TORSORS OVER THE PUNCTURED AFFINE LINE

V. CHERNOUSOV, P. GILLE, AND A. PIANZOLA

Abstract. We classify G-torsors over the punctured affine line Spec (k[t^{±}]) where G is a reductive algebraic group defined over a field k of good characteristic. Our classification is in terms of the Galois cohomology of the complete field k((t)) with values in G. Keywords: Linear algebraic group, group scheme, torsor, punctured affine line, non-abelian cohomology. MSC 2000 11E72, 14L30, 14E20.

Contents

1. Introduction 2 2. Statement of the two Main Theorems 4 3. Preliminary results 5 3.1. Notation and conventions 5 3.2. Generalities on torsors 6 3.3. Twisting 7 3.4. Reducibility and isotropy 11 3.5. Patching group schemes 12 3.6. Weil restriction considerations 15 4. Torsors over the affine line 19 5. Torsors over the punctured affine line 22 5.1. Cohomological exponent: Good and bad primes 23 5.2. Existence of maximal tori 25 5.3. Reformulation of Theorem 2.4 26 5.4. Reduction of structure group to N_{G}(T) 28 5.5. Proof of surjectivity 30 5.6. Proof of Injectivity 31 6. Appendix: The absolutely reduced subscheme attached to a scheme 38 References 42

V. Chernousov was partially supported by the Canada Research Chairs Program and an NSERC research grant.

A. Pianzola wishes to thank NSERC and CONICET for their continuous support.

1. Introduction

Let G be a reductive group over a field k. The main objective of our work is to give a description of the isomorphism classes of G-torsors over the punctured affine line 𝔸_{k}^{×} = Spec(k[t^{±1}]) in terms of the Galois cohomology of the complete field k((t)) with values in G. The relevance of this characterization is that the latter cohomology set H^{1}(k((t)), G) can be computed using the methods developed by Bruhat and Tits [BT3].

Many interesting objects defined over a rational function field k(t) can usually be described, or at least be partially understood, through their residues. Examples of this type of behaviour are the famous exact sequences related to Milnor K-theory, Brauer groups and Witt groups. The exact sequence for Milnor’s K-groups [Mn, §2] is given by

0 → K_{n}^{M}(k) → K_{M}^{n}(k(t)) ⟶^{∂} ⊕_{p} K_{n−1}^{M}(k(p)) → 0

where p runs over the closed points of the affine line. Similarly, if k is perfect, we have Faddeev’s exact sequence [GS, §6.4]

0 → Br(k) → Br(k(t)) ⟶^{∂} ⊕_{p} H^{1}(k(p), ℚ/ℤ) → 0

and the exact sequence for Witt groups

0 → W (k) → W (k(t)) ⟶^{∂} ⊕_{p} W (k(p)) → 0

in odd characteristic [Mn, §5].

Brauer groups and Witt groups are closely related to projective linear groups and orthogonal groups respectively. One is thus lead to consider the possibility that there may exist analogues of the last two of the above exact sequences for G-torsors over k(t). To study this problem the (traditional) local-global approach leads us to look in detail at the natural map

(1.1)  a = ∏ a_{p} : H^{1}(k(t), G) ⟶ ∏_{p∈ℙ_{k}^{1}} H^{1}(k(t)_{p}, G)

where p runs through the set of closed points of the projective line ℙ_{k}^{1}, k(t)_{p} is the completion of k(t) with respect to the discrete valuation v_{p} associated to p, and a_{p} : H^{1}(k(t), G) → H^{1}(k(t)_{p}, G) is the natural restriction map. Note that, unlike the case of Brauer groups and Witt groups, the set of isomorphism classes of G-torsors over k(t) is not a group. It is therefore unrealistic to expect that a unique short exact sequence with H^{1}(k(t), G) as middle term could describe all G-torsors over k(t) . The only reasonable hope is to try to describe the fibers of the map a in more or less acceptable terms. This leads to infinitely many exact sequences instead of just one, as we now explain.

We say that an element x of H^{1}(k(t), G) is unramified at p if a_{p}(x) is in the image of the natural map H^{1}(Ô_{p}, G) → H^{1}(k(t)_{p}, G), where Ô_{p} is the ring of integers of k(t)_{p}. Otherwise we say that x is ramified at at p. In the latter case, and if G/k is quasi-split, using Bruhat-Tits theory we can associate to x an element x_{p} in H^{1}(k(p), H_{p}) where k(p) is the residue of k(t)_{p}, and H_{p} is a proper subgroup of the k(p)-group G_{k(p)} which depends on x. It seems natural to call x_{p} the residue of x at p and call the finite set S = { p_{1}, …, p_{n}} where x is ramified the ramification locus of x. By Bruhat-Tits theory H^{1}(k(t)_{p}, G) is the disjoint union of “anisotropic parts” of H^{1}(k(p), H_{p}), namely of those classes that arise from [ξ] ∈ H^{1}(k(p), H_{p}) for which the twisted group _{ξ}H_{p} is anisotropic. This indicates that we could get a satisfactory description of H^{1}(k(t), G) in terms of sets H^{1}(k(p), H_{p}) if we knew the fibers of the map

H^{1}(k(t), G)_{S} ⟶ ∏_{p∈S} H^{1}(k(t)_{p}, G)

where H^{1}(k(t), G)_{S} = a^{−1}(∏_{p∈S} H^{1}(k(t)_{p}, G) × ∏_{p∉S} H^{1}(Ô_{p}, G)).

Note that H^{1}(k(t), G)_{S} is the image of the natural mapping

(1.2)  H^{1}(U, G) → H^{1}(k(t), G)

where U = ℙ_{k}^{1} \{ p_{1}, …, p_{n} } (see [H1]). Thus, out of necessity, we are forced to study the image of this map and its fibers.

Assume that characteristic of k is good (see §5.1 for details). If S = ∅, then H^{1}(k(t), G)_{S} = H^{1}(k, G) and the fibers of (1.2) are well understood [G1, §I.2]. If S = { ∞ } then by a theorem of Raghunathan–Ramanathan (see below) the natural maps H^{1}(k, G) → H^{1}(𝔸_{k}^{1}, G) is bijective and (1.2) is injective. In our paper we consider the next case, namely when S consists of two points S = { 0, ∞ }. It turns out that the map (1.2) is again injective, and we can describe its image in terms of local G-torsors.

***

Our work is unequivocally motivated by that of Raghunathan and Ramanathan [RR], where the case of the affine line 𝔸_{k}^{1} = Spec(k[t]) is considered. Even though the overlap between the two works is at times evident (extending group schemes to ℙ_{k}^{1}, the use Weil restrictions to address representability questions...), there are also substantial differences, notably the use of some key results from the theory of multiplicative group schemes developed in [SGA3], as well as techniques pertaining to fundamental domains in buildings due to Soulé (for the affine line) and to Abramenko (in the case of the punctured line). These methods allow us to establish–by different means than those used in [RR]–the existence of maximal tori on certain reductive group schemes. This is one of the crucial points within the main proof. Since our methods work equally well both for the affine and the punctured affine line, we have decided to include a (short) section with a proof of Raghunathan and Ramanathan’s original result.

One should observe that the nature of G-torsors over 𝔸_{k}^{1} and 𝔸_{k}^{×} are quite different, mostly owing to the fact that, unlike 𝔸1 ; \overline{k}, the scheme 𝔸× ; \overline{k} is not simply connected (in the algebraic sense). In good characteristic for example, G-torsors over the affine line are always constant, but this need not be the case for G-torsors over the punctured line.

***

The main body of the paper is divided into four parts. We begin by giving the statement of the two main theorems. This is followed by a section on “Preliminary results” that provides the conventions, notation and terminology that are used throughout the paper, as well as some general results (some of independent interest) that are used in the proofs of the two main theorems. The proofs of the main theorems themselves are the content of the last two sections. An Appendix with a technical observation about absolutely reduced algebraic groups is included at the end.

2. Statement of the two Main Theorems

Let X be a scheme, and G a group scheme over X. For any scheme Y over X we denote the Y -group scheme G ×_{X} Y by G_{Y} . Recall that a torsor over X under G 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. The set of isomorphisms classes of X-torsors under G is denoted by H^{1}(X, G).

Let G be a linear algebraic over a field k and X a k-scheme. The structural morphism X → Spec (k) yields a natural map

η : H^{1}(k, G) ⟶ H^{1}(X, G) := H^{1}(X, G_{X}).

Let E be a torsor over X under G_{X}. We say that E is constant if its isomorphism class belong to the image of the map η, and that E is geometrically separably trivial if E becomes trivial after the base change X ×_{k} k_{s} → X.

2.1. Theorem. [Raghunathan-Ramanathan] Let G be a linear algebraic kgroup whose connected component of the identity is reductive.^{1} Every torsor over the affine line 𝔸_{k}^{1} under G which is geometrically separably trivial is constant.

2.2. Remark. According to the bijection (3.3) described in Remark 3.2, the isomorphism classes of geometrically separably trivial 𝔸_{k}^{1}-torsors under G are classified by H^{1}(Gal(k_{s}/k), G(𝔸_{k_{s}}^{1})). The natural map η : H^{1}(k, G) ⟶ H^{1}(𝔸_{k}^{1}, G) corresponds to the natural map

(2.3)  H^{1}(Gal(k_{s}/k), G(k_{s})) ⟶ H^{1}(Gal(k_{s}/k), G(𝔸_{k_{s}}^{1}))

arising from the inclusion G(k_{s}) ⊂ G(𝔸_{k_{s}}^{1}). The map (2.3) is injective because of the existence of rational points on 𝔸_{k}^{1}. Theorem 2.1 is thus equivalent to (2.3) being bijective. In good characteristic, it is known that every G-torsor over 𝔸_{k}^{1} is geometrically separably trivial, hence constant. In bad characteristic, this is in general not true anymore, even for a semisimple simply connected group (see [G3, §2.4] for details).

1In [RR] the group G is assumed to be connected and reductive. The generalization we give here is not difficult to obtain from the reductive case. See §4.3 below.

The second part of our work gives a description of torsors over the punctured line 𝔸_{k}^{×} = Spec(k[t^{±1}]). It is not true that in this case geometrically separably trivial torsors are constant, even when G is semisimple. The correct parametrization is obtained by looking at the base change corresponding to the completion of the generic fiber. The inclusion k[t^{±1}] ⊂ k((t)) yields a natural map H^{1}(𝔸_{k}^{×}, G) → H^{1}(k((t)), G). Our result shows that, under a certain assumption on the characteristic of the base field, this map is bijective.

2.4. Theorem. Let G be a (connected) reductive algebraic group over k. Assume that the characteristic of k is good for G (see 5.1). Then the natural map H^{1}(𝔸_{k}^{×}, G) → H^{1}(k((t)), G) is bijective. In other words, the set of isomorphism classes of 𝔸_{k}^{×}-torsors under G is parametrized by the usual Galois cohomology of G over the complete field k((t)).^{2}

3. Preliminary results

This section contains the conventions, notation and terminology that are used throughout the paper. It also contains results that are common to the proofs of the main theorems.

3.1. Notation and conventions. Throughout k denotes a field, \overline{k} an algebraic closure of k, and k_{s} the separable closure of k in \overline{k}. Given a scheme X over k and a field extension k′/k we set

X_{k′} : = X ×_{k} k′.

For convenience we denote X_{k_{s}} simply by X_{s}.

Most of our work is related to group schemes over a given base scheme X. For convenience we will sometime refer to these simply as X-groups. If X = Spec(R) we use the terminology X-group and R-group indistinctively. We recall that given an X-group G and a scheme Y over X the Y -group G×_{X} Y is denoted by G_{Y} . That “Y is a scheme over X” is at times abbreviated by simply writing Y/X.

By a reductive X-group we will understand a reductive group scheme over X in the sense of [SGA3]. Accordingly, a reductive k-group is a connected reductive group defined over k in the sense of Borel [Bor]. If G is a reductive X-group, then the concept of maximal tori, parabolic subgroup, Levi subgroup... of G is again the one given by [SGA3].

Let F be a field extension of k. The additive and multiplicative groups over F will be denoted by G_{a,F} and G_{m,F} respectively, or simply by G_{a} and G_{m} when F = k. The separable closure of F, which will always be taken in some algebraic closure \overline{F} of F that contains \overline{k}, will be denoted by F_{sep}.

2The set H^{1}(k((t)), G) is well understood from the work of Bruhat-Tits [BT3].

Thus k_{sep} = k_{s}, but in general one should not confuse F_{sep} with F_{s} (which by definition equals F ⊗_{k} k_{s}, and plays no role in our work).

Given a field extension F of k and a torus T over F, there is a natural action of the Galois group Gal(F_{sep}/F) on the (abstract) group Hom_{F_{sep}−grp}(G_{m,F_{sep}}, T_{F_{sep}}). We call the resulting Galois module the group of cocharacters of T, and denote it by X(T)_{∗}.

3.2. Generalities on torsors. Let X be a scheme, and G a group scheme over X. As it is customary, for any scheme Y over X we denote by p_{i}, i = 1, 2, the corresponding projection Y ×_{X} Y → Y on the i-th component and by p_{ij}, i, j = 1, 2, 3, the projection Y ×_{X} Y ×_{X} Y → Y ×_{X} Y on the ij-th component. These projections naturally induce group homomorphisms

G(Y) → G(Y ×_{X} Y) and G(Y ×_{X} Y) → G(Y ×_{X} Y ×_{X} Y)

which we still denote by p_{i} and p_{ij} (instead of the more usual p_{i}^{∗} and p_{ij}^{∗} notation).

Assume now that Y/X is an fppf cover of X, that is Y → X is a scheme morphism which is faithfully flat and locally of finite presentation ([SGA3], IV.6.4).^{3} For such a covering Y → X, we define the corresponding set of cocycles

Ž^{1}(Y/X, G) := {g ∈ G(Y ×_{X} Y) | p_{23}(g)p_{12}(g) = p_{13}(g)}

and non-abelian cohomology

Ȟ^{1}(Y/X, G) := Ž^{1}(Y/X, G)/G(Y),

where G(Y) acts on Ž^{1}(Y/X, G) by g·z = p_{2}(g) z p_{1}(g)^{−1} (see [H1], §1.3 and [M] for details). For convenience we will at times use the notation Ž^{1}(X, G) to denote the totality of cocycles Ž^{1}(Y/X, G) with Y as above variable (within a given range, which would always be clear from the context, so as to avoid set-theoretical problems). We now define

Ȟ^{1}(X, G) := \underrightarrow{lim}_{Y}Ȟ^{1}(Y/X, G) ,

where the limit is taken over all (equivalence classes of) fppf covers Y → X.

3.1. Remark. One defines in an exact analogous fashion Ȟ^{1}(X, G) whenever G is a sheaf of groups for the fppf topology on X.

Recall that the set of isomorphisms classes of X-torsors under G is denoted by H^{1}(X, G). Thus H^{1}(X, G) is a pointed set; its distinguished class, which we denote by 1, is the class of the trivial torsor, namely the scheme G acting on itself by right translation. The subset of H^{1}(X, G) corresponding to torsors which are trivialized by a given (arbitrary) base change X′ → X is denoted by H^{1}(X′/X, G). If X′ → X is an fppf cover, H^{1}(X′/X, G) can be computed by means of cocycles (just as in Galois cohomology), so that H^{1}(X′/X, G) can be identified with a subset of Ȟ^{1}(X′/X, G). If G is affine and locally of finite presentation over X then H^{1}(X′/X, G) = Ȟ^{1}(X′/X, G), and the natural map

3An fppf cover is thus a covering morphism (morphism couvrant) for the fppf topology in the sense of [SGA3], but not conversely.

