UNRAMIFIED COHOMOLOGY OF QUADRICS, I

BRUNO KAHN, MARKUS ROST, AND R. SUJATHA

Abstract. Given a quadric X over a field F of characteristic ≠ 2, we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 Galois cohomology of F to the unramified mod 2 cohomology of F (X), when dim X > 10 and in several smaller-dimensional cases. Applications of these results to real quadrics and to the unramified Witt ring are given.

Contents

Introduction 1 Part I. Collected results on quadrics 8

1. Chow groups 8 2. K- and H-cohomology 10

Part II. Ker η_{2}^{4} for higher-dimensional quadrics 14 3. Generalities on Ker η_{2}^{i} 14 4. 6-dimensional spinors 15 5. Ker η_{2}^{4} and H^{2}(X, K_{3}) 17 6. Computations for dim X = 3 19

Part III. Coker η_{2}^{i} for higher-dimensional quadrics 21 7. A relationship between Ker η_{2}^{i} and Coker η_{2}^{i} 21 8. Constructing some maps 22 9. The case i = 2: Proof of theorem 4 28 10. The case i = 3: Proof of theorem 5 28 11. The case i = 4 31 12. Proof of theorems 6 and 7 33 Appendix A. Quadrics of dimension 0 and 1 35 Appendix B. Application to real quadrics 36 Appendix C. The unramified Witt ring 37 References 39

Introduction

Let F be a field, let X be a smooth, proper, and irreducible variety over F and let n be prime to char F. Consider the restriction map

H^{i}(F, μ_{n}^{⊗(i−1)}) → H^{i}(F (X), μ_{n}^{⊗(i−1)})

in Galois cohomology.

The kernel of this map has been studied in various cases, for unirational and notably for homogeneous varieties. In the case of the Brauer group (i = 2) this goes back to Witt [56] for conics and Amitsur [2] for Severi-Brauer varieties. For i = 3 the kernel has been determined by Arason [1] for quadrics. Other computations for i = 3 and homogeneous varieties (Suslin [49], Peyre [39, 40], Merkurjev [29], Esnault et al. [14]) rely on the Merkurjev-Suslin theorem (see below). The Merkurjev-Suslin/Rost theorem makes it possible to study also the case i = 4 and n = 2. Here certain quadrics have been considered by Jacob-Rost [16], Szyjewski [51], Rost and Merkurjev [27].

Date: November 12, 1997.

Unramified cohomology. The image of this map sits inside the unramified cohomology group

H_{nr}^{i}(F (X)/F, μ_{n}^{⊗(i−1)}) := Ker((H^{i}F (X), μ_{n}^{⊗(i−1)}) ⟶^{(∂_{x})} ⨁_{x∈X^{(1)}} H^{i−1}(F_{x}, μ_{n}^{⊗(i−2)})).

Here x runs through the codimension 1 points of X, and

∂_{x}: H^{i}(F (X), μ_{n}^{⊗(i−1)}) → H^{i−1}(F_{x}, μ_{n}^{⊗(i−2)})

is the residue map associated to the discrete valuation ring O_{X,x}.

We study the cokernel of the corresponding map

η_{n}^{i}: H^{i}(F, μ_{n}^{⊗(i−1)}) → H_{nr}^{i}(F (X)/F, μ_{n}^{⊗(i−1)}).

The group H_{nr}^{i}(F (X)/F, μ_{n}^{⊗(i−1)}) is a birational invariant and does not change when X is replaced by X × P^{1}. Moreover for the base field extension from F to the rational function field K = F (t) one has an exact sequence

(∗) 0 → H_{nr}^{i}(F (X)/F, μ_{n}^{⊗(i−1)}) → H_{nr}^{i}(K(X)/K, μ_{n}^{⊗(i−1)}) →

→ ⨁_{y∈A_{(0)}^{1}} H_{nr}^{i−1}(F_{y}(X)/F_{y}, μ_{n}^{⊗(i−2)}) → 0.

(cf. [7, §3.3, th. 4.1.1, 4.1.5], [42, (7.3),(12.10)].)

Saltman [43, 44], Bogomolov [3], Colliot Thélène-Ojanguren [8], and Peyre [38], inter alia, have studied the unramified cohomology of certain unirational varieties over an algebraically closed F (there are many more references available on this case, for which the reader can consult for instance [7, § 4]). The unramified cohomology of real varieties has been studied in high cohomological degree by Colliot Thélène-Parimala [9] and Scheiderer [46]. For quadrics X and i ≤ 3, the cokernel of η_{2}^{i} was studied by Colliot Thélène-Sujatha [10] (F = R), Peyre [39] (F arbitrary, X a conic), Sujatha [48] (F arbitrary, X defined by a neighbour of a 3-fold Pfister form) and Kahn [17] (F arbitrary, dim X > 2). It follows from [56, 1, 17] that η_{2}^{2} (resp. η_{2}^{3}) is bijective for dim X > 2 (resp. dim X > 6) and has kernel and cokernel of order at most 2 for all X (resp. for dim X > 2).

In this paper, we show that η_{2}^{4} is bijective if X is a quadric of dimension > 14, and that |Ker η_{2}^{4}| ≤ 2, |Coker η_{2}^{4}| ≤ 4 for dim X > 4. To our knowledge, this is the first non-trivial instance where unramified cohomology of degree 4 is determined over an arbitrary field.

The Milnor conjecture. Before a more detailed discussion we recall the Bloch-Kato conjecture, the Milnor conjecture, and the related conjecture about existence of cohomological invariants for the graded Witt ring.

The Bloch-Kato conjecture predicts that if n is invertible in F, then the norm residue homomorphism h_{i} of degree i,

h_{i}: K_{i}^{M}F/n → H^{i}(F, μ_{n}^{⊗i})

is an isomorphism for all i ≥ 0. Here K_{i}^{M}F denotes the i-th Milnor K-group of F [34]. If n is a power of 2, this is called the Milnor conjecture. The bijectivity of h_{i} holds trivially for i = 0 and by Kummer theory for i = 1. The Merkurjev-Suslin theorem states that h_{2} is an isomorphism [31] and Merkurjev-Suslin/Rost [32] proved the Milnor conjecture for i = 3. The second author announced a proof of the Milnor conjecture for i = 4. Voevodsky [54] recently released a proof of the Milnor conjecture for all i. In the present paper we use only the Milnor conjecture for i ≤ 3, except in appendix C for an application to the unramified Witt group which relies on the Milnor conjecture for i = 4.

Let W (F) denote the Witt ring of F and let (I^{i}F)_{i≥0} be the filtration on W (F) given by the higher powers of the fundamental ideal I(F) ([26, 45]). For a_{1}, a_{2}, …, a_{i} ∈ F^{∗} let ⟨⟨a_{1}, …, a_{i}⟩⟩ denote the i-fold Pfister form ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{i}⟩ (ibid.). A quadratic form q is a neighbour of an i-fold Pfister form φ if dim q > 2^{i−1} and q is similar to a subform of φ. For simplicity, write

H^{i}F = H^{i}(F, Z/2).

We define a symbol to be an element of H^{i}F of the form (a_{1}, …, a_{i}), where a_{j} ∈ F^{∗}, (a_{j}) is the class of a_{j} in H^{1}F by Kummer theory and (a_{1}, …, a_{i}) := (a_{1})·. . .·(a_{i}) is the cup-product. Let P_{i}(F) denote the set of classes of i-fold Pfister forms in W (F). There are maps

e^{i}: P_{i}(F) → H^{i}F

which send the class of an i-fold Pfister form ⟨⟨a_{1}, …, a_{i}⟩⟩ to the corresponding cohomological symbol (a_{1}, …, a_{i}). It is conjectured that these maps extend to well-defined isomorphisms

\overline{e}^{i}: I^{i}F/I^{i+1}F → H^{i}F

for all i ≥ 0. For i ≤ 2, it is easy to see that these maps are well-defined, being the rank mod 2, signed discriminant, and the Clifford invariant, respectively, for i = 0, 1, 2. Arason [1] proved that \overline{e}^{3} is well-defined and Jacob-Rost/Szyjewski [16, 51] proved that \overline{e}^{4} is well-defined. Granting the existence of \overline{e}^{i}, the Milnor conjecture in degree i implies the bijectivity of \overline{e}^{i}. Voevodsky has recently announced a proof of the existence and bijectivity of the homomorphisms \overline{e}^{i} for all i, jointly with Orlov and Vishik [35].

We now describe our results. Let q be a (regular) quadratic form over a field F with char F ≠ 2 and let X be the associated projective quadric. Unless mentioned otherwise, we assume dim q ≥ 3.

Note that if q is isotropic, then the extension F (X)/F is purely transcendental and therefore η_{2}^{i} is bijective for all i (cf. proposition 2.5).

Kernel of η_{2}^{i}. Here the major result is

Theorem 1 (cf. theorem 6.3). Let i ≤ 4. If dim q = 2^{i−2} + 1, then every element in Ker η_{2}^{i} is a symbol.

For i = 3 and, in case of a Pfister neighbour, for i = 4, this theorem follows easily from the computations of Ker η_{2}^{i} by Arason [1], Jacob-Rost [16], and Szyjewski [51, 6.1].

The proof of theorem 1 for i = 4 in general is an extension of the discussions in [16, 27] to arbitrary 5-dimensional quadratic forms. The method used there provides a very general description of Ker η_{2}^{4} for quadratic forms of dimension ≥ 5 (see [27]) as follows:

Ker η_{2}^{4} = Ker(H^{2}(X, K_{3}) → H^{2}(\overline{X}, K_{3})). (∗∗)

For an Albert form with corresponding biquaternion algebra A (see below) this yields an isomorphism SK_{1}(A) = Ker η_{2}^{4} (ibid.).

For a five dimensional form q, let q̂ = q ⊥ ⟨−d⟩, d = det(q) be the attached Albert form. In this case (∗∗) yields a surjective homomorphism

O(q̂)/[O(q̂), O(q̂)]O(q) → → Ker η_{2}^{4}.

Theorem 1 follows from a careful computation of this map.

Using this theorem and some quadratic form theory, we show in section 3:

Corollary 2. Let i ≤ 4. Then

(1) A symbol (a_{1}, …, a_{i}) ≠ 0 lies in Ker η_{2}^{i} if and only if q is similar to a subform of ⟨⟨a_{1}, …, a_{i}⟩⟩.

(2) Ker η_{2}^{i} consists of symbols for dim q > 2^{i−2}.

(3) |Ker η_{2}^{i}| ≤ 2 for dim q > 2^{i−1}.

(4) Ker η_{2}^{i} = 0 for dim q > 2^{i}.

Note that the bound dim q > 2^{i−2} in (2) is optimal: if q is an (i − 2)-fold Pfister form ⟨⟨a_{1}, …, a_{i−2}⟩⟩, then Ker η_{2}^{i} contains H^{2}F · (a_{1}, …, a_{i−2}), which obviously contains non-symbolic elements in general.

One may expect that theorem 1 also holds for higher degree i. Then corollary 2 would hold accordingly (using the existence and bijectivity of the \overline{e}^{i}).

Cokernel of η_{2}^{i}. Results on Coker η_{2}^{i} are best understood in terms of Coker η^{i}, where η^{i} is the restriction map

η^{i}: H^{i}(F, Q/Z(i − 1)) ⟶ H_{nr}^{i}(K/F, Q/Z(i − 1)).

Here Q/Z(i − 1) := \underrightarrow{lim} μ_{m}^{⊗(i−1)}, where m runs through all integers prime to char F. A consequence of Milnor’s conjecture in degree i − 1 is that H^{i}F → H^{i}(F, Q/Z(i − 1)) is injective; hence Ker η_{2}^{i} = Ker η^{i} since the latter group is killed by 2 (cf. lemma 7.2 a)). To relate Coker η_{2}^{i} and Coker η^{i}, define

(Ker η_{2}^{i})_{0} = { α ∈ Ker η_{2}^{i} | (−1) · α = 0 ∈ H^{i+1}F }.

We prove in section 7:

Proposition 3 (cf. proposition 7.4). For any quadric X and i ≤ 4, there is an exact sequence

0 → (Ker η_{2}^{i})_{0} ⟶ Coker η_{2}^{i} → Coker η^{i}.

This way, information on Ker η_{2}^{i} and Coker η^{i} immediately yields information on Coker η_{2}^{i}.

To describe our picture about Coker η^{i} we formulate:

Conjecture.

(1) Coker η^{i} = 0 if q is Pfister neighbour.

(2) Coker η^{i} = 0 for dim q ≤ 5.

(3) Coker η^{i} is finite for dim q > 6 · 2^{i−4}.

(4) Coker η^{i} = 0 for dim q > 6 · 2^{i−3}.

In appendix A we show Coker η^{i} = 0 for dim q = 3 and—after an appropriate modification of the target group—also for dim q = 2. Here the arguments rely on the Milnor conjecture in degree i − 2 (resp. i − 1).

For i ≤ 2 one has (cf. section 9):

Theorem 4. Coker η^{i} = 0 for i ≤ 2.

For i = 3 there appears a peak in the story at dim q = 6. For the reader’s convenience, we recall the various types of 6-dimensional quadratic forms. Let d_{±}q be the signed discriminant and let C(q) be the Clifford algebra of q [26]. The pure subform ⟨⟨a_{1}, …, a_{i}⟩⟩′ of the Pfister form ⟨⟨a_{1}, …, a_{i}⟩⟩ is defined by

⟨⟨a_{1}, …, a_{i}⟩⟩ = ⟨1⟩ ⊥ ⟨⟨a_{1}, …, a_{i}⟩⟩′.

There are four types of anisotropic quadratic forms q of dimension 6:

• d = d_{±}q ≠ 1, q_{F (\sqrt{d})} is anisotropic. For convenience, we shall call such forms virtual Albert forms.

• d_{±}q = 1, i.e., q ∈ I^{2}F. Such forms are called (anisotropic) Albert forms. An Albert form is similar to a form of type

⟨⟨a, b⟩⟩′ ⊥ −⟨⟨e, f⟩⟩′ a, b, e, f ∈ F^{∗}.

To an Albert form there is attached the biquaternion algebra A = (a, b) ⊗ (e, f) which is the central simple algebra with C(q) ≃ M_{2}(A). The form q is isotropic if and only if A is not a skewfield.

• d = d_{±}q ≠ 1, q_{F (\sqrt{d})} is hyperbolic. These forms are the 6-dimensional Pfister neighbours (cf. [24, p. 10]).

• d = d_{±}q ≠ 1, q_{F (\sqrt{d})} is isotropic without being hyperbolic. Such forms are of type aτ ⊥ b⟨1, −d⟩, where a, b ∈ F^{∗} and τ is a 2-fold Pfister form.

Now for i = 3 our result is (cf. section 10):

Theorem 5. Coker η^{3} = 0, except if q is an anisotropic Albert form, for which Coker η^{3} ≃ Z/2.

This result completes those of [17].

The main results about Coker η^{4} are:

Theorem 6 (cf. section 12).

(1) For dim q > 6 there is an exact sequence

0 → Coker η^{4} → (CH^{3}X)_{torsion} → (CH^{1}X ⊗ H^{3}(F, Q/Z(2))) ⊕ H^{5}(F, Q/Z(3)).

Here CH^{p}X denotes the p-th Chow group of X.

(2) |Coker η^{4}| ≤ 2 if dim q ≠ 4, 5, 6.

(3) Coker η^{4} = 0 if • q is a Pfister neighbour

• dim q = 8 and d_{±}q = 1

• dim q > 12.

In the proof of theorem 6 we use Karpenko’s results on CH^{3} of a quadric. The main computational tool is, as in [17], to mix the use of the Hochschild-Serre and Bloch-Ogus [4] spectral sequences, here with divisible coefficients. We use in particular the fact that these two spectral sequences are modules over the Galois cohomology of F. Divisible coefficients have been used also in [23, 6, 39, 40]. It is clear that these methods can also be applied to other homogeneous varieties.

Theorem 7. If q is an anisotropic Albert form or a virtual Albert form over F, then Coker η^{4} is infinite for q_{F (t)}.

This theorem is an easy consequence of theorem 5 and the exact sequence (∗). From corollary 2, proposition 3, and theorem 6 one easily deduces:

Corollary 8.

(1) Coker η_{2}^{2} = 0, unless q is a neighbour of a 2-fold Pfister form ⟨⟨a, b⟩⟩. In this case, it is 0 or Z/2 according as (−1, a, b) ∈ H^{3}F is nonzero or not.

