Galois cohomology and forms of algebras over Laurent polynomial rings

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

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

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

Contents

1 Introduction 2 1.1 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Brief description of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Torsors over k[t_{1}^{±1}, t_{2}^{±1}] and Serre’s Conjecture II 5 2.1 Cohomology of finite modules . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Serre’s conjecture II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Some evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Loop torsors 9 3.1 The Witt-Tits index of a loop torsor . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Almost commutative subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Almost commuting families of elements of finite order . . . . . . . . . . . . 16 3.5 Almost commuting pairs and their invariants . . . . . . . . . . . . . . . . . 17 3.6 Failure in the anisotropic case . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Twisted forms of algebras over rings 27 4.1 Multiplication algebras and centroids . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Forms of simple algebras over rings . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 Invariance of type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.4 Twisted automorphism groups . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.5 Graded considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5 Forms of algebras over Laurent polynomial rings 37

5.1 Multiloop Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 The case of Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.3 A characterization of multiloop algebras . . . . . . . . . . . . . . . . . . . . 45

6 Conjectures 46

1 Introduction

The main thrust of this work is the study of two seemingly unrelated questions: Non-abelian Galois cohomology of Laurent polynomial rings on the one hand, while on the other, a class of infinite dimensional Lie algebras which, as rough approximations, can be thought off as higher nullity analogues of the affine Kac-Moody Lie algebras.

Though the algebras in question are in general infinite dimensional over the given base field (say the complex numbers), they can be thought as being finite provided that the base field is now replaced by a ring (in this case the centroid of the algebras, which turns out to be a Laurent polynomial ring). This leads us to the theory of reductive group schemes as developed by M. Demazure and A. Grothendieck [SGA3]. Once this point of view is taken, Algebraic Principal Homogeneous Spaces^{1} and their accompanying non-abelian étale cohomology, arise naturally. It is this geometrical approach to infinite dimensional Lie theory that is one of the central themes of our work.

To illustrate these ideas, let us briefly look at the case of affine Kac-Moody Lie algebras over an algebraically closed field k of characteristic 0. Let L̂ be such an algebra, and let L be the derived algebra of L̂, modulo its centre. The Lie algebra L̂ can be recovered from L (by taking the universal central extension and then attaching a derivation), and we will now concentrate on the Lie algebra L itself. Recall that the centroid of L is the subring Ctd_{k}(L) ⊂ End_{k}(L) comprised of elements that commute with the Lie bracket of L. The k–Lie algebra L is infinite dimensional, but by viewing now L as an algebra over its centroid in the natural way, we find ourselves back on the finite world: There exists a finite dimensional simple Lie algebra 𝔤, and a finite Galois extension S of Ctd_{k}(L), such that L ⊗_{Ctd_{k}(L)} S and 𝔤 ⊗_{k} S are isomorphic as S–Lie algebras. Since the centroid of an affine algebra can be identified with the Laurent polynomial ring k[t^{±1}], we see that L is a twisted forms, for the étale topology on Spec(k[t^{±1}]), of the k[t^{±1}]–algebras 𝔤 ⊗_{k} k[t^{±1}]. Accordingly, we can attach to L a torsor X_{L} whose isomorphism class lives in H_{ét}^{1}(k[t^{±1}], Aut(𝔤)).

1Also called Torsors. This terminology is due to Giraud.

More generally, we could take L̂ to be an Extended Affine Lie Algebra (EALA for short), and L its centreless core ([AABFP], [N1] and [N2]). This is a beautiful class of infinite dimensional Lie algebras with striking connections to the classical finite dimensional landscape (simple Lie algebras of course, but also Jordan algebras, alternative algebras, quadratic forms...). The centroids are now Laurent polynomial rings R_{n} = k[t_{1}^{±1}, ⋯ t_{n}^{±1}] in finitely many variables. Just as in the affine Kac-Moody case, the resulting algebras L are twisted forms of algebras of the form 𝔤 ⊗_{k} R_{n}.

These examples hint towards a possible deep connection between some aspects of contemporary infinite dimensional Lie theory, and the Galois cohomology of the ring R_{n} = k[t_{1}^{±1}, ⋯ t_{n}^{±1}]. The present paper sets out to explore this possibility.

1.1 Notation and conventions

To help the reader we list below most of the notation and conventions used throughout the paper.

k denotes a field which, with the sole exception of §5, is assumed to be of characteristic 0. As usual, \overline{k} will denote an algebraic closure of k. The tensor product ⊗_{k} will be denoted by the unadorned symbol ⊗.

k–alg denotes the category of commutative associative unital k-algebras.

R_{n} = k[t_{1}^{±1}, ⋯ t_{n}^{±1}], R_{n,d} = k[t_{1}^{± \frac{1}{d}} , t_{2}^{± \frac{1}{d}} , ⋯ t_{n}^{± \frac{1}{d}}] and R_{n,∞} = \underrightarrow{lim}_{d}R_{n,d}.

K_{n} = k(t_{1}^{±1}, ⋯ t_{n}^{±1}), K_{n,d} = k(t_{1}^{± \frac{1}{d}} , t_{2}^{± \frac{1}{d}} , ⋯ t_{n}^{± \frac{1}{d}}) and K_{n,∞} = \underrightarrow{lim}_{d}K_{n,d}.

F_{n} = k((t_{1}))((t_{2}))...((t_{n})).

π_{1}(R_{n}) denotes the algebraic fundamental group of Spec(R_{n}) at the geometric point Spec(\overline{F_{n}}) for some (fixed) algebraic closure \overline{F_{n}} of F_{n}.

(ζ_{n})_{n≥1} is a set of compatible primitive n-roots of unity, i.e. ζ_{ℓn}^{ℓ} = ζ_{n} (in the case when k is algebraically closed of characteristic 0).

For a given R in k–alg, by an R–group we will understand an affine group scheme over Spec(R). If G is an R–group, the pointed set of non-abelian Čech cohomology on the étale site of X = Spec(R) with coefficients in G, is denoted by H_{ét}^{1}(X, G), or also by H_{ét}^{1}(R, G) (accordingly with customary usage depending on the context). At times, specially during proofs in order to cut down on notation, we write H^{1} instead of H_{ét}^{1}. When G is smooth, the set H_{ét}^{1}(R, G) measures the isomorphism classes of principal homogeneous spaces (torsors) over X under G (see Ch. IV §1 of [M] for basic definitions and references).

Given an R–group G and a morphisms R → S in k–alg, we let G_{S} denote the S-group G ×_{Spec(R)} Spec(S) obtained by base change. For convenience, we will under these circumstances denote most of the times H_{ét}^{1}(S, G_{S}) by H_{ét}^{1}(S, G).

The expression linear algebraic group (defined) over k, is to be understood in the sense of Borel [Bo]. For a k-group G, this is equivalent to requiring that G be smooth.^{2} The connected component of the identity of such group G, will be denoted by G^{0}.

A reductive R–group is to be understood in the sense of [SGA3]. In particular, a reductive k–group is a reductive connected algebraic group defined over k in the sense of Borel.

1.2 Brief description of contents

Section 2. This section is devoted to the étale cohomology of Laurent polynomials ring k[t_{1}^{±1}, t_{2}^{±1}] in two variables, and connections with Serre’s Conjecture II for the corresponding function field k(t_{1}, t_{2}) (the analogue of Conjecture I was dealt with in [P2]).

Section 3. Loop torsors, the main topic of this section, are the torsors that are of interest in infinite dimensional Lie theory. One of the main results of this section is the existence of an invariant, the Witt-Tits index, that can be attached to loop torsors. In the case of two variables, we defined another invariant with values in the Brauer group. One of the major results of the paper (Theorem 3.17) asserts that this Brauer invariant is fine enough to classify inner loop torsors. It is not however fine enough to distinguish torsors in general (§3.6), which leads to the failure of the analogue of Serre’s Conjecture II for the ring k[t_{1}^{±1}, t_{2}^{±1}].

Section 4. As explained in the Introduction, the study of Extended Affine Lie Algebras is intimately related to the study of k[t_{1}^{±1}, ⋯ t_{n}^{±1}]–forms of finite dimensional simple Lie algebras. This section contains general results about forms of arbitrary finite dimensional algebras (mostly assumed to be central and perfect) over rings.

Section 5. This is detailed study of the nature of forms of algebras in the case when the base ring is a Laurent polynomial ring in finitely many variables. Particular emphasis is put on the case when the base algebra is a finite dimensional simple Lie algebra, and the ensuing connections with Extended Affine Lie Algebras.

Section 6. This section contains several conjectures related to the Galois cohomology of k[t_{1}^{±1}, t_{2}^{±1}], and the classification of Extended Affine Lie Algebras in nullity 2.

2A smooth k–group is affine (by our convention on k–groups), and algebraic (since smooth schemes are by definition locally of finite type).

2 Torsors over k[t_{1}^{±1}, t_{2}^{±1}] and Serre’s Conjecture II

Throughout this section k is assumed to be algebraically closed. Torsors over k[t_{1}^{±1}] were studied in [P2], and behave according to an analogue of Serre’s Conjecture I (Steinberg’s theorem). We now look at the case of k[t_{1}^{±1}, t_{2}^{±1}]. The situation here is much more delicate, and some of the results perhaps unexpected. We shall come back to this section when we look at Extended Affine Lie Algebras of nullity 2 in §6.

We set R = k[t_{1}^{±1}, t_{2}^{±1}], K = k(t_{1}, t_{2}), and F = k((t_{1}))((t_{2})).

2.1 Cohomology of finite modules

We start by collecting some basic facts about the étale cohomology of R.

Proposition 2.1. 1. Gal(F) ≃ π_{1}(R) ≃ Ẑ × Ẑ.

2. H_{ét}^{1}(R,μ μ μ_{n}) ≃ H_{ét}^{1}(F,μ μ μ_{n}) ≃ (ℤ/nℤ)^{2}.

3. H_{ét}^{2}(R,μ μ μ_{n}) ≃ H_{ét}^{2}(F,μ μ μ_{n}) ≃ ℤ/nℤ.

4. Br(R) ≃ ℚ/ℤ and _{n} Br(R) ≃ ℤ/nℤ. The canonical maps Br(R) → Br(F) and nBr(R) →_{n} Br(F) are isomorphisms.

5. Let S/R be a finite connected étale cover of degree d. The restriction map Res_{S/R} : H^{2}(R,μ μ μ_{n}) → H^{2}(S,μ μ μ_{n}) is mutiplication by d, and the corestriction map Cor_{S/R} : H^{2}(S,μ μ μ_{n}) → H^{2}(R,μ μ μ_{n}) is the identity.

6. Given i ∈ ℤ, the class [i] ∈ ℤ/nℤ ≅_{n} Br(R) is represented by the R–Azumaya algebra A(i, n) with presentation

T_{1}^{n} = t_{1}, T_{2}^{n} = t_{2}^{i}, T_{2}T_{1} = ζ_{n} T_{1}T_{2}.

Proof. (1) Consider the Galois covering R_{2,d} = R[t_{1}^{±1/d}, t_{2}^{±1/d}] of R = R_{2}, with Galois group (Z/dZ)^{2} generated by τ_{i} (i = 1, 2) defined by

τ_{i}(t_{j}^{1/d}) = (ζ_{d})^{δ_{i,j}} t_{j}^{1/d}.

The crucial point is that every connected finite étale cover of R is dominated by one of these Galois extensions. Thus π_{1}(R_{2}) = \underleftarrow{lim}_{d} Gal(R_{2,d}/R) = \underleftarrow{lim}_{d} (ℤ/dℤ)^{2} = (ℤ̂)^{2} (see [GP1, cor. 3.2] and [GP2] for details).

(2) , (3) and (4) : see [GP2, §3 and §4.2]. Because of (4), the calculations of (5) can be carried on F instead of R. One can now reason as in proposition 6.3.9 of [GS]. By (3) and (4) we have ℤ/nℤ ≃ H^{2}(R,μ μ μ_{n}) ≃ _{n} Br(R) ≃ _{n} Br(F). So for establishing (6), it suffices to show that the element [i] ∈ ℤ/nℤ is represented by the F –algebra A(i, n) ⊗_{R} F. But this is Proposition 5.7.1 of [GS].

Remark 2.2. Note that the isomorphisms of (1) and (2) above depend on the choice of a compatible set of primitive roots of unity (ζ_{n})_{n≥1}.

Remark 2.3. Let Inv : Br(R) → ℚ/ℤ be the group isomorphism constructed in Part (4) of the above Proposition. Let A be an R–Azumaya algebra of degree n, and write Inv(A) = \frac{p}{q} with p, q coprime. Then the R–algebras A and A(p, q) are Brauer equivalent. We claim that q divides n. Since Br(R) ≃ Br(F), the statement can be checked for A ⊗_{R} F. But this central simple F -algebra has period (exposant) q, and we do know that q divides the degree n of A ⊗_{R} F (cf. [GS], § 4.5).

Remark 2.4. The class of 1 ∈ Z/nZ ≃ H^{2}(R, Z/nZ) ≃ _{n}Br(R) is nothing but the cup–product χ_{1} ∪ χ_{2}, where χ_{i} : Ẑ × Ẑ → Z/nZ stand for the unique continuous character satisfying χ_{i}(n_{1}, n_{2}) = n_{i} mod (n), for all n_{1}, n_{2}, ∈ ℤ ([GS] Remark 6.3.8).

2.2 Serre’s conjecture II

By theorem 2.1 of [CTGP], the cohomology and classification of semisimple groups over a two-dimensional geometric field with no factors of type E_{8}, is known. By adding some extra assumptions to the nature of the geometric field (assumptions which hold in the case of k(t_{1}, t_{2}) that we are interested in), the groups with factors of type E_{8} can also be covered. More precisely:

Theorem 2.5. Let K be a two-dimensional function field over k. Let G be a semisim-

ple, simply connected K–group, and G = ∏_{i=1}^{r} R_{K_{i}/K}(G_{i}) the decomposition of G in

almost simple factors. Assume that K_{i} = L_{i}(x_{i}), where L_{i} is a field of transcendence degree one over k for all i = 1, ..., r. Let

1 → μ μ μ → G → G^{ad} → 1

be the central isogeny associated to the centre μ μ μ of G. Then the boundary map

δ : H^{1}(K, G^{ad}) → H^{2}(K,μ μ μ)

is bijective.

Proof. Since groups of type E_{8} have trivial centre, we can consider separately groups with no E_{8} factors, and groups of type E_{8}. As it has already been mentioned, for groups G with no E_{8} factors, the Theorem at hand is a special case of theorem 2.1 of [CTGP]. Let E_{8} be the Chevalley k–group of type E_{8}. This group is simply connected and we have E_{8} ≅ Aut(E_{8}). Therefore by Shapiro’s formula, it will suffice to show that H^{1}(K_{i}, E_{8}) = 1. Let L be an extension of k of transcendence degree one. For any closed point x of the projective line ℙ_{L}^{1}, the completion of L(t) at x is isomorphic to L(x)((π_{x})). The field L(x) is a finite extension of L, so cd(L(x)) ≤ 1. By Bruhat-Tits theory, we have H^{1}(L(x)((π_{x})), E_{8}) = 1 ([BT], théorème 4.7). In other words, all classes of H^{1}(L(t), E_{8}) are unramified with respect to the closed points of the projective line P_{L}^{1}. By Harder’s lemma ([H], Lemma 4.1.3), the map H^{1}(ℙ_{L}^{1}, E_{8}) → H^{1}(L(t), E_{8}) is then surjective. According to the theorem of Grothendieck-Harder (cf. [Gi1], th. 3.8.a), we have the following exact sequence of pointed sets

1 → H_{Zar}^{1}(ℙ_{L}^{1}, E_{8}) → H^{1}(ℙ_{L}^{1}, E_{8}) ⟶^{ev_{∞}} H^{1}(L, E_{8}),

where ev_{∞} stands for the pull-back map defined by the point at infinity. But H^{1}(L, E_{8}) = 1 by Steinberg’s theorem. This implies that H_{Zar}^{1}(ℙ_{L}^{1}, E_{8}) ≅ H^{1}(ℙ_{L}^{1}, E_{8}). As Zariski torsors are rationally trivial, we conclude that H^{1}(L(t), E_{8}) = 1 as desired.

Based on this last Theorem, an optimistic outcome for the G–cohomology of R takes the following form.

Question 2.6. Let G be a semisimple R–group. Let 1 → μ μ μ → G̃ → G → 1 be its universal covering. Is the boundary map H_{ét}^{1}(R, G) → H_{ét}^{2}(R,μ μ μ) bijective ?

In particular, if G is simply connected, the question is whether H_{ét}^{1}(R, G) vanishes, namely a variant of Serre’s conjecture II.

2.3 Some evidence

Theorem 2.7. Let G be a semisimple R–group, and 1 → μ μ μ → G̃ → G → 1 its universal covering.

1. The boundary map H_{ét}^{1}(R, G) → H_{ét}^{2}(R,μ μ μ) is surjective.

2. If G is split, then H_{ét}^{1}(R, G̃) = 1.

Proof. (1) Let α ∈ H^{2}(R,μ μ μ). We consider the restriction maps

H^{2}(R,μ μ μ) → H^{2}(K,μ μ μ) → H^{2}(F,μ μ μ).