H^{1}(X, G) → Ȟ^{1}(X, G)

is bijective. This will be the situation that we will consider in our paper, and we will indistinctively think of (the isomorphism class) of a torsor as an element of H^{1} or the corresponding Ȟ^{1}. Along similar lines we write Z^{1} instead of Ž^{1}.

3.2. Remark. Assume that G is an algebraic group over k. Following standard practice, we will denote in what follows H^{1}(X, G_{X}) simply by H^{1}(X, G). Because G is of finite type, any X-torsor under G that becomes trivial over X_{s} already becomes trivial over X_{k′} for some finite Galois extension k′ ⊂ k_{s} of k (the extension k′ depends of course on the given torsor). As a consequence, the natural map

\underrightarrow{lim}_{k′/k Galois} H^{1}(X_{k′}/X, G) → H^{1}(X_{s}/X, G)

is bijective. Since X_{k′} is a Galois extension of X whose Galois group is naturally isomorphic to Gal(k′/k), we have H^{1}(X_{k′}/X, G) ≃ H^{1}(Gal(k′/k), G(X_{k′})). We thus have a bijection

(3.3)  H^{1}(X_{s}/X, G) ≃ H^{1}(Gal(k_{s}/k), G(X_{s}))),

where the H^{1} on the right denotes the “usual” Galois cohomology of the profinite group Gal(k_{s}/k) acting (continuously) on the (discrete) module G(X_{s}). The natural map η : H^{1}(k, G) ⟶ H^{1}(X, G) corresponds to the composition of maps in the sequence

H^{1}(Gal(k_{s}/k), G(k_{s})) → H^{1}(Gal(k_{s}/k), G(X_{s})) ≃ H^{1}(X_{s}/X, G) ⊂ H^{1}(X, G),

where the first map arises from the inclusion G(k_{s}) ⊂ G(X_{s}) obtained from the k_{s}-scheme structure of X_{s}.

3.3. Twisting. Throughout this section X will denote a k-scheme, and G a group scheme over X that we assume is affine and locally of finite presentation over X. Let Aut(G) be the X-group functor of automorphisms of G: For each Y/X

Aut(G)(Y) = Aut(G_{Y}).

The functor Aut(G) is always a sheaf of groups for the fppf topology on X, but it need not in general be representable (i.e. a group scheme).

Let Y → X be an fppf cover. To a cocycle z ∈ Ž^{1}(Y/X, Aut(G)) ⊂ Aut(G_{Y ×_{X}Y})) one can associate a twisted group scheme _{z}G over X whose functor of points is given by

(3.4)  _{z}G(S) = {x ∈ G(Y ×_{X} S) | z^{S}(p_{1}^{S}(x)) = p_{2}^{S}(x)}

for any X-scheme S. The notation in (3.4) is as follows. The morphisms p_{i}^{S} : Y ×_{X} Y ×_{X} S → Y ×_{X} S are given by p_{i}^{S} = p_{i} × id_{S}. Given x ∈ G(Y ×_{X} S) = Hom_{X}(Y ×_{X} S, G) we denote by p_{i}^{S}(x) the composition x∘p_{i}^{S}. The p_{i}^{S}(x) are thus elements of the abstract group G(Y ×_{X} Y ×_{X} S). Finally, since z is an automorphism of the Y ×_{X} Y -group G_{Y ×_{X}Y}, it induces an automorphism z^{S} of the abstract group G(Y ×_{X} Y ×_{X} S).

We can reinterpret this definition by saying that the sequence

(3.5)  \xymatrix{_{z}G(S) ⊂ G(Y ×_{X} S) \ar[r]^{z^{S}∘p_{1}^{S}} \ar[r]_{p_{2}^{S}} & G(Y ×_{X} Y ×_{X} S)}

is exact. If z = 1 then the twisted group _{z}G is isomorphic to G, and the sequence

(3.6)  \xymatrix{G(S) ⊂ G(Y ×_{X} S) \ar[r]^{p_{1}^{S}} \ar[r]_{p_{2}^{S}} & G(Y ×_{X} Y ×_{X} S)}

is exact. This allows us to identify G(S) with a subgroup G(Y ×_{X} S); an identification that we will henceforth use, whenever convenient, without further reference.

Since z ∈ Ž^{1}(Y/X, Aut(G)) the group schemes _{z}G_{Y} and G_{Y} are isomorphic. This isomorphism can be made explicit at the level of functor of points as we now explain for future reference.

If i, j, k ∈ {1, 2, 3} are different integers and j < k we set

z_{i} = p_{jk}(z) ∈ Aut(G_{Y ×_{X}Y ×_{X}Y}).

Given any scheme S over Y our automorphism z induces naturally an automorphism z^{S/Y} of the (abstract) group G(Y ×_{X} Y ×_{Y} S). Similarly the z_{i} induce automorphisms z_{i}^{S/Y} of the (abstract) group G(Y ×_{X} Y ×_{X} Y ×_{Y} S). By taking the composite map S → Y → X we may also view S as a scheme over X. The canonical isomorphism Y ×_{Y} S ≃ S yields canonical identifications

\begin{matrix}G(S) ≃ G(Y ×_{Y} S) \\ G(Y ×_{X} S) ≃ G(Y ×_{X} Y ×_{Y} S), \\ G(Y ×_{X} Y ×_{X} S) ≃ G(Y ×_{X} Y ×_{X} Y ×_{Y} S). \end{matrix} ; (3.7)

For i = 1, 2 we denote p_{i3} × id_{S} : Y ×_{X} Y ×_{X} Y ×_{Y} S → Y ×_{X} Y ×_{Y} S by p_{i3}^{S/Y} . Consider the diagram (3.8)

\xymatrix{_{z}G(S) \ar@{^{(}->}[r] & G(Y ×_{X} S) \ar[r]^{≃} & G(Y ×_{X} Y ×_{Y} S) \ar[r]^{z_{3}^{S/Y} ∘p_{13}^{S/Y}} \ar[r]_{p_{23}^{S/Y}} \ar[d]_{z^{S/Y}} & G(Y ×_{X} Y ×_{X} Y ×_{Y} S) \ar[d]^{z_{1}^{S/Y}} \\ G(S) \ar@{^{(}->}[r] & G(Y ×_{X} S) \ar[r]^{≃} & G(Y ×_{X} Y ×_{Y} S) \ar[r]^{p_{13}^{S/Y}} \ar[r]_{p_{23}^{S/Y}} & G(Y ×_{X} Y ×_{X} Y ×_{Y} S)}

The top row is the equalizer (3.5) combined with the identifications of (3.7), and the bottom row arises in the same fashion for the trivial cocycle as described in (3.6). Since z is a cocycle we have z_{2} = z_{1}z_{3}. This yields z_{2}^{S/Y} = z_{1}^{S/Y} z_{3}^{S/Y} which shows that the outermost right square is commutative. That the innermost right square is commutative is easy. This shows that z^{S/Y} induces a group isomorphism_{z}G(S) → G(S). This isomorphism is functorial on Y -schemes S and thus defines a Y -group scheme isomorphism which we still denote by z:

(3.9)  z : _{z}G_{Y} → G_{Y}

Finally, if λ : _{z}G(X) → _{z}G(Y) denotes the canonical inclusion corresponding to our fppf-cover Y/X, one can easily checks that the diagram

\xymatrix{_{z}G(X) \ar@{^{(}->}[r] \ar[d]_{λ} & G(Y) \ar[r]^{z∘p_{1}} \ar[r]_{p_{2}} \ar[d]_{p_{1}} & G(Y ×_{X} Y) \ar[d]^{p_{12}} \\ _{z}G(Y) \ar@{-->}[d] \ar@{^{(}->}[r] & G(Y ×_{X} Y) \ar[r]^{z_{3}∘p_{13}} \ar[r]_{p_{23}} \ar[d]_{z} & G(Y ×_{X} Y ×_{X} Y) \ar[d]^{z_{1}} \\ G(Y) \ar@{^{(}->}[r] & G(Y ×_{X} Y) \ar[r]^{p_{13}} \ar[r]_{p_{23}} & G(Y ×_{X} Y ×_{X} Y)}

commutes. Here the first row is the equalizer (3.6) for S = X, and the last two rows constitute (3.8) in the case Y = S.

3.10. Remark. Let Y → X be an fppf cover, and let z ∈ Ž^{1}(Y/X, G) ⊂ G(Y ×_{X} Y) be a cocycle. The element z defines an (inner) automorphism int(z) of the group G_{Y ×_{X}Y}. It is clear that int(z) ∈ Ž^{1}(Y/X, Aut(G)). We denote the corresponding twisted group _{int(z)}G by _{z}G.

We now turn our attention to maximal tori. For convenience we will denote the twisted group _{z}G by G′, and let T and T′ denote the scheme of maximal tori of G and G′ respectively [SGA3, XI.4]. We recall that T and T′ are affine schemes of finite type over X (ibid.). The cocycle z acts naturally on the scheme T_{Y ×Y} of maximal tori of G_{Y ×Y} . We denote (by a slight abuse of notation) the resulting automorphism of T_{Y ×Y} also by z. In this way we may view z as a cocycle in Ž^{1}(Y/X, Aut(T )). By [SGA3]Exp.

XXIV prop.4.2.1 T′ coincides with the twist of T by z, i.e., we may assume that the functor of points of T′ is given by

(3.11)  T′(S) = { T′′ ∈ T (Y × S) | z(p_{1}^{S}(T′′)) = p_{2}^{S}(T′′) }.

Arguing as above we get the commutative diagram

(3.12)  \xymatrix{T′(X) \ar@{^{(}->}[r] \ar[d]_{λ} & T (Y) \ar[r]^{z∘p_{1}} \ar[r]_{p_{2}} \ar[d]_{p_{1}} & T (Y ×_{X} Y) \ar[d]^{p_{12}} \\ T′(Y) \ar@{-->}[d] \ar@{^{(}->}[r] & T (Y ×_{X} Y) \ar[r]^{z_{3}∘p_{13}} \ar[r]_{p_{23}} \ar[d]_{z} & T (Y ×_{X} Y ×_{X} Y) \ar[d]^{z_{1}} \\ T (Y) \ar@{^{(}->}[r] & T (Y ×_{X} Y) \ar[r]^{p_{13}} \ar[r]_{p_{23}} & T (Y ×_{X} Y ×_{X} Y)}

Just as in (3.9), the automorphism z induces a group isomorphism (also denoted by z)

(3.13)  z : T′_{Y} → T_{Y}

The following result is inevitable.

3.14. Lemma. Let T′ be a maximal X-torus of G′ (i.e., T′ ∈ T′(X)), and let T′′ be the corresponding maximal torus of G_{Y} given by (3.11). Then z(T′_{Y}) = T′′.

Proof. We have λ(T′) = T′_{Y} , so we need to show that z(λ(T′)) = T′′. Let T′′′ ∈ T (Y) be the image of λ(T′) under z given by (3.12). The inclusion T (Y) ⊂ T (Y ×_{X} Y) in the bottom row of (3.12) is given by the projection p_{2}. Thus z(p_{1}(T′′)) = p_{2}(T′′′). By the top row of (3.12) we have z(p_{1}(T′′)) = p_{2}(T′′). It follows that T′′ = T′′′ as desired □

3.15. Lemma. Let G be a reductive group scheme over X, and T a maximal torus of G (assumed to exist). Let Y/X be an fppf cover of X, z ∈ Ž^{1}(Y/X, G) a cocycle. If the twisted X-group _{z}G admits a maximal torus T′ such that T′_{Y} and T_{Y} are conjugate by an element of G(Y) (where _{z}G_{Y} is identified with G_{Y} as described in (3.9) above), then

[z] ∈ Im [H^{1}(Y/X, N_{G}(T)) ⟶ H^{1}(X, G)].

Proof. To begin with we recall that N_{G}(T) is a closed smooth subgroup of G ([SGA3] Exp XI.5 and XIX.1) As before we denote the twisted group _{z}G by G′, and let T and T′ denote the scheme of maximal tori of G and G′ respectively.

According to 3.11 our torus T′ of G′ corresponds to a torus T′′ of G_{Y}. By Lemma 3.14 T′_{Y} coincides with T′′ via our identification G′_{Y} ≃ G_{Y} . Hence T′′ = g^{−1}T g for some g ∈ G(Y). Let z′ = p_{2}(g)zp_{1}(g)^{−1}. Then z′ ∈ G(Y ×_{X} Y) is a cocycle equivalent to z. We have

zp_{1}(T′′)z^{−1} = p_{2}(T′′)

because T′ ∈ T′(X). This implies that

zp_{1}(g^{−1}T_{Y} g)z^{−1} = p_{2}(g^{−1}T_{Y} g) or
zp_{1}(g)^{−1}p_{1}(T_{Y})p_{1}(g)z^{−1} = p_{2}(g)^{−1}p_{2}(T_{Y})p_{2}(g), or
z′p_{1}(T_{Y})(z′)^{−1} = p_{2}(T_{Y}).

Because T is defined over X this last yields

z′T_{Y ×_{X}Y} (z′)^{−1} = T_{Y ×_{X}Y}

Thus z′ ∈ N_{G}(T)(Y ×_{X} Y). □

3.4. Reducibility and isotropy. Let G be a reductive group scheme over a base scheme X. We recall two fundamental notions about G; one global (reducibility), and the other local (isotropy).

We say that G is reducible if G admits a proper parabolic subgroup, and irreducible otherwise. We denote by Par(G) the X-scheme of parabolic subgroup schemes of G [SGA3, XXVI.3.5]. This scheme is smooth and projective over X. Since by definition G is a parabolic subgroup of G, to say that G is reducible is to say that Par(G)(X) ≠ {G}.

3.16. Remark. If X is connected, to each parabolic subgroup P of G corresponds a “type” t = t(P) which is a subset of the Dynkin diagram of G. Given a type t the scheme Par_{t}(G) of parabolic subgroups of G of type t is also smooth and proper over X.

We know that if G contains a non-central split subtorus, then G is reducible (loc. cit., 6.3). As we will presently see, the converse is true locally.

Assume that the base scheme X is semilocal and connected. Following [SGA3, XXVI.6.13] we say that G is isotropic if (as a reductive X-group) G admits a non-trivial split subtorus. Otherwise we say that G is anisotropic. Recall that the radical torus rad(G) is the unique maximal torus of the center of G [SGA3, XXII.4.3.6].

3.17. Proposition. Assume that X is semilocal connected.

(1) [SGA3, XXVI.6.12] The following are equivalent:

(a) G is reducible.

(b) G admits a non-central split subtorus.

(2) [SGA3, XXVI.6.14] The following are equivalent:

(a) G is isotropic.

(b) G is reducible or rad(G) is isotropic.

3.5. Patching group schemes. A technical tool used by Raghunathan and Ramanathan in their proof falls within the content of the following useful result.

3.18. Lemma. Let X be an algebraic curve over k (i.e. a one-dimensional separated irreducible algebraic scheme over k), and let K denote its function field (i.e. the local ring of its generic point). Let x ∈ X be a closed point, and assume that the local ring D = O_{X,x} is a discrete valuation ring (for example, if X is smooth). Let D̂ and K̂ denote the corresponding completions.