(2) If q is an anisotropic Albert form, (Ker η_{2}^{3})_{0} = 0 and Coker η_{2}^{3} ≃ Z/2. In all other cases, Coker η_{2}^{3} ≃ (Ker η_{2}^{3})_{0}. In particular:

a) If q is a neighbour of a 3-fold Pfister form ⟨⟨a, b, c⟩⟩, then Coker η_{2}^{3} = 0 or Z/2 according as (−1, a, b, c) ∈ H^{4}F is nonzero or not.

b) If q is not an Albert form and does not embed into a 3-fold Pfister form

(up to similarity), then Coker η_{2}^{3} = 0.

(3) a) If dim q > 12 and q is not a neighbour of a 4-fold Pfister form, then

Coker η_{2}^{4} = 0.

b) If q is a neighbour of a 4-fold Pfister form ⟨⟨a, b, c, d⟩⟩, then Coker η_{2}^{4} = 0 or Z/2 according as (−1, a, b, c, d) ∈ H^{5}F is nonzero or not.

c) For dim q > 8, |Coker η_{2}^{4}| ≤ 4.

In the case F = R, these results immediately imply that Coker η_{2}^{i} = 0 for all quadrics when i ≤ 4. In appendix B, we also get partial results on the Hcohomology of such quadrics.

The unramified Witt ring. As in [17, corollary], we wish to apply the above results to the unramified Witt ring. Denote by W_{nr}(K/F) the unramified Witt ring of a finitely generated extension K of F (cf. [8]) and, for i ≥ 0, by I_{nr}^{i}(K/F) the intersection I^{i}K ∩ W_{nr}(K/F). Let

χ^{i}: I^{i}F/I^{i+1}F → I_{nr}^{i}(K/F)/I_{nr}^{i+1}(K/F)

be the natural map analogous to η_{2}^{i} and η^{i}. The homomorphism ē^{i} induces a homomorphism

I_{nr}^{i}K/I_{nr}^{i+1}K → H_{nr}^{i}K

and hence a homomorphism

Coker χ^{i} → Coker η_{2}^{i}.

Using the Milnor conjecture in degree i = 4, we find:

Theorem 9. Let i ≤ 4. For any quadric X, the composition

Coker χ^{i} → Coker η_{2}^{i} → Coker η^{i}

is injective.

From corollary 2, theorem 6, and theorem 9 we deduce:

Corollary 10.

(1) The map W (F)/I^{3}F → W_{nr}(F (X)/F)/I_{nr}^{3}(F (X)/F) is surjective. It is bijective for dim q > 4.

(2) The map W (F)/I^{4}F → W_{nr}(F (X)/F)/I_{nr}^{4}(F (X)/F) is surjective, except if q is an anisotropic Albert form. In this case, its cokernel is ≃ Z/2 (by [17, remark 3, p. 249]). It is bijective for dim q > 8.

(3) The map W (F)/I^{5}F → W_{nr}(F (X)/F)/I_{nr}^{5}(F (X)/F) is surjective in the fol-

lowing cases • q is a Pfister neighbour.

• dim q = 8 and d_{±}q = 1.

• dim q > 12. It is bijective for dim q > 16.

Remark. Voevodsky’s work yields also the computation of Ker η_{2}^{i} for i-fold Pfister forms. A consequence of [54] and [35] is the validity of proposition 3 and of theorem 9 for all i.

However these announcements do not imply a generalization of theorem 1 to higher degree or a proof of the conjecture in an obvious way.

In a sequel to this paper, we plan to complete the present results by determining Ker η^{4} and Coker η^{4} in the cases pending here, and by refining results on real quadrics. In particular, we shall show that Ker η_{2}^{4} is always generated by its symbols. For this, we shall use a method different from that of this paper, based on the following result:

Theorem ([19]). Let X be a quadric and let C be a conic such that there exist a morphism C → X. Let D be the quaternion algebra associated to C. Then there is a canonical isomorphism

Ker η_{2}^{4} ⟶^{∼} \frac{H^{0}(X, K_{2}^{D})}{K_{2}D}.

Here H^{0}(X, K_{2}^{D}) is the intersection of the kernels of the residue maps

K_{2}(D ⊗_{F} F (X)) → K_{1}(D ⊗_{F} F (x)),

where x runs through the points of codimension 1 in X.

This paper is organized as follows. In part I we recall some known results on Chow groups, in particular of Karpenko, and of K- and H-cohomology of quadrics, supplying proofs when they don’t appear in the literature. Part II is devoted to the proof of theorem 1 and corollary 2, and part III to the proof of proposition 3, theorem 5, and theorem 6. In appendix A we consider Coker η^{i} for quadrics of dimension 0 and 1. In appendix B we consider the case F = R. In appendix C we prove theorem 9.

Notation and conventions. For any scheme X, H^{i}X denotes the i-th étale cohomology group of X with coefficients Z/2. We use the abbreviation H^{i}(X, j) for H_{ét}^{i}(X, Q/Z(j)) and similarly H^{p}(X, H^{q}(j)) for the E_{2}-term of the coniveau spectral sequence with divisible coefficients. If X is smooth over a field F, we denote by CH^{i}X its i-th Chow group, the group of cycles of codimension i over X modulo rational equivalence. We denote by cl_{X}^{i} (resp. cl′_{X}^{i}) the cycle map with Z/2 coefficients CH^{i}X/2 → H^{2i}X (resp. with divisible coefficients CH^{i}X ⊗ Q/Z → H^{2i}(X, Q/Z(i))). We set \overline{X} = X ×_{F} F_{s}, where F_{s} is a separable closure of F. The word “quadric” means smooth, projective and absolutely irreducible quadric. If X is an even-dimensional quadric over F (assumed of characteristic ≠ 2), we denote by d(X) ∈ F^{∗}/F^{∗2} the (signed) discriminant d_{±}q of any quadratic form q defining X. If φ: A → B is a homomorphism between two abelian groups, we sometimes

allow ourselves to write B/A or \frac{B}{A} for Coker φ, even if φ is not injective.

Acknowledgements. The first author wishes to thank Andrei Suslin for a conversation which catalyzed the use of divisible coefficients in this paper, and gratefully acknowledges the hospitality of the Tata Institute of Fundamental Research, where this work was completed. The third author acknowledges the partial support of a NATO grant.

Part I. Collected results on quadrics

In this part, we state general results on Chow groups, H- and K-cohomology of quadrics, most of which are “classical”. The reader is advised to skip it and to use it only for reference purposes. We merely give references when published proofs are available, and provide proofs otherwise.

We say that a quadric X is anisotropic (resp. isotropic, split) if a quadratic form defining X is anisotropic (resp. is isotropic, has maximal index).

1. Chow groups

The following proposition is mainly a reformulation of results of N. Karpenko [20].

1.1. Proposition. Let X be a quadric over a field F of characteristic ≠ 2. Let h denote the class of a hyperplane section in CH^{1}X.

a) For any i ≥ 0, the natural map CH^{i}X → CH^{i}\overline{X} has kernel the torsion subgroup of CH^{i}X (which is 2-primary).

b) For i < dim X/2, CH^{i}X/ torsion is generated by h^{i}; CH^{i}X → CH^{i}\overline{X} is surjective.

c) For i > dim X/2, CH^{i}X/ torsion is generated by h^{i} if i ≤ dim X−n and by the class of a rational linear subspace of codimension i if i > dim X − n. Here n is the Witt index of a quadratic form defining X. Coker(CH^{i}X → CH^{i}\overline{X}) ≃ Z/2 if i ≤ dim X − n and CH^{i}X → CH^{i}\overline{X} is surjective if i > dim X − n.

d) For i = dim X/2 and d(X) = 1 one has

CH^{i}X/ torsion ≃ Z ⊕ Z,

with basis (h^{i}, 2^{t}l), where t ≥ 0 and l is the class of some maximal linear subspace of \overline{X}. Moreover Coker(CH^{i}X → CH^{i}\overline{X}) ≃ Z/2^{t}. We have t = 1 if i = 1 and X is anisotropic, and t = 2 if i = 2 and X is anisotropic.

e) For i = dim X/2 and d(X) ≠ 1 one has

CH^{i}X/ torsion ≃ Z

generated by h^{i}. The map CH^{i}X → (CH^{i}\overline{X})^{G_{F}} is surjective, where G_{F} is the absolute Galois group of F.

Proof. a), b), and c) follow easily from [20, (2.1), (2.2), and (2.7)]; d) and e) follow from the same references plus [20, (2.4) and (5.5)].

1.2. Corollary. Let X be a quadric over F, D a divisible group and i ≥ 0. If i ≠ dim X/2 or i = dim X/2 and d(X) = 1, there is an exact sequence

0 → Tor(D, Coker(CH^{i}X → CH^{i}\overline{X})) → D ⊗ CH^{i}X → D ⊗ CH^{i}\overline{X} → 0.

This is clear, since CH^{i}\overline{X} is torsion-free and the map CH^{i}X → CH^{i}\overline{X} has torsion kernel and cokernel.

1.3. Corollary. Let X be a quadric of dimension 2i over F such that d = d(X) ≠ 1, E = F (\sqrt{d}) and Γ = Gal(E/F). Then the Γ-module CH^{i}X_{E}/ torsion is isomorphic to Ind_{1}^{Γ} Z if X_{E} is split, and to Z⊕Z(ε) otherwise, where Z(ε) is the Γ-module of support Z with the nontrivial action of Γ. On the other hand, the G_{F} -module CH^{i}\overline{X} is always isomorphic to Ind_{G_{E}}^{G_{F}} Z. The maps

CH^{i}X/ torsion → (CH^{i}\overline{X})^{G_{F}}

and

CH^{i}X ⊗ Q/Z → H^{0}(F, CH^{i}\overline{X} ⊗ Q/Z)

are isomorphisms.

Proof. By prop. 1.1 d), CH^{i}X_{E}/ torsion is a free Z-module with basis (h^{i}, 2^{t}l) for some t ≥ 0. It is clear that t = 0 if and only if l is rational over E, i.e., iff X_{E} is split. In this case, another basis of CH^{i}X_{E}/ torsion is (l, l′) where l′ = h^{i} − l is the class of another linear section, and this basis is permuted by Γ, hence the first claim. If t > 0, another basis of CH^{i}X_{E}/ torsion is given by (h^{i}, 2^{t−1}h^{i} − 2^{t}l), and 2^{t−1}h^{i} − 2^{t}l is a −1-eigenvector for Γ. This proves the second claim. The isomorphism CH^{i}X/ torsion ⟶^{∼} (CH^{i}\overline{X})^{G_{F}} has been seen in prop. 1.1 a) and e). To get the other, we consider the short exact sequence of G_{F} -modules

0 → CH^{i}\overline{X} → CH^{i}\overline{X} ⊗ Q → CH^{i}\overline{X} ⊗ Q/Z → 0

obtained by tensoring the exact sequence 0 → Z → Q → Q/Z → 0 by the flat Z-module CH^{i}\overline{X}. Since CH^{i}\overline{X} ≃ Ind_{G_{E}}^{G_{F}} Z, we have H^{1}(F, CH^{i}\overline{X}) = 0 by Shapiro lemma, hence we get an isomorphism

CH^{i}X ⊗ Q/Z ⟶^{∼} (CH^{i}\overline{X})^{G_{F}} ⊗ Q/Z ⟶^{∼} H^{0}(F, CH^{i}\overline{X} ⊗ Q/Z).

1.4. Corollary. Let X be an anisotropic quadric over F, K = F (X) and i_{1}(X) the Witt index of q_{K}, where q is a quadratic form defining X.

a) For i < dim X/2, CH^{i}X/ torsion → CH^{i}X_{K}/ torsion is bijective.

b) For i > dim X/2, CH^{i}X/ torsion → CH^{i}X_{K}/ torsion is bijective if i ≤ dim X − i_{1}(X) and Coker(CH^{i}X/ torsion → CH^{i}X_{K}/ torsion) ≃ Z/2 if i > dim X − i_{1}(X).

c) For i = dim X/2, CH^{i}X/ torsion → CH^{i}X_{K}/ torsion is bijective if d(X) ≠ 1. If d(X) = 1, Coker(CH^{i}X/ torsion → CH^{i}X_{K}/ torsion) is cyclic of order 2^{s} for some s ≤ t, where t is as in prop. 1.1 d). For i = 1 or 2 we have s = 1.

d) For i = dim X/2 and d = d(X) ≠ 1, let E = F (\sqrt{d}). Then

CH^{i}X_{E}/ torsion → CH^{i}X_{KE}/ torsion

is bijective if X_{E} is isotropic and has cokernel as in c) otherwise.

Proof. Everything directly follows from proposition 1.1, except for the last sentence of c) and for d). Assume dim X = 2. Then X is defined by an anisotropic quaternion form, (= a 2-fold Pfister form). hence i_{1}(X) = 2, which implies s = 1 in this case by proposition 1.1 d). Assume dim X = 4. Then X is defined by an anisotropic Albert form φ, hence i_{1}(X) = 1. Let Y be the quadric over K given by the anisotropic part of φ_{K} (dim Y = 2). By [20, (2.2)] (cf. lemma 2.1 below), there is an isomorphism CH^{1}Y ⟶^{∼} CH^{2}X_{K}. This isomorphism is natural under change of base field, as can be seen from the proof of lemma 2.1. In particular, the diagram

CH^{1}Y ⟶^{∼} CH^{2}X_{K}
↓  ↓
CH^{1}\overline{Y} ⟶^{∼} CH^{2}\overline{X_{K}}

commutes, which shows that Coker(CH^{2}X_{K} → CH^{2}\overline{X_{K}}) ≃ Z/2. Moreover, since Coker(CH^{2}X ⟶ CH^{2}\overline{X}) ≃ Z/4 and CH^{2}\overline{X} ⟶^{∼} CH^{2}\overline{X_{K}}, this indeed shows that s = 1. Finally, d) is obvious since, if X_{E} is isotropic, the extension KE/E is purely transcendental (compare [20, proof of (3.12)]).

1.5. Proposition. Let X be a quadric over F, where F is separably closed (of characteristic ≠ 2). Then H^{j}X = 0 for j odd and the cycle maps

cl_{X}^{i}: CH^{i}X/n → H^{2i}(X, μ_{n}^{⊗i})

are isomorphisms for any n prime to char F and i ≥ 0.

Proof. This follows from proposition 2.2 a) and the computation of H_{ét}^{∗}(X, Z_{l}) [11, th. 3.3].

1.6. Proposition. Let X be a quadric over F.

a) (classical) CH^{1}X is torsion-free.

b) ([20, (6.1)]) (CH^{2}X)_{torsion} ≃ Z/2 if X is defined by a neighbour of an anisotropic 3-fold Pfister form, and is 0 otherwise.

c) ([20, (2.6)],[21],[22]) (CH^{3}X)_{torsion} ≃ 0 or Z/2. If dim X = 3 or dim X > 10, (CH^{3}X)_{torsion} = 0.

2. K- and H-cohomology

We denote by H^{i}(X, K_{j}) and H^{i}(X, H^{j}(μ_{m}^{⊗k})) the cohomology groups of the localization complexes with terms

⨁_{x∈X^{(i)}} K_{j−i}F (x)  resp.  ⨁_{x∈X^{(i)}} H^{j−i}(F (x), μ_{m}^{⊗(k−i)}).

These cohomology groups are defined for any equidimensional, not necessarily smooth variety. For the long exact excision sequences and for homotopy invariance we refer to [42]. The interpretation for smooth X as Zariski cohomology groups is not needed here.

2.1. Lemma. Let X be an isotropic quadric of dimension n ≥ 2 over a field F of characteristic ≠ 2; write q ≃ q′ ⊥ x_{n}x_{n+1} for a quadratic form defining X and Z ⊆ X for the codimension 2 subquadric with equation q′ = 0. Then, for 0 < i < n, j ≥ i and k ∈ Z, there are isomorphisms:

K_{j}F ⟶^{∼} H^{0}(X, K_{j}),
H^{i−1}(Z, K_{j−1}) ⟶^{∼} H^{i}(X, K_{j}),
K_{j−n}F ⟶^{∼} H^{n}(X, K_{j}),
H^{j}(F, μ_{m}^{⊗k}) ⟶^{∼} H^{0}(X, H^{j}(μ_{m}^{⊗k})),
H^{i−1}(Z, H^{j−1}(μ_{m}^{⊗(k−1)})) ⟶^{∼} H^{i}(X, H^{j}(μ_{m}^{⊗k})),
H^{j−n}(F, μ_{m}^{⊗(k−n)}) ⟶^{∼} H^{n}(X, H^{j}(μ_{m}^{⊗k})).

