COHOMOLOGICAL INVARIANTS OF SIMPLY CONNECTED GROUPS OF RANK 3

ALEXANDER MERKURJEV

Let G be a linear algebraic group defined over a field F. One can define an equivalence relation (called R-equivalence) on the group G(F) of points over F as follows (cf., [4], [9] and [14]). Two points g_{0}, g_{1} ∈ G(F) are R-equivalent, if there is a rational morphism f : 𝔸_{F}^{1} → G of algebraic varieties over F defined at points 0 and 1 such that f(0) = g_{0} and f(1) = g_{1}. The group of R-equivalence classes is denoted by G(F)/R. For example, if G = SL_{1}(A) is the special linear group of a central simple F -algebra A, then the group of R-equivalence classes G(F)/R is equal to the reduced Whitehead group in algebraic K-theory (cf., [27])

SK_{1}(A) = SL_{1}(A)/[A^{×}, A^{×}].

We say that the group G is R-trivial if G(E)/R = 1 for any field extension L/F (cf., [14]). The group G(F)/R measures complexity of G(F). One of the major properties of G(F)/R is that this group is rigid, i.e. G(F)/R = G(F (t))/R. In other words, any rational family of elements in G(F)/R is constant, whereas any R-trivial element (i.e. an element in the kernel of G(F) → G(F)/R) can be connected to the identity of the group by a rational family of elements in G.

It is known that an algebraic group G of rank (dimension of a maximal torus) at most 2 is rational (cf., [2, Th. 7.9], [27, Th.4.74]) and hence G is R-trivial by [4], [14, Prop. 1]. In the present paper we consider the group G(F)/R for a simple simply connected group of type A_{3} (rank 3). The starting point is the following two results:

Theorem 0.1. (M. Rost, [12, Th. 4], [7, §17]) Let A be a central simple algebra of dimension 16 over a field F (char F ≠ 2) and exponent 2 in the Brauer group of F. Then there is an exact sequence

1 ⟶ SK_{1}(A) ⟶^{r} H^{4}(F, ℤ/2ℤ) ⟶ H^{4}(F (X), ℤ/2ℤ),

where X is the Albert quadric of A.

Thus, Rost’s theorem gives a cohomological description of the group of equivalence classes SL_{1}(A)/R = SK_{1}(A) of the simple simply connected group SL_{1}(A) of rank 3.

Date: April, 1999.

Partially supported by the N.S.F.

Theorem 0.2. [10] Let A be a central division algebra of dimension 16 over a field F (char F ≠ 2). Then there exists a field extension E/F such that SK_{1}(A ⊗_{F} E) ≠ 1. In particular, the group SL_{1}(A) is not R-trivial and hence is not rational.

In other words, if A is a division algebra of exponent 2, the family of homomorphisms r⊗_{F} E in Theorem 0.1 for all field extensions E/F, being considered as a morphism of functors E ↦ SK_{1}(A ⊗_{F} E) and E ↦ H^{4}(E, ℤ/2ℤ) from the category of field extensions of F to the category of groups, is not trivial (whereas the group SK_{1}(A) can be trivial).

The homomorphism r can be considered as a dimension four cohomological invariant of the group SL_{1}(A) as defined in [17]. Note that cohomological invariants of dimension at most 3 of simply connected groups are trivial by [17, Prop. 4.9]. Thus the Rost’s invariant r is an example of a nontrivial invariant of the smallest dimension of a simply connected group of the smallest rank.

The main result of the paper (Theorem 4.5) is a generalization of the Rost’s Theorem 0.1 to the case of arbitrary simple simply connected group G of type A_{3}. It is known (cf., [29], [7, Th. 26.9]) that G is isomorphic to the special unitary group SU(B, τ) of a (uniquely determined up to isomorphism) semisimple algebra B of degree 4 over a quadratic extension L/F with a unitary involution τ. We define a dimension four invariant of G with coefficients in a certain cycle module and describe its image. If L splits, i.e. L = F × F, then B = A × A^{op} with the exchange involution τ and SU(B, τ) = SL_{1}(A). The special case exp(A) = 2 is the Rost’s Theorem 0.1.

Note that the root system A_{3} coincides with D_{3}, hence G is canonically isomorphic to the spinor group Spin(D, σ) for the discriminant algebra D of the pair (B, τ) with canonical involution σ (cf., [7, §26]).

We proceed in several steps. First we define a variety X = J(B, τ) which plays a role of the Albert quadric in Theorem 0.1. It is the variety of right ideals J in B of L-dimension 8 such that τ(J) · J = 0. Let Γ_{1}(F) be the inverse image of K_{1}(F) ⊂ K_{1}(L) under the reduced norm homomorphism Nrd : K_{1}(B) → K_{1}(L). Denote by Σ_{1}(F) the subgroup in Γ_{1}(F) generated by the classes of the elements represented by b ∈ B^{×} such that τ(b) = b. We use a computation of G(F)/R in [3, Th. 5.4] as the factor group Γ_{1}(F)/Σ_{1}(F). Using a computation of algebraic K-theory of projective homogeneous varieties given in [22] we relate the group Γ_{1}(F)/Σ_{1}(F) to the factor group K_{1}(X)^{(2/3)} of the topological filtration on K_{1}(X). Then, considering the Brown-Gersten-Quillen spectral sequence for X we prove that K_{1}(X)^{(2/3)} is isomorphic to the K-cohomology group H^{2}(X, K_{3}) using computation of the Chow group CH^{4}(X) given in [11]. Finally we use a theorem of B. Kahn relating the latter group to the degree four Galois cohomology groups.

We use the following notation.

For an algebraic variety X defined over a field F and a field extension L/F, by X_{L} we denote X×_{Spec F} Spec L and by X_{sep} the variety X_{F_{sep}} over a separable closure F_{sep} of F.

By H^{p}(X, K_{q}) we denote the K-cohomology groups and by CH^{p}(X) = H^{p}(X, K_{p}) the Chow groups (cf., [24, §7.5]).

For a field F, charF ≠ 2, denote by H^{p}(F) the Galois cohomology group H^{p}(Gal(F_{sep}/F), ℤ/2ℤ). For an element a ∈ F^{×}, (a) is the element of H^{1}(F) corresponding to a under the canonical isomorphism H^{1}(F) ≃ F^{×}/F^{×2}.

We denote by μ_{n} the Galois module of n-th roots of unity and for any q ∈ ℤ and by ℚ/ℤ(q) the direct limit of μ_{n}^{⊗q} over all n prime to charF.

