ON THE GALOIS COHOMOLOGY OF SPIN(14)

MARKUS ROST

                       Preliminary Notes

Note from May/June 2006

I am very grateful to Skip Garibaldi for comments. They led to several corrections and additions.

In the version from 1999 I had claimed without proof ed(Spin_{13}) = 6. I have now added a new section (Section 10) containing a proof.

Abstract

Let k be a field with char k ≠ 2. For i = 6, 7 we define invariants

h_{i}: H^{1}(k, Spin(14)) → H^{i}(k, Z/2)/(−1)H^{i−1}(k, Z/2).

Further we show that the natural map

H^{1}(k, (G_{2} × G_{2}) ⋊ μ_{8}) → H^{1}(k, Spin(14))

is surjective.

One concludes that the essential dimension of Spin(14) is equal to 7. Similar considerations are done for Spin(12). We also present the list of essential

dimensions of the split groups Spin(n) for n ≤ 14.

Contents

Note from May/June 2006 1 Abstract 1 1. The Arason invariant 2 2. Reduced squares 3 3. Lambda operations 3 4. Multiplicative transfer 5 5. The invariants h_{6} and h_{7} 5 6. A reduction lemma 7 7. 14-dimensional spinors 8 8. The essential dimension of Spin(14) 9 9. On the cohomology of Spin(12) 9 10. On the cohomology of Spin(13) 11 11. The essential dimension of split Spin(n) for n ≤ 14 16 References 16

Date: March 18, 1999; June 6, 2006.

1. The Arason invariant

1.1. The invariants e_{i}, i ≤ 3. Let

e_{i}: I^{i}(k)/I^{i+1}(k) → H^{i}(k, Z/2), i = 0, …, 3

be the first invariants on the graded Witt ring given by dimension, discriminant, the Hasse-Witt invariant, and Arason’s invariant, cf. [1, 26].

1.2. The split groups of type D_{n}. We denote by SO(n, n) the automorphism group of the quadratic form

∑_{1}^{n}(x_{i}^{2} − y_{i}^{2}).

Furthermore, Spin(n, n) denotes the universal cover of SO(n, n) and

PSO(n, n) = SO(n, n)/{±1} = Spin(n, n)/μ

denotes the corresponding adjoint group. Here μ is the center of Spin(n, n). One has μ = μ_{2} × μ_{2} if n is even and μ = μ_{4} if n is odd.

If n is odd, every split group of type D_{n} is isomorphic to one of Spin(n, n), SO(n, n), PSO(n, n).

1.3. Galois cohomology of SO(n, n). The set H^{1}(k, SO(n, n)) consists of the isomorphism classes of 2n-dimensional quadratic forms with trivial discriminant. We consider H^{1}(k, SO(n, n)) as a subset of I^{2}(k) ⊂ W (k).

The image of

H^{1}(k, SO(n, n)) → H^{1}(k, PSO(n, n))

consists of the similarity classes of the quadratic forms in H^{1}(k, SO(n, n)). For u ∈ H^{1}(k, Spin(n, n)) let q_{u} be the corresponding quadratic form.

The image of

H^{1}(k, Spin(n, n)) → H^{1}(k, SO(n, n))

consists of those classes in H^{1}(k, SO(n, n)) with trivial Hasse-Witt invariant.

1.4. The invariant ẽ_{3} in K_{3}^{M}/2. Let K_{n}^{M}k be Milnor’s K-group [18].

By Merkurjev’s theorem [2, 16, 31] the invariant e_{2} is bijective. Furthermore, Milnor’s homomorphism

s_{3} : K_{3}^{M}k/2 → I^{3}(k)/I^{4}(k)

is bijective (cf. [11, 17, 18, 25]).

Putting things together yields natural maps

ẽ_{3} : H^{1}(k, Spin(n, n)) → K_{3}^{M}k/2.

For u ∈ H^{1}(Spin(n, n)) the class ẽ_{3}(u) depends alone on q_{u}. For u ∈ H^{1}(Spin(8, 8)) the corresponding quadratic form q_{u} is a 3-fold Pfister form (cf. [5, 15, 20, 26]); if q_{u} = ⟨⟨a, b, c⟩⟩, then ẽ_{3}(u) = {a, b, c}. Furthermore, the maps ẽ_{3} behave additively with respect to the natural inclusions

Spin(n, n) × Spin(m, m) → Spin(n + m, n + m).

These properties determine the family of maps ẽ_{3} uniquely.

2. Reduced squares

It has been observed by Serre that for any n ≥ 2 there is a natural map

P: K_{n}^{M}k/2 → K_{2n}^{M}k/(2K_{2n}^{M}k + {−1}^{n−1}K_{n+1}^{M}k)

characterized by

P (∑_{i} x_{i}) = ∑_{i<j} x_{i}x_{j} mod (2K_{2n}^{M}k + {−1}^{n−1}K_{n+1}^{M}k)

where x_{i} are symbols. (An element x ∈ K_{n}^{M}k/2 is called a symbol if it is of the form x = {a_{1}, …, a_{n}} for some a_{i} ∈ k^{∗}.)

To define the operation P one checks that the right hand side of this formula does not depend on the presentation of an element as a sum of symbols. This follows easily from the definition of Milnor’s K-theory and the identity {a, a} = {a, −1}, cf. [18].

Let

α_{n}: K_{n}^{M}k/2 → H^{n}(F, Z/2)

be the norm residue homomorphism [18]. Milnor’s conjecture (cf. [30]) asserts that α_{n} is bijective. With Milnor’s conjecture, the operations P give rise to corresponding maps

H^{n}(k, Z/2) → H^{2n}(k, Z/2)/(−1)^{n−1}H^{n+1}(k, Z/2).

Combining this with the fact that (−1)H^{2n−1}(k, Z/2) is in the kernel of the natural maps H^{2n}(k, Z/2) → H^{2n}(k, Z/4), one obtains operations

H^{n}(k, Z/2) → H^{2n}(k, Z/4).

In the case n = 2 this operation is nothing else than the Pontryagin square, cf. [3, 4, 32, 33]. For n > 2 I don’t know any explanation of the operations P by an operation defined on the cohomology of topological spaces.

