On the Group H^{3}(F (ψ, D)/F)

Oleg T. Izhboldin and Nikita A. Karpenko

Received: August 14, 1997

Communicated by Ulf Rehmann

Abstract. Let F be a field of characteristic different from 2, ψ a quadratic F -form of dimension ≥ 5, and D a central simple F -algebra of exponent 2. We denote by F (ψ, D) the function field of the product X_{ψ} × X_{D}, where X_{ψ} is the projective quadric determined by ψ and X_{D} is the Severi-Brauer variety determined by D. We compute the relative Galois cohomology group H^{3}(F (ψ, D)/F, ℤ/2ℤ) under the assumption that the index of D goes down when extending the scalars to F (ψ). Using this, we give a new, shorter proof of the theorem [23, Th. 1] originally proved by A. Laghribi, and a new, shorter, and more elementary proof of the assertion [2, Cor. 9.2] originally proved by H. Esnault, B. Kahn, M. Levine, and E. Viehweg.

1991 Mathematics Subject Classification: 19E15, 12G05, 11E81.

Let ψ be a quadratic form and D be an exponent 2 central simple algebra over a field F (always assumed to be of characteristic not 2). Let X_{ψ} be the projective quadric determined by ψ, X_{D} the Severi-Brauer variety determined by D, and F (ψ, D) the function field of the product X_{ψ} × X_{D}.

A computation of the relative Galois cohomology group

H^{3}(F (ψ, D)/F) =^{def} ker (H^{3}(F, ℤ/2ℤ) → H^{3}(F (ψ, D), ℤ/2ℤ))

plays a crucial role in obtaining the results of [8] and [10] concerning the problem of isotropy of quadratic forms over the function fields of quadrics.

The group H^{3}(F (ψ, D)/F) is closely related to the Chow group CH^{2}(X_{ψ} × X_{D}) of 2-codimensional cycles on the product X_{ψ} × X_{D}. The main result of this paper is the following theorem, where both groups are computed assuming dim ψ ≥ 5 and the index of D goes down when extending the scalars to the function field of ψ:

Theorem 0.1. Let D be a central simple F -algebra of exponent 2. Let ψ be a quadratic form of dimension ≥ 5. Suppose that ind D_{F (ψ)} < ind D. Then Tors CH^{2}(X_{ψ} × X_{D}) = 0 and H^{3}(F (ψ, D)/F) = [D] ∪ H^{1}(F).

A proof is given in §8. The essential part of the proof is Theorem 6.9, dealing with the special case where D is a division algebra of degree 8. This theorem has two applications in the theory of quadratic forms. The first one is a new, shorter proof of the following assertion, originally proved by A. Laghribi ([23, Th. 1]):

Corollary 0.2. Let ϕ ∈ I^{2}(F) be an 8-dimensional quadratic form such that ind C(ϕ) = 8. Let ψ be a quadratic form of dimension ≥ 5 such that ϕ_{F (ψ)} is isotropic. Then there exists a half-neighbor ϕ^{∗} of ϕ such that ψ ⊂ ϕ^{∗}.

The other application we demonstrate is a new, shorter, and more elementary proof of the assertion, originally proved by H. Esnault, B. Kahn, M. Levine, and E. Viehweg ([2, Cor. 9.2]):

Corollary 0.3. Let ϕ ∈ I^{2}(F) be any quadratic form such that ind C(ϕ) ≥ 8. Let A be a central simple F -algebra Brauer equivalent to C(ϕ) and let F (A) be the function field of the Severi-Brauer variety of A. Then ϕ_{F (A)} ∉ I^{4}(F (A)). In particular, ϕ_{F (A)} is not hyperbolic. Moreover, if dim ϕ = 8 then ϕ_{F (A)} is anisotropic.

Our proofs of Corollaries 0.2 and 0.3 are given in §7.

An important part in the proof of Theorem 6.9 is played by the formula of Proposition 4.5, which is in fact applicable to a wide class of algebraic varieties.

A computation of the group H^{3}(F (ψ, D)/F) in some cases not covered by Theorem 0.1 is given in [8] and [10].

1. Terminology, notation, and backgrounds

1.1. Quadratic forms. Mainly, we use notation of [24] and [30]. However there is a slight difference: we denote by ⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩ the n-fold Pfister form

⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩ .

The set of all n-fold Pfister forms over F is denoted by P_{n}(F); GP_{n}(F) is the set of forms similar to a form from P_{n}(F).

We recall that a quadratic form ψ is called a (Pfister) neighbor (of a Pfister form π), if it is similar to a subform in π and dim ϕ > \frac{1}{2} dim π. Two quadratic forms ϕ and ϕ^{∗} are half-neighbors, if dim ϕ = dim ϕ^{∗} and there exists s ∈ F^{∗} such that the sum ϕ⊥sϕ^{∗} is similar to a Pfister form.

For a quadratic form ϕ of dimension ≥ 3, we denote by X_{ϕ} the projective variety given by the equation ϕ = 0 and we set F (ϕ) = F (X_{ϕ}).

1.2. Generic splitting tower. Let γ be a non-hyperbolic quadratic form over F. Put F_{0} =^{def} F and γ_{0} =^{def} γ_{an}. For i ≥ 1 let F_{i} =^{def} F_{i−1}(γ_{i−1}) and γ_{i} =^{def} ((γ_{i−1})_{F_{i}})_{an}. The smallest h such that dim γ_{h} ≤ 1 is called the height of γ. The sequence F_{0}, F_{1}, …, F_{h} is called the generic splitting tower of γ ([21]). We need some properties of the fields F_{s}:

Lemma 1.3 ([22]). Let M/F be a field extension such that dim(γ_{M})_{an} = dim γ_{s}. Then the field extension MF_{s}/M is purely transcendental.

The following proposition is a consequence of the index reduction formula [25].

Proposition 1.4 (see [6, Th. 1.6] or [5, Prop. 2.1]). Let ϕ ∈ I^{2}(F) be a quadratic form with ind(C(ϕ)) ≥ 2^{r} > 1. Then there is s (0 ≤ s ≤ h(ϕ)) such that dim ϕ_{s} = 2r + 2 and ind C(ϕ_{s}) = 2^{r}.

Corollary 1.5. Let ϕ ∈ I^{2}(F) be a quadratic form with ind(C(ϕ)) ≥ 8. Then there is s (0 ≤ s ≤ h(ϕ)) such that dim ϕ_{s} = 8 and ind C(ϕ_{s}) = 8.

