Tim Wouters - March 18, 2010

Comparing invariants of SK_{1}

Abstract In this text, we compare several invariants of the reduced Whitehead group SK_{1}

of a central simple algebra.

For biquaternion algebras, we compare a generalised invariant of Suslin as constructed by the author in [Wou] to an invariant introduced by Knus-Merkurjev-Rost-Tignol [KMRT]. Using explicit computations, we prove these invariants are essentially the same.

We also prove the non-triviality of an invariant introduced by Kahn [Kah2]. To obtain this result, we compare Kahn’s invariant to an invariant introduced by Suslin in 1991 [Sus1] which is non-trivial for Platonov’s examples of non-trivial SK_{1} [Pla]. We also give a formula for the value on the centre of the tensor product of two symbol algebras which generalises a formula of Merkurjev for biquaternion algebras [Mer1].

1 Introduction

Let k be a field and A a central simple k-algebra. The triviality of the reduced Whitehead group SK_{1}(A) (which is isomorphic to SL_{1}(A)/[A^{×}, A^{×}]) is a long studied question. Tannaka and Artin posed the question in the 1930’s [NM, Wan]. For more than 30 years, one tried to prove the triviality of SK_{1}(A) in full generality. In 1976, Platonov gave a counterexample using discrete valuation rings [Pla, Thm. 5.19]. Wang, however, did prove the triviality of SK_{1}(A) if ind_{k}(A) is square-free [Wan]. This inspired Suslin to conjecture that this can be the only case of triviality [Sus1]. This would give a sufficient answer to the question of Tannaka and Artin. Merkurjev proved it is true when 4 | ind_{k}(A) [Mer3]; and Rehmann-Tikhonov-Yanchevskiĭ proved it is sufficient to prove the conjecture for the tensor product of two symbol algebras [RTY, Thm. 0.19].

In order to study his conjecture, Suslin conjectured in 1991 the existence of a cohomological invariant of SK_{1}(A) with values in Galois cohomology (n = ind_{k}(A) ∈ k^{×}):

ρ : SK_{1}(A)(k) → H^{4}(k, μ_{n}^{⊗3})/(H^{2}(k, μ_{n}^{⊗2}) ∪ [A]),  (1.1)

where [A] stands for the class of A in _{n}Br(k) ≅ H^{2}(k, μ_{n}) [Sus1, Conj. 11.6]. There are various definitions of invariants of this flavour. In 1991, Suslin defined twice his conjectured invariant (ibid., §2). For biquaternion algebras Rost gave a closely related invariant [Mer1, Thm. 4], and in 2006 Suslin defined his conjectured invariant in full generality [Sus2, §6]. Kahn even generalised this invariant to a range of new invariants [Kah2, Cor. 8.4 & Def. 11.3].

K.U.Leuven – Departement Wiskunde, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium – tim@wouters.in 2010 Mathematics Subject Classification: 19B99 (12G05, 16K50, 17B20)

Keywords: Reduced Whitehead Group – Suslin’s conjecture – Cohomological invariants – Symbol algebras

The restriction to central simple algebras with n = ind_{k}(A) ∈ k^{×} is a natural one, since otherwise the cohomology groups can be trivial and _{n}Br(k) does not have to be isomorphic to H^{2}(k, μ_{n}). Using Kato’s cohomology of logarithmic differentials [Kat1], the author generalised any of the aforementioned invariants to all central simple algebras using a lift from positive characteristic to characteristic 0 [Wou]. We recall the definitions of the invariants in more detail in Section 2. It is generally assumed that all defined invariants are essentially the same, but very few results exist on this subject. In this paper, we compare some of them.

For biquaternion algebras, Knus-Merkurjev-Rost-Tignol constructed a cohomological invariant of SK_{1}(A) without the condition on the index [KMRT, §17]. They use Witt groups, Witt rings, and a involution on the biquaternion algebra to define it. If char(k) ≠ 2, they prove the invariant is essentially the same as Suslin’s invariant for biquaternions. Using the construction of the generalisation of Suslin’s invariant, we prove that for base fields of characteristic 2 their invariant essentially equals Suslin’s generalised invariant (Section 3).

In Section 4, we compare a new invariant of Kahn with all of the other existing invariants (or more correctly, we compare the invariants to Kahn’s invariant). This allows us to prove the non-triviality of Kahn’s invariant for Platonov’s examples of non-trivial SK_{1}. We also prove a formula for the value on the centre of the tensor product of two symbol algebras under Kahn’s invariant which generalises a formula of Merkurjev for biquaternion algebras ([Mer1, Ex. p.70] – see also [KMRT, Ex. 17.23]).

Notations – Let us fix the following notations throughout this text.

• If k is a field, then k_{s} denotes a separable closure and Γ_{k} = Gal(k_{s}/k) its absolute Galois group. Furthermore, denote 𝔾_{m} = Spec(ℤ[t, t^{−1}]).

• A prime factorisation p_{1}^{e_{1}} · . . . · p_{r}^{e_{r}} of a (positive) integer m is always supposed to primitive (i.e. m = p_{1}^{e_{1}} · . . . · p_{r}^{e_{r}}, with p_{i} primes, e_{i} ≥ 1 integers for 1 ≤ i ≤ n and p_{i} ≠ p_{j} for any 1 ≤ i < j ≤ r)

• We use standard notations for the following categories: the category 𝔖𝔢𝔱𝔰 of set, the category k-𝔣𝔦𝔢𝔩𝔡𝔰 of field extensions of a field k, the category 𝔊𝔯𝔬𝔲𝔭𝔰 of groups, and the category 𝔄𝔟 of abelian groups.

• If m > 0 is an integer (prime to char(k)), then μ_{m} denotes the Γ_{k}-module of consisting of m-th roots of unity of k_{s}. If we want to stress the field in use, we write μ_{m}(k) (so that this can be viewed as the k-rational points of the appropriate sheaf).

• The appearing cohomology groups are Galois cohomology groups (unless mentioned otherwise).

• _{m}Br(k) is the m-th torsion part of the Brauer group of k (m > 0 an integer). If K is a field extension of k, we denote by Br(K/k) the kernel of the base extension morphism Br(k) → Br(K).

• If F is a discrete valuation field (with valuation v), then the valuation ring is denoted by O_{v} and the residue field by κ(v). If x ∈ O_{v}, we denote by x̄ its class in κ(v). We also use this notation for other objects for which we can define (canonical) residues. A discrete valuation is supposed to be non-trivial (of rank 1). By F_{nr} we denote the maximal unramified extension of F.

• If A is a central simple k-algebra and if F is a field extension of k, then A_{F} = A⊗_{k} F is the central simple F -algebra obtained from A by base extension to F. More generally, for a ring R, a commutative R-algebra S, and an Azumaya R-algebra A, we denote A_{S} = A⊗_{R} S, the Azumaya S-algebra obtained from A by base extension to S. By [A] we denote the Brauer class of a central simple algebra/Azumaya algebra A.

• For any central simple algebra A and a ∈ A, the reduced norm of a is denoted as Nrd_{A/k}(a). In the same way, Trd_{A/k}(a) is the reduced trace and Prd_{A,a/k}(X) is the reduced characteristic polynomial. If

Prd_{A,a/k}(X) = X^{n} − s_{1}(a)X^{n−1} + … + (−1)^{n}s_{n}(a),

then we know Nrd_{A/k}(a) = s_{n}(a) and Trd_{A/k}(a) = s_{1}(a).

• For a central simple k-algebra, we denote by SL_{1}(A) the usual linear algebraic group scheme. If F is a field extension of k, the F -rational points of SL_{1}(A) are given by SL_{1}(A)(F) = {x ∈ A_{F} | Nrd_{A/F} (a) = 1}. Furthermore, by SK_{1}(A) we denote the group functor

k-𝔣𝔦𝔢𝔩𝔡𝔰 → 𝔊𝔯𝔬𝔲𝔭𝔰 : F ↦ SK_{1}(A)(F) = SK_{1}(A_{F}) ≅ SL_{1}(A)(F)/[A^{×}, A^{×}],

called the reduced Whitehead group of A.

Acknowledgements – The author thanks Jean-Louis Colliot-Thélène, Philippe Gille, Bruno Kahn, Marc Levine, and Jean-Pierre Tignol for interesting comments and discussions leading to this article. The author expresses his gratitude to the École Normale Supérieure (Paris), K.U.Leuven, and Research Foundation - Flanders (G.0318.06) for financial support.

2 Existing invariants and results

In this section, we recall the various invariants of SK_{1} introduced by several authors. All differ a little bit on the value groups. It takes some time to introduce all of them in quite a rigorous way. The author excuses to the reader who is aware of all the definitions and results and hopes for his tolerance (and endurance). He believes it makes the text more accessible for the non-expert. Experts can eventually skip this section at first and come back to it if necessary to find e.g. a particular definition. Before recalling the invariants, we recall Merkurjev’s viewpoint on invariants and Platonov’s examples of non-trivial SK_{1} as both are used later on.

2.1 Invariants à la Merkurjev

For two group functors G, H : k-𝔣𝔦𝔢𝔩𝔡𝔰 → 𝔊𝔯𝔬𝔲𝔭𝔰, an invariant of G in H is a natural transformation of functors of G into H. Typically H equals the degree j part M_{j} of a cycle module M (à la Rost [Ros1]), such an invariant is called an invariant of G in M of degree j. It is clear that all invariants of G in H form a group (abelian if H has images in 𝔄𝔟). In case of degree j invariants of G in a cycle module M, we denote this group by Inv^{j}(G, M). We can define the same terminology if M is any functor of graded groups.

(a) Cycle modules – A cycle module M having a field k as base is a formal object having the shared properties of certain Galois cohomology groups, Milnor’s K-groups, … It associates with any field extension F of k a graded abelian group (M_{j})_{j≥0} endowed with four data (functoriality, reciprocity, K-theory module structure, and residues – D1-D4 in ibid., Def. 1.1) satisfying some homological and geometrical rules (R1a-R3e, FD, and C - ibid., Def. 1.1 & 2.1). For a field k, a central simple k-algebra A of n = ind_{k}(A) ∈ k^{×}, and an integer m ∈ k^{×}, we use the following cycle modules (for any integer r):

H_{m}^{∗} : k-𝔣𝔦𝔢𝔩𝔡𝔰 ↦ 𝔄𝔟 : F ↦ (H_{m}^{j}(F))_{j>0}  (2.1)
H_{n,A^{⊗r}}^{∗} : k-𝔣𝔦𝔢𝔩𝔡𝔰 ↦ 𝔄𝔟 : F ↦ (H_{n,A^{⊗r}}^{j}(F))_{j>1},  (2.2)

with

H_{m}^{j}(F) = H^{j}(F, μ_{m}^{⊗(j−1)}) and H_{n,A^{⊗r}}^{j}(F) = H_{n}^{j}(F)/(H^{j−2}(F, μ_{n}^{⊗(j−2)}) ∪ r[A]). (2.3)

Remark that if r ≡ 0 mod per_{k}(A), then H_{n,A^{⊗r}}^{j}(F) = H_{n}^{j}(F). So the second cycle module is actually a generalisation of the first one.

(b) Gersten complex – Given a k-variety X and a cycle module M with base k, we have a Gersten complex (ibid., §3.3) (for integers i, j ≥ 0):

… →^{∂^{i−2}} ⨁_{x∈X^{(i−1)}} M_{j−i+1}(k(x)) →^{∂^{i−1}} ⨁_{x∈X^{(i)}} M_{j−i}(k(x)) →^{∂^{i}} ⨁_{x∈X^{(i+1)}} M_{j−i−1}(k(x)) →^{∂^{i+1}} …,

induced by the residues of the cycle module. Here, X^{(i)} is the set of points of X of codimension i, k(x) is the function field of a point of codimension i, and any appearance of negative degree of the cycle module is to be interpreted as the trivial group. The homology of this complex on spot i is denoted A^{i}(X, M_{j}).

(c) Merkurjev’s link – Let G be an algebraic k-group which we view as a group functor associating to a field extension F of k, the group G(F) of F -rational points of G. If M is of bounded exponent, then Merkurjev gives an isomorphism

Inv^{j}(G, M) →^{∼} A^{0}(G, M_{j})_{mult} ⊂ A^{0}(G, M_{j}) : ρ ↦ ρ_{K}(ξ),  (2.4)

where K = k(G) is the function field of G and ξ ∈ G(K) the generic point of G [Mer2, Lem. 2.1 & Thm. 2.3]. The image A^{0}(G, M_{j})_{mult} consists of the multiplicative elements of A^{0}(G, M_{j}). These are those elements x such that p_{1}^{∗}(x) + p_{2}^{∗}(x) = m^{∗}(x) for p_{1}^{∗}, p_{2}^{∗}, m^{∗} induced by the projection p_{1}, p_{2} : G × G → G and multiplication m : G × G → G.