Assume we are given a triple (G, F, τ) consisting of:

(i) An affine group scheme G over U = X − {x} of finite type.

(ii) An affine and finitely presented group scheme F over D̂.

(iii) A K̂-group scheme isomorphism τ : G ×_{U} K̂ ≃ F ×D̂ K̂.

Then there exists a group scheme H, affine and of finite type over X, such that H ×_{X} U = G and H ×_{X} D̂ = F. Furthermore, if G and F are smooth, then so is H.

Proof. By [BLR] §6.2 proposition D.4(b) applied to our isomorphism τ : (G ×_{U} K) ×_{K} K̂ ≃ F ×D̂ K̂ there exists a group scheme F_{D} over D together with isomorphisms

(a) F_{D} ×_{D} D̂ ≃ F and

(b) F_{D} ×_{D} K ≃ G ×_{U} K. which are compatible with τ. Note that since D → D̂ is faithfully flat the descended group F_{D} is finitely presented.

Fix an affine open neighborhood Spec(S) of x. Since D = S_{x} and F_{D} is finitely presented, there exists f ∈ S with f(x) ≠ 0 [i.e. x ∈ Spec(S_{f})] and a finitely presented S_{f}-group F_{f} such that F_{f} ×_{S_{f}} D ≃ F_{D}.

Choose g ∈ S such that Spec(S_{fg}) ⊂ Spec(S)∩U. If we set G_{fg} = G×_{U} S_{fg} and F_{fg} = F_{f} ×_{S_{f}} S_{fg} then (b) above yields an isomorphism of K-groups

G_{fg} ×_{S_{fg}} K ≃ F_{fg} ×_{S_{fg}} K.

Because both G_{fg} and F_{fg} are of finite type and K is the field of quotients of S_{fg}, there exists some h ∈ S such that

G ×_{U} S_{fgh} ≃ F_{f} ×_{S_{f}} S_{fgh}.

Let V = Spec(S_{fgh}) ∪ {x} ⊂ Spec(S_{f}). Then the U-group G and the V - group F_{f} ×_{S_{f}} V are isomorphic on the overlap U ∩ V = Spec(S_{fgh}). We can thus glue these two groups to obtain a group H over X such that H_{U} ≃ G and HD̂ ≃ F.

Let Y = U ⨆ Spec D̂. The natural map Y → X is faithfully flat and quasicompact. The assertions about H being affine and of finite type, and smooth if G and F are smooth, now follow from descent ([EGA4] II Prop. 2.7.1 and IV Cor. 17.7.3(ii)). □

The group H constructed above is said to correspond to the given triple (G, F, τ). The following two lemmas easily follow from the above argument and properties of descent in the faithfully flat quasicompact topology.

3.19. Lemma. Assume we are given two triples (G, F, τ) and (G′, F′, τ′) as in the previous Lemma. If ϕ_{1} : G′ → G and ϕ_{2} : F′ → F are isomorphisms over U and D̂ respectively such that the diagram

G′ ×_{U} K̂ ⟶^{τ′} F′ ×D̂ K̂
ϕ_{1}×id↓  ^{ϕ_{2}×id}↓
G ×_{U} K̂ ⟶^{τ} F ×D̂ K̂

commutes, then the group schemes H and H′ over X corresponding to (G, F, τ) and (G′, F′, τ′) are isomorphic.

3.20. Lemma. Assume that we are given a triple (G, F, τ) as in Lemma 3.18. If G′ < G and F′ < F are closed subgroup schemes such that τ(G′×_{U} K̂) = F′×D̂ K̂, then the group scheme H′ corresponding to the triple (G′, F′, τ_{|_{G′×_{U} K̂}}) admits a natural closed immersion into the group scheme H corresponding to (G, F, τ).

3.21. Example. The following example, which gives a procedure for extending certain group schemes over 𝔸_{k}^{1} to the projective line ℙ_{k}^{1}, is of fundamental importance to the proof of the main theorems.

We maintain the general notation of the previous Lemmata, but look at the particular case when X is the projective line ℙ_{k}^{1} over k, U = 𝔸_{k}^{1} = Spec(k[t]) is the affine line over k, and {x} = X \ U is the point at infinity of X. We have D = O_{P_{k}^{1},x} = Spec(k[\frac{1}{t}]_{( \frac{1}{t})}), K = k(t) = k(\frac{1}{t}) where t is our

coordinate function on U, D̂ = k[[\frac{1}{t}]] and K̂ = k((\frac{1}{t})).

(a) Let G = Spec(k[t][G]) be a semisimple simply connected group scheme over U. Consider the generic fiber G_{K} of G and pass to the completion GK̂. Let p be a point of the building B = B_{GK̂} corresponding to GK̂. Recall that G(K̂) acts on B, and that the stabilizer Stab_{G(K̂)}(p) of p in G(K̂) is the

parahoric subgroup of G(K̂) associated to p. This parahoric subgroup, in turn, gives rise to a smooth group scheme F over D̂ whose generic fiber is canonically isomorphic to GK̂ [BT2, §5.1.8]:

(3.22)  τ : G ×_{k[t]} K̂ → F ×D̂ K̂.

More precisely, Bruhat-Tits theory (ibid.) shows that F = Spec(D̂[F]) where

(3.23)  D̂[F] = {f ∈ K̂[G] | f(x) ∈ D̂ for all x ∈ Stab_{G(K̂)}(p)}.

Moreover D̂[F] ⊗D̂ K̂ = K̂[G], and this gives rise to the isomorphism τ of (3.22). The triple (G, F, τ) corresponds then to the diagram

(3.24)  \xymatrix{ & K̂[G] \\ k[t][G] \ar@{^{(}->}[ur] &  & D̂[F] \ar@{^{(}->}[ul]}

Of course, by Lemma 3.18, this data gives rise to a group scheme H = H(p) which is affine and smooth over ℙ_{k}^{1}.

(b) For future use we note that the above construction of F (and hence of H) is compatible with any finite Galois field extension k′/k [BT2, §5]. More precisely, if K′ = k′(\frac{1}{t}) then the extension K̂′/K̂ is unramified, so it gives rise to a canonical embedding B_{GK̂} ↪ B_{GK̂′} of the buildings corresponding to GK̂ and GK̂′ respectively. Let F′ denote the group scheme over D̂′ = k′[[\frac{1}{t}]] associated to p viewed now as an element p′ of B_{GK̂′}. Then we have F ×D̂ D̂′ ≃ F′ via D̂′[F′] = D̂[F] ⊗D̂ D̂′. It follows that H(p′) ≃ H(p) ×_{ℙ_{k}^{1}} ℙ_{k′}^{1} ≃ H(p) ×_{k} k′.

(c) Assume additionally that G is of the form G = G_{0} ×_{k} 𝔸_{k}^{1} where G_{0} is a k-split group and that p is contained in the apartment A ⊂ B corresponding to a maximal k-split torus T_{0} ⊂ G_{0}. Then T_{0}(k) acts trivially on p and this gives rise to the canonical closed embedding

T_{0,D̂} = T_{0} ×_{k} D̂ ↪ F

of D̂-groups. Since T_{0} ⊂ G_{0} we also have the canonical closed embedding

T_{0,k[t]} = T_{0} ×_{k} k[t] ↪ G = G_{0} ×_{k} k[t].

Both of the above embeddings are compatible with the isomorphism τ of (3.22), i.e.

τ(T_{0,k[t]} ×_{k[t]} K̂) = T_{0,D̂} ×D̂ K̂ ⊂ F ×D̂ K̂.

If we denote by τ′ the restriction of τ to T_{0,k[t]} ×_{k[t]} K̂ ≤ G ×_{k[t]} K̂, then by Lemma 3.20, the triple (T_{0,k[t]}, T_{0,D̂}, τ′) corresponds to a group subscheme T ⊂ H over ℙ_{k}^{1}. Moreover, from the diagram

(3.25)  \xymatrix{ & K̂[T_{0,k[t]}] = K̂[T_{0}] \\ k[t][T_{0,k[t]}] \ar@{^{(}->}[ur] &  & D̂[T_{0,F}] \ar@{^{(}->}[ul] \\  & k[T_{0}] \ar@{^{(}->}[ur] \ar@{^{(}->}[ul]}

it follows that T ≃ T_{0} ×_{k} ℙ_{k}^{1}. In particular, T is split.

(d) Let g ∈ G(k[t]) ⊂ G(K̂). Then g yields an (inner) automorphism int(g) ∈ Aut(GK̂). Let g^{⋆} : K̂[G] → K̂[G] be the comorphism corresponding to int(g). Set p′ = g(p) and let F′ be the group scheme over D̂ corresponding to p′. Then we clearly have

g^{⋆} : D̂[F] → D̂[F′] = g^{⋆}(D̂[F]).

Thus g^{⋆} gives rise to an isomorphism ϕ : F′ → F. Set

ϕ_{1} = int(g) : G = G′ → G,

and τ′ = (ϕ × idK̂)^{−1}∘τ∘(int(g)×idK̂). By Lemma 3.18, the triple (G, F′, τ′) gives rise to a group scheme H′ over ℙ_{k}^{1}, and by Lemma 3.19 we have H′ ≃ H. Tracing through all these constructions and identifications we see that τ′ corresponds to the diagram

\xymatrix{ & K̂[G] \\ k[t][G] \ar@{^{(}->}[ur] &  & D̂[F′] = g^{⋆}(D̂[F]) \ar@{^{(}->}[ul]}

3.6. Weil restriction considerations. Throughout this section X will denote a projective variety over k, and H a group scheme which is affine and of finite type over X.

For a given k-scheme S we denote by h_{S} the corresponding functor of points:

h_{S} : Sch/k ⟶ Sets
Y ⟶ Hom_{k}(Y, S).

Recall Grothendieck’s definition of the Weil restriction of the X-scheme H to k. ^{4} [Gr, exp. 221, Remarque 3.9.c]. This is the functor

∏_{X/k} H : Sch/k ⟶ Sets

defined by

Y ⟶ Hom_{X}(X ×_{k} Y, H)

According to loc. cit. (see also [H2, page 121]), this functor is representable by an affine k-scheme of finite type, say S/k, which we henceforth assume is fixed in our discussion. Fix an isomorphism

(3.26)  α : h_{S} → ∏_{X/k} H

By definition our map α is thus a family of bijections

(3.27)  Hom_{k}(Y, S) ⟶^{α_{Y}} Hom_{X}(Y ×_{k} X, H)

which is functorial on k-schemes Y. The identity map id_{S} : S → S defines a morphism

(3.28)  ev_{α} : S ×_{k} X ⟶ H

4Called the functor of global sections in [RR].

called the evaluation map. By Yoneda considerations the bijections of (3.27) are then given by

(3.29)  α_{Y} : f → ev_{α} ∘ (f × id_{X}).

The algebraic k-scheme S can be made into a k-group by transport of structure via α. More precisely, the group structure on S(Y) is as follows: Given a, b ∈ S(Y) then ab = c where c satisfies

α_{Y} (c) = α_{Y} (a) α_{Y} (b)

where the right-hand side is the multiplication on the group H(Y ×_{k} X). We denote this group by S^{α}, or simply by S if no confusion is possible. One checks that ev_{α}, when viewed as a map from S^{α} ×_{k} X to H is in fact a morphism of group schemes over X.

3.30. Remark. Assume that the X-group H is obtained by base change from an affine k-group G_{0}, i.e., that H = G_{0} ×_{k} X. Then we may take S to be G_{0}. Indeed, since X/k is projective the canonical map O(Y) → O(X ×_{k} Y) is an isomorphism. Thus

Hom_{k}(Y, G_{0}) ≃ Hom_{k}(k[G_{0}], O(Y)) ≃ Hom_{k}(k[G_{0}], O(Y ×_{k} X)) ≃
Hom_{k}(Y ×_{k} X, G_{0}) ≃ Hom_{X}(Y ×_{k} X, G_{0} ×_{k} X).

Note that the resulting isomorphism α : h_{G_{0}} → ∏_{X/k}(G_{0} ×_{k} X) is such that ev_{α} = id_{G_{0}×_{k}X}.

3.31. Remark. Let k′ be a field extension of k and set X′ = X ×_{k} k′ and H′ = H ×_{k} k′. Then the Weyl restriction ∏_{X′/k′} H′ is represented by S′ = S ×_{k} k′. A natural isomorphism

α′ : h_{S′} ⟶ ∏_{X′/k′} H′

is obtained from α via the following functorial identifications on schemes Y over k′:

h_{S′}(Y) = Hom_{k′}(Y, S′) ≃ Hom_{k}(Y, S) = h_{S}(Y)

≃^{α_{Y}} Hom_{X}(Y ×_{k} X, H)

≃ Hom_{k′×_{k}X}(Y ×_{k} X, k′ ×_{k} H)

≃ Hom_{k′×_{k}X}(Y ×_{k′} (k′ ×_{k} X), k′ ×_{k} H)

= Hom_{X′}(Y ×_{k′} X′, H′).

It follows from this explicit description that the evaluation map commutes with base field extension, namely that

ev_{α′} = ev_{α} × id_{k′}

or, more precisely, that the diagram

\xymatrix{(S ×_{k} X) ×_{k} k′ \ar[r]^{ev_{α}×id_{k′}} \ar[d]_{≃} & H ×_{k} k′ \ar[d]^{id} \\ S′ ×_{k′} X′ \ar[r]^{ev_{α′}} & H ×_{k} k′}

commutes.

3.32. Remark. If ϕ : H_{1} → H_{2} is a morphism of X-groups we have a natural k-group morphism

∏ ϕ : ∏_{X/k} H_{1} ⟶ ∏_{X/k} H_{2}

If ϕ is a monomorphism (i.e., if ϕ has trivial kernel) then ∏ ϕ is also a monomorphism.

Assume our ∏_{X/k} H_{i} are representable by algebraic k-schemes S_{i} as explained above, and that we are given natural isomorphisms α_{i} : h_{S_{i}} ≃ ∏_{X/k} H_{i}. These yields a morphism of k-groups h_{ϕ} : S_{1}^{α_{1}} → S_{2}^{α_{2}} which, at the level of functor of points, is given by

h_{ϕ} = α_{2}^{−1} ∘ (∏ ϕ) ∘ α_{1} : h_{S_{1}} ⟶ h_{S_{2}}

Note that if ϕ is a monomorphism then so is h_{ϕ}.

By Yoneda considerations h_{ϕ} corresponds to the k-algebra homomorphism h_{ϕ}(id_{S_{1}}) : S_{1} → S_{2}. Chasing through the definitions we see that the diagram

\xymatrix{S_{1} ×_{k} X \ar[r]^{ev_{α_{1}}} \ar[d]_{h_{ϕ}(id_{S_{1}})×id_{X}} & H_{1} \ar[d]^{ϕ} \\ S_{2} ×_{k} X \ar[r]^{ev_{α_{2}}} & H_{2}}

commutes. Of particular interest is the case when H_{1} = T ×_{k} X for some k-group T. As we saw in Remark 3.30, we may take S_{1} to be T and the evaluation map ev_{α_{1}} : T ×_{k} X → T ×_{k} X to be the identity. The above then reads

ϕ = ev_{α_{2}} ∘ (h_{ϕ}(id_{T}) × id_{X}).