By Theorem 2.5 δ_{K} : H^{1}(K, G) → H^{2}(K,μ μ μ) is an isomorphism, so there exist a nonempty open subvariety U ⊂ Spec(R), and a γ ∈ H^{1}(U, G), such that δ_{U}(γ) = α_{U} ∈ H^{2}(U,μ μ μ). For every point x ∈ Spec(R)^{(1)} \ U, the completed field K̂_{x} is a field of Laurent series over the residue field k(x), which is the field of functions of a k–curve. So

H^{1}(K̂_{x}, G) ≅ H^{2}(K̂_{x},μ μ μ)

(Theorem 2.5). On the other hand, consider the residue map

∂_{x} : H^{2}(K̂_{x},μ μ μ) → H^{1}(k(x),μ μ μ(−1))

where μ μ μ(−1) stands for the Tate twist μ μ μ(−1) = Hom(ℤ̂,μ μ μ) ≃ Hom(μ μ μ_{d},μ μ μ) for some d >> 0 (see [GMS] §I.7.8). Since cd(k(x)) = 1, ∂_{M} is an isomorphism. We summarize the previous facts in the commutative diagram

H^{1}(U, G) ⟶^{δ} H^{2}(U,μ μ μ)
↓  ↓
H^{1}(K̂_{x}, G) ⟶_{≃}^{δ} H^{2}(K̂_{x},μ μ μ) ⟶_{≃}^{∂_{x}} H^{1}(k(x),μ μ μ(−1)).

We have ∂_{x}(α_{U}) = 0, so δ(γ̂_{x}) = 0. It follows that γK̂_{x} = 1. By Harder’s lemma [H, lemma 4.1.3], the class γ extends on codimension 1 points of Spec(R) \ U. The variety Spec(R) is a smooth affine surface, so γ extends to Spec(R) [CTS, théorème 6.13], i.e there exists some class γ̃ ∈ H^{1}(R, G) such that γ̃_{U} = γ. By construction, δ_{F} (γ̃_{F}) = α_{F} ∈ H^{2}(F,μ μ μ). Since the restriction H^{2}(R,μ μ μ) → H^{2}(F,μ μ μ) is an isomorphism (Proposition 2.1(4)), we conclude that δ_{R}(γ) = α.

(2) Assume that G is split and simply connected. In particular, G is defined over k. We see R as the localisation of k[t_{1}, t_{2}] at t_{1}t_{2} ≠ 0. Let γ ∈ H^{1}(R, G). By Bruhat-Tits ([BT], cor 3.15), we have H^{1}(k(t_{1})((t_{2})), G) = 1 and H^{1}(k(t_{2})((t_{1})), G) = 1. Harder’s lemma implies that γ extends at the generic points of the subvarieties t_{1} = 0 and t_{2} = 0 of A_{k}^{2}. In other words, there exists an open subvariety U ⊂ A_{k}^{2} such that codim_{X}(U) = 2 and γ extends to U. Again by [CTS, théorème 6.13], γ extends to a class γ̃_{A_{k}^{2}}. Since H^{1}(A_{k}^{2}, G) = 1 ([Rg], cf. [CTO], proposition 2.4), it follows that γ = γ̃_{U} = 1 ∈ H^{1}(R, G) as desired.

Corollary 2.8. Every semisimple R–group scheme G of type E_{8}, F_{4} or G_{2} is split. Furhermore H^{1}(R, G) = 1.

Remark 2.9. Serre’s conjecture on rationally trivial torsors proven by Colliot-Thélène and Ojanguren ([CTO], théorème 3.2) permits to give a shorter proof of the second statement of Theorem 2.7. Taking into account Theorem 2.5, all R–torsors under G are rationally trivial. Thus H_{Zar}^{1}(R, G̃) = H^{1}(R, G̃), and therefore H^{1}(R, G̃) = 1 by [GP2, corollary 2.3].

One more piece of evidence towards a positive answer of Question 2.6 comes from reduced norms (in analogy with the theorem of Merkurjev and Suslin which characterizes fields of cohomological dimension 2 by the surjectivity of their reduced norms. See [Se], II.3.2).

Lemma 2.10. Let A(i, n) be the standard Azumaya R–algebra defined in Proposition 2.1. Then the reduced norm map ([K2], I.7.3.1.5) N : A(i, n)^{×} → R^{×} is surjective.

Proof. The algebra A = A(i, n) has presentation T_{1}^{n} = t_{1}, T_{2}^{n} = t_{2}^{i}, T_{2}T_{1} = ζ_{n} T_{1}T_{2}. In particular, it contains R[T_{1}] as a maximal commutative R–subalgebra. As in the field case, the restriction of the reduced norm to R[T_{1}]^{×} is nothing but the norm map N_{1} : R[T_{1}]^{×} → R^{×} of the cyclic Galois R–algebra R[T_{1}]. Since N_{1}(T_{1}) = (−1)^{n−1}t_{1}, we have (−1)^{n−1}t_{1}.R^{×}^{n} ⊂ N(A^{×}). So the group R^{×}/N(A^{×}) is cyclic generated by the class of t_{2}. Similarly, R[T_{2}] is also a cyclic Galois R–algebra and N_{2}(T_{2}) = (−1)^{n−1}t_{2}. We conclude that A(i, n)^{×} → R^{×} is surjective.

3 Loop torsors

Throughout this section the base field k is assumed to be algebraically closed and of characteristic 0.

Let G be a reductive k–group. We fix a maximal torus T of G, and a base Δ of the root system Φ = Φ(G, T). Let 𝔱 denote the Lie algebra of T. For any subset I ⊂ Δ ⊂ Φ(G, T) we set

𝔱_{I} = ⋂_{a∈I} ker(a) ⊂ 𝔱,

and let T_{I} be the subtorus of T with Lie algebra 𝔱_{I}. Define L_{I} = Z_{G}(T_{I}); this is a Levi subgroup of the standard parabolic group P_{I}.

In what follows, we will often encounter the following situation: H is a subgroup of G, and x x x = (x_{1}, x_{2}, ...., x_{n}) is an n–tuple of elements of H. We summarize this by simply saying that “x x x lies in H”.^{3}

Let x x x = (x_{1}, x_{2}, ...., x_{n}) be an n–tuple of commuting elements of finite order of G. Let d be an integer such that x_{1}^{d} = ⋯ = x_{n}^{d} = 1. Recall the Galois covering R_{n,d} = R[t_{1}^{±1/d}, ⋯, t_{2}^{±1/d}] of R_{n}, with Galois group (Z/mZ)^{n} generated by τ_{i} (i = 1, ..., n) defined by

τ_{i}(t_{j}^{1/d}) = (ζ_{d})^{δ_{i,j}} t_{j}^{1/d}.

This enables us to define the cocycle α(x x x) ∈ Z^{1}(Gal(R_{n,d}/R_{n}), G(R_{n,d})) as follows^{4}

α(x x x) : Gal(R_{n,d}/R_{n}) → G(k) → G(R_{n,d}), τ_{1}^{i_{1}} ⋯ τ_{n}^{i_{n}} ↦ x_{1}^{−i_{1}}x_{2}^{−i_{2}} ⋯ x_{n}^{−i_{n}}.

As it is customary, we denote the class of α(x x x) in H^{1}(R_{n}, G) by [α(x x x)]. Observe that this class is independent of the choice of the common period d.

3Strictly speaking, the x_{i} are elements of H(k). This abuse of notations, whenever harmless, will be used throughout.

4The somehow unnatural appearance of inverses in the definition of α(x x x), is consequence of the way α(x x x) arises from multiloop algebras. See §6 below for details.

Classes of the form [α(x x x)] as above are called loop classes. They form a subset H_{loop}^{1}(R_{n}, G) ⊂ H^{1}(R_{n}, G). The elements of these classes are called loop torsors.

There is another useful way of looking at loop torsors in terms of the algebraic fundamental group of R. Recall that π_{1}(R_{n}) ≃ (ℤ̂)^{n} under our fixed choice of compatible roots of unity, and that H_{ct}^{1}(π_{1}(R_{n}), G(R_{n,∞})) ⟶^{∼} H^{1}(R_{n}, G) (see [GP1, cor. 3.2], or [GP2, cor. 2.15]) The cocycle α(x x x) corresponding to x x x can thus be thought as the unique (continuous) map α(x x x) : (ℤ̂)^{n} → G(k) for which

α(x x x) : (i_{1}, ..., i_{n}) ↦ x_{1}^{−i_{1}}x_{2}^{−i_{2}} ⋯ x_{n}^{−i_{n}}

for all (i_{1}, ..., i_{n}) ∈ ℤ^{n}.

3.1 The Witt-Tits index of a loop torsor

The following Theorem is the crucial ingredient which will allow us to attach an index set I ⊂ Δ to a loop torsor. This result is also related to the work of Reichstein-Youssin [RY] linking non-toral abelian sugroups and the essential dimension of G. Our proof is a higher dimensional version in characteristic 0 of [Gi2] (e.g. proposition 3).

Theorem 3.1. Let x x x = (x_{1}, x_{2}, ...., x_{n}) be an n–tuple of commuting elements of finite order of G. For a subset I ⊂ Δ, the following conditions are equivalent:

1. x x x normalizes a parabolic subgroup P of G of type I,

2. [α(x x x)_{K_{n}}] ∈ im (H^{1}(K_{n}, P_{I}) → H^{1}(K_{n}, G)),

2^{bis}. The twisted group _{α(x x x)}G_{K_{n}} admits a K_{n}–parabolic subgroup of type I.

3 [α(x x x)_{F_{n}}] ∈ im (H^{1}(F_{n}, P_{I}) → H^{1}(F_{n}, G)),

3 ^{bis}. The twisted group _{α(x x x)}G_{F_{n}} admits a F_{n}–parabolic subgroup of type I.

In particular, if G is semisimple, then x x x is irreducible (i.e it does not lie in any proper parabolic subgroup of G), if and only if the corresponding twisted F_{n}–group α(x x x)G_{F_{n}} is anisotropic.

Proof. By a theorem of Chevalley ([Bo] theorem 11.6) P_{I} is its own normalizer in G. The equivalence (2) ⟺ (2^{bis}) and (3) ⟺ (3^{bis}) then follows from a classical lemma ([Se], III.2.2, lemme 1).

(1) ⟹ (2): Up to conjugacy by a suitable element g ∈ G(k), we may assume that the x_{i}’s lie in P_{I}(k). Because Gal(K_{n}) acts trivially on g, the g^{−1}α(x x x)g define a cocycle which is cohomologous to α(x x x). Now (2) is clear.

(2) ⟹ (3): Obvious.

(3) ⟹ (1): Consider the k–variety Y = G/P_{I} of parabolic subgroups of G of type I, as well as the corresponding twisted F_{n}–variety

X = _{α(x x x)}Y_{F_{n}}.

Our hypothesis is that X(F_{n}) ≠ ∅. We have

X(F_{n}) = {y ∈ Y(F_{n,m}) | α(x x x)(σ).^{σ}y = y for all σ ∈ Gal(F_{n,m}/F_{n})}.

Since Y is complete, we have

Y(F_{n−1,m}[[ \sqrt[m]{t}_{n}]]) = Y(F_{n,m}).

Now, by specializing at t_{n} = 0, we obtain

{y ∈ Y(F_{n−1,m}) | α(x x x)(σ).^{σ}y = y for all σ ∈ Gal(F_{n,m}/F_{n})} ≠ ∅,

where the Galois action of Gal(F_{n,m}/F_{n}) on Y(F_{n−1,m}), is induced by the canonical projection Gal(F_{n,m}/F_{n}) → Gal(F_{n−1,m}/F_{n−1}). Repeating the same process, we finally get

{y ∈ Y(k) | α(x x x)(σ).^{σ}y = y for all σ ∈ Gal(F_{n,m}/F_{n})} ≠ ∅,

and therefore

{y ∈ Y(k) | α(x x x)(σ).y = y ∀σ ∈ Gal(F_{n,m}/F_{n}))} ≠ ∅

since Gal(F_{n,m}/F_{n}) acts trivially on Y(k). But this means that all of the x_{i} normalize a k-parabolic subgroup of type I, hence (1).

Example 3.2. For the split group E_{8} and its standard non-toral abelian subgroup (ℤ/2ℤ)^{9}, the corresponding loop torsor is studied by Chernousov-Serre [CS]. Our result gives another proof that the associated twisted group defined over k(t_{1}, ..., t_{9}) is anisotropic.

Since assertions (2) and (3) of Theorem 3.1 are satisfied for a unique maximal index I, namely the Witt-Tits index ^{5} of _{α(x x x)}G_{K_{n}}, we get the following interesting fact.

5Let K/k be a field and [z] ∈ H^{1}(K, G). The twisted group _{z}G admits a single G(K)–conjugacy classe of minimal parabolic subgroups, and any such minimal P is G(\overline{K})–conjugated to a unique minimal standard parabolic subgroup P_{I} ⊂ G. This I is called the Witt-Tits index of _{z}G, and it depends only on [z] ∈ H^{1}(K, G) ([BoT], § 6.5). In terms of Galois cohomology, P_{I} is the unique minimal standard parabolic subgroup of G such that [z] ∈ Im(H^{1}(K, P_{I}) → H^{1}(K, G)).

Corollary 3.3. The minimal elements (with respect to inclusion) of the set of parabolic subgroups of G normalized by x_{1}, ..., x_{n} are all conjugate under G(k). The type I(x x x) of this conjugacy class is the Witt-Tits index of the F_{n}–group _{α(x x x)}G_{F_{n}}.

We call I(x x x) the Witt-Tits index of x x x. As we shall see later, this invariant plays a crucial role in the classification of loop torsors.

Remark 3.4. One can also define an index in the linearly reductive case by taking into account the star action of x x x on Δ. More precisely, assume F is a linear algebraic group whose connected component F^{0} := G is reductive. We have an exact sequence of algebraic groups

1 → G^{ad} → Aut(G) → Out(G) → 1

([SGA3] XXV théorème 1.3). Let x x x be an n–tuple of commuting elements of finite order of F. The above exacts sequence yields an action of < x x x > on Out(G). This is the first part of the invariant attached to x x x. The second part is defined as in the connected case, by replacing P_{I} by N_{F}(P_{I}) in Theorem 3.1. If G is semisimple and adjoint, then Out(G) can be identified with the group Aut(Δ) of automorphisms of Δ. By the uniqueness of type for outer forms, the action of x x x on Aut(Δ) leaves the index set I stable.

3.2 Reducibility

Recall that a subgroup H of a reductive k–group G is called G–irreducible if H is not contained in any proper parabolic subgroup P of G.

Recall also that, by definition, a reductive k–group is connected.^{6} There is a weaker notion, that of linearly reductive subgroup H of G, that we now need (see [BMR] for details). A linear algebraic group H is linearly reductive if every rational representation of H is semisimple. Because our algebraically closed base field is of characteristic 0, for H to be linearly reductive it is necessary and sufficient that H^{0} be reductive.

Lemma 3.5. Let G be a reductive k–group, and H a subgroup of G. If S is a maximal torus of Z_{G}(H), then S = Z(Z_{G}(S))^{0} and Z_{G}(S) is a Levi subgroup of a parabolic subgroup of G.

Proof. Let S′ be the identity component of the centre of Z_{G}(S). Clearly S ⊂ S′. Since H ⊂ Z_{G}(S), we have S′ ⊂ Z_{G}(H). Since S is a maximal torus of Z_{G}(H), we have S = S′. So S = Z(Z_{G}(S))^{0}. That Z_{G}(S) is the Levi subgroup of a parabolic subgroup of G is well known ([Bo] proposition 20.4).

6See the conventions in §1.1

Recall that if H is linearly reductive, then the centralizer Z_{G}(H) of H in G is also linearly reductive ([R], proposition 10.1.5). Corollary 3.5 of [BMR] and its proof state the following.

Theorem 3.6. Let H be a linearly reductive subgroup of a reductive k–group G, and let S be a maximal torus of Z_{G}(H). Then

1. H is irreducible in Z_{G}(S).

2. If P be a parabolic subgroup of G for which Z_{G}(S) is a Levi subgroup, then P is a minimal element (with respect to inclusion) of the set of parabolic subgroups of G that contain H.

3. If P is a minimal element (with respect to inclusion) of the set of parabolic subgroups of G that contain H, then H is L–irreducible for some Levi subgroup L of P.

Corollary 3.7. Let H be a linearly reductive subgroup of a reductive k–group G. For a parabolic parabolic subgroup P of G, the following conditions are equivalent:

1. P is a minimal element of the set of parabolic subgroups containing H.

2. There exists a maximal torus S of Z_{G}(H) such that Z_{G}(S) is a Levi subgroup of P.

Proof. (2) ⟹ (1): This is given by the second assertion of Theorem 3.6.

(1) ⟹ (2): Let L be a Levi subgroup of P for which H is L–irreducible (Theorem 3.6(3)). For S := Z(L)^{0}, we have L = Z_{G}(S) ([Bo] proposition 11.23 and corollary 14.19), and S ⊂ Z_{G}(H) ( since H ⊂ L). We claim that S is a maximal torus of Z_{G}(H). Let S′ ⊂ Z_{G}(H) be a torus containing S. Since S commutes with S′, we have S′ ⊂ Z_{G}(S) = L. If S ⊊ S′, then S′ is not central in L. Thus Z_{L}(S′) is a Levi subgroup of a proper parabolic subgroup Q of L (ibid. proposition 20.6). But since H commutes with S′, we have H ⊂ Z_{L}(S′) ⊂ Q. This contradicts our assumption on L–irreducibility.

The minimality of parabolic subgroups containing H can be tested on elements of finite order of H.

Corollary 3.8. Let H be a linearly reductive subgroup of a reductive group G. There exists elements x_{1}, ..., x_{n} ∈ H of finite order such that

1. Z_{G}(H) = Z_{G}(x_{1}, ..., x_{n}).

2. The abstract group < x_{1}, ..., x_{n} > is linearly reductive (i.e. the Zariski closure of < x_{1}, ..., x_{n} > in G is linearly reductive).

