Comparing invariants of SK1

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-Yanchevskiȋ 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 × ): 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]. 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 e 1 1 • . . . • p er r of a (positive) integer m is always supposed to primitive (i.e. m = p e 1 1 • . . . • p er 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 x 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 . 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 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 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. 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. 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): ) and 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): . . . 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 (2.4) where 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 ⊕ n-1 i=0 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 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 [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 4 m (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]: (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 4 m (k) ∼ = Z/m by applying the splitting to the valuations defined by t 1 and t 2 . 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)): The maps ∂ 3 t 1 , ∂ 4 t 3 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: 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 is a commutative diagram, where is r A is the morphism induced on Galois cohomology by the map µ ⊗3 4 → µ 2 : a → a 2 and where X and Y are as above. Hence ρ S06 is injective for biquaternion algebras and 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 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 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 e 1 1 • . . . • p er r , we denote by n the integer p e 1 -1 Ch. 4,Ex. 9). Recall also that SK 1 (D i ) has p e i i -torsion [Dra,§23,Lem. 3]. Then the result follows immediately from the primary result of Kahn and the isomorphism 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 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 For a central simple k-algebra A with n = ind k (A) ∈ k × and per k (A) = n/n, there is a similar statement using a Brauer decomposition. Take a prime factorisation n = p e 1 1 •. . .•p er 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 f 1 1 • . . . • p fr r , where Then it is clear that ρ r has m-torsion. 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 q+1 p l (k) are defined as where W l (k) are the Witt vectors of length l on k and I is the ideal generated by with w ∈ W l (k), b 1 , . . . , b q ∈ k × , and w (p) = (a p 1 , . . . , a p l ) if w = (a 1 , . . . , a l ). For l = 1, we can view H q+1 p (k) as the cokernel of hence the terminology "logarithmic differentials". (For l = 0, set H q+1 p l (k) = 0.) In general, for an integer n = p l m > 0 (l, m ≥ 0 integers with p ∤ m), we define 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 2 p l (k). So for any integer n > 0 we get n Br(k) ∼ = H 2 n (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 q+1 p l (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 M r (k) (for an integer r ≥ 0) are defined as 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 M r (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 M r (k) on (H q+1 p l (k)) q≥0 is given by 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 for any m ∈ k × gives the Galois symbol by taking the cup-product: (2.9) The Bloch-Kato conjecture (proved by Voevodsky-Rost-Weibel [BK,Voe,Ros2,Wei2]) even says it is surjective with kernel mK M r (k). Hence we get a scalar multiplication of For arbitrary n, this defines in total a K M r (k)-module structure on (H q n (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: By the remarks above, this is clearly a generalisation of the moderate case. If r ≡ 0 mod per k (A), then clearly H q+1 n,A ⊗r (F ) = H q+1 n (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) : [Pla,Thm. 3.12] -see also [Wou,Cor. 3.3]. Furthermore, there exists an injection H i+1 n (k) → H i+1 n (K); on the prime-to-p parts of H i+1 n (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 i+1 n,A ⊗r (k) → H i+1 n,B ⊗r K (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 4 n,A ⊗r for r a fixed integer. Then we say ρ is a moderate invariant of SK 1 with values in H 4 ⊗r . 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 ⊗r K ) (for r any integer). There exists a unique invariant ρ ∈ 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: (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 ρ. 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 = ρA . 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 (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. 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. 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 : ) is an automorphism of M n (S), we can choose u ∈ GL n (S) such that σ(x) = ux t u -1 for all x ∈ M n (S). Because σ2 = 1, we get u t = ǫu for some ǫ ∈ µ 2 (S). Then ǫ 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): (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 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 We denote the graded quotients by 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 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 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 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). 