3. Lambda operations

Let Ŵ(k) be the Grothendieck (-Witt) ring of quadratic forms over k. One defines λ-operations

λ^{i}: Ŵ(k) → Ŵ(k)

in the usual fashion (see for instance [13]):

For a quadratic form φ: V → k let λ^{i}φ: ⋀^{i}V → k be its i-th exterior power. One has λ^{0}φ = ⟨1⟩ and λ^{1}φ = φ. The form λ^{2} is also given by the Killing form on the Lie algebra so(φ) (at least if Q̄ ⊂ k).

One forms the formal power series

λ_{t}φ = ∑_{i≥0} t^{i}λ^{i}φ.

Then

λ_{t}(φ ⊥ ψ) = λ_{t}φ ⊗ λ_{t}ψ.

The series λ_{t} extends to Ŵ(k) by

λ_{t}(φ − ψ) = λ_{t}φ ⊗ (λ_{t}ψ)^{−1}

and the operations λ^{i} on Ŵ(k) are defined by

λ_{t}(x) = ∑_{i≥0} t^{i}λ^{i}(x)

for x ∈ Ŵ(k).

We are mainly interested in λ^{2}. Note that

λ^{0}(x) = 1,
λ^{1}(x) = x,
y^{2} = dim y + 2λ^{2}(y),
λ^{2}(x + y) = λ^{2}(x) + xy + λ^{2}(y),
λ^{2}(x − y) = λ^{2}(x) − y(x − y) − λ^{2}(y),
λ^{2}(x − y) = λ^{2}(x) − xy + dim y + λ^{2}(y),
λ^{2}(⟨a⟩x) = λ^{2}(x)

for x, y ∈ Ŵ(k) and a ∈ k^{∗}.

Let Î(k) ⊂ Ŵ(k) be the fundamental ideal of zero dimensional virtual quadratic forms. The projection Ŵ(k) → W (k) induces identifications Î^{n}(k) = I^{n}(k) for n > 0. Î^{n}(k) is additively generated by elements of the form

⟨⟨a_{1}, …, a_{n}⟩⟩ − ⟨⟨1⟩⟩^{n} = ⟨⟨a_{1}, …, a_{n−1}⟩⟩ − ⟨a_{n}⟩⟨⟨a_{1}, …, a_{n−1}⟩⟩.

Lemma 3.1. Let φ be an n-fold Pfister form and x = φ − ⟨⟨1⟩⟩^{n}. Then

λ^{2}(x) = ⟨⟨−1⟩⟩^{n−1}x.

Proof. Write φ = ψ⟨⟨a⟩⟩ where ψ is an (n − 1)-fold Pfister form and where a ∈ k^{∗}. Then x = ψ − ⟨a⟩ψ and one finds

λ^{2}(x) = λ^{2}(ψ − ⟨a⟩ψ)
= λ^{2}(ψ) − ⟨a⟩ψx − λ^{2}(ψ)
= −⟨a⟩⟨⟨−1⟩⟩^{n−1}x
= ⟨⟨−1⟩⟩^{n−1}⟨−a⟩x = ⟨⟨−1⟩⟩^{n−1}x

Here one uses ψ^{2} = ⟨⟨−1⟩⟩^{n−1}ψ, ⟨−a⟩x = −⟨a⟩x if dim x = 0, and ⟨−a⟩⟨⟨a⟩⟩ = ⟨⟨a⟩⟩. □

Corollary 3.2. Let φ be an n-fold Pfister form. Then

λ^{2}(φ) ≃ φ′⟨⟨−1⟩⟩^{n−1},
λ^{2}(φ′) ≃ φ′(⟨⟨−1⟩⟩^{n−1})′.  □

We define operations

P′: I^{n}(k) → I^{2n}(k),
P′(x) = λ^{2}(x) − ⟨⟨−1⟩⟩^{n−1}x.

It follows from Lemma 3.1 and λ^{2}(x + y) = λ^{2}(x) + xy + λ^{2}(y) that indeed P′(x) ∈ I^{2n}(k).

These operations lift the operations P to the Witt ring.

4. Multiplicative transfer

Let L/F be separable field extension. In addition to the restriction map

r_{L/F} : W (F) → W (L), [φ] ↦ [φ_{L}]

and the corestriction map

c_{L/F} : W (L) → W (F), [ψ] ↦ [trace_{L/F} φ]

one may define a multiplicative transfer map

N_{L/F} : W (L) → W (F).

This map is analogous to the multiplicative transfer in cohomology, cf. [6, 12, 29].

We are interested in the case [L : F] = 2. Let σ denote the generator of the Galois group. Then for a quadratic form ψ: W → L the form N_{L/F} (ψ) is given by the restriction of ψ ⊗^{σ}ψ: W ⊗^{σ}W → L to the subspace of invariants (W ⊗^{σ}W)^{σ}.

Suppose L = F (\sqrt{a}). One has the following rules

dim_{F} (N_{L/F} (ψ)) = (dim_{L} ψ)^{2},
N_{L/F} (⟨α⟩) = ⟨N_{L/F} (α)⟩,
N_{L/F} (x + y) = N_{L/F} (x) + c_{L/F} (xσ(y)) + N_{L/F} (y),
N_{L/F} (x − y) = N_{L/F} (x) − c_{L/F} (xσ(y)) + N_{L/F} (y),
λ^{2}(c_{L/F} (x)) = c_{L/F} (λ^{2}(x)) + aN_{L/F} (x),
N_{L/F} (⟨⟨α⟩⟩) = ⟨⟨a⟩⟩ + \begin{cases}⟨⟨trace α, −aN_{L/F} (α)⟩⟩ & if trace α ≠ 0, \\ 0 & if trace α = 0, \end{cases}

N_{L/F} (⟨⟨α_{1}, …, α_{n}⟩⟩) = ⟨⟨a⟩⟩^{n} + \begin{cases}∏_{i}⟨⟨trace α_{i}, −aN_{L/F} (α_{i})⟩⟩ & if trace α_{i} ≠ 0, \\ 0 & else. \end{cases}

In particular, if −1 is a square in F, then

N_{L/F} (I^{n}(L)) ⊂ I^{2n}(F)

for n ≥ 2.

5. The invariants h_{6} and h_{7}

For this section it is assumed for simplicity that \sqrt{−1} ∈ k.

We define

h_{6} : H^{1}(k, Spin(7, 7)) → H^{6}(k, Z/2),
h_{6}(u) = α_{6} ∘ P ∘ ẽ_{3}(u).

The invariant h_{6}(u) depends only on q_{u}.