In this case, the minimal elements of the set of parabolic subgroups containing H are precisely the minimal elements of the set of parabolic subgroups containing {x_{1}, ..., x_{n}}.

Proof. Let H_{ss} (respectively H_{fin}), denote the subset of H(k) consisting of those elements which are semisimple (respectively of finite order).

We claim that Z_{G}(H) = ∩_{x∈H_{ss}}Z_{G}(x). For since the finite constant group H/H^{0} is diagonalizable, there exists a set of coset representatives of H^{0} in H comprised of semisimple elements. This, together with the fact that the semisimple elements of H^{0} are dense in H^{0} ([Bo] theorem 11.10), establishes the claim.

If x ∈ H_{ss}, then \overline{< x >} ⊂ H is a closed subgroup of a torus of G. In particular, the elements of finite order of \overline{< x >}, form a dense subset of \overline{< x >}. Together with our previous claim, this shows that Z_{G}(H) = ∩_{x∈H_{fin}}Z_{G}(x). Given that each Z_{G}(x) is closed and k[G] is noetherian, we conclude that Z_{G}(H) = Z_{G}(x_{1})∩....∩Z_{G}(x_{m}) := Z_{G}(x_{1}, ..., x_{m}) for some x_{1}, ..., x_{m} ∈ H_{fin}.

Observe that Z_{G}(x_{1}, ..., x_{m}) = Z_{G}(x_{1}, ..., x_{m}, x) for all x ∈ H_{fin}. To finish the proof of (1) and (2) therefore, it will suffice to show that there exists x_{m+1}, ..., x_{n} ∈ H_{fin} such that the abstract group < x_{1}, ..., x_{n} > is linearly reductive.

Let H_{1} = \overline{< x_{1}, ..., x_{m} >}. If H_{1}^{0} is reductive we are done. If not, the H_{1}^{0} = U_{1}.L_{1} where 1 ≠ U_{1} (resp. L_{1}) is the unipotent radical (resp. a Levi subgroup) of H_{1}^{0}. Since H^{0} has trivial unipotent radical, there exists x_{m+1} ∈ H_{fin}^{0} such that x_{m+1}U_{1}x_{m+1}^{−1} ≠ U_{1} (recall that H_{fin}^{0} is dense in H^{0}). Let H_{2} = \overline{< x_{1}, ..., x_{m}, x_{m+1} >}. If H_{1}^{0} = H_{2}^{0}, these two groups would have the same unipotent radical, namely U_{1}. But they do not: x_{m+1} does not normalize U_{1}. Thus H_{1}^{0} ⊊ H_{2}^{0}. One now considers the unipotent radical of H_{2}^{0} and repeats the above argument. Since the dimension of the resulting groups H_{1}^{0} ⊊ H_{2}^{0} ⊊ ... are bounded by H^{0}, there exists elements x_{m+1}, ..., x_{n} ∈ H_{fin} as desired.

Finally, let P be a parabolic subgroup of G which is minimal among those containing {x_{1}, ..., x_{n}}. Then P is minimal among those parabolic subgroups containing the group K = \overline{< x_{1}, ..., x_{n} >} . Since K is linearly reductive, there exists a maximal torus S of Z_{G}(K) such that L = Z_{G}(S) is a Levi subgroup of P (Corollary 3.7). By construction, Z_{G}(K) = Z_{G}(H). In particular, S commutes with H, and therefore H ⊂ L.

3.3 Almost commutative subgroups

Following [BFM], we say that a subgroup H of G is almost commutative if (H.Z(G))/Z(G) is abelian. The last assertion of the following result generalises lemma 2.1.2 of loc. cit. about compact Lie groups (see also [KS]).

Proposition 3.9. Let H be an almost commutative subgroup of G which is linearly reductive, and let S be a maximal torus of Z_{G}(H).

1. The k–parabolic subgroups of G which are minimal among those containing H are all conjugate under G(k). Furthermore, the type I(H) of this conjugacy class is the Witt–Tits index of some twisted inner form of G.

2. If J ⊂ Δ is such that S is conjugated to T_{J}, then J = I(H).

Proof. (1) After replacing G by G^{ad}, we may assume that H is an abelian subgroup of G. Corollary 3.8 provides an n–tuple x x x = (x_{1}, ..., x_{n}) of elements of finite order of G, such that the parabolic subgroups which are minimal among those containing H, are precisely those which are minimal among those containing {x_{1}, .., x_{n}}. We can now apply Corollary 3.3.1.

(2) Up to conjugacy, we may assume that T_{J} is a maximal torus of Z_{G}(H). Corollary 3.7 implies that P_{J} is a minimal parabolic subgroup containing H. By (1), we get that J = I(H).

Lemma 3.10. Let λ : G̃ → G be an isogeny of reductive k–groups. Let H be an almost commutative subgroup of G which is linearly reductive, and set H̃ = λ^{−1}(H). Then

1. H̃ is an almost commutative and linearly reductive subgroup of G̃.

2. The morphism ZG̃(H̃)^{0} → Z_{G}(H)^{0} is a central isogeny.

3. Via the natural identification of Δ with a base of the root system Φ(G̃, T̃), we have I(H̃) = I(H).

Proof. We begin by recalling certain relevant facts about isogenies (see [Bo]§22 for details). Because λ is separable, it is central. In particular, for a subgroup S of G to be a maximal torus, it is necessary and sufficient that λ^{−1}(S) be a maximal torus of G̃. Let T̃ = λ^{−1}(T). This is a maximal torus of G̃, and the standard map λ^{∗} : Φ(G, T) → Φ(G̃, T̃) is an isomorphism.

(1) The isogeny λ maps the unipotent radical of H̃^{0} injectively into the unipotent radical of H^{0}, which is trivial by assumption. This shows that H̃^{0} is reductive, hence that H is linearly reductive. Given that λ is central, that H̃ is almost commutative follows from the almost commutativity of H, together with Z(G̃) = ∩T̃ = ∩λ^{−1}(T) = λ^{−1}(∩T) = λ^{−1}(Z(G)) (the intersections being taken over all maximal torus of G̃ and G respectively).

(2) The induced map λ : ZG̃(H̃)^{0} → Z_{G}(H)^{0} has finite central kernel, so only surjectivity must be checked. It is enough to do this at the Lie algebra level. But here the situation is clear. Indeed

Lie(ZG̃(H̃)) = Lie(G̃)^{Ad H̃}, Lie(Z_{G}(H)) = Lie(G)^{Ad H},

and

Ad λ(x̃)(dλ(ỹ)) = dλ(Ad x̃(ỹ))

for all x̃ and ỹ in G̃, were dλ := Lie(λ) is the differential of λ ([DG] II §4.1 and §5 5.7).

(3) By (2) and the opening remarks on isogenies, the maximal tori of ZG̃(H̃)^{0} are precisely of the form λ^{−1}(S) for S a maximal torus of Z_{G}(H)^{0}. Now (3) follows from Proposition 3.9.

3.4 Almost commuting families of elements of finite order

Recall that an n–tuple x x x = (x_{1}, ..., x_{n}) of elements of G is said to almost commutes if [x_{i}, x_{j}] ∈ Z(G) for i, j = 1, .., n. This is equivalent to require that the group < x x x > generated by the x_{i} be almost commutative. Define the rank of such a family x x x to be the dimension of the centralizer of Z_{G}(x x x) = Z_{G}(x_{1}, ..., x_{n}) of x x x in G. Notice that x x x is of rank zero if and only if Z_{G}(x x x) is a finite group.

If in addition the x_{i} are all of finite order, there is a more subtle invariant of x x x that can be defined in terms of any maximal torus S of Z_{G}(x x x).

Proposition 3.11. Let x x x = (x_{1}, ..., x_{n}) be an almost commuting n–tuple of elements of finite order of a semisimple k–group G.

1. The group < x x x > generated by the x_{i} is finite. In particular, this group is linearly reductive and all its elements are semisimple.

2. There is an unique subset I = I(x x x) ⊂ Δ for which any maximal torus S of Z_{G}(x) is conjugated to T_{I}.

3. Let S be a maximal torus of Z_{G}(x). Then x x x belongs to Z_{G}(S) and its image in Z_{G}(S)/S is a rank zero n–tuple of the semisimple k–group Z_{G}(S)/S.

4. Let G̃ → G be the universal covering of G. There exist liftings x̃ x x = (x̃_{1}, ..., x̃_{n}) of x x x in G̃, i.e. such that λ(x̃_{i}) = x_{i} for i = 1, ..., n. Moreover, < x̃ x x > is finite and I(x̃ x x) = I(x x x).

5. If S is a maximal torus of Z_{G}(x x x), there exists a maximal torus T of G containing S such that < x x x > ⊂ N_{G}(T). Furthermore, S = (T ∩ Z_{G}(< x x x >))^{0}.

Proof. (1) If G is adjoint, then x x x is a commuting family of elements of finite order, and the result is clear. The general case follows from the fact the kernel of the canonical morphism G → G^{ad} = G/Z(G) is finite and central.

(2) This is nothing but Proposition 3.9 applied to the case H =< x x x >.^{7}

(3) Since S is the connected centre of Z_{G}(S), the group Z_{G}(S)/S is semisimple (possibly trivial). By construction, any maximal torus of Z_{G}(S)/S containing the images x_{i}S of the x_{i}’s must be finite. By (2) the resulting n–tuple is of rank zero.

(4) Liftings do exist since the map G̃(k) → G(k) is surjective (k is algebraically closed). Since the universal covering G̃ → G is an isogeny, the remaining assertions of (3) follow from (1) and (2) with the help of Lemma 3.10.

(5) We have < x x x > ⊂ Z_{G}(S). By the main theorem of [BM], there exists a maximal torus T of Z_{G}(S) such that < x x x > ⊂ N_{G}(T). Since S is the connected centre of Z_{G}(S), we conclude that T contains S. Finally (T ∩ Z_{G}(< x x x >))^{0} is a torus of Z_{G}(< x x x >)^{0} containing S. But S is a maximal torus of Z_{G}(< x x x >)^{0}, hence S = (T ∩ Z_{G}(< x x x >)^{0}).

3.5 Almost commuting pairs and their invariants

We now concentrate on the case n = 2. For convenience, as in §2, we set R = k[t_{1}^{±1}, t_{2}^{±1}], K = k(t_{1}, t_{2}), F = k((t_{1}))((t_{2})).

Proposition 3.12. If a semisimple k–group G has a rank zero pair of almost commuting elements, then G is of type A_{r_{1}} × ⋯ × A_{r_{l}}.

Proof. Let x x x = (x_{1}, x_{2}) be an almost commuting rank zero pair of elements of G. The group Z_{G}(x x x) is finite, both x_{1}, x_{2} are of finite order. By Lemma 3.11, we may assume that G is simply connected. First case: k = ℂ: Let G_{an} be the real anisotropic form of G. The group G_{an}(ℝ) is a maximal compact subgroup of G(ℂ). The finite group Z_{G}(x x x) of G(ℂ) lies in a maximal compact subgroup of G(ℂ). By Cartan’s theorem, up to conjugacy by an element of G(ℂ), we may assume that Z_{G}(x x x) ⊂ G_{an}(ℝ). Then x x x is a rank zero pair of commuting elements of the compact Lie group G_{an}(ℝ). Proposition 4.1.1 of [BFM] then states that G_{an} is a product of groups of type A. Second case : k is a subfield of ℂ: Since the notion of rank zero pair is algebraic, it follows that the proposition holds also in this case. General case: The field k contains the algebraic closure \overline{ℚ} of ℚ in k. The group G is defined over ℚ. Since the elements of Z_{G}(x x x) are of finite order, it follows that Z_{G}(x x x) ⊂ G(\overline{ℚ}) ⊂ G(k). Therefore x x x is a rank zero pair of G(\overline{ℚ}) and the second case shows that G is product of groups of type A.

7According to [BFM], the uniqueness of the index set I “is clear”. This uniqueness is not clear to us however, at least not without the arguments explained herein.

For adjoint groups, the last result has the following useful consequence.

Lemma 3.13. Assume that G is adjoint and simple. Let x x x = (x_{1}, x_{2}) be a commuting pair of elements of G of rank zero. Then G = PGL_{n}, and there exists d ≥ 1 relatively prime to n for which x x x is G(k)–conjugated to the pair (Ad Y_{1}, Ad Y_{2}^{d}), where Y_{1} and Y_{2} in GL_{n}(k) are given by

(3.1) Y_{1} = (\begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 1 & 0 & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & & 0 \\ & ⋯ & & 1 & 0 \end{matrix}) , Y_{2} = (\begin{matrix}1 & 0 & ⋯ \\ 0 & ζ_{n} & 0 & ⋯ & 0 \\ 0 & ⋯ & & & 0 \\ 0 & ⋯ & & 0 & ζ_{n}^{n−1} \end{matrix}) .

Proof. The group G is adjoint simple and, by the last Lemma, of type A. Thus G = PGL_{n} for some n ≥ 2. Let x x x = (x_{1}, x_{2}) be a rank zero pair of PGL_{n}(k). Let (X_{1}, X_{2}) be any lift of (x_{1}, x_{2}) to a pair of elements of GL_{n}(k). Then the commutator of these two elements satisfies [X_{2}, X_{1}] = diag(ζ_{n}, ..., ζ_{n})^{d}, for some d ≥ 0.

By looking at the infinitesimal centralizer of x_{1} and x_{2} ([Bo], corollary to theorem 9.2), we see that the centraliser (under the adjoint action) of {x_{1}, x_{2}} in 𝔰𝔩_{n}(k) is trivial. From this it follows that the centralizer of {X_{1}, X_{2}} in M_{n}(k) = k Id ⊕𝔰𝔩_{n}(k) consists only of the homotheties k Id. So X_{1} and X_{2} do not commute (for otherwise x_{1} = 1 = x_{2} and (x_{1}, x_{2}) is not a rank zero pair). This forces d ∉ nZ. Let m be the order of d in Z/nZ. We have (ζ_{n}^{d})^{m} = 1, so [X_{1}^{m}, X_{2}] = 1. Thus X_{1}^{m} commutes with X_{1} and X_{2}, and therefore X_{1}^{m} = z_{1} Id ∈ k^{×} Id. Similarly, X_{2}^{m} = z_{2} Id ∈ k^{×} Id. Choose a_{i} ∈ k such that a_{i}^{n} = z_{i}^{−1}, and set Z_{i} = a_{i} X_{i}. Then

Z_{1}^{m} = 1, Z_{2}^{m} = 1, and Z_{2}Z_{1} = ωZ_{1}Z_{2}

where ω = ζ_{n}^{d}. Since ω is a primitive m–root of unity, the k–algebra {Z_{1}, Z_{2}} is a non trivial quotient of the standard cyclic central simple k–algebra (1, 1)_{ω} ([GS], §2.5) of degree m. Since (1, 1)_{ω} is simple, {Z_{1}, Z_{2}} ≃ (1, 1)_{ω} ≃ M_{m}(k). But the centralizer of {Z_{1}, Z_{2}} in M_{n}(k) is k, so m = n by the double centralizer theorem (cf. [Sc], corollary 8.4.8). So d is relatively prime to n. On the other hand, the elements Y_{1}, Y_{2} of (3.1) above satisfy

(3.2)  Y_{1}^{n} = 1, (Y_{2}^{d})^{m} = 1, and Y_{2}^{d} Y_{1} = ζ_{n}^{d} Y_{1} Y_{2}^{d}.

We can now apply lemma 1.2 of [Ro] (with ω = ζ_{n}^{d} as primitive n–root of unity), to conclude that (Z_{1}, Z_{2}) is GL_{n}(k)–conjugated to (Y_{1}, Y_{2}^{d}). Since Ad Z_{i} = Ad X_{i}, the Lemma now follows by pushing into PGL_{n}(k).

Remark 3.14. One checks that det(Y_{i}) = (−1)^{n+1}. When n is even, replacing Y_{i} by Ỹ_{i} := ζ_{2n}Y_{i} produces elements of SL_{n}(k) with the desired properties.

These examples of pairs enables us to recover the following useful result:

Lemma 3.15. Let G be a semisimple k–group. If c ∈ Z(G), then c = [x_{1}, x_{2}] for some elements of finite order x_{1} and x_{2} of G .

Proof. It is well known that G admits a semisimple subgroup H of type A_{r_{1}} ×...×A_{r_{l}} such that rank(H) = rank(G). Such a subgroup H contains a maximal torus of G, so it contains Z(G). This reduces the problem to the case of a group G of type A_{r_{1}} × ... × A_{r_{l}}. Let G̃ be the simply connected covering of G. Since the map Z(G̃) → Z(G) is surjective, it will suffice to establish the Lemma for G̃. Furthermore, since G̃ is the product of almost simple semisimple simply connected group, we may also assume that G̃ is almost simple, i.e G̃ = SL_{n}. But then, from the calculations in the proof of previous Proposition, and with the notation of the last Remark, we obtain Z(SL_{n}) = {[Ỹ_{1}^{m}, Ỹ_{2}] : 0 ≤ m < n}.

Let x x x = (x_{1}, x_{2}) be a commuting pair of elements of finite order of G, and [α(x x x)] ∈ H^{1}(R, G) the corresponding loop class (see the beginning of this section). Consider now the simply connected covering

1 → μ μ μ → G̃ ⟶^{λ} G → 1

of G. By Proposition 3.11(4), any lifting x̃ x x = (x̃_{1}, x̃_{2}) of x x x to G̃ is a pair of almost commuting elements of finite order. Since the exact sequence 1 → μ μ μ(k) → G̃(k) → G(k) → 1 is central, the commutator [x̃_{1}, x̃_{2}] ∈ G̃ does not depend on the lifting x̃ x x. We denote this commutator by μ(x x x). As we shall see, μ(x x x) is an important invariant encoded in the cohomology class α(x x x).