2.2 Platonov’s examples

Among the examples of non-trivial SK_{1} of Platonov, we concentrate on the tensor product of two cyclic algebras.

(a) Cyclic algebras – Let k be a field and K a cyclic field extension of degree n. Take furthermore a generator σ ∈ Gal(K/k) ≅ ℤ/n. Then for b ∈ k^{×}, we denote by (K/k, σ, b) the so-called cyclic k-algebra generated by K and a variable x satisfying x^{n} = b and xc = σ(c)x for any c ∈ K. Then clearly deg_{k}(K/k, σ, b) = n and we can also write this cyclic algebra as ⊕_{i=0}^{n−1}Kx^{i} with multiplication defined as above [Dra, §7, Def. 4]. Furthermore, K is a splitting field of (K/k, σ, b) (see [GS, §2.5]).

If k contains an n-th primitive root of unity and if K = k(\sqrt[n]{a}) for a ∈ k^{×}, then (K/k, σ, b) ≅ (a, b)_{n} as k-algebras (if σ is well chosen). Here (a, b)_{n} is the usual symbol kalgebra generated over k by variables x and y satisfying x^{n} = b, y^{n} = b, and xy = ξ_{n}yx for a well chosen primitive n-th root of unity ξ_{n} ∈ k. In case n = p = char(k) and if K is the cyclic Galois extension defined by x^{p} − x − a, then (K/k, σ, b) ≅ [a, b)_{p} as k-algebras (for a well chosen σ). Here [a, b)_{p} is the usual p-algebra: generated as k-algebra by variables x and y satisfying x^{p} − x = a, y^{p} = b, and xy = y(x + 1) (loc. cit.).

If n = 2, a symbol algebra or p-algebra is more commonly called a quaternion algebra. The product of two quaternion algebras is a biquaternion algebra; it is a central simple algebra of degree 4 and period 1 or 2. It is know that biquaternion algebras are in fact the only central simple algebras of degree 4 and period 1 or 2 [Alb, p. 369].

(b) Non-trivial SK_{1} – Let k be a local field (e.g. ℚ_{p} or 𝔽_{p}((x))) and let K_{1}, K_{2} be two cyclic extensions of degree n over k which are linearly disjoint. Let σ_{1} (resp. σ_{2}) be a generator of Gal(K_{1}/k) (resp. Gal(K_{2}/k)). Now let F = k((t_{1}))((t_{2})), F_{1} = K_{1}((t_{1}))((t_{2})), and F_{2} = K_{2}((t_{1}))((t_{2})). Then Platonov proves that

A = (F_{1}/F, σ_{1}, t_{1}) ⊗ (F_{2}/F, σ_{2}, t_{2})

is a division F -algebra and furthermore SK_{1}(A) ≅ Br(K/k)/(Br(K_{1}/k)Br(K_{2}/k)) ≅ ℤ/n for K = K_{1} ⊗ K_{2} [Pla, Thms. 4.7 & 5.9].

(c) Galois cohomology of ℚ_{p}((t_{1}))((t_{2})) – To study the invariants later on, we encounter the fourth Galois cohomology groups H_{m}^{4}(k) for k = ℚ_{p}((t_{1}))((t_{2})). These can be calculated using a splitting for a complete discrete valuation field K with residue field κ(v) and with m ∈ κ(v)^{×} (hence also m ∈ K^{×}) [GMS, 7.11]:

H_{m}^{i+1}(K) ≅ H_{m}^{i+1}(κ(v)) ⊕ H_{m}^{i}(κ(v)).  (2.5)

Using the fact that cd(ℚ_{p}) = 2 and Br(ℚ_{p}) = ℚ/ℤ [Ser, Ch. II, §5.1 & Prop. 15], we find H_{m}^{4}(k) ≅ ℤ/m by applying the splitting to the valuations defined by t_{1} and t_{2}.

2.3 Suslin’s invariants

We recall the invariants of Suslin and an invariant for biquaternion algebras introduced by Rost. Let us first give the motivation why these invariants can help to explain Platonov’s counterexamples.

(a) Suslin 1991 – By constructing his invariant ρ_{A} ∈ Inv^{4}(SK_{1}(A), H_{m,A}^{∗}) (for m = ind_{k}(A) ∈ k^{×}), Suslin hoped to be able to complete the following diagram (for A as in §2.2 (b)):

\xymatrix{SK_{1}(A) \ar[r]^{≅} \ar[d]_{ρ_{A,F}} & Br(K/k)/(Br(K_{1}/k)Br(K_{2}/k)) \ar[d] \\ H_{n^{2},A}^{4}(F) \ar[r]^{∂_{t_{1}}^{3}∘∂_{t_{2}}^{4}} & H_{n^{2}}^{2}(k)/∂_{t_{1}}^{3} ∘ ∂_{t_{2}}^{4}(H^{2}(k, μ_{n^{2}}^{⊗2}) ∪ [A])}  (2.6)

The maps ∂_{t_{1}}^{3}, ∂_{t_{3}}^{4} are residues induced by the discrete valuation associated with t_{1} and t_{2}, i.e. the projection maps of degree −1 in (2.5). At the time he conjectured the existence of such an invariant, he could not yet give a definition. He was however able to define an invariant ρ_{S91,A} ∈ Inv^{4}(SK_{1}(A), H_{m,A^{⊗2}}^{∗}) which he proves to be non-trivial for Platonov’s examples of non-trivial SK_{1}.

(b) Biquaternion algebras – In the case of biquaternion algebras, Rost was able to define a related invariant of SK_{1}(A). Suppose A is a biquaternion algebra over a field k of char(k) ≠ 2. Then Rost’s invariant ρ_{Rost,A} is an invariant sitting in Inv^{4}(SK_{1}(A), H_{2}^{∗}) [Mer1, Thm. 4]. Moreover, it fits into an exact sequence:

0 → SK_{1}(A)(k) → H^{4}(k, ℤ/2ℤ) → H^{4}(k(Y), ℤ/2ℤ),

where Y an Albert form of A. This invariant was generalised in [KMRT, §17] to biquaternion algebras in any characteristic using Witt groups and Witt rings. We come back to this generalised invariant in Section 3 as its definition requires a lot of terminology related to involutions.

(c) Suslin 2006 – Using Voevodsky’s motivic étale cohomology, Suslin was able to define his conjectured invariant in 2006 [Sus2, §3]. We denote this invariant by ρ_{S06,A}. It is however not clear whether (2.6) commutes for this invariant. It is clear that this (and also the other invariants) become trivial after base extension to the function field of X = SB(A) (it is a splitting field of A). Suslin hence proves his invariant is essentially the same as Rost’s invariant ρ_{Rost,A} for a biquaternion algebra over a field k of char(k) ≠ 2. He does this by proving that

\xymatrix{SK_{1}(A)(k) \ar[d]_{=} \ar[r]^{ρ_{S06}} & ker[H_{4,A}^{4}(k) → H_{4,A}^{4}(k(X))] \ar[d]^{r_{A}} \\ SK_{1}(A)(k) \ar[r]_{ρ_{Rost}} & ker[H_{2}^{4}(k) → H_{2}^{4}(k(Y))],}  (2.7)

is a commutative diagram, where is r_{A} is the morphism induced on Galois cohomology by the map μ_{4}^{⊗3} → μ_{2} : a ↦ a^{2} and where X and Y are as above. Hence ρ_{S06} is injective for biquaternion algebras and

SK_{1}(A)(k) ≅ ker[H_{4,A}^{4}(k) → H_{4,A}^{4}(k(X))].

2.4 Kahn’s invariants

Let k be a field and A a central simple algebra with n = ind_{k}(A) ∈ k^{×}. We recall the inspiring results on invariants of SK_{1}(A) as obtained by Kahn in [Kah2].

(a) Cyclicity of invariant group – By calculations with motivic étale cohomology, Kahn shows A^{0}(SL_{1}(A), H_{n}^{4})_{mult} is finite cyclic [Kah2, Def. 11.3]. So by Merkurjev’s isomorphism (2.4), Inv^{4}(SL_{1}(A), H_{n}^{∗}) is finite cyclic. As the canonical projection SL_{1}(A) → SK_{1}(A) induces an injective morphism

Inv^{4}(SK_{1}(A), H_{n}^{∗}) → Inv^{4}(SL_{1}(A), H_{n}^{∗}),  (2.8)

we also find Inv^{4}(SK_{1}(A), H_{n}^{∗}) to be cyclic. Using Kahn’s calculations (loc. cit.), we can pick a canonical generator that we call Kahn’s invariant ρ_{Kahn,A} of SK_{1}(A).

(b) Bounds on invariant group – Kahn also argues the size of Inv^{4}(SL_{1}(A), H_{n}^{∗}) is bounded by ind(A)/l if n = ind_{k}(A) is the power of a prime l (ibid., Lem. 12.1). Hence the same holds for Inv^{4}(SK_{1}(A), H_{n}^{∗}) by (2.8). For general n, Kahn’s bound is retrieved using Brauer’s decomposition theorem [GS, Ch. 4, Prop. 4.5.16]. For any integer n with prime factorisation p_{1}^{e_{1}} · . . . · p_{r}^{e_{r}}, we denote by \overline{n} the integer p_{1}^{e_{1}−1} · . . . · p_{r}^{e_{r}−1}.

Lemma 2.1. Let k be a field and A a central simple algebra of ind_{k}(A) = n ∈ k^{×}. Then

|Inv^{4}(SK_{1}(A), H_{n}^{∗})| ≤ \overline{n}.

Proof. Let p_{1}^{e_{1}} · . . . · p_{r}^{e_{r}} be a prime decomposition of n, then Brauer’s decomposition theorem (loc.cit.) gives division k-algebras D_{1}, …, D_{r} of ind_{k}(D_{i}) = p_{i}^{e_{i}} such that A is Brauer-equivalent to D_{1} ⊗ … ⊗ D_{r}. This even gives rise to a decomposition SK_{1}(A) ≅ SK_{1}(D_{1}) ⊕ … ⊕ SK_{1}(D_{r}) (ibid., Ch. 4, Ex. 9). Recall also that SK_{1}(D_{i}) has p_{i}^{e_{i}}-torsion [Dra, §23, Lem. 3]. Then the result follows immediately from the primary result of Kahn and the isomorphism H_{n}^{4}(k) ≅ H_{p_{1}^{e_{1}}}^{4}(k) ⊕ … ⊕ H_{p_{r}^{er}}^{4} (k). ■

Remark 2.2 – As Kahn mentions, this bound is sharp for biquaternion division algebras [Kah2, §12]. This follows from [Mer2, Prop. 4.9 & Thm. 5.4]. In particular, ρ_{Kahn} is not trivial for biquaternion division algebras. In §4.1 (c), we generalise this result.

(c) Generalisation of Suslin’s invariant – Apart from using Merkurjev’s viewpoint to define a new invariant, Kahn also generalises ρ_{S06} to invariants

ρ_{r} ∈ Inv^{4}(SK_{1}(A), H_{n,A^{⊗r}}^{∗})

with n ∈ ind_{k}(A) ∈ k^{×} and r = 1, …, per_{k}(A) − 1. Suslin’s invariant ρ_{S06} is retrieved setting r = 1. It is not clear whether ρ_{S91} equals ρ_{2}. As mentioned in §2.3 (c), ρ_{S06} has its image in ker[H_{n,A}^{4}(F) → H_{n,A}^{4}(F (X))] for F a field extension of k and X the Severi-Brauer variety of A [Sus2, §3]. Kahn generalises this to ρ_{r} replacing X by the generalised Severi-Brauer variety SB(r, A) (ibid., §8.B).

He also gives a bound on the torsion of these invariants inside Inv^{4}(SK_{1}(A), H_{n,A}^{∗}) if l = per_{k}(A) is a prime. Indeed from (ibid., Thm. 7.1(c) & Cor. 12.10) it follows that they have

• l-torsion if ind_{k}(A) = per_{k}(A) = l > 2,

• l^{2}-torsion if per_{k}(A) > ind_{k}(A) = l > 2, and

• 2-torsion if per_{k}(A) = 2.

For a central simple k-algebra A with n = ind_{k}(A) ∈ k^{×} and per_{k}(A) = n/\overline{n}, there is a similar statement using a Brauer decomposition. Take a prime factorisation n = p_{1}^{e_{1}}·. . .·p_{r}^{e_{r}} and let D_{1} ⊗…⊗D_{r} be a Brauer decomposition of A as in the proof of Lemma 2.1. Then put m = p_{1}^{f_{1}} · . . . · p_{r}^{f_{r}}, where f_{i} = 1 if p_{i} = 2 or if ind_{k}(D_{i}) = per_{k}(D_{i}) = p_{i} > 2, and f_{i} = 2 if ind_{k}(D_{i}) > per_{k}(D_{i}) = p_{i} > 2. Then it is clear that ρ_{r} has m-torsion.