We will consider a central semisimple algebra B with a unitary involution τ over a quadratic Galois extension L/F, i.e. τ acts on L as the generator of the Galois group of L over a field F. If L is a field, then B is a simple algebra over L. If L splits, i.e. L ≃ F × F, then B ≃ A × A^{op} for a central simple F -algebra A with the exchange involution τ (cf., [7, Prop. 2.14]). By index ind B we mean the standard index of the central simple algebra B if L is a field and index of A if L splits.

For (B, τ) as above, SU(B, τ) denotes the special unitary group of a central simple algebra B with a unitary involution τ (cf., [7, §23]). It is a simple simply connected algebraic group of type A_{n−1}, where n is the degree of B over L. The group SU(B, τ) of its F -points consists of all b ∈ B^{×} such that τ(b) · b = 1 and Nrd(b) = 1.

1. K-theory of X

In this section we introduce a variety X, which replaces the Albert quadric in the Rost’s theorem and compute its algebraic K-theory.

1.1. Split case. Let V be a vector space of dimension 4 over a field F. Consider the Grassmannian variety X = Gr(2, V) of 2-dimensional subspaces U ⊂ V. Denote by E the locally free sheaf of sections of the canonical vector bundle on X (of the subvariety in V × X consisting of all pairs (v, U) such that v ∈ U). We denote by ρ the class of E in K_{0}(X) and by ξ the class of the second exterior power Λ^{2}E, so that ξ = λ^{2}ρ, where λ^{2} is the second exterior power operation on K_{0}(X). By [21, 5.9 and 7.1], K_{∗}(X) is a free module over K_{∗}(F) with the basis

(1)  {1, ξ, ξ^{2}, ρ, ρξ, s^{2}ρ},

where s^{2} is the symmetric square.

We will need a slightly different basis. The second exterior power gives a closed embedding

X ↪ ℙ_{F} (Λ^{2}V),

identifying X with a projective quadric in ℙ_{F} (Λ^{2}V). Let Y ⊂ X be a hyperplane section in X. Since the class of the sheaf O_{X}(−1) in K_{0}(X) equals ξ, the existence of an exact sequence

0 → O_{X}(−1) → O_{X} → i_{∗}O_{Y} → 0,

where i : Y ↪ X is the closed embedding, implies that the class h ∈ K_{0}(X) of i_{∗}O_{Y} is equal to 1 − ξ.

A linear form φ ∈ V^{∗} can be viewed as a section of the sheaf E^{∗} and hence as a morphism of sheaves E → O. The variety P ⊂ X of zeros of a section φ ≠ 0 consists of all 2-dimensional subspaces U ⊂ V such that φ(U) = 0 and hence is equal to the projective plane Gr(2, Ker φ) ⊂ X. There is an exact sequence

0 → Λ^{2}E → E ⟶^{φ} O_{X} → j_{∗}O_{P} → 0,

where j : P ↪ X is the closed embedding. Denote the class of j_{∗}O_{P} in K_{0}(X) by p, so that

(2)  p = 1 + ξ − ρ.

Dualizing the natural inclusion E ↪ V ⊗_{F} O_{X} we get an exact sequence

(3)  0 → E′ → V^{∗} ⊗_{F} O_{X} → E^{∗} → 0.

The class of E′ in K_{0}(X) we denote by ρ′. A vector v ∈ V can be viewed as a section of the sheaf E′^{∗} and hence as a morphism E′ → O. The variety P′ ⊂ X of zeros of a section v ≠ 0 consists of all 2-dimensional subspaces U ⊂ V such that v ∈ U and hence is isomorphic to another projective plane ℙ(V/F v) in the quadric X.

There is an exact sequence

0 → Λ^{2}E′ → E′ →^{v} O_{X} → j′_{∗}O′_{P} → 0,

where j′ : P′ ↪ X is the closed embedding. Denote the class of j′_{∗}O_{P′} in K_{0}(X) by p′, so that

(4)  p′ = 1 + ξ − ρ′.

It follows from (3) that λ^{2}ρ′ · λ^{2}ρ^{∗} = 1, hence λ^{2}ρ′ = λ^{2}ρ = ξ. The natural pairing

E′ ⊗ E′ → Λ^{2}(E′)

induces an isomorphism between sheaves E′^{∗} and E′ ⊗ Λ^{2}(E′)^{⊗−1}, hence ρ′^{∗} = ρ′ξ^{−1}. The exact sequence dual to (3) gives ρ′^{∗} = 4 − ρ, hence

(5)  ρ′ = 4ξ − ρξ.

Thus, the element ρξ in the basis (1) can be replaced by ρ′.

Similarly, the sequence (3) gives ρ^{∗} = 4 − ρ′ and hence

(6)  ρ = 4ξ − ρ′ξ.

If φ ∈ V^{∗} and v ∈ V are chosen so that φ(v) ≠ 0, then P ∩ P′ = ∅, hence pp′ = 0 ∈ K_{0}(X). Using (2), (4), (5) and (6) we then get

(7)  ρ^{2}ξ^{−1} = 1 − 6ξ + ξ^{2} + 4ρ.

Since ρ^{2} = s^{2}ρ + λ^{2}ρ = s^{2}ρ + ξ, by (5) and (7),

s^{2}ρ = −6ξ^{2} + ξ^{3} + 4ρξ = 16ξ − 6ξ^{2} + ξ^{3} − 4ρ′.

Thus, we can replace s^{2}ρ in the basis (1) by ξ^{3}.

We have proved Proposition 1.1. The group K_{∗}(X) is a free module over K_{∗}(F) with the basis {1, ξ, ξ^{2}, ξ^{3}, ρ, ρ′}. □

We will also need a basis of K_{∗}(X) which is compatible with the topological filtration K_{∗}(X)^{(i)} by codimension of support (cf., [24, §5]). Clearly, p and p′ are the elements of codimension 2 and h^{i} is of codimension i. The variety X is cellular (cf., [5]), hence

H^{i}(X, K_{∗+i}) = K_{∗}(F) ⊗ CH^{i}(X).