Proposition 3.16. The image of [α(x x x)] under the connecting map

δ : H_{ét}^{1}(R, G) → H_{ét}^{2}(R,μ μ μ) ≅ μ μ μ

is given by the formula

δ([α(x x x)]) = μ(x x x)^{−1}.

Proof. ¿From x̃_{1}x̃_{2} = μ(x x x)x̃_{2}x̃_{1} it easily follows that

(3.3)  x̃_{1}^{p_{1}} x̃_{2}^{p_{2}} = μ(x x x)^{p_{1}p_{2}} x̃_{2}^{p_{2}} x̃_{1}^{p_{1}}.

for all p_{1} and p_{2} in ℤ. Let α̃ : π_{1}(R) ≃ ℤ̂ × ℤ̂ → μ μ μ be the unique (continuous) map satisfying α̃ : (i_{1}, i_{2}) ↦ α̃_{(i_{1},i_{2})} = x̃_{1}^{−i_{1}} x̃_{2}^{−i_{2}} for all (i_{1}, i_{2}) ∈ ℤ×ℤ. By the consideration explained in §5 above, we see that δ([α(x x x)]) corresponds, under the connecting map δ : H^{1}(R, G) → H^{2}(R,μ μ μ), to the class of the 2–cocycle c : ℤ̂ × ℤ̂ → μ μ μ given by

c_{(i_{1},i_{2}),(j_{1},j_{2})} : = α̃_{(i_{1},i_{2})}^{(i_{1},i_{2})}α̃_{(j_{1},j_{2})}α̃_{(i_{1}+j_{1},i_{2}+j_{2})}^{−1} = α̃_{(i_{1},i_{2})}α̃_{(j_{1},j_{2})}α̃_{(i_{1}+j_{1},i_{2}+j_{2})}^{−1}
= x̃_{1}^{−i_{1}} x̃_{2}^{−i_{2}} x̃_{1}^{−j_{1}} x̃_{2}^{−j_{2}}(x̃_{1}^{−i_{1}−j_{1}} x̃_{2}^{−i_{2}−j_{2}})^{−1}
= x̃_{1}^{−i_{1}} x̃_{2}^{−i_{2}} x̃_{1}^{−j_{1}} x̃_{2}^{i_{2}} x̃_{1}^{i_{1}+j_{1}}
= μ μ μ(x x x)^{−j_{1}i_{2}}.

The class of this cocycle in H^{2}(ℤ̂ × ℤ̂,μ μ μ) is precisely the cup product −θ_{1} ∪ θ_{2} of the homomorphisms θ_{1}, θ_{2} : ℤ̂ × ℤ̂ → μ μ μ given by (i_{1}, i_{2}) ↦ μ(x x x)^{i_{1}} and (i_{1}, i_{2}) ↦ μ(x x x)^{i_{2}} respectively (see Remark 2.4). Thus δ(α(x x x)) = μ(x x x)^{−1} ∈ H^{2}(R,μ μ μ) ≅ μ μ μ as stated.

Theorem 3.17. Let 1 → μ μ μ → G̃ ⟶^{λ} G → 1 be the simply connected covering of a semisimple k–group G. Then the boundary map δ : H^{1}(R, G) → H^{2}(R,μ μ μ) induces a bijection

H_{loop}^{1}(R, G) ⟶^{∼} H_{ét}^{2}(R,μ μ μ).

In other words, to an R–loop torsor X under G, we can attach a “Brauer invariant” in H^{2}(R,μ μ μ) ⊂ Br(R) which characterizes the isomorphism class of X. The Brauer invariant can be easily computed (Proposition 3.16). We will come back to this in §6, when we classify inner multiloop algebras of nullity 2.

We begin with two preliminary results needed for the proof.

Lemma 3.18. 1. If G = G_{1} × G_{2}, Theorem 3.17 holds for G iff it holds for G_{1} and G_{2};

2. ker(H_{ét}^{1}(R, G) → H_{ét}^{2}(R,μ μ μ)) = ker(H_{ét}^{1}(R, G) → H^{2}(K,μ μ μ))

= ker(H_{ét}^{1}(R, G) → H^{1}(K, G)).

3. Given [z] ∈ H_{loop}^{1}(R, G), assume that

(a) H_{loop}^{1}(R, G^{ad}) → H^{2}(R, Z(G̃)) has trivial fiber at [z^{ad}];

(b) ker(H^{1}(R, _{z}G̃_{R}) → H^{1}(R, _{z}G_{R}^{ad})) = 1.

Then H_{loop}^{1}(R, G) → H^{2}(R,μ μ μ) has trivial fiber at [z].

Proof. (1) This is obvious.

(2) By Proposition 2.1(1) ( or by general properties of the Brauer group. See [M, Ch. IV cor 2.6], H^{2}(R,μ μ μ) injects in H^{2}(K,μ μ μ). On the other hand, H^{1}(K, G) ≃ H^{2}(K,μ μ μ) by Theorem 2.5. Now (2) follows.

(3) Consider the commutative diagram

(3.4)  1 ; ↓ ; 1  Z(G) ; ↓  ↓ ; 1 ⟶ μ μ μ ⟶ G̃ ⟶^{λ} G ⟶ 1 ; ↓  ^{≀}↓  ^{p}↓ ; 1 ⟶ Z(G̃) ⟶ G̃ \underrightarrow{p̃} G^{ad} ⟶ 1 ; ↓  ↓ ; Z(G)  1 ; ↓ ; 1

Passing to cohomology yields

H_{loop}^{1}(R, G) ⊂ H^{1}(R, G) ⟶ H^{2}(R,μ μ μ)
p_{∗}↓  ^{p_{∗}}↓  ↓
H_{loop}^{1}(R, G^{ad}) ⊂ H^{1}(R, G^{ad}) ⟶ H^{2}(R, Z(G̃)).

By diagram chasing and assumption (a), it will suffice to establish the triviality of the fiber of p_{∗} at [z] ∈ H_{loop}^{1}(R, G). This fiber is controled by the diagram of torsion maps of the twisted R–group _{z}G_{R} := _{z}G ([DG], III.4.3.4), namely

H^{1}(R, G) ⟶^{p_{∗}} H^{1}(R, G^{ad})
θ_{z}↓^{≀}  ^{θ_{z}}↓^{≀}
H^{1}(R, Z(G)) ⟶ H^{1}(R, _{z}G_{R}) ⟶^{_{z}p_{∗}} H^{1}(R, _{z}G_{R}^{ad})

In other words, p_{∗}^{−1}([z]) ≅ _{z}p_{∗}^{−1}(1). Since Z(G̃) and Z(G) are both constant and finite, the map H^{1}(R, Z(G̃)) → H^{1}(R, Z(G)) is surjective. We then see that in the commutative diagram

H^{1}(R, Z(G̃)) ⟶ H^{1}(R, _{z}G̃_{R}) ⟶^{_{z}p̃_{∗}} H^{1}(R, _{z}G_{R}^{ad})
↓  ↓  ↓^{≀}
H^{1}(R, Z(G)) ⟶ H^{1}(R, _{z}G_{R}) ⟶^{_{z}p_{∗}} H^{1}(R, _{z}G_{R}^{ad}).

the map ker(_{z}p̃_{∗}) → ker(_{z}p_{∗}) is surjective. Since by assumption (b) we have ker(_{z}p̃_{∗}) = 1, we conclude that _{z}p_{∗}^{−1}(1) = 1, hence that p_{∗}^{−1}([z]) = [z] as desired.

We also need the following fairly general cohomological result.

Lemma 3.19. Let I ⊂ Δ be a subset, and A and object in k–alg. Then.

1. The canonical map H^{1}(A, L_{I}) → H^{1}(A, P_{I}) is bijective.

2. If Pic(A) = 0, the canonical map H^{1}(A, L_{I}) → H^{1}(A, L_{I}/T_{I}) is injective.

Proof. (1) This is a special case of corollary 2.3 of exp. XXVI of [SGA3].

(2) The sequence of algebraic groups 1 → T_{I} → L_{I} → L_{I}/T_{I} → 1 is central. According to § III.4.5.3 of [DG], the fibers of H^{1}(A, L_{I}) → H^{1}(A, L_{I}/T_{I}) are suitable quotients of H^{1}(A, T_{I}). Since T_{I} is a split torus, the hypothesis Pic(A) = 0 implies that H^{1}(A, T_{I}) = 0. Thus the map H^{1}(A, L_{I}) ⟶ H^{1}(A, L_{I}/T_{I}) is injective.

Proof of Theorem 3.17: We may asume without loss of generality that G ≠ 1.

To establish the injectivity of the map H_{loop}^{1}(R, G) → H^{2}(R,μ μ μ), we will reason according to the index I of the relevant pairs. Since by Theorem 2.5, the map H^{1}(K, G) → H^{2}(K,μ μ μ) is a bijection, it will suffice to show that the map H_{loop}^{1}(R, G) → H^{1}(K, G) is injective. Suppose then that we are given two pairs x x x = (x_{1}, x_{2}) and x x x′ = (x′_{1}, x′_{2}) of commuting elements of finite order of G for which [α(x x x)]_{K} = [α(x x x′)]_{K} ∈ H^{1}(K, G). Since the twisted groups _{α(x x x)}G_{K} and _{α(x x x′)}G_{K} are then isomorphic, they have same Witt-Tits index I. Theorem 3.1 implies that I(x x x) = I(x x x′) = I. Step 1: Injectivity for G adjoint when x x x, x x x′ are of rank zero:

By Lemma 3.18 we reduce to the case of a simple adjoint group. Since G is not trivial, we have G = PGL_{n} with n ≥ 2. According to Lemma 3.13, we may assume (up to conjugacy) that x x x = (Ad(Y_{1}), Ad(Y_{2})^{d}), x x x′ = (Ad(Y_{1}), Ad(Y_{2})^{d′}), with d, d′ prime to n. The corresponding twisted K–algebras A(d, n) ⊗_{R} K and A(d′, n) ⊗_{R} K are isomorphic if and only if d′ = d modulo n. Thus Ad(Y_{2})^{d} = Ad(Y_{2})^{d′}, and therefore x x x and x x x′ are conjugated under G(k). Hence [α(x x x)] = [α(x x x′)] ∈ H_{loop}^{1}(R, G) as desired.

Step 2: Injectivity when x x x, x x x′ are of rank zero: Again the simple factors of G^{ad} are all of type A. Assumption (a) of Lemma 3.18.(4) holds by step 1. Let us check that assumption (b), namely that H^{1}(R, _{z}G̃_{R}) → H^{1}(R, _{z}G_{R}^{ad}) has trivial kernel for any [z] ∈ H_{loop}^{1}(R, G^{ad}) corresponding to a rank zero pair. To check this, we may assume that G^{ad} is simple, i.e. G = PGL_{n} with n ≥ 2. As in Step 1, the class [z] corresponds to the R–Azumaya algebra A(d, n) for some d prime to n. The corresponding twisted adjoint and simply connected groups are _{z}G_{R}^{ad} = PGL_{1}(A(d, n)) and _{z}G̃_{R} = SL_{1}(A(d, n)) respectively (the latter being the R–group scheme of elements of reduced norm 1 of A(d, n)). The exact sequence of twisted R–groups

1 → μ μ μ_{n} → SL_{1}(A(d, n)) → PGL_{1}(A(d, n)) → 1

gives rise to the exact sequence of pointed sets

1 → μ μ μ_{n}(R) → SL_{1}(A(d, n))(R) → PGL_{1}(A(d, n))(R) →
→ R^{×}/(R^{×})^{×n} → H^{1}(R, SL_{1}(A(d, n))) → H^{1}(R, PGL_{1}(A(d, n))).

Since Pic(R) = 1, we have PGL_{1}(A(d, n))(R) = GL_{1}(A(d, n))(R)/R^{×}, and the boundary map PGL_{1}(A(d, n))(R) → R^{×}/R^{×}^{n} is nothing but the reduced norm modulo R^{×}^{n}. But since this reduced norm map is surjective (Lemma 2.10), the map H^{1}(R, SL_{1}(A(d, n))) → H^{1}(R, PGL_{1}(A(d, n))) is injective. This completes the proof that assumption (b) of Lemma 3.18 holds, hence also the proof of Step 2. Accordingly, the map H_{loop}^{1}(R, G) → H^{2}(R,μ μ μ) has trivial fiber at loop classes corresponding to rank zero pairs. Step 3: Injectivity, the general case: By Proposition 3.11(2) we may assume, after conjugating x x x and x x x′ by two elements of G(k), that both x x x and x x x′ are pairs of L_{I} = Z_{G}(T_{I}) inducing rank zero pairs \overline{x x x} and \overline{x x x}′ of Z_{G}(T_{I})/T_{I}. Consider the exact sequence of groups

1 → T_{I} → L_{I} → L_{I}/T_{I} → 1.

Recall that H^{1}(K, P_{I}) injects in H^{1}(K, G) ([BoT], théorème 4.13.a). The map H^{1}(K, L_{I}) → H^{1}(K, G) is therefore injective by Lemma 3.19(1). Using again this same Lemma, but now for R, we obtain the following commutative diagram of pointed sets.

H_{loop}^{1}(R, G) ⊂ H^{1}(R, G) ⟶ H^{1}(K, G)
↑  ↑  ∪
H_{loop}^{1}(R, L_{I}) ⊂ H^{1}(R, L_{I}) ⟶ H^{1}(K, L_{I})
∩  ∩  ∩
H_{loop}^{1}(R, L_{I}/T_{I}) ⊂ H^{1}(R, L_{I}/T_{I}) ⟶ H^{1}(K, L_{I}/T_{I})

The elements [α(x x x)] and [α(x x x′)] actually belong to H_{loop}^{1}(R, L_{I}), and map to [α(\overline{x x x})] and [α(\overline{x x x′})] in H_{loop}^{1}(R, L_{I}/T_{I}). By chasing the above diagram, we see that [α(\overline{x x x})]_{K} = [α(\overline{x x x′})]_{K} in H^{1}(K, L_{I}/T_{I}), and that the rank zero pairs \overline{x x x} and \overline{x x x′} of L_{I}/T_{I} have the same image in H^{1}(K, L_{I}/T_{I}). By Step 2 then [α(\overline{x x x})] = [α(\overline{x x x′})] in H_{loop}^{1}(R, L_{I}/T_{I}). A second diagram chase enables us to conclude that [α(x x x)] = [α(x x x′)] in H_{loop}^{1}(R, L_{I}).

Step 4: surjectivity: Given c ∈ μ μ μ ≃ H^{2}(R,μ μ μ), Lemma 3.15 provides an almost commuting pair x x x = (x_{1}, x_{2}) of elements of finite order of G̃(k) such that μ(x x x) = c^{−1}. By Proposition 3.16 we have δ(α(x x x)) = μ(x x x)^{−1} = c. □

3.6 Failure in the anisotropic case

The following example is analogous to the construction by Ojanguren-Sridharan of reduced rank one projective quaternion modules over the real affine plane, which are not free [OS].

Proposition 3.20. Let n ≥ 2 be an integer, and let A = A(1, n) the standard Azumaya algebra over R = k[t_{1}^{±1}, t_{2}^{±1}] with generators T_{1}, T_{2}, and relations T_{1}^{n} = t_{1}, T_{2}^{n} = t_{2} and T_{2}T_{1} = ζ_{n}T_{1}T_{2}. The equation

(1 + T_{1})λ = (1 + T_{2})μ

defines an invertible A–module L which is not free. The Azumaya algebra End_{A}(L) has same class than A in the Brauer group, but is not isomorphic to A.

The proof is based on a valuation argument on the division algebra A_{K}. We equip the additive group Γ := ℝ ⊕ ℝ with the lexicographical order, and define a valuation v : R \ {0} → Γ by

v(∑_{i,j} a_{i,j} t_{1}^{i}t_{2}^{j}) = Min_{Γ}{ (j, i) | a_{i,j} ≠ 0} ∈ Γ.

We denote by P ↦ \overline{P} the specialisation map k[t_{1}, t_{2}] → k at the point (0, 0). As before, N : A → R denotes the reduced norm of A ([K2], I.7.3.1.5).

Lemma 3.21. Given a non zero element

x = ∑ a_{i,j} T_{1}^{i}T_{2}^{j} ∈ A, a_{i,j} ∈ k,

define v_{A}(x) = \frac{1}{n} Min_{Γ}\{ (j, i) | a_{i,j} ≠ 0\} ∈ Γ.

1. v_{A} is a valuation on A which extends v on R. Furthermore

v_{A}(x) = \frac{1}{n} v(N(x)).

2. If v_{A}(x) ≥ 0, then \overline{N(x)} = a_{0,0}^{n}.

For valuation theory on division algebras, we refer the reader to the nice survey by Wadsworth [W2].

Proof. (1) The completion of K = k(t_{1}, t_{2}) with respect to v is the field F = k((t_{1}))((t_{2})). Extend the function v_{A} to A_{F}^{×} by the formula

w( ∑_{j=−p}^{∞} ∑_{i=p_{j}}^{∞} a_{i,j}T_{1}^{i}T_{2}^{j}) = Inf_{Γ}{(j, i) | a_{i,j} ≠ 0} ∈ Γ.

Since A_{F} is a cyclic division algebra, example 2.7 of [W2] states that the map w is a valuation on A_{F}.^{8} So v_{A} is a valuation. The formula v_{A} = \frac{1}{n} v ∘ N was proven independently by Ershov [E1][E2] and Wadsworth [W1].