3.33. Remark. The algebraic k-scheme S need not be reduced. Following the Appendix we associate to S a closed k-subscheme S_{r} which is absolutely reduced. The k-group structure of S^{α} carries to S_{r}, and we denote the resulting algebraic k-group by S_{r}^{α}, or simply by S_{r} when no confusion is possible. If Y is an absolutely reduced scheme over k then every morphism Y → S factors through the closed immersion S_{r} ↪ S (see the Appendix for details). We call the composition

S_{r} ×_{k} X ↪ S ×_{k} X ⟶^{ev_{α}} H

the (reduced) evaluation map and denote it by the symbol ev_{α}^{r}.

The main result of this section concern the rank of the algebraic k-group S_{r}.

3.34. Proposition. Let H and S be as above. Assume that X ×_{k} k_{s} is connected. Let M_{0} be a k-subgroup of multiplicative type of S.

(1) The morphism of X-group schemes M_{0} ×_{k} X → H induced by ev_{α} is a closed immersion.

(2) For all x ∈ X, we have rank(S_{r}) ≤ rank(H_{x}).

Proof. (1) Since the evaluation map commutes with base change (Remark 3.31) we may assume that k = k_{s}. Let μ denote the kernel of the X-group morphism ev_{α} : M_{0} ×_{k} X → H induced by our evaluation map. According to theorem IX.6.8 of [SGA3] μ is a group scheme of multiplicative type and of finite type over X. Furthermore, to establish (1) it suffices to show that μ is trivial.

Since k is separably closed M_{0} is a diagonalisable k-group of finite type (prop. X.1.4 of [SGA3]). By Proposition IX.2.11.i of [SGA3], it follows that μ is diagonalisable as well (it is here that we use the assumption that X_{s} is connected). Hence μ = μ_{0} ×_{k} X where μ_{0} is a diagonalisable k-group of finite type. But by rigidity of diagonalisable groups ([SGA3], VIII.1.6), we have

Hom_{k−gr}(μ_{0}, M_{0}) ⟶^{∼} Hom_{X−gr}(μ_{0} ×_{k} X, M_{0} ×_{k} X).

So we are given actually a morphism ι_{0} : μ_{0} → M_{0} which is a closed immersion.

Recall that ∏_{X/k}(μ_{0} ×_{k} X) is represented by μ_{0} with the identity map for evaluation map (see Remark 3.30). Similarly for M_{0} ×_{k} X and S ×_{k} X. By Remark 3.32) if we take the Weil restriction for the X-group morphism ev_{α} : S ×_{k} X → H, then the corresponding k-group morphism S → S is the identity map. If we apply these considerations to the composite X-group morphism

(3.35)  f : μ_{0} ×_{k} X ⟶^{ι=ι_{0}×id} M_{0} ×_{k} X → S ×_{k} X ⟶^{ev_{α}} H

we obtain

(3.36)  f_{0} : μ_{0} ⟶^{ι_{0}} M_{0} ↪ S ⟶^{id} S,

which is a closed immersion. Since μ = μ_{0} ×_{k} X the morphism f of (3.35) is trivial. This forces the closed immersion f_{0} of (3.36) to be trivial. Thus μ_{0} = 1, and consequently μ = 1 as desired.

(2) Let T be a maximal k-torus of S_{r}. By (1), the X-group morphism T ×_{k}X → H is a closed immersion. It follows that the morphism T ×_{k}k(x) → H_{x} is a closed immersion for all points x ∈ X. Thus rank(S_{r}) = rank(T) ≤ rank(H_{x}). □

4. Torsors over the affine line

Let E be a geometrically separably trivial 𝔸_{k}^{1}-torsor under the action of a k-linear algebraic group G whose connected component of the identity is reductive. We must show that E is constant.

We begin our proof by reducing to the case when G is reductive.

4.1. Lemma. Let G be a reductive k-group and T a maximal torus of G. Suppose X is a geometrically irreducible k-scheme for which the canonical map k_{s}^{×} → O_{X_{s}}(X_{s})^{×} is an isomorphism. Then every geometrically separably trivial X-torsor under N_{G}(T) is constant.

Proof. Let N = N_{G}(T). The bijection (3.3) of Remark 3.2 shows that to a geometrically separably trivial X-torsor E under N corresponds the class of a (continuous) cocycle u ∈ Z^{1}(Gal(k_{s}/k), N(X_{s})).

Since T_{s} is split the underlying scheme structure of N_{s} is given by N_{s} = ⨆_{w∈W} G_{m}^{l} where G_{m} denotes the multiplicative group over k, l = rk (G) and W is the (abstract) Weyl group of (G_{s}, T_{s}). Since X is geometrically irreducible we obtain

N(X_{s}) = N_{s}(X_{s}) = Hom(X_{s}, N_{s}) =

⨆_{w∈W} G_{m}^{l}(X_{s}) = ⨆_{w∈W} G_{m}^{l}(k_{s}) = N(k_{s}).

Thus u ∈ Z^{1}(Γ, N(k_{s})) and E is constant. □

By considering the case when X = 𝔸_{k}^{1} and G = T we obtain a stronger version of Theorem 2.1 for tori.

4.2. Corollary. Let T be a k-torus. Every 𝔸_{k}^{1}-torsor under T is constant.

Proof. Since T splits over k_{s} and 𝔸_{k}^{1} has trivial Picard group, every 𝔸_{k}^{1}-torsor under T is geometrically separably trivial. □

4.3. Proposition. Assume that every geometrically separably trivial 𝔸_{k}^{1}- torsor under a reductive k-group is constant. If G is an algebraic k-group whose connected component G^{∘} is reductive then every geometrically separably trivial 𝔸_{k}^{1}-torsor under G is also constant.

Proof. Let E be a G-torsor over 𝔸_{k}^{1} which is trivial over 𝔸_{s}^{1}. Consider the fiber E_{0} of E at the origin 0 ∈ 𝔸_{k}^{1}, and its class [E_{0}] ∈ H^{1}(k, G). We view E_{0} as a constant torsor over 𝔸_{k}^{1} under G, and consider the twisted group E_{0}G which, for convenience, we will denote by G_{0}. We have a canonical bijection (the twisting map. See [DG])

η : H^{1}(𝔸_{k}^{1}, G_{0}) → H^{1}(𝔸_{k}^{1}, G)

which maps the trivial class of H^{1}(𝔸_{k}^{1}, G_{0}) to [E_{0}]. Furthermore η maps constant (resp. geometrically separably trivial) torsors into constant (resp. geometrically separably trivial) torsors. By replacing G by G_{0} we may thus assume without loss of generality that E_{0} is trivial.

Let C = G/G^{∘}. This is a twisted constant group ([DG] II §5). Consider the commutative diagram

H^{0}(𝔸_{k}^{1}, C) ⟶ H^{1}(𝔸_{k}^{1}, G^{∘}) ⟶^{ϕ_{1}} H^{1}(𝔸_{k}^{1}, G) ⟶^{ϕ_{2}} H^{1}(𝔸_{k}^{1}, C)
↓  ^{ψ_{1}}↓  ^{ψ_{2}}↓  ^{ψ_{3}}↓
H^{0}(𝔸_{s}^{1}, C) ⟶ H^{1}(𝔸_{s}^{1}, G^{∘}) ⟶^{γ} H^{1}(𝔸_{s}^{1}, G) ⟶ H^{1}(𝔸_{s}^{1}, C)

Since by assumption ψ_{2}([E]) = 1, we have ψ_{3}(ϕ_{2}([E])) = 1. By Remark 3.2 we have ϕ_{2}([E]) ∈ H^{1}(Gal (k_{s}/k), C(𝔸_{s}^{1})) ⊂ H^{1}(𝔸_{s}^{1}, C). But since C_{s} is a constant group C(𝔸_{s}^{1}) = C(k_{s}). Thus

H^{1}(Gal (k_{s}/k), C(𝔸_{s}^{1})) = H^{1}(Gal(k_{s}/k), C(k_{s})).

In other words, ϕ_{2}([E]) is a constant class, hence trivial because of our assumption on the fiber E_{0} of E.

Let E^{∘} be an 𝔸_{k}^{1}-torsor under G^{∘} such that [E^{∘}] ∈ H^{1}(𝔸_{k}^{1}, G^{∘}) satisfies ϕ_{1}([E^{∘}]) = [E]. It remains to show that ψ_{1}([E^{∘}]) = 1. As before, it suffices to prove that γ has trivial kernel or, equivalently, that G(𝔸_{s}^{1}) → C(𝔸_{s}^{1}) is surjective. But this is clear. Indeed, we have already observed above that C(𝔸_{s}^{1}) = C(k_{s}). □

We now turn to the proof of Theorem 2.1. By Proposition 4.3 we may assume that G is reductive. Let _{E}G be the corresponding twisted 𝔸_{k}^{1}-group. Let k′ ⊂ k_{s} be a finite Galois extension such that G_{k′} is split and E_{𝔸_{k′}^{1}} is trivial. By the Lemmata of §3 applied to Y = 𝔸_{k′}^{1} and Lemma 4.1 it will suffice to show that the extension k′ above can be chosen so that

(a) There exists a maximal torus T′ of _{E}G and a maximal torus T of G such that T′_{Y} is conjugate to T_{Y} = T ×_{k} Y under G(Y).

Let G̃ → G be the simply connected cover of the derived group of G. We can then construct the twisted group _{E}G̃ by considering the adjoint action of G on G̃, and this coincides with the simply connected cover of the derived group of the twisted group _{E}G [SGA3]. There exists a natural correspondence between the maximal tori of_{E}G̃ and those of_{E}G (ibid.) This shows that in order to establish that (a) above we may (and henceforth do) assume that G is simply connected.

By choosing a point p in the building B of _{E}GK̂ we obtain a group H(p) over ℙ_{k}^{1} extending _{E}G. Let S be an algebraic k-scheme representing the Weil restriction of H(p) to k, and S_{r} its corresponding separably reduced version. We have the evaluation maps

ev_{α}^{r} : S_{r} ×_{k} ℙ_{k}^{1} ↪ S ×_{k} ℙ_{k}^{1} ⟶^{ev_{α}} H(p).

Since ev_{α} : S ×_{k} ℙ_{k}^{1} → H(p) is a morphism over ℙ_{k}^{1} we see that the restriction of ev_{α} to S ×_{k} 𝔸_{k}^{1} maps into _{E}G. This gives the commutative diagram

S_{r} × ℙ_{k}^{1} ⟶^{ev_{α}^{r}} H(p)
∪  ∪
S_{r} × 𝔸_{k}^{1} ⟶^{ev_{α}^{r}} _{E}G

By Proposition 3.34.1 we obtain

(b) If T_{r} is a maximal torus of S_{r} (which exists since S_{r} is an algebraic group over k) then T′ = ev_{α}^{r}(T_{r} × 𝔸_{k}^{1}) is a torus of _{E}G.

Since the evaluation map commutes with arbitrary base field change (Remark 3.31) and the construction of S_{r} commutes with any separable field extension k′/k base change (Appendix, Proposition 6.12) we may replace k by k′ when trying to show that the torus T′ of (b) satisfies the conditions of (a). Indeed, if α′ : h_{S′} → ∏_{X′/k′}(H(p) ×_{k} k′) is as in Remark 3.31, then the closed immersion

(T_{r} ×_{k} 𝔸_{k}^{1}) ×_{k} k′ ≃ (T_{r} ×_{k} k′) ×_{k′} 𝔸_{k′}^{1} ⟶^{ev_{α′}^{r}} _{E}G ×_{k} k′

makes T′ ×_{k} k′ ≃ T′ ×_{𝔸_{k}^{1}} 𝔸_{k′}^{1} into a torus of _{E}G ×_{k} k′ ≃ _{E}G ×_{𝔸_{k}^{1}} 𝔸_{k′}^{1}. If this last torus is maximal, then so is our original T′ (since the maximality at the level of the geometric fibers is preserved under our base change).

We may therefore assume that _{E}G = G ×_{k} 𝔸_{k}^{1} = G_{𝔸_{k}^{1}}. Let T be a maximal split torus of G. By Soulé’s theorem [So] there exists g ∈ G(k[t]) such that q = g(p) is a point in the apartment of BK̂ corresponding to TK̂. As explained in Example 3.21 (d) we have an isomorphism g̃ : H(p) → H(q) of ℙ_{k}^{1}-schemes. The pullback of g̃ along 𝔸_{k}^{1} ⊂ ℙ_{k}^{1} is an inner automorphism of G_{𝔸_{k}^{1}}. Thus, by replacing T′ by g̃(T′) and by taking Remark 3.21(b) into consideration we may assume without loss of generality that p = q,

By Example 3.21 (c) the torus T yields a closed immersion

T ×_{k} ℙ_{k}^{1} ⟶ H(q)

which by Remark 3.32 is nothing but the restriction of the reduced evaluation map, namely

(4.4)  ev_{α}^{r} : T ×_{k} ℙ_{k}^{1} → S_{r} ×_{k} ℙ_{k}^{1} → S ×_{k} ℙ_{k}^{1} →^{ev_{α}} H(q).

By pulling back along 𝔸_{k}^{1} we see that the reduced evaluation map induces a closed immersion

(4.5)  ev_{α}^{r} : T ×_{k} 𝔸_{k}^{1} ⟶ G_{𝔸_{k}^{1}}

By Remark 3.30 we know that the functor of global sections of T ×_{k} ℙ_{k}^{1} is represented by T. Thus from the closed immersion (4.4) we obtain a canonical embedding T → S (see Remark 3.32) which factors through S_{r} since T is reduced. The resulting map T → S_{r} is an injective morphism of algebraic groups, hence a closed embedding. This allows us to identify T with a torus of S_{r}. By Proposition 3.34 this torus is necessarily maximal. We record this important fact for future reference.

(c) If T_{r} is a maximal torus of S_{r} then T′ = ev_{α}^{r}(T_{r} ×𝔸_{k}^{1}) is a maximal torus of _{E}G. In particular the 𝔸_{k}^{1}-group _{E}G admits a maximal torus.

We will see that T′ actually satisfies the conditions of (a). This will finish the proof of Theorem 2.1

Let k′ ⊂ k_{s} be a finite extension of k such that T_{r} ×_{k} k′ and T ×_{k} k′ are conjugate under an element s ∈ S_{r}(k′) ⊂ S(k′). We again replace k by k′. Think of s as an 𝔸_{k}^{1}-point of S ×_{k} 𝔸_{k}^{1} (which we denote by s̃). Then the inner automorphism int(s̃) of S ×_{k} 𝔸_{k}^{1} is given by int(s) × id. Consider the element

ev_{α}(s̃) ∈ G_{𝔸_{k}^{1}}(𝔸_{k}^{1}) = G(𝔸_{k}^{1}).

We have the commutative diagram

S_{r} ×_{k} 𝔸_{k}^{1} ⟶^{ev} G_{𝔸_{k}^{1}}
int(s̃)↓  ↓^{int(ev_{α}^{r}(s̃))}
S_{r} ×_{k} 𝔸_{k}^{1} ⟶^{ev_{α}^{r}} G_{𝔸_{k}^{1}}

Since int(s)(T_{r}) = T this last diagram shows that T′ = ev_{α}^{r}(T_{r} ×_{k} 𝔸_{k}^{1}) is conjugate to ev_{r}(T ×_{k} 𝔸_{k}^{1}) under the element int(ev_{α}(s̃)) ∈ G(𝔸_{k}^{1}). Since ev_{α}(T ×_{k} 𝔸_{k}^{1}) is a maximal torus of G_{𝔸_{k}^{1}} the proof of (a), hence also of Theorem 2.1, is complete. □

By reasoning as in Steps (a), (b) and (c) of the above proof we obtain the following important fact.