Also for i = 0, 1, the composition

K_{∗}(F) ⟶^{h^{i}} K_{∗}(X)^{(i/i+1)} ↪ H^{i}(X, K_{∗+i})

is then injective since CH^{i}(X) is an infinite cyclic group generated by the i-th power of the class of a hyperplane section. A simple computation using (2) and (4) then gives

Corollary 1.2. The set of elements {1, h, h^{2}, h^{3}, p, p′} is a basis of K_{∗}(X) over K_{∗}(F), {h, h^{2}, h^{3}, p, p′} is a basis of K_{∗}(X)^{(1)} and {h^{2}, h^{3}, p, p′} is a basis of K_{∗}(X)^{(2)}. □

1.2. Nonsplit case. We consider in this section algebraic K-theory of twisted forms of Gr(2, 4).

Let B be a central semisimple algebra of degree 4 with a unitary involution τ over a quadratic extension L of a field F of characteristic different from 2 (we don’t exclude the split case L = F × F). Consider the variety X = J(B, τ) of right ideals J in B of L-dimension 8 such that τ(J)·J = 0. This is a projective homogeneous variety of the special unitary group SU(B, τ) of dimension 4 being a twisted form of the Grassmannian variety Gr(2, 4) considered in section (1.1) (cf., [19]). In other words, X is isomorphic to Gr(2, 4) in the split case, i.e. when B and L split.

Let D be the discriminant algebra of (B, τ) (cf., [7, §10]). It is a central simple algebra of degree 6 over F with canonical orthogonal involution σ. The algebra with involution (B, τ) can be reconstructed from (D, σ) as the Clifford algebra with canonical involution [7, §15 D]. The variety X is isomorphic to a unique projective homogeneous variety of dimension 4 of the spinor group Spin(D, σ) which is canonically isomorphic to SU(B, τ) (cf., [7, Prop. 15.27]). Thus, X is canonically isomorphic to the involution variety I(D, σ) of right ideals I ⊂ D of dimension 6 such that σ(I) · I = 0. In particular, D splits over any field extension E/F such that X(E) ≠ ∅, for example, over the function field F (X). In the split case X is isomorphic to a quadric in ℙ^{5}.

Consider the following locally free sheaves on X. Denote by J the sheaf of sections of the canonical vector bundle on X (of the subvariety in B × X consisting of all pairs (b, J) such that b ∈ J). It is a sheaf of rank 16 having a natural structure of a right B-module.

In the split case, B = End_{F} (V) × End_{F} (V^{∗}) for a vector space V over F of dimension 4. Any rank 8 over L = F × F right ideal J ⊂ B such that τ(J) · J = 0 is of the form

J = Hom(V, U) × Hom(V^{∗}, (V/U)^{∗})

for some 2-dimensional subspace U ⊂ V. Hence X = Gr(2, V) and

(8)  J ≃ (V^{∗} ⊗_{F} E) ⊕ (V ⊗_{F} E′).

Viewing X as the involution variety I(D, σ) we can consider the sheaf I of sections of the canonical vector bundle on I(D, σ) (of the subvariety in D × I(D, σ) consisting of all pairs (a, I) such that a ∈ I). Clearly, I is a sheaf of rank 6 and there is a natural structure of a right D-module on I.

In the split case, D = End_{F} (Λ^{2}V) and hence any 6-dimensional right ideal I ⊂ D is of the form Hom(Λ^{2}V, W) for a 1-dimensional subspace W ⊂ Λ^{2}V. Hence

(9)  I ≃ Λ^{2}V^{∗} ⊗_{F} Λ^{2}E = Λ^{2}(V^{∗} ⊗_{F} E).

The exact functor j : M ↦ J ⊗_{B} M from the category P(B) of finitely generated left B-modules to the category P(X) of locally free sheaves on X of finite rank induces a graded group homomorphism

g : K_{∗}(B) ⟶ K_{∗}(X).

In the split case the category P(B) is equivalent to P(F) × P(F) and we identify K_{∗}(B) with K_{∗}(F) ⊕ K_{∗}(F). By (8), the functor j is then identified with

N × N′ ↦ (E ⊗_{F} N) ⊕ (E′ ⊗_{F} N′)

and hence

(10)  g(a, a′) = a · ρ + a′ · ρ′.

Let D^{⊗i} be the i-th tensor power of D over F, so that the tensor power I^{⊗i} is a right D^{⊗i}-module. The exact functor i : P(D^{⊗i}) → P(X), taking a module N to I^{⊗i} ⊗_{D^{⊗i}} N, induces a graded group homomorphism

f_{i} : K_{∗}(D^{⊗i}) ⟶ K_{∗}(X).

In the split case the category P(D) is equivalent to P(F) and we identify K_{∗}(D) with K_{∗}(F). By (9), the functor i is then identified with

N ↦ (Λ^{2}E)^{⊗i} ⊗_{F} N

and hence

(11)  f_{i}(a) = a · ξ^{i}.

Proposition 1.3. The map

h = g + ∑_{i=0}^{i=3} f_{i} : K_{∗}(B) ⊕ K_{∗}(F) ⊕ K_{∗}(D) ⊕ K_{∗}(D^{⊗2}) ⊕ K_{∗}(D^{⊗3}) ⟶ K_{∗}(X)

is an isomorphism.

Proof. By Proposition 1.1, (10) and (11), h is an isomorphism in the split case. The general case follows from the split one by the splitting principle (cf., [22], [16, 2.8]). □

We’d like to find a similar presentation of K_{n}(X) for n = 0, 1, which is compatible with the topological filtration as in Corollary 1.2 in the split case. Consider the reduced norm homomorphism

N = Nrd : K_{n}(D) ⟶ K_{n}(F).

If n = 0, N takes the canonical generator of K_{0}(D) to ind(D) times the generator of K_{0}(F). Also consider the composition

Ñ : K_{n}(B) ⟶^{Nrd} K_{n}(L) ⟶^{N_{L/F}} K_{n}(F).

The algebra D carries the involution σ of the first kind, hence the index of D is a power of 2. Since D is of degree 6, it is isomorphic to the matrix algebra M_{3}(Q) for a quaternion algebra Q over F. Denote by m the composition

K_{n}(F) →^{∼} K_{n}M_{3}(F) ⟶ K_{n}M_{3}(Q) →^{∼} K_{n}(D).

The reduced norm homomorphism Nrd : K_{n}(D) → K_{n}(F) is injective (if n = 1, for all algebras D of squarefree index by [28]) and its image is generated by the norms in finite field extensions of F which split D. The group K_{n}(B) is also generated by the norms in finite field extensions of L. If B and its center L split, then D is also split since by [7, Prop. 10.30] D ⊗_{F} L is similar to B^{⊗2}. Hence the image of Ñ is contained in the image of N. Thus we have a well defined homomorphism

j : K_{n}(B) ⟶ K_{n}(D)

such that N ∘ j = Ñ.

The exponent of the algebra D is at most 2, hence D^{⊗2} ∼ F and D^{⊗3} ∼ D. We identify K_{n}(D^{⊗2}) with K_{n}(F) and K_{n}(D^{⊗3}) with K_{n}(D).