(2) This is a special case of a formula due to Ershov ([E1], Corollary 2).

Proof of Proposition 3.20: Put ζ = ζ_{n} and ϵ = ∏_{i=0}^{n−1} ζ^{i} = (−1)^{n−1}. The module

L is the kernel of the map f : A ⊕ A → A, (λ, μ) ↦ (1 + T_{1})λ − (1 + T_{2})μ. Since f(1 + T_{2}, 1 + T_{1}) = T_{1}T_{2} − T_{2}T_{1} = (1 − ζ)T_{1}T_{2} is invertible, f is split surjective. Thus L is a projective module of reduced rank 1. Assume that L is free, i.e L = (λ_{0}, μ_{0})A with (λ_{0}, μ_{0}) ∈ A^{2}. By taking reduced norms, we have

(3.5)  N(λ_{0})N(1 + T_{1}) = N(μ_{0})N(1 + T_{2}).

For a non zero element a ∈ A, by Galois descent, there exists a unique element N′(a) ∈ A such that N′(a)a = aN′(a) = N(a) ∈ A. Since

f(N′(1 + T_{1})(1 + T_{2}), N(1 + T_{1})) = N(1 + T_{1})(1 + T_{2}) − (1 + T_{2})N(1 + T_{1}) = 0,

we have

(N′(1 + T_{1})(1 + T_{2}), N(1 + T_{1})) = (λ_{0}, μ_{0})α, α ∈ A. (3.6)

By comparing norms on (3.6), we obtain

N(λ_{0}) | N(1 + T_{1})^{n−1}N(1 + T_{2}) and N(μ_{0}) | N(1 + T_{1})^{n}.

8Schilling’s theorem states actually that w is the unique valuation on A_{F} extending v (loc. cit, corollary 2.2).

Similarly, since f(N(1+T_{2}), N′(1+T_{1})(1+T_{2})) = 0, we have N(λ_{0}) | N(1+T_{2})^{n} and N(μ_{0}) | N(1 + T_{1})N(1 + T_{2})^{n−1}. We also have

N(1 + T_{1}) = ∏_{i=0,...,n−1}(1 + ζ^{i}T_{1}) = 1 + ϵt_{1},

and similarly N(1 + T_{2}) = 1 + ϵt_{2}. The above yield the following three identities.

1. N(λ_{0}) (1 + ϵt_{1}) = N(μ_{0}) (1 + ϵt_{2});

2. N(λ_{0}) | (1 + ϵt_{2})^{n} and N(λ_{0}) | (1 + ϵt_{1})^{n−1} (1 + ϵt_{2});

3. N(μ_{0}) | (1 + ϵt_{1})^{n} and N(μ_{0}) | (1 + ϵt_{1}) (1 + ϵt_{2})^{n−1}.

Thus N(λ_{0}) = (1 + ϵt_{2})u and N(μ_{0}) = (1 + ϵt_{1})u for some u ∈ R^{×}. Since the reduced norm N : A^{×} → R^{×} is surjective (Lemma 2.10), we may henceforth assume with no loss of generality that N(λ_{0}) = 1 + ϵt_{2} and N(μ_{0}) = 1 + ϵt_{1}. We have v(1 + ϵt_{1}) = 0, so Lemma 3.21 shows that v_{A}(μ_{0}) = 0 and similarly we have v_{A}(λ_{0}) = 0. Hence

λ_{0} = a_{0,0} + ∑_{(i,j)>(0,0)} a_{i,j} T_{1}^{i}T_{2}^{j}, a_{i,j} ∈ k,

μ_{0} = b_{0,0} + ∑_{(i,j)>(0,0)} b_{i,j} T_{1}^{i}T_{2}^{j}, b_{i,j} ∈ k.

with a_{0,0}, b_{0,0} ≠ 0. Lemma 3.21(2) enables us to specialize at (0, 0) the equality N(λ_{0}) = 1 + ϵt_{2}. This yields a_{0,0}^{n} = 1 and b_{0,0}^{n} = 1. On the other hand, the equation (1 + T_{1})λ_{0} = (1 + T_{2})μ_{0} now implies that a_{0,0} = b_{0,0}. Thus, after multiplying λ_{0} and μ_{0} by the n–root of unity a_{0,0}^{−1}, we may assume that a_{0,0} = b_{0,0} = 1 while still keeping the identities N(λ_{0}) = 1 + ϵt_{2} and N(μ_{0}) = 1 + ϵt_{1}.

We now look at the behavior at infinity by considering the valuation v_{∞} on R which is the highest bidegree with respect to t_{2}, t_{1}. We extend v_{∞} to a valuation v_{∞,A} on A as in Lemma 3.21. Since v_{∞}(N(λ_{0})) = v_{∞}(1 + ϵt_{2}) = (1, 0), it follows that v_{∞,A}(λ_{0}) = \frac{1}{n}(1, 0). Hence λ_{0} = 1 + ζ_{n}^{i}T_{2} for some i. Similarly μ_{0} = 1 + ζ_{n}^{j}T_{1}. In all cases, we have (1 + T_{1})λ_{0} ≠ (1 + T_{2})μ_{0} which is a contradiction.

We conclude that L is not free. The Azumaya algebra End_{A}(L) has the same class than A in the Brauer group. Since Pic(R) = 0, lemma 4.1.(3) of [GP2] enables to conclude that End_{A}(L) is not isomorphic to A. □

Corollary 3.22. There exists an R–Azumaya algebra^{9} M which is a non trivial Zariski form of A(1, n). Moreover, [M] and [A(1, n)] ∈ H^{1}(R, PGL_{n}) have same connecting invariant 1 ∈ Z/nZ ≅ H^{2}(R,μ μ μ_{n}).

9The Margaux algebra.

Remark 3.23. By twisting by A the exact sequence 1 → μ μ μ_{n} → SL_{n} → PGL_{n} → 1, the last assertion can be rephrased by saying that H^{1}(R, SL_{1}(A(1, n))) ≠ 1. This shows that, in contrast to the nullity 1 case of k[t_{1}^{±1}], the analogue of Serre conjecture II for k[t_{1}^{±1}, t_{2}^{±1}] fails (see Question 2.6).

4 Twisted forms of algebras over rings

In this section the base field k is of arbitrary characteristic. Throughout A will denote a finite dimensional k–algebra (not necessarily unital or associative; for example a Lie algebra). All rings are assumed to be commutative and unital.

4.1 Multiplication algebras and centroids

Let R be a ring. For an arbitrary R–algebra L, recall that the multiplication algebra Mult_{R}(L) of L, is the unital subalgebra of End_{R}(L) generated by {1, l_{x}, r_{x}}, where l_{x} (resp. r_{x}) denotes the left (resp. right) multiplication operator by the element x ∈ L. The abelian group L has a natural left Mult_{R}(L)–module structure, and an algebra structure thereof if the ring Mult_{R}(L) is commutative.

The centroid C_{R}(L) of L, is the centralizer of Mult_{R}(L) in End_{R}(L). Thus C_{R}(L) is the subalgebra of End_{R}(L) consists of all the endomorphisms of the R–module L that commute with right and left multiplication by elements of L, i.e.

C_{R}(L) = {χ ∈ End_{R}(L) : χ(xy) = χ(x)y = xχ(y) for all x, y in L}.

For r ∈ R, define λ_{r} ∈ End_{R}(L) by λ_{r}(x) = rx. Then λ_{r} ∈ C_{R}(L), and the map λ_{L} : r → λ_{r} is a ring homomorphism from R into C_{R}(L). Recall that L is called central if λ_{L} is an isomorphism. If L is a faithful R–module, the map λ_{L} is injective and we may, and at times will, identify R with a subring λ_{L}(R) of C_{R}(L).

Let L′ = {∑ x_{i}y_{i} : x_{i}, y_{i} ∈ L} (finite sums of course). L′ is a two-sided ideal of L, and we recall that L is called perfect if L = L′.

Lemma 4.1. Let L be an R–algebra. Let R_{0} → R by any (unital) ring homomorphism, and denote by L_{0} the resulting R_{0}–algebra structure on L. Then.

1. L is perfect as an R–algebra if and only if L_{0} is perfect as an R_{0}–algebra.

2. Assume L is a perfect R–algebra. Then C_{R}(L) is commutative and the canonical map C_{R}(L) → C_{R_{0}}(L_{0}) is a ring isomorphism.

Proof. (1) is clear. That C_{R}(L) is commutative is proved in[J, Ch. X lemma 1]. The remaining point is that, because L is perfect, any endomorphism of the additive underlying group of L that commutes with right and left multiplication, is automatically R–linear (see [ABP2, lemma 4.1] for details).

Remark 4.2. We will be interested in looking at L not only as an R–algebra, but also as a k–algebra. The loop algebra attached to an affine Kac-Moody case is a good example to have in mind. These are infinite dimensional algebras over ℂ, but much can be gained by looking at them as algebras over their centroids (which are Laurent polynomial rings). In view of this last result, if L is perfect, its centroid is independent of which of these two base rings one uses.

4.2 Forms of simple algebras over rings

Recall that A is a finite dimensional k–algebra. Throughout R denotes an object of k–alg.

Definition 4.3. An R–form of A is an algebra L over R for which there exists a faithfully flat and finitely presented extension S/R in k–alg such that

(4.1)  L ⊗_{R} S ≃_{S} A ⊗ S

(isomorphism of S–algebras).

Remark 4.4. Since A⊗S ≃ (A⊗R)⊗_{R}S, the R–algebra L is nothing but an R–form (trivialized by Spec(S) in the f.p.p.f. topology of Spec(R)) of the R–algebra A ⊗ R. Since Spec(R) is affine, the isomorphism classes of such R–algebras are parametrized by H_{fppf}^{1}(R, Aut(A_{R})); the pointed set of Čech cohomology on the (small) f.p.p.f. site of Spec(R) with coefficients on Aut(A_{R}). The R–group sheaf Aut(A_{R}) is in fact an affine R–group scheme (because A is finite dimensional). We have Aut(A_{R}) = Aut(A)_{R}. If Aut(A) is smooth (for example if char(k) = 0), then S in (4.1) may be assumed to be an étale cover. (see [SGA3] and [M] for details).

Example 4.5. (1) If A is the matrix algebra of rank n, the R–forms of A are the Azumaya algebras over R of constant rank n.

(2) If 𝔤 is a finite dimensional simple Lie algebra over an algebraically closed field k of characteristic zero, then the k[t^{±1}]–forms of 𝔤 are precisely the affine Kac-Moody Lie algebras (derived modulo their centres) over k. This is consequence of an analogue of Serre Conjecture I for the Dedekind ring k[t^{±1}]. See [P2] for details.

Lemma 4.6. Let L be an R–form of a finite dimensional perfect and central finite dimensional k–algebra A.

1. L is perfect. In particular, its centroid C_{R}(L) is commutative and coincides with C_{k}(L).

2. As an R–module, L is faithfully projective (in particular of finite type).

3. The canonical map λ_{L} : R→ C_{R}(L) is a ring isomorphism. In particular, L is central as an R–algebra.

Proof. (1) We have

\begin{aligned}(L/L′) ⊗_{R} S & ≃ L ⊗_{R} S/L′ ⊗_{R} S ≃ A ⊗ S/(L ⊗_{R} S)′ \\ & ≃ A ⊗ S/(A ⊗ S)′ = 0 \end{aligned}

(the first isomorphism because S/R is flat, the last equality because A, hence A ⊗ S, is perfect). Since S/R is faithfully flat, L/L′ = 0. The assertions about the centroid now follow from Lemma 4.1.

(2) After a faithfully flat base change L becomes free of finite rank. By descent properties then, L is a projective R–module of finite type. To see that this module is faithful, observe that if r ∈ R annihilates L, then the image s of r in S annihilates the S–module L ⊗_{R} S ≃ A ⊗ S. Thus s = 0 (the k–algebra A, being central, cannot be zero dimensional). Since S/R is faithfully flat, r = 0 as desired.

(3) Since L is faithful, the canonical map λ_{L} : R → C_{R}(L) is injective, and we may thereof identify R with a subring λ_{L}(R) of the (commutative) ring C_{R}(L). Since L is projective of finite type, the canonical map ϕ : End_{R}(L) ⊗_{R} S → End_{S}(L ⊗_{R} S) ≃ End_{S}(A⊗S) is an S–algebra isomorphism. Clearly ϕ(C_{R}(L)⊗_{R} S) ⊂ C_{S}(A⊗S) ≃ S (the latter by [ABP2.5] lemma 2.3(a)). It follows that ϕ(λ_{L}(R) ⊗_{R} S) = ϕ(C_{R}(L) ⊗_{R} S). Thus the inclusion λ_{L}(R) ⊗_{R} S ⊂ C_{R}(L) ⊗_{R} S is an equality (being an equality after applying ϕ). By faithful flatness, λ_{L}(R) = C_{R}(L) as desired.

Remark 4.7. Let L_{1} and L_{2} be R–forms of a finite dimensional perfect and central k–algebra A. If ϕ : L_{1} → L_{2} is an isomorphism of k –algebras, then C(ϕ) : C_{k}(L_{1}) → C_{k}(L_{2}) given by χ ↦ ϕχϕ^{−1} is an isomorphism in k–alg. By part (3) of the last Lemma, there exists a unique ϕ̂ ∈ Aut_{k}(R) such that the diagram

R  \underrightarrow{ϕ̂}  R
λ_{L_{1}} ↓  ↓ λ_{L_{2}}
C_{k}(L_{1}) ⟶_{C(ϕ)} C_{k}(L_{2})

commutes. We have

ϕ(rx) = ϕ̂(r)ϕ(x)

for all r ∈ R and x ∈ L_{1}. To say that ϕ is R–linear, is to say that ϕ̂ = 1. If ψ : L_{2} → L_{3} is another isomorphism as above, then { \atop ψ ∘ ϕ} = ψ̂ ∘ ϕ̂.

¿From the foregoing we obtain the following useful exact sequence of groups

(4.2)  1 → Aut_{R}(L) → Aut_{k}(L) ⟶^{∧} Aut_{k}(R)

for any R–form L of A. This last map need not be surjective, and even when it is, the sequence need not split. Notwithstanding all of these problems, (4.2) is enough to completely describe the group Aut_{k}(L) of automorphisms of the k–algebra L in many interesting cases (see Examples 4.11 and 5.5 below, and also [PPS]).

Remark 4.8. Let 𝔤 be a finite dimensional simple Lie algebra over k, with k algebraically closed of characteristic zero. Let R_{n} = k[t_{1}^{±1}, ..., t_{n}^{±1}]. As we saw in Example 4.5(2) , the case n = 1 corresponds to the affine Kac-Moody algebras. In general, there is a delicate connection between R_{n}–forms of 𝔤 and centerless cores of Extended Affine Lie Algebras (EALA’s for short). As their name suggest, these algebras are higher nullity analogues of the affine algebras (see [AABFP] for details).

Neher has described ([N1] and [N2]) a precise procedure for building up a (tame) EALA out of centreless cores. It is not our intention to go into details about this construction, and the affine case will suffice to illustrate the spirit of how this goes. The centreless core L is in this case a “loop algebra” L(𝔤, σ) (see §6.2). One builds an EALA out of this by considering the universal central extension of L (which happens to be one dimensional), and tacking on a degree derivation. The algebra obtained is in this case an affine Kac-Moody Lie algebra.

The above showcases the crucial role that centreless cores play in the theory of EALAs. One of the main Theorems of [ABFP] shows that centreless cores which are finitely generated as modules over their centroids,^{10} are always k[t_{1}^{±1}, ..., t_{n}^{±1}]–forms of a finite dimensional simple Lie algebra 𝔤. Centreless cores thus fall within the present language of forms. We believe that the approach described herewith is a new useful tool for the study of EALAs.

Our approach also sheds insight into some of the fundamental results of EALA theory. For example, Neher has shown that the centerless cores L under consideration, are always free module of finite rank over their centroid. But since L is a form of some 𝔤 as above, the previous Lemma tells that the R–module L is finitely generated and projective. Given that in the present situation R = k[t_{1}^{±1}, ..., t_{n}^{±1}], freeness follows from a well-known theorem of Quillen and of Suslin.

10The ones which are not are fully understood.

4.3 Invariance of type

Let α ∈ Aut_{k}(R). Define a new R–module structure on L by r ·x = α(r)x. We denote the resulting R–algebra structure by _{α}L to avoid confusion. (The multiplication in _{α}L coincides with that of L. It is the R–module structure that has changed.)

Assume i : R → S is an fppf base change for which there exists an S–algebra isomorphism ψ : L ⊗_{R} S ≃ A ⊗ S. Then i_{α} = i ∘ α : R → S is also fppf, and there exists a unique S–algebra isomorphism

ψ_{α} : _{α}L ⊗_{R} S → A ⊗ S

satisfying ψ_{α}(x⊗s) = ψ(x⊗s) (where on the left hand ⊗_{R} we view S as an R–algebra via i_{α}). Thus _{α}L is also an R–form of A, said to be obtained from L by twisting by α. Note that id : _{α}L → L is a k–algebra isomorphism, and that îd = α (see Remark 4.7).

Theorem 4.9. [Invariance of type] Let L_{1} (resp. L_{2}) be an R–form of some finite dimensional perfect central k–algebra A_{1} (resp. A_{2}). If L_{1} and L_{2} are isomorphic as k-algebras, there exists a finite field extension K/k for which A_{1} ⊗ K and A_{2} ⊗ K are isomorphic as K–algebras. In particular, if k is algebraically closed then A_{1} and A_{2} are isomorphic.

