Order of Torsion in CH^{4} of Quadrics

Nikita A. Karpenko^{1}

Received: November 24, 1995 Revised: January 30, 1996

Communicated by Alexander S. Merkurjev

Abstract. It is shown that the order of the torsion subgroup in the 4codimensional Chow group CH^{4}(X_{φ}) of a projective quadric X_{φ} is at most 4 provided that the dimension of the corresponding quadratic form φ is greater than 8.

Consider a non-degenerate quadratic form φ over a field F of characteristic different from 2 and the corresponding projective quadric X_{φ}. We always assume that dim X_{φ} ≥ 1, i.e. dim φ ≥ 3. It is an open problem to describe the torsion subgroup of the Chow group CH^{∗}(X_{φ}) (this is the group of algebraic cycles on X_{φ} modulo rational equivalence graded by co-dimension of cycles [1, 2]).

Generally speaking computation of the Chow group of an algebraic variety is an interesting and important problem of algebraic geometry. However the class of varieties for which this problem is solved is rather small. Chow groups and K-theory of quadrics were studied first by R. Swan. Although the K-theory was completely computed [13] the question on the Chow group remained open.

A new motivation grew out of the attempts to solve the norm residue homomorphism problem. During the work on this problem it became clear that a decisive progress could be achieved by computation of the so called K-cohomology groups [10, 12] for quadrics and in particular of their Chow groups.

In [4] Chow groups of small-dimensional quadrics were computed. An interesting phenomenon was found: some Chow groups have torsion and the problem of computing the whole Chow group reduces to finding the torsion.

Let us consider some first gradation components. The group CH^{1}(X_{φ}) is always torsion-free. The next group — CH^{2}(X_{φ}) is computed in [4]. In particular, it turns out that # Tors CH^{2}(X_{φ}) ≤ 2 for any form φ; moreover, Tors CH^{2}(X_{φ}) = 0 if dim φ > 8 [4, theorem (6.1)]. In co-dimension 3 one has: # Tors CH^{3}(X_{φ}) ≤ 2 for any φ [5, theorem] and Tors CH^{3}(X_{φ}) = 0 if dim φ > 12 [6, theorem 6.1]. As to co-dimension 4, it is known today that Tors CH^{4}(X_{φ}) = 0 if dim φ > 24 [6, theorem 8.5]; however, one has an example of a 7-dimensional form φ (defined over an appropriate F) with infinite Tors CH^{4}(X_{φ}) [7, theorem 6.5].

1Support and hospitality of Sonderforschungsbereich 343 (Bielefeld University) are gratefully acknowledged.

Here we prove that

# Tors CH^{4}(X_{φ}) ≤ 4

for any φ of dimension greater than 8 (4.1). Notice that Tors CH^{4}(X_{φ}) = 0 if dim φ = 6 [14], [4, (2.6)] and CH^{4}(X_{φ}) = 0 if dim φ < 6; so, the “exceptional” dimensions are only 7 and 8. We also reproduce (with small simplifications) the proof that # Tors CH^{3}(X_{φ}) ≤ 2.

This note grew out from a remark of B. Kahn that Tors CH^{4}(X_{φ}) is finite if dim φ > 8.

1. An exact sequence

We consider (Quillen’s) K-cohomology H^{p}(X_{φ}, K_{q}) and the Grothendieck group K′_{0}(X_{φ}) which we denote simply by K(X_{φ}) and supply with the so called topological filtration

K(X_{φ}) = K(X_{φ})^{(0)} ⊃ K(X_{φ})^{(1)} ⊃ ….

We denote by φ̃ the form φ over a field extension F̃ of F which completely (so much as possible by the dimension reason) splits φ.

Proposition 1.1. One has an exact sequence

Ker (H^{2}(X_{φ}, K_{3}) → H^{2}(Xφ̃, K_{3})) → CH^{4}(X_{φ}) → K(X_{φ})^{(4/5)} → 0 .