These isomorphisms commute with change of base field, transfer and cup-product by K_{∗}F (resp. H^{∗}(F, μ_{m}^{⊗l})).

Remark. If n = 2, the “subquadric” Z is Spec E, with E = F [t]/t^{2} − d, where d = d(X) is the (signed) discriminant of the 2-dimensional form q′. E is a quadratic extension of F if d ≠ 1 and F × F otherwise.

Proof. (compare [16, lemma 3.3], [51, prop. 3.2.1] for dim X = 3, and [20, (2.2)] for i = j). We do it only for K-cohomology, the case of H-cohomology being identical. Let Y be the hyperplane section x_{n+1} = 0 and P = (0 : ⋯: 0 : 1 : 0). We have excision exact sequences:

⋯ → H^{i−1}(Y, K_{j−1}) → H^{i}(X, K_{j}) → H^{i}(X \ Y, K_{j}) → H^{i}(Y, K_{j−1}) → ⋯

⋯ → H^{i−n}(P, K_{j−n}) → H^{i−1}(Y, K_{j−1}) → H^{i−1}(Y \ P, K_{j−1}) →

→ H^{i−n+1}(P, K_{j−n}) → ⋯

On the other hand, X \ Y is isomorphic to the affine space and P makes the projection Y \ P → Z a vector bundle with affine line fibres. Homotopy invariance of K-cohomology yields new exact sequences and isomorphisms:

H^{i−1}(Y, K_{j−1}) ⟶^{∼} H^{i}(X, K_{j}) if i > 1, (2.1)

0 → H^{0}(X, K_{j}) → K_{j}F → H^{0}(Y, K_{j−1}) → H^{1}(X, K_{j}) → 0 (2.2)

and

H^{i−1}(Z, K_{j−1}) ⟶^{∼} H^{i−1}(Y \ P, K_{j−1}) if i < n − 2, (2.3)

0 → H^{n−2}(Y, K_{j−1}) → H^{n−2}(Z, K_{j−1}) → K_{j−n}F → H^{n−1}(Y, K_{j−1}) → 0. (2.4)

The composition K_{j}F → H^{0}(X, K_{j}) → K_{j}F is the identity, so that the exact sequence (2.2) splits as two isomorphisms. Similarly, for n > 0, K_{j−n}F → H^{n−1}(Y, K_{j−1}) ⟶^{∼} H^{n}(X, K_{j}) is induced by the inclusion of the closed point P in X and composition with the norm H^{n}(X, K_{j}) ⟶^{N} K_{j−n}F is the identity, so the exact sequence (2.4) also splits as two isomorphisms. Putting all this together, we get the isomorphisms of lemma 2.1. The naturality statements are clear from the construction of these isomorphisms.

2.2. Proposition. Let X be a split quadric over a field F of characteristic ≠ 2, and h ∈ CH^{1}X the class of a hyperplane section. a) CH^{i}X ≃ Z for i ≠ dim X/2, and CH^{i}X ≃ Z ⊕ Z if i = dim X/2; h^{i} is a

generator of CH^{i}X for i < dim X/2 and twice a generator for i > dim X/2. For i = dim X/2, a basis of CH^{i}X is given by the classes l, l′ of two maximal linear subspaces of X such that h^{i} = l + l′.

b) The natural maps

K_{j−i}F ⊗ CH^{i}X → H^{i}(X, K_{j})

given by the Brown-Gersten-Quillen spectral sequence and cup-product are isomorphisms.

c) The natural maps

H^{j−i}(F, μ_{m}^{⊗k}) ⊗ CH^{i}X → H^{i}(X, H^{j}(μ_{m}^{⊗(i+k)}))

given by the coniveau spectral sequence and cup-product are isomorphisms for all m ≥ 1.

Proof. a) follows from proposition 1.1. Let us prove b) (as above, the proof of c) is identical and omitted). (See [29, prop. 1 and prop. 3] for another approach.) The quadric X is isomorphic to

X_{n} : \begin{cases}x_{0}x_{1} + ⋯ + x_{n}x_{n+1} = 0 & if n is even \\ x_{0}^{2} + x_{1}x_{2} + ⋯ + x_{n}x_{n+1} = 0 & if n is odd \end{cases}

where n = dim X. As a consequence of lemma 2.1, the diagram

\begin{aligned}H^{i−1}(X_{n−2}, K_{j−1}) & ⟶^{∼} & H^{i}(X_{n}, K_{j}) \\ ^{∪}↑ & & ^{∪}↑ \\ K_{j−i}F ⊗ CH^{i−1}X_{n−2} & ⟶^{∼} & K_{j−i}F ⊗ CH^{i}X_{n} \end{aligned} for 0 < i < n

commutes for i ≤ j. Inductively we get the commutative diagrams

H^{0}(X_{n−2i}, K_{j−i}) ⟶^{∼} H^{i}(X_{n}, K_{j})
∪↑  ^{∪}↑  for 2i ≤ n,
K_{j−i}F ⊗ CH^{0}X_{n−2i} ⟶^{∼} K_{j−i}F ⊗ CH^{i}X_{n}
H^{2i−n}(X_{2i−n}, K_{j+i−n}) ⟶^{∼} H^{i}(X_{n}, K_{j})
∪↑  ^{∪}↑  for 2i ≥ n.
K_{j−i}F ⊗ CH^{2i−n}X_{2i−n} ⟶^{∼} K_{j−i}F ⊗ CH^{i}X_{n}

Still by lemma 2.1, K_{j−i}F ⊗ CH^{0}X_{n−2i} ⟶^{∪} H^{0}(X_{n−2i}, K_{j−i}) and

K_{j−i}F ⊗ CH^{2i−n}X_{2i−n} ⟶^{∪} H^{2i−n}(X_{2i−n}, K_{j+i−n})

are isomorphisms (note that for 2i = n, X_{0} is a disjoint union of two points so CH^{0}X_{0} ≃ Z ⊕ Z). The claim follows.

2.3. Lemma. Let i > 0 and X be a quadric over F, with dim X ≠ 2i or dim X = 2i and d(X) = 1. Then, for all j ≥ i, the map K_{j−i}F ⊗ CH^{i}X → H^{i}(X, K_{j}) has 2-primary torsion kernel and cokernel. If dim X = 2i and d(X) = d ≠ 1, the composition

(K_{j−i}E ⊗ CH^{i}X_{E})_{Γ} → H^{i}(X_{E}, K_{j})_{Γ} ⟶^{N} H^{i}(X, K_{j})

has 2-primary torsion kernel and cokernel, where E = F (\sqrt{d}), Γ = Gal(E/F) and N is the transfer in K-cohomology.

Proof. Assume first that dim X ≠ 2i or dim X = 2i and d(X) = 1. Let K be a splitting field for X that is finite and Galois over F, with [K: F] a power of 2 (some multiquadratic extension will do) and G = Gal(K/F). In the commutative diagram

K_{j−i}F ⊗ CH^{i}X ⟶ H^{i}(X, K_{j})
Res↓  ^{Res}↓
(K_{j−i}K ⊗ CH^{i}X_{K})^{G} ⟶ H^{i}(X_{K}, K_{j})^{G}

the bottom horizontal map is an isomorphism by proposition 2.2 and the right vertical map has 2-primary torsion kernel by the usual transfer argument. Therefore, to prove the claim, it is enough to show that the left vertical map has 2-primary torsion kernel and cokernel. But this map decomposes as

K_{j−i}F ⊗ CH^{i}X → K_{j−i}F ⊗ CH^{i}X_{K} →

→ (K_{j−i}K)^{G} ⊗ CH^{i}X_{K} = (K_{j−i}K ⊗ CH^{i}X_{K})^{G}

since CH^{i}X_{K} ≃ Z or Z ⊕ Z with trivial Galois action by propositions 1.1 and 2.2. The same proposition shows that the left map has 2-primary torsion kernel and cokernel, while the transfer argument shows the same for the right map. This implies the same for their composition.

Assume now that dim X = 2i and d(X) = d ≠ 1. By the above, the left map in the composition of lemma 2.3 has 2-primary torsion kernel and cokernel, and the same holds for the right one still by a transfer argument. This concludes the proof of lemma 2.3.

2.4. Proposition. Let i > 0 and X be a quadric over F, with dim X ≠ 2i − 2 or dim X = 2i − 2 and d(X) = 1. Then there is an exact sequence:

H^{1}(F, 1) ⊗ CH^{i−1}X → H^{i−1}(X, H^{i}(i)) → (CH^{i}X)_{torsion} → 0.

If dim X = 2i − 2 and d(X) = d ≠ 1, there is an exact sequence:

(H^{1}(E, 1) ⊗ CH^{i−1}X_{E})_{Γ} → H^{i−1}(X, H^{i}(i)) → (CH^{i}X)_{torsion} → 0,

where E = F (\sqrt{d}), Γ = Gal(E/F) and the left map is cup-product followed by transfer.

Proof. Tensoring the maps of lemma 2.3 with Q/Z we get surjections:

K_{1}F ⊗ CH^{i−1}X ⊗ Q/Z ⟶ H^{i−1}(X, K_{i}) ⊗ Q/Z

if dim X ≠ 2i − 2 or dim X = 2i − 2 and d(X) = 1,

(K_{1}E ⊗ CH^{i−1}X_{E} ⊗ Q/Z)_{Γ} = (K_{1}E ⊗ CH^{i−1}X_{E})_{Γ} ⊗ Q/Z

⟶ H^{i−1}(X, K_{i}) ⊗ Q/Z

if dim X = 2i − 2 and d(X) = d ≠ 1.

By Kummer theory, K_{1}F ⊗ Q/Z ≃ H^{1}(F, 1). The claim now follows from the short exact sequence [5, 3.6]

0 → H^{i−1}(X, K_{i}) ⊗ Q/Z → H^{i−1}(X, H^{i}(i)) → (CH^{i}X)_{torsion} → 0.

2.5. Proposition.

a) ([8, prop. 1.2]) If the quadric X is isotropic over F, then η_{2}^{i} and η^{i} are isomorphisms for all i ≥ 0.

b) In general, Ker η^{i} and Coker η^{i} have exponent ≤ 2.

c) (cf. [10, lemma 1.3]) Suppose X and X′ are two quadrics such that X′ becomes isotropic over F (X). Then there is a natural commutative diagram for all i, k:

H_{nr}^{i}(F (X)/F, k)
η_{X}^{i} ↗
H^{i}(F, k)  ↑
η_{X′}^{i} ↘
H_{nr}^{i}(F (X′)/F, k)

so that Ker η_{X′}^{i} ⊆ Ker η_{X}^{i} and Coker η_{X′}^{i} maps to Coker η_{X}^{i}. If X also becomes isotropic over F (X′), then Ker η_{X}^{i} = Ker η_{X′}^{i} and Coker η_{X}^{i} ≃ Coker η_{X′}^{i}.

Proof. a) comes from the fact that the extension F (X)/F is purely transcendental if X is isotropic (compare [7, th. 4.1.5]). b) follows from a) and a transfer argument, using a quadratic extension of F which makes X isotropic. We give a proof of c) which is more elementary than [10, lemma 1.3]. Consider the diagram of fields

F (X)

↗  ↘
F  F (X ×_{F} X′)
↘  ↗
F (X′)

There are natural maps

H_{nr}^{i}(F (X)/F, k) → H_{nr}^{i}(F (X ×_{F} X′)/F, k)

and

H_{nr}^{i}(F (X′)/F, k) → H_{nr}^{i}(F (X ×_{F} X′)/F, k).

If X′ becomes isotropic over F (X), then F (X ×_{F} X′)/F (X) is a purely transcendental extension. Therefore, by [8, prop. 1.2] or [7, th. 4.1.5], the first quoted map is an isomorphism, which proves the first claim. If X is also purely transcendental over F (X′), then H_{nr}^{i}(F (X′)/F, k) → H_{nr}^{i}(F (X)/F, k) is clearly the inverse of H_{nr}^{i}(F (X)/F, k) → H_{nr}^{i}(F (X′)/F, k), hence the second claim.

Part II. Ker η_{2}^{4} for higher-dimensional quadrics

3. Generalities on Ker η_{2}^{i}

We fix a number i and assume that the invariant \overline{e}_{F}^{r} is is well-defined for r ≤ i−1 and an isomorphism for r ≤ i − 2 (for all fields F).

3.1. Lemma. A symbol (a_{1}, …, a_{i}) ≠ 0 lies in Ker η_{2}^{i} if and only if q is similar to a subform of ⟨⟨a_{1}, …, a_{i}⟩⟩.

Proof. Let φ = ⟨⟨a_{1}, …, a_{i}⟩⟩. Since (a_{1}, …, a_{i})_{F (X)} = 0, [16, remark p. 555] implies φ_{F (X)} ∼ 0, and the claim follows by the Cassels-Pfister theorem [26, th. IX.1.3].

As observed in the introduction, for dim q ≤ 2^{i−2} the group Ker η_{2}^{i} does not consist of symbols in general. The following proposition and corollary indicate that this bound on dim q might be sharp. Recall that two n-fold Pfister forms φ, ψ are linked if there exist a_{1}, …, a_{n−1}, a_{n}, b_{n} ∈ F^{∗} such that φ ≃ ⟨⟨a_{1}, …, a_{n−1}, a_{n}⟩⟩ and ψ ≃ ⟨⟨a_{1}, …, a_{n−1}, b_{n}⟩⟩. We define the notion of linked symbols (in Galois cohomology) similarly.

3.2. Proposition. Let dim q ≥ 2^{i−2} + 1, let X be the projective quadric with equation q = 0 and let φ, ψ ∈ Ker(W (F) → W (F (X))) be two i-fold Pfister forms. Then φ and ψ are linked, hence φ ⊥ −ψ is Witt-equivalent to a third i-fold Pfister form (up to similarity).

Proof. (cf. Fitzgerald [15, proof of th. 2.1]) The forms φ and ψ are multiplicative and split over F (X). Therefore q is isomorphic to subforms of φ and of ψ (assuming q represents 1). Hence the split form q ⊥ −q is a subform of φ ⊥ −ψ and the Witt index n = i(φ ⊥ −ψ) is ≥ 2^{i−2} + 1. By [13, th. 4.5], n is a power of 2, hence n ≥ 2^{i−1}. Therefore φ and ψ are linked, again by loc. cit..

3.3. Lemma. For dim q ≥ 2^{i−2} + 1 the following are equivalent:

a) Ker η_{2}^{i} consists of symbols;

b) Ker η_{2}^{i} is additively generated by symbols.

Proof. Let α, β ∈ Ker η_{2}^{i} be two symbols and α̃, β̃ Pfister forms such that e^{i}(α̃) = α, e^{i}(β̃) = β. By [16, remark p. 555], α̃, β̃ ∈ Ker(W (F) → W (F (X))). By proposition 3.2, α̃ and β̃ are linked, so α and β are linked and α+β is a symbol.

3.4. Proposition. Assume that for dim q = 2^{i−2} + 1 the group Ker η_{2}^{i} consists of symbols. Then

(1) Ker η_{2}^{i} consists of symbols for dim q > 2^{i−2}.

(2) |Ker η_{2}^{i}| ≤ 2 for dim q > 2^{i−1}.

(3) Ker η_{2}^{i} = 0 for dim q > 2^{i}.

Remark. The hypothesis of proposition 3.4 is satisfied for i ≤ 3 by [1]. For i = 4 it will be proved in the next three sections (theorem 6.3).

Proof. (1) Let dim q′ ≥ 2^{i−2} +1 and let q be a subform of q′ of dimension 2^{i−2} +1. The claim follows now from proposition 2.5 c) applied to the associated quadrics X′ resp. X.

(2) + (3) If dim q > 2^{i−1}, for any nonzero symbol (a_{1}, …, a_{i}) ∈ Ker η_{2}^{i}, q is a neighbour of the Pfister form ⟨⟨a_{1}, …, a_{i}⟩⟩ by lemma 3.1. But q can be a neighbour of at most one Pfister form, up to isomorphism, cf. [24, 7.4]. Moreover if ⟨⟨a_{1}, …, a_{i}⟩⟩ ≃ ⟨⟨b_{1}, …, b_{i}⟩⟩, then (a_{1}, …, a_{i}) = (b_{1}, …, b_{i}) [1, Satz 1.6]. Hence |Ker η_{2}^{i}| ≤ 2, and if it is nontrivial then dim q ≤ 2^{i}.