On the Group H^{3}(F (ψ, D)/F) 299

1.6. Central simple algebras. We are working with finite-dimensional associative algebras over a field. Let D be a central simple F -algebra. We denote by X_{D} the Severi-Brauer variety of D and by F (D) the function field F (X_{D}).

For another central simple F -algebra D′ and for a quadratic F -form ψ of dimension ≥ 3, we set F (D′, D) =^{def} F (X_{D′} × X_{D}) and F (ψ, D) =^{def} F (X_{ψ} × X_{D}).

1.7. Galois cohomology. By H^{∗}(F) we denote the graded ring of Galois cohomology

H^{∗}(F, ℤ/2ℤ) = H^{∗}(Gal(F_{sep}/F), ℤ/2ℤ).

For any field extension L/F, we set H^{∗}(L/F) =^{def} ker(H^{∗}(F) → H^{∗}(L)).

We use the standard canonical isomorphisms H^{0}(F) = ℤ/2ℤ, H^{1}(F) = F^{∗}/F^{∗2}, and H^{2}(F) = Br_{2}(F).

We also work with the cohomology groups H^{n}(F, ℚ/ℤ(i)), i = 0, 1, 2 (see e.g.

[12] for the definition). For any field extension L/F, we set

H^{∗}(L/F, ℚ/ℤ(i)) =^{def} ker (H^{∗}(F, ℚ/ℤ(i)) → H^{∗}(L, ℚ/ℤ(i))) .

For n = 1, 2, 3, the group H^{n}(F) is naturally identified with

Tors_{2} H^{n}(F, ℚ/ℤ(n − 1)) .

1.8. K-theory and Chow groups. We are mainly working with smooth algebraic varieties over a field, although the smoothness assumption is not always essential.

Let X be a smooth algebraic F -variety. The Grothendieck ring of X is denoted by K(X). This ring is supplied with the filtration “by codimension of support” (which respects multiplication); the adjoint graded ring is denoted by G^{∗}K(X). There is a canonical surjective homomorphism of the graded Chow ring CH^{∗}(X) onto G^{∗}K(X); its kernel consists only of torsion elements and is trivial in the 0-th, 1-st and 2-nd graded components ([32, §9]). In particular we have the following

Lemma 1.9. The homomorphism CH^{i}(X) → G^{i}K(X) is bijective if at least one of the following conditions holds:

• i = 0, 1, or 2,

• CH^{i}(X) is torsion-free.

Let X be a variety over F and E/F be a field extension. We denote by i_{E/F} the restriction homomorphism K(X) → K(X_{E}). We use the same notation for the restriction homomorphisms CH^{∗}(X) → CH^{∗}(X_{E}) and G^{∗}K(X) → G^{∗}K(X_{E}). Note that for any projective homogeneous variety X, the homomorphism i_{E/F} : K(X) → K(X_{E}) is injective by [27].

1.10. Other notations. We denote by F̄ a separable closure of the field F. The order of a set S is denoted by |S| (if S is infinite, we set |S| =^{def} ∞).

2. The group Tors G^{∗}K(X)

Lemma 2.1. Let X be a variety over F and E/F be a field extension such that the homomorphism i_{E/F} : K(X) → K(X_{E}) is injective and the factor group K(X_{E})/i_{E/F} (K(X)) is finite. Then

| ker(G^{∗}K(X) → G^{∗}K(X_{E})| = \frac{|G^{∗}K(X_{E})/i_{E/F} (G^{∗}K(X))|}{|K(X_{E})/i_{E/F} (K(X))|}

Proof. The proof is the same as the proof of [15, Prop. 2].

Lemma 2.2. Let X be a variety, i be an integer, and E/F be a field extension such that the group G^{i}K(X_{E}) is torsion-free. Then

ker(G^{i}K(X) → G^{i}K(X_{E})) = Tors G^{i}K(X) .

Proof. Since G^{i}K(X_{E}) is torsion-free, one has ker(G^{i}K(X) → G^{i}K(X_{E})) ⊃ Tors G^{i}K(X).

To prove the inverse inclusion, let us take an intermediate field E_{0} such that the extension E_{0}/F is purely transcendental while the extension E/E_{0} is algebraic. The specialization argument shows that the homomorphism G^{i}K(X) → G^{i}K(X_{E_{0}}) is injective; the transfer argument shows that ker(G^{i}K(X_{E_{0}}) → G^{i}K(X_{E})) ⊂ Tors G^{i}K(X_{E_{0}}). Therefore ker(G^{i}K(X) → G^{i}K(X_{E})) ⊂ Tors G^{i}K(X).

Lemma 2.3. Let X be a smooth variety, i be an integer, and E/F be a field extension such that the group CH^{i}(X_{E}) is torsion-free. Then

• CH^{i}(X_{E}) ≃ G^{i}K(X_{E}) (and hence the group G^{i}K(X_{E}) is torsion-free),

• CH^{i}(X_{E})/i_{E/F} (CH^{i}(X)) ≃ G^{i}K(X_{E})/i_{E/F} (G^{i}K(X)).

Proof. The first assertion is contained in Lemma 1.9. The canonical homomorphism CH^{i}(X_{E}) → G^{i}K(X_{E}) induces a homomorphism

CH^{i}(X_{E})/i_{E/F} (CH^{i}(X)) → G^{i}K(X_{E})/i_{E/F} (G^{i}K(X))

which is bijective since CH^{i}(X_{E}) → G^{i}K(X_{E}) is bijective and CH^{i}(X) → G^{i}K(X) is surjective.

Proposition 2.4. Suppose that a smooth F -variety X and a field extension E/F satisfy the following three conditions:

• the homomorphism i_{E/F} : K(X) → K(X_{E}) is injective,

• the factor group K(X_{E})/i_{E/F} (K(X)) is finite,

• the group CH^{∗}(X_{E}) is torsion-free.

Then

| Tors G^{∗}K(X)| = \frac{|G^{∗}K(X_{E})/i_{E/F} (G^{∗}K(X))|}{|K(X_{E})/i_{E/F} (K(X))|} = \frac{|CH^{∗}(X_{E})/i_{E/F} (CH^{∗}K(X))|}{|K(X_{E})/i_{E/F} (K(X))|}

Proof. It is an obvious consequence of Lemmas 2.1, 2.2, and 2.3.