2.5 Generalising invariants

In [Wou], the author introduced a way of generalising the invariants of SK_{1}(A) to any central simple k-algebra A (so also when ind_{k}(A) ∉ k^{×}). This is done using a lift from a field of positive characteristic to a field of zero characteristic where the invariants are always defined. In this subsection, let k be a field of char(k) = p > 0. We first explain Kato’s cohomology of logarithmic differentials which are used in (loc. cit.) to generalise H_{n}^{∗} when p | n. This allows us to perform lifts from positive characteristic to characteristic 0.

(a) Logarithmic differentials – For any integer l > 0, the cohomology groups H_{p^{l}}^{q+1}(k) are defined as

(W_{l}(k) ⊗ \underbrace{k^{×} ⊗ … ⊗ k^{×}}_{q times})/I,

where W_{l}(k) are the Witt vectors of length l on k and I is the ideal generated by

(i) w ⊗ b_{1} ⊗ … ⊗ b_{q}, satisfying b_{i} = b_{j} for 1 ≤ i < j ≤ q,

(ii) (0, …, 0, a, 0, …, 0) ⊗ a ⊗ b_{2} ⊗ … ⊗ b_{q},

(iii) (w^{(p)} − w) ⊗ b_{1} ⊗ … ⊗ b_{q},

with w ∈ W_{l}(k), b_{1}, …, b_{q} ∈ k^{×}, and w^{(p)} = (a_{1}^{p}, …, a_{l}^{p}) if w = (a_{1}, …, a_{l}).

For l = 1, we can view H_{p}^{q+1}(k) as the cokernel of

F −1 : Ω_{k}^{q} → Ω_{k}^{q}/dΩ_{k}^{q−1}, defined by x\frac{dy_{1}}{y_{1}} ∧…∧\frac{dy_{q}}{y_{q}} ↦ (x^{p}−x)\frac{dy_{1}}{y_{1}} ∧…∧\frac{dy_{q}}{y_{q}} mod dΩ_{k}^{q−1},

hence the terminology “logarithmic differentials”. (For l = 0, set H_{p^{l}}^{q+1}(k) = 0.) In general, for an integer n = p^{l}m > 0 (l, m ≥ 0 integers with p ∤ m), we define

H_{n}^{q+1}(k) = H_{p^{l}}^{q+1}(k) ⊕ H_{m}^{q+1}(k).

This is a generalisation of Galois cohomology, since this theory fills in some gaps in Galois cohomology. It gives for example a description of the p^{l}-th torsion part of the Brauer group, compatible with the prime-to-p part: _{p^{l}}Br(k) ≅ H_{p^{l}}^{2}(k). So for any integer n > 0 we get _{n}Br(k) ≅ H_{n}^{2}(k). We can also define H_{n}^{∗} in the same way as in (2.1). It is however not a cycle module, but rather a functor of graded groups. To obtain a cycle module we have to tweak it a little bit. For this paper we do not need a cycle module, so we rather work with this functor of graded groups to ease the discussion (see [Wou, §4.1 (d)] for more details – see also Remark 2.6 infra).

Using this isomorphism, together with a scalar multiplication by Milnor’s K-groups on (H_{p^{l}}^{q+1}(k))_{q≥0}, we can generalise the definition of H_{n,A}^{∗} for a central simple k-algebra A with arbitrary index. Recall that Milnor K-groups K_{r}^{M}(k) (for an integer r ≥ 0) are defined as

\underbrace{k^{×} ⊗ … ⊗ k^{×}}_{r times}/J,

where J is the ideal generated by x_{1} ⊗ … ⊗ x_{i} with x_{i} + x_{j} = 1 for some 1 ≤ i < j ≤ r. Elements of K_{r}^{M}(k) are called symbols and the generators x_{1} ⊗ … ⊗ x_{r} are called pure symbols, commonly denoted {x_{1}, …, x_{r}}. The scalar multiplication of K_{r}^{M}(k) on (H_{p^{l}}^{q+1}(k))_{q≥0} is given by

· : K_{r}^{M}(k) × H_{p^{l}}^{q+1}(k) → H_{p^{n}}^{r+q+1}(k), defined by
({x_{1}, …, x_{r}}, w ⊗ b_{1} ⊗ … ⊗ b_{q}) ↦ w ⊗ x_{1} ⊗ … ⊗ x_{r} ⊗ b_{1} ⊗ … ⊗ b_{q}.

This allows us to define a relative version. Before doing so, we recall that also the cupproduct definition of (2.3) can be generalised using K-theory. Indeed, the isomorphism k^{×}/(k^{×})^{m} ≅ H^{1}(k, μ_{m}) for any m ∈ k^{×} gives the Galois symbol by taking the cup-product:

h_{m,k}^{r} : K_{r}^{M}(k) → H^{r}(k, μ_{m}^{⊗r}).  (2.9)

The Bloch-Kato conjecture (proved by Voevodsky-Rost-Weibel [BK, Voe, Ros2, Wei2]) even says it is surjective with kernel mK_{r}^{M}(k). Hence we get a scalar multiplication of K_{r}^{M}(k) on (H_{m}^{q+1}(k))_{q≥0}:

· : K_{r}^{M}(k) × H_{m}^{q+1}(k) → H_{m}^{r+q+1}(k) defined by (a, b) ↦ h_{m,k}^{r}(a) ∪ b.

For arbitrary n, this defines in total a K_{r}^{M}(k)-module structure on (H_{n}^{q}(k))_{q>0}. If A is a central simple k-algebra of ind_{k}(A) = n, we can then define for any field extension F of k and integers q ≥ 0 and r:

H_{n,A^{⊗r}}^{q+1}(F) = H_{n}^{q+1}(F)/(K_{q−1}^{M}(F) · r[A_{F}]).

By the remarks above, this is clearly a generalisation of the moderate case. If r ≡ 0 mod per_{k}(A), then clearly H_{n,A^{⊗r}}^{q+1}(F) = H_{n}^{q+1}(F) (cfr. §2.1 (a)). In the same way as in (2.2), we obtain a functor of graded groups H_{n,A^{⊗r}}^{∗}.

(b) Lifts – We now consider k to be the residue field of a complete discrete valuation ring R with fraction field K of char(K) = 0. The specialisation map Br(R) → Br(k) : [A] ↦ [A ⊗_{R} k] is bijective [Gro2, Cor. 6.2] and Br(R) → Br(K) : [A] ↦ [A ⊗_{R} K] is injective [AG, Thm. 7.2]. So we have an inclusion Br(k) → Br(K); given a central simple algebra A over k, we get a lifted Azumaya algebra B over R and an associated central simple algebra B_{K} over K. Because of the definition, ind_{k}(A) = ind_{K}(B_{K}) and by a theorem of Platonov we get SK_{1}(A)(k) ≅ SK_{1}(B_{K})(K) [Pla, Thm. 3.12] – see also [Wou, Cor. 3.3].

Furthermore, there exists an injection H_{n}^{i+1}(k) → H_{n}^{i+1}(K); on the prime-to-p parts of H_{n}^{i+1}(k) defined by (2.5), for general n see [Kat1, Proof of Prop. 2] and [Izh, Prop. 6.8] (see also Remark 3.7 for 2-primary n). This injection also continues to the relative cohomology groups; i.e. there exists an injection H_{n,A^{⊗r}}^{i+1}(k) → H_{n,B_{K}^{⊗r}}^{i+1}(K) for any integer r and A and B as above [Wou, Prop. 4.10].

This allows us to define an invariant for any central simple k-algebra, using the existence in the characteristic 0 case. In order to stay functorial, we have to use p-rings. A p-ring is a complete discrete valuation ring R with residue field k of char(k) = p > 0 and whose maximal ideal is generated by p. For a reference see e.g. [Mat, §23] where p-rings can also be not complete, we however always suppose them to be complete. For a p-ring R, the fraction field K is of characteristic 0. Moreover, Cohen proved that given a field k of char(k) > 0, there always exists a p-ring with k as residue field [Coh].

For sake of convenience, we also use the following terminology.

Definition 2.3. Suppose ρ is an invariant of SK_{1} which is defined for any central simple algebra A with index n not divisible by the characteristic of its base field and which has values in the Galois cohomology group H_{n,A^{⊗r}}^{4} for r a fixed integer. Then we say ρ is a moderate invariant of SK_{1} with values in H_{⊗r}^{4}. We denote by ρ_{A} the invariant for a central simple algebra A.

In [Wou, Thm.4.20], the author proves the following theorem.

Theorem 2.4. Let k be a field of char(k) = p > 0 and A a central simple k-algebra of n = ind_{k}(A). Take R a p-ring with residue field k and fraction field K. Let B be the lifted Azumaya R-algebra of A and let ρ ∈ Inv^{4}(SK_{1}(B_{K}), H_{n,B_{K}^{⊗r}}^{∗}) (for r any integer). There

exists a unique invariant ρ̃ ∈ Inv^{4}(SK_{1}(A), H_{n,A^{⊗r}}^{∗}) such that for any field extension k′, p-ring R′ with residue field k′, and fraction field K′, we have a commutative diagram:

\xymatrix{SK_{1}(A)(k′) \ar[r]^{ρ̃_{k′}} & H_{n,A^{⊗r}}^{4}(k′) \ar[d] \\ SK_{1}(B_{K})(K′) \ar[u]^{≅} \ar[r]_{ρ_{K′}} & H_{n,B_{K}^{⊗r}}^{4}(K′).}  (2.10)

Remark 2.5 – The invariants obtained by this theorem are the wild generalisations of their moderate variants (hence the terminology moderate versus wild). If ρ is a moderate invariant of SK_{1}, we denote the wild generalisation by ρ̃. If A is a central simple k-algebra of ind_{k}(A) ∈ k^{×} (with char(k) = p > 0), it is in general not clear whether ρ_{A} = ρ̃_{A}. By the uniqueness of the theorem, to prove such an equality it suffices to verify that ρ satisfies a lifting property as in (2.10).

Remark 2.6 (for the reader who takes the effort to look at the original paper.) – In the original statement, the author treats just the case r = 1. The proof does not depend on r, so it can easily be generalised to any r. If r = 0, we can also use (ibid., Cor. 4.14) straightaway to prove the theorem. Also an extra field extension L of k is used. This is to be sure H_{n,L}^{∗} of (ibid., Def. 4.3) is a cycle module with base R. We do not explicitly need this here. Even more, the statement over here is not weaker as by functoriality any invariant has images in H_{n,L,A^{⊗r}}^{∗}.

Remark 2.7 – Note that the theorem actually defines an injective morphism

Inv^{4}(SK_{1}(B_{K}), H_{n,B_{K}^{⊗r}}^{∗}) → Inv^{4}(SK_{1}(A), H_{n,A^{⊗r}}^{∗}).  (2.11)

As the invariants ρ_{Kahn} and ρ_{S06} are non-trivial for biquaternion algebras in characteristics different from 2, this induces their wild generalisations to be non-trivial for biquaternion algebras in characteristic 2.

3 Biquaternion algebras

In [KMRT, §17], Knus-Merkurjev-Rost-Tignol construct an invariant of the reduced Whitehead group of biquaternion algebras in any characteristic. For sake of brevity we call it KMRT’s invariant. If the characteristic of the base field is not equal to 2, it is known that this invariant essentially equals Suslin’s invariant. In this section, we prove in the characteristic 2 case it is essentially equal to Suslin’s generalised invariant.

3.1 Definition

We start by giving the concrete definition of KMRT’s invariant. This needs the notion of involutions on Azumaya algebras and Witt groups.

(a) Involutions on Azumaya algebras – In order to define the invariant, a symplectic involution σ on the biquaternion algebra is used. We recall the definition of a symplectic involution on an Azumaya algebra (so in particular on a central simple algebra). We treat this in this general setting of Azumaya algebras, because we need this for our purposes later on. We refer to [Knu, Ch. III, §8] for more details on involutions on Azumaya algebras.

Definition 3.1. Let R be a ring and A an Azumaya algebra over R with an R-linear involution σ. Suppose α : A ⊗_{R} S →^{∼} M_{n}(S) is a faithfully flat splitting of A. Then σ̃ = α(1 ⊗ σ)α^{−1} is an involution on M_{n}(S). Since x ↦ σ̃(x^{t}) is an automorphism of M_{n}(S), we can choose u ∈ GL_{n}(S) such that σ̃(x) = ux^{t}u^{−1} for all x ∈ M_{n}(S). Because σ̃^{2} = 1, we get u^{t} = ϵu for some ϵ ∈ μ_{2}(S). Then ϵ is called the type of σ (it is well defined and independent of the choice of a faithfully flat splitting [Knu, Ch. III, 8.1.1.]). An involution of type 1 is called orthogonal and an involution of type -1 is called symplectic.