4. 6-dimensional spinors

For generalities on Clifford algebras and the special Clifford group see [45]. Let q: V → F be 6-dimensional, let C_{0}(q) be the even Clifford algebra of q, and

let Z(q) be the center of C_{0}(q). Then Z(q) is a quadratic extension of F and C_{0}(q) is an Azumaya algebra of degree 4 over Z(q). The standard involution x ↦ x^{t} is an involution of second kind. Denote by

Nrd: C_{0}(q) → Z(q)

the reduced norm.

Let

SΓ(q) = { x ∈ C_{0}(q)^{∗} | xV x^{−1} = V }

be the special Clifford group and let

GΓ(q) = { x ∈ C_{0}(q)^{∗} | xx^{t} ∈ F^{∗} }.

Then SΓ(q) ⊂ GΓ(q) and there is an exact sequence

1 → Spin(q) → SΓ(q) ⟶^{sn} F^{∗}

where sn(x) = xx^{t} is the spinor norm.

Suppose d_{±}(q) = 1. Then Z(q) = F × F and C_{0}(q) = A × A′ where A and A′ are algebras of degree 4 over F. The standard involution identifies A′ with A^{op}, so that there is an isomorphism

C_{0}(q) = A × A^{op} (4.1)

under which the standard involution reads as (a, b)^{t} = (b, a). The reduced norm C_{0}(q) → Z(q) is then componentwise the reduced norm on A. The group GΓ(q) becomes the group of pairs

{ (a, a^{−1}f) ∈ A × A^{op} | a ∈ A^{∗}, f ∈ F^{∗} }.

Let π: C_{0}(q) → A be the projection onto the first factor. Then

GΓ(q) = A^{∗} × F^{∗}

via x ↦ (π(x), xx^{t}).

We call an element x ∈ SΓ(q) plane if x ∈ SΓ(q̃) where q̃ is a regular 2dimensional subform of q. If q is anisotropic, then x is plane if and only if x is the product vw of two vectors v, w ∈ V \ 0.

4.1. Lemma. Nrd(x) = (xx^{t})^{2} for x ∈ SΓ(q).

Proof. The action of SΓ(q) on V induces an exact sequence

1 → F^{∗} → SΓ(q) → SO(q) → 1. (4.2)

Since O(q) is generated by reflections, it follows that SΓ(q) is generated by products x = vw where v, w ∈ V are anisotropic vectors. For generic v, w, the subspace of V generated by v, w is regular of dimension 2. Therefore it suffices to check the claim for plane elements.

So suppose x ∈ C_{0}(q̃) for a regular 2-dimensional subform q̃. Write L = C_{0}(q̃). The standard involution leaves L invariant and restricts to the canonical involution of L/F. Hence x ↦ xx^{t} restricts on L to the norm of L/F. The claim follows now from the fact that C_{0}(q) is of degree 4 and L ⊗ Z(q) ⊂ C_{0}(q) is of degree 2 over Z(q).

4.2. Proposition.

a) SΓ(q) = { x ∈ C_{0}(q)^{∗} | xx^{t} ∈ F^{∗}, Nrd(x) = (xx^{t})^{2} },

b) Spin(q) = { x ∈ C_{0}(q)^{∗} | xx^{t} = 1, Nrd(x) = 1 }.

Proof. The inclusions “⊂” are clear from lemma 4.1.

We may assume that F is algebraically closed. In this case it easy to see that SΓ(q) = Spin(q) · F^{∗} and similarly for the righthand sides. Hence b) ⇒ a).

The righthand side of b) becomes under (4.1) the group of pairs

{ (a, a^{−1}) ∈ A × A^{op} | a ∈ A^{∗}, Nrd(a) = 1 } = SL(1, A).

This way b) reads as the classical isomorphism Spin(6) = SL(4) [55, 12, 25].

4.3. Corollary. If d_{±}(q) = 1 and A is as above, there is an exact sequence

1 → SΓ(q) → A^{∗} × F^{∗} ⟶^{ω} F^{∗}

where ω(a, f) = Nrd(a)/f^{2}.

4.4. Corollary. Let A be a biquaternion skewfield and let a ∈ A^{∗} and f ∈ F^{∗} with

Nrd(a) = f^{2}.

Then there exist a_{1}, a_{2}, a_{3} ∈ A^{∗} with the following properties:

• a = a_{1}a_{2}a_{3},

• Every a_{i} lies in a quadratic subfield L_{i} ⊂ A,

• N_{L_{1}/F} (a_{1})N_{L_{2}/F} (a_{2})N_{L_{3}/F} (a_{3}) = f.

Proof. If A = (a, b) ⊗ (c, d), let q = ⟨−a, −b, ab, c, d, −cd⟩. We are then in the situation of corollary 4.3. The form q is anisotropic. Since every element in SO(q) is a product of 6 reflections, it follows that every element in SΓ(q) is a product of 3 plane elements x_{i}. Their images a_{i} = π(x_{i}) in A^{∗} do the job.

We will have to make use of the following variant of the Cartan-Dieudonné theorem for anisotropic forms.

4.5. Lemma. Let q: V → F be anisotropic of dimension n and let V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{n} = V be a sequence of subspaces with dim V_{i} = i. Then every α ∈ O(q) is a product

α = τ_{i_{k}} ⋯ τ_{i_{1}}

where 1 ≤ i_{1} < ⋯ < i_{k} ≤ n and τ_{j} is a reflection in O(q|V_{j}).

Proof. By induction on n. Let v ≠ 0 be orthogonal to V_{n−1}. If α ∉ O(q|V_{n−1}), let τ be the reflection at the hyperplane orthogonal to α(v) − v. Then τα ∈ O(q|V_{n−1}).

5. Ker η_{2}^{4} and H^{2}(X, K_{3})

The following theorem reduces the computation of Ker η_{2}^{4} for dim q ≥ 5 to a Kcohomology computation. The statement is essentially a reformulation of arguments in [27], where the result has been applied to Albert forms. The principal idea has already been used in [16] for 5-dimensional Pfister neighbours.

5.1. Theorem. Let X be a smooth quadric of dimension ≥ 3. Then there exist isomorphisms

ρ_{3} : Ker(H^{2}(X, K_{2}) → H^{2}(\overline{X}, K_{2})) ⟶^{∼} Ker(H^{3}F → H^{3}F (X)),
ρ_{4} : Ker(H^{2}(X, K_{3}) → H^{2}(\overline{X}, K_{3})) ⟶^{∼} Ker(H^{4}F → H^{4}F (X)).

These are functorial with respect to base change F → F′ and with respect to transfer for finite extensions F′/F.

Moreover they commute with multiplication with K_{1} in the following sense:

ρ_{4}({a} · x) = (a) · ρ_{3}(x)

for a ∈ F^{∗} and x ∈ Ker(H^{2}(X, K_{2}) → H^{2}(\overline{X}, K_{2})).

Proof. The (inverse of the) map ρ_{4} has been described in [27, p. 71] in terms of a diagram chase. It is also proven that the map is an isomorphism (it should be noted that condition (7) in [27, p. 71] is superfluous). From this definition the compatibility with respect to base change and transfer is easy to see.

One proceeds in a similar way for ρ_{3} (see also [16, diagrams (3.5), (4.2)]), just by repeating the diagram chase in one degree less. The product formula is then obvious.

Remark. It can be shown that the maps ρ_{i} are functorial with respect to linear morphisms j: X ⊂ X′ of quadrics. Namely then Ker η_{X′}^{i} ⊂ Ker η_{X}^{i} (proposition 2.5 c)) and the maps ρ_{i} commute with these inclusions and with the pull back maps j^{∗} on H^{2}( , K_{i−1}).

Our aim is to compute the map ρ_{4} for dim X = 3.

Let q be a form defining X and let A = C_{0}(q) be its even Clifford algebra (a biquaternion algebra). We denote by h ∈ CH^{1}(X) the class of a hyperplane section. If A is split, then X is split. In this case CH^{2}(X) = Z, generated by the class l of a line and one has h^{2} = 2l.

5.2. Theorem. For dim X = 3 there exist surjections

θ_{3} : K_{0}A ⊕ K_{0}F → → H^{2}(X, K_{2}),
θ_{4} : K_{1}A ⊕ K_{1}F → → H^{2}(X, K_{3}),

functorial with respect to base change and transfer, compatible with multiplication with K_{1}, and such that θ_{3} has the following properties:

a) θ_{3}(0, 1) = −h^{2}.

b) If A is split, let y ∈ K_{0}A be the element with Nrd(y) = 1. Then θ_{3}(y, 0) = l.

Proof. All the essential arguments are in [27, p. 74–76]. In fact, in the 5-dimensional case the situation is somewhat simpler. By [27, prop. 2],

H^{2}(X, K_{3}) = K_{1}(X)^{(2)}/K_{1}(X)^{(3)}

where the superscripts refer to the filtration coming from the Brown-Gersten-Quillen (BGQ) spectral sequence [41]. In our case one finds K_{1}(X)^{(2)} = K_{1}A⊕K_{1}F. (We don’t have to care about a computation of K_{1}(X)^{(3)}).

Let

Res_{\overline{F} /F} : H^{2}(X, K_{i}) → H^{2}(\overline{X}, K_{i})

be the restriction to a separable closure. Theorem 5.2 a), b) shows

Res_{\overline{F} /F} ∘ θ_{3}(y, m) = (Nrd(y) − 2m) · l ∈ CH^{2}(\overline{X}) = Z · l.

The compatibility of θ_{3}, θ_{4} with multiplication with K_{1} yields

Res_{\overline{F} /F} ∘ θ_{4}(z, \{f\}) = (Nrd(z) − 2\{f\}) · l ∈ H^{2}(\overline{X}, K_{3}) = K_{1}\overline{F} · l.

This together with theorem 5.1 and theorem 5.2 yields 5.3. Corollary. There are surjections

ω_{3} : Ker(K_{0}A ⊕ K_{0}F ⟶^{(Nrd,−2)} K_{0}F) → → Ker η_{2}^{3},
ω_{4} : Ker(K_{1}A ⊕ K_{1}F ⟶^{(Nrd,−2)} K_{1}F) → → Ker η_{2}^{4},

compatible with base change, transfer and multiplication with K_{1}.

6. Computations for dim X = 3

Let q: W → F be 5-dimensional and A = C_{0}(q). We assume that the determinant of q is −1. Put

q̂: F ⊕ W → F
q̂(t, w) = t^{2} + q(w).

Then C_{0}(q̂) = A × A^{op} as in (4.1).

We first compute ω_{3}, ω_{4} in the case when q̂ is isotropic. In this case q represents −1 and therefore

q ≃ ⟨−1⟩ ⊥ c⟨⟨a, b⟩⟩

for some a, b, c ∈ F^{∗}. Moreover A = M_{2}(F)⊗D where D is the quaternion algebra associated to a, b.

6.1. Proposition. Let y ∈ K_{0}A = K_{0}D be the element represented by the Dmodule D (i.e., Nrd(y) = 2) and let ε = (y, 1).

(1) The sequences

K_{0}F ⟶^{· ε} K_{0}A ⊕ K_{0}F ⟶^{(Nrd,−2)} K_{0}F,

K_{1}F ⟶^{· ε} K_{1}A ⊕ K_{1}F ⟶^{(Nrd,−2)} K_{1}F

are exact.

(2) One has

ω_{3}(ε) = (a, b, c),
ω_{4}({e} · ε) = (e, a, b, c),  e ∈ F^{∗}.

Proof. (1) follows from the injectivity of the reduced norm K_{i}D → K_{i}F for i = 0, 1.

Corollary 5.3 shows that ω_{3}(ε) generates Ker η_{2}^{3}. Since (a, b, c) ∈ Ker η_{2}^{3}, we must have the first formula in (2). (If (a, b, c) = 0, then q is isotropic and Ker η_{2}^{3} = 0.) The second formula follows from the compatibility with multiplication.

We now assume that q̂ is anisotropic. Via corollary 4.3, the map ω_{4} of corollary 5.3 reads as a map

ω̃_{4} : SΓ(q̂) → → Ker η_{2}^{4}.

We compute this map for plane elements x ∈ SΓ(q̂).

Suppose x ∈ L = C_{0}(q̃) for a 2-dimensional subform q̃ of q̂. The map

L^{∗} = C_{0}(q̃)^{∗} = SΓ(q̃) ⊂ SΓ(q̂) → A^{∗} × F^{∗}

coincides with the map

K_{1}L → K_{1}A ⊕ K_{1}F

given by the inclusion L ⊂ A and the norm for L/F.

The form q̂_{L} is isotropic. Denote by

ε_{L} ∈ Ker(K_{0}(A ⊗ L) ⊕ K_{0}L ⟶^{(Nrd,−2)} K_{0}L)

the generator considered above.

By compatibility with transfer for L/F one finds

ω̃_{4}(x) = Cor_{L/F} ((x) · ω_{3}(ε_{L}))

Note that if q̃ is a subform of q, then q_{L} is isotropic and therefore ω_{3}(ε_{L}) = 0.

We compute ω_{4}(u) for arbitrary u ∈ SΓ(q̂). If u ∈ SΓ(q), then u is a product of plane elements in SΓ(q) and ω̃_{4}(u) = 0 by the last remark. In general we have

6.2. Lemma. For any u ∈ SΓ(q), there exist a one-dimensional subspace H of W and a plane element x ∈ SΓ(q̂|(F ⊕ H)) such that u ∈ x · SΓ(q).

Proof. There exist by lemma 4.5 and the exact sequence (4.2) a vector v_{1} ∈ F ⊕ W and vectors v_{2}, …, v_{6} ∈ W such that u = v_{1} ⋯ v_{6}.

Write v_{1} = s(1, 0) + th with s, t ∈ F^{∗}, 0 ≠ h ∈ W, and let H be the subspace generated by h. Then x = v_{1}h has the desired properties.

Since ω_{4}| SΓ(q) = 0, we are reduced to computing ω̃_{4} on plane elements x ∈ SΓ(q̂|F ⊕ H) as in lemma 6.2. Let h be a basis vector of H and let a = −q(h). Then L = C_{0}(q̃) = F (\sqrt{a}) and q is of the form

q ≃ ⟨−a, a_{1}, a_{2}, a_{3}, aa_{1}a_{2}a_{3}⟩.

Over L this gives

q_{L} ≃ ⟨−1, a_{1}, a_{2}, a_{3}, a_{1}a_{2}a_{3}⟩ ≃ ⟨−1⟩ ⊥ a_{1}⟨⟨−a_{1}a_{2}, −a_{1}a_{3}⟩⟩.

Hence

ω_{3}(ε_{L}) = Res_{L/F} ((a_{1}, −a_{1}a_{2}, −a_{1}a_{3}))

comes from the ground field (although ε_{L} does not). So

ω̃_{4}(x) = Cor_{L/F} ((x) · ω_{3}(ε_{L})) = (N_{L/F} (x), a_{1}, −a_{1}a_{2}, −a_{1}a_{3})

is a symbol.

We have proved:

6.3. Theorem. If dim X = 3, then Ker η_{2}^{4} consists of symbols.

Remark. The homomorphism ω̃_{4} is obviously trivial on the commutator subgroup [SΓ(q̂), SΓ(q̂)]. Moreover it is easy to see that the map

SΓ(q̂)/ SΓ(q) → O(q̂)/ O(q)

is bijective. Hence the map ω̃_{4} actually induces a surjective homomorphism

Ω: O(q̂)/[O(q̂), O(q̂)]O(q) → → ker(H^{4}F → H^{4}F (X))

Merkurjev and Tignol have described this map in terms of symplectic involutions [28, 52].

The kernel of the spinor norm

O(q̂)/[O(q̂), O(q̂)] → F^{∗}/(F^{∗})^{2},

can be identified with SK_{1}A. The restriction of Ω to SK_{1}A depends only on q̂ (and not on the choice of 5-dimensional subform q). It fits into an exact sequence [27]

0 → SK_{1}A ⟶^{Ω} H^{4}F → H^{4}F (X̂)

where X̂ is the quadric for q̂.

Further details will appear in [25].