Documenta Mathematica 2 (1997) 297–311

3. Auxiliary lemmas

For an Abelian group A we use the notation rk(A) = dim_{ℚ}(A ⊗_{ℤ} ℚ).

Lemma 3.1. Let A_{0} ⊂ A, B_{0} ⊂ B be free Abelian groups such that rk A_{0} = rk A = r_{A}, rk B_{0} = rk B = r_{B}. Then

|\frac{A ⊗_{ℤ} B}{A_{0} ⊗_{ℤ} B_{0}} | = |\frac{A}{A_{0}} |^{r_{B}} · |\frac{B}{B_{0}} |^{r_{A}} .

Proof. One has

(A ⊗ B)/(A_{0} ⊗ B) ≃ (A/A_{0}) ⊗ B ≃ (A/A_{0}) ⊗ ℤ^{r_{B}} ≃ (A/A_{0})^{r_{B}},
(A_{0} ⊗ B)/(A_{0} ⊗ B_{0}) ≃ A_{0} ⊗ (B/B_{0}) ≃ ℤ^{r_{A}} ⊗ (B/B_{0}) ≃ (B/B_{0})^{r_{A}} .

Therefore,

|\frac{A ⊗ B}{A_{0} ⊗ B_{0}} | = |\frac{A ⊗ B}{A_{0} ⊗ B} | · |\frac{A_{0} ⊗ B}{A_{0} ⊗ B_{0}} | = |\frac{A}{A_{0}} |^{r_{B}} · |\frac{B}{B_{0}} |^{r_{A}} .

The following lemma is well-known.

Lemma 3.2. Let A be an Abelian group with a finite filtration A = F^{0}A ⊃ F^{1}A ⊃ ⋯ ⊃ F^{k}A = 0. Let B be a subgroup of A with the filtration F^{p}B = B ∩ F^{p}A. Let G^{∗}A = ⨁_{p≥0} F^{p}A/F^{p+1}A and G^{∗}B = ⨁_{p≥0} F^{p}B/F^{p+1}B. Then

• |A/B| = |G^{∗}A/G^{∗}B|,

• if A is a finitely generated group then rk G^{∗}A = rk A.

In the following lemma the term “ring” means a commutative ring with unit.

Lemma 3.3. Let A and B be rings whose additive groups are finitely generated Abelian groups. Let I be a nilpotent ideal of A such that A/I ≃ ℤ. Let R be a subring of A⊗_{ℤ} B and A_{R} be a subring of A such that A_{R} ⊗1 ⊂ R. Then the following inequality holds

|\frac{A ⊗_{ℤ} B}{R} | ≤ |\frac{A}{A_{R}} |^{r_{B}} · |\frac{A ⊗_{ℤ} B}{R + (I ⊗_{ℤ} B)}|^{r_{A}}

where r_{A} = rk A and r_{B} = rk B.

Proof. Let us denote by B_{R} the image of R under the following composition A⊗B → (A/I) ⊗ B ≃ ℤ ⊗ B ≃ B. Obviously,

|\frac{A ⊗_{ℤ} B}{R + (I ⊗_{ℤ} B)}| = |\frac{B}{B_{R}} | .

For any p ≥ 0 we set F^{p}A = {a ∈ A | ∃m ∈ ℕ such that ma ∈ I^{p}}. Clearly, Tors(A/F^{p}A) = 0, and so A/F^{p} is a free Abelian group. Therefore all factor groups F^{p}A/F^{p+1}A (p = 0, 1, …) are free Abelian. Since A/I ≃ ℤ, it follows that F^{1}A = I. Thus A/F^{1}A ≃ ℤ. Since I is a nilpotent ideal of A, there exists k such that I^{k} = 0. Then F^{k}A = 0. Thus the filtration A = F^{0}A ⊃ F^{1}A ⊃ F^{2}A ⊃ … is finite and results of Lemma 3.2 can be applied.

Let F^{p}A_{R} =^{def} R ∩ F^{p}A, F^{p}(A ⊗ B) =^{def} im(F^{p}A ⊗ B → A ⊗ B), and F^{p}R =^{def} R ∩

F^{p}(A⊗B). If K is one of the rings A, A_{R}, A⊗B, or R, we set G^{p}K =^{def} F^{p}K/F^{p+1}K

and G^{∗}K =^{def} ⨁_{p≥0} F^{p}K/F^{p+1}K. Obviously, F^{p}K · F^{q}K ⊂ F^{p+q}K for all p and q.

Therefore, K = F^{0}K ⊃ F^{1}K ⊃ ⋯ ⊃ F^{p}K ⊃ … is a ring filtration. Hence, the adjoint graded group G^{∗}K has a graded ring structure. Since the additive group of B is free, we have a natural ring isomorphism G^{∗}A ⊗ B ≃ G^{∗}(A ⊗ B).

Since A_{R} ⊗ 1 ⊂ R, we have G^{∗}A_{R} ⊗ 1 ⊂ G^{∗}R. Clearly G^{0}(A ⊗ B) = (A/I) ⊗ B, and G^{0}R coincides with the image of the composition R → A ⊗ B → (A/I) ⊗ B. By definition of B_{R}, one has G^{0}R = 1_{G^{∗}A} ⊗ B_{R} (here 1_{G^{∗}A} denotes the unit of the ring G^{∗}A). Therefore 1_{G^{∗}A} ⊗ B_{R} ⊂ G^{∗}R. Since G^{∗}A_{R} ⊗ 1 ⊂ G^{∗}R, 1_{G^{∗}A} ⊗ B_{R} ⊂ G^{∗}R, and G^{∗}R is a subring of G^{∗}A ⊗ B, we have G^{∗}A_{R} ⊗ B_{R} ⊂ G^{∗}R. Therefore |G^{∗}(A ⊗ B)/G^{∗}R| ≤ |(G^{∗}A ⊗ B)/(G^{∗}A_{R} ⊗ B_{R})|. Applying Lemmas 3.1 and 3.2, we have

|\frac{A ⊗ B}{R} | = |\frac{G^{∗}(A ⊗ B)}{G^{∗}R}| ≤ |\frac{G^{∗}A ⊗ B}{G^{∗}A_{R} ⊗ B_{R}} | = |\frac{G^{∗}A}{G^{∗}A_{R}} |^{r_{B}} · |\frac{B}{B_{R}} |^{r_{A}} =
= |\frac{A}{A_{R}} |^{r_{B}} · |\frac{B}{B_{R}} |^{r_{A}} = |\frac{A}{A_{R}} |^{r_{B}} · |\frac{A ⊗_{ℤ} B}{R + (I ⊗_{ℤ} B)}|^{r_{A}} .