Remark 3.2 – If R is an integral domain, then an involution on an Azumaya algebra can only have type 1 or −1. When k is a field, a central simple k-algebra of odd degree can only have orthogonal involutions, while a central simple algebra of even degree can have involutions of both types [KMRT, Cor. 2.8].

If A is a central simple algebra over k of degree 2n with a symplectic involution σ, we can refine the definition of reduced norm, trace and characteristic polynomial. Indeed, if a ∈ Symd(A, σ) = {a + σ(a) | a ∈ A}, the reduced characteristic polynomial Prd_{A,a/k}(X) is a square [KMRT, Prop. 2.9]. Take Prp_{σ,a/k}(X) the unique monic polynomial such that Prd_{A,a/k}(X) = (Prp_{σ,a/k}(X))^{2}; this is the Pfaffian characteristic polynomial. The Pfaffian trace Trp_{σ/k}(a) and the Pfaffian norm Nrp_{σ/k}(a) are defined as coefficients of Prp_{σ,a/k}(X), compatible with the expression of Nrd_{A/k}(a) and Trd_{A/k}(a) as coefficients of Prd_{A,a/k}(X) (see standard notations in §1):

Prp_{σ,a/k}(X) = X^{n} − Trp_{σ/k}(a)X^{n−1} + … + (−1)^{n}Nrp_{σ/k}(a).

So Nrd_{A/k}(a) = (Nrp_{σ/k}(a))^{2} and Trd_{A/k}(a) = 2Trp_{σ/k}(a).

(b) Witt groups – To explain the value group of KMRT’s invariant, we need Witt groups and rings.^{1} The Witt group W_{q}(k) is the group of Witt-equivalence classes of non-singular quadratic spaces over k with addition defined by the orthogonal sum ⊥. The Witt ring W (k) is the ring of Witt-equivalence classes of non-singular symmetric bilinear spaces with addition given by the orthogonal sum ⊥ and multiplication by the tensor product ⊗.

Remark 3.3 – If char(k) ≠ 2, we know that as groups (with the orthogonal sum) W_{q}(k) and W (k) are isomorphic; not as rings, since one can not come up with a direct definition of multiplication of quadratic forms. For our purposes we are however interested in the characteristic 2 case, so we have to make a clear distinction. For more information on Witt groups and Witt rings in this general case, we refer to [Bae, Ch. I] and [Kah1, Ch. 1] (including the discussion on the characteristic 2 case by Laghribi in [Kah1, App. E]).

We can equip W_{q}(k) with a W (k)-module structure. If (V, B) is a non-singular symmetric bilinear space on k and (V′, q) is a non-singular quadratic space on k, then (V ⊗ V′, B ⊗ q) is a quadratic space on k with B ⊗ q defined by

(B ⊗ q)(v ⊗ v′) = B(v, v)q(v′), for v ∈ V, v′ ∈ V′.

Let I(k) be the fundamental ideal of W (k) (generated by the non-singular bilinear spaces of even dimension). For any integer n ≥ 0, we set I^{n}(k) = (I(k))^{n} (with I^{0}(k) = W (k)) and I^{n}W_{q}(k) = I^{n}(k) ⊗ W_{q}(k). This clearly defines a filtration

W_{q}(k) = I^{0}W_{q}(k) ⊃ I^{1}W_{q}(k) ⊃ I^{2}W_{q}(k) ⊃ …

We denote the graded quotients by \overline{I^{n}W_{q}(k)} = I^{n}W_{q}(k)/I^{n+1}W_{q}(k).

Remark 3.4 – Set W′_{q}(k) the subgroup of W_{q}(k) consisting of equivalence classes of evendimensional non-singular quadratic spaces over k and I^{n}W′_{q}(k) = I^{n}(k) ⊗ W′_{q}(k). If char(k) ≠ 2, we have I^{n}W′_{q}(k) = I^{n+1}(k) by the equivalence of symmetric bilinear and quadratic spaces. Again, in general we are not able to use this fact.

(c) Definition – Suppose A is a biquaternion algebra over k (see §2.2 (a)) and suppose furthermore that σ is a symplectic involution on A. Knus-Merkurjev-Rost-Tignol construct an explicit map

SL_{1}(A) → \overline{I^{3}W′_{q}(k)} : a ↦ \begin{cases}0 & if σ is hyperbolic, \\ Φ_{v} + I^{4}W′_{q}(k) & if σ is not hyperbolic, \end{cases}

with kernel equal to [A^{×}, A^{×}]. Recall that an involution is called hyperbolic if there exists an idempotent e ∈ A such that σ(e) = 1 − e. Furthermore, Φ_{v} is the quadratic form

A → k : x ↦ Φ_{v}(x) = Trp_{σ}(σ(x)vx),

where v ∈ Symd(A, σ) ∩ A^{×} satisfies v(Trp_{σ}(v) − v)^{−1} = −σ(a)a. There always exists a v satisfying this condition [KMRT, Lem. 17.3]. This definition is well defined and independent of the choice of v and σ. Moreover the construction is functorial so that we get an invariant ρ_{BI,A} of SK_{1}(A).

1Do not mix up the Witt group and Witt ring with W_{n}(k) consisting of the Witt vectors on a field k - see §2.5 (a).

3.2 Comparison, moderate case

In this section, we recall why ρ_{BI,A} and ρ_{S06,A} are equal if A is a biquaternion algebra over k with char(k) ≠ 2. This is because both Suslin and Knus-Merkurjev-Rost-Tignol proved their invariant of SK_{1}(A) equals ρ_{Rost,A}. We already recalled the commutative diagram (2.7) giving us the equality of ρ_{S06,A} and ρ_{Rost,A}.

To compare ρ_{BI} to ρ_{Rost}, famous isomorphisms are used, most of them recently proved. Indeed, there are isomorphisms ψ_{F}^{1} : K_{4}^{M}(F)/2 → \overline{I^{4}(F)} = I^{4}(F)/I^{5}(F) for any F of char(F) ≠ 2 (Milnor’s conjecture for quadratic forms [Mil, Q. 4.3], proved by Orlov-Vishik-Voevodsky [OVV, Thm 4.1]) and ψ_{F}^{2} : H^{4}(F, μ_{2}) → K_{4}^{M}(F)/2 (Milnor’s conjecture [Mil, §6] or a special case of the Bloch-Kato conjecture (2.9)).

So the obvious way of comparing ρ_{BI} and ρ_{Rost} is by the composed isomorphism ψ_{F} = ψ_{F}^{1} ∘ψ_{F}^{2} . Indeed, Knus-Merkurjev-Rost-Tignol prove that the following diagram commutes [KMRT, Notes §17]:

\xymatrix{0 \ar[r] & SK_{1}(A)(F) \ar[d]_{=} \ar[r]^{ρ_{Rost,A,F}} & H_{2}^{4}(F) \ar[d]_{≅}^{ψ} \ar[r] & H_{2}^{4}(F (Y)) \ar[d]^{≅} \\ 0 \ar[r] & SK_{1}(A)(F) \ar[r]^{ρ_{BI,A,F}} & \overline{I^{4}(F)} \ar[r] & \overline{I^{4}(F (Y))},}  (3.1)

for F any field extension of k and Y the Albert form attached to A from §2.3 (b).

So combining (2.7) and (3.1), it follows that ρ_{S06} and ρ_{BI} are the same for biquaternion algebras in characteristic different from 2.

3.3 Comparison, wild case

We first explain how to lift central simple algebras with a symplectic involution. We do this for general central simple algebras and later on use the result for biquaternion algebras.

(a) Lifting algebras with involution – Let k be a field of char(k) = p > 0 and R a p-ring with residue field k and fraction field K. Take an Azumaya algebra A over R of degree 2n with symplectic involution σ. Define the R-group scheme PGSp(A, σ) = Aut(A, σ), defined for any R-algebra S by

Aut(A, σ)(S) = Aut(A_{S}, σ_{S}) = {φ ∈ Aut_{S}(A_{S}) | φ ∘ σ_{S} = σ_{S} ∘ φ},

with σ_{S} = σ ⊗ id the canonical extension of σ to A_{S}. All Azumaya algebras of degree 2n with symplectic involutions up to isomorphism are classified by H_{ét}^{1}(R, PGSp(A, σ)) [KMRT, 29.22]. Since PGSp(A, σ) is a smooth group scheme (proof as in the field case [KMRT, p. 347]), we can use Hensel’s lemma à la Grothendieck to get an isomorphism [SGA, Exp. XXIV, Prop. 8.1]:

H_{ét}^{1}(R, PGSp(A, σ)) ≅ H^{1}(k, PGSp(\overline{A}, σ̄)),

where \overline{A} = A ⊗_{R} k is the reduced central simple k-algebra and σ̄ = σ ⊗ id is the reduced involution on \overline{A}, which is also symplectic. On the other hand, we have an inclusion

H_{ét}^{1}(R, PGSp(A, σ)) ↪ H^{1}(K, PGSp(A_{K}, σ_{K})).

So in total, we have an inclusion

H^{1}(k, PGSp(\overline{A}, σ̄)) ↪ H^{1}(K, PGSp(A_{K}, σ_{K})).

Remark 3.5 – Note that this lift coincides with lifting central simple algebras as explained in §2.5 (b). Over there we actually used the same arguments for the smooth R-group scheme PGL_{R,∞} in order to prove

Br(k) = H^{1}(k, PGL_{k,∞}) ↪ H^{1}(K, PGL_{K,∞}) = Br(K).

So starting with a central simple k-algebra A with symplectic involution σ, we find a lifted Azumaya algebra B over R with symplectic involution τ and hence a central simple K-algebra B_{K} with symplectic involution τ_{K}. In particular, deg_{k}(A) = deg_{K}(B_{K}) and per_{k}(A) = per_{K}(B_{K}). Since biquaternion algebras are exactly the central simple algebras of degree 4 and period 1 or 2, we see that a biquaternion algebra over k with symplectic involution lifts to a biquaternion algebra with symplectic involution over K.

(b) Preparing the ingredients – We now continue the work of §3.2 in the wild case. Throughout this section, let k be a field of characteristic 2, R a 2-ring with residue field k and fraction field K, and A a biquaternion algebra over k with lifted Azumaya algebra B over R. As ρ̃_{S06} and ρ_{BI} have different value groups, we first give some remarks on how they relate and how we can use the uniqueness statement of Theorem 2.4 to compare the invariants.

By a theorem of Kato, we have an isomorphism ψ_{k} : H_{2}^{4}(k) → \overline{I^{3}W_{q}(k)} [Kat2]. Similar to Suslin’s construction (2.7), we can also give a morphism H_{4,A}^{4}(k) → H_{2}^{4}(k). Indeed, the projection

π_{1}^{2} : W_{2}(k) → W_{1}(k) : (a_{0}, a_{1}) → (a_{0})

gives a morphism r : H_{4}^{4}(k) → H_{2}^{4}(k). Since π_{1}^{2} sends elements of order 2 to 0, r does exactly the same. Hence we get a morphism r_{A} : H_{4,A}^{4}(k) → H_{2}^{4}(k) because any element of K_{2}^{M}(k)·[A] is of order 2. Now we can compare the different groups with a commutative diagram.

Proposition 3.6. Let k′ be a field extension of k and R′ a 2-ring (containing R) with residue field k′ and fraction field K′, then the following diagram commutes:

\xymatrix{H_{4,A}^{4}(k′) \ar[d]^{i^{∗}} \ar[r]^{r_{A}} & H_{2}^{4}(k′) \ar[r]_{≅}^{ψ_{k′}} \ar[d]^{i^{∗}} & \overline{I^{3}W_{q}(k′)} \ar[d]^{j} \\ H_{4,B_{K}}^{4}(K′) \ar[r]_{r_{B}} & H_{2}^{4}(K′) \ar[r]^{≅}_{ψ_{K′}} & \overline{I^{3}W_{q}(K′)}.}  (3.2)

Remark 3.7 – The morphisms r_{B} = r_{B_{K′}} and ψ_{K′} are as in (2.7) and (3.1), while r_{A} = r_{A_{k′}} and ψ_{k′} are as above. The morphism j on Witt groups is as in [Bae, Ch. V, Cor. 1.5]; it is the composition of a bijection of W_{q}(R′) ≅ W_{q}(k′) induced by the residual morphism R′ → k′ and an injection W_{q}(R′) → W_{q}(K′). Here W_{q}(R′) is the Witt group of quadratic spaces of constant rank over R′. See [Bae, Ch. I and V] for more information.