By the remarks of Section 3 one can lift this invariant to I^{6}(k).

In some cases the invariant h_{6} can be described explicitly. For a Pfister form φ one denotes by φ′ its pure subform (one has φ = ⟨1⟩ ⊥ φ′). Let a_{i}, b_{i}, c ∈ k^{∗}, i = 1, 2, 3, and put

(1)  q = c(⟨⟨a_{1}, a_{2}, a_{3}⟩⟩′ ⊥ −⟨⟨b_{1}, b_{2}, b_{3}⟩⟩′)

Then q = q_{u} for some u ∈ H^{1}(k, Spin(7, 7)) and for any such u one finds

(2)  h_{6}(u) = (a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}).

Lemma 5.1. Let u ∈ H^{1}(k, Spin(7, 7)). If q_{u} is isotropic, then h_{6}(u) = 0.

Proof. If q_{u} is isotropic, the q_{u} has a representation (1) with a_{1} = b_{1}, see [19, Satz 14, Zusatz] or [24]. The claim follows from (2). □

Proposition 5.2. Let u ∈ H^{1}(k, Spin(7, 7)) and let c be a nonzero value of q_{u}. The element

h_{6}(u) ∪ (c) ∈ H^{7}(k, Z/2)

does not depend on the choice of c.

Proof (Variant 1). Write q = q_{u}. If q is isotropic, then h_{6}(u) = 0 by Lemma 5.1. We may therefore assume that q is anisotropic. Let c = q(v) and c′ = q(v′) be two values of q with v, v′ linearly independent. Then c/c′ is a norm from the quadratic extension L splitting the 2-dimensional subform q|(vk + v′k). Say c/c′ = N_{L/k}(λ). Then

h_{6}(u) ∪ (c) − h_{6}(u) ∪ (c′) = h_{6}(u) ∪ (c/c′)
= h_{6}(u) ∪ N_{L/k}((λ))
= N_{L/k}(h_{6}(u_{L}) ∪ (λ))
= N_{L/k}(0 ∪ (λ)) = 0

since q_{L} is isotropic and by Lemma 5.1. □

Proof (Variant 2). Write q = q_{u} as q: V → k. Then any x = [v] ∈ PV determines an element

q(x) ∈ κ(x)/(κ(x)^{∗})^{2}.

Let ξ ∈ PV be the generic point and consider

ω = h_{6}(u) ∪ (q(ξ)) ∈ H^{7}(k(PV), Z/2).

The element ω is unramified on PV, except possibly at the divisor

Z = {q = 0} ⊂ PV

Here the residue is a multiple of (in fact, equal to)

h_{6}(u)_{k(Z)} ∈ H^{6}(k(Z), Z/2)

But the quadratic form q_{k(Z)} is isotropic, whence h_{6}(u)_{k(Z)} = 0 by Lemma 5.1. Hence ω is unramified everywhere on PV and therefore ω = (ω_{0})_{k(PV)} for some ω_{0} ∈ H^{7}(k, Z/2). The claim follows by specialization. □

Proposition 5.2 gives rise to an invariant

h_{7} : H^{1}(k, Spin(7, 7)) → H^{7}(k, Z/2),
h_{7}(u) = h_{6}(u) ∪ (q_{u}(v))

where q_{u}(v) is any nonzero value of q_{u}.

As for h_{6}, the invariant h_{7}(u) depends only on q_{u}. If q_{u} = q with q as in (1), then

h_{7}(u) = (a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, c).

This computation shows that the invariant h_{7} is non-trivial.

In the next two statements (Proposition 5.3, Lemma 5.4) we assume that k contains the algebraic closure of Q. This assumption is made to be sure that we can neglect some universal constants arising in decompositions of Killing forms and of λ^{2}(q). I have not tried to figure out the best possible conditions.

Proposition 5.3. Assume Q̄ ⊂ k. Any value of h_{6} and of h_{7} is a symbol.

Proof. It suffices to consider h_{6}. Let u ∈ H^{1}(k, Spin(7, 7)) and write q = q_{u}. Then h_{6}(u) is represented by 92-dimensional form

λ^{2}q ⊥ ⟨1⟩.

The form λ^{2}q is also given by the Killing form on so(q).

We may assume that u is induced from an element x ∈ H^{1}(k, (G_{2} × G_{2}) ⋊ μ_{8}), see Corollary 7.3. Let 𝔤 ⊂ so(q) be the Lie algebra of type G_{2} + G_{2} corresponding to x. Its Killing form is the trace of the Killing form of a Lie algebra of type G_{2} over some quadratic extension. In view of the next Lemma, this form is hyperbolic.

Therefore the 92-dimensional form λ^{2}q⊥⟨1⟩ contains a 28-dimensional hyperbolic subform. Thus h_{6}(u) is represented by a 92 − 28 = 64-dimensional quadratic form, which therefore must be a multiple of a 6-fold Pfister form. □

This proof indicates that one may represent h_{6}(u) by a form on the spinor representation S, cf. below. In fact there is a natural way to represent h_{6}(u) as N_{L/k}(ψ) on S, where ψ/L is the 3-fold Pfister form corresponding to a reduction x ∈ H^{1}(k, (G_{2} × G_{2}) ⋊ μ_{8}) of u, cf. [24].

Lemma 5.4. Assume Q̄ ⊂ k. Let 𝔤 be a Lie algebra of type G_{2} and let φ be the associated 3-fold Pfister form. Then the Killing form on 𝔤 is hyperbolic.

Proof. Let V be the 7-dimensional representation of 𝔤. Then

𝔤 ⊥ V = ⋀^{2}V.

Let further ψ denote the Killing form on 𝔤 and let φ be the associated 3-fold Pfister form. Then

ψ ⊥ φ′ = λ^{2}(φ′) = φ′⟨⟨−1, −1⟩⟩′

by Corollary 3.2. The claim follows. □

Our considerations in the construction of the invariants h_{6}, h_{7} may be also applied to the group SO(6). This leads to invariants

H^{1}(k, SO(6)) → H^{i}(k, Z/2)

for i = 4, 5, given by

c(⟨⟨a_{1}, a_{2}⟩⟩′ ⊥ ⟨⟨b_{1}, b_{2}⟩⟩′) ↦ (a_{1}, a_{2}, b_{1}, b_{2}),
c(⟨⟨a_{1}, a_{2}⟩⟩′ ⊥ ⟨⟨b_{1}, b_{2}⟩⟩′) ↦ (a_{1}, a_{2}, b_{1}, b_{2}, c).