4. On the group CH^{∗}(X × Y)

Let X be a smooth variety. We denote by F^{p}CH^{∗}(X) the group

⨁_{i≥p} CH^{i}(X) .

Let Y be another smooth variety. For a subgroup A of CH^{∗}(X) and a subgroup B of CH^{∗}(Y), we denote by A ⊠ B the image of the composition A ⊗ B → CH^{∗}(X) ⊗ CH^{∗}(Y) → CH^{∗}(X × Y).

The following assertion is evident (see also [20, §3] or [11]).

Proposition 4.1. Let X and Y be smooth varieties over F. Then

• the natural homomorphism CH^{∗}(X × Y) → CH^{∗}(Y_{F (X)}) is surjective,

• the kernel of the homomorphism CH^{∗}(X ×Y) → CH^{∗}(Y_{F (X)}) contains the group F^{1}CH^{∗}(X) ⊠ CH^{∗}(Y).

Corollary 4.2. If the natural homomorphism CH^{∗}(X) ⊗ CH^{∗}(Y) → CH^{∗}(X × Y) is bijective and CH^{∗}(Y) is torsion-free, then the homomorphism CH^{∗}(X × Y) → CH^{∗}(Y_{F (X)}) induces an isomorphism

\frac{CH^{∗}(X × Y)}{F^{1}CH^{∗}(X) ⊠ CH^{∗}(Y)} → CH^{∗}(Y_{F (X)}).

Proof. Since CH^{∗}(X)⊗CH^{∗}(Y) ≃ CH^{∗}(X ×Y) and CH^{∗}(X)/F^{1}CH^{∗}(X) ≃ CH^{0}(X), the factor group CH^{∗}(X × Y)/(F^{1}CH^{∗}(X) ⊠ CH^{∗}(Y)) is isomorphic to CH^{0}(X) ⊗_{ℤ} CH^{∗}(Y) ≃ ℤ ⊗_{ℤ} CH^{∗}(Y) ≃ CH^{∗}(Y). Thus, it is sufficient to prove that the homomorphism CH^{∗}(Y) → CH^{∗}(Y_{F (X)}) is injective. This is obvious since CH^{∗}(Y) is torsion-free.

Corollary 4.3. Let X and Y be smooth varieties and E/F be a field extension such that the natural homomorphism CH^{∗}(X_{E}) ⊗ CH^{∗}(Y_{E}) → CH^{∗}(X_{E} × Y_{E}) is bijective and CH^{∗}(Y_{E}) is torsion-free. Then there exists an isomorphism

\frac{CH^{∗}(X_{E} × Y_{E})}{i_{E/F} (CH^{∗}(X × Y)) + F^{1}CH^{∗}(X_{E}) ⊠ CH^{∗}(Y_{E})} ≃ \frac{CH^{∗}(Y_{E(X)})}{i_{E(X)/F (X)}(CH^{∗}(Y_{F (X)}))}

Proof. Obvious in view of Corollary 4.2.

Remark 4.4. It was noticed by the referee that the conditions of Corollary 4.3 (which appear also in Proposition 4.5) hold, if the variety Y_{E} possess a cellular decomposition (see e.g. [13, Def. 3.2] for the definition of cellular decomposition). In the case of complete varieties X and Y, this statement follows e.g. from [19, Th. 6.5]. In the present paper, we shall apply Corollary 4.3 only to the case where Y_{E} is isomorphic to a projective space.

Proposition 4.5. Let X and Y be smooth varieties over F and E/F be a field extension such that the following conditions hold

• CH^{∗}(X_{E}) is a free Abelian group of rank r_{X},

• CH^{∗}(Y_{E}) is a free Abelian group of rank r_{Y},

• the canonical homomorphism CH^{∗}(X_{E}) ⊗_{ℤ} CH^{∗}(Y_{E}) → CH^{∗}(X_{E} × Y_{E}) is an

isomorphism. Then

|\frac{CH^{∗}(X_{E} × Y_{E})}{i_{E/F} (CH^{∗}(X × Y))}| ≤ |\frac{CH^{∗}(X_{E})}{i_{E/F} (CH^{∗}(X))}|^{r_{Y}} · |\frac{CH^{∗}(Y_{E(X)})}{i_{E(X)/F (X)}(CH^{∗}(Y_{F (X)}))}|^{r_{X}} .

Proof. Let A = CH^{∗}(X_{E}), A_{R} = i_{E/F} (CH^{∗}(X)) and I = ⨁_{p>0} CH^{p}(X_{E}) = F^{1}CH^{∗}(X_{E}). Let B = CH^{∗}(Y_{E}). By our assumption, we have CH^{∗}(X_{E} × Y_{E}) ≃ A ⊗_{ℤ} B. We denote by R the image of the composition CH^{∗}(X × Y) → CH^{∗}(X_{E} ⊗ Y_{E}) ≃ A ⊗_{ℤ} B. Clearly, all conditions of Lemma 3.3 hold. Moreover,

|\frac{CH^{∗}(X_{E} × Y_{E})}{i_{E/F} (CH^{∗}(X × Y))}| = |\frac{A ⊗_{ℤ} B}{R} |  and  |\frac{CH^{∗}(X_{E})}{i_{E/F} (CH^{∗}(X))}| = |\frac{A}{A_{R}} | .

By Corollary 4.3 we have

|\frac{A ⊗_{ℤ} B}{R + (I ⊗_{ℤ} B)}| = |\frac{CH^{∗}(Y_{E(X)})}{i_{E(X)/F (X)}(CH^{∗}(Y_{F (X)}))}| .

To complete the prove it suffices to apply Lemma 3.3.

5. The group Tors CH^{2}(X_{ψ} × X_{D})

The aim of this section is Corollary 5.6.