4.6. Theorem. Let G be a reductive group scheme over 𝔸_{k}^{1}. Assume that G is “geometrically separably split”, i.e. that G ×_{𝔸_{k}^{1}} 𝔸_{k_{s}}^{1} is split. Then G has a maximal torus. □

5. Torsors over the punctured affine line

In what follows we will denote by X = Spec k[t^{±1}] the punctured affine line and by K = k(t) the rational function field of X. We will denote by K̃ the maximal unramified extension of K̂ = k((t)). Recall that K̃ is the subfield of k_{s}((t)) consisting of those elements f = ∑_{n≥N} c_{n}t^{n} for which the set {c_{n} : n ≥ N} belong to a finite separable extension of k. The natural map Gal (k_{s}/k) → Gal (K̃/K̂) is an isomorphism of profinite groups. We henceforth identify these two groups.

Let Ô = k[[t]] and Õ = k_{s}[[t]] ∩ K̃. The residue map ρ : Õ → k_{s} (which is also the specialization map at t = 0) induces a group homomorphism G(ρ) : G(Õ) → G(k_{s}) whose kernel we denote by G(Õ)^{(1)} : These are “the elements of G(Õ) that are congruent to 1 modulo (t)”. We denote G_{m}(Õ)^{(1)} simply by Õ^{(1)}. Thus O^{(1)} = {∑_{n≥0} c_{n}t^{n} ∈ Õ : c_{0} = 1}.

Our goal is to prove Theorem 2.4 by comparing the étale cohomology of X with the Galois cohomology of k((t)). As in the case of the affine line, the existence of maximal tori will play a crucial role.

5.1. Cohomological exponent: Good and bad primes. Theorem 2.4 for the punctured line has an assumption on the “good characteristic” on the base field. In this section we give the relevant definitions and basic results concerning this point.

5.1. Lemma. Let G be a reductive k-group. There exists a positive integer n with the property that for every field extension F/k and for every maximal torus T of the reductive F -group G_{F}

(1) nH^{1}(F, T) = 1. Moreover, if X(T)_{∗} denotes the Gal(F_{sep}/F)-module of cocharacters of T (see 3.1), then

(2) nH^{1}(F, X(T)_{∗}) = 1.

Proof. (1) Let T be a maximal torus of G_{F} . Consider a minimal splitting extension L/F of T. Since H^{1}(L, T) = 1 we have H^{1}(F, T) = H^{1}(L/F, T (L)). In particular, if m = [L : F] then mH^{1}(F, T) = 1. We now show that there is a positive integer n which does not depend on T and F such that m divides n. This would complete the proof of (a).

Recall that G is the almost direct product G = C · G′ of its central torus C and derived group G′. This yields T = C_{F} · T′ where T′ is the maximal torus G′_{F} given by T′ = G′_{F} ∩ T.

Let l/k be the minimal field extension of k splitting C. The decomposition T = C_{F} ·T′ implies that L is the composition L = L_{1} ·L_{2} (taken inside some fixed separable closure F_{s} of F containing k_{s}) of two fields L_{1} and L_{2} where L_{1} = l · F and L_{2} is the minimal splitting extension of the F -torus T′. It is well known that Gal (L_{2}/F) admits an embedding into the automorphism group Aut (Σ) of the root system Σ = Σ(G′_{F} , T′) of G′_{F} with respect to T′. It is easy to see that the positive integer n = [l : k]·|Aut (Σ)| is then divisible by m = [L : F]. Observe that n depends neither on T nor F.

(2) Let T be a maximal F -torus of G_{F}. As will be explained in Lemma 5.16(3) X(T)_{∗} is a Galois stable direct summand of T (F̃ (t)). From this it follows that H^{1}(F, X(T)_{∗}) is a direct summand of H^{1}(Gal(F_{sep}/F), T (F̃ (t)) ⊂ H^{1}(F̂ (t), T), hence that nH^{1}(F, X(T)_{∗}) = 0. ^{5} □

Let G be a reductive k-group. The smallest positive integer satisfying the conditions of Lemma 5.1 is called the cohomological toral exponent of G. It will be denoted by cte(G).

We will use cte(G) to define the concept of good and bad primes for G. We first define the relevant concepts in the semisimple case. Let G be a semisimple k-group. There exists a unique Chevalley form of G, that is a Chevalley group H = H_{G} such that H ⊗_{ℤ} k_{s} ≃ G ⊗_{k} k_{s}. Recall [SGA3] Exp. XXIV.3 that we have a split exact sequence of ℤ-groups

5Since T splits over F_{sep} the inclusion H^{1}(Gal(F_{sep}/F), T (F̃ (t)) ⊂ H^{1}(F̂ (t), T) is in fact an equality.

(5.2)  1 → H_{ad} → Aut(H) → Out(H) → 1,

where H_{ad} is the adjoint group of H and Out(G) is the “group of outer automorphisms” of H. The group Out(H) is a finite constant ℤ-group whose underlying abstract group is a subgroup of the group of symmetries of the Dynkin diagram of H. The sequence (5.2) comes equipped with a natural section Out(H) → Aut(H) that arises from the fixed choice of “épinglage” used in defining H. After applying the base change ℤ → k and passing to cohomology (5.2) yields the exact sequence of pointed sets

(5.3)  H^{1}(k, H_{ad}) → H^{1}(k, Aut(H)) → H^{1}(k, Out(H)) → 1.

Let z ∈ Z^{1}(k, Aut(H)) be such that G ≃ _{z}H. Let [z′] be the image of [z] under the morphism H^{1}(k, Aut(H)) → H^{1}(k, Out(H)). There is a canonical section H^{1}(k, Out(H)) → H^{1}(k, Aut(H)) which is obtained from the given section in (5.2). Let [z′′] be the image of [z′] with respect to this last mapping. Clearly, the twisted group G_{qs} = _{z′′}H_{k} is quasi-split. We call G_{qs} the (Chevalley) quasi-split form of G. It has the following characteristic properties:

(a) G_{qs} is a k-form of G, i.e. G_{qs} and G are isomorphic over k_{s};

(b) for a field extension F/k the group G_{F} is an inner form of H_{F} if and only if G_{qs} ×_{k} F is split. In particular, the star-action of Gal(k_{s}/k) on the Dynkin diagrams of G_{qs} and G is the same.

5.4. Lemma. If P is a parabolic subgroup of G, then G_{qs} contains a parabolic subgroup of type t(P).

Proof. Let t denote the type of P. Since P is k-defined the quotient variety G/P is k-defined as well. In particular t is stable with respect to the star action of Gal(k_{s}/k) on the Dynkin diagram. Let Q be a parabolic subgroup of the k_{s} reductive group G_{qs}_{s} = G_{qs} ×_{k} k_{s} of type t. Since t is Galois stable, the variety G_{qs}_{s}/Q is k-defined, hence isomorphic to the variety Par_{t}(G_{qs}) of parabolic subgroups in G_{qs} of type t (c.f. [MPW, prop. 1.3]).

Analogously, if B is a Borel subgroup of G_{qs} (which exists, since G_{qs} is quasisplit) the variety G_{qs}/B is isomorphic to the variety Bor(G_{qs}) of Borel subgroups in G_{qs}. Without loss of generality we may assume that Q contains B. The canonical morphism G_{qs}/B → G_{qs}/Q is k-defined. Since (G_{qs}/B)(k) ≠ ∅ we have Par_{t}(G_{qs})(k) = (G_{qs}/Q)(k) ≠ ∅ and the Lemma follows.

□

Next we define the concept of quasisplit form in the reductive case. Let G be a reductive k-group, and let C denote its radical torus [SGA3, XXII.4.3.6].

Recall that C is the unique maximal torus of the center of G. It is not difficult to see that up to isomorphism there exists a unique reductive k-group G_{qs}, called the (Chevalley) quasisplit form of G, with the following two properties:

(1) The central torus of G and G_{qs} are isomorphic.

(2) Der(G_{qs}) = Der(G)_{qs}. That is, the quasisplit form of the (semisimple) derived group of G coincides with the derived group of G_{qs}.

We can now state the definition of good and bad primes: Let G be a reductive k-group and G_{qs} its quasisplit form. The prime divisors of the cohomological toral exponent cte(G_{qs}) of G_{qs} are called bad primes for G. Prime numbers which are not bad are called good. We say that the characteristic p of the base field k is good for G if either p = 0 or p is a good prime for G.

5.5. Remark. From Steinberg’s work we know that every maximal k-torus of G admits a k-embedding into G_{qs}. From the definition it follows that cte(G) divides cte(G_{qs}). In particular the characteristic of k does not divide cte(G).

5.6. Remark. The cohomological toral exponent “depends on the base field”. If G is a trialitarian k-group, then 3 divides cte(G) but not cte(G_{k_{s}}) (this last since G_{k_{s}} is a classical group of type D_{4}).

We now state and prove two stability properties of the set of good primes which will be used while proving the main result on torsors over the punctured line.

5.7. Lemma. Let G be a reductive k-group. Let P ⊂ G be a parabolic subgroup. If H is a Levi subgroup of P then the good primes for G are also good primes for H. In particular, if the characteristic of k is good for G, then it is also good for H.

Proof. Let G_{qs} be the quasi-split form of G, and let P_{0} be a parabolic subgroup of G_{qs} of the same type as P (see Lemma 5.4). Clearly any Levi subgroup H_{0} of P_{0} is isomorphic to the quasisplit form H_{qs} of H. Since for any field extension F/k any maximal torus of H_{0,F} is also a maximal torus of G_{qs,F} we see that cte(H_{0}) divides cte(G_{qs}). The result now follows. □

5.8. Lemma. Let η ∈ Z^{1}(k, G) be a cocycle and _{η}G the corresponding twisted group. The set of good primes for G and _{η}G coincide.

Proof. Indeed, the quasisplit forms of G and_{η}G are isomorphic, so the result follows by definition. □

5.2. Existence of maximal tori. The following result yields the existence of maximal tori that will be used in the proof of Theorem 2.4.

5.9. Proposition. Let G be a reductive group scheme over X. Assume that G is geometrically separably split; that is there exists an isomorphism f : G_{0} ×_{ℤ} X_{k′} ⟶^{∼} G ×_{X} X_{k′} where G_{0}/ℤ is a Chevalley reductive group and k′/k is a finite Galois extension.

Denote by T_{0} the standard maximal split torus of the ℤ-group G_{0}. Then G admits a maximal X-torus M such that M ×_{X} X_{s} is G(X_{s})-conjugated to f(T_{0} ×_{ℤ} X_{s}).

Proof. By [SGA3, XII.4.7.c] there is a one-to-one correspondence between the maximal tori of G, those of its adjoint group G_{ad} and those of the simply connected covering of G_{ad}. We may thus assume without loss of generality that our X-group G is semisimple and simply connected. Following Tits [Ti3], we consider the twin building B = B_{+}×B_{−} of G_{K} over the completions of K at 0 and ∞.

Consider a point p = (p_{+}, p_{−}) ∈ B, as well as the two associated parahoric group schemes F_{+}/k[[t]] and F_{−}/k[[\frac{1}{t}]] corresponding to p_{+} and p_{−} respectively. The patching process of §5 (applied twice) produces a smooth group scheme H(p) over ℙ_{k}^{1} extending G/X. Let S be the k-group representing the Weil restriction of H(p) to k, and S_{r} the corresponding absolutely reduced group.

Let T_{r} ⊂ S_{r} be a maximal torus. By Proposition 3.34.1 ev(T_{r} ×_{k} ℙ_{k}) is a torus of H(p). Pulling back to X we obtain that M = ev(T_{r} ×_{k} X)is a torus of G. We will show that M has the desired properties. The same reasoning given for the affine shows that we may replace k by k′.

There is a canonical embedding B → B′ where B′ is the twin building associated to G ×_{k((t))} k′((t)) and G ×_{k((t))} k′((\frac{1}{t})). This allows us to view our chosen point p as an element of the twin building B′. The construction of H(p) is compatible with this identification [see Example 3.21(b)].

We now use the splitting G_{0} ×_{k} X_{k′} ⟶^{∼} G ×_{X} X_{k′}. By Abramenko’s result [A, Proposition 5], there exists g ∈ G(X_{k′}) such that q = g(p) lives in the canonical twin apartment corresponding to the torus T_{0}. Clearly the group schemes H(q) and H(p) ×_{ℙ^{1}} ℙ_{k′}^{1} are isomorphic, so we may assume that p = q. The torus T_{0} gives rise to a canonical subtorus T_{0} ×_{ℤ}ℙ_{k′}^{1} ⊂ H(p), as one can see by applying the reasoning of Example 3.21(c) twice.

The proof can now be finished along the exact same reasoning given in the proof of the affine line. □

5.3. Reformulation of Theorem 2.4. Henceforth G will denote a reductive k-group where the characteristic of k is good for G.

By Remark 2.1 the isomorphism classes of geometrically separably split torsors over 𝔸_{k}^{×} under G are parametrized by H^{1}(Gal(k_{s}/k), G(k_{s}[t^{±1}])). Along similar lines we see that the classes of H^{1}(K̂, G) corresponding to torsors that are trivialized by the base change K̂ → K̃ are parametrized by H^{1}(Gal(K̃/K̂), G(K̃)). Theorem 2.4 can thus be stated as follows:

5.10. Proposition. Under the natural identification of Γ := Gal (k_{s}/k) with Gal (K̃/K̂) we have.

(1) Every X-torsor under G is geometrically separably trivial. In particular the canonical map H^{1}(Γ, G(k_{s}[t^{±1}])) → H^{1}(X, G) is bijective.

(2) The canonical map H^{1}(Γ, G(K̃)) → H^{1}(K̂, G) is bijective: Every K̂- torsor under G is split by the base change K̂ → K̃.

(3) The thesis of Theorem 2.4 is equivalent to the following assertion: The canonical map

H^{1}(Γ, G(k_{s}[t^{±1}])) → H^{1}(Γ, G(K̃))

is bijective.

We begin with a useful general fact.

5.11. Lemma. Let G be a split reductive group over a field F. Let z ∈ Z^{1}(F, G) be a cocycle. There exists a maximal torus T ⊂ G such that [z] is in the image of the natural map H^{1}(F, T) → H^{1}(F, G).

Proof. The result essentially follows from arguments in Steinberg’s paper [St], which show that the Lemma holds for quasi-split simple group (see the prelude to Theorem 3.1 of [Chr] for details). In particular, the Lemma holds if our G is semisimple and of adjoint type. The reduction to this case is done along standard lines as follows.