The maps i^{∗} are defined by Kato as in §2.5 (b). We recall the exact definition of this morphism which we need in the proof: for any integer n > 0

i^{∗} : H_{2^{n}}^{q+1}(k′) → H_{2^{n}}^{q+1}(K′) is defined by w ⊗ b̄_{1} ⊗ … ⊗ b̄_{q} ↦ i(w) ∪ h_{2^{n},K′}^{q}({b_{1}, …, b_{q}}), where b_{1}, …, b_{q} ∈ R′, the morphism h_{2^{n},K′}^{q} is the Galois symbol (2.9) and i(w) is the composition

W_{n}(k′)/{w^{(p)} − w|w ∈ W_{n}(k′)} ≅ H^{1}(k′, ℤ/2^{n}ℤ) ↪^{ι} H^{1}(K′, ℤ/2^{n}ℤ),

where the isomorphism is induced by the additive form of Hilbert 90 for W_{n}(k′_{s}) applied to Witt’s short exact sequence [Wit, §5]:

\xymatrix{0 \ar[r] & ℤ/2^{n}ℤ \ar[r] & W_{n}(k′_{s}) \ar[r]^{x^{(2)}−x} & W_{n}(k′_{s}) \ar[r] & 0.}

The injection ι is defined in a similar way as one can get an injection from the splitting (2.5). It can be proved that i^{∗} behaves well by going to the relative cohomology groups H_{4,A}^{4}(k′) and H_{4,B_{K}}^{4}(K′) [Wou, Prop. 4.10].

Proof. Let R′_{nr} be a 2-ring with residue field k′_{s} and fraction field K′_{nr}. So R′_{nr} is the integral closure of R′ in K′_{nr}.

We first prove i^{∗} ∘ r_{A} = r_{B} ∘ i^{∗}. This follows merely by the definition of i^{∗}. Let (a_{0}, a_{1}) ⊗ x_{1} ⊗ x_{2} ⊗ x_{3} ∈ H_{4,A}^{4}(k′) and take (b_{0}, b_{1}) ∈ W_{2}(k′_{s}) such that (b_{0}^{2}, b_{1}^{2}) − (b_{0}, b_{1}) = (a_{0}, a_{1}). Then (a_{0}) = (b_{0})^{2} − (b_{0}) ∈ W_{1}(k′) and

i^{∗} ∘ r_{A}((a_{0}, a_{1}) ⊗ x_{1} ⊗ x_{2} ⊗ x_{3}) = (σ̄(b_{0}) − b_{0})_{σ∈Γ_{K′}} ∪ h_{2}^{3}({x_{1}, x_{2}, x_{3}}),

where we consider σ̄(b_{0})−b_{0} as an element of ℤ/2ℤ for any σ ∈ Γ_{K′} (with residue σ̄ ∈ Γ_{k′}). On the other hand,

r_{B} ∘ i^{∗}((a_{0}, a_{1}) ⊗ x_{1} ⊗ x_{2} ⊗ x_{3}) = r_{B} [(σ̄(b_{0}, b_{1}) − (b_{0}, b_{1}))_{σ∈Γ_{K′}} ∪ h_{4}^{3}({x_{1}, x_{2}, x_{3}})]
= (σ̄(b_{0}) − (b_{0}))_{σ∈Γ_{K′}} ∪ h_{2}^{3}({x_{1}, x_{2}, x_{3}}).

The commutativity of the right square is essentially due to Kato [Kat2, Lem. 11]; he proves the existence of a commutative diagram

\xymatrix{H_{2}^{n}(k′) \ar[r]^{≅} \ar[d]^{φ} & \overline{I^{3}W_{q}(k′)} \ar[d]^{j} \\ K_{n}^{M}(K′)/2K_{n}^{M}(K′) \ar[r]_{ψ_{K′}^{1}}^{≅} & \overline{I^{3}W_{q}(K′)}}

where ψ_{K′}^{1} is the isomorphism of Milnor’s conjecture on quadratic forms (see §3.2) and φ is defined by

b̄\frac{dā_{1}}{ā_{1}} ∧ \frac{dā_{2}}{ā_{2}} ∧ \frac{dā_{3}}{ā_{3}} mod I ↦ \{1 + 4b, a_{1}, a_{2}, a_{3}\} mod 2K_{n}^{M}(K′),

for a_{1}, a_{2}, a_{3}, b ∈ R′. Since the isomorphism ψ_{K′} : H_{2}^{4}(K′) → \overline{I^{3}W_{q}(K′)} is defined as composition of ψ_{K′}^{1} with the Galois symbol h_{2,K′}^{4}, it suffices to check i(b̄) = h_{2,k′}^{1}(4b + 1) for any b ∈ R′. So take c ∈ k′_{s} such that c^{2} − c = b̄. Then

i(b̄) = (σ̄(c) − c)_{σ∈Γ_{K′}} ∈ H^{1}(K′, ℤ/2)

under the standard identification of ℤ/2 and μ_{2}(K′). Take c̃ to be a lift of c in R_{nr}. By eventually changing the representant of b̄ in R′, we can assume c̃^{2} − c̃ = b. Then 4b + 1 = (2c̃ + 1)^{2} and

h_{2,K′}^{1}(4b + 1) = (σ(2c̃ + 1)/(2c̃ + 1))_{σ∈Γ_{K′}} ∈ H_{2}^{1}(K′).

So if σ(2c̃+1)/(2c̃+1) = 1, we have σ(c̃) = c̃. On the other hand, if σ(2c̃+1)/(2c̃+1) = −1, we get σ(c̃) = −c̃ − 1. This gives indeed the desired equality. ■

(c) Cooking up the result – Using Theorem 2.4 and Proposition 3.6, we can prove the main theorem.

Theorem 3.8. Let k be a field of characteristic 2 and A a biquaternion algebra over k, then for any field extension k′ of k

ρ_{BI,A,k′} = ψ_{k′} ∘ r_{A} ∘ ρ̃_{S06,A,k′}

with ψ_{k′} and r_{A} as in (3.2).

Proof. Let k′ be a field extension of k and R (resp. R′) a 2-ring with residue field k (resp. k′) and fraction field K (resp. K′). Suppose σ is a symplectic involution on A and take B a lifted Azumaya R-algebra with lifted symplectic involution τ. Use the same notations as in (3.2). We know j is injective, i^{∗} ∘ ρ̃_{S06,A} = ρ_{S06,B_{K}} (by definition of ρ̃_{S06,A}) and ρ_{BI,B_{K}} = ψ_{K′} ∘ r_{B} ∘ ρ_{S06,B_{K}}. So it suffices to prove that ρ_{BI,B_{K}} = j ∘ ρ_{BI,A}, which merely follows from the definition.

Let us first explain the isomorphism SK_{1}(B_{K})(K′) ≅ SK_{1}(A)(k′). We can suppose that SK_{1}(A)(k′) ≠ 0 so that A_{k′} and B_{K′} are division algebras by Wang’s theorem [Wan]. Then B_{K′} is equipped with a valuation w that extends the valuation v′ of K′, namely \frac{1}{4}v′ ∘ Nrd_{B_{K′}/K′}. The associated valuation ring is B_{R′} and the reduced k-algebra is A_{k′}. Even more, SL_{1}(B_{K′}) is part of B_{R′} and the isomorphism SK_{1}(B_{K})(K′) ≅ SK_{1}(A)(k′) is induced by the residue map on SL_{1}(B_{K′}) [Pla, Cor. 3.13] – see also [Wou, Cor. 3.3].

The involutions σ and τ can not be hyperbolic due to [KMRT, Prop. 6.7 (3)]. Take a ∈ SK_{1}(A)(k′) with lift b ∈ SK_{1}(B_{K})(K′). Then by definition it follows that Prd_{A_{k′},a/k′}(X) = \overline{Prd_{B_{K′},b/K′}(X)}, where the residue is the canonical residue on R′[X]. So we also get Prp_{σ_{k′},a/k′}(X) = \overline{Prp_{τ_{K′},b/K′}(X)} and Trp_{σ_{k′}/k′}(a) = \overline{Trp_{τ_{K′}/K′}(b)}. Then take y ∈ Symd(B_{K′}, τ_{K′}) ∩ B_{K′}^{×} satisfying y(Trp_{τ_{K′}/K′}(y) − y)^{−1} = −τ(b)b. We can assume w(y) ≥ 0, since if w(y) < 0, i.e. Nrd_{B_{K′}/K′}(y) = λ/μ ∈ K′ with λ, μ ∈ R′, then w(μy) = v(λ) ≥ 0 and

μy (Trp_{τ_{K′}/K′}(μy) − μy)^{−1} = y(Trp_{τ_{K′}/K′}(y) − y)^{−1}.

Then we get ȳ(Trp_{σ′_{K}/K′}(ȳ) − ȳ)^{−1} = −σ(a)a as b is a lift of a and moreover ȳ ∈ Symd(A, σ). Hence

ρ_{BI,A,k′}(a) = Φȳ : A_{k′} → k′ : x ↦ Trp_{σ_{k′}/k′}(σ_{k′}(x)ȳx) and
ρ_{BI,B_{K′},K′}(b) = Φ_{y} : B_{K′} → K′ : x ↦ Trp_{τ_{K′}/K′}(τ_{K′}(x)yx).

As \overline{Trp_{τ_{K′}/K′}(τ_{K′}(x)yx)} = Trp_{σ_{k′}/k′}(σ_{k′}(x̄)ȳx̄) for x ∈ B_{R′}, the required compatibility holds. ■

3.4 Non-triviality of the invariant

Because the invariants for biquaternions in odd or zero characteristic are injective, they are also injective in characteristic 2 due to the lifting property (Theorem 2.4). As SK_{1} is not trivial for Platonov’s examples (§2.2 (b)) and in general for biquaternion algebras of index 4 [Mer3], we find non-trivial invariants in characteristic 2.

Another argument for non-triviality of ρ_{BI} in characteristic different from 2 is given by a formula of Merkurjev for the value on the centre of the biquaternion algebra [Mer1, Ex. p. 70] – see also [KMRT, Ex. 17.23]. Using this formula and the lift from characteristic 2 to characteristic 0, one could hope to prove the non-triviality of ρ_{BI} (and hence of ρ_{S06}) in the case when char(k) = 2, but this fails. Let us comment on this fact.

Say k is a field of characteristic 2, R a p-ring with residue field k and fraction field K, and let A = [ā, b̄) ⊗_{k} [c̄, d̄) be a biquaternion k-algebra for a, c ∈ R and b, d ∈ R^{×}. Then the lifted Azumaya R-algebra is B = [a, b)⊗_{R} [c, d) where e.g. [a, b) is the R-algebra generated by u, v satisfying slightly different relations than usual: u^{2} + u = a, v^{2} = b, and uv = −v(u + 1). We can rewrite it as B = (4a + 1, b)_{R} ⊗_{R} (4c + 1, d)_{R}, where (4a + 1, b)_{R} is the R-algebra generated by i, j with i^{2} = 4a + 1, j^{2} = b, and ij = −ji. Indeed, an isomorphism is given by i = 2u + 1 and j = v. Suppose K contains a primitive fourth root of unity ζ, then by (loc. cit.) we have

ρ_{BI,B_{K},K}([ζ]) = ⟨⟨4a + 1, b, 4c + 1, d⟩⟩ + I^{4}W′_{q}(K),

where [ζ] is the class of ζ in SK_{1}(B_{K})(K) and where ⟨⟨4a + 1, b, 4c + 1, d⟩⟩ is an n-fold Pfister quadratic K-form [Kah1, Lem. 2.1.1].

Let π be the isomorphism SK_{1}(B_{K})(K) ≅ SK_{1}(A)(k), then π([ζ]) = [1] because k contains no non-trivial fourth roots of unity. By the proof of Theorem 3.8, we have j ∘ ρ_{BI,B_{K},K}([ζ]) = ρ_{BI,A,k} ∘ π([ζ]) = 0 ∈ \overline{I^{3}W′_{q}(k)}. Because the map j from Proposition 3.6 is injective, we get that ⟨⟨4a + 1, b, 4c + 1, d⟩⟩ = 0 ∈ \overline{I^{3}W′_{q}(K)}. We can also verify this by calculating with Pfister forms. Define Q = (4a + 1, b)_{R} and let X be the natural affine R-scheme with

X (R) = {x ∈ Q | Nrd_{Q_{K}/K}(x) = 4c + 1},

where Q_{K} = Q⊗_{R} K. Then X is an R-torsor under SL_{1}(Q), where SL_{1}(Q) is the natural affine R-scheme so that SL_{1}(Q)(R) = SL_{1}(Q_{K})(K) ∩ Q. The special fibre X_{k} = X ×_{R} k clearly has a rational point, so its class [X_{k}] ∈ H^{1}(k, SL_{1}(Q_{k})) is trivial. By Hensel’s lemma [SGA, Exp. XXIV, Prop. 8.1], we get [X] = 0 ∈ H_{ét}^{1}(R, SL_{1}(Q)). Hence X (as well as the generic fibre X_{K}) has a rational point, but then by theory of Pfister forms we get ⟨⟨4a + 1, b, 4c + 1⟩⟩ = 0 ∈ W′_{q}(K) [Kah1, Cor. 2.1.10]. Indeed, Nrd_{Q_{K}/K}(x) corresponds with a value of ⟨⟨4a + 1, b⟩⟩. So a fortiori ⟨⟨4a + 1, b, 4c + 1, d⟩⟩ = 0 ∈ \overline{I^{3}W′_{q}(k)}.