Proposition 5.1 (see [14, §2.1]). Let ψ be a (2n + 1)-dimensional quadratic form over a separably closed field. Set X =^{def} X_{ψ} and d =^{def} dim X = 2n − 1. Then for all 0 ≤ p ≤ d the group CH^{p}(X) is canonically isomorphic to ℤ (for other p the group CH^{p}(X) is trivial). Moreover,

• if 0 ≤ p < n, then CH^{p}(X) = ℤ · h^{p}, where h ∈ CH^{1}(X) denotes the class of a hyperplane section of X; • if n ≤ p ≤ d, then CH^{p}(X) = ℤ · l_{d−p}, where l_{d−p} denotes the class of a linear

subspace in X of dimension d − p, besides 2l_{d−p} = h^{p}.

Corollary 5.2. Let ψ be a (2n + 1)-dimensional quadratic form over F and let X = X_{ψ}. Then

• CH^{∗}(XF̄) is a free Abelian group of rank 2n,

• if 0 ≤ p < n then |CH^{p}(XF̄)/i_{F̄/F} (CH^{p}(X))| = 1,

• if n ≤ p ≤ 2n − 1 then |CH^{p}(XF̄)/i_{F̄/F} (CH^{p}(X))| ≤ 2,

• |CH^{∗}(XF̄)/i_{F̄/F} (CH^{∗}(X))| ≤ 2^{n}.

Proposition 5.3. Let D be a central simple F -algebra of exponent 2 and of degree 8. Let E/L/F be field extensions such that ind D_{L} = 4 and ind D_{E} = 1. Let Y = SB(D). For any 0 ≤ p ≤ dim Y = 7, the group CH^{p}(Y_{E}) is canonically isomorphic to ℤ. Moreover, the image of the homomorphism i_{E/L} : CH^{p}(Y_{L}) → CH^{p}(Y_{E}) ≃ ℤ contains 1 if p = 0, 4; 2 if p = 1, 2, 5, 6; 4 if p = 3, 7.

Proof. Since deg D = 8 and ind D_{E} = 1, Y_{E} is isomorphic to ℙ_{E}^{7}. Hence, the group CH^{p}(Y_{E}) ≅ CH^{p}(ℙ_{E}^{7}) (where p = 0, …, 7) is generated by the class h^{p} of a linear subspace ([4]).

The rest part of the proposition is contained in [16, Th.]. For the reader’s convenience, we also give a direct construction of the elements required. The class of Y_{L} itself gives 1 ∈ i_{E/L}(CH^{0}(Y_{L})). Let ξ be the tautological line bundle on the projective space ℙ_{E}^{7} ≃ Y_{E}. Since exp D = 2, the bundle ξ^{⊗2} is defined over F and, in particular, over L. Its first Chern class gives 2 ∈ i_{E/L}(CH^{1}(Y_{L})). Since ind D_{L} = 4, the bundle ξ^{⊕4} is defined over L. Its second Chern class gives 6 ∈ i_{E/L}(CH^{2}(Y_{L})).^{1} Thus 2 ∈ i_{E/L}(CH^{2}(Y_{L})). The third Chern class of ξ^{⊕4} gives 4 ∈ i_{E/L}(CH^{3}(Y_{L})). The fourth Chern class of ξ^{⊕4} gives 1 ∈ i_{E/L}(CH^{4}(Y_{L})). Finally, taking the product of the cycles constructed in codimensions 1, 2, and 3 with the cycle of codimension 4, one gets the cycles of codimensions 5, 6, and 7 required.

Corollary 5.4. Under the condition of Proposition 5.3, we have

|CH^{∗}(Y_{E})/i_{E/L}(CH^{∗}(Y_{L}))| ≤ 256 .

Proof. ∏_{p=0}^{7} |CH^{p}(Y_{E})/i_{E/L}(CH^{p}(Y_{L}))| ≤ 1 · 2 · 2 · 4 · 1 · 2 · 2 · 4 = 256 .

Proposition 5.5. Let D be a central division F -algebra of degree 8 and exponent 2. Let ψ be a 5-dimensional quadratic F -form. Suppose that D_{F (ψ)} is not a skewfield. Then Tors G^{∗}K(X_{ψ} × X_{D}) = 0.

Proof. Let X = X_{ψ} and Y = X_{D}. Corollary 5.2 shows that CH^{∗}(XF̄) is a free abelian group of rank r_{X} = 4 and |CH^{∗}(XF̄)/i_{F̄/F} (CH^{∗}(X))| ≤ 2^{2} = 4.

Since D is a division algebra of degree 8 and D_{F (ψ)} is not division algebra, it follows that ind D_{F (X)} = 4. Applying Corollary 5.4 to the case L = F (X), E = F̄(X), we have |CH^{∗}(Y_{F̄(X)})/i_{F̄(X)/F (X)}(CH^{∗}(Y_{F (X)}))| ≤ 256.

1In fact, it is enough only to know that the Grothendieck classes of the bundles ξ^{⊗2} and ξ^{⊕4} are in K(Y_{L}) what can be also seen from the computation of the K-theory.

Since YF̄ = SB(DF̄) ≃ ℙ7 ; F̄, the group CH^{∗}(YF̄) is a free Abelian of rank r_{Y} = 8 and CH^{∗}(XF̄)⊗CH^{∗}(YF̄) ≃ CH^{∗}(XF̄ ×YF̄) (see [3, Prop. 14.6.5]). Thus all conditions of Proposition 4.5 hold for X, Y, E = F̄ and we have

|\frac{CH^{∗}(XF̄ × YF̄)}{i_{F̄/F} (CH^{∗}(X × Y))}| ≤ 4^{8} · 256^{4} = 2^{48}.

Using [29, Th. 4.1 of §8] and [33, Th. 9.1], we get a natural (with respect to extensions of F) isomorphism

K(X × Y) ≃ K((F^{×3} × C) ⊗_{F} (F^{×4} × D^{×4})) ≃

≃ K(F^{×12} × C^{×4} × D^{×12} × (C ⊗_{F} D)^{×4})

where C =^{def} C_{0}(ψ) is the even Clifford algebra of ψ. Note that C is a central simple F -algebra of the degree 2^{2}. Since D_{F (ψ)} is not a skew field, [25, Th. 1] states that D ≃ C ⊗_{F} B with some central division F -algebra B. Therefore, ind C = deg C = 2^{2} and ind C ⊗ D = ind B = deg B = 2. Hence

|\frac{K(XF̄ × YF̄)}{i_{F̄/F} (K(X × Y))}| = (ind C)^{4} · (ind D)^{12} · (ind C ⊗ D)^{4} = 2^{2·4+3·12+1·4} = 2^{48} .