Proof. Let ϕ : L_{1} → L_{2} be a k–algebra isomorphism, and let ϕ̂ ∈ Aut_{k}(R) be the corresponding automorphism at the centroid level (Remark 4.7). Denote ϕ̂^{−1} by α, and consider the twisted algebra _{α}L_{1}. Then the k–algebra isomorphism ϕ_{α} = ϕ ∘ id : αL_{1} → L_{2} satisfies ϕ̂_{α} = id. It follows from this that we may (and henceforth do ) assume with no loss of generality that our ϕ is an R–algebra isomorphism.

Consider the “switch” σ : S_{1} ⊗ S_{2} → S_{2} ⊗ S_{1}. Fix S_{i}–algebra isomorphisms ψ_{i} : L_{i} ⊗_{R} S_{i} → A_{i} ⊗ S_{i}. The composite map

σψ_{2} : L_{2} ⊗_{R} S_{1} ⊗ S_{2} ⟶^{id⊗σ} L_{2} ⊗_{R} S_{2} ⊗ S_{1} ⟶^{ψ_{2}⊗id} A_{2} ⊗ S_{2} ⊗ S_{1} ⟶^{id⊗σ} A_{2} ⊗ S_{1} ⊗ S_{2}

is an isomorphism of S_{1} ⊗ S_{2}–algebras. Thus the composite map

ψ : A_{1} ⊗ S_{1} ⊗ S_{2} ⟶^{ψ_{1}^{−1}⊗id} L_{1} ⊗_{R} S_{1} ⊗ S_{2} ⟶^{ϕ⊗id⊗id} L_{2} ⊗_{R} S_{1} ⊗ S_{2} ⟶^{^{σ}ψ_{2}} A_{2} ⊗ S_{1} ⊗ S_{2}

is also an isomorphism of S_{1} ⊗ S_{2}–algebras.

Let 𝔪 be a maximal ideal of S_{1} ⊗ S_{2}, and let F = (S_{1} ⊗ S_{2})/𝔪 the corresponding quotient field. Our ψ : A_{1}⊗S_{1}⊗S_{2} → A_{2}⊗S_{1}⊗S_{2} induces, upon reduction modulo 𝔪, an F –algebra isomorphism \overline{ψ} : A_{1} ⊗ F → A_{2} ⊗ F. Since the A_{i} are finite dimensional, we may replace F by a subfield K which is finite dimensional over k.

4.4 Twisted automorphism groups

As before, A denotes a finite dimensional k–algebra, and L an R–form of A. Our next objective is to look in some detail at the group of automorphisms of L. For simplicity we will henceforth assume that the extension S/R trivializing L is finite and Galois. By the Isotriviality Theorem of [GP1], this assumption is superfluous in the case we are interested in, namely Laurent polynomials in finitely many variables over a field of characteristic zero.^{11}

Let G = Aut(A). This is a linear algebraic group over k whose functor of points is given by

(4.3)  G(S) = Mor_{k−sch}( Spec(S), G) = Aut_{S}(A ⊗ S),

where the latter is the abstract group of automorphisms of the S–algebra A ⊗ S. Each γ ∈ Γ = Gal(S/R) induces an automorphism γ^{∗} of Spec(S) as a scheme over Spec(R), and a fortiori also as a scheme over Spec(k). For γ ∈ Γ and g ∈ G(S) = Mor_{k−sch}( Spec(S), G_{R}), define

(4.4)  ^{γ}g = g ∘ γ^{∗}.

This yields an action of Γ on G(S). (If one thinks of g as a matrix with entries in S, then ^{γ}g is nothing but the matrix obtained by applying γ to each entry of g).

We now look at the k–group functor Aut(G) of automorphisms of G. By definition, for any S in k–alg

(4.5)  Aut(G)(S) = Aut(G_{S}).

Some care is needed not to misunderstand this definition. The right hand side is the (abstract) group of automorphisms of the S–group G_{S} obtained by the base change S/k, and not the group of automorphisms of the group G(S).

We will henceforth assume that the k–group Aut(G) is representable (for example G reductive). Just as in (4.4) above, the group Γ acts on Aut(G)(S) = Mor_{k−sch}( Spec(S), Aut(G)) by means of Γ^{∗}.

Next consider the group homomorphism

int : G → Aut(G)

given by conjugation. Fix an element g ∈ G(S). Then the composite sequence

Spec(S) ⟶^{γ^{∗}} Spec(S) ⟶^{g} G ⟶^{int} Aut(G),

11The industrious reader may rewrite this section in the general case.

together with the definition of the action of Γ on Aut(G)(S), readily yield the equality

(4.6)  ^{γ}int g = int ^{γ}g for all g ∈ G(S) and γ ∈ Γ.

We now return to our twisted form L. Up to R–isomorphism, we may assume that

(4.7)  L = {x ∈ A ⊗ S : u_{γ} ^{γ}x = x for all γ ∈ Γ}

for some fixed cocycle u = (u_{γ})_{γ∈Γ} ∈ Z^{1}(Γ, G(S)). From (4.6) it follows that ũ_{γ} := int u_{γ} defines a cocycle ũ ∈ Z^{1}(Γ, Aut(G)(S)). Let ũG_{R} be the corresponding twisted group. This is an affine R–group scheme which becomes G_{S} = Aut(A_{S}) after the base change S/R.

Proposition 4.10. With the above notation and asumptions we have

Aut_{R}(L) = (ũG_{R})(R).

That is, the automorphisms of the twisted R–algebra L, are precisely the R–points of the corresponding twisted group ũG_{R}.

Proof. By definition

(ũG_{R})(R) = {θ ∈ Aut_{S} (A ⊗ S) : ũ_{γ} ^{γ}θ = θ for all γ ∈ Γ}.

Thus, if θ ∈ ũG_{R}(R) and x ∈ L ⊂ A ⊗ S (see (4.7)), we have by (4.4) that

\begin{aligned}u_{γ} ^{γ}(θ(x)) & = u_{γ} ^{γ}θ ^{γ}x = u_{γ} ^{γ}θu_{γ}^{−1}u_{γ} ^{γ}x \\ & = (ũ_{γ} ^{γ}θ)(x) = θ(x), \end{aligned}

thereby showing that θ stabilizes L.

Conversely, let τ ∈ Aut_{R}(L). Then θ := τ ⊗1 ∈ Aut_{S}(L⊗_{R} S) ≃ Aut_{S}(A⊗S) = G(S). For x ∈ L we have as above, that θ(x) = (ũ_{γ} ^{γ}θ)(x). Thus θ = ũ_{γ} ^{γ}θ when restricted to L. Since L spans A ⊗ S as an S–module, the result follows.

This last Proposition, together with Remark 4.7, can be put to good use to study the group of automorphisms of many important families of infinite dimensional Lie algebras.^{12} We will illustrate how this goes via two interesting examples; quantum tori (immediately below) and affine Kac-Moody Lie algebras (Example 5.5).

12Most notably Extended Affine Lie Algebras.

Example 4.11. (Quantum tori) Assume k contains a primitive n–th root of unity

ζ_{n}, and consider the quantum torus A_{q} where q = (\begin{matrix}1 & ζ_{n} \\ ζ_{n}^{−1} & 1 \end{matrix}) . By definition, A_{q} is

the associative unital k–algebra generated by T_{1}^{±1} and T_{2}^{±1} subject to the relations

T_{i}T_{i}^{−1} = 1 = T_{i}^{−1}T_{i} and T_{2}T_{1} = ζ_{n}T_{1}T_{2}.

The center of A_{q} is R = k[t_{1}^{±1}, t_{2}^{±1}] where t_{ı} = T_{i}^{n}. Thus A_{q}, as an R–algebra, is nothing but the cyclic algebra (t_{1}, t_{2}, ζ)_{R}. In particular, A_{q} is an R–Azumaya algebra of constant rank n. Furthermore

A_{q} ⊗_{R} S ≃ M_{n}(S)

with S = R[T_{1}].

Since the Picard group of R is trivial, every R–linear automorphism of A_{q} is inner. By Proposition 4.10, the R–group of such automorphisms is a twisted form ũ PGL_{n,R} of PGL_{n,R} . In particular (ũPGL_{n,R})(R) ≃ A_{q}^{×}/R^{×} is a finite group. These are the R–linear automorphisms of the k–algebra A_{q}. To complete the picture we can use Remark 4.7. For example if k^{×}^{n} = k^{×}, an easy calculation shows that the canonical map

: Aut_{k}(A_{q}) → Aut_{k}(R) ≃ (k^{×})^{2} ⋊ GL_{2}(ℤ)

is surjective if n ≤ 2, and has image (k^{×})^{2} ⋊ SL_{2}(ℤ) if n > 2 (cf. [OP] and [N])

4.5 Graded considerations

The material in this section will play a crucial role in the Recognition Theorem for multiloop algebras in §6.4. The reader may want to postpone reading this section until then.

Let Λ be an abelian group (denoted additively). A Λ–grading on a k–algebra B is a collection of subspaces {B^{λ}}_{λ∈Λ} indexed by Λ such that

(i) The sum ∑_{λ∈Λ}B^{λ} is direct and equals B.

(ii) B^{λ}B^{μ} ⊂ B^{λ+μ} for all λ, μ ∈ Λ.

All of this information will be summarized by the expression “ B is a Λ–graded k–algebra”. For a Λ–graded k–algebra B, the following two conditions are easily shown to be equivalent.

(GSI) B ≠ {0} and the only homogeneous ideals of B are {0} and B.

(GSII) B ≠ {0} and Mult_{k}(B)x = B for all nonzero homogeneous x in B.

We then say that B is graded simple.

Remark 4.12. Let B be graded simple. Then B is a monogenic Mult_{k}(B)–module (in particular, finitely generated). By lemma 4.2.3 of [ABFP], the centroid of B is also Λ–graded. Thus, if χ ∈ C_{k}(B)\{0}, there exist unique λ_{1}, …, λ_{h} ∈ Λ and χ_{1}, …, χ_{h} ∈ C_{k}(B) such that

χ = χ_{1} + ⋯ + χ_{n}, and
χ_{i}(B^{λ}) ⊂ B^{λ+λ_{i}} for all λ ∈ Λ and 1 ≤ i ≤ h.

Definition 4.13. Let S be a Λ–graded object of k–alg. An S/R–form L of A is said to be Λ–graded, if there exists a Λ–graded structure on the k–algebra L, and an S–algebra isomorphism ψ : L ⊗_{R} S → A ⊗ S which are compatible with each other, that is

ψ(L^{λ} ⊗ 1) ⊂ A ⊗ S^{λ}

for all λ ∈ Λ.

Proposition 4.14. Let A be a finite dimensional central simple k–algebra, and S = ⊕_{λ∈Λ}S^{λ} a graded simple object of k–alg.

1. With the natural Λ–grading on A⊗S given by (A⊗S)^{λ} = A⊗S^{λ}, the k–algebra A ⊗ S is graded simple.

2. Let L be a Λ–graded S/R–form of A. Then L is a graded simple k–algebra. Furthermore, R is naturally identified with a graded k–subalgebra of S, and the canonical map λ_{L} : R → C_{k}(L) is a Λ–graded isomorphism.

Proof. (1) Let x ∈ (A ⊗ S)^{λ} be a nonzero element, and set M = Mult_{k}(A ⊗ S)x. We must show that M = A ⊗ S. Write x = ∑ a_{i} ⊗ s_{i} with a_{i} ∈ A linearly independent, and s_{i} ∈ S^{λ} nonzero. By Jacobson’s Density theorem (see the proof of Ch.X, theorem 3 of [J]), there exists α ∈ Mult_{k}(A) such that αa_{1} = a_{1} and αa_{i} = 0 if i > 1. This shows that a_{1} ⊗ s_{1} ∈ M. Let s ∈ S. Since S is graded-simple, there exists s′ ∈ S such that s′s_{1} = s. Similarly since A is simple, there exist a′ ∈ A such that a′a_{1} ≠ 0. Then

a′a_{1} ⊗ s = (a′ ⊗ s′)(a_{1} ⊗ s_{1}) ∈ M.

Thus A ⊗ s = (Mult_{k} (A) ⊗ 1)(a′a_{1} ⊗ s) ⊂ M for all s ∈ S. It follows that M = A ⊗ S as desired.

(2) Fix an S–algebra isomorphism ψ : L ⊗_{R} S → A ⊗ S which is compatible with the Λ–gradings. We begin with three general observations that will be used in the proof.

(a) Let α : R → S be the underlying map to the R–algebra structure of S, and let β : L → L ⊗ 1 ⊂ L ⊗ S the canonical map. Because α is faithfully flat, both α and β are injective.

(b) Every element of A ⊗ S is a sum of elements of the form sψ(y ⊗ 1) with y ∈ L and s ∈ S.

(c) The associative unital k–algebra Mult_{k}(L) has a natural Λ–graded structure. Right and left multiplication by elements of L generate a graded (two-sided) ideal Mult_{k}(L)^{+} of Mult_{k}(L).

Fix a nonzero homogeneous element x of L. Let N = Mult_{k}(L)^{+}x. This is a submodule (in fact an ideal) of the R–algebra L which is graded as a subspace of L. We claim that N ≠ 0. For otherwise, for all s ∈ S and y ∈ L we have

ψ(x ⊗ 1)(sψ(y ⊗ 1)) = sψ(xy ⊗ 1) = 0 = sψ(yx ⊗ 1) = (sψ(y ⊗ 1))ψ(x ⊗ 1).

It now follows from (b) above, that ψ(x ⊗ 1) is killed by left and right multiplication by all elements of A ⊗ S. Thus the one–dimensional space kψ(x ⊗ 1) ⊂ A ⊗ S is a nonzero graded ideal of A⊗S. By part (1) we obtain kψ(x⊗1) = A⊗S. Since A⊗S is perfect and nonzero, we must have x^{2} ⊗ 1 ≠ 0. By (a) then, x^{2} ≠ 0. But this is impossible since x^{2} ∈ N. Thus N ≠ 0 as claimed.

Let M be the S–submodule of A ⊗ S generated by ψ(N ⊗ 1). We claim that M = A ⊗ S. By part (1), it will suffice to show that M is a nonzero graded ideal of A ⊗ S. That M ≠ 0 follows from (a) and the fact that N ≠ 0. Our assumption on ψ implies that the subspace ψ(N ⊗ 1) of A ⊗ S is graded, hence M is also a graded subspace of A ⊗ S. To show that M is an ideal of A ⊗ S, observe that for s ∈ S and y ∈ L we have

(sψ(y ⊗ 1))ψ(N ⊗ 1) = s(ψ(y ⊗ 1)ψ(Mult_{k}(L)^{+}x ⊗ 1))
⊂ s(Mult_{k}(L)^{+}x ⊗ 1) = sN ⊂ M.

By (b) above M is a left ideal of A ⊗ S. Similarly, M is a right ideal. The claim follows. As a consequence, we conclude that

ψ(N ⊗_{R} S) = A ⊗ S = ψ(L ⊗_{R} S).

Thus the canonical injective map N ⊗_{R}S ↪ L⊗_{R}S is in fact an equality. By faithfully flat descent N = L as desired. This finishes the proof that L is graded–simple.

It remains to show that the induced map at the centroid level preserves the Λ– gradings (see Remark 4.12). Let r ∈ R. Write α(r) = ∑ s_{i} with s_{i} ∈ S^{λ_{i}}. For x ∈ L^{λ}

ψ(rx ⊗ 1) = ψ(x ⊗ ∑ s_{i}) = ∑ s_{i}ψ(x ⊗ 1).

Since rx ∈ L, we conclude by graded considerations that s_{i}ψ(L ⊗ 1) ⊂ ψ(L ⊗ 1). Consider χ_{i} : L → L defined by χ_{i}(x) = β^{−1}ψ^{−1}λ_{s_{i}}ψβ(x). Then

χ_{i}(xy) = β^{−1}ψ^{−1}λ_{s_{i}}(ψ(x ⊗ 1)ψ(y ⊗ 1))
= β^{−1}ψ^{−1}(ψ(x ⊗ 1)s_{i}ψ(y ⊗ 1))
= xχ_{i}(y).

Similarly χ_{i}(xy) = χ_{i}(x)y. Thus χ_{i} ∈ C_{k}(L). By Lemma 4.6.3 there exists a unique r_{i} ∈ R such that χ_{i} = λ_{r_{i}}. It follows from the definitions that α(r_{i}) − s_{i} kills ψ(x ⊗ 1). Since the S module A⊗S is free, we obtain α(r_{i}) = s_{i}. This shows that R is identified, via α, with a graded k–subalgebra of S. Finally, one checks that R^{λ}L^{μ}⊗1 ⊂ L^{λ+μ} ⊗ 1. This immediately shows that the isomorphism λ_{L} : R → C_{k}(L) is Λ–graded.

5 Forms of algebras over Laurent polynomial rings

Throughout this section k is assumed to be algebraically closed and of characteristic 0. Recall that R_{n} = k[t_{1}^{±1}, … t_{n}^{±1}], that R_{n,d} = k[t_{1}^{± \frac{1}{d}} , t_{2}^{± \frac{1}{d}} , ⋯ t_{n}^{± \frac{1}{d}}], and that R_{n,∞} = \underrightarrow{lim}_{d} R_{n,d}. At the field level we have K_{n,d} = k(t_{1}^{± \frac{1}{d}} , t_{2}^{± \frac{1}{d}} , ⋯ t_{n}^{± \frac{1}{d}}) and K_{n,∞} = \underrightarrow{lim}_{d} K_{n,d}.

5.1 Multiloop Algebras

.

Throughout this section A is a finite dimensional algebra over k. Recall that (ζ_{n})_{n>0} in k^{×} is a compatible family of primitive roots of unity.