4 Comparing to Kahn’s invariant

We compare now all defined invariants of SK_{1}(A) to ρ_{Kahn,A} in the moderate case, i.e. as they are originally defined. The results can be generalised to the wild invariants, but with some loss of information. We also generalise the formula of Merkurjev for the value on the centre of SK_{1}(A) (§3.4).

4.1 Moderate case

We explain two natural ways of comparing Inv^{4}(SK_{1}(A), H_{n}^{∗}) and Inv^{4}(SK_{1}(A), H_{n,A^{⊗r}}^{∗}). Let A be a central simple k-algebra with ind_{k}(A) = n ∈ k^{×} and m = per_{k}(A).

(a) Ways of looking – For any field extension F of k and any integer r, we can look at the composition

m_{r} : H_{n,A^{⊗r}}^{4}(F) →^{·m} H_{n/m}^{4}(F) ↪ H_{n}^{4}(F)

and at the projection

π_{r} : H_{n}^{4}(F) → H_{n,A^{⊗r}}^{4}(F).

These induce respectively maps

m̃_{r} : Inv^{4}(SK_{1}(A), H_{n,A^{⊗r}}^{∗}) → Inv^{4}(SK_{1}(A), H_{n}^{∗}) and
π̃_{r} : Inv^{4}(SK_{1}(A), H_{n}^{∗}) → Inv^{4}(SK_{1}(A), H_{n,A^{⊗r}}^{∗}).

The maps π̃_{r} where introduced by Kahn [Kah2, Rem. 11.6], but we rather consider the maps m̃_{r} to compare because of the special definition of Kahn’s invariant as generator of the target group. We could also refine m̃_{r} if H^{2}(k, μ_{n}^{⊗2})∪r[A] has m′-torsion for an integer 0 ≤ m′ < m. A good comprehension of both maps actually relies, as Kahn mentions, on a good comprehension of the cup product with the class of A (loc. cit.).

By the cyclicity of Inv^{4}(SK_{1}(A), H_{n}^{∗}) (§2.4 (a)), we certainly find the following relations.

Proposition 4.1. Let A be a central simple k-algebra of ind_{k}(A) ∈ k^{×}. Then for any integer r and any ρ ∈ Inv^{4}(SK_{1}(A), H_{n,A^{⊗r}}^{∗}) there exists an integer d_{A} ∈ ℤ/\overline{n} such that

m̃_{r}(ρ) = d_{A} ρ_{Kahn,A} ∈ Inv^{4}(SK_{1}(A), H_{n}^{∗}) ⊂ ℤ/\overline{n}.

Proof. Use the definition of ρ_{Kahn} and the bounds on Inv^{4}(SK_{1}(A), H_{n}^{∗}) (see §2.4 (b)). ■

Kahn also raises the issue whether π̃_{r} is surjective or not (loc. cit.). We can prove it to be non-surjective for biquaternion division algebras à la Platonov.

Proposition 4.2. Let k = ℚ_{p}((t_{1}))((t_{2})) for a prime p. Suppose A = (a, t_{1}) ⊗ (b, t_{2}) is a biquaternion division k-algebra for a, b ∈ ℚ_{p}^{×}. Then π̃_{1} is not surjective.

Proof. In §§2.2 (b) and (c) we saw that SK_{1}(A) ≅ ℤ/2 and H_{4}^{4}(k) ≅ ℤ/4. We can also add a fourth primitive root of unity to k as this does not change the Brauer group. In this case we have the Bloch-Kato isomorphism H_{4}^{4}(k) ≅ K_{4}^{M}(k)/4.

We now prove H_{4,A}^{4}(k) ≅ ℤ/2. Under the Bloch-Kato-isomorphism K_{2}^{M}(k)/2 ≅ 2Br(k), the class of A corresponds to {a, t_{1}} + {b, t_{2}} ∈ K_{2}^{M}(k)/2 [GS, Prop. 4.7.1] so that H^{2}(k, μ_{4}^{⊗2}) ∪ [A] is isomorphic to (K_{2}^{M}(k)/4) · (2{a, t_{1}} + 2{b, t_{2}}). As the isomorphism H_{4}^{4}(k) ≅ ℤ/4 is retrieved by taking two residues ∂_{t_{2}}^{3} and ∂_{t_{1}}^{4}, it suffices to determine the group

∂_{t_{2}}^{3} ∘ ∂_{t_{1}}^{4}((K_{2}^{M}(k)/4) · (2{a, t_{1}} + 2{b, t_{2}})).

By the definition of residues on Milnor K-groups [Mil, §2], it is clear that this equals (K_{1}^{M}(k)/4) · 2{a} + (K_{1}^{M}(k)/4) · 2{b}. As we assumed that SK_{1}(A) is not trivial, a can not be a square otherwise A would have been Brauer-trivial. This means that (K_{1}^{M}(k)/4)· 2{a}+(K_{1}^{M}(k)/4)·2{b} is not trivial. On the other hand it has 2-torsion inside K_{2}^{M}(k)/4 ≅ ℤ/4 so that indeed H_{4,A}^{4}(k) ≅ ℤ/2.

Then π_{1} : ℤ/4 → ℤ/2 is the “modulo 2” map and m_{1} : ℤ/2 → ℤ/4 is canonical injection. Suslin proves ρ_{S06,A,k} : SK_{1}(A)(k) → H_{4,A}^{4}(k) is not trivial (2.7), so it is the identity map on ℤ/2. It is then clear that this can never factor through H_{4}^{4}(k) so that π̃_{1} is clearly not surjective. ■

(b) Determining factors – We prove that for the product of two symbol algebras of degree n the factor d_{A} appearing in Proposition 4.1 only depends on the invariant ρ and the characteristic of k.

Proposition 4.3. Let ρ be a moderate invariant of SK_{1} with values in H_{⊗r}^{4}. Let furthermore p be equal to zero or to any prime and let m be an integer not divisible by p. Then there exist an integer i(p, m) ∈ ℤ/\overline{m^{2}} such that for any field k of char(k) = p containing a primitive m-th root of unity ξ_{m} and for any product A = (a, b)_{m} ⊗ (c, d)_{m} of two symbol k-algebras

m̃_{r}(ρ_{A}) = i(p, m) ρ_{Kahn,A} ∈ Inv^{4}(SK_{1}(A), H_{m^{2}}^{∗}) ⊂ ℤ/\overline{m^{2}}.

Remark 4.4 – Although i(p, m) is in general not uniquely determined, we can take a canonical representant as we know Inv^{4}(SK_{1}(A), H_{m^{2}}^{∗}) is cyclic. This comes down to taking the class in ℤ/\overline{m^{2}} satisfying the required relation and such that the representant in {0, …, m^{2} − 1} is as low as possible. It also of course depends on the invariant. We add an index if necessary to stress which invariant is compared to Kahn’s invariant. Moreover, it also depends on the exact definition of the injection Inv^{4}(SK_{1}(A), H_{m^{2}}^{∗}) ⊂ ℤ/\overline{m^{2}}. For the remainder of the paper, we fix this injection.

Proof. Take k the prime field of characteristic p and set k′ = k(ξ_{m}) for an m-primitive root of unity ξ_{m} ∈ k_{s}. Denote by T = (t_{1}, t_{2})_{m} ⊗ (t_{3}, t_{4})_{m} the product of two Azumaya symbol algebra over R = k′[t_{1}^{±1}, t_{2}^{±1}, t_{3}^{±1}, t_{4}^{±1}] where t_{1}, t_{2}, t_{3}, t_{4} are variables and where Azumaya symbol algebras are defined using the same relations as used for symbol algebras over a field. Take K = k′(t_{1}, t_{2}, t_{3}, t_{4}) and T = T_{K} = (t_{1}, t_{2})_{m} ⊗ (t_{3}, t_{4})_{m}, the product of the respective symbol algebras over K. By Proposition 4.1, we find a unique d_{T} ∈ ℤ/\overline{m^{2}} such that

m̃_{r}(ρ_{T}) = d_{T} ρ_{Kahn,T} .  (4.1)

We prove d_{T} only depends on m and p.

So suppose F is a field of characteristic p containing an m-th primitive root of unity so that k′ ⊂ F. Take any product A = (a, b)_{m} ⊗ (c, d)_{m} of two symbol algebras of degree m over F. Now A can be obtained from T_{F} = T ⊗_{R} F by specialising t_{1}, t_{2}, t_{3}, t_{4} to a, b, c, d respectively.

Furthermore, (a, b, c, d) defines a k-rational point x of Spec(F [t_{1}^{±1}, t_{2}^{±1}, t_{3}^{±1}, t_{4}^{±1}]). Take O_{x} to be the local ring of Spec(F [t_{1}^{±1}, t_{2}^{±1}, t_{3}^{±1}, t_{4}^{±1}]) in x with maximal ideal M. It is clear that the completion Ô_{x} of O_{x} with respect to the M-adic topology is F -isomorphic to R′ = F [[u_{1}, u_{2}, u_{3}, u_{4}]] where u_{1} = t_{1} − a, u_{2} = t_{2} − b, u_{3} = t_{3} − c, and u_{4} = t_{4} − d (see also [Gro1, Thm. 19.6.4]). Under the isomorphism Br(R′) ≅ Br(F) from §2.5 (b), it is clear that A_{R′} = A ⊗ R′ is an Azumaya R′-algebra mapping to A. Furthermore, the F -isomorphism of Ô_{x} with R′ gives an isomorphism Br(Ô_{x}) ≅ Br(R′). In its turn, this gives an isomorphism Br(Ô_{x}) → Br(F) with inverse given by taking the tensor product over F with Ô_{x}. By construction it sends the class of TÔ_{x} to the class of A.

Let K′ = F ((u_{1}))((u_{2}))((u_{3}))((u_{4})), then A⊗_{F} K′ is Brauer-equivalent to TÔ_{x} ⊗Ô_{x} K′ ≅ T_{K′}. We find SK_{1}(A_{K′}) ≅ SK_{1}(T_{K′}) (as in §2.5 (b)). Furthermore, (2.5) gives an injection H_{m^{2}}^{4}(F) → H_{m^{2}}^{4}(K′). By functoriality of invariants, the diagram

\xymatrix{SK_{1}(A) \ar[d]_{≅} \ar[r]^{ρ} & H_{m^{2}}^{4}(F) \ar[d] \\ SK_{1}(T_{K′}) \ar[r]_{ρ} & H_{m^{2}}^{4}(K′)}

commutes both for m̃_{r}(ρ) and ρ_{Kahn}. Then by (4.1), we get m̃_{r}(ρ_{A}) = d_{T} ρ_{Kahn,A}. ■

In particular, we find such relations for ρ = ρ_{S91}, ρ_{S06}, and the ρ_{r}’s.

(c) Non-triviality of Kahn’s invariants – As mentioned in Remark 2.2, ρ_{Kahn} is not-trivial for biquaternion algebras (of index 4). We generalise this to the product of two cyclic algebras à la Platonov (§2.2). Therefore, we compare ρ_{Kahn} to ρ_{S91} as this invariant is non-trivial for Platonov’s examples (§2.3 (a)). This means we have to work with H_{n,A^{⊗2}}^{∗} for suitable n and A. (In the same way as in Proposition 4.2, these give also examples of non-trivial π̃_{2}.)

Theorem 4.5. Let k be p-adic field containing a n^{3}-th primitive root unity. Suppose A = (a, t_{1})_{n} ⊗ (c, t_{2})_{n} is a division k((t_{1}))((t_{2}))-algebra, then ρ_{Kahn,A} is not trivial. If n = q_{1} · . . . · q_{r} for different primes q_{i}, then

Inv^{4}(SK_{1}(A), H_{n^{2}}^{∗}) ≅ ℤ/n.

Moreover if n is odd, the integer i_{S91}(0, n) ∈ ℤ/\overline{n^{2}} defined in Proposition 4.3 for ρ_{S91} is not trivial.

Proof. We know SK_{1}(A) ≅ ℤ/n by §2.2. Furthermore H_{n^{2}}^{4}(k) = ℤ/n^{2} as the results in §2.2 (c) hold also when one replaces ℚ_{p} by a finite extension of it.

