G_{m} → G. We are interested in the twisted ℙ_{k}^{1}–group scheme 𝔈_{λ}(G) = \underline{Isom}_{G}(𝔈_{λ}, 𝔈_{λ}), as well as the abstract group 𝔈_{λ}(G)(ℙ_{k}^{1}). This group is the group of global automorphisms of the G–torsor 𝔈_{λ} over ℙ_{k}^{1}. It has a concrete description.

Lemma 16.1. 𝔈_{λ}(G)(ℙ_{k}^{1}) = G(k[t]) ∩ λ(t)G(k[t^{−1}])λ(t^{−1}).

Proof. We recover ℙ_{k}^{1} by two affine lines U_{0} = Spec(k[t]) and U_{1} = Spec(k[t^{−1}]). The Hopf bundle is isomorphic to the twist of G_{m} by the cocycle z ∈ Z^{1}(U_{0} ⊔U_{1}/ℙ_{k}^{1}, G_{m}) where z_{0,0} = 1, z_{0,1} = t^{−1}, z_{1,0} = t, z_{1,1} = 1. Then λ(z) ∈ Z^{1}(U_{0} ⊔ U_{1}/ℙ_{k}^{1}, G_{m}) is the cocycle of 𝔈_{λ}. Hence

𝔈_{λ}(G)(ℙ_{k}^{1}) = {(g_{0}, g_{1}) ∈ G(U_{0}) × G(U_{1}) | λ^{−1}(z_{0,1}).g_{1} = g_{0} }
= G(k[t]) ∩ λ(t)G(k[t^{−1}])λ(t^{−1}).

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} = ∏_{ℙ_{k}^{1}/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 this allows us to define the Z(λ)–torsor S_{λ} := (−λ)_{∗}(O(−1)) over ℙ_{k}^{1}. We can twist the morphism Z_{G}(λ) → G by S_{λ}, so we get a morphism Z_{G}(λ) ×_{k} ℙ_{k}^{1} → 𝔈_{λ}(G) and then a morphism Z_{G}(λ) → H_{λ}.

Proposition 16.2. The homomorphisms of k–groups

(∏_{ℙ_{k}^{1}/k} S_{λ}(U(λ))) ⋊ Z_{G}(λ) → ∏_{ℙ_{k}^{1}/k} S_{λ}(P(λ)) → H_{λ}.

are isomorphisms. Furthermore, ∏_{ℙ_{k}^{1}/k} S_{λ}(U(λ)) is a unipotent k–group.

40This is note the usual way to state the theorem (see [Gi0, II.2.2.1]), but it is easy to derive the formulation that we are using.

Proof. Write P, U for P(λ), U(λ).

∏_{ℙ_{k}^{1}/k} S_{λ}(P) = ∏_{ℙ_{k}^{1}/k} S_{λ}(U) ⋊ ∏_{ℙ_{k}^{1}/k} Z_{G}(λ) = ∏_{ℙ_{k}^{1}/k} S_{λ}(U) ⋊ Z_{G}(λ)

so it remains to show that ∏_{ℙ_{k}^{1}/k} S_{λ}(P) ⟶^{∼} H_{λ}. Consider a faithfull representation

ρ : G → G′ = GL_{n}. Denote by P′ the parabolic subgroup of G′ attached to λ. We have P = G ×_{G}′ P′, hence S_{λ}(P) = 𝔈_{λ}(G) ×_{𝔈_{λ}(G′)} S_{λ}(P′). It follows that

∏_{ℙ_{k}^{1}/k} S_{λ}(P) = ∏_{ℙ_{k}^{1}/k} 𝔈_{λ}(G) ×_{∏_{ℙ^{1}_{k}/k} 𝔈_{λ}(G′)} ∏_{ℙ_{k}^{1}/k} S_{λ}(P′)

as can be seen by reducing to the case of GL_{n} already done in Example 15.1. This case also shows that ∏_{ℙ_{k}^{1}/k} S_{λ}(U(λ)) is a unipotent k–group.