Let C be the centre of G and let G′ = G/C. Let z′ ∈ Z^{1}(F, G′) be the image of z under the canonical map Z^{1}(F, G) → Z^{1}(F, G′). Since G′ is an adjoint group there exists a maximal torus T′ ⊂ G′ such that [z′] ∈ Im[H^{1}(F, T′) → H^{1}(F, G′)]. Let T ⊂ G be the inverse image of T′ under G → G′. Since the image of [z′] under the map H^{1}(F, G′) → H^{2}(F, C) vanishes, the class [z′] lifts to a class [u] ∈ H^{1}(F, T) as one can see by considering the exact sequence 1 → C → T → T′ → 1. We now pass to the twisted group _{u}G. Under the twisting bijection H^{1}(F, G) → H^{1}(F,_{u}G) the class [z] goes into some class, say [w], whose image under H^{1}(F,_{u}G) → H^{1}(F,_{u}G′) is zero. Hence we may assume that [w] ∈ H^{1}(F,_{u}C). Note that uC = C is the centre of _{u}G and is contained in _{u}T = T. It follows that [z] = [u + w] where the sum [u + w] = [u] + [w] is taken inside the group H^{1}(F, T). □

We now turn to the proof of Proposition 5.10. We begin by showing that

(5.12)  H^{1}(k_{s}(t), G) = 1.

For convenience we denote k_{s}(t) by F. The main fundamental property of F we are going to use is that it is a field of q-cohomological dimension 1 for all primes q different than the characteristic p of k [Se1, §II.4.2].

Let z ∈ Z^{1}(F, G) be a cocycle. Since G_{F} is split, the previous Lemma reduces the problem to showing that H^{1}(F, T) = 1 for any maximal torus T of G_{F} .

Let L/F be a minimal Galois extension splitting T. Since T is split over L we have H^{1}(L, T) = 1, hence H^{1}(L/F, T (L)) = H^{1}(F, T). Because of the definition of good characteristic the abelian group H^{1}(L/K, T (L)) is the direct sum of its q-Sylow subgroups H^{1}(L/F, T (L))_{q} where q runs through the set of primes other than p that divide n = [L : F].

Let Γ be the Galois group of L/F. For each prime q dividing the order of Γ we fix a q-Sylow subgroup Γ_{q} ⊂ Γ. We have the tower of fields F ⊂ L_{q} ⊂ L where L_{q} is the subfield in L corresponding to Γ_{q}. The standard restrictioncorestriction argument shows that the equality H^{1}(L/F, T (L))_{q} = 1 follows immediately from that of H^{1}(L/L_{q}, T (L)) = 1, or equivalently, from H^{1}(L_{q}, T) = 1. To establish this last equality we consider an exact sequence of L_{q}-tori

1 → T_{1} → P → T → 1

where P is a permutation torus ([CTS2] lemme 3), and its corresponding Galois cohomology sequence. From this it follows that it will suffice to show that the q-torsion part of H^{2}(L_{q}, T_{1}) is trivial. Let A ⊂ T_{1} be the q^{m}-torsion part of T_{1} where m is suitably large. Every element in H^{2}(L_{q}, T_{1}) of order a power of q comes from H^{2}(L_{q}, A). Since L_{q} is of q-cohomological dimension 1 we have H^{2}(L_{q}, A) = 1. This finishes the proof of (5.12).

We now turn to the proof of (1). Let E be a G-torsor over X_{s} and let _{E}G be the corresponding twisted X_{s}-group scheme. The idea of the proof is to show that E admits a reduction of structure group to a Borel subgroup B of G. In other words, we want to prove that the isomorphism class of E is in the image of

(5.13)  H^{1}(X_{s}, B) → H^{1}(X_{s}, G).

Recall that the class of E is in the image of the map (5.13) if and only if the X_{s}-group scheme _{E}G has a Borel subgroup.

By (5.12) the generic fiber of _{E}G is split. In particular this generic fiber has a Borel subgroup over k_{s}(t). Since X_{s} = Spec(k_{s}[t^{±1}]) has dimension 1, it follows from a standard argument (see Remark 5.31 below) that _{E}G itself has a Borel subgroup as required. It remains to show that H^{1}(X_{s}, B) = 1, but this is clear by devissage since H^{1}(X_{s}, G_{a}) = 1 and the Picard group of X_{s} is trivial.

(2) In analogy to (5.12) we have

(5.14)  H^{1}({ \atop k_{s}(t)}, G) = 1.

This follows by reasoning as in (1) by taking into consideration that { \atop k_{s}(t)} is of q-cohomological dimension 1 for all primes q other than p (see [Se2] theorem 4.4). The same devissage reasoning used in (1) completes the proof.

(3) This is a direct consequence of (1) and (2). □

5.4. Reduction of structure group to N_{G}(T). Let T be a maximal torus of our reductive k-group G. By combining Lemma 3.15 and Proposition 5.9 we obtain.

5.15. Lemma. The map

H^{1}(Γ, N_{G}(T)(k_{s}[t^{±1}])) ⟶ H^{1}(Γ, G(k_{s}[t^{±1}]))

is surjective. □

We now look in detail at H^{1}(Γ, N_{G}(T)(k_{s}[t^{±1}]). Let X(T)_{∗} be the group of cocharacters of T. Recall that X(T)_{∗} comes equipped with a natural Gal(k_{s}/k)-module structure (see 3.1). Since the underlying scheme of G_{m} is X = Spec(k[t^{±1}]), a cocharacter of T is naturally an element of T (k_{s}[t^{±1}]). This allows us to henceforth identify X(T)_{∗} with a Galois submodule of T (k_{s}[t^{±1}]).

5.16. Lemma. Let N = N_{G}(T). There exists natural Galois modules isomorphisms:

(1) T (k_{s}) × X(T)_{∗} ⟶^{∼} T (k_{s}[t^{±1}]);

(2) T (Õ) = T (k_{s}) × T (Õ)^{(1)};

(3) T (k_{s}) × T (Õ)^{(1)} × X(T)_{∗} ⟶^{∼} T (K̃);

(4) X(T)_{∗} ⋊ N(k_{s}) ⟶^{∼} N(k_{s}[t^{±1}]);

(5) X(T)_{∗} × T (Õ)^{(1)} ⋊ N(k_{s}) ⟶^{∼} N(K̃).

Proof. (1) T_{s} corresponds to the ring k_{s}[x_{1}^{±1}, …, x_{l}^{±1}] with a given action of Γ. An element a ∈ T (k_{s}[t^{±1}]) corresponds to a k_{s}-algebra homomorphism

a^{∗} : k_{s}[x_{1}^{±1}, …, x_{l}^{±1}] ⟶ k_{s}[t^{±1}]

which is entirely defined by the values a^{∗}(x_{i}). We have a^{∗}(x_{i}) = λ_{i}t^{n_{i}}. It is clear that there exists χ ∈ X(T)_{∗} such that χ : x_{i} → t^{n_{i}}. This shows that T (k_{s}[t^{±1}]) is generated by T (k_{s}) and X(T)_{∗}. That the natural map T (k_{s}) × X(T)_{∗} → T (k_{s}[t^{±1}]) is injective and compatible with the action of Γ is clear.

(2) and (3) Every element of K̃^{×} can uniquely be written in the form λft^{n} with λ ∈ k_{s}^{×}, f ∈ G_{m}(Õ)^{(1)} = Õ^{(1)} and n ∈ ℤ. Moreover, if the element is in Õ then n = 0. One now reasons mutatis mutandis as in (1).

(4) We know that

N_{s} = ∐_{w̄∈W} wT_{s}

where the w ∈ N(k_{s}) are representatives of the elements of the Weyl group W. Since Spec(k_{s}[t^{±1}]) is connected it is then clear that

N(k_{s}[t^{±1}]) = ∐ wT (k_{s}[t^{±1}]).

If y ∈ wT (k_{s}[t^{±1}]) ⊂ N(k_{s}[t^{±1}]) then

w^{−1}y ∈ T (k_{s}[t^{±1}]) = T (k_{s}) × X(T)_{∗}.

Since N(k_{s}) = ∐ wT (k_{s}) the result follows (after checking compatibility with the action of Γ).

(5) The reasoning is similar to (3) and (4) above. □

5.17. Lemma. Let Γ denote the Galois group Gal (k_{s}/k). For all i ≥ 1, we have (1) H^{i}(Γ, T (Õ)^{(1)}) = 0;

(2) H^{i}(Γ, T (k_{s})) ⟶^{∼} H^{i}(Γ, T (Õ));

(3) H^{i}(Γ, T (k_{s}[t^{±1}])) ⟶^{∼} H^{i}(Γ, T (K̃)).

Proof. (1)We have a natural isomorphism

(5.18)  Õ^{(1)} ⊗_{ℤ} X(T)_{∗} ≃ T (Õ)^{(1)}

This allows us to generalize the classical proof for T = G_{m} (e.g. [GS, §6.3]) in an obvious way to yield H^{i}(Γ, T (Õ)^{(1)}) = 0.

(2) Follows from (1) and Lemma 5.16.2

(3) Follows from (2) and Lemma 5.16.1,3 □

5.5. Proof of surjectivity. The following result, together with Proposition 5.10.3 and Lemma 5.15, establishes the surjectivity part of Theorem 2.4.

5.19. Proposition. In the commutative diagram

H^{1}(Γ, N_{G}(T)(k_{s}[t^{±1}])) ⟶^{β_{N_{G}(T)}} H^{1}(Γ, N_{G}(T)(K̃))
ρ_{Xs} ↓  ^{ρK̃} ↓
H^{1}(Γ, G(k_{s}[t^{±1}])) ⟶^{β_{G}} H^{1}(Γ, G(K̃))

obtained by the base change and change of structure group, the maps ρ_{X_{s}}, ρK̃ and β_{N_{G}(T)} are surjective (and as a consequence so is β_{G}).

Proof. That ρ_{X_{s}} is surjective follows from Lemma 5.15. The same reasoning shows that ρK̃ is surjective since we know that for all z ∈ Z^{1}(Γ, G(K̃)) the twisted K̂-group_{z}GK̂ contains a maximal K̂-torus which is split by K̃ [BT2, cor. 5.1.12], and any two such tori are conjugate by an element of G(K̃).

It remains to show that β_{N_{G}(T)} is surjective. For convenience we will denote N_{G}(T) simply by N. Let z ∈ Z^{1}(Γ, N(K̃)). Then z = (z_{γ})_{γ∈Γ} for some z_{γ} ∈ N(K̃) which according to Lemma 5.16 can be written in the form

z_{γ} = z′_{γ}z_{γ}′′z_{γ}′′′

with

z_{γ}′ ∈ N(k_{s}), z_{γ}′′ ∈ X(T)_{∗}, z_{γ}′′′ ∈ T (Õ)^{(1)}.

Moreover,

z′ = (z_{γ}′)_{γ∈Γ} ∈ Z^{1}(Γ, N(k_{s})).

One can consider the twisted k-group _{z′} T, and it is well known that the family z′′′ = (zγ^{′′′})_{γ∈Γ} is in fact an element of Z^{1}(Γ,_{z′}T (Õ)^{(1)}). By Lemma 5.17 H^{1}(Γ,_{z′}T (Õ)^{(1)}) = 1. Choose x ∈ T (Õ)^{(1)} such that

x^{−1}zγ^{′′′}z′_{γ}^{γ}x(z′_{γ})^{−1} = 1.

An immediate calculation shows that

x^{−1}z_{γ}^{γ}x ∈ X(T)_{∗} ⋊ N(k_{s}).

By Lemma 5.16(4) it now follows that [z] is in the image of our map

β_{N} : H^{1}(Γ, N(k_{s}[t^{±1}])) ⟶ H^{1}(Γ, N(K̃))

as desired. □

5.6. Proof of Injectivity. The proof that the natural map

H^{1}(Γ, G(k_{s}[t^{±1}])) → H^{1}(Γ, G(K̃)).

is injective is the most delicate part of the argument. For convenience we will divide the reasoning into several steps. By Lemma 5.15 the injectivity result we want to prove can be reformulated as follows:

5.20. Proposition. Let z_{1} = (a_{σ})_{σ∈Γ} and z_{2} = (b_{σ})_{σ∈Γ} be two cocycles in Z^{1}(Γ, N(k_{s}[t^{±1}])) whose image in H^{1}(Γ, G(K̃)) coincide. Then z_{1} and z_{2} have the same image in H^{1}(Γ, G(k_{s}[t^{±1}])).

The rest of this section is devoted to the proof of this result.

5.6.1. Linear and translation parts of a cocycle. We begin with a straightforward application of Lemma 5.16 that will be used in the main proof.

5.21. Lemma. There is a unique decomposition a_{σ} = a′_{σ}(t)aσ^{′′} and b_{σ} =

b′_{σ}(t)bσ^{′′} where a′_{σ}(t), b′_{σ}(t) ∈ X(T)_{∗} and aσ^{′′}, bσ^{′′} ∈ N(k_{s}). □

We will call the families z′_{1} = (a′_{σ}(t)) and z′_{2} = (b′_{σ}(t)) [resp. z1^{′′} = (aσ^{′′}) and z2^{′′} = (bσ^{′′})] the translation [resp. linear] parts of the cocycles z_{1} and z_{2}. Since X(T)_{∗} is a normal subgroup of N(k_{s}[t^{±1}]), one can easily check that the z1^{′′} is a cocycle in Z^{1}(Γ, N(k_{s})). If we identify (_{z′′_{1}} T)(k_{s}[t^{±1}]) = T (k_{s}[t^{±1}]) then one checks that the translation parts z′_{1}is a cocycle with coefficients in the twisted tori _{z′′_{1}} T. Similar considerations apply to z_{2}.

5.22. Remark. The linear part is constant, in the sense that it takes values in N(k_{s}). The use of the parameter t within the notation a′_{σ}(t) of the translation part of a_{σ} is used to emphasize that a_{σ} is being thought as a morphism from Spec(k_{s}[t^{±1}]) to T. This point will become relevant later on when we show that z_{1} and z_{2} are rationally equivalent.

5.6.2. Equality of generic fibers. The first step of the proof of injectivity is to show that the classes [z_{1}] and [z_{2}] coincide generically, i.e. that the images of [z_{1}] and [z_{2}] coincide in H^{1}(Γ, G(k_{s}(t))). The proof is mainly based on the following theorem of Bruhat-Tits.

5.23. Theorem. (Bruhat-Tits, [BT3, 3.15]) Let H be a reductive algebraic group defined over a field l. Then the canonical map H^{1}(l, H) → H^{1}(l((t)), H) is injective. □

A much easier statement of similar flavour is the following.

5.24. Lemma. Let H be a reductive algebraic group over a field l. Then the following are equivalent:

(1) H is irreducible (resp. anisotropic) over l;

(2) H is irreducible (resp. anisotropic) over l(t);

(3) H is irreducible (resp. anisotropic) over l((t)).

Proof. We have only to prove that i) ⟹ iii) in both cases.

Assume first that H is irreducible over l. If H_{l((t))} is reducible then it has a parabolic subgroup of type t different than H (see Remark 3.16), and therefore Par_{t}(H)(l((t))) = Par_{t}(H_{l((t))})(l((t))) ≠ ∅. Since Par_{t}(H) is proper, we have Par_{t}(H)(l[[t]]) ≠ ∅ by the valuative criterion of properness. But then Par_{t}(H)(l) ≠ ∅, which is a contradiction. Thus H_{l((t))} is irreducible.