The latter coincides with the invariant

⟨a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}⟩ ↦ (a_{1}, a_{2}, a_{3}, a_{4}, a_{5})

defined by Serre.

6. A reduction lemma

Let G be an algebraic group over k and let i: H ⊂ G be a subgroup. For x ∈ H^{1}(k, G) we denote by P_{x} a corresponding G-torsor.

Lemma 6.1. Let x ∈ H^{1}(k, G). Then x is in the image of

i_{∗}: H^{1}(k, H) → H^{1}(k, G)

if and only if the variety P_{x}/H has a k-rational point.

Proof. Indeed, if x = i_{∗}(y), then P_{x} ≃ P_{y} ×_{H} G and P_{x}/H has the k-rational point given by [P_{y}, 1] mod H.

Conversely, if z ∈ P_{x}/H is k-rational, then the fiber of z under P → P_{x}/H is an H-torsor Q with Q ×_{H} G ≃ P_{x}. □

This simple lemma is the basis of many structure theorems on quadratic forms and algebras. It applies usually when there is a “small” representation of G, i.e., a representation G → GL(V) with dim V < dim G.

A fairly simple example is given by G = O(n) and H = O(n − 1) × μ_{2}: Let x ∈ H^{1}(k, O(n)); if q_{x}: V → k is the corresponding quadratic form, then P_{x}/H is naturally isomorphic to U = PV \{q_{x} = 0}. Since U has a rational point, it follows that x has a reduction to H.

Her majesty E_{8} does not have a small representation.

7. 14-dimensional spinors

Let Spin(7, 7) → GL(S) be one of the spinor representations (dim S = 64) and let PSO(7, 7) → PGL(S) be the induced homomorphism. We denote G = PSO(7, 7) and define H ⊂ G as the image of

(G_{2} × G_{2}) ⋊ Z/2 → PSO(7, 7)

given by

(g, h)ϵ^{n} ↦ (\begin{matrix}ρ(g) & 0 \\ 0 & ρ(h) \end{matrix}) (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})^{n}

where ρ: G_{2} → Spin(7) is the standard representation.

We need the following fact, see [7, 9, 21, 24].

Proposition 7.1. The action of G on PS has an open and dense orbit U. If k is algebraically closed, then the isotropy group H_{u} of u ∈ U is conjugate to H. In particular, U = G/H.

Now let x ∈ H^{1}(k, G). Then X_{x} = P_{x} ×_{G} PS is a Brauer-Severi variety whose Brauer class coincides with the Tits class t(x) ∈ H^{2}(k, μ_{4}) of x. Further, the variety U_{x} = P_{x} ×_{G} U = P_{x}/H is a dense open subscheme of X_{x}. It follows that P_{x}/H has k-rational points if and only if t(x) = 0 (to be sure, let us assume that k is infinite). Lemma 6.1 shows

Corollary 7.2. An element x ∈ H^{1}(k, G) has an H-reduction if and only if t(x) = 0. □

Let H̃ be the preimage of H under Spin(7, 7) → PSO(7, 7). One finds (see [24])

H̃ = (G_{2} × G_{2}) ⋊ μ_{8}

where μ_{8} ⊂ Spin(7, 7) is the normalizer of G_{2} × G_{2}.

Corollary 7.3. The homomorphism

H^{1}(k, (G_{2} × G_{2}) ⋊ μ_{8}) → H^{1}(k, Spin(7, 7))

is surjective.

Proof. This follows from a diagram chase in

H^{1}(k, μ_{4}) ⟶ H^{1}(k, (G_{2} × G_{2}) ⋊ μ_{8}) ⟶ H^{1}(k, (G_{2} × G_{2}) ⋊ Z/2) ⟶ H^{2}(k, μ_{4})
∥  ↓  ↓  ∥
H^{1}(k, μ_{4}) ⟶ H^{1}(k, Spin(7, 7)) ⟶  H^{1}(k, PSO(7, 7))  ⟶^{t} H^{2}(k, μ_{4})
□

It can be shown that there exist a field k and x ∈ H^{1}(k, Spin(7, 7)) such that x has no reduction to the subgroup (G_{2} × G_{2}) × μ_{4}. This means that the appearing forms of G_{2} × G_{2} are necessarily of type R_{ℓ/k}(G_{2}) with ℓ/k a quadratic field extension. Examples have been provided in [8] using residue arguments and in [10] using computations of the K-theory of certain homogeneous varieties.

8. The essential dimension of Spin(14)

We denote by ed(G) the essential dimension of G, see [22].

Proposition 8.1. ed(Spin(14)) = 7.

Proof. ed(Spin(14)) ≥ 7 follows from the non-triviality of the invariant h_{7}.

It remains to show ed(Spin(14)) ≤ 7. By Corollary 7.3 it suffices show ed(H̃) ≤ 7. To describe any H̃-torsor one needs one parameter to describe a class (a) ∈ H^{1}(k, μ_{8}) = k^{∗}/(k^{∗})^{8} and 3 · 2 parameters to describe an octonion algebra

O(a_{1} + \sqrt{a}b_{1}, a_{2} + \sqrt{a}b_{2}, a_{3} + \sqrt{a}b_{3})

over k(\sqrt{a}). □

9. On the cohomology of Spin(12)

We briefly sketch a proof of ed(Spin(6, 6)) = 6.

We define H ⊂ SO(6, 6) as the image of

SL(6) ⋊ Z/2 → SO(6, 6)

given by

gϵ^{n} ↦ (\begin{matrix}g & 0 \\ 0 & (g^{t})^{−1} \end{matrix}) (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})^{n} .

Here we understand coordinates (x, y) with respect to the quadratic form ∑_{i} x_{i}y_{i}. The preimage H̃ of H in Spin(6, 6) is

H̃ = SL(6) ⋊ μ_{4}.

By the mentioned theorem of Pfister ([19, Satz 14, Zusatz] or [24]), any Spin(6, 6)- torsor admits an H̃-reduction. Since any hermitian form can be diagonalized, the map

H^{1}(k, SO(6) × μ_{4}) → H^{1}(k, H̃)

is surjective. Hence

Corollary 9.1. ed(Spin(6, 6)) ≤ 6.