References

[A] P. Abramenko, Group actions on twin buildings, Bull. Belg. Math. Soc. Simon Stevin 3 (1996), 391–406.

[Am] S.A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408–420.

[AABFP] S. Azem, B. Allison, S. Berman, Y. Gao and A. Pianzola, Extended affine Lie algebras and their root systems, Mem. Amer. Math. Soc. 603 (1997) 128 pp.

[ABP2] B.N. Allison, S. Berman and A. Pianzola, Covering algebras II: Isomorphism of loop algebras, J. Reine Angew. Math. 571 (2004) 39-71.

[ABP2.5] B.N. Allison, S. Berman and A. Pianzola, Iterated loop algebras, Pacific Jour. Math. 227 (2006), 1–41.

[ABP3] B.N. Allison, S. Berman and A. Pianzola, Covering algebras III: Multiloop algebras, 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 gradedsimple algebras as loop algebras, Forum Mathematicum 20 (2008), 395–432.

[BMR] M. Bate, B. Martin and G. Röhrle, A geometric approach to complete reducibility, Invent. math 161, 177-218 (2005).

[BL] C. Birkenhake and H. Lange, Complex abelian varieties, Gründlehren der mathematischen Wissenschaften 302 (2004), Springer.

[BFM] A. Borel, R. Friedman and J.W. Morgan, Almost commuting elements in compact Lie groups Mem. Amer. Math. Soc. 157, (2002), no. 747.

[Bo] A. Borel, Linear Algebraic Groups (Second enlarged edition), Graduate text in Mathematics 126 (1991), Springer.

[BM] A. Borel and G.D Mostow, On semi-simple automorphisms of Lie algebras, Ann. Math. 61 (1955), 389-405.

[BoT] A. Borel and J. Tits, Groupes réductifs, Pub. Math. IHES 27, (1965), 55–152.

[BoT2] A. Borel and J. Tits, Théorèmes de structure et de conjugaison pour les groupes algébriques linéaires, C. R. Acad. Sci. Paris Sér. A-B 287 (1978),55-57.

[Bbk] N. Bourbaki, Groupes et algèbres de Lie, Ch. 7 et 8, Masson.

[BT1] F. Bruhat and J. Tits, Groupes algébriques sur un corps local, Pub. IHES 41 (1972), 5–251.

[BT2] F. Bruhat and J. Tits, Groupes algébriques sur un corps local II. Schémas en groupes. Existence d’une donnée radicielle valuée, Pub. IHES 60 (1984), 197–376.

[BT3] F. Bruhat and J. Tits, Groupes algébriques sur un corps local III. Compléments et application à la cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo 34 (1987), 671–698.

[Br] E. S. Brussel, The division algebras and Brauer group of a strictly Henselian field, J. Algebra 239 (2001) 391–411.

[CGP] V. Chernousov, P. Gille and A. Pianzola, Torsors on the punctured line, to appear in American Journal of Math. (2011).

[CGR1] V. Chernousov, P. Gille and Z. Reichstein, Resolution of torsors by abelian extensions, Journal of Algebra 296 (2006), 561–581.

[CGR2] V. Chernousov, P. Gille and Z. Reichstein, Resolution of torsors by abelian extensions II, Manuscripta Mathematica 126 (2008), 465-480.

[CTGP] J.–L. Colliot–Thélène, P. Gille and R. Parimala, Arithmetic of linear algebraic groups over two-dimensional geometric fields, Duke Math. J. 121 (2004), 285-321.

[CTS1] J.–L. Colliot–Thélène and J.–J. Sansuc, Fibrés quadratiques et composantes connexes réelles, Math. Annalen 244 (1979), 105–134.

[CoGP] B. Conrad, O. Gabber and G. Prasad, Pseudo-reductive groups, book available at http://math.stanford.edu/ conrad/

[D] M. Demazure, Schémas en groupes réductifs, Bull. Soc. Math. France 93 (1965), 369–413.

[DG] M. Demazure and P. Gabriel, Groupes algébriques, Masson (1970).