We begin by introducing the ingredients needed in the definition of multiloop algebras. Let σ σ σ = (σ_{1}, …, σ_{n}) be a commuting family of finite order automorphisms of the k–algebra A. Let d be a common period of the σ_{i}, i.e. σ_{i}^{d} = 1.

For each (i_{1}, ..., i_{n}) ∈ ℤ^{n}, consider the simultaneous eigenspaces

A_{i_{1}...i_{n}} := {x ∈ A : σ_{j}(x) = ζ_{d}^{i_{j}}x for all 1 ≤ j ≤ n}

(which of course depend only on the i_{j} modulo the d). The multiloop algebra associated to this data is the k–subalgebra L of A ⊗ R_{n,∞} defined as follows:

(5.1)  L = L(A,σ σ σ) = ⊕A_{i_{1}...i_{n}} ⊗ t_{1}^{i_{1}/d} … t_{n}^{i_{n}/d} ⊂ A ⊗ R_{n,d} ⊂ A ⊗ R_{n,∞}.

Remark 5.1. Observe that L does not depend on the choice of period d, and that L has a natural R_{n}–algebra structure.

One easily verifies that

L ⊗_{R_{n}} R_{n,d} ≃_{R_{n,d}} A ⊗ R_{n,d}.

Since R_{n,d}/R_{n} is free of finite rank (hence fppf), L is an R_{n}–form of A which is trivialized by the extension R_{n,d}/R_{n} (in the sense of §5).

From this last Remark it follows that to a multiloop algebra L as above, corresponds an R_{n}–torsor X_{L} under the group Aut(A).^{13} By descent theory, X_{L} is representable by an affine R_{n}–scheme. The functor of points of X_{L} is easily described: For all S ∈ R_{n}–alg

X_{L}(S) = Hom_{S–alg}(L ⊗_{R_{n}} S, A ⊗ S).

The isomorphism class of the R_{n}–torsor X_{L} will be denoted by [X_{L}]. Thus

(5.2)  [X_{L}] ∈ H_{ét}^{1}(R_{n,d}/R_{n}, Aut(A)) ⊂ H_{ét}^{1}(R_{n}, Aut(A)).

Recall the exact sequence of algebraic k–groups

(5.3)  1 → Aut^{0}(A) → Aut(A) →^{}¯ Out(A) → 1,

where Out(A) = Aut(A)/Aut^{0}(A) is the finite constant k–group of connected components of Aut(A). Let Out(A) = Out(A)(k), and let¯ : Aut_{k}(A) = Aut(A)(k) → Out(A) be the canonical map. The kernel of ¯ is Aut^{0}(A)(k).

The extension R_{n,d}/R_{n} is finite Galois. Its Galois group Γ_{n,d} is henceforth identified with (ℤ/dℤ)^{n} acting naturally on R_{n,d} via our fixed choice of compatible roots of unity; namely ē(t_{i}^{1/d}) = ζ_{d}^{e_{i}}t_{i}^{1/d} for all e = (e_{1}, ..., e_{n}) ∈ ℤ^{n}. If we now let Γ acts on Aut(A)(R_{n,d}) = Aut_{R_{n,d}−alg}(A⊗R_{n,d}) by conjugation, i.e. ^{γ}σ = (1⊗γ)σ(1⊗(−γ)), we have a natural correspondence H^{1}(Γ_{n,d}, Aut(A)(R_{n,d})) ≃ H_{ét}^{1}(R_{n,d}/R_{n}, Aut(A)). In terms of H^{1}(Γ_{n,d}, Aut(A)(R_{n,d})), the loop torsor X_{L} corresponds to the cocycle α_{L} ∈ Z^{1}(Γ_{n,d}, Aut(A)(R_{n,d})) given by (α_{L})ē = σ_{1}^{−e_{1}} … σ_{n}^{−e_{n}} ⊗ id ∈ Aut_{R_{n,d}–alg} (A⊗R_{n,d}). We have a natural correspondence ([SGA1, Exp XI.5]).

(5.4)  H_{ét}^{1}(R_{n}, Out(A)) ≃ H_{ct}^{1}(π_{1}(R_{n}), Out(A))

Remark 5.2. Since the fundamental algebraic group π_{1}(R_{n}) of Spec(R_{n}) is identified with (ℤ̂)^{n} via our choice of compatible primitive roots of unity (Corollary 2.11 and Remark 2.13 of [GP2]), the set H_{ct}^{1}(π_{1}(R_{n}), Out(A)) is nothing but the set of conjugacy classes of commuting n–tuples of elements of finite order automorphisms of Out(A). This interpretation plays a crucial role in the classification of multiloop algebras by cohomological methods.

13Strictly speaking, the structure group is the affine R_{n}–group Aut(A ⊗ R_{n}). As already mentioned, this harmless slight abuse of notation is used throughout.

Under the canonical map

H_{ét}^{1}(R_{n}, Aut(A)) →^{}¯ H_{ét}^{1}(R_{n}, Out(A)) ≃ H_{ct}^{1}(π_{1}(R_{n}), Out(A))

arising from (5.3) and (5.4), our [X_{L}] = [α_{L}] maps to the class [\overline{α}_{L}] of the unique cocycle (= continuous homomorphism) \overline{α}_{L} : π_{1}(R_{n}) → Out(A) satisfying

e ↦ \overline{σ} _{1}^{−e_{1}} … \overline{σ} _{n}^{−e_{n}}

for all e = (e_{1}, ..., e_{n}) ∈ ℤ^{n}.

Define an action of the group GL_{n}(ℤ) on the set of n–tuples of commuting elements of finite order of Aut_{k}(A) as follows: for aaa = (a_{ij}) ∈ GL_{n}(ℤ) and σ σ σ = (σ_{1}, …, σ_{n}) as above, set

(^{a a a}σ σ σ)_{i} = ∏_{j=1}^{n} σ_{j}^{a_{ij}} and ^{a a a}σ σ σ = ((^{a a a}σ σ σ)_{1}, …, (^{a a a}σ σ σ)_{n}).

Non-abelian cohomology allows us to classify isomorphism classes of multiloop algebras as algebras over R_{n}. One is however interested in the classification of these as algebras over k. The following lemma is therefore most useful.

Lemma 5.3. Let A be finite dimensional perfect and central algebra over k. If σ σ σ = (σ_{1}, …, σ_{n}) and τττ = (τ_{1}, …, τ_{n}) are two n–tuples of commuting automorphisms of finite order of A, the following conditions are equivalent:

(1) L(A,σ σ σ) ≃_{k} L(A,τττ).

(2) L(A,^{a a a} σ σ σ) ≃_{R_{n}} L(A,τττ) for some aaa ∈ GL_{n}(ℤ).

Proof. Let ϕ : L(A,σ σ σ) → L(A,τττ) be an isomorphism of k–algebras. Identify R_{n} with the corresponding centroids (Lemma 4.6), and consider ϕ̂ ∈ Aut_{k}(R_{n}) such that ϕ(rx) = ϕ̂(r)ϕ(x) for all r ∈ R_{n} and x ∈ A (see Remark 4.7). We have ϕ̂(t_{i}) = λ_{i}t_{1}^{a_{1i}} … t_{n}^{a_{ni}} for some aaa = (a_{ij}) ∈ GL_{n}(ℤ) and some λ_{i} ∈ k^{×}.

Identify L(A,σ σ σ) with a k–subalgebra of A ⊗ R_{n,d} for a suitable d (Remark 5.1). Fix γ_{i} ∈ k^{×} such that γ_{i}^{d} = λ_{i}. Let α be the unique element of Aut_{k}(R_{n,d}) satisfying t_{i}^{1/d} ↦ γ_{i}t_{1}^{a_{1i}/d} … t_{n}^{a_{ni}/d}. A straightforward calculation shows that the automorphism ψ = 1⊗α of the k–algebra A⊗R_{n,d}, induces an isomorphism of L(A,σ σ σ) onto L(A, ^{a a a}σ σ σ). At the centroid level we have ψ̂ = ϕ̂. It follows that the k–linear isomorphism ϕ∘ψ^{−1} : L(A, ^{a a a}σ σ σ) → L(A,τττ) is in fact R_{n}–linear.

The converse is clear since L(A,σ σ σ) and L(A, ^{a a a}σ σ σ) are evidently isomorphic as k–algebras as explained above.

Corollary 5.4. Let σ and τ be two finite order automorphisms of a finite dimensional perfect central k–algebra A. Then L(A, σ) ≃_{k} L(A, τ) if and only if L(A, σ) ≃_{R_{1}} L(A, τ) or L(A, σ^{−1}) ≃_{R_{1}} L(A, τ).

5.2 The case of Lie algebras

The algebra A is now a finite dimensional (split) simple Lie algebra over k. To follow standard practices, we will denote A by 𝔤. The relevant exact sequence of algebraic groups is

(5.5)  1 → G^{ad} → Aut(𝔤) → Out(𝔤) → 1,

where Out(𝔤) is the finite constant group corresponding to the (abstract) finite group Out(𝔤) of automorphisms of the Dynkin diagram of 𝔤 ([SGA3] Exp. XXV théorème 1.3). The sequence (5.5) is split. We fix a section Out(𝔤) → Aut(𝔤) and identify thereof in what follows Out(𝔤) with a subgroup of Aut(𝔤)(k) = Aut_{k}(𝔤).

Example 5.5. (Affine Kac-Moody Lie algebras) Let L̂ be an affine Kac-Moody Lie algebra over k, and L be the derived algebra of L̂ modulo its centre. As shown by Kac, there exists a finite dimensional simple Lie algebra 𝔤, and an automorphism π of the corresponding Dynkin diagram, such that

(5.6)  L ≃ L(𝔤, π).

As already mentioned, here and elsewhere π ∈ Out(𝔤) is viewed as an element of Aut_{k}(𝔤) via our fixed section of the split exact sequence (5.5) above. Thus loop algebras (in nullity 1) provides us with concrete realizations of the affine algebras.^{14}

The structure of the group of automorphisms of an arbitrary symmetrizable Kac-Moody Lie algebra (derived modulo its centre), was determined in [PK]: It is generated by the “adjoint” Kac-Moody group, together with a “Cartan-like” subgroup H̃, the symmetries of the extended Dynkin diagram, and the so-called Chevalley involution. In the case of an untwisted affine algebra , namely π = 1 in (5.6), it is known that the adjoint Kac-Moody group is nothing but G^{ad}(k[t^{±1}]) above.

The present cohomological viewpoint yields a new concrete realization of the automorphism group in both the twisted and untwisted affine case. To illustrate, consider the case of the twisted algebra L of type BC_{n−2}^{(2)}, n > 2 (type A_{n−1}^{(2)} in Kac’s notation). Here 𝔤 = 𝔰𝔩_{n}(k), π(X) = −X^{tr}, R = k[t^{± 1}], S = k[t^{± 1/2}]. The Galois group Γ of the extension S/R will throughout be identified with ℤ/2ℤ via \overline{1}(t^{1/2}) = −t^{1/2}. The cocycle u ∈ Z^{1}(Γ, Aut_{S}(𝔰𝔩_{n} ⊗ S)) defining L is given by u\overline{1} = π^{−1} = π. The R–linear automorphisms of L are the R–points of the twisted R–group that fits into the split exact sequence

1 → ũ PGL_{n,R} → Aut_{R}(L) → ℤ/2ℤ → 1.

14A priori, the affine algebras are given by generators and relations à la Chevalley. This presentation does not provide much insight into the nature of the algebras.

This sequence is obtained by first applying the base change R/k to (5.5), and then twisting by ũ as prescribed by Proposition 4.10.

We can give an explicit description of the R–points of the twisted group ũ PGL_{n,R} . Indeed. The exact sequence of R–groups schemes

1 → G_{m,R} → GL_{n,R} → PGL_{n,R} → 1

can be twisted by ũ. The obstruction to the map ũ GL_{n,R}(R) → ũ PGL_{n,R}(R) being surjective lies on H^{1}(R, ũG_{m,R}). But this H^{1} vanishes. One can see by general consideration [P2], or by direct computation by interpreting ũG_{m,R} as a kernel of a Weil restriction, namely

1 → ũG_{m,R} → R_{S/R}G_{m,S} →^{N_{S/R}} G_{m,R} → 1,

and then passing to cohomology (N_{S/R} is surjective because k is algebraically closed, and H^{1}(R, R_{S/R}G_{m,S}) = H^{1}(S, G_{m,S}) = Pic (S) = 0 by Shapiro’s Lemma).

The outcome is that R–linear inner automorphisms of our Kac-Moody algebra L, i.e. the R–points of the image of ũ PGL_{n,R} inside Aut_{R}(L), are given by conjugation by a matrix X ∈ GL_{n}(S) satisfying ũ \overline{1}X := u\overline{1}Xu^{−1} = X.

It remains to look at the map : Aut_{k}(L) → Aut_{k}(R). By definition

(5.7)  L = \{X ∈ 𝔰𝔩_{n} ⊗ S : −(\overline{1}X)^{tr} = X\},

where Γ = ℤ/2ℤ acts on the second coordinate of the matrix X ∈ 𝔰𝔩_{n} ⊗ S. On the other hand

Aut_{k}(R) = Aut_{k}(k[t^{±1}]) ≃ k^{×} ⋊ ℤ/2ℤ.

We claim that the map is surjective. Clearly any element θ of Aut_{k}(R) induces an element θ̃ of Aut_{k}(𝔰𝔩_{n} ⊗ S). So we must show that θ̃ stabilizes L for θ ∈ k^{×} and θ ∈ ℤ/2ℤ. If θ ∈ k^{×}, this is clear because k is algebraically closed. If θ is the generator of ℤ/2ℤ, we may assume θ(t) = t^{−1}. Again using (5.7) one sees that θ̃(L) = L. We thus have the split exact sequence

1 → Aut_{R}(L) → Aut_{k}(L) → Aut_{k}(R) → 1.

This finishes the description of the group of automorphisms of the affine Lie algebra L.

We now return to our general discussion. In terms of cohomology, the affine Lie algebras as in (5.6) account for H^{1}(R_{1}, Out(𝔤)) ≃ {conjugacy classes in Out(𝔤)}. The classes of all R_{1}–forms of 𝔤 on the other hand, are measured by H^{1}(R_{1}, Aut(𝔤)). As it turns out, the canonical map H^{1}(R_{1}, Aut(𝔤)) → H^{1}(R_{1}, Out(𝔤)) is bijective. For this one needs to know that H^{1}(R_{1}, −) vanishes for quasisplit R_{1}–groups of adjoint type. This was established in [P1] with the aid of Harder’s work. More generally we have Theorem 5.6. [P2] Let G be a reductive group scheme over k[t^{±1}]. Then H^{1}(k[t^{±1}], G) = 1.

This Theorem shows that the k[t^{±1}]–forms of 𝔤 are precisely the affine Kac-Moody algebras (derived modulo their centres) as it was mentioned in Example 4.5. In particular over k[t^{±1}], all forms of 𝔤 are loop algebras. By the invariance of type (Theorem 4.9), 𝔤 is unique up to isomorphism. For a fixed 𝔤, two k[t^{±1}]–forms are isomorphic over k if and only if they are isomorphic over k[t^{±1}]. This follows from Corollary 5.4, together with the fact that in Out(𝔤), every element is conjugate to its inverse. Combining all of the above we recover, by purely theoretical considerations, the existence of exactly 16 non-isomorphic classes of affine Lie algebras.^{15}

Similar considerations apply to R_{1}–forms of an arbitrary finite dimensional k– algebras [P2].

We now turn our attention to the case n = 2. Here we find that some interesting and unexpected behavior arises. Theorem 5.6 ought to be thought as the validity of “Serre conjecture I” for R_{1} = k[t_{1}^{±1}]. Since Serre’s Conjecture II holds for K = k(t_{1}, t_{2}), one is lead to raise the following inevitable question.

Let G be a semisimple group scheme over R_{2} = k[t_{1}^{±}, t_{2}^{±1}]. Assume G is of simply connected type. Is H^{1}(R_{2}, G) trivial?

Somehow surprisingly perhaps, the answer to this question is negative (as we have seen in §4.6). The next example recalls the reason for this failure, and the implications that this has for the classification of EALA by cohomological methods.

Example 5.7. (𝔰𝔩_{2} in nullity 2) Let 𝔤 = 𝔰𝔩_{2} = 𝔰𝔩_{2}(k). Then Aut(𝔤) = PGL_{2} and we have the exact sequence 1 → μ μ μ_{2} → SL_{2} → PGL_{2} → 1. Relativizing at R = R_{2} = k[t_{1}^{±1}, t_{2}^{±1}] and passing to cohomology yields

H_{ét}^{1}(R, SL_{2}) → H_{ét}^{1}(R, PGL_{2}) →^{δ} H_{ét}^{2}(R,μ μ μ_{2}) ⊂ _{2}Br(R).

We have H_{ét}^{2}(R,μ μ μ_{2}) ≃ ℤ/2ℤ = \{\overline{0}, \overline{1}\} (Proposition 2.1). The kernel of δ is trivial (straightforward for SL_{2}, but also consequence of Theorem 2.7), so the fiber δ^{−1}(\overline{0}) is comprised of one isomorphism class, namely the isomorphism class of the trivial R–Lie double loop algebra

L_{0} = 𝔰𝔩_{2} ⊗ R = L(𝔰𝔩_{2}, id, id).

The Lie algebra L_{0} is the centreless core of an EALA of nullity 2.

15As far as we know, nowhere in the usual Kac-Moody literature it is actually shown that the algebras of these sixteen families are non isomorphic.