We define the following homomorphisms:

\overline{g} : K_{n}(B) ⟶ K_{n}(X), \overline{g} = f_{0} ∘ Ñ + f_{1} ∘ j − g
\overline{f}_{0} : K_{n}(F) ⟶ K_{n}(X), \overline{f}_{0} = f_{0};
\overline{f}_{1} : K_{n}(D) ⟶ K_{n}(X), \overline{f}_{1} = f_{0} ∘ N − f_{1};
\overline{f}_{2} : K_{n}(F) ⟶ K_{n}(X), \overline{f}_{2} = f_{0} − f_{1} ∘ m + f_{2};
\overline{f}_{3} : K_{n}(D) ⟶ K_{n}(X), \overline{f}_{3} = f_{0} ∘ N − 3f_{1} + 3f_{2} ∘ N − f_{3}.

Thus, we have a group homomorphism

\overline{h} = \overline{g} + ∑_{i=0}^{i=3} \overline{f}_{i} : K_{n}(B) ⊕ K_{n}(F) ⊕ K_{n}(D) ⊕ K_{n}(F) ⊕ K_{n}(D) ⟶ K_{n}(X).

If B and L split, we identify Ñ : K_{∗}(F) ⊕ K_{∗}(F) → K_{∗}(F) and j with (a, a′) ↦ a + a′. It follows from (2), (4), (10) that

\overline{g}(a, a′) = (a + a′) · (1 + ξ) − a · ρ − a′ · ρ′ = a · p + a′ · p′.

In particular,

(12)  Im(\overline{g}) ⊂ K_{n}(X)^{(2)}.

If just D splits, we identify N with the identity endomorphism of K_{∗}(F) and m with the twice of the identity of K_{∗}(F). It follows from (11) that

\overline{f}_{0}(a) = a · 1;
\overline{f}_{1}(a) = a · 1 − a · ξ = a · h;
\overline{f}_{2}(a) = a · 1 − 2a · ξ + a · ξ^{2} = a · h^{2};
\overline{f}_{3}(a) = a · 1 − 3a · ξ + 3a · ξ^{2} − a · ξ^{3} = a · h^{3}.

In particular,

(13)  Im(\overline{f}_{i}) ⊂ K_{n}(X)^{(i)}.

We claim that the inclusions (12) and (13) also hold in the nonsplit case. It is trivial for \overline{f}_{0}. Since the homomorphisms \overline{f}_{i} (resp. \overline{g}) commute with the norms in finite field extensions and the groups K_{n}(D) (resp. K_{n}(B)) for n = 0, 1 are generated by the norms in field extensions which split D (resp. split B and L), the inclusions hold also for \overline{f}_{1}, \overline{f}_{3}, \overline{g} and it remains to show the statement in the case i = 2 and n = 0. It follows from the fact that the natural homomorphisms of factor groups of the terms of the topological filtration (for k = 0, 1)

K_{0}(X)^{(k/k+1)} = CH^{k}(X) ⟶ CH^{k}(X_{sep}) = K_{0}(X_{sep})^{(k/k+1)}

are injective.

Proposition 1.4. The following homomorphisms: \overline{h},

\overline{h}_{1} = \overline{g} + \overline{f}_{1} + \overline{f}_{2} + \overline{f}_{3} : K_{n}(B) ⊕ K_{n}(D) ⊕ K_{n}(F) ⊕ K_{n}(D) → K_{n}(X)^{(1)},

\overline{h}_{2} = \overline{g} + \overline{f}_{2} + \overline{f}_{3} : K_{n}(B) ⊕ K_{n}(F) ⊕ K_{n}(D) → K_{n}(X)^{(2)}

are isomorphisms for n = 0, 1. The image of \overline{f}_{3} is contained in K_{n}(X)^{(3)}.

Proof. The homomorphisms h and \overline{h} are related by the invertible matrix

(\begin{matrix}−1 & Ñ & j & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & N & −1 & 0 & 0 \\ 0 & 1 & −m & 1 & 0 \\ 0 & N & −3 & 3N & −1 \end{matrix})

hence by Proposition 1.3, \overline{h} is an isomorphism.

It suffices to prove surjectivity of \overline{h}_{1} and \overline{h}_{2}. The factor group K_{n}(X)^{(0/1)} is canonically embedded into K_{n}(F (X)). The composition

K_{n}(F) ⟶^{\overline{f}_{0}} K_{n}(X) ⟶ K_{n}(X)^{(0/1)} ↪ K_{n}(F (X))

is the natural injection, whence the surjectivity of \overline{h}_{1}.

In order to prove surjectivity of \overline{h}_{2} it suffices to show that the composition

l : K_{n}(D) ⟶^{\overline{f}_{1}} K_{n}(X)^{(1)} ⟶ K_{n}(X)^{(1/2)}

is injective. By Corollary 1.2, l is an isomorphism over a splitting field extension E/F. Now the statement follows from the fact that l is compatible with field extensions and the natural homomorphism K_{n}(D) → K_{n}(D ⊗_{F} E) is injective since D is similar to a quaternion algebra. □

It follows from the Proposition 1.4 that the composition

(14)  ε_{n} : K_{n}(B) ⊕ K_{n}(F) ⟶^{\overline{g}+\overline{f}_{2}} K_{n}(X)^{(2)} ⟶ K_{n}(X)^{(2/3)}

is surjective for n = 0, 1.

For n = 0, 1, define a subgroup Γ_{n}(F) in K_{n}(B):

Γ_{n}(F) = {b ∈ K_{n}(B) such that Nrd(b) ∈ K_{n}(F) ⊂ K_{n}(L)},

where Nrd : K_{n}(B) → K_{n}(L) is the reduced norm homomorphism and we identify K_{n}(F) with a subgroup in K_{n}(L). Clearly, if B splits, then Γ_{n}(F) = K_{n}(F) ⊂ K_{n}(L) = K_{n}(B).

We define injective homomorphism

ρ_{n} : Γ_{n}(F) ⟶ K_{n}(B) ⊕ K_{n}(F), ρ_{n}(b) = (b, − Nrd(b)).

Lemma 1.5. In the split case, the sequence

0 ⟶ Γ_{n}(F) ⟶^{ρ_{n}} K_{n}(B) ⊕ K_{n}(F) ⟶^{ε_{n}} K_{n}(X)^{(2/3)}

is exact.