[EGA IV] A. Grothendieck (avec la collaboration de J. Dieudonné), Eléments de Géométrie Algébrique IV, Publications mathématiques de l’I.H.É.S. no 20, 24, 28 and 32 (1964 - 1967).

[F] M. Florence, Points rationnels sur les espaces homogènes et leurs compactifications, Transformation Groups 11 (2006), 161-176.

[FSS] B. Fein, D. Saltman and M. Schacher, Crossed products over rational function fields, J. Algebra 156 (1993), 454-493.

[GMS] R.S. Garibaldi, A.A. Merkurjev and J.-P. Serre, Cohomological invariants in Galois cohomology, University Lecture Series, 28 (2003). American Mathematical Society, Providence.

[Gi0] P. Gille, Torseurs sur la droite affine et R-équivalence, Thèse de doctorat (1994), Université Paris-Sud, on author’s URL.

[Gi1] P. Gille, Torseurs sur la droite affine, Transform. Groups 7 (2002), 231-245, errata dans Transform. Groups 10 (2005), 267–269.

[Gi2] P. Gille, Unipotent subgroups of reductive groups of characteristic p > 0, Duke Math. J. 114 (2002), 307–328.

[Gi3] P. Gille, Serre’s conjecture II: a survey, Proceedings of the Hyderabad conference “Quadratic forms, linear algebraic groups, and cohomology”, Springer, 41–56.

[Gi4] P. Gille, Sur la classification des schémas en groupes semi-simples, see author’s URL.

[GM] G. van der Geer and B. Moonen, Abelian varieties, book in preparation, second author’s URL.

[GiQ] P. Gille and A. Quéguiner-Mathieu, Formules pour l’invariant de Rost, to appear 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 340 (2005) 725–729.

[GP2] P. Gille and A. Pianzola, Galois cohomology and forms of algebras over Laurent polynomial rings, Mathematische Annalen 338 (2007), 497-543.

[GP3] P. Gille and A. Pianzola, Isotriviality and étale cohomology of Laurent polynomial rings, J. Pure Appl. Algebra 212 (2008), 780–800.

[GR] P. Gille and Z. Reichstein, A lower bound on the essential dimension of a connected linear group, Commentarii Mathematici Helvetici 84 (2009), 189-212.

[GS] P. Gille and T. Szamuely, Lectures on the Merkurjev-Suslin’s theorem, Cambridge Studies in Advanced Mathematics 101 (2006), Cambridge University Press.

[Gi] J. Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften 179 (1971), Springer-Verlag.

[Gs] R. Griess, Elementary abelian p-subgroups of algebraic groups, Geom. Ded. 39 (1991), 253–305.

[Gr1] A. Grothendieck, Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire Bourbaki (1958-1960), Exposé No. 190, 29 p.

[Gr2] A. Grothendieck, Le groupe de Brauer. II, Théorie cohomologique (1968), Dix Exposés sur la Cohomologie des Schémas 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-Mills theories with any gauge group. The many Faces of the Superworld, pp. 185234. World Sci. Publishing, River Edge (2000) J. Algebra 13 (1969), 535–543.

[K2] M. A. Knus, Quadratic and hermitian forms over rings, Grundlehren der mat. Wissenschaften 294 (1991), Springer.

[L] E. Landvogt, Some functorial properties of the Bruhat-Tits building, J. reine angew. math. 518 (2000), 213–241.

[Mg] B. Margaux, Passage to the limit in non-abelian Čech cohomology, Journal of Lie Theory 17 (2007), 591–596.

[Mg2] B. Margaux, Vanishing of Hochschild Cohomology for Affine Group Schemes and Rigidity of Homomorphisms between Algebraic Groups, Doc. Math. 14 (2009), 653-672.

[Mt] G. D. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (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. 1 (1966), 287–354.

[1] 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 homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 5–8.

[Ne] K.H. Neeb, On the classification of rational quantum tori and the structure of their automorphism groups, Canad. Math. Bull. 51 (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, Academic Press (1993).