Since PGL_{2} is also the group of automorphisms of the matrix algebra M_{2}(k), there is a natural correspondence between R–forms of 𝔤 and M_{2}(k): Given an R–form A of M_{2}(k), view A as a Lie algebra Lie(A) with bracket given by the commutator [x, y] = xy − yx. The derived Lie algebra Lie(A)′ is then an R–form of 𝔰𝔩_{2}.

We now apply this to the quaternion R–algebra A = A(1, n), to obtain an R–Lie algebra L_{1} = Lie (A)′. This is also a double loop algebra. In fact

L_{1} = L(𝔰𝔩_{2}, σ_{1}, σ_{2}),

where σ_{1} and σ_{2} in Aut(𝔰𝔩_{2})(k) are given by conjugation by (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) and (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})

respectively. The Lie algebra L_{1} has anisotropic generic fiber, hence cannot be the centreless core of an EALA.

Because of the classification of commuting pair of elements of finite order of PGL_{2}(k), we know that L_{0} and L_{1} are the only two k–isomorphism classes of double loop algebras based on 𝔰𝔩_{2}.

Following the same procedure, we can attach to the Margaux algebra M of §4.6 the Lie algebra

L_{2} = Lie (M)′.

Recall that M = End_{A}(M), where M is a rank one faithfully projective A–module which is not free. We have M ⊗_{R} S ≃ M_{2}(S) for S = R[t_{1}^{1/2}]. However A and M are not isomorphic R–algebras. Thus L_{1} and L_{2} are non-isomorphic R–forms of 𝔰𝔩_{2}. The algebra L_{2} is neither a multiloop algebra, nor the centreless core of an EALA.

The algebras L_{1} and L_{2} are part of the fiber δ^{−1}(\overline{1}) of the boundary map δ : H_{ét}^{1}(R, PGL_{2}) → H_{ét}^{2}(R,μ μ μ_{2}). This fiber, which is measured by H_{ét}^{1}(R, SL_{1}(A)), has therefore at least two elements. However L_{1} is the only class of this fiber which is a multiloop algebra. In particular, L_{1} is not isomorphic to L_{2} as a k–algebra.

Remark 5.8. 𝔰𝔩_{2} ⊗ k[t_{1}^{±1}] corresponds to the affine Kac-Moody Lie algebra L̂ of type A_{1}^{(1)}. This algebra has Dynkin diagram ∘ <=> ∘. The nontrivial symmetry of this diagram “lifts” to an automorphism σ̂ of L̂, which in turn induces an automorphism σ of the k–Lie algebra 𝔰𝔩_{2} ⊗ k[t_{1}^{±1}]. We have L_{1} ≃ L(𝔰𝔩_{2} ⊗ k[t_{1}^{±1}], σ). Thus our double loop algebra L_{1} = L(𝔰𝔩_{2}, σ_{1}, σ_{2}) can be obtained as a single loop algebra of an affine algebra, but the automorphism σ of the affine algebra cannot be obtained from an automorphism of 𝔰𝔩_{2} by the base change k[t_{1}^{±1}]/k. This point of view for the classification of double loop algebras (i.e. as single loop algebras of affine Kac-Moody algebras) will be described in [ABP3].

Remark 5.9. It is interesting to observe that unlike the nullity 1 case where inner automorphisms always lead to trivial loop algebras, two inner automorphisms may lead to non-trivial loop algebras. For type E_{7}, Theorem 3.17 shows that there must necessarily exists two commuting inner automorphisms whose corresponding loop algebra is not trivial. This was first empirically discovered by van de Leur with the aid of a computer [vdL].

If x x x = (x_{1}, ..., x_{n}) is an n–tuple of commuting elements of finite order of G^{ad}, then Adx x x = (Ad x_{1}, ..., Ad x_{n}) is an n–tuple of commuting automorphisms of 𝔤. For convenience, we will denote the corresponding loop algebra L(𝔤, Adx x x) simply by L(𝔤,x x x).

Theorem 5.10. Let x x x = (x_{1}, x_{2}) and yyy = (y_{1}, y_{2}) be two commuting pairs of elements of finite order of G^{ad}. Then

1. L(𝔤,x x x) ≃_{R− Lie} L(𝔤,yyy) ⟺ μ(x x x) = μ(yyy).

2. L(𝔤,x x x) ≃_{k− Lie} L(𝔤,yyy) ⟺ μ(x x x) = μ(yyy)^{±1}.

Proof. (1) This is consequence of the torsor interpretation of L(𝔤,x x x) and L(𝔤,yyy); namely that the boundary map H^{1}(R, G^{ad}) → H^{2}(R,μ μ μ) is bijective (Theorem 3.17), and determined by μ(x x x)’s (Proposition 3.16).

(2) By Lemma 5.3, we are reduced to comparing μ(x x x) and μ(^{a a a}x x x). Consider then

a a ax x x = (x_{1}^{a_{11}}x_{2}^{a_{12}} , x_{1}^{a_{21}}x_{2}^{a_{22}}) where aaa = (\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}) ∈ GL_{2}(ℤ).

By definition

x̃_{1}^{a_{11}} x̃_{2}^{a_{12}} x̃_{1}^{a_{21}} x̃_{2}^{a_{22}} = μ(^{a a a}x x x) x̃_{1}^{a_{21}} x̃_{2}^{a_{22}} x̃_{1}^{a_{11}} x̃_{2}^{a_{12}}.

By repeated use of (3.3) we obtain μ(^{a a a}x x x) = μ(x x x)^{det^{a a a}}. Now (2) follows from (1) and Lemma 5.3.

Remark 5.11. Clearly μ(x_{1}, x_{2}) = μ(x_{2}, x_{1})^{−1}. As a consequence we recover the obvious fact that L(𝔤, x_{1}, x_{2}) ≃_{k− Lie} L(𝔤, x_{2}, x_{1}). By contrast, we also obtain the following not entirely intuitive result: L(𝔤, x_{1}, x_{2}) ≃_{R− Lie} L(𝔤, x_{2}, x_{1}) if and only if μ(x x x) is of period 2.

Remark 5.12. Similar considerations apply to any perfect and central finite dimensional algebra A for which Aut(A)^{0} is semisimple.

5.3 A characterization of multiloop algebras

The main theorem of [ABFP] asserts that centreless cores of EALAs which are modules of finite type over their centroids, are always multiloop algebras. The Lie algebra L_{2} described in Example 5.7, shows that for n > 1 there exists R_{n}–forms which need not be multiloop algebras. It thus seem important to give a criterion that distinguishes multiloop algebras among all forms.

The k–algebra R_{n,∞} has a very natural interpretation: As a k–space R_{n,∞} has basis (t_{1}^{q_{1}}...t_{n}^{q_{n}})_{q_{i}∈ℚ}, and the multiplication is given by bilinear extension of t_{i}^{p}t_{i}^{q} = t_{i}^{p+q}. We give R_{n,∞} a ℚ^{n}–grading in a natural fashion, by assigning t_{1}^{q_{1}} … t_{n}^{q_{n}} degree (q_{1}, …, q_{n}). Each R_{n,d} is a graded subalgebra of R_{n,∞} whose homogeneous elements have degrees in \frac{1}{d} ℤ^{n}. Note that a homogeneous element r ∈ R_{n}, when viewed as an element of R_{n,d}, is homogeneous of degree dλ for some λ in the grading group \frac{1}{d} ℤ^{n} of R_{n,d}.

Theorem 5.13. [Recognition of multiloop algebras] Let A be a finite dimensional simple algebra over an algebraically closed field k of characteristic 0. Let n ≥ 0 be an integer. For a k–algebra L the following conditions are equivalent.

1. L ≃_{k} L(A,σ σ σ) for some n–tuple σ σ σ = (σ_{1}, …, σ_{n}) of commuting finite order automorphisms of A.

2. There exists a ℚ^{n}–grading of L and an R_{n,∞}–algebra isomorphism ψ : L⊗_{R_{n}} R_{n,∞} → A⊗R_{n,∞} which are compatible, namely ψ(L^{λ}⊗1) ⊂ A⊗R_{n,∞}^{λ} for all λ ∈ ℚ^{n}.

Proof. A multiloop algebra as in (1) clearly satisfies the conditions of (2). This follows at once form the very definition (5.1) of L(A,σ σ σ), and the nature of the gradings.

Assume L and ψ are as in (2). We show that L ≃ L(A, σ_{1}, …, σ_{n}) by appealing to the realization theorem 8.3.2 of [ABFP]. To this end, it will suffice to give L a Λ–grading for which the following three conditions hold.

(a) Λ is finitely generated and torsion free.

(b) L is graded simple.

(c) There exists d > 0 such that C(L) = ⊕_{λ∈dΛ}C(L)^{λ}. Furthermore, C(L)^{0} = k.

Fix an R_{n,∞}–algebra isomorphism ψ : L ⊗_{R_{n}} R_{n,∞} → A ⊗ R_{n,∞} such that ψ(L^{λ} ⊗ 1) ⊂ A ⊗ R_{n,∞}^{λ} for all λ ∈ ℚ^{n}.

Since A is finite dimensional, there exists d_{1} > 0 such that A ⊗ 1 ⊂ ψ(L ⊗ R_{n,d_{1}}). On the other hand, since L as an R_{n}–module is of finite type (Lemma 4.6(2)), there exists d_{2} > 0 such that ψ(L ⊗ 1) ⊂ A ⊗ R_{n,d_{2}}. If we now set d = d_{1}d_{2}, the fact that ψ is R_{n,∞}–linear shows that ψ induces, by restriction of the base ring, an R_{n,d}–algebra isomorphism

ψ : L ⊗_{R_{n}} R_{n,d} → A ⊗ R_{n,d}

By assumption, for all λ ∈ ℚ^{n} we have

ψ(L^{λ} ⊗ 1) ⊂ A ⊗ R_{n,∞}^{λ} ∩ A ⊗ R_{n,d}.

As a consequence L^{λ} ⊗ 1 ≠ {0} ⟹ dλ ∈ ℤ^{n}. But since the extension R_{n,d}/R_{n} is faithfully flat, the canonical map L → L ⊗_{R} R_{n,d} given by x → x ⊗ 1 is injective. In particular, then ℚ^{n}–grading of L has support inside the subgroup Λ = \frac{1}{d} ℤ^{n}.

As a Λ–graded algebra, we may identify L with a graded k–subalgebra of A⊗R_{n,d} (where R_{n,d} is given the standard Λ–grading). Since R_{n,d} is graded simple, L is graded simple by Proposition 4.14. This establishes (a) and (b) above.

Since L is graded simple, the centroid C_{k}(L) of L inherits a Λ–grading

C_{k}(L) = ⨁_{λ∈Λ}C_{k}(L)^{λ},

where

C_{k}(L)^{λ} := {χ ∈ C_{k}(L) : χ(L^{μ}) ⊂ L^{λ+μ} for all μ ∈ Λ}.

(see [ABFP] lemma 4.2.3). Let r ∈ R_{n} be homogeneous. Then r, as an element of R_{n,d}, is homogeneous of degree dλ for some λ ∈ Λ. If x ∈ L^{μ} we have

ψ(rx ⊗ 1) = ψ(x ⊗ r) = rψ(x ⊗ 1) ∈ r(A ⊗ R_{n,d}^{μ}) ⊂ A ⊗ R_{n,d}^{μ+λd}.

This forces rx ∈ L^{μ+λd}, hence that C_{k}(L) = ⊕_{λ∈Λ}C_{k}(L)^{dλ}. Finally, if dλ = 0, then r is

a homogeneous element of R_{n} of degree 0, i.e. r ∈ k. Thus (c) above holds and the proof of the Theorem is now complete.

Remark 5.14. By the Isotriviality Theorem [GP1, cor. 3.3] every R_{n}–form of A is split by R_{n,∞}. The last Theorem therefore gives a way of recognizing multiloop algebras among all forms. The crucial ingredient is the existence of a ℚ^{n}–grading compatible with that of R_{n,∞}. Note how the defining relations of the algebra M of Example 5.7 “breaks” the compatibility between the gradings of the quaternion algebra A(1, n) and R_{n,∞}.

6 Conjectures

Throughout this section k is assumed to be algebraically closed and of characteristic 0. Let R = k[t_{1}^{±1}, t_{2}^{±1}], and K = k(t_{1}, t_{2}).

If the semisimple R–group G is isotropic (after the base change K/R), it is reasonable to expect that H^{1}(R, G̃) = 1. Indeed theorem 2.1 of [CTGP] states that the anisotropic kernel of a semisimple group defined over K is always of type A. This is somehow analogous to the fact that almost commuting rank zero pairs only appear in type A. In the anisotropic case however, this is not true: We have seen that H^{1}(R, SL_{1}(A(1, n)) ≠ 1 (Remark 3.23).

Conjecture 6.1. Let G be a semisimple almost simple R–group with no factors of type A. Then the connecting map H^{1}(R, G) → H^{2}(R,μ μ μ) is bijective. In particular, H_{loop}^{1}(R, G) = H^{1}(R, G) (see Theorem 3.17).

By Theorem 2.7(2), the conjecture is fully established for groups of type G_{2}, F_{4} and E_{8} (because these groups are their own automorphism groups). The case of special orthogonal groups can be understood from the classification of R–quadratic forms.

Theorem 6.2. (Parimala, [Pa])

1. Cancellation holds for rational R–isotropic forms : if q_{1}, q_{2} are rationally isotropic R–forms such that q_{1} ⊥ q ≅ q_{2} ⊥ q for some quadratic R–form q, then q_{1} ≅ q_{2}.

2. Let q_{1}, q_{2} be isotropic quadratic R–forms. If q_{1} ⊗_{R} K ≅ q_{2} ⊗_{R} K, then q_{1} ≅ q_{2}.

3. Let q be a R–quadratic form of rank ≥ 5. Then q is diagonalizable and isotropic.

Proof. 1) This is part of the proof of the classification of isotropic R–forms (proposition 3.4 of [Pa]).

2) Since the field K is of class C_{2}, the quadratic form q_{K} is isotropic. By theorem 3.5 of loc. cit., there exists two k[t_{1}^{±1}]–quadratic forms q_{0} and q_{1} such that q = q_{0} ⊥ ⟨t_{2}⟩q_{1}. Since quadratic forms over k[t_{1}^{±1}] are diagonalizable (Harder, cf. [Kn] §13.4.4), it follows that q is diagonalisable, i.e is a sum of rank one R–forms in the following list ⟨1⟩ ⟨t_{1}⟩, ⟨t_{2}⟩, ⟨t_{1}t_{2}⟩. Thus q contains an orthogonal summand ⟨z, z⟩ which is hyperbolic. We conclude that q is an isotropic R–quadratic form.

Corollary 6.3. Let q be an R–quadratic form of rank ≥ 5. Then H^{1}(R, Spin(q)) = 1 and H^{1}(R, SO(q)) ≃ Z/2Z.

Proof. Theorem 6.2 shows that q is diagonalisable and contains an hyperbolic summand. The exact sequence 1 → μ μ μ_{2} → Spin(q) → SO(q) → 1 of reductive R–groups induces the following exact commutative diagram of pointed sets

SO(q)(R) ⟶^{Ns} R^{×}/(R^{×})^{2} ⟶ H^{1}(R, Spin(q)) ⟶ H^{1}(R, SO(q))
↓  ↓
1 = H^{1}(K, Spin(q)) ⟶ H^{1}(K, SO(q)).

Recall that H^{1}(R, O(q)) classifies isomorphism classes of quadratic modules of rank equal to rank(q). Parimala’s theorem states that H^{1}(R, O(q)) injects in H^{1}(K, O(q)). By diagram chasing, it follows that the map

H^{1}(R, Spin(q)) → H^{1}(R, SO(q)) → H^{1}(R, O(q))

is trivial. On the other hand, since q is isotropic (over R), the sequence 1 → SO(q) → O(q) → ℤ/2ℤ → 1 is split exact ([K2], proposition 5.2.2 page 225). It follows that the map H^{1}(R, SO(q)) → H^{1}(R, O(q)) has trivial kernel, hence the triviality of the map H^{1}(R, Spin(q)) → H^{1}(R, SO(q)). On the other hand, the spinor norm SO(q)(R) → R^{×}/(R^{×})^{2} contains even products of invertible values of q ([K2], page 232). Since q contains a hyperbolic summand, it follows that the spinor norm is surjective. By diagram chasing, we conclude that H^{1}(R, Spin(q)) = 1. This implies that the kernel of the connecting map H^{1}(R, SO(q)) → H^{2}(R, Z/2Z) = Z/2Z has trivial kernel. Now a classical twisting argument shows that the connecting map is indeed injective. Since it is also surjective by Theorem 2.7.(1), we conclude that H^{1}(R, SO(q)) ≃ Z/2Z as desired.

In other words, Conjecture 6.1 holds for special orthogonal and spinor groups of quadratic forms of rank ≥ 5.

Our final conjecture states that, outside of type A, double loop algebras are completely determined by their Witt-Tits index.

Conjecture 6.4. Let x x x = (x_{1}, x_{2}) and yyy = (y_{1}, y_{2}) be two commuting pairs of automorphisms of a finite dimensional simple Lie algebra 𝔤 over k. Assume 𝔤 is not of type A. Then L(𝔤,x x x) ⊗_{R} K is an isotropic finite dimensional simple Lie algebra over K. Furthermore,

L(𝔤,x x x) ≃_{k− Lie} L(𝔤,yyy) ⟺ I(x x x) = I(yyy).

The assumption that 𝔤 is not of type A is necessary. For in type A_{n}, with n >> 1, there are anisotropic loop algebras which are not isomorphic (Theorem 5.10 and Proposition 2.1(6)). The conjecture holds if x x x and yyy are in G^{ad}, for in this case the Brauer invariant and the Witt-Tits index determine each other.

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