Proof. The kernel of the canonical epimorphism CH^{4}(X_{φ}) →→ K(X_{φ})^{(4/5)} is controlled by certain differentials of the BGQ-spectral sequence E_{2}^{p,q} = H^{p}(X_{φ}, K_{−q}) [10, §7]. Since CH^{4}(X_{φ}) = E_{2}^{4,−4}, the differentials in question start from E_{4}^{0,−1}, E_{3}^{1,−2} and E_{2}^{2,−3}. Since

E_{2}^{0,−1} = H^{0}(X_{φ}, K_{1}) = F^{×} and
E_{2}^{1,−2} = H^{1}(X_{φ}, K_{2}) = F^{×} (if dim φ > 4) [4, theorem (4.1)]

all the differentials starting from E_{r}^{0,−1} and E_{r}^{1,−2} with r ≥ 2 are 0. Hence we have an exact sequence

H^{2}(X_{φ}, K_{3}) →^{d} CH^{4}(X_{φ}) → K(X_{φ})^{(4/5)} → 0 .

Using pull-back with respect to the embedding of X_{φ} in the enveloping projective space ℙ, one can define a homomorphism

F^{×} = H^{2}(ℙ, K_{3}) → H^{2}(X_{φ}, K_{3})

which is easily checked to be an isomorphism in the case when φ splits and dim φ > 6. For an arbitrary φ we obtain a commutative square

F^{×} ⟶ H^{2}(X_{φ}, K_{3})
↓  ↓
F̃^{×} ⟶ H^{2}(Xφ̃, K_{3})

which produce a decomposition

H^{2}(X_{φ}, K_{3}) = F^{×} ⊕ Ker (H^{2}(X_{φ}, K_{3}) → H^{2}(Xφ̃, K_{3}))

provided that dim φ > 6. Since d|_{F^{×}} = 0, we are done in this case. The case dim φ ≤ 6 is trivial and not of use for the consequent.

Order of Torsion in CH^{4} of Quadrics 59

2. The left-hand side term

There is a description of the left-hand side term of (1.1).

Proposition 2.1 ([8, prop. 1]). Suppose that dim φ ≥ 5 and φ is not a 3-Pfister neighbor (i.e. not similar to a subform of an anisotropic 3-Pfister form). The kernel of the restriction

H^{2}(X_{φ}, K_{3}) ⟶ H^{2}(Xφ̃, K_{3})

is naturally isomorphic to the kernel of the Galois cohomology map

H^{4}(F, Z/2) ⟶ H^{4}(F (φ), Z/2) .

Remark 2.2. The assumption that φ is not a 3-Pfister neighbor is likely superfluous.

Definition 2.3. Denote by P_{4}(φ) the subset of H^{4}(F, ℤ/2) consisting of 0 and all cup-products (a, b, c, d) with a, b, c, d ∈ F^{×} such that φ is similar to a subform of the 4-Pfister form ⟨⟨a, b, c, d⟩⟩ (the latter means as usual the product ⟨1, −a⟩ ⊗ ⟨1, −b⟩ ⊗ ⟨1, −c⟩ ⊗ ⟨1, −d⟩).

Proposition 2.4 ([3]). If φ is any quadratic form with dim φ ≥ 5 then

Ker (H^{4}(F, Z/2) → H^{4}(F (φ), Z/2)) = P_{4}(φ) .

Corollary 2.5. If dim φ > 8 one can rewrite the sequence (1.1) as follows:

P_{4}(φ) → CH^{4}(X_{φ}) → K(X_{φ})^{(4/5)} → 0 .

3. The right-hand side term

In order to control the right-hand side term of (2.5), we need some general facts on the subsequent quotients of the topological filtration on the Grothendieck group of a quadric. Most results of this § are from [5].

We are going to use the following notation.

We put for shortness K = K(X_{φ}).

The quotient K^{(p/p+1)} will be denoted by G^{p}K.

We put forever d = dim X_{φ} = dim φ − 2.

Sometimes it is more convenient to use the lower indexes for the topological filtration by meaning dimension instead of co-dimension, i.e. K_{(p)} = K^{(d−p)}. All the graded groups appearing in this § are graded “by co-dimension”; by that reason the asterisk stays always as a superscript. However, sometimes it is more convenient to refer to a component of a graded group by giving its “dimension”; in this case we use the subscript. For instance, G_{p}K will stay for the p-dimensional component of the graded group G^{∗}K; it is the same as G^{d−p}K.