Proof. We identify K_{n}(B) with K_{n}(F)⊕K_{n}(F). For an element c = (b, b′, a) ∈ K_{n}(B) ⊕ K_{n}(F) we have

ε_{n}(c) = b̄ · p + b̄′ · p′ + ā · h^{2}

(“bar” means the residue modulo K_{n}(X)^{(3)}). By [6, §3.2],

K_{n}(X)^{(2/3)} = K_{n}(F) · \overline{p} ⊕ K_{n}(F) · \overline{p}′

and p + p′ − h^{2} in K_{0}(X) is the class of a projective line in the split quadric X, hence \overline{h} = \overline{p} + \overline{p}′. Therefore, c ∈ Ker ε_{n} if and only if b = b′ = −a if and only if c ∈ Im ρ_{n}. □

In the general case denote by θ_{n} the composition ε_{n} ∘ ρ_{n}, (n = 0, 1).

Corollary 1.6. In the general case, the sequence

Γ_{n}(F) ⟶^{θ_{n}} K_{n}(X)^{(2/3)} ⟶ K_{n}(X_{sep})^{(2/3)}

is exact.

Proof. We have the following commutative diagram

K_{n}(B) ⊕ K_{n}(F) ⟶^{ε_{n}} K_{n}(X)^{(2/3)}
↓i  ↓
K_{n}(B_{sep}) ⊕ K_{n}(F_{sep}) ⟶^{ε_{n}} K_{n}(X_{sep})^{(2/3)}.

The statement follows now from Lemma (1.5) and the obvious equality i^{−1}(Γ_{n}(F_{sep})) = Γ_{n}(F). □

2. Brown-Gersten-Quillen spectral sequence for X

In this section we compare certain K-cohomology groups with factors of the topological filtration on the K-groups of the variety X = J(B, τ) as defined in the first section. We still assume that charF ≠ 2.

Consider the Brown-Gersten-Quillen spectral sequence [24, Th. 5.4]

E_{2}^{p,q} = H^{p}(X, K_{−q}) ⟹ K_{−p−q}(X).

The composition of the canonical homomorphism K_{1}(F) → K_{1}(X), with the edge map i_{1} : K_{1}(X) → H^{0}(X, K_{1}) is clearly an isomorphism, hence i_{1} is surjective and therefore all differentials starting at E^{0,−1} are trivial. Therefore, all differential arriving in E^{2,−2} are trivial. Hence

CH^{2}(X) = H^{2}(X, K_{2}) = E_{2}^{2,−2} = E_{∞}^{2,−2} = K_{0}(X)^{(2/3)}.

The composition of the canonical homomorphism K_{2}(F) → K_{2}(X) with the edge map i_{2} : K_{2}(X) → H^{0}(X, K_{2}) is an isomorphism by [26], hence i_{2} is surjective and therefore all differentials starting at E^{0,−2} are trivial. Therefore, all differential arriving in E^{2,−3} are trivial.

The only possibly nontrivial differential starting at E^{2,−3} is

d : E_{2}^{2,−3} → E_{2}^{4,−4} = H^{4}(X, K_{4}) = CH^{4}(X).

The image of this homomorphism is torsion (by Grothendieck’s theorem, the image of d has exponent 3! = 6). It is shown in [11] that the Chow group of zero-dimensional cycles CH_{0}(X) = CH^{4}(X) is torsion free (it is true for all involution varieties X for central simple algebras of index at most 2 with orthogonal involution). Hence all the differentials starting at E^{2,−3} are trivial. Thus, we have proved

Proposition 2.1. There are canonical isomorphisms

CH^{2}(X) ≃ K_{0}(X)^{(2/3)}, H^{2}(X, K_{3}) ≃ K_{1}(X)^{(2/3)}. □

We will need an information about the torsion part of CH^{2}(X). Since X_{sep} is split, the group CH^{2}(X_{sep}) is torsion free, hence the standard transfer argument (cf., the proof of Corollary 2.3) shows that

CH^{2}(X)_{tors} = Ker(CH^{2}(X) ⟶ CH^{2}(X_{sep})).

Lemma 2.2. Assume that B splits (over L) and the involution τ is isotropic. Then CH^{2}(X)_{tors} = 0.

Proof. By Propositions 1.6 and 2.1, there is canonical surjective homomorphism

Γ_{0}(F) ⟶^{θ_{0}} Ker(K_{0}(X)^{(2/3)} → K_{0}(X_{sep})^{(2/3)}) →^{∼} CH^{2}(X)_{tors}.

The group Γ_{0}(F) is infinite cyclic with the canonical generator γ which corresponds to 1 under isomorphism Γ_{0}(F) = K_{0}(F) ≃ ℤ. Note that Γ_{0}(F) does not change under field extensions since B is split over L.

We will prove that the image of γ under θ_{0} is trivial in K_{0}(X)^{(2/3)}. In the split case, θ_{0}(γ) = \overline{p}+\overline{p}′−\overline{h}^{2}, and by [6, §3.2], p+p′−h^{2} = l, there l is the class of a projective line in the quadric X. Thus, since the map K_{0}(X) ⟶ K_{0}(X_{sep}) is injective, it suffices to show that l belongs to the image of the natural homomorphism

K_{0}(X)^{(3)} ⟶ K_{0}(X_{sep})^{(3)}.

The algebra B splits, i.e. B = End_{L}(V), where V is a vector space over L of dimension 4. The involution τ is then the adjoint involution τ_{h} with respect to a nondegenerate hermitian form h on V uniquely determined up to a scalar multiple by [7, Th. 4.2]. The variety X can be identified with the variety of isotropic (with respect to h) subspaces U ⊂ V of L-dimension 2. Since τ is isotropic, then so is h. Let W ⊂ V be a 1-dimensional isotropic subspace. Then the orthogonal complement W^{⊥} is of dimension 3 and contains W. Consider the subvariety Y ⊂ X consisting of isotropic subspaces U ⊂ V of dimension 2 such that W ⊂ U ⊂ W^{⊥}. In the split case, Y is the subvariety of 2-dimensional subspaces U in Gr(2, 4) such that W_{1} ⊂ U ⊂ W_{3} where W_{1} and W_{3} are some fixed subspaces of dimensions 1 and 3 respectively. It is easy to check that Y corresponds to a projective line on X considered as a quadric, hence the class [O_{Y}] ∈ K_{0}(X) is equal to l over F_{sep}. □

Corollary 2.3. If B splits, then | CH^{2}(X)_{tors}| ≤ 2.

Proof. Since Γ_{0}(F) is a cyclic group, CH^{2}(X)_{tors} is also cyclic. On the other hand, by [11], there is a quadratic extension E/F such that τ is isotropic. By Lemma (2.2), the composition of the multiplication by 2

CH^{2}(X)_{tors} ⟶ CH^{2}(X_{E})_{tors} ⟶^{N_{E/F}} CH^{2}(X)_{tors}

is zero. □

3. A theorem of B. Kahn

In this section we compare K-cohomology groups H^{2}(X, K_{3}) for X = J(B, τ) and dimension four Galois cohomology groups of F.