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 1 F : K M 4 (F )/2 → 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 2 F : H 4 (F, µ 2 ) → K M 4 (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 Indeed, Knus-Merkurjev-Rost-Tignol prove that the following diagram commutes [KMRT, Notes §17]: 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. 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 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 1 ét (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]: where A = A ⊗ R k is the reduced central simple k-algebra and σ = σ ⊗ id is the reduced involution on A, which is also symplectic. On the other hand, we have an inclusion So in total, we have an inclusion 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 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 ρS06 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 4 2 (k) → I 3 W q (k) [Kat2]. Similar to Suslin's construction (2.7), we can also give a morphism Since π 2 1 sends elements of order 2 to 0, r does exactly the same. Hence we get a morphism r 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: (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 where b 1 , . . . , b q ∈ R ′ , the morphism h q 2 n ,K ′ is the Galois symbol (2.9) and i(w) is the composition 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]: 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 Proof. Let R ′ nr be a 2-ring with residue field k ′ s and fraction field , where we consider σ(b 0 )-b 0 as an element of Z/2Z for any σ ∈ Γ K ′ (with residue σ ∈ Γ k ′ ). On the other hand, The commutativity of the right square is essentially due to Kato [Kat2,Lem. 11]; he proves the existence of a commutative diagram where ψ 1 K ′ is the isomorphism of Milnor's conjecture on quadratic forms (see §3.2) and ϕ is defined by bdā under the standard identification of Z/2 and µ 2 (K ′ ). Take c to be a lift of c in R nr . By eventually changing the representant of b in R ′ , we can assume c2 -c = b. Then 4b + 1 = (2c + 1) 2 and h 1 2,K ′ (4b + 1) = (σ(2c + 1)/(2c + 1)) σ∈Γ K ′ ∈ H 1 2 (K ′ ). So if σ(2c+1)/(2c+1) = 1, we have σ(c) = c. On the other hand, if σ(2c+1)/(2c+1) = -1, we get σ(c) = -c -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 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 * • ρS06,A = ρ S06,B K (by definition of ρS06,A ) and 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 [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 , where the residue is the canonical residue on R ′ [X]. So we also get Prp 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 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 the required compatibility holds. 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 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 because k contains no non-trivial fourth roots of unity. By the proof of Theorem 3.8, we have j Because the map j from Proposition 3.6 is injective, we get that 4a + 1, b, 4c + 1, d = 0 ∈ 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 ) is trivial. By Hensel's lemma [SGA,Exp. XXIV,Prop. 8.1], we get [X ] = 0 ∈ H 1 ét (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 ∈ I 3 W ′ q (k). 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). 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). (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 4 ⊗r . 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/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 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/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/m 2 . For the remainder of the paper, we fix this injection. Proof. Take k the prime field of characteristic p and set k 4 ] 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/m 2 such that mr (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 ±1 2 , t ±1 3 , t ±1 4 ]). Take O x to be the local ring of Spec(F [t ±1 1 , t ±1 2 , t ±1 3 , t ±1 4 ]) in x with maximal ideal M. It is clear that the completion Ôx 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 Ôx with R ′ gives an isomorphism Br( Ôx ) ∼ = Br(R ′ ). In its turn, this gives an isomorphism Br( Ôx ) → Br(F ) with inverse given by taking the tensor product over F with Ôx . By construction it sends the class of T Ôx to the class of A. Let K ′ = F ((u 1 ))((u 2 ))((u 3 ))((u 4 )), then A⊗ F K ′ is Brauer-equivalent to T Ôx ⊗ Ôx K ′ ∼ = T K ′ . We find SK 1 (A K ′ ) ∼ = SK 1 (T K ′ ) (as in §2.5 (b)). Furthermore, (2.5) gives an injection by T (k)/R the R-equivalence classes of T (k). The dual T 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 1 K/k (G m ) is then J Γ , the cokernel of the norm: The dual of R K/k (G m )/G m is the kernel I Γ of the augmentation map: Recall that a k-torus F is called flabby (flasque) if F is a flabby Γ k -module, i.e. Ext 1 ( F , 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 for ζ an n 2 -th primitive root of unity and an integer λ ≡ 0 mod n 2 (and ϕ as in Proposition 4.8). A fortiori, j(0, n) ≡ 0 mod n 2 for any n. Proof. We know by Theorem 4.5 that ρ Kahn,A : SK 1 (A)(k) → H 4 n 2 (k) is not trivial and moreover SK 1 (A)(k) = Z/n and H 4 n 2 (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 4 n 2 (k) (and in H 4 n 2 (k) ∼ = Z/n 2 ) so that j(0, n) ≡ 0 mod n 2 . To prove the statement, let L = k( n √ a, 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 1 L/k (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]: 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. Do 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).