Part III. Coker η_{2}^{i} for higher-dimensional quadrics

7. A relationship between Ker η_{2}^{i} and Coker η_{2}^{i}

In this section, we prove a general result which we then apply to quadrics. To start with, let F have arbitrary characteristic exponent p and X be a smooth variety over F. We denote by δ = δ(X/F) the prime-to-p part of the gcd of degrees of closed points of X. Let n be some integer divisible by δ and prime to p, i ≥ 1 and let

η_{n}^{i}: H^{i}(F, μ_{n}^{⊗(i−1)}) → H^{0}(X, H^{i}(μ_{n}^{⊗(i−1)}))
η^{i}: H^{i}(F, i − 1) → H^{0}(X, H^{i}(i − 1))

be the natural maps.

7.1. Proposition. Let m be another integer divisible by δ and prime to p. For i ≤ 3, there is an exact sequence

0 → Ker η_{m}^{i} ∩ Im (H^{i}(F, μ_{mn}^{⊗(i−1)}) ⟶^{n} H^{i}(F, μ_{m}^{⊗(i−1)})) → Coker η_{n}^{i} → Coker η^{i}

If m, n are powers of 2, this result also holds for i = 4.

Proof. We shall freely use the following fact, which is a consequence of Kummer theory for i = 2 (resp. the Merkurjev-Suslin theorem for i = 3, the Merkurjev-Suslin/Rost theorem for i = 4): the map H^{i}(F, μ_{n}^{⊗(i−1)}) ⟶^{ι_{∗}} H^{i}(F, i−1) is injective, where ι′: μ_{n}^{⊗(i−1)} → Q/Z(i − 1) is the natural inclusion. We need:

7.2. Lemma. Under the assumptions of proposition 7.1, a) Ker η_{n}^{i} and Ker η_{mn}^{i} are killed by δ.

b) Ker η_{n}^{i} → Ker η_{mn}^{i} is an isomorphism.

c) Ker(Coker η_{n}^{i} → Coker η_{mn}^{i}) = Ker(Coker η_{n}^{i} → Coker η^{i}).

Proof. a) If X(F) ≠ ∅, this follows from a classical specialization argument; in general, it follows from this case and a transfer argument.

b) follows from a).

c) We have a commutative diagram with exact rows:

0→ H^{i}(F, μ_{n}^{⊗(i−1)}) → H^{i}(F, i − 1) ⟶^{n} H^{i}(F, i − 1)
η_{n}^{i}↓  ^{η^{i}}↓  ^{η^{i}}↓
0→H^{0}(X, H^{i}(μ_{n}^{⊗(i−1)}))→H^{0}(X, H^{i}(i − 1)) ⟶^{n} H^{0}(X, H^{i}(i − 1))

where the exactness of the bottom row follows from a diagram chase and the injectivity of H^{i−1}(F (x), μ_{n}^{⊗(i−2)}) → H^{i−1}(F (x), i−2) for all codimension 1 points of X. Let x ∈ Ker(Coker η_{n}^{i} → Coker η^{i}), x̃ a lift of x in H^{0}(X, H^{i}(μ_{n}^{⊗(i−1)})) and pick α ∈ H^{i}(F, i − 1) such that η^{i}α = image of x̃. Since nα ∈ Ker η^{i}, we have δnα = 0 by a), hence mnα = 0. If β is the corresponding element of H^{i}(F, μ_{mn}^{⊗(i−1)}), it is clear that η_{mn}^{i}β is the image of x̃ in H^{0}(X, H^{i}(μ_{mn}^{⊗(i−1)})).

Let ι: μ_{n}^{⊗(i−1)} → μ_{mn}^{⊗(i−1)} (resp. π: μ_{mn}^{⊗(i−1)} → μ_{m}^{⊗(i−1)}) denote the natural inclusion (resp. projection). Consider the commutative diagram with exact rows:

0 ⟶ H^{i}(F, μ_{n}^{⊗(i−1)}) ⟶^{ι_{∗}} H^{i}(F, μ_{mn}^{⊗(i−1)}) ⟶^{π_{∗}} H^{i}(F, μ_{m}^{⊗(i−1)})_{0} ⟶ 0
η_{n}^{i}↓  ^{η_{mn}^{i}}↓  ^{η_{m}^{i}}↓
0 ⟶ H^{0}(X, H^{i}(μ_{n}^{⊗(i−1)})) ⟶^{ι_{∗}} H^{0}(X, H^{i}(μ_{mn}^{⊗(i−1)})) ⟶^{π_{∗}} H^{0}(X, H^{i}(μ_{m}^{⊗(i−1)}))

where H^{i}(F, μ_{m}^{⊗(i−1)})_{0} = Im π_{∗}. From lemma 7.2 b) and the snake lemma applied to this diagram, we derive an exact sequence

0 → Ker η_{m}^{i} ∩ Im (H^{i}(F, μ_{mn}^{⊗(i−1)}) ⟶^{n} H^{i}(F, μ_{m}^{⊗(i−1)})) → Coker η_{n}^{i} → Coker η_{mn}^{i}.

Proposition 7.1 now follows from lemma 7.2 c).

7.3. Proposition. Let char F ≠ 2, i ≥ 0 and α ∈ H^{i}F be a decomposable element (i.e., a sum of symbols). Then α ∈ Im(H^{i}(F, μ_{4}^{⊗(i−1)}) ⟶^{p} H^{i}F) if and only if (−1) · α = 0 ∈ H^{i+1}F. Here p is the map induced by the surjective homomorphism of Galois modules μ_{4}^{⊗(i−1)} → μ_{2}^{⊗(i−1)} ≃ Z/2.

Proof. Let ∂ (resp. ∂′) be the boundary map H^{i}F → H^{i+1}F corresponding to the short exact sequence 0 → Z/2 → μ_{4}^{⊗i} → Z/2 → 0 (resp. 0 → Z/2 → μ_{4}^{⊗(i−1)} → Z/2 → 0). It follows from [18, lemma 1] that ∂α − ∂′α = (−1) · α. If α is decomposable, then it obviously comes from H^{i}(F, μ_{4}^{⊗i}) (reduce to a symbol (a_{1}) · . . . · (a_{i}), where a_{k} ∈ F^{∗}, and observe that each (a_{k}) comes from the Kummer class of a_{k} in H^{1}(F, μ_{4})), hence ∂α = 0 and the lemma follows.

7.4. Proposition. Suppose char F ≠ 2 and δ = 2 in proposition 7.1 (e.g., X is a quadric). For i ≤ 4, the group

Ker η_{2}^{i} ∩ Im (H^{i}(F, μ_{4}^{⊗(i−1)}) ⟶^{2} H^{i}F )

coincides with the group (Ker η_{2}^{i})_{0} of the introduction. We have an exact sequence

0 → (Ker η_{2}^{i})_{0} → Coker η_{2}^{i} → Coker η^{i}.

Proof. Let q be a quadratic form defining X and q′ = ⟨a, b⟩ be any binary subform of q. Then Ker η_{2}^{i} ⊆ Ker(H^{i}F → H^{i}F (\sqrt{−ab})) = (−ab) · H^{i−1}F. Since H^{i−1}F is generated by symbols [31, 32], we see that every element of Ker η_{2}^{i} is decomposable. Finally, the last claim follows from this and proposition 7.1.

Remark. The arguments in this section show in fact that proposition 7.4 follows for a given i from the Milnor conjecture in degree i (see the remark at the end of introduction).

8. Constructing some maps

In this section, we show how the Chow ring acts on étale and H-cohomology with divisible coefficients. This action will be repeatedly used in the sequel.

Let X be a smooth variety over F, r ≥ 1, i, j ≥ 0 and k, k′ ∈ Z. We first recall that Galois cohomology of F acts on the Hochschild-Serre and coniveau spectral sequences

HS_{2}^{p,q}(Z/2^{r}(k)) := H^{p}(F, H^{q}(\overline{X}, Z/2^{r}(k))) ⇒ H^{p+q}(X_{ét}, Z/2^{r}(k))

BO_{1}^{p,q}(Z/2^{r}(k)) := ∐_{x∈X^{(p)}} H^{q−p}(F (x), Z/2^{r}(k − p)) ⇒ H^{p+q}(X_{ét}, Z/2^{r}(k))

in the following sense. There are pairings

H^{n}(F, Z/2^{r}(j)) ⊗ HS_{2}^{p,q}(Z/2^{r}(k)) → HS_{2}^{p+n,q}(Z/2^{r}(j + k))
H^{n}(F, Z/2^{r}(j)) ⊗ BO_{1}^{p,q}(Z/2^{r}(k)) → BO_{1}^{p,q+n}(Z/2^{r}(j + k))

coming from the cup-product in Galois cohomology (and the restriction maps H^{∗}(F) → H^{∗}(F (x)) for the coniveau spectral sequence). These pairings commute with differentials and induce compatible pairings

H^{n}(F, Z/2^{r}(j)) ⊗ HS_{s}^{p,q}(Z/2^{r}(k)) → HS_{s}^{p+n,q}(Z/2^{r}(j + k))

H^{n}(F, Z/2^{r}(j)) ⊗ BO_{s}^{p,q}(Z/2^{r}(k)) → BO_{s}^{p,q+n}(Z/2^{r}(j + k)) for s ≤ ∞

which are also compatible at the E_{∞}-level with the pairings on the abutments

H^{n}(F, Z/2^{r}(j)) ⊗ H^{p+q}(X_{ét}, Z/2^{r}(k)) → H^{p+q+n}(X_{ét}, Z/2^{r}(j + k))

(obtained from the maps H^{i}(F, Z/2^{r}(k)) → H^{i}(X, Z/2^{r}(k)) and cup-product in the étale cohomology of X). This is classical for the Hochschild-Serre spectral sequence; for the coniveau spectral sequence, it is immediate from its exact couple construction out of étale cohomology with supports (this supersedes the sophisticated argument of [17, end of introduction]).

If E/F is a finite extension, there are also compatible transfer maps.

We now play with these pairings a little bit. Using the cycle map

CH^{j}X/2^{r} → H_{ét}^{2j}(X, Z/2^{r}(j))

we get composite pairings

H^{i}(F, Z/2^{r}(k)) ⊗ CH^{j}X → H^{i}(F, Z/2^{r}(k)) ⊗ H_{ét}^{2j}(X, Z/2^{r}(j))

⟶^{∪} H_{ét}^{i+2j}(X, Z/2^{r}(k + j))

hence, at the inductive limit:

H^{i}(F, k) ⊗ CH^{j}X → H^{i+2j}(X, k + j). (8.1)

Recall now that BO_{2}^{j,j}(Z/2^{r}(j)) ≃ CH^{j}X/2^{r}. We get therefore a pairing (similar to those in proposition 2.2):

H^{i}(F, Z/2^{r}(k)) ⊗ CH^{j}X → H^{j}(X, H^{i+j}(Z/2^{r}(k + j))).

Passing to the inductive limit, we get pairings:

H^{i}(F, k) ⊗ CH^{j}X → H^{j}(X, H^{i+j}(k + j)). (8.2)

It is not difficult to check that the maps of proposition 2.4 are special cases of (8.2).

We have:

8.1. Lemma. Let (H^{∗}(X, k+j)^{(m)})_{m≥0} be the decreasing filtration on H^{∗}(X, k+j) defined by the coniveau spectral sequence. Then

a) The image of the map (8.1) is contained in H^{i+2j}(X, k + j)^{(j)}.

b) The image of the map (8.2) consists of universal cycles. c) The diagram

H^{i}(F, k) ⊗ CH^{j}X ⟶ H^{j}(X, H^{i+j}(k + j))_{d=0}
↓  ↓
H^{i+2j}(X, k + j)^{(j)} ⟶  BO_{∞}^{j,i+j}

commutes, where the subscript d = 0 denotes universal cycles and BO_{∞}^{j,i+j} denotes the E_{∞}-term of the coniveau spectral sequence.

d) For i = 0, 1, the triangle

H^{i}(F, k) ⊗ CH^{j}X ⟶ H^{j}(X, H^{i+j}(k + j))
↓  ↙
H^{i+2j}(X, k + j)

commutes, where the vertical map is (8.1), the horizontal map is (8.2) and the diagonal map is the edge homomorphism from the coniveau spectral sequence.

Recall that, in a spectral sequence (E_{r}), an element x ∈ E_{r}^{p,q} is called a universal cycle if d_{s}^{p,q}x = 0 for all s ≥ r (this definition makes sense inductively).

Proof. It is enough to prove this for the maps defined at the finite coefficients level. By the multiplicativity of the coniveau spectral sequence, we are reduced to the case i = k = 0. Then a), b), and c) follow from the fact that BO_{1}^{p,q} = 0 for p > q. Finally, d) follows from c) and the definition of the edge homomorphism.

Let A be a discrete topological G_{F} -module and M an abelian group. There is a pairing

H^{i}(F, A) ⊗ M → H^{i}(F, A ⊗ M)

for all i ≥ 0, coming from the natural map M → Hom_{G_{F}} (A, A ⊗ M).

8.2. Lemma.

a) If M is torsion-free, the above pairing is an isomorphism.

b) If A is divisible, it is surjective.

c) If A and H^{i}(F, A) are divisible, it is bijective.

Proof. a) Reduce to the case where M is finitely generated, hence free. b) Reduce to the case where M is torsion-free by using the commutative diagram:

H^{i}(F, A) ⊗ M ⟶ H^{i}(F, A) ⊗ (M/ torsion)
↓  ^{≀}↓
H^{i}(F, A ⊗ M) ⟶^{∼} H^{i}(F, A ⊗ (M/ torsion)).

c) Use the same diagram, noting that now the top horizontal map is bijective as well.

Define a map

α = α^{i,j,k}: H^{i}(F, k) ⊗ CH^{j}X ⟶ H^{i}(F, H^{2j}(\overline{X}, j + k)) (8.3) as the composite

H^{i}(F, k) ⊗ CH^{j}X → H^{i}(F, CH^{j}X ⊗ Q/Z(k))

→ H^{i}(F, CH^{j}\overline{X} ⊗ Q/Z(k)) ⟶^{H^{i}(F,cl′j ; \overline{X}(k))} H^{i}(F, H_{ét}^{2j}(\overline{X}, j + k)).

We have:

8.3. Lemma. Let (H^{∗}(X, k+j)^{[m]})_{m≥0} be the decreasing filtration on H^{∗}(X, k+j) defined by the Hochschild-Serre spectral sequence. Then

a) The image of the map (8.1) is contained in H^{i+2j}(X, k + j)^{[i]}.

b) The image of α^{i,j,k} consists of universal cycles.

c) The diagram

H^{i}(F, k) ⊗ CH^{j}X ⟶^{α^{i,j,k}} H^{i}(F, H^{2j}(\overline{X}, k + j))_{d=0}
↓  ↓
H^{i+2j}(X, k + j)^{[i]} ⟶  HS_{∞}^{i,2j}

commutes, where the subscript d = 0 denotes universal cycles and HS_{∞}^{i,2j} denotes the E_{∞}-term of the Hochschild-Serre spectral sequence.

d) Suppose that H^{2j+1}(\overline{X}, k + j) = 0. Then, for i = 1, the triangle

H^{1}(F, k) ⊗ CH^{j}X ⟶^{α^{1,j,k}} H^{1}(F, H^{2j}(\overline{X}, k + j))
↓  ↗
H^{1+2j}(X, k + j)

commutes, where the vertical map is (8.1) and the diagonal map is the edge homomorphism from the Hochschild-Serre spectral sequence.

Proof. We argue as above: it is enough to prove this for the maps defined at the finite coefficients level (we could have defined α as a limit, as before). By the multiplicativity of the Hochschild-Serre spectral sequence, we are reduced to the case i = k = 0. Then a) is trivial, and b) and c) hold because α^{0,j,0} factors through the (F -rational) cycle map. Finally, d) follows as in the proof of lemma 8.1 from c) and the definition of the edge homomorphism.

The map α is the most important of the three maps introduced above in (8.1), (8.2), and (8.3). We are interested in its kernel and cokernel.

8.4. Lemma. If the geometric divisible cycle map is bijective, then

Coker α^{i,j,k} ≃ Coker(H^{i}(F, CH^{j}X ⊗ Q/Z(k)) → H^{i}(F, CH^{j}\overline{X} ⊗ Q/Z(k))).

If moreover G_{F} acts trivially on CH^{j}\overline{X}, this cokernel is a quotient of

Coker(H^{i}(F, k) ⊗ CH^{j}X → H^{i}(F, k) ⊗ CH^{j}\overline{X}).