Now we assume that H is anisotropic, namely that H is irreducible and its radical torus C is anisotropic (Proposition 3.17.2). This same Proposition, together with the first step, reduces the proof to showing that C_{l((t))} is anisotropic. The torus C is the twist of G_{m}^{r} by a continuous morphism ϕ : Gal(l_{s}/l) → GL_{r}(ℤ). To say that C is anisotropic is equivalent to (ℤ^{r})^{ϕ} = 0. Since the map Gal(l((t))_{s}/l((t))) → Gal(l_{s}/l) is split, it follows that (ℤ^{r})^{ϕ_{l((t))}} = 0 where ϕ_{l((t))} : Gal(l((t))_{s}/l((t))) → Gal(l_{s}/l) → GL_{r}(ℤ) is the composite map. Thus the torus C ×_{l} l((t)) is anisotropic. □

By our hypothesis on the good characteristic of the base field k there exists a positive integer n not divisible by char(k) with the property that nH^{1}(F, X(T)_{∗}) = 0 for all field extension F/k. The trick to establish equality of the generic fibers of z_{1} and z_{2} is to add an independent variable x over k, and work over the fields l = k(x) and l_{s} = k_{s}(x).

Let L = l(t), L̂ = l((t)) and denote by L̃ the unramified extension of L̂ having residue field k_{s}(x)/k(x) = l_{s}/l.

Consider the l_{s}-algebra endomorphism of l_{s}[t^{±1}] given by t → xt^{n}. It gives rise to a commutative diagram of fields extensions

l_{s}(t) ⟶^{ϕ} l_{s}(t)
↓  ↓
L̃ \underrightarrow{ϕ̃} L̃

that commutes with the action of the Galois group Γ. Passing to cohomology we thus obtain the commutative diagram

\begin{matrix}H^{1}(Γ, G(l_{s}[t^{±1}])) & ⟶^{ϕ^{∗}} & H^{1}(Γ, G(l_{s}[t^{±1}])) \\ ↓ & & ↓ \\ H^{1}(Γ, G(L̃)) & \underrightarrow{ϕ̃^{∗}} & H^{1}(Γ, G(L̃)) \end{matrix} (5.25)

Let z_{3} = ϕ^{∗}(z_{1}) and z_{4} = ϕ^{∗}(z_{2}). These are cocycles in Z^{1}(Γ, N(X_{l_{s}})). Recall that z_{1} = (a_{σ}), z_{2} = (b_{σ}) and that a_{σ} and b_{σ} can be decomposed into their translation and linear parts:

a_{σ} = a′_{σ}(t)aσ^{′′}, b_{σ} = b′_{σ}(t)bσ^{′′}

such that the linear parts are constant, and the translation parts live in X(T)_{∗} ⊂ T (k_{s}[t^{±1}]) (see Remark 5.22). It follows that

(5.26)  ϕ^{∗}(a_{σ}) = a′_{σ}(xt^{n})aσ^{′′}, ϕ^{∗}(b_{σ}) = b′_{σ}(xt^{n})bσ^{′′}.

From the way the embedding X(T)_{∗} ↪ T (k_{s}[t^{±1}]) is defined it easily follows that

(5.27)  a′_{σ}(xt^{n}) = a′_{σ}(x)a′_{σ}(t^{n}) = a′_{σ}(x)(a′_{σ}(t))^{n}, and
(5.28)  b′_{σ}(xt^{n}) = b′_{σ}(x)b′_{σ}(t^{n}) = b′_{σ}(x)(b′_{σ}(t))^{n}.

5.29. Lemma. The cocycle z_{3} = ϕ^{∗}(z_{1}) is equivalent to (a′_{σ}(x)aσ^{′′}). Similarly z_{4} = ϕ^{∗}(z_{2}) is equivalent to (b′_{σ}(x)bσ^{′′}).

Proof. Since a_{σ} = a′_{σ}(t)aσ^{′′} it is clear that u = (a′_{σ}(x)aσ^{′′})_{σ∈Γ} is a cocycle in Z^{1}(Γ, N(X_{l_{s}})). The statement of the Lemma is equivalent to the fact that the image of z_{3} under the canonical bijection H^{1}(Γ, N(X_{l_{s}})) → H^{1}(Γ, _{u}N(X_{l_{s}})) is trivial. By (5.26)–(5.28) this image is (a′_{σ}(t))^{n}, and it takes values in the cocharacter group X(_{u}T)_{∗} of the twisted torus _{u}T. By our assumption on n all elements of H^{1}(Γ, X(_{u}T)_{∗}) have exponent n. The result follows. □

The lemma shows that z_{3} and z_{4} are equivalent to the cocycles (a′_{σ}(x)aσ^{′′}) and (b′_{σ}(x)bσ^{′′}) respectively. Since the last two cocycles are constant, that is they belong to the image Im[Z^{1}(Γ, G(k_{s}(x))) → Z^{1}(Γ, G(k_{s}(x)[(t)^{±1}]))], and since the images of the classes of z_{3} and z_{4} in H^{1}(Γ, G(L̃)) ⊂ H^{1}(L̂, G) are equal [because of the commutativity of (5.25)], we can conclude with the aid of Theorem 5.23 that the classes of (a′_{σ}(x)aσ^{′′}) and (b′_{σ}(x)bσ^{′′}) are in fact equal in H^{1}(Γ, G(k_{s}(x))) ⊂ H^{1}(l, G) = H^{1}(k(x), G). This of course completes the proof that z_{1} and z_{2} are generically equivalent, that is [z_{1}]_{K} = [z_{2}]_{K} in H^{1}(K, G), as one can see “by replacing x by t”.

For later use let us formulate the following by-product of the above trick of introducing the new variable x.

5.30. Corollary. Assume that the twisted K-group_{z_{1}}G_{K} is irreducible. Then the twisted K̂-group _{z_{1}}GK̂ is also irreducible.

Proof. We reason by contradiction. For convenience let us denote _{z_{1}}G_{K} by H. Consider the commutative diagram

K = k(t) ⟶ k(x)(t) ⟶^{ϕ} k(x)(t)
↓  ↓  ↓
K̂ = k((t)) ⟶ k(x)((t)) ⟶^{ϕ} k(x)((t))

of field extensions. The two maps denoted by ϕ are the k(x)-morphisms given by t → xt^{n}, and the rest of the maps are natural inclusions. Assume HK̂ is reducible. By base change along the two bottom arrows we obtain a reducible k(x)((t))-group which for convenience we denote by A. On the other hand since H is irreducible, by Lemma 5.24 H_{k(t,x)} = H ×_{k(t)} k(t)(x) is an irreducible k(t)(x) = k(x, t) = k(x)(t)-group. We now apply to H_{k(t,x)} the base change given by ϕ. The resulting k(x)(t)-group, call it B, is the twist of G_{k(x,t)} by the cocycle z_{3}, which by Lemma 5.29 is a cocycle with values in G(k_{s}(x)). Since z_{1} becomes z_{3} if we change the variable t to x, the twisted group _{z_{3}}G_{k(x)} is irreducible. This implies that B is irreducible. But then A, which is obtained form B by the base change k(x)(t) → k(x)((t)), is according to Lemma 5.24 also irreducible–a contradiction. □

5.31. Remark. Let z ∈ Z^{1}(X, G). If the X-group _{z}G_{X} is irreducible then the K-group _{z}G_{K} is irreducible. For since the scheme P_{t(P)} of parabolic subgroups of _{z}G_{X} of type t(P) is proper over X and dim X = 1, we have P_{t(P)}(K) ≠ ∅ if and only if P_{t(P)}(X) ≠ ∅. By taking Corollary 5.30 into account we see that if _{z}G_{X} is irreducible then the twisted K̂-group _{z}GK̂ is also irreducible.

5.6.3. Reduction to the irreducible case. The third step of the proof is the reduction to the case when our twisted groups _{z_{i}}G_{X} are irreducible. We begin with a simple result.

5.32. Lemma. Let G be a reductive k-group and let η ∈ Z^{1}(k, G) be an arbitrary cocycle. If the injectivity assertion of Theorem 2.4 holds for the twisted k-group _{η}G then it holds for G.

Proof. Twisting by η induces a commutative diagram

H^{1}(X, G) ⟶ H^{1}(X,_{η}G)
f↓  ^{g}↓
H^{1}(K̂, G) ⟶ H^{1}(K̂,_{η}G)

whose horizontal arrows are the twisting bijections. It follows that f is injective if and only if g is injective. □

Assume now that our given cocycle z_{1} ∈ Z^{1}(Γ, G(k_{s}[t^{±1}])) is such that the twisted X-group _{z_{1}}G_{X} is reducible. Let P_{1} be a proper parabolic subgroup of _{z_{1}}G_{X}. Choose a k-point x ∈ X(k) and let η = z_{1}(x) ∈ Z^{1}(Γ, G(k_{s})) be the fiber of z_{1} at x. We set G′ = _{η}G. Since G′ is the fiber of _{z_{1}}G_{X} at x we conclude that G′ contains a parabolic subgroup P′ of the same type as P_{1}.

It follows easily from the definition that if z′_{1} = z_{1}η^{−1} is the cocycle corresponding to z_{1} under the twisting bijection H^{1}(X, G) → H^{1}(X, G′) then the twisted X-groups _{z_{1}}G_{X} and _{z′_{1}}G′_{X} are isomorphic; in particular z′_{1}G′_{X} has a parabolic subgroup of the same type as P′. By taking the last Lemma into consideration, and using the fact that the cohomological toral exponent of G and G′ coincide (Lemma 5.8), we conclude that to establish Proposition 5.20 we may assume without loss of generality that the k-group G and the X-group G_{1} = _{z_{1}}G_{X} have parabolic subgroups P and P_{1} of the same type. Under this assumption we then have.

5.33. Lemma. [z_{1}] ∈ Im [H^{1}(X, P) → H^{1}(X, G)].

Proof. Recall the general fact that the isomorphism class of cocycle z ∈ Z^{1}(X, G) is contained in the image of H^{1}(X, P) if and only if the twisted quotient scheme_{z}(G_{X}/P_{X}) has an X-point. On the other hand by reasoning as in [SGA3, XXIV Prop 4.2.1] we see that _{z}(G_{X}/P_{X}) is isomorphic to the scheme of parabolic subgroups of type t(P) of the twisted group scheme zG_{X} (see Remark 3.16). The result follows. □

The last Lemma shows that we may assume that z_{1} takes values in P. Let H be a Levi subgroup of P. Since X is affine, the map H^{1}(X, H) → H^{1}(X, P) is bijective [SGA3, XXVI.2.3] so we may additionally assume that [z_{1}] ∈ H^{1}(X, H).

We next remark that since z_{1} and z_{2} are rationally equivalent the twisted group scheme _{z_{2}}G_{K} also has a parabolic subgroup of type t(P). By Remark 5.31 the X-group_{z_{2}}G_{X} itself contains a parabolic subgroup of type t(P). The foregoing reasoning shows that we may also assume that [z_{2}] ∈ Z^{1}(X, H).

To complete the reduction to the irreducible case it will suffice to prove that the images of [z_{1}] and [z_{2}] under H^{1}(X, H) → H^{1}(K̂, H) are equal (for then if injectivity fails for G, it also fails for H, and we can assume from the outset that G was chosen of smallest possible dimension so that injectivity fails). For this in turn it suffices to show that the composition (of natural maps)

H^{1}(K̂, H) → H^{1}(K̂, P) → H^{1}(K̂, G)

is injective. This follows from the following two results.

5.34. Lemma. The mapping H^{1}(K̂, H) → H^{1}(K̂, P) is bijective.

Proof. This is a special case of [SGA3, XXVI.2.3]. □

5.35. Lemma. The mapping H^{1}(K̂, P) → H^{1}(K̂, G) is injective.

Proof. Let μ_{1}, μ_{2} ∈ Z^{1}(K̂, P) be such that f([μ_{1}]) = f([μ_{2}]). After twisting with μ_{2} we may assume that μ_{2} = 1 and f([μ_{1}]) = 1. Consider the exact sequence

G(K̂) ⟶^{g} (G/P)(K̂) ⟶ H^{1}(K̂, P) ⟶^{f} H^{1}(K̂, G).

By Borel-Tits theorem [BT, th. 4.13], the map g is surjective. Hence Ker f = 1 and therefore [μ_{1}] = 1 as desired. One may also quote [SGA3, XXVI.5.10]

□ 5.6.4. Reduction to anisotropic k-loop torsors. Before we can finish the proof of injectivity we need one more result related to a type of torsors, called “kloop torsors” in [GP3], that arise in connections with infinite dimensional Lie theory (see [GP1], [GP2] and [Pi] for details). According to the reformulation given in Proposition 5.20 we may assume that our cocycles z_{1} and z_{2} are of the form z_{1} = (a_{σ})_{σ∈Γ′} and z_{2} = (b_{σ})_{σ∈Γ′} where Γ′ = Gal(k′/k) is the Galois group of a finite extension k ⊂ k′ ⊂ k_{s}, and both the a_{σ} and the b_{σ} belong to N(k′[t^{±1}]). After further extending k′ if necessary, we may further assume that k′ contains a primitive n-root of unity (recall that n is prime to the characteristic of k) and splits our fixed maximal torus T ⊂ G. Consider a chain of finite field extensions

k(t) ⊂ k′(t) ⊂ L = k′(t′)

where t′ = t^{1/n}. The extensions L/k′(t) and L/k(t) are Galois. Let Λ = Gal (L/k(t)) and Δ = Gal (L/k′(t)). Clearly we have Λ = Δ ⋊ Γ′ where Γ′ = Gal(k′/k). The natural mappings Λ → Γ′, G(k′) → G(k′(t′)) and G(k′[t^{±1}]) → G(k′[(t′)^{±1}]) induce a diagram

H^{1}(Λ, G(k′))
g↓
H^{1}(Γ′, G(k′[t^{±1}])) ⟶^{f} H^{1}(Λ, G(k′[(t′)^{±1}])

5.36. Lemma. f([z_{1}]) and f([z_{2}]) are in the image of g.

Proof. Recall that z_{1} = (a_{σ}) where σ ∈ Γ′ and a_{σ} ∈ N(k′[t^{±1}]) ⊂ G(k′[t^{±1}]). Recall also that a_{σ} = a′_{σ}(t)aσ^{′′} where z′_{1} = (a′_{σ}(t)) and z1^{′′} = (aσ^{′′}) are the translation and linear parts of z_{1}.

Assume first that the linear part of z_{1} is trivial. Since Λ = Δ ⋊ Γ′, every λ ∈ Λ can be written uniquely in the form λ = δσ with δ ∈ Δ and σ ∈ Γ′. With this notation, and according to Remark 5.22, f([z_{1}]) is given by a family (a′_{λ}(t′)) where

a′_{λ}(t′) = a_{σ}(t) = a_{σ}((t′)^{n}) = a_{σ}(t′)^{n}.

Let u(t) ∈ X(T)_{∗} ⊂ T (k′[t^{±1}]) be such that

a_{σ}(t)^{n} = u(t)^{1−σ}. (5.37)

We claim that the cocycle (u(t′)^{−1}a′_{λ}(t′)u(t′)^{λ}) takes values in G(k′). Indeed, let λ = δσ. Given that δ(t′) = ζ_{n}t′ where ζ_{n} is a root of unity of degree n we have

(u(t′))^{δ} = vu(t′)

where v ∈ T (k′) has entries consisting of roots of unity of degree n [where we identify T_{k′} with G_{m}^{rk(G)}]. Furthermore, since σ acts trivially on t′

u(t′)^{λ} = (u(t′)^{δ})^{σ} = (vu(t′))^{σ} = v^{σ}u(t′)^{σ}

and equation (5.37) implies

a_{σ}(t′)^{n} = u(t′)^{1−σ}.