We define invariants in H^{5}(Z/2), H^{6}(Z/2) by a variant of the previous method. It is based on the following facts:

Lemma 9.2. Let a ∈ k^{∗}. Then the kernel of

W (k) → W (k), x ↦ ⟨⟨a⟩⟩x

is generated by 2-dimensional forms of the form ⟨⟨N_{ℓ/k}(α)⟩⟩ with α ∈ ℓ^{∗}, ℓ = k(\sqrt{a}).

Proof. Well known… □

Lemma 9.3. Let a, b ∈ k^{∗} and let x, y ∈ W (k). If

⟨⟨a⟩⟩x = ⟨⟨b⟩⟩y,

then there exist z ∈ W (k) with

⟨⟨a⟩⟩x = ⟨⟨a⟩⟩z = ⟨⟨b⟩⟩z = ⟨⟨b⟩⟩y.

Moreover, any such z may be written as a sum of 2-dimensional forms of the form ⟨⟨N_{ℓ/k}(α)⟩⟩ with α ∈ ℓ^{∗}, ℓ = k(\sqrt{ab}).

Proof. Let φ be a quadratic form representing x, let K = k(\sqrt{b}), and suppose that ⟨⟨a⟩⟩φ_{K} is split.

Since ⟨⟨a⟩⟩φ_{K} is isotropic, one has ⟨⟨a⟩⟩φ = ⟨⟨a⟩⟩(c⟨⟨d⟩⟩ + φ′) such that ⟨⟨a, d⟩⟩_{K} is isotropic. To see this, let φ = ⟨a_{1}, …, a_{n}⟩ and let

q: V = L^{n} → k

q(λ_{1}, …, λ_{n}) = ∑_{i} a_{i}N_{L/k}(λ_{i})

with L = k(\sqrt{a}). Note that q = ⟨⟨a⟩⟩φ and that q(λv) = N_{L/k}(λ)q(v) for λ ∈ L. If q_{K} is isotropic, there exists a 2-dimensional L-submodule W of V such that q|W is isotropic over K. Next note that q|W = c⟨⟨a, d⟩⟩ for some c, d.

We may assume φ = c⟨⟨d⟩⟩ ⊥ φ′. There exists e such that

⟨⟨a, d⟩⟩ = ⟨⟨a, e⟩⟩ = ⟨⟨b, e⟩⟩

Then ⟨⟨a⟩⟩φ = c⟨⟨a, e⟩⟩ + ⟨⟨a⟩⟩φ′. The claim follows by induction on dim φ and Lemma 9.2. □

Let I_{2}(k) ⊂ I(k) be the subset of elements which are split over some quadratic extension. One defines an operation

Q: I_{2}(k) → I_{2}(k),
Q(⟨⟨a⟩⟩x) = ⟨⟨a⟩⟩λ^{2}(x).

This map is well defined by Lemma 9.2 and Lemma 9.3.

We assume that −1 is a square. Let u ∈ H^{1}(k, Spin(6, 6)). Then

q_{u} = a⟨⟨b⟩⟩(⟨⟨c, d⟩⟩′ − ⟨⟨e, f⟩⟩′)

and

Q(q_{u}) = ⟨⟨b, c, d, e, f⟩⟩

Hence we an invariant

k_{5} : H^{1}(k, Spin(6, 6)) → H^{5}(k, Z/2).

If q_{u} is isotropic, then q_{u} = a⟨⟨b, c′, d′⟩⟩ ⊥ ⟨1, −1⟩. This shows Q(q_{u}) = 0. By the same argument as in the proof of Proposition 5.2 we get an invariant

k_{6} : H^{1}(k, Spin(6, 6)) → H^{6}(k, Z/2),
k_{6}(u) = k_{5}(u) ∪ (q_{u}(v))

where q_{u}(v) is any nonzero value of q_{u}.

If q_{u} = a⟨⟨b⟩⟩(⟨⟨c, d⟩⟩′ − ⟨⟨e, f⟩⟩′), then

k_{6}(u) = (a, b, c, d, e, f).

This shows that k_{6} is nontrivial.

Corollary 9.4. ed(Spin(6, 6)) ≥ 6.

10. On the cohomology of Spin(13)

Let

q: V = k^{13} → k
q(x_{1}, …, x_{13}) = (x_{1}^{2} + x_{2}^{2} + x_{3}^{2}) − (x_{4}^{2} + x_{5}^{2} + x_{6}^{2})
− (x_{7}^{2} + x_{8}^{2} + x_{9}^{2}) + (x_{10}^{2} + x_{11}^{2} + x_{12}^{2}) − x_{13}^{2}

An element of H^{1}(k, SO(q)) is given by a 13-dimensional quadratic form q′ with

q′ ⊥ ⟨1⟩ ∈ H^{1}(k, SO(7, 7)) ⊂ I^{2} ⊂ W (k)

Let G be the subgroup of SO(q) generated by (matrix notation with respect to k^{13} = k^{3} × k^{3} × k^{3} × k^{3} × k)

U(g, h) = (\begin{matrix}g & 0 & 0 & 0 & 0 \\ 0 & g & 0 & 0 & 0 \\ 0 & 0 & h & 0 & 0 \\ 0 & 0 & 0 & h & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

and

V (α, β) = (\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & α & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & β & 0 \\ 0 & 0 & 0 & 0 & αβ \end{matrix}) , W (η) = (\begin{matrix}0 & 0 & η & 0 & 0 \\ 0 & 0 & 0 & η & 0 \\ η & 0 & 0 & 0 & 0 \\ 0 & η & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

with g, h ∈ SO(3), α, β ∈ μ_{2} and η ∈ μ_{4}. One has

G = (SO(3) × μ_{2})^{2} ⋊ μ_{4} ⊂ SO(q)

with μ_{4} acting via the projection μ_{4} → μ_{2} = Z/2 by permutation of the factors. We consider the commutative diagram

(3)  \begin{aligned}1 & ⟶ & μ_{2} & ⟶ & G̃ & ⟶^{π_{G}} & G & ⟶ & 1 \\ & & ∥ & & ↓j̃ & & ↓^{j} \\ 1 & ⟶ & μ_{2} & ⟶ & Spin(q) & ⟶^{π} & SO(q) & ⟶ & 1 \end{aligned}

where G̃ ⊂ Spin(q) is the preimage of G under the projection π: Spin(q) → SO(q) and j̃, j are the inclusions.