Proof. The first claim follows from the definition of α^{i,j,k} and lemma 8.2. The second one follows from the commutative diagram

H^{i}(F, k) ⊗ CH^{j}X⟶ H^{i}(F, CH^{j}X ⊗ Q/Z(k))
↓  ↓
H^{i}(F, k) ⊗ CH^{j}\overline{X}⟶H^{i}(F, CH^{j}\overline{X} ⊗ Q/Z(k)).

8.5. Lemma. Let X be a quadric and k = i ≤ 3. Then a) If dim X ≠ 2j or dim X = 2j and d(X) = 1, α^{i,j,i} is surjective.

b) Suppose dim X = 2j and d(X) = d ≠ 1. Let E = F (\sqrt{d}). Then the composition

H^{i}(E, i) ⊗ CH^{j}X_{E} ⟶^{α_{E}^{i,j,i}} H^{i}(E, H^{2j}(\overline{X}, i + j)) ⟶^{Cor_{E/F}} H^{i}(F, H^{2j}(\overline{X}, i + j))

is surjective. For i = 0, α^{0,j,0} is itself surjective.

c) If dim X > 2j, α^{i,j,i} is bijective.

Proof. In case a), the geometric cycle map is bijective by proposition 1.5 and G_{F} acts trivially on CH^{j}\overline{X} by proposition 1.1, hence, by lemma 8.4, it is enough to see that the map

H^{i}(F, i) ⊗ CH^{j}X → H^{i}(F, i) ⊗ CH^{j}\overline{X}

is surjective. But, since i ≤ 3, the group H^{i}(F, i) is divisible by bijectivity of the 2-primary Galois symbol in this case. Surjectivity now follows from corollary 1.2, applied with D = H^{i}(F, i).

In case b), α_{E} is surjective by a). It remains to prove that

H^{i}(E, H^{2j}(\overline{X}, i + j)) ⟶^{Cor_{E/F}} H^{i}(F, H^{2j}(\overline{X}, i + j))

is surjective. By the geometric cycle map, the coefficients can be identified with CH^{j}\overline{X} ⊗ Q/Z(i) ≃ Ind_{G_{E}}^{G_{F}} Q/Z(i) by corollary 1.3. Using Shapiro’s lemma, the corestriction map can therefore be identified with

H^{i}(E, i) ⊕ H^{i}(E, i) ⟶^{Σ} H^{i}(E, i)

which is obviously surjective. Finally, the last claim is corollary 1.3.

In case c), the three maps out of which α is defined are bijective: the first one is by lemma 8.2 c), the second one by proposition 1.1 b) and the last one by proposition 1.5.

In section 11 we shall need the following lemma and its corollary for (j, k) = (2, 1).

8.6. Lemma. Let X be a quadric over F. Then:

a) If F contains all 2-primary roots of unity, α^{0,j,k} is surjective for any j, k. The same holds if k is even and F (\sqrt{−1}) contains all 2-primary roots of unity.

b) In general, Coker α^{0,j,k} is (i) 0 if dim X > 2j; (ii) cyclic of order ≤ 2 if dim X < 2j; (iii) cyclic of order ≤ 2^{t} if dim X = 2j and d(X) = 1, where t is the integer

of proposition 1.1 d).

Proof. By lemma 8.4,

Coker α^{0,j,k} ≃ Coker(H^{0}(F, CH^{j}X ⊗ Q/Z(k)) → H^{0}(F, CH^{j}\overline{X} ⊗ Q/Z(k))).

Assume first that we are not in the case dim X = 2j and d = d(X) ≠ 1. Applying proposition 1.1 and corollary 1.2 with D = Q/Z(k), we see that the map on coefficients

CH^{j}X ⊗ Q/Z(k) → CH^{j}\overline{X} ⊗ Q/Z(k)

can be identified with multiplication by |Coker(CH^{j}X → CH^{j}\overline{X})| on Q/Z(k), except if dim X = 2j and d(X) = 1, where it is diag(1, 2^{t}) on Q/Z(k) ⊕ Q/Z(k) with t as in prop. 1.1 d). If F contains all 2-primary roots of unity, this shows that α^{0,j,k} is surjective, and similarly if k is even and F (\sqrt{−1}) contains all 2-primary roots of unity. For arbitrary F, if dim X > 2j then CH^{j}X → CH^{j}\overline{X} is surjective (prop. 1.1) hence α^{0,j,k} is an isomorphism. If dim X < 2j (resp. dim X = 2j and d(X) = 1), the map CH^{j}X → CH^{j}\overline{X} has cokernel cyclic of order ≤ 2 (resp. ≤ 2^{t}), so that Coker α^{0,j,k} is cyclic of order ≤ 2 (resp. ≤ 2^{t}).

Assume now that dim X = 2j and d = d(X) ≠ 1; let E = F (\sqrt{d}). By corollary 1.3, α^{0,j,k} can be identified with the map H^{0}(F, k) → H^{0}(E, k), so is surjective when F and E have the same number of 2-primary roots of unity, in particular if F contains all 2-primary roots of unity; similarly if k is even and F (\sqrt{−1}) contains all 2-primary roots of unity.

8.7. Corollary. Let X be a quadric over F. Consider the composition

γ: H^{j}(X, H^{j}(j + k)) → H^{2j}(X, j + k) → H^{0}(F, H^{2j}(\overline{X}, j + k))

where the first (resp. second) map is an edge homomorphism of the coniveau (resp. Hochschild-Serre) spectral sequence. Then:

a) If F contains all 2-primary roots of unity or if k is even and F (\sqrt{−1}) contains all 2-primary roots of unity, γ is surjective.

b) In general, Coker γ is: (i) 0 if dim X > 2j; (ii) cyclic of order ≤ 2 if dim X < 2j; (iii) cyclic of order ≤ 2^{t} if dim X = 2j and d(X) = 1, where t is the integer

of proposition 1.1 d). (iv) cyclic of order ≤ 2^{t_{E}} if dim X = 2j and d(X) = d ≠ 1, where t_{E} is the

integer of proposition 1.1 d) for X_{E}, with E = F (\sqrt{d}).

Proof. By lemmas 8.1 and 8.3, the composition

H^{0}(F, k) ⊗ CH^{j}X → H^{j}(X, H^{j}(j + k)) ⟶^{γ} H^{0}(F, H^{2j}(\overline{X}, j + k))

coincides with α^{0,j,k}. Therefore corollary 8.7 follows directly from lemma 8.6, except when dim X = 2j, d(X) = d ≠ 1 and F does not contain enough roots of unity. But in this case, we have a commutative diagram:

H^{0}(E, k) ⊗ CH^{j}X_{E} ⟶ H^{j}(X_{E}, H^{j}(j + k)) ⟶^{γ_{E}} H^{0}(E, H^{2j}(\overline{X}, j + k))
Cor^{E/F} ↓  ^{Cor}^{E/F} ↓
H^{0}(F, k) ⊗ CH^{j}X ⟶ H^{j}(X, H^{j}(j + k)) ⟶^{γ} H^{0}(F, H^{2j}(\overline{X}, j + k))

where the left Cor is best described from the E_{1}-term of the coniveau spectral sequence. By lemma 8.6, Coker γ_{E} is cyclic of order ≤ 2^{t_{E}}. But

Cor_{E/F} : H^{0}(E, H^{2j}(\overline{X}, j + k)) → H^{0}(F, H^{2j}(\overline{X}, j + k))

is surjective: this follows from the bijectivity of the geometric cycle map and the fact that the Galois module CH^{j}\overline{X} is isomorphic to Ind_{G_{E}}^{G_{F}} Z. This concludes the proof of corollary 8.7.

9. The case i = 2: Proof of theorem 4

Let X be a smooth variety over F such that H^{1}(\overline{X}, Q/Z) = 0. We have a cross of exact sequences

0

↓

H^{2}(F, 1)
↓  ↘ ^{η^{2}}
0 ⟶ H^{1}(X, H^{1}(1)) ⟶  H^{2}(X, 1)  ⟶ H^{0}(X, H^{2}(1)) ⟶ 0
α^{0,1,0} ↘  ↓
H^{0}(F, H^{2}(\overline{X}, 1))

↓

H^{3}(F, 1)

↓

H^{3}(X, 1)

where the horizontal (resp. vertical) line is given by the coniveau (resp. Hochschild-Serre) spectral sequence with coefficients Q/Z(1). The group H^{1}(X, H^{1}(1)) can be identified with CH^{1}X ⊗ Q/Z and H^{1}(X, H^{1}(1)) → H^{2}(X, 1) with cl′_{X}^{1}. The composite

H^{1}(X, H^{1}(1)) ⟶ H^{2}(X, 1) → H^{0}(F, H^{2}(\overline{X}, 1))

defined by the above cross can therefore be identified with the map α^{0,1,0} (8.3) of section 8.

9.1. Proposition. For X as above, there is an exact sequence

0 → Coker η^{2} → Coker α^{0,1,0} → H^{3}(F, 1) → H^{3}(X, 1).

Proof. This follows from the “lemma of the 700th” [33].

9.2. Corollary. For any (projective) quadric X over F, η^{2} is surjective.

Proof. α^{0,1,0} is surjective for all quadrics by lemma 8.5.

10. The case i = 3: Proof of theorem 5

Let X be a smooth variety over F such that H^{1}(\overline{X}, Q/Z) = H^{3}(\overline{X}, Q/Z) = 0. In the following cross of exact sequences the (broken) horizontal sequence comes from the coniveau spectral sequence and the vertical sequence from the Hochschild-Serre spectral sequence with coefficients Q/Z(2).

H^{3}(F, 2)
↓  ↘ ^{η^{3}}
0 ⟶ H^{1}(X, H^{2}(2)) ⟶  H^{3}(X, 2)  ⟶ H^{0}(X, H^{3}(2))
β ↘  ↓  ↓
H^{1}(F, H^{2}(\overline{X}, 2))  H^{2}(X, H^{2}(2))
↓  ↓
H^{4}(F, 2)  H^{4}(X, 2)
↓
H^{4}(X, 2).

In this diagram, one can identify the group H^{2}(X, H^{2}(2)) with CH^{2}X ⊗ Q/Z and accordingly the map H^{2}(X, H^{2}(2)) → H^{4}(X, 2) with cl′_{X}^{2}. The above cross defines a composite homomorphism:

β: H^{1}(X, H^{2}(2)) ⟶ H^{3}(X, 2) → H^{1}(F, H^{2}(\overline{X}, 2)).

10.1. Proposition. For X as above, there is a natural surjection

Coker η^{3} → Ker cl′_{X}^{2} → 0.

Denote by (Coker η^{3})^{1} the kernel of this surjection. Then there is a natural exact sequence

0 → (Coker η^{3})^{1} → Coker β → H^{4}(F, 2) → H^{4}(X, 2).

Proof. The first claim is obvious and the second again follows from the lemma of the 700th [33].

10.2. Corollary. For any (projective) quadric X over F, there is a natural isomorphism Coker η^{3} ⟶^{∼} Ker cl′_{X}^{2}.

Proof. We prove that β is surjective for all quadrics. For this, we remark that the composition

H^{1}(F, 1) ⊗ CH^{1}X → H^{1}(X, H^{2}(2)) ⟶^{β} H^{1}(F, H^{2}(\overline{X}, 2)),

where the first map is (8.2), coincides with α^{1,1,1} by lemmas 8.1 and 8.3. If dim X ≠ 2 or dim X = 2 and d(X) = 1, lemma 8.5 says that α^{1,1,1} is surjective, hence β is surjective. If dim X = 2 and d = d(X) ≠ 1, we use the same trick as in the proof of corollary 8.7: the composite Cor_{E/F} ∘α_{E}^{1,1,1} is surjective, still by lemma 8.5, where

E = F (\sqrt{d}). But there is a transfer map H^{1}(X_{E}, H^{2}(2)) ⟶^{Cor_{E/F}} H^{1}(X, H^{2}(2)) making the following diagram commute:

H^{1}(E, 1) ⊗ CH^{1}X_{E} ⟶ H^{1}(X_{E}, H^{2}(2)) ⟶^{β_{E}} H^{1}(E, H^{2}(\overline{X}, 2))
Cor^{E/F} ↓  ^{Cor}^{E/F} ↓
H^{1}(F, 1) ⊗ CH^{1}X ⟶ H^{1}(X, H^{2}(2)) ⟶^{β} H^{1}(F, H^{2}(\overline{X}, 2)),

which concludes the proof in this case.

10.3. Proposition (cf. [17, th. 3]). For a quadric X over a field of characteristic ≠ 2, the cycle map

cl′_{X}^{2} : CH^{2}X ⊗ Q/Z → H^{4}(X, 2)

is injective, except if X is defined by an anisotropic Albert form. In this case, Ker cl′_{X}^{2} ≃ Z/2.

Proof. In view of corollary 10.2, we may assume X anisotropic (prop. 2.5). If dim X > 4 or dim X = 4 and d(X) ≠ 1, the map

CH^{2}X ⊗ Q/Z → H^{0}(F, CH^{2}\overline{X} ⊗ Q/Z)

is an isomorphism by proposition 1.1 and the claim follows from the injectivity of the geometric cycle map (proposition 1.5). If dim X = 1, CH^{2}X = 0 and the claim is trivial. If dim X = 3 and Y is a hyperplane section of X, then the restriction map CH^{2}X ⊗ Q/Z → CH^{2}Y ⊗ Q/Z is an isomorphism by proposition 1.1 again, so that injectivity for dim X = 2 implies injectivity for dim X = 3. Let us assume dim X = 2. As in [17], we may assume d(X) = 1 since, if d = d(X) ≠ 1, the map CH^{2}X → CH^{2}X_{F (\sqrt{d})} is bijective (prop. 1.1 and 1.6). In this case, CH^{2}X is generated by h^{2}, where h ∈ CH^{1}X is the class of a hyperplane section (ibid.). Since Ker cl′_{X}^{2} is killed by 2 (cor. 10.2 and prop. 2.5), it is enough to prove:

10.4. Lemma. If X is an anisotropic quaternion quadric (= defined by a 2-fold Pfister form), the image of cl_{X}^{2}(h^{2}) ∈ H^{4}X in H^{4}(X, 2) is ≠ 0.

Proof. Indeed, cl_{X}^{2}(h^{2}) = c · cl_{X}^{1}(h) by [17, lemma 2], where c is the Brauer class of the quadric, and cl_{X}^{1}(h) ≠ 0 by [51, lemma 5.3.2 c)]. Therefore cl_{X}^{2}(h^{2}) maps to a nonzero element of H^{2}(F, H^{2}\overline{X}) ≃ H^{2}F. Its image in H^{2}(F, H^{2}(\overline{X}, 2)) ≃ H^{2}(F, 1) is not 0 since the map H^{2}F → H^{2}(F, 1) is injective (compare beginning of proof of prop. 7.1). Therefore the image of cl_{X}^{2}(h^{2}) in H^{4}(X, 2) cannot be 0.

It remains to deal with the case of an anisotropic Albert quadric. In this case we have Ker η_{2}^{3} = 0 [1, Satz 5.6] and Coker η_{2}^{3} ≃ Z/2 [17, theorem 2 d)], hence from proposition 7.4 and corollary 10.2 we get an injection

(10.1) Z/2 ≃ Coker η_{2}^{3} ↪ Ker cl′_{X}^{2}.

But propositions 1.5 and 1.1 d) yield a commutative diagram with exact row

0 ⟶ Z/4 ⟶ CH^{2}X ⊗ Q/Z ⟶ CH^{2}\overline{X} ⊗ Q/Z ⟶ 0
cl′_{X}^{2} ↓  ^{cl′2 ; \overline{X}} ↓^{≀}
H^{4}(X, 2) ⟶ H^{4}(\overline{X}, 2)

It follows that Ker cl′_{X}^{2} is cyclic; as it is of exponent 2 by corollary 10.2 and proposition 2.5 b), the map (10.1) is an isomorphism.

Theorem 5 follows from corollary 10.2 and proposition 10.3.

UNRAMIFIED COHOMOLOGY OF QUADRICS, I 31

11. The case i = 4

This case is similar to the former two, but more complicated.