Applying Proposition 2.4 to the variety X × Y and E = F̄, we have

| Tors G^{∗}K(X × Y)| = \frac{|CH^{∗}(XF̄ × YF̄)/i_{F̄/F} (CH^{∗}(X × Y))|}{|K(XF̄ × YF̄)/i_{F̄/F} (K(X × Y))|} ≤ \frac{2^{48}}{2^{48}} = 1 .

Therefore, Tors G^{∗}K(X × Y) = 0.

Applying Lemma 1.9 we get the following

Corollary 5.6. Under the condition of Proposition 5.5, the group CH^{2}(X_{ψ} × X_{D}) is torsion-free.

6. A special case of Theorem 0.1

In this section we prove Theorem 0.1 in the special case where D is a division algebra of degree 8.

Proposition 6.1 ([1, Satz 5.6]). Let ψ be a quadratic F -form of dimension ≥ 5. The group H^{3}(F (ψ)/F) is non-trivial iff ψ is a neighbor of an anisotropic 3-Pfister form.

Proposition 6.2 (see [28, Prop. 4.1 and Rem. 4.1]). Let D be a central division F - algebra of exponent 2. Suppose that D is decomposable (in the tensor product of two proper subalgebras). Then H^{3}(F (D)/F) = [D] ∪ H^{1}(F).

Proposition 6.3. If D and D′ are Brauer equivalent central simple F -algebras, then the function fields F (D) and F (D′) are stably equivalent.^{2} Proof. Since the algebras D_{F (D′)} and D′_{F (D)} are split, the field extensions

2Two field extensions E/F and E′/F are called stably equivalent, if some finitely generated purely transcendental extension of E is isomorphic (over F) to some finitely generated purely transcendental extension of E′.

F (D, D′)/F (D′) and F (D, D′)/F (D)

are purely transcendental. Therefore each of the field extensions F (D)/F and F (D′)/F is stably equivalent to the extension F (D, D′)/F.

Corollary 6.4. Fix a quadratic F -form ψ and integers i, j ∈ ℤ. For any central simple F -algebra D, the groups H^{i}(F (D)/F), H^{i}(F (D)/F, ℚ/ℤ(j)), H^{i}(F (ψ, D)/F), H^{i}(F (ψ, D)/F, ℚ/ℤ(j)) only depend on the Brauer class of D.

Proposition 6.5. Let D be a central simple F -algebra of exponent 2 and let ψ be a quadratic F -form. The group H^{3}(F (ψ, D)/F, ℚ/ℤ(2)) is annihilated by 2.

Proof. Let ψ_{0} be a 3-dimensional subform of ψ. Clearly,

H^{3}(F (ψ, D)/F, ℚ/ℤ(2)) ⊂ H^{3}(F (ψ_{0}, D)/F, ℚ/ℤ(2)) .

Therefore, it suffices to show that the latter cohomology group is annihilated by 2. Replacing ψ_{0} by the quaternion algebra C_{0}(ψ_{0}), we come to a statement covered by [7, Lemma A.8].

Corollary 6.6. In the conditions of Proposition 6.5, one has H^{3}(F (ψ, D)/F, ℚ/ℤ(2)) = H^{3}(F (ψ, D)/F) .

Proposition 6.7. Let D be a central simple F -algebra of exponent 2 and let ψ be a quadratic F -form of dimension ≥ 3. Suppose that ind D_{F (ψ)} < ind D. Then ψ is not a 3-Pfister neighbor and there is an isomorphism

\frac{H^{3}(F (ψ, D)/F)}{H^{3}(F (ψ)/F) + [D] ∪ H^{1}(F)} ≃ Tors CH^{2}(X_{ψ} × X_{D}) .

Proof. By [9, Prop. 2.2], there is an isomorphism

\frac{H^{3}(F (ψ, D)/F, ℚ/ℤ(2))}{H^{3}(F (ψ)/F, ℚ/ℤ(2)) + H^{3}(F (D)/F, ℚ/ℤ(2))} ≃

≃ \frac{Tors CH^{2}(X_{ψ} × X_{D})}{pr_{ψ}^{∗} Tors CH^{2}(X_{ψ}) + pr_{D}^{∗} Tors CH^{2}(X_{D})} .

By Corollary 6.6, we have H^{3}(F (ψ, D)/F, ℚ/ℤ(2)) = H^{3}(F (ψ, D)/F); by [9, Lemma 2.8], we have H^{3}(F (ψ)/F, ℚ/ℤ(2)) = H^{3}(F (ψ)/F); and by [7, Lemma A.8], we have H^{3}(F (D)/F, ℚ/ℤ(2)) = H^{3}(F (D)/F).

Let D′ be a division algebra Brauer equivalent to D. By Corollary 6.4, we have H^{3}(F (D)/F) = H^{3}(F (D′)/F); by [18, Prop. 1.1], we have Tors CH^{2}(X_{D}) ≃ Tors CH^{2}(X_{D′}). Since D′_{F (ψ)} is no more a skew field, there is a homomorphism of F - algebras C_{0}(ψ) → D′ ([34, Th. 1], see also [26, Th. 2]). Although the algebra C_{0}(ψ) is not always central simple, it always contains a non-trivial subalgebra central simple over F. Therefore, D′ is decomposable, what implies H^{3}(F (D′)/F) = [D] ∪ H^{1}(F) (Proposition 6.2) and Tors CH^{2}(X_{D′}) = 0 ([17, Prop. 5.3]). Finally, the existence of a homomorphism C_{0}(ψ) → D′ implies that ψ is not a 3-Pfister neighbor; therefore Tors CH^{2}(X_{ψ}) = 0 ([14, Th. 6.1]).

Corollary 6.8. Let D be a central division F -algebra of degree 8 and exponent 2. Let ψ be a 5-dimensional quadratic F -form. Suppose that D_{F (ψ)} is not a skew field. Then H^{3}(F (ψ, D)/F) = [D] ∪ H^{1}(F).

Proof. It is a direct consequence of Proposition 6.7, Corollary 5.6, and Proposition 6.1.

Theorem 6.9. Theorem 0.1 is true if D is a division algebra of degree 8.