To calculate H_{n^{2},A^{⊗2}}^{4}(k), we use a analogous argument as in the proof of Proposition 4.2. If n is odd, we also find H_{n^{2},A^{⊗2}}^{4}(k) = ℤ/n as in this case per_{k}(A^{⊗2}) = per_{k}(A). If n is even, per_{k}(A^{⊗2}) = n/2 so that H_{n^{2},A^{⊗2}}^{4}(k) = ℤ/(2n). In either case, m_{2} : H_{n^{2},A^{⊗2}}^{4}(k) → H_{n^{2}}^{4}(k) is the canonical injection (m_{2} is the multiplication by m for m = n if n odd and m = n/2 if n even).

Suslin proves ρ_{S91,A} is not trivial (on the field k) [Pla, Thm. 4.8]. If n is odd, ρ_{Kahn,A} is not trivial (on k) by Proposition 4.1 and hence by definition i_{S91}(0, n^{2}) ≠ 0 ∈ ℤ/\overline{n^{2}}. If n is even, a similar argument as in the proof of Proposition 4.1 gives the non-triviality of ρ_{Kahn,A} (mutatis mutandis m by n/2).

By the bound on the invariant group (§2.4 (b)) and a Brauer decomposition of A with a related decomposition of invariants in primary parts, the isomorphism statement follows. ■

4.2 Wild case

We continue the comparison in the wild case. Using a lift, we can generalise the statement to any central simple algebra with some loss of information. This does let us prove a relation between the several i(p, n)’s.

Let A be a central simple k-algebra of ind_{k}(A) = n and per_{k}(A) = m. We again have morphisms for any integer r

m̃_{r} : Inv^{4}(SK_{1}(A), H_{n,A^{⊗r}}^{∗}) → Inv^{4}(SK_{1}(A), H_{n}^{∗})

induced by the multiplication for any field extension F of k:

m_{r} : H_{n,A^{⊗r}}^{4}(F) →^{·m} H_{n/m}^{4}(F) ↪ H_{n}^{4}(F).

Note that we can also define maps π̃_{r} as in §4.1 (a).

Proposition 4.6. Let ρ be a moderate invariant of SK_{1} with values in H_{⊗r}^{4}. Suppose k is a field of char(k) = p > 0 and let A = [a, b)_{p} ⊗ [c, d)_{p} be the product of two p-algebras over k, then

m̃_{r}(ρ̃_{A}) = i(0, p)ρ̃_{Kahn,A}.

Proof. Take a p-ring R with residue field k and fraction field K. Remark first that the lifted Azumaya R-algebra B of A is (after base extension to K) a product of two symbol algebras of degree p. This follows from [GS, Prop. 4.7.1 & Prop. 9.2.5] and the injection H_{p^{2}}^{2}(k) → H_{p^{2}}^{2}(K) [Kat1, Proof of Thm. 1 & 3].

The result follows immediately from the injections

Inv^{4}(SK_{1}(B_{K}), H_{p^{2}}^{∗}) → Inv^{4}(SK_{1}(A), H_{p^{2}}^{∗}) and
Inv^{4}(SK_{1}(B_{K}), H_{p^{2},B_{K}^{⊗r}}^{∗}) → Inv^{4}(SK_{1}(A), H_{p^{2},A^{⊗r}}^{∗})

defined by lifting invariants and the relations for ρ_{B_{K}} and ρ_{Kahn,B_{K}} (Proposition 4.3). ■

Remark 4.7 – In the view of Remark 2.5, we could even refine the statement in the moderate case. If k is a field of char(k) = p > 0 and if A = (a, b)_{n} ⊗ (c, d)_{n} is the product of two symbol algebras for n ∈ k^{×}, then a similar statement holds as A lifts to the product of two symbol algebras of degree n in characteristic 0. If ρ̃_{A} = ρ_{A}, then i(p, n) is a multiple of i(0, n) in ℤ/\overline{n}. Indeed, ρ_{Kahn,A} is a generator of Inv^{4}(SK_{1}(A), H_{n}^{∗}) ⊂ ℤ/\overline{n} and for some integer d

i(p, n)ρ_{Kahn,A} = m̃_{r}(ρ_{A}) = i(0, n) ρ̃_{Kahn,A} = i(0, n) d ρ_{Kahn,A}.

4.3 Formula on the centre

We can now generalise the formula of Merkurjev on the centre of a biquaternion algebra ([Mer1, Ex. p.70] – see also [KMRT, Ex. 17.23] and §3.4) to the tensor product of two symbol algebras. We first prove a general formula and later we prove a finer result using Theorem 4.5.

(a) General result – We again use cohomological invariants, however not invariants of algebraic groups as in §2.1, but rather invariants as introduced in [GMS, Ch. I]. These are also natural transformations of functors, but rather a transformation of a functor B : k-𝔣𝔦𝔢𝔩𝔡𝔰 → 𝔖𝔢𝔱𝔰 into a functor H : k-𝔣𝔦𝔢𝔩𝔡𝔰 → 𝔊𝔯𝔬𝔲𝔭𝔰.

Proposition 4.8. Let p be equal to 0 or to any prime and let n > 0 be an integer not divisible by p. Let φ be the canonical map H_{m}^{4}(k) → H_{n^{2}}^{4}(k) (for m = \overline{n^{2}}). There exists an integer j(p, n) such that the following formula holds for any field k of char(k) = p containing a primitive n^{2}-th root of unity ζ and for any product A = (a, b)_{n} ⊗ (c, d)_{n} of two symbol k-algebras:

ρ_{Kahn,A,k}([ζ]) = φ [j(p, n) h_{m}^{4}({a, b, c, d})] ∈ H_{n^{2}}^{4}(k)

Remark 4.9 – Remark that as k contains an n^{2}-th primitive root of unity, μ_{n^{2}}^{⊗i} ≅ ℤ/n^{2} for any i > 0. Note also that φ[h_{m}^{4}({a, b, c, d})] = (n/m) h_{n^{2}}^{4}({a, b, c, d}).

This expression is compatible to the biquaternion case keeping in mind diagrams (2.7) and (3.1). Also, the integer j(p, n) in the theorem is not uniquely determined, but can be picked canonically by taking the smallest positive integer satisfying the relation. Moreover, j(p, n) depends on the n-th primitive root of unity used in the definition of the symbol algebra and of the choice of n^{2}-th primitive root of unity ζ. We are interested in the invertibility of j(p, n) modulo m and therefore the exact choices do not matter, so we do not incorporate these in the notation.

Proof. As ρ_{Kahn} has m-torsion (Lemma 2.1), we can assume ρ_{Kahn,A,k}([ζ]) to have values in H_{m}^{4}(k).

Let k be the prime field of characteristic p and set k′ = k(ζ) for ζ ∈ k̄ a primitive n^{2}-th root of unity. Take T = (t_{1}, t_{2})_{n} ⊗ (t_{3}, t_{4})_{n} over F = k′(t_{1}, t_{2}, t_{3}, t_{4}). We prove the formula for T. The proof ends by specialising to A as in the proof of Proposition 4.3.

Let B : k-𝔣𝔦𝔢𝔩𝔡𝔰 → 𝔖𝔢𝔱𝔰 be the functor attaching to a field extension F of k the Galois cohomology group H^{1}(F, μ_{m})^{4} and H associating H^{4}(F, μ_{m}^{⊗4}) with F. Now ρ_{Kahn} induces a cohomological invariant of B into H. Indeed, using the isomorphism H^{1}(F, μ_{n}) ≅ F^{×}/(F^{×})^{n}, we associate with any four representants a, b, c, d ∈ F^{×} of classes in H^{1}(F, μ_{m}) the value ρ_{Kahn,A,F} ([ζ]) ∈ H_{m}^{4}(F) ≅ H^{4}(F, μ_{m}^{⊗4}) ≅ K_{4}^{M}(F)/m (for A = (a, b)_{n} ⊗ (c, d)_{n}).

Using a full description of all possibles invariants of B into H of [Gar, Prop. 2.1 & §3.1] and [GMS, Ex. 16.5], we find that ρ_{Kahn,T,F} ([ζ]) can be written in K_{4}(F)/m as sum of pure symbols of the form λ{z_{1}, z_{2}, z_{3}, z_{4}} where λ is an integer and each z_{i} is either a t_{j} either an element of k. We prove that only {t_{1}, t_{2}, t_{3}, t_{4}} occurs. By specializing t_{1} to 1, we obtain T_{1} = (1, t_{2})_{n} ⊗ (t_{3}, t_{4})_{n} from T. But then SK_{1}(T_{1}) = 0 by Wang’s theorem so that ρ_{Kahn,T_{1},F} ([ζ]) = 0. This induces that for all (non-trivial) pure symbols {z_{1}, z_{2}, z_{3}, z_{4}} appearing in ρ_{Kahn,T,F} ([ζ]) one of the z_{i} has to equal t_{1} (as the other ones are zero by the specialisation above). Three other specialisations give the result. ■

Remark 4.10 – In wild characteristics (i.e. when p | n), a formula as above does not make sense as there are no non-trivial p^{2}-th roots of unity. So similar as in §3.4, we cannot generalise this formula to wild invariants by means of a lift.

(b) Non-triviality of factor – We prove the non-triviality of the factor appearing in Proposition 4.8. This uses the non-triviality of ρ_{Kahn} for Platonov’s examples (Theorem 4.5). First we recall some notions related to tori. See [CTS] as a reference for more details.

Denote for a finite separable field extension K of k by R_{K/k}(𝔾_{m}) the torus obtained by Weil restriction of scalars from K to k. Denote furthermore the kernel of the multiplication map R_{K/k}(𝔾_{m}) → 𝔾_{m,k} by R_{K/k}^{1}(𝔾_{m}) and the cokernel of the injection 𝔾_{m,k} → R_{K/k}(𝔾_{m}) by R_{K/k}(𝔾_{m})/𝔾_{m}. Furthermore for any k-torus T, we denote by T (k)/R the R-equivalence classes of T (k). The dual T̂ of a k-torus T is the character group Hom(T, 𝔾_{m}). The dual of R_{K/k}(𝔾_{m}) is clearly the free abelian group ℤ[Γ] for Γ = Gal(K/k). The dual of R_{K/k}^{1}(𝔾_{m}) is then J_{Γ}, the cokernel of the norm:

ℤ → ℤ[Γ] : a ↦ ∑_{γ_{i}∈Γ} aγ_{i}.

The dual of R_{K/k}(𝔾_{m})/𝔾_{m} is the kernel I_{Γ} of the augmentation map:

ℤ[Γ] → ℤ : ∑_{γ_{i}∈Γ} n_{i}γ_{i} ↦ ∑_{γ_{i}∈Γ} n_{i}.

Recall that a k-torus F is called flabby (flasque) if F̂ is a flabby Γ_{k}-module, i.e. Ext^{1}(F̂, P) = 0 for any permutation Γ_{k} module P (for equivalent definitions see ibid., Lem. 1). A flasque resolution of a k-torus T is an exact sequence of k-tori

0 → S → E → T → 0

with E quasi-trivial (i.e. Ê is a permutation module) and S flabby. This always exists and if T is split by a field extension K, then E and S can also be chosen to be split by K.

Theorem 4.11. Let k be a p-adic field containing a n^{3}-th primitive root unity. Suppose A = (a, t_{1})_{n} ⊗ (c, t_{2})_{n} is a division k((t_{1}))((t_{2}))-algebra, then

ρ_{Kahn,A,k}([ζ]) = φ [λ h_{n}^{4}({a, t_{1}, c, t_{2}})] ∈ H_{n^{2}}^{4}(k)

for ζ an n^{2}-th primitive root of unity and an integer λ ≢ 0 mod \overline{n^{2}} (and φ as in Proposition 4.8). A fortiori, j(0, n) ≢ 0 mod \overline{n^{2}} for any n.

Proof. We know by Theorem 4.5 that ρ_{Kahn,A} : SK_{1}(A)(k) → H_{n^{2}}^{4}(k) is not trivial and moreover SK_{1}(A)(k) = ℤ/n and H_{n^{2}}^{4}(k) ≅ ℤ/n^{2}. We prove that the image of μ_{n^{2}}(k) ≅ ℤ/n^{2} inside SK_{1}(A)(k) is all of SK_{1}(A)(k). In that case, ρ_{Kahn,A}([ζ]) is not trivial in H_{n^{2}}^{4}(k) (and in H4 ; \overline{n^{2}}(k) ≅ ℤ/\overline{n^{2}}) so that j(0, n) ≢ 0 mod \overline{n^{2}}.