We describe the image

J = j_{∗}(H^{1}(k, G)) ⊂ H^{1}(k, SO(q))

Lemma 10.1. The set J consists exactly of the (isomorphism classes of) quadratic forms q′ of the type

(4)  q′ = q̃ ⊥ ⟨− det(q̃)⟩

with

(5)  q̃ = (T_{K/k})_{∗}(⟨s⟩⟨1, −λ⟩⟨−μ_{1}, −μ_{2}, μ_{1}μ_{2}⟩)

with K = k[s]/(s^{2} − b) for some b ∈ k^{×} and λ, μ_{1}, μ_{2} ∈ K^{×}.

Proof. Note that G ⊂ SO(q) leaves the subspace V′ = k^{12} ×{0} ⊂ V invariant. Let

ℓ: G → O(q|V′)
ℓ(g) = j(g)|V′

Then

j(g) = (ℓ(g), det(ℓ(g))) ∈ O(q|V′) × O(1) ⊂ O(q)

This yields the decomposition (4).

It remains to show that ℓ_{∗}(H^{1}(k, G)) ⊂ H^{1}(O(q|V′)) consists of the forms q̃ as in (5).

Elements of H^{1}(k, G) are the isomorphism classes of triples (K′, φ, φ_{1}), where K′ = k[t]/(t^{4} − b) is a Galois μ_{4}-algebra and where φ, φ_{1} are quadratic forms over the quadratic subextension K = k[s] ⊂ K′, s = t^{2} with φ of rank 3 and determinant 1 and with φ_{1} of rank 1. Let

H = (O(1) × O(1) × O(1)) ∩ SO(3) ≃ μ_{2} × μ_{2}

and

G′ = (H × μ_{2})^{2} ⋊ μ_{4} ⊂ G

Since quadratic forms (over K) can be diagonalized, it follows that H^{1}(k, G′) → H^{1}(k, G) is surjective.

The claim follows from Corollary 10.3 below. □

Lemma 10.2. Let

G′′ = (μ_{2})^{2} ⋊ μ_{4}

generated by μ_{4} and elements α, β with the relations

α^{2} = β^{2} = (αβ)^{2} = 1, ζαζ^{−1} = β, ζβζ^{−1} = α

for a generator ζ of μ_{4}.

Let

q_{0} : k^{2} → k
q_{0}(x, y) = x^{2} − y^{2}

and let

φ: G′′ → O(q_{0})

be the homomorphism with

φ(α) = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}) , φ(β) = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) , φ(η) = (\begin{matrix}0 & η \\ η & 0 \end{matrix})

with α, β ∈ μ_{2} and η ∈ μ_{4}.

Let ξ ∈ H^{1}(k, G′′) and write the corresponding Galois G′′-algebra as

E_{ξ} = k[t, s, x, y]/(t^{4} − b, s − t^{2}, x^{2} − u − sv, y^{2} − u + sv)

with b, u, v ∈ k, b ≠ 0, u^{2} − bv^{2} ≠ 0. Here the action of G′′ is given by

ζ(t) = ζt, ζ(s) = −s, ζ(x) = y, ζ(y) = x
α(t) = t, α(s) = s, α(x) = −x, α(y) = y
β(t) = t, β(s) = s, β(x) = x, β(y) = −y

Then the associated quadratic form q_{ξ} = φ_{∗}(ξ) ∈ H^{1}(k, O(q_{0})) is given by

q_{ξ} = (T_{K/k})_{∗}(⟨s⟩⟨u + sv⟩)

with K = k[s] ⊂ E_{ξ}.

Proof. One has (more or less by definition)

q_{u} = (q_{0} ⊗_{k} E)|(k^{2} ⊗_{k} E)^{G′′}

with G′′ acting on k^{2} via O(q_{0}) and on E as Galois algebra, respectively.

The claim follows from the following explicit computation (for a related consideration see Garibaldi’s Lens notes from May 2006, Example 16.5):

One finds that (k^{2} ⊗_{k} E)^{G′′} is the free k-module with basis

X = (xt, −yt), Y = (xt^{3}, yt^{3}) = (xts, yts)

For c, d ∈ k one has with λ = x^{2} = u + sv and λ̄ = y^{2} = u − sv

q_{0}(cX + dY) = (xt(c + ds))^{2} − (yt(−c + ds))^{2}
= λs(c + ds)^{2} + λ̄(−s)(c − ds)^{2}
= T_{K/k}(λs(c + ds)^{2})
□

Corollary 10.3. Let n, m ≥ 0, let U = (μ_{2})^{n} and let

Φ: U → O(1)^{m} ⊂ O(m)

be some homomorphism. Let

G′′ = U^{2} ⋊ μ_{4}

with μ_{4} acting via the projection μ_{4} → μ_{2} = Z/2 by permutation of the factors and let

φ: G′′ → O(m, m)

φ(u_{1}, u_{2}) = (\begin{matrix}Φ(u_{1}) & 0 \\ 0 & Φ(u_{2}) \end{matrix})

φ(ζ) = (\begin{matrix}0 & ζ \\ ζ & 0 \end{matrix})

for (u_{1}, u_{2}) ∈ U^{2} and a generator ζ of μ_{4}.

Let ξ ∈ H^{1}(k, G′′) and write the corresponding Galois G′′-algebra as

E_{ξ} = k[t, s, x_{i}, y_{i}; i = 1, …, n]/(t^{4} − b, s − t^{2}, x_{i}^{2} − u_{i} − sv_{i}, y_{i}^{2} − u_{i} + sv_{i})

with b, u_{i}, v_{i} ∈ k, b ≠ 0, u_{i}^{2} − bv_{i}^{2} ≠ 0 (with obvious G′′ action, see Lemma 10.2). Then the associated quadratic form q_{ξ} = φ_{∗}(ξ) ∈ H^{1}(k, O(m, m)) is given by

q_{ξ} = (T_{K/k})_{∗}(⟨s⟩⟨μ_{1}, …, μ_{m}⟩)

with K = k[s] ⊂ E_{ξ} and with

μ_{j} = ∏_{i=1}^{n} λ_{i}^{Φ_{ij}} ∈ K, j = 1, …, m

where

λ_{i} = u_{i} + sv_{i}

and where Φ_{ij} = 0, 1 is defined by

Φ(α_{1}, …, α_{n}) = (∏_{i=1}^{n} α_{i}^{Φ_{ij}})_{j=1,...,m}

Proof. One easily reduces to the case m = 1, n = 1 and Φ = id, which is treated in Lemma 10.2. □

Proposition 10.4. The natural map j̃_{∗}: H^{1}(G̃) → H^{1}(Spin(q)) is surjective.