Since X has a closed point x of degree 2, the transfer argument for the quadratic field extension F (x)/F shows the kernel of the natural homomorphism

H^{p}(F, ℚ/ℤ(q)) ⟶ H^{p}(F (X), ℚ/ℤ(q))

is of exponent 2. Since H^{p}(F) is naturally isomorphic to the subgroup of elements of exponent 2 in H^{p}(F, ℚ/ℤ(p − 1)) for p ≤ 4 (cf., [20, 25]), the kernels of two homomorphisms

H^{p}(F, ℚ/ℤ(p − 1)) ⟶ H^{p}(F (X), ℚ/ℤ(p − 1)),
H^{p}(F) ⟶ H^{p}(F (X))

are naturally isomorphic. We denote the kernel of the second homomorphism by H̃^{p}(F).

Since D splits over F (X), the cup-product with the class [D] ∈ H^{2}(F) gives a well defined homomorphism

i : H^{p−2}(F) ⟶^{[D]} H̃^{p}(F).

It is proved in [18] that if p = 2, the homomorphism i is surjective. A generalization of this result to the case p = 3 is given in the following theorem. The original formulation of this theorem involves Galois cohomology groups with coefficients in ℚ/ℤ(2). Using remark above we formulate it in terms of the group H̃^{3}(F).

Theorem 3.1. [13], [23] There is an exact sequence

H^{1}(F) ⟶^{i} H̃^{3}(F) ⟶ CH^{2}(X)_{tors}. □

This Theorem and Corollary 2.3 imply

Corollary 3.2. If B splits, then |H̃^{3}(F)/[D] ∪ H^{1}(F)| ≤ 2. □

B. Kahn found a further generalization to the case p = 4. He uses a theory of V. Voevodsky which requires resolution of singularities. That is why he assumes that F is of characteristic zero.

Theorem 3.3. [5, Cor. 5.5] Assume that charF = 0. Then there is an exact sequence

H^{2}(F) ⟶^{i} H̃^{4}(F) ⟶ H^{2}(X, K_{3}) ⟶ H^{2}(X_{sep}, K_{3}). □

Thus, by Corollary 1.6, Proposition 2.1, Theorems 3.1 and 3.3, the composition

φ_{n} : Γ_{n}(F) ⟶^{θ_{n}} Ker(K_{n}(X)^{(2/3)} ⟶ K_{n}(X_{sep})^{(2/3)}) →^{∼}

Ker(H^{2}(X, K_{n+2}) ⟶ H^{2}(X_{sep}, K_{n+2})) →^{∼} H̃^{n+3}(F)/[D] ∪ H^{n+1}(F)

is a surjective homomorphism for n = 0, 1. We have proved

Proposition 3.4. For n = 0, 1 there is an exact sequence

Γ_{n}(F) ⟶^{φ_{n}} H^{n+3}(F)/[D] ∪ H^{n+1}(F) ⟶ H^{n+3}(F (X)). □

4. Invariant e_{3}(B, τ)

Let (B, τ) be a central semisimple algebra of degree 4 with a unitary involution τ over a quadratic extension L/F. We assume that the algebra B is split, i.e. B = End_{L}(V) for a 4-dimensional vector space V over L. Hence τ = τ_{h} is the adjoint involution with respect to some nondegenerate hermitian form h on V (uniquely determined up to a scalar multiple) with respect to the quadratic extension L/F. Consider the quadratic form q on V (viewed now as a vector space of dimension 8 over F) defined by q(v) = h(v, v). The Clifford algebra of q is equivalent to the quaternion algebra Q = (L, d)_{F}, where d is the discriminant of h (cf., [8, Ch. V]) and hence, by [7, Cor. 10.35], to the discriminant algebra D of (B, τ). Denote by g be the 2-fold Pfister form corresponding to the quaternion algebra Q. Then for any a ∈ F^{×} the form f = q ⊥ ag has split Clifford algebra and hence belongs to I^{3} where I is the fundamental ideal in the Witt ring of F. One can choose a such that f is represented by a 8-dimensional form which is then necessarily similar to a 3-fold Pfister form. Its Arason invariant in H^{3}(F) we denote by e_{3}(B, τ). A different choice of an element a leads to the change of e_{3}(B, τ) by a multiple of the class [Q] = [D] ∈ H^{2}(F). Thus, we have a well defined invariant

e_{3}(B, τ) ∈ H^{3}(F)/[D] ∪ H^{1}(F).

The invariant e_{3}(B, τ) reflects splitting properties of the involution τ.

Lemma 4.1. The invariant e_{3}(B, τ) is trivial if and only if the involution τ is isotropic. In particular, e_{3}(B, τ) is trivial over F (X).

Proof. If the invariant e_{3}(B, τ) is trivial, then q ⊥ ag + ⟨1, −b⟩g ∈ I^{4} for some b ∈ F^{×} since I^{3}/I^{4} is canonically isomorphic to H^{3}(F) (cf., [20, 25]). Then the form f′ = q ⊥ abg also belongs to I^{4}. Since dim(f′) < 16, f′ is hyperbolic by [8, Th. X.3.1], and hence the form q is equal to (−ab)g in the Witt ring W (F) and hence is isotropic. Therefore the hermitian form h is isotropic, i.e. the involution τ = τ_{h} is also isotropic.

Conversely, if τ is isotropic, then q is isotropic and since the Clifford algebras of q and g are similar, q is a scalar multiple of g in W (X). Thus, e_{3}(B, τ) = 0.

□

One can describe the invariant e_{3}(B, τ) in terms of the homomorphism

φ_{0} : Γ_{0}(F) ⟶ H̃^{3}(F)/[D] ∪ H^{1}(F).

Lemma 4.2. If B splits, then φ_{0}(γ) = e_{3}(B, τ).

Proof. If τ is isotropic, then Lemma 2.2 shows that φ_{0}(γ) = 0. By Lemma 4.1, e_{3}(B, τ) is also trivial.

Assume that τ is anisotropic. Again by Lemma 4.1, e_{3}(B, τ) is a nontrivial element in the image of φ_{0}. On the other hand, by Corollary 3.2, | Im(φ_{0})| ≤ 2. Since γ is a generator of Γ_{0} and φ_{0} is surjective, we must have φ_{0}(γ) = e_{3}(B, τ). □

Corollary 4.3. If B splits, then for any a ∈ F^{×}, considered as an element of Γ_{1}(F) = K_{1}(F), one has φ_{1}(a) = (a) ∪ e_{3}(B, τ) ∈ H^{4}(F).