To prove the statement, let L = k(\sqrt[n]{a}, \sqrt[n]{b}) and Γ = Gal(L/k) ≅ ℤ/n×ℤ/n. Then by taking residues on k((t_{1}))((t_{2})) with respect to t_{1} and t_{2}, Platonov proves SK_{1}(A)(k) ≅ Ĥ^{−1}(Γ, L^{×}) where the cohomology group is a Tate cohomology group (see e.g. [Wei1, Def. 6.2.4]) - also use [Pla, Thms. 4.17 & 5.7] and [Wad, (6.15)]). On the other hand, Ĥ^{−1}(Γ, L^{×}) = T (k)/R for T = R_{L/k}^{1}(𝔾_{m}) [CTS, Prop. 15]. The resulting isomorphism SK_{1}(A)(k) ≅ T (k)/R is a specialisation morphism (in t_{1} and t_{2}) [Wad, (6.9) & (6.10)] so that the composite μ_{n^{2}}(k) → SK_{1}(A)(k) ≅ T (k)/R is the canonical morphism μ_{n^{2}}(k) → T (k)/R. It suffices to prove that the latter is surjective.

First take a flabby resolution 1 → S → E → T → 1 of L-split tori, then H^{1}(k, S) = T (k)/R (loc. cit., Thm. 2). The evaluation morphism S × Ŝ → 𝔾_{m} induces a perfect pairing [Nak, Tat]:

H^{1}(k, S) × H^{1}(k, Ŝ) → H^{2}(k, 𝔾_{m}) ≅ ℚ/ℤ.

Moreover H^{1}(k, S) ≅ H^{1}(Γ, S(L)) as this follows from the inflation-restriction exact sequence [GS, 3.3.14] and H^{1}(L, S) = 0. The pairing above can be modified to a pairing

H^{1}(Γ, S(L)) × H^{1}(Γ, Ŝ(L)) → Br(L/k) ≅ ℤ/n^{2}ℤ.

Note that μ_{n^{2}}(k) ⊂ T so that we get a dual map T̂ → ℤ/n^{2}ℤ. Using the flabby resolution and the pairing T (k)×T̂(L) → L^{×}, we get the following commutative diagram of pairings:

[figure: H^{1}(k, S); ×; H^{1}(k, Ŝ); ≅; H^{2}(k, 𝔾_{m}) ≅ ℚ/ℤ; H^{1}(Γ, S(L)); ≅; ×; H^{1}(Γ, Ŝ(L)); Br(L/k); T (k); ×; H^{2}(Γ, T̂(L)); Br(L/k); μ_{n^{2}}(k); ×; H^{2}(Γ, ℤ/n^{2}); Br(L/k).]

The bottom pairing is perfect as μ_{n^{2}}(k) ≅ ℤ/n^{2}; note that the bottom square comes from the compatibility of the pairings

[figure: T (k); ×; T̂(L); L^{×}; μ_{n^{2}}(k); ×; ℤ/n^{2}; L^{×}.]

As H^{1}(k, S) = T (k)/R ≅ ℤ/n, to prove the surjectivity of μ_{n^{2}} → T (k)/R it suffices to prove the injectivity of H^{1}(k, Ŝ) → H^{2}(Γ, ℤ/n^{2}). Since H^{1}(Γ, Ê(L)) = 0, this comes down to proving the injectivity of H^{2}(Γ, T̂) → H^{2}(Γ, ℤ/n^{2}). This morphism fits into an exact sequence

H^{2}(Γ, I_{Γ}) → H^{2}(Γ, T̂) → H^{2}(Γ, ℤ/n^{2})

because of the exact sequence of group functors

0 → μ_{n^{2}} → T → R_{L/k}(𝔾_{m})/𝔾_{m} → 0.

Clearly T → R_{L/k}(𝔾_{m})/𝔾_{m} factors through R_{L/k}(𝔾_{m}), so that H^{2}(Γ, I_{Γ}) → H^{2}(Γ, T̂) factors through H^{2}(Γ, ℤ[Γ]) which is trivial by Shapiro’s Lemma. This proves the desired injectivity. ■

Remark 4.12 – Note that the proof also defines an invariant of the torus T with values inside H_{n^{2}}^{4}(k).

From this we get the following corollary.

Corollary 4.13. Let k be a field containing an l^{2}-th primitive root of unity (for l ≠ char(k) any prime) and let A = (a, b)_{l} ⊗ (c, d)_{l} be a product of two symbol algebras. If {a, b, c, d} ≠ 0 ∈ K_{4}^{M}(k)/l, then SK_{1}(A) ≠ 0.

Proof. For a field k of characteristic 0, the corollary follows from the previous theorem.

Let k be a field of char(k) = p > 0 and let l ≠ p be a prime and assume k to contain an l^{2}-th primitive root ζ ∈ \overline{k}. Take any product of two symbol k-algebras A = (a, b)_{l} ⊗ (c, d)_{l} for a, b, c, d ∈ k^{×}. Let R be a p-ring with residue field k and fraction field K. Then A lifts to the central simple K-algebra B = (ã, b̃)_{l} ⊗ (c̃, d̃)_{l} where ã, b̃, c̃, d̃ are lifts from a, b, c, d ∈ R. Under the injection H_{l^{2}}^{4}(k) → H_{l^{2}}^{4}(K) induced by (2.5), φ[h_{l,k}^{4}({a, b, c, d})] is sent to φ[h_{l,K}^{4}({ã, b̃, c̃, d̃})] (with an abuse of notation for φ from Proposition 4.8). This follows from a same splitting for Milnor’s K-Theory [Mil, Lem. 2.6].

As ρ_{Kahn,B,K}([ζ̃]) = φ[j(0, l) h_{l,K}^{4}({ã, b̃, c̃, d̃})] (for a lift ζ̃ ∈ μ_{l^{2}}(K) of ζ), we find from the construction in Theorem 2.4 that ρ̃_{Kahn,A,k}([ζ]) = φ[j(0, l) h_{l,k}^{4}({a, b, c, d})]. On the other hand, as Kahn’s invariant generates the invariant group (§2.4 (a)), there is an integer d such that ρ̃_{Kahn,A} = d ρ_{Kahn,A}. From this the result follows. ■

By the proof, it even suffices to prove the non-triviality of the symbols in characteristic zero to obtain the non-triviality of the symbols in moderate positive characteristic. (Use a p-ring and the splitting in Milnor’s K-Theory (loc. cit.)).

References

[AG] Maurice Auslander and Oscar Goldman. The Brauer group of a commutative ring. Trans. Amer. Math. Soc., 97:367–409, 1960.

[Alb] Adrian Albert. Normal division algebras of degree four over an algebraic field. Trans. Amer. Math. Soc., 34(2):363–372, 1932.

[Bae] Ricardo Baeza. Quadratic forms over semilocal rings. Lecture Notes in Mathematics, Vol. 655. Springer-Verlag, Berlin, 1978.

[BK] Spencer Bloch and Kazuya Kato. p-adic étale cohomology. Publ. Math. Inst. Hautes Études Sci., (63):107–152, 1986.

[Coh] Irvin Cohen. On the structure and ideal theory of complete local rings. Trans. Amer. Math. Soc., 59:54–106, 1946.

[CTS] Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc. La R-équivalence sur les tores. Ann. Sci. École Norm. Sup. (4), 10(2):175–229, 1977.

[Dra] Peter Draxl. Skew Fields, volume 81 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1983.

[Gar] Skip Garibaldi. Cohomological invariants: exceptional groups and spin groups. Mem. Amer. Math. Soc., 200(937):xii+81, 2009. With an appendix by Detlev W. Hoffmann.

[GMS] Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre. Cohomological invariants in Galois cohomology, volume 28 of University Lecture Series. Amer. Math. Soc., 2003.

[Gro1] Alexander Grothendieck. Éléments de Géométrie Algébrique IV, Étude locale des schémas et des morphismes de schémas, Première Partie, volume 20 of Publ. Math. Inst. Hautes Études Sci. Bures-sur-Yvette, 1964.

[Gro2] Alexander Grothendieck. Le groupe de Brauer : I. Algèbres d’Azumaya et interprétations diverses. Séminaire Bourbaki, 9:199–219, 1964-1966. Exposé No. 290.

[GS] Philippe Gille and Tamás Szamuely. Central Simple Algebras and Galois Cohomology, volume 101 of Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge, 2006.

[Izh] Oleg Izhboldin. On the cohomology groups of the field of rational functions. In Mathematics in St. Petersburg, volume 174 of Amer. Math. Soc. Transl. Ser. 2, pages 21–44. Amer. Math. Soc., Providence, RI, 1996.

[Kah1] Bruno Kahn. Formes quadratiques sur un corps, volume 15 of Cours Spécialisés. Société Mathématique de France, 2008.

[Kah2] Bruno Kahn. Cohomological approaches to SK_{1} and SK_{2} of central simple algebras. Preprint, 2009.

[Kat1] Kazuya Kato. Galois cohomology of complete discrete valuation fields. In Algebraic K-Theory, volume 967 of Lecture notes in mathematics, pages 215–238, Berlin, 1982.

[Kat2] Kazuya Kato. Symmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two. Invent. Math., 66(3):493–510, 1982.

[KMRT] Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol. The book of involutions, volume 44 of Amer. Math. Soc. Colloq. Publ. 1998.

[Knu] Max-Albert Knus. Quadratic and Hermitian forms over rings, volume 294 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1991.

[Mat] Hideyuki Matsumura. Commutative ring theory, volume 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid.

[Mer1] Alexander Merkurjev. K-theory of simple algebras. In K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), volume 58 of Proc. Sympos. Pure Math., pages 65–83. Amer. Math. Soc., Providence, RI, 1995.

[Mer2] Alexander Merkurjev. Invariants of algebraic groups. J. reine angew. Math., 508:127–156, 1999.

[Mer3] Alexander Merkurjev. The group SK_{1} for simple algebras. K-Theory, 37(3):311–319, 2006.

[Mil] John Milnor. Algebraic K-theory and quadratic forms. Invent. Math., 9:318–344, 1969/1970.

[Nak] Tadasi Nakayama. Cohomology of class field theory and tensor product modules. I. Ann. of Math. (2), 65:255–267, 1957.

[NM] Tadasi Nakayama and Yozô Matsushima. Über die multiplikative Gruppe einer p-adischen Divisionsalgebra. Proc. Imp. Acad. Tokyo, 19:622–628, 1943.

[OVV] Dmitri Orlov, Alexander Vishik, and Vladimir Voevodsky. An exact sequence for K_{∗}^{M}/2 with applications to quadratic forms. Ann. of Math., 165(1):1–13, 2007.

[Pla] Vladimir Platonov. The Tannaka-Artin problem and reduced K-theory. Math. USSR Izv., 10(2):211–243, 1976. English translation.

[Ros1] Markus Rost. Chow Groups with Coefficients. Doc. Math. J. DMV, 1:319–393, 1996.

[Ros2] Markus Rost. The basic correspondence of a splitting variety. 1998. Notes downloadable from his website.

[RTY] Ulf Rehmann, Sergey Tikhonov, and Vyacheslav Yanchevskiĭ. Symbols and cyclicity of algebras after a scalar extension. Fundam. Prikl. Mat., 14(6):193–209, 2008.

[Ser] Jean-Pierre Serre. Galois Cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002.

[SGA] Schémas en groupes. III: Structure des schémas en groupes réductifs. Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 153. Springer-Verlag, Berlin, 1962/1964.

[Sus1] Andrei Suslin. SK_{1} of division algebras and Galois cohomology. In Algebraic K-theory, volume 4 of Adv. Soviet Math., pages 75–99. Amer. Math. Soc., Providence, RI, 1991.

[Sus2] Andrei Suslin. SK_{1} of division algebras and Galois cohomology revisited. In Proceedings of the St. Petersburg Mathematical Society. Vol. XII, volume 219 of Amer. Math. Soc. Transl. Ser. 2, pages 125–147, Providence, RI, 2006. Amer. Math. Soc.

[Tat] John Tate. The cohomology groups of tori in finite Galois extensions of number fields. Nagoya Math. J., 27:709–719, 1966.

[Voe] Vladimir Voevodsky. On Motivic Cohomology with ℤ/l coefficients. Preprint, 2009.

[Wad] Adrian Wadsworth. Valuation theory on finite dimensional division algebras. In Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), volume 32 of Fields Inst. Commun., pages 385–449. Amer. Math. Soc., Providence, RI, 2002.

[Wan] Shianghaw Wang. On the commutator group of a simple algebra. Amer. J. Math., 72:323–334, 1950.

[Wei1] Charles Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997.

[Wei2] Charles Weibel. The norm residue isomorphism theorem. J. Topol., 2(2):346–372, 2009.

[Wit] Ernst Witt. Zyklische Körper und Algebren der Charakteristic p vom Grad p^{n}. J. reine angew. Math., 176:126–140, 1937.

[Wou] Tim Wouters. L’invariant de Suslin en caractéristique positive. Preprint, 2009.