Thus we have

u(t′)^{−1}a′_{λ}(t′)u(t′)^{λ} = u(t′)^{−1}a_{σ}(t′)^{n}u(t′)^{λ} =
u(t′)^{−1}(u(t′)u(t′)^{−σ})(v^{σ}u(t′)^{σ}) = v^{σ}

This shows that the cocycle (u(t′)^{−1}a′_{λ}(t′)u(t′)^{λ}) takes values in G(k)′ as claimed. Since the class of this cocycle equals f([z_{1}])] the proof of the Lemma in the case when the linear part of z_{1} is trivial is complete.

In general since the linear part is constant, we can consider the twisted k-group _{z′′_{1}} G which for convenience we denote by G′. We then have the diagram

H^{1}(Λ, G′(k′))
g′↓
H^{1}(Γ′, G′(k′[t^{±1}])) ⟶^{f′} H^{1}(Λ, G′(k′[(t′)^{±1}])

One checks that the three relevant twisting bijections are compatible with the maps f and f′, and g and g′. This reduces the general problem to the case when the linear part is trivial. □

Before concluding the proof of injectivity we need one more final reduction.

5.38. Lemma. Let C = rad(G) the radical torus of G and consider its maximal split subtorus C_{d}. If the natural map H^{1}(X, G/C_{d}) → H^{1}(K̂, G/C_{d}) is injective, then the natural map H^{1}(X, G) → H^{1}(K̂, G) is also injective.

Proof. Since C_{d} is central the fibers of the natural map H^{1}(X, G) → H^{1}(X, G/C_{d}) arise as quotients of H^{1}(X, C_{d}) = 1, so our map is injective. □

5.6.5. Proof of injectivity. We finally come to the proof of Proposition 5.20. By Lemma 5.38 we may assume that the connected centre of our reductive group G is an anisotropic k-torus, and that, furthermore, the twisted Xgroups _{z_{1}}G_{X} and _{z_{2}}G_{X} are irreducible. By Remark 5.31 the K̂-groups _{z_{1}}GK̂ and _{z_{2}}GK̂ are irreducible. Since the radical tori of the _{z_{i}}GK̂ are isomorphic to rad(G) ×_{k} K̂, it follows from Proposition 3.17.2 that the K̂-groups _{z_{1}}GK̂ and _{z_{2}}GK̂ are anisotropic.

By Lemma 5.36 we may also assume that z_{1} = (a_{λ})_{λ∈Λ} and z_{2} = (b_{λ})_{λ∈Λ} are cocycles in Z^{1}(Λ, G(k′)) where Λ is the Galois group of the extension L = k′(t′) of k(t) described in 5.6.4. We will finish the proof by showing that there exists g ∈ G(k′) such that a_{λ} = g^{−1}b_{λ} g^{λ} for all λ ∈ Λ.

To this end we consider the two extended Bruhat-Tits buildings [BT2, §4.2.16] BK̂ and BL̂ of G over K̂ and L̂ = k′((t′)) respectively, as well as the apartment AL̂ corresponding to TL̂. Recall that AL̂ = X(T)_{∗} ⊗_{ℤ} ℝ and denote by o its origin. The group G(L̂) acts on BL̂. We also have a canonical action of Λ = Gal (L/K) ≃ Gal (L̂/K̂) on BL̂ and a canonical identification BK̂ = BΛ ; L̂. We also observe that the action of G(L̂) on BL̂ is Λ-equivariant. By construction, the point o is fixed by Λ. Furthermore the k′[[t′]]-parahoric group scheme attached to o is nothing but the Chevalley group scheme G ×_{k} k′[[t′]] [BT2, 3.2.13]. Hence the the stabilizer of o in G(L̂) is G(k′[[t′]]).

The two cocycles z_{1} and z_{2} give rise to two twisted actions of Λ on BL̂, namely λ_{1}(x) = a_{λ}(λ(x)) and λ_{2}(x) = b_{λ}(λ(x)) for all x ∈ BL̂. The invariant subsets in BL̂ with respect to these two twisted actions of Λ are buildings of the twisted K̂-groups _{z_{1}}GK̂ and _{z_{2}}GK̂. Since these twisted K̂-groups are anisotropic, by the Bruhat-Tits-Rousseau’s theorem ([Ro] and [Pr]) the fixed point sets under consideration consist of unique points. Since z_{1} and z_{2} take values in G(k′) ⊂ G(k′[[t′]]) and G(k′[[t′]]) is the stabilizer of o, these fixed points are necessary the origin o.

Since z_{1} and z_{2} are cocycles in Z^{1}(Λ, G(k′)) ⊂ Z^{1}(Gal(L/k(t)), G(L)) which are rationally equivalent, there exists ĝ ∈ G(L) ⊂ G(L̂) such that

a_{λ} = g^{−1} b_{λ} g^{λ}. (5.39)

It is easy to see that g^{−1}o is invariant with respect to the second twisted action of Λ. Hence g^{−1}o = o, which shows that ĝ ∈ G(k′[[t′]]). We now “evaluate (5.39) at t′ = 0”, namely we apply the base change given by the residue map k′[[t′]] → k′. Since this evaluation map commutes with the action of the Galois group Λ, for g = ĝ(0) ∈ G(k′) we finally obtain a_{λ} = g^{−1} b_{λ} g^{λ} as desired. □

6. Appendix: The absolutely reduced subscheme attached to a scheme

Throughout the appendix k denotes a field, \overline{k} an algebraic closure of k and k_{s} the separable closure of k is \overline{k}. The nilradical of a (commutative unital) ring A will be denoted by 𝔫(A). We fix a basis (α_{i})_{i∈I} of k̄ viewed as a k-space. The category of commutative associative and unital algebras over k will be denoted by k-alg.

Let A be an object of k-alg. For convenience we set A ⊗_{k} k̄ = \overline{A}. For a subspace V of A we let \overline{V} = V ⊗_{k} k̄ ⊂ \overline{A}. We will make repeated use of the following elementary fact.

6.1. Lemma. (Basic Lemma) Let V be a subspace of A. Assume x = ∑ x_{i} ⊗ α_{i} belongs to \overline{V} ⊂ \overline{A}. Then x_{i} ∈ V for all i.

Proof. Extend a k-basis of V to a basis of A. □

Following [RR] we consider the following ideal of A:

\overline{𝔫}(A) = ⋂_{\begin{matrix}I⊲A \\ \overline{I}⊃𝔫(\overline{A}) \end{matrix}} I,

namely the intersection of all ideals I of A for which \overline{I} = I ⊗_{k} k̄ contain the nilradical of \overline{A}. Note that \overline{I} = I\overline{A}.

Recall that given x ∈ \overline{A} we can uniquely write x = ∑ x_{i} ⊗ α_{i} where x_{i} ∈ A.

6.2. Lemma. \overline{𝔫}(A) = ⟨ x_{i} | x ∈ 𝔫(\overline{A}) ⟩.

Proof. For convenience let us denote the ideal ⟨ x_{i} | x ∈ 𝔫(A) ⟩ by J. It is clear from the definition that 𝔫(\overline{A}) ⊂ J ⊗_{k} k̄. Thus, it will suffices to show that J ⊂ I whenever I is an ideal of A for which I ⊗_{k} k̄ ⊃ 𝔫(\overline{A}). But this is an immediate consequence of the basic Lemma. □

6.3. Remark. It follows from the Lemma (but also from elementary linear algebra considerations) that J does not depend of the choice of basis (α_{i})_{i∈I}.

6.4. Corollary. For A to be absolutely reduced it is necessary and sufficient that \overline{𝔫}(A) = 0.

6.5. Remark. It is incorrectly asserted in [RR] that the ring A/\overline{𝔫}(A) is absolutely reduced, namely that A/\overline{𝔫}(A) ⊗_{k} k̄ is reduced. The following counterexample is due to A. Merkurjev.

6.6. Example. Let k be a separably closed field of characteristic 2 such that k̄ ≠ k. Consider the polynomial f(x, y) = x^{4} + ay^{2} ∈ k[x, y] where a ∈ k is not a square. It is easy to see that f is irreducible over k and hence A = k[x, y]/(f(x, y)) is a domain. However in k̄[x, y] we have f(x, y) = (x^{2}+by)^{2}, where b = \sqrt{a}, whence 𝔫(\overline{A}) = (x̄^{2} + bȳ). From Lemma 6.2 it follows that \overline{𝔫}(A) = ⟨x̄^{2}, ȳ⟩. This implies that the ring

A/\overline{𝔫}(A) ≃ k[x, y]/⟨x, y⟩ ≃ k[x]/(x^{2})

is not reduced.

6.7. Proposition. Let ψ : A → B be a morphism in k-alg. If B is absolutely reduced then ψ factors through A/\overline{𝔫}(A).

Proof. Since 𝔫(\overline{B}) = 0 we have 𝔫(\overline{A}) ⊂ ker (ψ ⊗ 1) = ker (ψ) ⊗ k̄. By taking Lemma 6.2 into consideration, this yields \overline{𝔫}(A) ⊂ ker ψ as desired. □

6.8. Lemma. Assume A is a Hopf algebra. Then \overline{𝔫}(A) is a Hopf ideal of A.

Proof. Fix a k-basis {a_{s}}_{s∈S} of A where S = S_{1} ⨆ S_{2} and {a_{s}}_{s∈S_{1}} is a basis of \overline{𝔫}(A). We have the commutative diagram

A ⟶^{Δ} A ⊗_{k} A
↓  ↓
\overline{A} ⟶_{\overline{Δ}} \overline{A} ⊗k̄ \overline{A}

and the canonical isomorphism

ψ : \overline{A} ⊗k̄ \overline{A} → (A ⊗_{k} A) ⊗_{k} k̄.

Let x ∈ 𝔫(\overline{A}) and write x = ∑ x_{i} ⊗ α_{i}. By Lemma 6.2 it will suffice to show that

Δ(x_{i}) ∈ \overline{𝔫}(A) ⊗ A + A ⊗ \overline{𝔫}(A),

namely that if we write

Δ(x_{i}) = ∑_{s,t∈S} c_{st}^{(i)} a_{s} ⊗ a_{t}

then c_{st}^{(i)} = 0 whenever both s and t belong to S_{2}.

Recall that by Lemma 6.2 we have 𝔫(\overline{A}) ⊂ \overline{𝔫}(A) ⊗_{k} k̄. Since 𝔫(\overline{A}) is a Hopf ideal of \overline{A}

\overline{Δ}(x) ∈ 𝔫(\overline{A}) ⊗k̄ \overline{A} + \overline{A} ⊗k̄ 𝔫(\overline{A}).

Thus

(6.9)  ψ(\overline{Δ}(x)) ∈ (\overline{𝔫}(A) ⊗_{k} A) ⊗_{k} k̄ + (A ⊗_{k} \overline{𝔫}(A)) ⊗_{k} k̄.

On the other hand

ψ(\overline{Δ}(x)) = ψ (\overline{Δ} (∑_{i} x_{i} ⊗ α_{i}))

= ∑_{i} ψ(\overline{Δ}(x_{i} ⊗ α_{i}))

= ∑_{i} (∑_{s,t} c_{s,t}^{(i)} a_{s} ⊗ a_{t}) ⊗ α_{i}

= ∑_{i,s,t} a_{s} ⊗ a_{t} ⊗ c_{st}^{(i)}α_{i}.

By (6.9) we have c_{s,t}^{(i)} = 0 whenever both s and t belong to S_{2}. □

Define a sequence j_{0}(A) ⊂ j_{1}(A) ⊂ ... of ideals of A inductively as follows:

j_{0}(A) = \overline{𝔫}(A)

j_{i+1}(A) = \{x ∈ A : x + j_{i}(A) ∈ \overline{𝔫}(A/j_{i}(A))\}.

6.10. Proposition. Let j(A) = ⋃_{i≥0} j_{i}(A).

(i) If A is a Hopf algebra then j(A) is a Hopf ideal of A.

(ii) If A is noetherian then A/j(A) is absolutely reduced.

(iii) If ψ : A → B is a morphism in k-alg and B is absolutely reduced, then ψ factors through A/j(A).

Proof. The first assertion follows by induction on i with the aid of Lemma 6.8. By Corollary 6.4 we see that if j_{i}(A) = j_{i+1}(A) then A/j_{i}(A) is absolutely reduced. This establishes (ii). Finally if ψ is as in (iii), then by Proposition 6.7 ψ factors through A/\overline{𝔫}(A), and one concludes by the inductive definition of j(A). □

We assume for the remainder of this section that the k-algebra A is of finite type, and denote A/j(A) by A_{r}. If k′/k is a field extension we have a natural inclusion \overline{𝔫}(A⊗_{k}k′) ⊂ \overline{𝔫}(A)⊗_{k}k′. This yields j(A⊗_{k}k′) ⊂ j(A)⊗_{k}k′, hence a canonical surjective k′-algebra homomorphism

χ_{k′} : (A ⊗_{k} k′)_{r} ⟶ A_{r} ⊗_{k} k′

It will be convenient to reformulate the result under consideration in terms of k-schemes. Set X = Spec(A) and X_{r} = Spec(A_{r}). Then χ_{k′} corresponds to the closed immersion

(6.11)  Spec(χ_{k′}) : Spec(A_{r} ⊗_{k} k′) ⟶ Spec((A ⊗_{k} k′)_{r}).

6.12. Proposition. If k′ ⊂ k_{s} then χ_{k′} is an isomorphism.

Proof. We first prove the useful

6.13. Lemma. Let \overline{X(k_{s})} denote the closure of X(k_{s}) ⊂ X(k̄) with respect to the Zariski topology of X(k̄). Then X_{r}(k̄) = \overline{X(k_{s})}.

Proof. Since X_{r} is absolutely reduced the inclusion \overline{X_{r}(k_{s})} ⊂ X_{r}(k̄) is an equality [Bor, AG 13.3]. Thus

X_{r}(k̄) = \overline{X_{r}(k_{s})} ⊂ \overline{X(k_{s})}.

For the reverse inclusion it will suffice to show that X(k_{s}) ⊂ X_{r}(k̄) since X_{r}(k̄) is a closed subset of X(k̄). Now X(k_{s}) = Hom (Spec(k_{s}), X). Since Spec(k_{s}) is absolutely irreducible any morphism ϕ : Spec(k_{s}) → X factors through X_{r}. Thus the inclusion X_{r}(k_{s}) ⊂ X(k_{s}) is in fact an equality. □

We now finish the proof of the Proposition. Assume that k′ ⊂ k_{s}. Then to show that χ_{k′} is an isomorphism we may pass to k̄. This corresponds to the closed immersion

(6.14)  Spec(A_{r} ⊗_{k} k̄) ↪ Spec ((A ⊗_{k} k′)_{r} ⊗_{k′} k̄).

By the last Lemma both of these affine varieties have the same k̄-points, so their underlying topological subspaces agree ([DG] I §3 prop. 6.8). This means that A_{r} ⊗_{k} k̄ is defined by an ideal inside the nilradical of (A ⊗_{k} k′)_{r} ⊗_{k′} k̄. Since this last k̄-algebra is reduced (Proposition 6.10) the result follows. □

A more functorial approach to the type of problem we have considered in this Appendix can be found in B. Conrad’s recent preprint [Crd].

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Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

E-mail address: chernous@math.ualberta.ca

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

E-mail address: gille@ens.fr

Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

E-mail address: a.pianzola@math.ualberta.ca