Proof. Let u ∈ H^{1}(k, Spin(q)) and let q_{u} ∈ H^{1}(k, SO(q)) be the associated quadratic form. Then

q_{u} ⊥ ⟨1⟩ ∈ I^{3}

By the results on 14-dimensional forms in I^{3} one has

q_{u} ⊥ ⟨1⟩ = (T_{K/k})_{∗}(⟨s⟩φ′)

with K = k[s]/(s^{2} − b) for some b ∈ k^{×} and with φ a 3-fold Pfister form over K (and with φ = ⟨1⟩ ⊥ φ′). Since the left hand side represents 1, there exists a value −λ of φ′ with T_{K/k}(−sλ) = 1. As for any (invertible) value −λ of φ′, one has φ = ⟨⟨λ, μ_{1}, μ_{2}⟩⟩ for some μ_{1}, μ_{2} ∈ K^{×}. Note that

(T_{K/k})_{∗}(⟨−sλ⟩) = ⟨1, −N_{K/k}(λ)⟩

Thus

q_{u} = (T_{K/k})_{∗}(⟨s⟩⟨⟨λ⟩⟩⟨⟨μ_{1}, μ_{2}⟩⟩′) ⊥ ⟨−N_{K/k}(λ)⟩

By Lemma 10.1 it follows that q_{u} ∈ J. A diagram chase (see diagram (3)) involving the coboundary maps H^{1}(k, G), H^{1}(k, SO(q)) → H^{2}(k, μ_{2}) shows that there exists ũ ∈ H^{1}(k, G̃) such that j̃(ũ), u ∈ H^{1}(k, Spin(q)) have the same image in H^{1}(k, SO(q)). Another diagram chase shows that we can arrange j̃(ũ) = u. □

We next compute G̃ ⊂ Spin(q) ⊂ C(q) inside the Clifford algebra. Let e_{1}, …, e_{13} be the standard base of V.

Let ζ be a primitive 4-th root of unity.

For v, w ∈ V with q(v) = 1, q(w) = −1 and v ⊥ w let

ω(v, w) = \frac{1 + ζwv}{\sqrt{2}}

Then ω(v, w)ω(w, v) = 1 and therefore ω(v, w) ∈ Spin(q). Moreover ω(v, w)^{2} = ζwv and ω(v, w)^{4} = −1. Furthermore ω(v, w)v = vω(v, w)^{−1} and ω(v, w)w = wω(v, w)^{−1}. Also ω(v, w)vω(v, w)^{−1} = ζw and ω(v, w)wω(v, w)^{−1} = ζv.

Consider the element

ω = ω(e_{1}, e_{7})ω(e_{2}, e_{8})ω(e_{3}, e_{9})ω(e_{10}, e_{4})ω(e_{11}, e_{5})ω(e_{12}, e_{6}) ∈ Spin(q)

Its image in SO(q) is π(ω) = W (ζ). Moreover

ω^{4} = 1

Next let

α̃ = e_{4}e_{5}e_{6}e_{13}, β̃ = −ζe_{10}e_{11}e_{12}e_{13}

Both elements are in Spin(q) and π(α̃) = V (−1, 1) and π(β̃) = V (1, −1). Moreover

α̃^{2} = 1
β̃^{2} = 1
α̃β̃ = −β̃α̃
(α̃β̃)^{2} = −1
ωα̃ω^{−1} = β̃
ωβ̃ω^{−1} = −α̃
ωα̃β̃ω^{−1} = α̃β̃
α̃ωα̃^{−1} = α̃β̃ω
α̃ω^{2}α̃^{−1} = −ω^{2}

Let H be the subgroup generated by ω and α̃. Then β̃ ∈ H and

H = (μ_{4} × μ_{4}) ⋊ μ_{2}

with the μ_{2} generated by α̃ and μ_{4} × μ_{4} generated by ω and α̃ωα̃^{−1}.

Note further that the diagonal embedding SO(3) → SO(3, 3) lifts to Spin(3, 3). Thus the connected component of G lifts (uniquely) to Spin(q). This yields:

Lemma 10.5. One has

G̃ ≃ (SO(3))^{2} ⋊_{φ} H

where H acts by permutation of the factors via φ: H → Z/2, φ(α̃) = 0, φ(ω) = 1.

(I was surprised about the simple structure of H. There ought to be a better approach to the subgroup G̃ of Spin(13) than just by a computation starting from G.)

Proposition 10.6. ed(G̃) ≤ 6

Proof. Elements of H^{1}(k, H) are given by Galois H-algebras which can be written as

L = k[z, x, y]/(z^{2} − a, x^{4} − u − sv, y^{4} − u + sv)

with a, u, v ∈ k, a ≠ 0, u^{2}−av^{2} ≠ 0. For the generic case we may assume v ≠ 0 and replace s by sv and a by av^{2}. Then v = 1. Therefore H-torsors are parameterized by a and u and we have ed(H) ≤ 2.

Thus an element of H^{1}(k, G̃) is given by a Galois H-algebra

L = k[z, x, y]/(z^{2} − a, x^{4} − u − s, y^{4} − u + s)

and a quadratic form of rank 3 and determinant 1 over K = k[t] ⊂ L with t = (xy)^{2} and t^{2} = u^{2} − a. Thus ed(G̃) ≤ ed(H) + 2 · 2. □

Corollary 10.7. ed(Spin(q)) ≤ 6

Proof. This is clear from Proposition 10.4 and Proposition 10.6. □

11. The essential dimension of split Spin(n) for n ≤ 14

Let Spin_{n} denote a split form of Spin(n).

Theorem.

ed(Spin_{n}) = 0 for n ≤ 6,
ed(Spin_{7}) = 4,
ed(Spin_{8}) = 5,
ed(Spin_{9}) = 5,
ed(Spin_{10}) = 4,
ed(Spin_{11}) = 5,
ed(Spin_{12}) = 6,
ed(Spin_{13}) = 6,
ed(Spin_{14}) = 7.

Proof. (Sketch) The cases n = 12, 14 have been just considered. It is not difficult to extend our considerations to the case n = 11.