Proof. The statement follows from Lemma 4.2 and the commutativity of the diagram

K_{0}(F) = Γ_{0}(F) ⟶^{φ_{0}} H̃^{3}(F)/[D] ∪ H^{1}(F)
↓(a)  ↓(a)
K_{1}(F) = Γ_{1}(F) ⟶^{φ_{1}} H̃^{4}(F)/[D] ∪ H^{2}(F)

where the vertical homomorphisms are the products with the element a. □

Denote by H^{4}(B, τ) the subgroup in H^{4}(F) generated by [D] ∪ H^{2}(F) and the norms N_{E/F} (E^{×} ∪ e_{3}(B_{E}, τ_{E})) for all finite field extensions E/F such that the algebra B_{E} = B ⊗_{F} E is split. Also let Σ_{1}(F) be the subgroup in Γ_{1}(F) generated by the classes of all b ∈ B^{×} such that τ(b) = b.

Lemma 4.4. Assume that ind B = 4. Then

1. φ_{1}(Σ_{1}(F)) ⊂ H^{4}(B, τ).

2. H^{4}(B, τ) splits over F (X).

Proof. 1. Let b ∈ B^{×} such that τ(b) = b. Choose a τ-invariant subfield E ⊂ B of degree 4 over F containing b. The algebra B_{E} splits and the element b ∈ Γ_{1}(F) is the image of b̃ = 1 ⊗ b under the norm map

K_{1}(L ⊗_{F} E) = K_{1}(B_{E}) ⟶^{N_{E/F}} K_{1}(B).

Corollary 4.3 and the commutativity of the diagram

Γ_{1}(E) ⟶^{φ_{1}} H̃^{4}(E)/[D ⊗_{F} E] ∪ H^{2}(E)
↓N_{E/F}  ↓N_{E/F}
Γ_{1}(F) ⟶^{φ_{1}}  H̃^{4}(F)/[D] ∪ H^{2}(F)

implies that

φ_{1}(b) = N_{E/F} ((b̃) ∪ e_{3}(B_{E}, τ_{E})) ∈ H^{4}(B, τ).

The second statement follows from Lemma 4.1 since τ is isotropic over F (X) and D is split over X. □

Now we can prove the main result of the paper. If ind(B) ≤ 2, then by [3, Cor. 6.5], the group SU(B, τ) is rational, hence SU(B, τ)/R = 1. Thus we may assume that ind B = 4.

Theorem 4.5. Let (B, τ) be a central semisimple algebra of index 4 over a quadratic extension L of a field F of characteristic zero, X the variety of right ideals in B of L-dimension 8 such that τ(J) · J = 0. Then there is an exact sequence

SU(B, τ)/R ⟶ H^{4}(F)/H^{4}(B, τ) ⟶ H^{4}(F (X)).

Proof. By Lemma 4.4 and Proposition 3.4 there is an exact sequence

Γ_{1}(F)/Σ_{1}(F) ⟶ H^{4}(F)/H^{4}(B, τ) ⟶ H^{4}(F (X)).

On the other hand, there is a natural isomorphism [3, Th. 5.4]

SU(B, τ)/R →^{∼} Γ_{1}(F)/Σ_{1}(F).

□

Remark 4.6. One can derive from the proof that injectivity of the first homomorphism of the exact sequence in Theorem 4.5 is equivalent to the fact that Ker θ_{1} ⊂ Σ_{1}(F). It is an easy consequence of the following theorem of Rost (unpublished, private communication) applied to X: If for a smooth proper variety X of dimension d over a perfect field F the degree map CH^{d}(X_{E}) → K_{0}(X_{E}) is injective for any field extension E/F, then the group H^{d−1}(X, K_{d}) is generated by the images of the compositions

E^{×} ⊗ CH^{d−1}(X_{E}) ⟶ H^{d−1}(X_{E}, K_{d}) ⟶^{N_{E/F}} H^{d−1}(X, K_{d})

for all finite extensions E/F.

Consider two special cases.

1. Assume that the quadratic extension L/F splits. Then B = A × A^{op} for some central simple algebra A of degree 4 over F with the exchange involution τ and G = SU(B, τ) = SL_{1}(A) is the special linear groups of A. The variety X is the generalized Severi-Brauer variety SB(2, A) (cf., [7, 1.16]).

The group of R-equivalence classes G(F)/R is naturally isomorphic to SK_{1}(A). The exchange involution τ is isotropic (even hyperbolic), hence by Lemma 4.1, the invariant e_{3}(B, τ) is trivial. The discriminant algebra D is similar to A^{⊗2} by [7, 10.31]. The exact sequence in the Theorem 4.5 looks as follows:

SK_{1}(A) ⟶ H^{4}(F)/(2[A]) ∪ H^{2}(F) ⟶ H^{4}(F (X)).

Exactness of this sequence (including injectivity of the first homomorphism) has been proven in [17]. In particular, if A is biquaternion algebra, we get Rost’s exact sequence in Theorem 0.1.

2. Assume that the discriminant algebra D splits, i.e. SU(B, τ) = Spin(q) for some 6-dimensional quadratic form q over F unique up to a scalar multiple. The algebra B is the even Clifford algebra of q by [7, Th. 15.24]. The variety X is the projective quadric associated to q.

Assume that B is split, i.e. q is similar to a subform of a 3-fold Pfister form p. If q is isotropic, then the kernel of i : H^{3}(F) → H^{3}(F (X)) is trivial and hence the Arason class e_{3}(p) and e_{3}(B, τ) in H^{3}(F) are trivial. If q is anisotropic, then e_{3}(p) is the only nontrivial element of the kernel of i (cf., [1]) and so is e_{3}(B, τ) by Corollary 3.2. Hence in any case e_{3}(B, τ) = e_{3}(p).

Thus the subgroup H^{4}(B, τ) ⊂ H^{4}(F) in the Theorem 4.5 is generated by the norms N_{E/F} (E^{×} ∪ e_{3}(p)) for all finite field extensions E/F such that the form q_{E} is similar to a subform of a 3-fold Pfister form p. Denote this subgroup by H^{4}(q). Theorem 4.5 reduces to Corollary 4.7. For a quadratic form q of dimension 6 over F there is an exact sequence

Spin(q)/R ⟶ H^{4}(F)/H^{4}(q) ⟶ H^{4}(F (X)),

where X is the projective quadric of q. □

If the discriminant of q is trivial (q is an Albert form), then q_{E} cannot be anisotropic Pfister neighbor over any field extension E/F, hence H^{4}(q) = 0 in this case and we again get back to the Rost’s sequence.