Proof. Let ψ_{0} be a 5-dimensional subform of ψ. Applying Corollary 6.8, we have [D] ∪ H^{1}(F) ⊂ H^{3}(F (ψ, D)/F) ⊂ H^{3}(F (ψ_{0}, D)/F) = [D] ∪ H^{1}(F). Hence H^{3}(F (ψ, D)/F) = [D] ∪ H^{1}(F).

The assertion on Tors CH^{2}(X_{ψ} × X_{D}) is Corollary 5.6.

Corollary 6.10. Let ϕ ∈ I^{2}(F) be a 8-dimensional quadratic form such that ind C(ϕ) = 8. Let D be a degree 8 central simple algebra such that c(ϕ) = [D]. Let ψ be a quadratic form of dimension ≥ 5 such that ϕ_{F (ψ)} is isotropic. Then

1) D is a division algebra;

2) D_{F (ψ)} is not a division algebra;

3) H^{3}(F (ψ, D)/F) = [D] ∪ H^{1}(F).

7. Proof of Corollaries 0.2 and 0.3

We need several lemmas.

Lemma 7.1. Let ϕ ∈ I^{2}(F) be a 8-dimensional quadratic form and let D be an algebra such that c(ϕ) = [D]. Then ϕ_{F (D)} ∈ GP_{3}(F (D)).

Proof. We have c(ϕ_{F (D)}) = c(ϕ)_{F (D)} = [D_{F (D)}] = 0. Hence ϕ_{F (D)} ∈ I^{3}(F (D)). Since dim ϕ = 8, we are done by the Arason-Pfister Hauptsatz.

Lemma 7.2. Let ϕ, ϕ^{∗} ∈ I^{2}(F) be 8-dimensional quadratic forms such that c(ϕ) = c(ϕ^{∗}) = [D], where D is a triquaternion division algebra.^{3} Suppose that there is a quadratic form ψ of dimension ≥ 5 such that the forms ϕ_{F (ψ,D)} and ϕ_{F (ψ,D)}^{∗} are isotropic. Then ϕ and ϕ^{∗} are half-neighbors.

Proof. Lemma 7.1 implies that ϕ_{F (ψ,D)}, ϕ_{F (ψ,D)}^{∗} ∈ GP_{3}(F (ψ, D)). By the assumption of the lemma, ϕ_{F (ψ,D)} and ϕ_{F (ψ,D)}^{∗} are isotropic. Hence ϕ_{F (ψ,D)} and ϕ_{F (ψ,D)}^{∗} are hyperbolic. Thus ϕ, ϕ^{∗} ∈ W (F (ψ, D)/F).

Let τ = ϕ ⊥ ϕ^{∗}. Clearly τ ∈ W (F (ψ, D)/F). Since c(τ) = c(ϕ) + c(ϕ^{∗}) = [D] + [D] = 0, we have τ ∈ I^{3}(F). Thus e^{3}(τ) ∈ H^{3}(F (ψ, D)/F). It follows from Corollary 6.10 that e^{3}(τ) ∈ [D] ∪ H^{1}(F). Hence there exists s ∈ F^{∗} such that e^{3}(τ) = [D] ∪ (s). We have e^{3}(τ) = [D] ∪ (s) = c(ϕ) ∪ (s) = e^{3}(ϕ ⟨ ; ⟨s⟩ ; ⟩). Since ker(e^{3} : I^{3}(F) → H^{3}(F)) = I^{4}(F), we have τ ≡ ϕ ⟨ ; ⟨s⟩ ; ⟩ (mod I^{4}(F)). Therefore

ϕ + ϕ^{∗} = τ ≡ ϕ ⟨ ; ⟨s⟩ ; ⟩ = ϕ − sϕ (mod I^{4}(F)). Hence ϕ^{∗} + sϕ ∈ I^{4}(F). Hence ϕ and ϕ^{∗}

are half-neighbors.

The following statement was pointed out by Laghribi ([23]) as an easy consequence of the index reduction formula [25].

3An F -algebra is called triquaternion, if it is isomorphic to a tensor product of three quaternion F -algebras.

Lemma 7.3. Let ψ be a quadratic form of dimension ≥ 5 and D be a division triquaternion algebra. Suppose that D_{F (ψ)} is not a division algebra. Then there exists an 8-dimensional quadratic form ϕ^{∗} ∈ I^{2}(F) such that ψ ⊂ ϕ^{∗} and c(ϕ^{∗}) = [D].

Proof of Corollary 0.2. Let D be triquaternion algebra such that c(ϕ) = [D]. Since ind C(ϕ) = 8, it follows that D is a division algebra. Since ϕ_{F (ψ)} is isotropic, D_{F (ψ)} is not a division algebra. It follows from Lemma 7.3 that there exists an 8-dimensional quadratic form ϕ^{∗} ∈ I^{2}(F) such that ψ ⊂ ϕ^{∗} and c(ϕ^{∗}) = [D]. Obviously, all conditions of Lemma 7.2 hold. Hence ϕ and ϕ^{∗} are half-neighbors.

Lemma 7.4. Let D be a division triquaternion algebra over F. Then there exist a field extension E/F and an 8-dimensional quadratic form ϕ^{∗} ∈ I^{2}(E) with the following properties:

(i) D_{E} is a division algebra, (ii) c(ϕ^{∗}) = [D_{E}], (iii) ϕ_{E(D)}^{∗} is anisotropic.

Proof. Let ϕ ∈ I^{2}(F) be an arbitrary F -form such that c(ϕ) = [D]. Let K =

F (X, Y, Z) and γ = ϕ_{K} ⊥ ⟨ ; ⟨X, Y, Z⟩ ; ⟩ be a K-form. Let K = K_{0}, K_{1}, …, K_{h};

γ_{0}, γ_{1}, …, γ_{h} be a generic splitting tower of γ.

Since γ ≡ ϕ_{K} (mod I^{3}(K)), we have c(γ) = c(ϕ_{K}) = [D_{K}]. Since K/F is purely transcendental, ind D_{K} = ind D = 8. Hence ind C(γ) = 8. It follows from Corollary 1.5 that there exists s such that dim γ_{s} = 8 and ind C(γ_{s}) = 8. We set E = E_{s}, ϕ^{∗} = γ_{s}.

We claim that the condition (i)–(iii) of the lemma hold. Since c(ϕ^{∗}) = c(γ_{E}) = c(ϕ_{E}) = [D_{E}], condition (ii) holds. Since [D_{E}] = c(ϕ^{∗}) = c(γ_{s}), we have ind D_{E} = ind C(γ_{s}) = 8 and thus condition (i) holds.