As for n = 13: By corollary 10.7 one has ed(Spin_{13}) ≤ 6. The invariant h_{6} restricted to Spin_{13} is nontrivial, for example for

q ⊥ ⟨1⟩ = b_{1}(⟨⟨a_{1}, a_{2}, a_{3}⟩⟩′ − ⟨⟨b_{1}, b_{2}, b_{3}⟩⟩′)

Hence ed(Spin_{13}) ≥ 6.

For n = 7, 10 one uses that any Spin_{n}-torsor admits a reduction to G_{2} ×μ_{2} resp. to G_{2} × μ_{4}. For n = 8, 9 one may use the fact that

Spin_{8} → Spin_{9} → F_{4}

induce surjections on H^{1} at the prime 2 and Serre’s H^{5}(Z/2)-invariant for F_{4}, cf. [27, III. Annexe, § 3.4] or [28, III. Appendix 2, 3.4] and [14, § 40], [23]. For n ≤ 6 note that any n-dimensional quadratic form with trivial e_{1}-, e_{2}-invariants is split. □

References

[1] J. K. Arason, Cohomologische Invarianten quadratischer Formen, J. Algebra 36 (1975), no. 3, 448–491.

[2] , A proof of Merkurjev’s theorem, Quadratic and Hermitian forms (Hamilton, Ont., 1983) (C. R. Riehm and I. Hambleton, eds.), CMS Conf. Proc., vol. 4, Amer. Math. Soc., Providence, R.I., 1984, pp. 121–130.

[3] S. Eilenberg and S. Mac Lane, On the groups H(π, n). II. Methods of computation, Ann. of Math. (2) 60 (1954), 49–139.

[4] , On the groups H(π, n). III, Ann. of Math. (2) 60 (1954), 513–557.

[5] R. Elman and T. Y. Lam, Pfister forms and K-theory of fields, J. Algebra 23 (1972), 181–213.

[6] L. Evens, A generalization of the transfer map in the cohomology of groups, Trans. Amer. Math. Soc. 108 (1963), 54–65.

[7] V. Gatti and E. Viniberghi, Spinors of 13-dimensional space, Adv. in Math. 30 (1978), no. 2, 137–155.

[8] D. W. Hoffmann and J.-P. Tignol, On 14-dimensional quadratic forms in I^{3}, 8-dimensional forms in I^{2}, and the common value property, Doc. Math. 3 (1998), 189–214 (electronic).

[9] J. Igusa, On a certain class of prehomogeneous vector spaces, J. Pure Appl. Algebra 47 (1987), no. 3, 265–282.

[10] O. Izhboldin and N. Karpenko, Some new examples in the theory of quadratic forms, Preprint, 1997, http://wwwmath.uni-muenster.de/math/u/karpenko/Publications/.

[11] B. Jacob and M. Rost, Degree four cohomological invariants for quadratic forms, Invent. Math. 96 (1989), no. 3, 551–570.

[12] B. Kahn, Classes de Stiefel-Whitney de formes quadratiques et de représentations galoisiennes réelles, Invent. Math. 78 (1984), no. 2, 223–256.

[13] M. Karoubi, K-theory, Springer-Verlag, Berlin, 1978, An introduction, Grundlehren der Mathematischen Wissenschaften, Band 226.

[14] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998, With a preface in French by J. Tits.

[15] T. Y. Lam, The algebraic theory of quadratic forms, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1980, Revised second printing, Mathematics Lecture Note Series.

[16] A. S. Merkurjev, On the norm residue symbol of degree 2, Dokl. Akad. Nauk SSSR 261 (1981), no. 3, 542–547 (Russian), [Soviet Math. Dokl. 24 (1981), no. 3, (1982), 546–551].

[17] A. S. Merkurjev and A. A. Suslin, Norm residue homomorphism of degree three, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 2, 339–356 (Russian), [Math. USSR Izv. 36 (1991), no. 2, 349–367], also: LOMI-preprint (1986).

[18] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318–344.

[19] A. Pfister, Quadratische Formen in beliebigen Körpern, Invent. Math. 1 (1966), 116–132.

[20] , Quadratic forms with applications to algebraic geometry and topology, Cambridge University Press, Cambridge, 1995.

[21] V. L. Popov, A classification of spinors of dimension fourteen, Trudy Moskov. Mat. Obshch. 37 (1978), 173–217, 270, [Trans. Mosc. Math. Soc. (1980), Issue 1, 181–232].

[22] Z. Reichstein, On the notion of essential dimension for algebraic groups, preprint, 1998, http://ucs.orst.edu/˜ reichstz/pub.html.

[23] M. Rost, A descent property for Pfister forms, preprint, 1998, http://www.physik.uniregensburg.de/˜ rom03516/descent.html, submitted to Journal of the Ramanujan Mathematical Society.

[24] , On 14-dimensional quadratic forms, their spinors, and the difference of two octonion algebras, notes, 1994, http://www.physik.uni-regensburg.de/˜ rom03516/14-dim-abstract.html.

[25] , On Hilbert Satz 90 for K_{3} for quadratic extensions, preprint, 1988, http://www. physik.uni-regensburg.de/˜ rom03516/K3-88.html.

[26] W. Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften, vol. 270, Springer-Verlag, Berlin, 1985.

[27] J.-P. Serre, Cohomologie galoisienne, fifth ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994.

[28] , Galois cohomology, Springer-Verlag, Berlin, 1997, Translated from the French by Patrick Ion and revised by the author.

[29] J. M. Shapiro, A Riemann-Roch type theorem for the Witt and Milnor rings of a field, J. Pure Appl. Algebra 15 (1979), no. 3, 293–304.

[30] V. Voevodsky, The Milnor conjecture, preprint, 1996, Max-Planck-Institute for Mathematics, Bonn, http://www.math.uiuc.edu/K-theory/0170.

[31] A. R. Wadsworth, Merkurjev’s elementary proof of Merkurjev’s theorem, Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) (S. J. Bloch, R. K. Dennis, E. M. Friedlander, and M. R. Stein, eds.), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, R.I., 1986, pp. 741–776.

[32] J. H. C. Whitehead, A certain exact sequence, Ann. of Math. (2) 52 (1950), 51–110.

[33] , On the theory of obstructions, Ann. of Math. (2) 54 (1951), 68–84.

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

E-mail address: rost@math.uni-bielefeld.de
URL: http://www.math.uni-bielefeld.de/~rost