Remark 4.8. If ind(B) = 4, then the group SU(B, τ) is not R-trivial (and hence is not stably rational) by [15, Th. 9.1], i.e. the group SU(B_{E}, τ_{E}) is not trivial for some field extension E/F. Thus, in general the kernel of the natural homomorphism H^{4}(F) → H^{4}(F (X)) in larger than the subgroup of “obvious” elements H^{4}(B, τ).

Remark 4.9. We have considered in the paper simply connected groups of types A_{3} = D_{3}. Other types of rank 3 are B_{3} and C_{3}. A simply connected group G of type C_{3} (symplectic groups of a central simple algebra of degree 6 with a symplectic involution) is rational by [29] and hence G(F)/R = 1. The only case left is the one of a simply connected group G of type B_{3}. By [29] and [7, Th. 26.12], G = Spin(q) for a nondegenerate quadratic form q of dimension 7.

Conjecture 4.10. For a non-degenerate quadratic form q of dimension 7 there is an exact sequence

1 ⟶ Spin(q)/R ⟶ H^{4}(F)/H^{4}(q) ⟶ H^{4}(F (X)),

where X is a quadric of q and H^{4}(q) is the subgroup in H^{4}(F) generated by the norms N_{E/F} (E^{×} ∪ e_{3}(p)) for all finite field extensions E/F such that the form q_{E} is similar to a subform of a 3-fold Pfister form p.

References

[1] J. Arason. Cohomologische Invarianten quadratischer Formen. J. Algebra 36 (1975), 448–491.

[2] A. Borel, T.A. Springer. Rationality properties of linear algebraic groups II. Tôhoku Math. J. 20 (1968), 443–497.

[3] V.I. Chernousov, A.S. Merkurjev. R-equivalence and special unitary groups. Journal of algebra 209 (1998), no. 1, 175–198.

[4] J.-L. Colliot-Thélène, J.-J. Sansuc. La R-équivalence sur les tores. Ann. Scient. Éc. Norm. Sup., 4^{e} série. 10, (1977), 175–230.

[5] B. Kahn. Motivic cohomology of smooth geometrically cellular varieties. To appear in the Proceedings of the Seattle K-theory conference (Proc. Symp. Pure Math., AMS).

[6] N.A. Karpenko. Algebro-geometric invariants of quadratic forms. (Russian) Algebra i Analiz 2 (1990), no. 1, 141–162, translation in St. Petersburg Math. J. 2 (1991), 119–138.

[7] M.-A. Knus, A.S. Merkurjev, M. Rost, J.-P. Tignol. The Book of Involutions. American Mathematical Society Colloquium Publications, 44. American Mathematical Society, Providence, RI, 1998.

[8] T.-Y. Lam. The Algebraic theory of Quadratic Forms. Benjamin, Reading, Massachusetts, 1973.

[9] Yu.I. Manin. Cubic forms. North-Holland, Amsterdam, 1974.

[10] A.S. Merkurjev. Generic element in SK_{1} for simple algebras. K-theory. 7 (1993), 1–3.

[11] A.S. Merkurjev. Zero-dimensional cycles on some involution varieties. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 227 (1995), Voprosy Teor. Predstav. Algebr i Grupp. 4, 93–105, 158.

[12] A.S. Merkurjev. K-theory of simple algebras. Proceedings of the 1992 Summer Research Institute on Quadratic Forms and Division Algebras, (W. Jacob and A. Rosenberg, eds), Symposia Pure Math. 58.1 (1995) 65–83, Amer. Math. Soc., Providence, R.I.

[13] A.S. Merkurjev. The group H^{1}(X, K_{2}) for projective homogeneous varieties. (Russian) Algebra i Analiz. 7 (1995), 136–164. translation in St.Petersburg Math.J. 7 (1996), no. 3, 421–444.

[14] A.S. Merkurjev. R-equivalence and rationality problem for semisimple adjoint classical algebraic groups. Inst. Hautes Études Sci. Publ. Math. 84 (1996), 189–213 (1997).

[15] A.S. Merkurjev. K-theory and algebraic groups. Proceedings of the Second European Congress of Mathematics, Progress in Math. 169, (1998) Birkhauser Verlag Basel/Switzerland, 43–72.

[16] A.S. Merkurjev. Comparison of the equivariant and the standard K-theory of algebraic varieties. (Russian) Algebra i Analiz 9 (1997), no. 4, 175–214, translation in St. Petersburg Math. J. 9 (1998), no. 4, 815–850.

[17] A.S. Merkurjev. Invariants of algebraic groups. J. Reine Angew. Math. 508 (1999), 127–156.

[18] A.S. Merkurjev, J.-P. Tignol. The multipliers of similitudes and the Brauer group homogeneous varieties. J. Reine Angew. Math. 461 (1995), 13–47.

[19] A.S. Merkurjev, I.A. Panin, A. Wadsworth. Index reduction formulas for twisted flag varieties I. K-Theory 10 (1996)), no. 6, 517–596.

[20] A.S. Merkurjev, A.A. Suslin. The norm residue homomorphism of degree 3. Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), 339–356.

[21] I.A. Panin. On algebraic K-theory of generalized flag fibre bundles and some of their twisted forms. Algebraic K-Theory (ed. A. A. Suslin), Advances in Soviet Math., Vol. 4, Amer. Math. Soc., Providence, R.I., 1991, 21–46.

[22] I.A.Panin. On the algebraic K-theory of twisted flag varieties. K-Theory, 8, (1994), 541–585.

[23] E. Peyre. Corps de fonctions de variétés homogènes et cohomologie galoisienne. C. R. Acad. Sci. Paris 321 (1995), 891–896.

[24] D. Quillen. Higher algebraic K-theory I. Lect. Notes Math., Springer-Verlag, Berlin 341, (1972), 85–147.

[25] M. Rost. Hilbert 90 for K_{3} for degree-two extensions. Preprint Regensburg, 1986.

[26] A.A. Suslin. Algebraic K-theory and norm-residue homomorphism. J. Soviet Math. 30 (1985), 2556–2611.

[27] V.E. Voskresenskiĭ. Algebraic tori. Nauka, Moscow, 1977 (Russian).

[28] S. Wang. On the commutator group of a simple algebra. Amer. J. Math. 72 (1950), 323–334.

[29] A. Weil. Algebras with involutions and the classical groups. J. Indian Math. Soc. 24 (1960), 589–623.

Alexander Merkurjev, Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

E-mail address: merkurev@math.ucla.edu