Now we only need to verify that (iii) holds. Let M_{0}/F be an arbitrary field extension such that ϕ_{M_{0}} is hyperbolic. Let M = M_{0}(X, Y, Z). We have γ_{M} = ϕ_{M} ⊥ ⟨ ; ⟨X, Y, Z⟩ ; ⟩_{M}. Clearly ⟨ ; ⟨X, Y, Z⟩ ; ⟩ is anisotropic over M. Since ϕ_{M} is hyperbolic, we have (γ_{M})_{an} = ⟨ ; ⟨X, Y, Z⟩ ; ⟩_{M} and hence dim(γ_{M})_{an} = 8. Therefore dim(γ_{M})_{an} = dim γ_{s}. By Lemma 1.3, we see that the field extension ME/M = MK_{s}/M is purely transcendental. Hence dim(γ_{ME})_{an} = dim(γ_{M})_{an} = 8. Since (ϕ_{ME}^{∗})_{an} = (γ_{ME})_{an}, we see that ϕ_{ME}^{∗} is anisotropic. Since ϕ_{M} is hyperbolic, it follows that [D_{M}] = c(ϕ_{M}) = 0. Hence [D_{ME}] = 0 and therefore the field extension ME(D)/ME is purely transcendental. Hence ϕ_{ME(D)}^{∗} is anisotropic. Therefore ϕ_{E(D)}^{∗} is anisotropic.

Lemma 7.5. Let ϕ, ϕ^{∗} ∈ I^{2}(F) be 8-dimensional quadratic forms such that c(ϕ) = c(ϕ^{∗}) = [D], where D is a triquaternion division algebra. Suppose that ϕ_{F (D)}^{∗} is anisotropic. Then ϕ_{F (D)} is anisotropic.

Proof. Suppose at the moment that ϕ_{F (D)} is isotropic. Then letting ψ =^{def} ϕ^{∗}, we see that all conditions of Lemma 7.2 hold. Hence ϕ and ϕ^{∗} are half-neighbors, i.e., there exists s ∈ F^{∗} such that ϕ^{∗} + sϕ ∈ I^{4}(F). Therefore ϕ_{F (D)}^{∗} + sϕ_{F (D)} ∈ I^{4}(F (D)). Since ϕ_{F (D)} is isotropic, it is hyperbolic and we see that ϕ_{F (D)}^{∗} ∈ I^{4}(F (D)). By the Arason-Pfister Hauptsatz, we see that ϕ_{F (D)}^{∗} is hyperbolic. So we get a contradiction to the assumption of the lemma.

Proposition 7.6. Let ϕ ∈ I^{2}(F) be an 8-dimensional quadratic form such that ind C(ϕ) = 8. Let A be an algebra such that c(ϕ) = [A]. Then ϕ_{F (A)} is anisotropic.

Proof. Let D be a triquaternion algebra such that c(ϕ) = [D]. Since ind C(ϕ) = 8, D is a division algebra. Let E/F and ϕ^{∗} be such that in Lemma 7.4. All conditions of Lemma 7.5 hold for E, ϕ_{E}, ϕ^{∗}, and D_{E}. Therefore ϕ_{E(D)} is anisotropic. Hence ϕ_{F (D)} is anisotropic. Since [A] = c(ϕ) = [D], the field extension F (A)/F is stably isomorphic to F (D)/F (Proposition 6.3). Therefore ϕ_{F (A)} is anisotropic.

Proof of Corollary 0.3. Suppose at the moment that ϕ_{F (A)} ∈ I^{4}(F (A)). Since ind C(ϕ) ≥ 8, it follows that dim ϕ ≥ 8. By Corollary 1.5 there exists a field extension E/F such that dim(ϕ_{E})_{an} = 8, ind C(ϕ_{E}) = 8. Since dim(ϕ_{E})_{an} = 8 and ϕ_{E(A)} ∈ I^{4}(E(A)), the Arason-Pfister Hauptsatz shows that ((ϕ_{E})_{an})_{E(A)} is hyperbolic. We get a contradiction to Proposition 7.6.

8. Proof of Theorem 0.1

By Proposition 6.7, there is a surjection

\frac{H^{3}(F (ψ, D)/F)}{[D] ∪ H^{1}(F)} ↠ Tors CH^{2}(X_{ψ} × X_{D}) .

Thus, it suffices to prove the second formula of Theorem 0.1.

Proving the second formula, we may assume that dim ψ = 5 (compare to the proof of Theorem 6.9) and D is a division algebra (Corollary 6.4). Under these assumptions, we can write down D as the tensor product C_{0}(ψ) ⊗_{F} B (using [25, Th. 1]). In particular, we see that C_{0}(ψ) is a division algebra, i.e. ind C_{0}(ψ) = deg C_{0}(ψ) = 4.

If deg D < 8, then D ≃ C_{0}(ψ). In this case, ψ_{F (D)} is a 5-dimensional quadratic form with trivial Clifford algebra; therefore ψ_{F (D)} is isotropic; by this reason, the field extension F (ψ, D)/F (D) is purely transcendental and consequently H^{3}(F (ψ, D)/F (D)) = 0. It follows that

H^{3}(F (ψ, D)/F) = H^{3}(F (D)/F) = [D] ∪ H^{1}(F) ,

where the last equality holds by Proposition 6.2.

If deg D > 8, then ind B ≥ 4. Applying the index reduction formula [31, Th. 1.3], we get

ind C_{0}(ψ)_{F (D)} = min{ind C_{0}(ψ), ind B} = 4 .

Therefore ψ_{F (D)} is not a 3-Pfister neighbor and by Proposition 6.1 the group H^{3}(F (ψ, D)/F (D)) is trivial. Thus once again

H^{3}(F (ψ, D)/F) = H^{3}(F (D)/F) = [D] ∪ H^{1}(F) .

Finally, if deg D = 8, then we are done by Theorem 6.9 and Proposition 6.7.

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Oleg Izhboldin Department of Mathematics and Mechanics St.-Petersburg State University Petrodvorets, 198904 Russia oleg@izh.usr.pu.ru Nikita Karpenko Westfälische Wilhelms-Universität Mathematisches Institut Einsteinstraße 62 D-48149 Münster Germany karpenk@math.uni-muenster.de 312

Documenta Mathematica 2 (1997)