For any smooth variety X over F define

\overline{H}^{4}(X, 3) = \frac{H^{4}(X, 3)}{H^{4}(F, 3) + H^{2}(X, H^{2}(3)) + ⨁_{[E:F]<+∞} H^{2}(E, 2) ⊗ CH^{1}X_{E}} ,

where H^{2}(X, H^{2}(3)) → H^{4}(X, 3) is the edge homomorphism of the coniveau spectral sequence and the map ⨁_{[E:F]<+∞} H^{2}(E, 2) ⊗ CH^{1}X_{E} → H^{4}(X, 3) comes from (8.1) via transfer.

Suppose that H^{1}(\overline{X}, Q/Z) = H^{3}(\overline{X}, Q/Z) = 0. We then have two exact sequences

(11.1) 0 → H^{1}(X, H^{3}(3)) → \frac{H^{4}(X, 3)}{H^{2}(X, H^{2}(3))} → H^{0}(X, H^{4}(3)) ⟶

⟶^{d_{2}^{0,4}} H^{2}(X, H^{3}(3)) → H^{5}(X, 3)

(coniveau)

(11.2) 0 → Ker(H^{2}(F, H^{2}(\overline{X}, 3)) ⟶^{d_{3}^{2,2}} H^{5}(F, 3)) →

→ \frac{H^{4}(X, 3)}{H^{4}(F, 3)} → H^{0}(F, H^{4}(\overline{X}, 3))

(Hochschild-Serre).

The exact sequence (11.2) identifies Ker d_{3}^{2,2} to a subgroup of \frac{H^{4}(X, 3)}{H^{4}(F, 3)}. For the purpose of stating proposition 11.1 below, let H^{2}(X, H^{2}(3))_{0} denote its intersection with Im (H^{2}(X, H^{2}(3)) → H^{4}(X, 3) → \frac{H^{4}(X, 3)}{H^{4}(F, 3)}) (the first map in this composition being an edge homomorphism of the coniveau spectral sequence). 11.1. Proposition. For X as above, there are exact sequences

\overline{H}^{4}(X, 3) → Coker η^{4} → H^{2}(X, H^{3}(3)) → H^{5}(X, 3)

and

\frac{Ker(H^{2}(F, H^{2}(\overline{X}, 3)) ⟶^{d_{3}^{2,2}} H^{5}(F, 3))}{H^{2}(X, H^{2}(3))_{0} + ⨁_{[E:F]<+∞} H^{2}(E, 2) ⊗ CH^{1}X_{E}} → \overline{H}^{4}(X, 3) → \frac{H^{0}(F, H^{4}(\overline{X}, 3))}{H^{2}(X, H^{2}(3))},

where the factor

⨁_{[E:F]<+∞} H^{2}(E, 2) ⊗ CH^{1}X_{E}

comes from

∑_{[E:F]<+∞} Cor_{E/F} ∘ α_{E}^{2,1,2}.

Proof. This follows from the exact sequences (11.1) and (11.2), using lemma 8.3.

Consider the cross of exact sequences

H^{1}(F, 1) ⊗ CH^{2}X
↓  ν ↘
\overline{H}^{4}(X, 3) ⟶ Coker η^{4} ⟶ H^{2}(X, H^{3}(3)) ⟶ H^{5}(X, 3)

↓

(CH^{3}X)_{torsion}

↓

0 (11.3)

coming from propositions 2.4 (for i = 3) and 11.1. From diagram (11.3) and the lemma of the 700th, we deduce a complex

(11.4) \overline{H}^{4}(X, 3) → Coker η^{4} → (CH^{3}X)_{torsion} → Coker ν

which is exact at (CH^{3}X)_{torsion} and whose homology at Coker η^{4} is a quotient of Ker ν.

11.2. Lemma. Let X be a quadric. Then \overline{H}^{4}(X, 3) is cyclic of order ≤ 4; it is 0 if dim X = 1, dim X > 4 or if F contains all 2-primary roots of unity.

Proof. By lemma 8.5, α^{2,1,2} is surjective, except if dim X = 2 and d = d(X) ≠ 1; in this case, Cor_{E/F} ∘α_{E}^{2,1,2} is surjective. It follows that for any quadric, the group on the left of \overline{H}^{4}(X, 3) in the second exact sequence of proposition 11.1 is 0. Therefore

\overline{H}^{4}(X, 3) injects into \frac{H^{0}(F, H^{4}(\overline{X}, 3))}{H^{2}(X, H^{2}(3))}. But the latter group is cyclic of order as

in the statement of lemma 11.2, by corollary 8.7 and proposition 1.1.

11.3. Lemma. If dim X = 1 or dim X > 4, the map ν of diagram (11.3) is injective. If dim X > 4, there is a canonical isomorphism

Coker ν ≃ H^{5}(F, 3) ⊕ (H^{3}(F, 2) ⊗ CH^{1}X).

Proof. The case dim X = 1 is obvious, since then CH^{2}X = 0. Suppose dim X > 4. By lemma 8.1 d), the map ν coincides with (8.1). By lemma 8.5 c), α^{1,2,1} is bijective. Therefore, by lemma 8.3 d), ν is (split-) injective. This argument shows that, in the Hochschild-Serre spectral sequence for coefficients Q/Z(3), the term E_{2}^{1,4} is isomorphic to H^{1}(F, 1) ⊗ CH^{2}X and consists of universal cycles. The same holds for all the terms E_{2}^{p,2} ≃ H^{p}(F, 2) ⊗ CH^{1}X by the same argument, since CH^{1}X ⟶^{∼} CH^{1}\overline{X} for dim X ≥ 3. Finally the term E_{2}^{0,4} also consists of universal cycles by lemma 11.2. Hence we get a canonical isomorphism:

H^{5}(X, 3) ≃ H^{5}(F, 3) ⊕ (H^{3}(F, 2) ⊗ CH^{1}X) ⊕ (H^{1}(F, 1) ⊗ CH^{2}X)

and the second part of the lemma follows.

12. Proof of theorems 6 and 7

Proof of theorem 6. Claim (1) follows from (11.4), lemma 11.2 and lemma 11.3. We now check (2) and (3). Let q be a quadratic form defining the quadric X.

a) If dim q = 3, Coker η^{4} = 0 by (11.4), lemma 11.2 and lemma 11.3.

b) If dim q > 6, part (1) of theorem 6 implies that Coker η^{4} injects into the group (CH^{3}X)_{torsion}, hence has order ≤ 2 by Karpenko’s results (proposition 1.6). If dim q > 12, (CH^{3}X)_{torsion} = 0 by the same results, and so Coker η^{4} = 0.

c) If dim q = 8, d_{±}q = 1, and q is not similar to a Pfister form, then CH^{3}X is torsion-free by [20, th. 8.3] and Coker η^{4} = 0 by (1).

The case of a 2-fold Pfister neighbour and that of an i-fold Pfister neighbour for i ≥ 4 is covered by a) and b) respectively.

It remains to prove claim (3) in the case of a 3-fold neighbour. We may assume the quadric to be of dimension at least 5 by proposition 2.5 c). Then the claim follows from:

12.1. Lemma. Suppose dim X > 4 and X is defined by a neighbour of a 3-fold Pfister form. Then the map of theorem 6 (1)

(CH^{3}X)_{torsion} → H^{5}(F, 3) ⊕ (H^{3}(F, 2) ⊗ CH^{1}X)

is injective.

Proof. By the same argument as in the proof of lemma 11.3, we get a canonical isomorphism

H^{3}(X, 2) ≃ H^{3}(F, 2) ⊕ (H^{1}(F, 1) ⊗ CH^{1}X)

compatible with the maps of section 8. The diagram analogous to (11.3)

H^{1}(F, 1) ⊗ CH^{1}X

↓

0 ⟶ H^{1}(X, H^{2}(2)) ⟶ H^{3}(X, 2)

↓

(CH^{2}X)_{torsion}

↓

0

yields an injection

(CH^{2}X)_{torsion} ⟶ H^{3}(F, 2).

Tensoring this injection by CH^{1}X yields another injection (since CH^{1}X is torsion-free)

(CH^{2}X)_{torsion} ⊗ CH^{1}X ⟶ H^{3}(F, 2) ⊗ CH^{1}X.

The way these maps are defined shows that the diagram

(CH^{2}X)_{torsion} ⊗ CH^{1}X ⟶  H^{3}(F, 2) ⊗ CH^{1}X
↓  ↓
(CH^{3}X)_{torsion}  ⟶ H^{5}(F, 3) ⊕ (H^{3}(F, 2) ⊗ CH^{1}X)

commutes, where the left vertical map is multiplication, the right vertical map is inclusion, the top horizontal map has just been defined and the bottom horizontal map comes from theorem 6 (1). Since CH^{1}X ≃ Z and (CH^{2}X)_{torsion} ≃ Z/2 by propositions 1.1 and 1.6, this shows that the latter map is nonzero. Since, by proposition 1.6, (CH^{3}X)_{torsion} has order at most 2, this map is injective, which is what we wanted (moreover, we get that CH^{1}X ⊗ (CH^{2}X)_{torsion} → (CH^{3}X)_{torsion} is bijective, which fits with what Karpenko’s method gives [21]).

Proof of theorem 7. In the case where X is defined by an anisotropic Albert form q, we define a homomorphism

ρ: H^{1}F ⟶ Coker η_{2}^{4}

by cup-product with a representative of Coker η_{2}^{3} ≃ Z/2.

Recall [17] that such a representative e is constructed as follows: let K = F (X) and q_{K} ∼ aτ, where a ∈ K^{∗} and τ is a 2-fold Pfister form. Then e = (a) · e^{2}(τ).

The composition H^{1}F → Coker η_{2}^{4} → Coker η^{4} has infinite image if we pass from F to the rational function field F (t). Indeed, F is infinite since q is anisotropic. Then, for any α ∈ F, the element

(t − α) · e ∈ H^{4}F (X)(t) ↪ H^{4}(F (X)(t), 3)

is unramified over F (t), and its residue at α is e ∈ H^{3}(F (X), 2). This shows that the classes of the (t − α) · e in Coker η_{F (t)}^{4} are linearly independent.

In the case of a virtual Albert form q of discriminant d, we define a homomorphism

ρ′: E^{∗} ⟶ Coker η_{2}^{4}

simply as the composite

E^{∗} ⟶^{ρ} Coker η_{2,q_{E}}^{4} ⟶^{Cor_{E/F}} Coker η_{2}^{4}

where Coker η_{2,q_{E}}^{4} is the cokernel attached to the form q_{E} (E = F (\sqrt{d})). We show just as above that Coker η_{F (t)}^{4} is infinite, now using rational points of A_{E}^{1} not defined over F.

These arguments can be summarized by noting that the groups Coker η^{i} are cycle modules in the sense of [42] and using the exact sequence

0 → Coker η_{F}^{4} → Coker η_{F (t)}^{4} → ⨁_{y∈A_{(0)}^{1}} Coker η_{F_{y}}^{3} → 0

cf. the exact sequence (∗) in the introduction.

Exercise. Let X be an anisotropic quadric, a ∈ F^{∗} and E = F (\sqrt{a}) a quadratic extension such that X_{E} is isotropic.

a) Show that extension of scalars defines a homomorphism

θ_{E}^{n}: H^{0}(X, H^{n}) → H^{n}E

for all n ≥ 0.

b) Define a 0-sequence

H^{3}F ⟶^{·(a)} Ker θ_{E}^{4} ⟶ Ker(CH^{3}X → CH^{3}X_{E})

(proceed by a diagram chase, as in [27] for the definition of the map Ker η_{2}^{4} → Ker(H^{2}(X, K_{3}) → H^{2}(X_{E}, K_{3}))). c) Compare the map Ker θ_{E}^{4} → Ker(CH^{3}X → CH^{3}X_{E}) of b) with the map

Coker η_{2}^{4} → (CH^{3}X)_{torsion} deduced from theorem 6.

Appendix A. Quadrics of dimension 0 and 1

A.1. Proposition. Let dim X = 1 and suppose the Milnor conjecture holds in degree i − 2. Then Coker η^{i} = 0.

Proof. It follows from the cross of exact sequences

H^{i}(F, i − 1)
↓
0 → H^{1}(X, H^{i−1}(i − 1)) → H^{i}(X, i − 1) → H^{0}(X, H^{i}(i − 1)) → 0
↓
H^{i−2}(F, H^{2}(\overline{X}, i − 2))

(coniveau and Hochschild-Serre), by using the fact that the composition

H^{i−2}(F, i − 2) ⊗ CH^{1}X → H^{1}(X, H^{i−1}(i − 1)) → H^{i−2}(F, H^{2}(\overline{X}, i − 2))

is surjective. As above, this is deduced from the divisibility of H^{i−2}(F, i − 2) and corollary 1.2.

In the following we present analogues of proposition A.1 and proposition 3 for 0-dimensional quadrics. Let L = F (\sqrt{d}) be a quadratic extension with Galois group G = Gal(L/F).

A.2. Proposition. Suppose the Milnor conjecture holds in degree i − 1. Then a) The restriction map

H^{i}(F, i − 1) → (H^{i}(L, i − 1))^{G}

is surjective.

b) The restriction map

H^{i}F → (H^{i}L)^{G}

has kernel (d) · H^{i−1}F and cokernel isomorphic to

{ x ∈ (d) · H^{i−1}F | (−1) · x = 0 }.

Proof. a) Let σ be the generator of G and consider the exact sequence of Galoismodules

0 → Z → Z[G] ⟶^{1−σ} Z[G] → Z → 0.

Tensoring with Q/Z(i − 1) gives a spectral sequence converging to 0, with E_{1}- term:

0 ⟶ H^{0}(F,i−1) ⟶^{Res} H^{0}(L,i−1) ⟶^{1−σ} H^{0}(L,i−1) ⟶^{Cor} H^{0}(F,i−1) ⟶ 0
⋯
0 ⟶ H^{i−1}(F,i−1) ⟶^{Res} H^{i−1}(L,i−1) ⟶^{1−σ} H^{i−1}(L,i−1) ⟶^{Cor} H^{i−1}(F,i−1) ⟶ 0
0 ⟶ H^{i}(F,i−1) ⟶^{Res} H^{i}(L,i−1) ⟶^{1−σ} H^{i}(L,i−1) ⟶^{Cor} H^{i}(F,i−1) ⟶ 0
⋯

Here the d^{2}-differentials go 2 steps right and 1 step up. The norm map

H^{i−1}(L, i − 1) ⟶^{Cor} H^{i−1}(F, i − 1)

is computed as the norm map

K_{i−1}^{M}L ⊗ Q/Z ⟶^{N} K_{i−1}^{M}F ⊗ Q/Z,

which is obviously surjective. Hence

H^{i}(F, i − 1) ⟶^{Res} H^{i}(L, i − 1) ⟶^{1−σ} H^{i}(L, i − 1)

is exact, i.e., η^{i} is surjective.

This discussion holds for any cyclic extension L/F with the degree [L: F] prime to char F.

b) The statement on the kernel is classical [1]; the statement on the cokernel follows from this one and a), by using proposition 7.4, which is shown to apply in this case just as in section 7.

Appendix B. Application to real quadrics

In this section, we examine the case when the ground field is the field R of real numbers. The real anisotropic quadric of dimension d, denoted Q_{d}, is defined by the (d + 2)-dimensional form (d + 2)⟨1⟩. As all anisotropic quadratic forms over R are Pfister neighbours, the anisotropic quadrics can be grouped according to the level of their function fields (cf. [10, §3]). As observed in [10, §3], the unramified cohomology groups are invariant for the set of quadrics grouped in this manner. In [10], it was proved that the unramified cohomology group H^{0}(Q_{d}, H^{3}) is 0 for d ≤ 6 and that the map Z/2 ≃ H^{3}(R) → H^{0}(Q_{d}, H^{3}) is bijective for d > 7. This result was used to show that the cycle map in codimension two

CH^{2}Q_{d}/2 → H^{4}Q_{d}

is injective for all quadrics Q_{d} (compare also [17, th. 3]).

We extend these results in this section, and use them to get extra information on the cohomology of real quadrics. Concerning unramified cohomology, results can be expressed concisely in a striking way:

B.1. Theorem. Let F = R and i ≤ 4. Then, for any d > 0, the map H^{i}(R) → H^{0}(Q_{d}, H^{i}) is surjective. If d > 2^{i} −2 it is bijective; if d ≤ 2^{i} −2, H^{0}(Q_{d}, H^{i}) = 0.

Proof. For i = 2 it is classical and for i = 3 this is the main result of [10]. Note however that these cases also follow directly from proposition 7.4, theorem 6 and the fact that there are no anisotropic Albert quadrics over R in view of

B.2. Lemma. With notation as above, if F = R then (Ker η_{2}^{i})_{0} = 0 for all quadrics X and all i ≥ 1.