[P] P. Pansu, Superrigidité géométrique et applications harmoniques, Séminaires et congrès 18 (2008), Soc. Math. France, 375-422.

[P1] A. Pianzola, Affine Kac-Moody Lie algebras as torsors over the punctured line, Indagationes Mathematicae N.S. 13(2) (2002) 249-257.

[P2] A. Pianzola, Vanishing of H^{1} for Dedekind rings and applications to loop algebras, C. R. Acad. Sci. Paris, Ser. I 340 (2005), 633-638.

[P3] A. Pianzola, On automorphisms of semisimple Lie algebras Algebras, Groups, and Geometries 2 (1985) 95–116.

[Pr] G. Prasad, Galois-fixed points in the Bruhat-Tits building of a reductive group, Bull. Soc. Math. France 129, (2001), 169–174.

[Ra] A. Ramanathan, Deformations of principal bundles on the projective line, Invent. Math. 71 (1983), 165–191.

[RZ] L. Ribes and P. Zaleski, Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete vol. 40, Springer (2000).

[Ri] R. W. Richardson, On orbits of algebraic groups and Lie groups, Bull. Austral. Math. Soc. 25 (1982), 1–28.

[Ro] G. Rousseau, Immeubles des groupes réductifs sur les corps locaux, Thèse, Université de Paris-Sud (1977).

[RY1] Z. Reichstein and B. Youssin Essential Dimensions of Algebraic Groups and a Resolution Theorem for G-varieties , with an appendix by J. Kollár and E. Szabó, Canada Journal of Math. 52 (2000), 265-304.

[RY2] Z. Reichstein and B. Youssin A birational invariant for algebraic group actions, Pacific J. Math. 204 (2002), 223–246.

[Sa] J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math. (Crelle) 327 (1981), 12–80.

[Sc] W. Scharlau, Quadratic and hermitian forms, Grundlehren der math. Wiss. 270 (1985), Springer.

[Se] J.-P. Serre, Cohomologie Galoisienne, cinquième édition révisée et complétée, Lecture Notes in Math. 5, Springer-Verlag.

[Se2] J.-P. Serre, Cohomologie galoisienne: progrès et problèmes, Séminaire Bourbaki Exp. No. 783 (1994), Astérisque 227 (1995), 229257.

[Sp] T.-A. Springer, Linear algebraic groups, second edition (1998), Birkhaüser.

[SZ] A Steinmetz-Zikesch, Algèbres de Lie de dimension infinie et théorie de la descente, to appear in Mémoire de la Société Mathématique de France.

[SGA1] Séminaire de Géométrie algébrique de l’I.H.E.S., Revêtements étales et groupe fondamental, dirigé par A. Grothendieck, Lecture Notes in Math. 224. Springer (1971).

[SGA3] Séminaire de Géométrie algébrique de l’I.H.E.S., 1963-1964, schémas en groupes, dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Math. 151-153. Springer (1970).

[SGA4] Théorie des topos et cohomologie étale des schémas. Tome 2, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964, dirigé par M. Artin, A. Grothendieck et J. L. Verdier, Lecture Notes in Mathematics 270 (1972), Springer-Verlag.

[Ti] J.-P. Tignol, Algèbres indécomposables d’exposant premier, Adv. in Math. 65 (1987), 205 228.

[TiW] J.-P. Tignol and A. R. Wadsworth Totally ramified valuations on finitedimensional division algebras, Trans. Amer. Math. Soc. 302 (1987), 223–250.

[T1] J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous 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 conference on L- functions etc., Proc. Symp. Pure Math. 33 (1979), part 1, 29-69.

[T3] J. Tits, Twin buildings and groups of Kac-Moody type, Groups, combinatorics and geometry (Durham, 1990), 249–286, London Math. Soc. Lecture Note Ser. 165 (1992), Cambridge Univ. Press.

[V] A. Vistoli, Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, 1–104, Math. Surveys Monogr. 123 (2005), Amer. Math. Soc.

[W] C. Weibel, An introduction to homological algebra, Cambridge studies in a advanced mathematics 38 (1994), Cambridge University Press.