Proof. Cup-product by (−1) is injective on H^{∗}R.

Let us now prove theorem B.1 in the case i = 4. For d > 14 and for 6 < d ≤ 14, this follows from theorem 3.4 and corollary 8. For 3 < d ≤ 6, we may assume d = 3; then cd(R(Q_{3})) = 3 [47], hence H^{4}(R(Q_{3})) = 0 and a fortiori H^{0}(Q_{3}, H^{4}) = 0. The argument is the same for d ≤ 2.

Theorem B.1 implies in particular that, in the coniveau spectral sequence

E_{2}^{p,q} = H^{p}(Q_{d}, H^{q}) ⇒ H^{p+q}Q_{d},

E_{2}^{0,q} consists of universal cycles for q ≤ 4. This in turn implies:

B.3. Theorem.

a) For any d ≥ 1, we have:

dim H^{1}(Q_{d}, H^{2}) + dim H^{0}(Q_{d}, H^{3}) = dim H^{3}Q_{d};

dim CH^{2}Q_{d} ⊗ Z/2 + dim H^{1}(Q_{d}, H^{3}) + dim H^{0}(Q_{d}, H^{4}) = dim H^{4}Q_{d}.

b) The “edge homomorphism” H^{2}(Q_{d}, H^{3}) → H^{5}Q_{d} is injective.

B.4. Corollary. We have:

dim H^{1}(Q_{d}, H^{2}) = \begin{cases}0 & for d = 1 \\ 1 & for d = 2 \\ 2 & for 3 ≤ d ≤ 6 \\ 1 & for d > 6 \end{cases}

and

dim H^{1}(Q_{d}, H^{3}) = \begin{cases}0 & for d ≤ 3 \\ 1 & for 4 ≤ d ≤ 6 \\ 2 & for 7 ≤ d ≤ 14 \\ 1 & for d > 14. \end{cases}

Sketch of proof. An inductive argument allows one to compute dim H^{∗}Q_{d}. The method is to first compute the cohomology of the corresponding affine quadric by reduction to the complex case. The answer is [19]:

dim H^{n}Q_{d} = \begin{cases}[\frac{n}{2}] + 1 & if d ≥ n \\ [\frac{2d − n}{2}] + 1 & if d ≤ n. \end{cases}

The corollary follows from this fact, from theorem B.1, theorem B.3, and from (for the value of dim CH^{2}Q_{d} ⊗ Z/2) propositions 1.1 and 1.6.

Further results on H^{2}(Q_{d}, H^{3}), H^{1}(Q_{d}, H^{4}) and the cycle map cl_{Q_{d}}^{3} can be obtained at a higher cost [19].

Appendix C. The unramified Witt ring

In this appendix, we prove theorem 9, which we now recall:

C.1. Theorem. Let i ≤ 4. For any quadric X, the composition

Coker χ^{i} → Coker η_{2}^{i} → Coker η^{i}

is injective, where

χ^{i}: I^{i}F/I^{i+1}F ⟶ I_{nr}^{i}(F (X)/F)/I_{nr}^{i+1}(F (X)/F)

is the map induced by extension of scalars and the first map in the composition is induced by the cohomological invariant ē^{i}.

C.2. Lemma. Let C be a conic over F and K = F (C). Let c ∈ H^{2}F be the class of the quaternion algebra corresponding to C. Then, for i ≤ 4,

a) Ker η_{2}^{i} = c · H^{i−2}F.

b) The natural map

I^{i+1}F → I_{nr}^{i+1}(K/F)

is surjective.

Proof. a) has been given earlier in the paper for i ≤ 3 and is proven in [30, prop. 3.15] for i = 4. Let us prove b). By a result of Parimala [37, 53, 10], the map W (F) → W_{nr}(K/F) is surjective. Let φ ∈ I_{nr}^{i+1}(K/F) and, by induction on i, let ψ ∈ I^{i}F be such that ψ_{K} = φ. We have e^{i}(ψ) ∈ Ker η_{2}^{i}, hence e^{i}(ψ) = c · x for some x ∈ H^{i−2}F by a). By the surjectivity of e^{i−2} and the injectivity of ē^{i}, we can write ψ = τ ⊗ θ ⊥ ψ′, where τ is the quaternion form corresponding to c, e^{i−2}(θ) = x and ψ′ ∈ I^{i+1}F. Then ψ′_{K} = φ, as desired.

Notation. We write Ī^{i}F = I^{i}F/I^{i+1}F and, for a finitely generated extension L/F, Ī_{nr}^{i}(L/F) = I_{nr}^{i}(L/F)/I_{nr}^{i+1}(L/F).

C.3. Lemma. Let X be a quadric (of dimension > 0), and let C be a conic lying on X. Let τ be the quaternion form associated to C. Then, for i ≤ 4, there is a surjection

τĪ^{i−2}F (X) ∩ Ī_{nr}^{i}(F (X)/F) ⟶ Coker χ^{i}.

Proof. Let K = F (C). Consider the commutative diagram

Ī_{nr}^{i}(F (X)/F) ⟶ Ī_{nr}^{i}(K(X)/F)
χ^{i}↑  ^{≀}↑
Ī^{i}F  ⟶ Ī_{nr}^{i}(K/F)

In this diagram, the lower horizontal map is surjective by lemma C.2 b) and the right vertical one is an isomorphism because K(X)/K is purely transcendental. A diagram chase then yields a surjection

Ker(Ī_{nr}^{i}(F (X)/F) ⟶ Ī_{nr}^{i}(K(X)/F)) → → Coker χ^{i}.

To conclude the proof, it suffices to observe that Ker(Ī^{i}F (X) ⟶ Ī^{i}K(X)) = τĪ^{i−2}F (X) by lemma C.2 a) and the bijectivity of ē^{i−2} and ē^{i}.

Proof of theorem C.1. Consider the commutative diagram

Coker χ^{i}  ⟶^{ē^{i}} Coker η_{2}^{i}  ⟶^{λ}  Coker η^{i}
↑  ↑  ↑
Ī_{nr}^{i}(F (X)/F) ⟶^{ē^{i}} H_{nr}^{i}(F (X)/F) ⟶^{λ} H_{nr}^{i}(F (X)/F, 2)
χ^{i}↑  ^{η_{2}^{i}}↑  ^{η^{i}}↑
Ī^{i}F  ⟶^{ē^{i}}  H^{i}F  ⟶^{λ}  H^{i}(F, 2)

where λ is the natural map. Let x ∈ Ī_{nr}^{i}(F (X)/F). Choose C as in lemma C.3. By this lemma, we can modify x by an element from I^{i}F so that x = τ ⊗ θ. Let y ∈ H^{i}(F, 2) be such that λē^{i}(x) = η^{i}(y). Let E/F be a quadratic extension such that C_{E}, hence X_{E}, is isotropic. Then

τ_{E} ∼ 0

hence

x_{E(X)} = 0

and

η^{i}(y)_{E(X)} = η^{i}(y_{E}) = 0.

Since X_{E} is isotropic, E(X)/E is purely transcendental and y_{E} = 0 (prop. 2.5 a)). Hence 2y = 0 ∈ H^{i}(F, 2) by the usual transfer argument, and y = λ(z) for some z ∈ H^{i}F. By the surjectivity of ē^{i}, write z = ē^{i}(t) for t ∈ Ī^{i}F. Then

λē^{i}(x) = η^{i}λē^{i}(t) = λē^{i}(χ^{i}(t))

hence x = χ^{i}(t) by the injectivity of ē^{i} and λ.

References

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

[2] S.A. Amitsur, Generic splitting fields and central simple algebras, Ann. of Math. 62 (1955), 8–43.

[3] F.A. Bogomolov, The Brauer group of quotient spaces by linear group actions, Izv. Akad. Nauk. SSSR 51 (1987), 485–516. English translation: Math. USSR Izv. 30 (1988), 455–485.

[4] S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. Ec. Norm. 7 (1974), 181–202.

[5] J.-L. Colliot-Thélène, Cycles algébriques de torsion et K-théorie algébrique, Arithmetical algebraic geometry (Trento, 1991), E. Ballico, ed., Lect. Notes in Math. 1553, Springer, Berlin-Heidelberg-New York, 1993, 1–49.

[6] J.-L. Colliot-Thélène, On the reciprocity sequence in higher class field theory of function fields, in Algebraic K-theory and algebraic topology (P.G. Goerss and J.F. Jardine, eds), NATO ASI Series, Ser. C 407 (1993), 35–55.

[7] J.-L. Colliot-Thélène, Birational invariants, purity and the Gersten conjecture, in: W. Jacob and A. Rosenberg (ed.), K-theory and algebraic geometry: connections with quadratic forms and division algebras, Proceedings of Symposia in Pure Mathematics 58 (I) (1995), 1–64.

[8] J.-L. Colliot-Thélène and M. Ojanguren, Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford, Invent. Math. 97 (1989), 141–158.

[9] J.-L. Colliot-Thélène and R. Parimala, Real components of algebraic varieties and étale cohomology, Invent. Math. 105 (1991), 221–245.

[10] J.-L. Colliot-Thélène and R. Sujatha, Unramified Witt groups of real anisotropic quadrics, in: W. Jacob and A. Rosenberg (ed.), K-theory and algebraic geometry: connections with quadratic forms and division algebras, Proceedings of Symposia in Pure Mathematics 58 (II) (1995), 127–147.

[11] P. Deligne, Quadriques, Groupes de Monodromie en géométrie algébrique (SGA 7), exposé XII, Lect. Notes in Math. 340, Springer, Berlin-Heidelberg-New York (1973), 62–81.

[12] J. Dieudonné, Les extensions quadratiques des corps non commutatifs et leurs applications, Acta Math. 87 (1952), 175–242.

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

[14] H. Esnault, B. Kahn, M. Levine, E. Viehweg, The Arason invariant and mod 2 algebraic cycles, to appear in J. Amer. Math. Soc.

[15] R.W. Fitzgerald, Witt kernels of function field extensions, Pacific J. Math. 109 (1983), 89– 106.

[16] W. Jacob and M. Rost, Degree four cohomological invariants for quadratic forms, Invent. Math. 96 (1989), 551–570.

[17] B. Kahn, Lower H-cohomology of higher-dimensional quadrics, Archiv Math. (Basel) 65 (1995), 244–250.

[18] B. Kahn, On the Scharlau transfer, Rocky Mountain J. Math. 19 (1989), 741–747.

[19] B. Kahn, R. Sujatha, Unramified cohomology of quadrics, II, in preparation.

[20] N. Karpenko, Algebro-geometric invariants of quadratic forms (in Russian), Algebra-i-Analiz 2 (1990), 141–162, 262. English translation: Leningrad Math. J. 2 (1991), 119–138.

[21] N. Karpenko, Cycles of codimension 3 on a projective quadric (in Russian), Zapiski Nauch. Sem. 191 (1991), 114–123. English translation: Journal of Soviet Mathematics 63 (1993), 678–682.

[22] N. Karpenko, Chow groups of quadrics and index reduction formula, Nova J. of Algebra and Geometry 3 (1995), 357–379.

[23] K. Kato, A Hasse principle for two-dimensional global fields, J. reine angew. Math. 366 (1986), 142–183.

[24] M. Knebusch Generic splitting of quadratic forms, II, Proc. London Math. Soc. 34 (1977), 1–31.

[25] M. Knus, A.S. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, in preparation.

[26] T.Y. Lam, The Algebraic theory of quadratic forms (2nd edition), Benjamin, New York, 1980.

[27] A.S. Merkurjev, K-theory of simple algebras, in: W. Jacob and A. Rosenberg (ed.), K-theory and algebraic geometry: connections with quadratic forms and division algebras, Proceedings of Symposia in Pure Mathematics 58 (I) (1995), 65–83.

[28] A.S. Merkurjev, SK_{1} for biquaternion algebras, preprint, 1994.

[29] A.S. Merkurjev, The group H^{1}(X, K_{2}) for projective homogeneous varieties, Algebra i Analiz. English translation: Leningrad (Saint-Petersburg) Math. J. 7 (1995), 136–164.

[30] A.S. Merkurjev, On the norm residue homomorphism for fields, Mathematics in St. Petersburg, 49–71, Amer. Math. Soc. Transl. Ser. 2, 174, Amer. Math. Soc., Providence, RI, 1996.

[31] A.S. Merkurjev and A.A. Suslin, K-cohomology of Severi-Brauer varieties and norm residue homomorphism (in Russian), Izv. Akad. Nauk SSSR 46 (1982), 1011–1046. English translation: Math USSR Izv. 21 (1983), 307–340.

[32] A.S. Merkurjev and A.A. Suslin, The norm residue homomorphism of degree 3 (in Russian), Izv. Akad. Nauk SSSR 54 (1990), 339–356. English translation: Math. USSR Izv. 36 (1991), 349–368.

[33] A.S. Merkurjev and J.-P. Tignol, Galois cohomology of biquadratic extensions, Comment. Math. Helv. 68 (1993), 138–169.

[34] J.W. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344.

[35] D. Orlov, A. Vishik, V. Voevodsky, Motivic cohomology of Pfister quadrics, in preparation.

[36] I. Panin, Applications of K-theory in algebraic geometry, 1984, Thesis, LOMI, Leningrad.

[37] R. Parimala, Witt groups of anisotropic conics, elliptic and hyperelliptic curves, J. Number Theory 28 (1988), 69–93.

[38] E. Peyre, Unramified cohomology and rationality problems, Math. Ann. 296 (1993), 247–268.

[39] E. Peyre, Products of Severi-Brauer varieties and Galois cohomology, in: W. Jacob and A. Rosenberg (ed.), K-theory and algebraic geometry: connections with quadratic forms and division algebras, Proceedings of Symposia in Pure Mathematics 58 (2) (1995), 369–401.

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

[41] D. Quillen, Higher algebraic K-theory, I, Lect. Notes in Math. 341 (1973), Springer, Berlin, 83–147.

[42] M. Rost, Chow groups with coefficients, Documenta Math. 1 (1996), 319–393.

[43] D.J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984), 71–84.

[44] D.J. Saltman, Brauer groups and the center of generic matrices, J. of Alg. 97 (1985), 53–67.

[45] W. Scharlau Quadratic and hermitian forms, Grundl. Math. Wiss. 294, Springer, Berlin, 1985.

[46] C. Scheiderer, Real and étale cohomology, Lect. Notes in Math. 1588, Springer, Berlin, 1994.

[47] J-P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413–420.

[48] R. Sujatha, Unramified cohomology and Witt groups of anisotropic Pfister quadrics, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2341–2358.

[49] A. Suslin, K-theory and K-cohomology of certain group varieties, Algebraic K-theory (A. Suslin, ed.), Adv. Soviet Math. 4, AMS, Providence, 1991, 53–74.

[50] R.G. Swan, Zero cycles on quadric hypersurfaces, Proc. Amer. Math. Soc. 107 (1989), 43–46.

[51] M. Szyjewski, The fifth invariant of quadratic forms (in Russian), Algebra Anal. 2 (1990), 213–234. English translation: Leningrad Math. J. 2 (1991), 179–198.

[52] J.-P. Tignol, SK_{1} for biquaternion algebras, after M. Rost and A. Merkurjev, secret paper, 1994.

[53] J. van Geel, Applications of the Riemann-Roch theorem for curves to quadratic forms and division algebras, preprint, Université catholique de Louvain, 1991.

[54] V. Voevodsky, The Milnor conjecture, preprint, Max-Planck-Institute for Mathematics, Bonn,

    1996, http://www.math.uiuc.edu/K-theory/0170,

[55] B.L. van der Waerden, Gruppen von linearen Transformationen, Ergebnisse Math. Wiss., 4, Springer-Verlag, Berlin, 1935.

[56] E. Witt, Über ein Gegenbeispiel zum Normensatz, Math. Zeit. 39 (1935), 462–467.

Institut de Mathématiques de Jussieu, Equipe Théories Géométriques, Université Paris 7, Case 7012, 75251 Paris Cedex 05, France

   E-mail address: kahn@math.jussieu.fr

Fakultät für Mathematik, Universität Regensburg, Universitätsstraße 31, D-93040 Regensburg, Germany

   E-mail address: markus.rost@mathematik.uni-regensburg.de

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

   E-mail address: sujatha@math.tifr.res.in