Let h ∈ K be the class of a hyperplane section of X_{φ}. This h does not depend on the choice of the hyperplane, moreover h = 1 − [O_{X_{φ}}(−1)].

For any x ∈ K we define dimension dim x of x as the infimum of p such that x ∈ K_{(p)}. For instance, dim 0 = −∞, dim h = d − 1. Any 0 ≠ x ∈ K determines an element 0 ≠ x̄ ∈ G^{∗}K, namely the residue class in G_{dim x}K.

The subring of K generated by h will be denoted by H. It contains [O(n)] for all integers n. As a group, H is freely generated by 1, h, h^{2}, …, h^{d}. The filtration on H induced from K is just the “filtration by powers of h”. In particular, the adjoint graded group G^{∗}H is torsion-free.

Definition 3.1. Let us define an integer s = s(φ) in the following way. If φ ∉ I^{2}(F) (where I(F) stays for the ideal of the even-dimensional forms in the Witt ring of F) then the even Clifford algebra C_{0}(φ) is simple, so it is isomorphic to the algebra M_{n}(D) of (n×n)-matrices over a skew-field D; in this case we take s such that n = 2^{s}. If φ ∈ I^{2}(F), we take s such that C_{0}(φ) ≃ M_{2^{s}}(D) × M_{2^{s}}(D).

There is a trivial observation

Lemma 3.2. If φ ∉ I^{2}(F) then K(C_{0}(φ)) is freely generated by the class of a (unique up to an isomorphism) simple C_{0}(φ)-module P; moreover,

[C_{0}(φ)] = 2^{s(φ)} · [P] ∈ K(C_{0}(φ)) .

If φ ∈ I^{2}(F) then K(C_{0}(φ)) is freely generated by the classes of two non-isomorphic simple C_{0}(φ)-modules P and P′; moreover,

[C_{0}(φ)] = 2^{s(φ)} · ([P] + [P′]) ∈ K(C_{0}(φ)) .

Lemma 3.3 ([4, lemma (3.6)]). Let U be the Swan’s sheaf on X_{φ} [13, p. 126]. Then in K

[U(d)] = h^{d} + 2h^{d−1} + ⋯ + 2^{d−1}h + 2^{d} .

Since the sheaf U has a (right) action of C_{0}(φ) the class [U] ∈ K is divisible by 2^{s} (3.2), so the following definition is correct (take also in account that the group K is torsion-free by [13, theorem 1] and (3.2)).

Definition 3.4. For any 0 ≤ i < s we define an element l_{i} ∈ K as

l_{i} = \frac{1}{2^{i+1}} (h^{d} + 2h^{d−1} + ⋯ + 2^{i}h^{d−i}) ;

for a certain convenience reason we also put l_{−1} = 0.

What these elements are explains the following

Lemma 3.5. The element l_{i} is equal to the class of an i-dimensional linear subspace on X_{φ} if such a subspace lies on X_{φ} (i.e. if the form φ contains an (i+1)-dimensional totally isotropic subspace, i.e. if the Witt index of φ is at least i + 1).

Proof. Let L_{i} ⊂ X_{φ} be an i-dimensional linear subspace of X_{φ} and in : X_{φ} ↪ ℙ the embedding of X_{φ} into the projective space as a hypersurface. First assume that dim φ is odd. Then using [13, theorem 1] it is easy to see that the push-forward in_{∗} : K(X_{φ}) → K(ℙ) is injective, so it would be enough to check that in_{∗}([L_{i}]) = in_{∗}(l_{i}). The left-hand side is just [L_{i}] ∈ K(ℙ) while the right-hand side can be rewritten with using the projection formula as

\frac{1}{2^{i+1}} (l^{d} + 2l^{d−1} + ⋯ + 2^{i}l^{d−i}) · [X_{φ}]

where l^{i} denotes the class of an i-co-dimensional linear subspace of ℙ. Computing

[X_{φ}] = 1 − [O_{ℙ}(−2)] = 2l^{1} − l^{2}

and multiplying we get l^{d−i+1} what is the same as the required [L_{i}] because dim ℙ = d + 1.

Now assume that dim φ is even. Take any non-singular hyperplane section Y of X_{φ} containing L_{i} (it is really possible to find such a Y because i ≠ d/2 (3.4)). Since Y is an odd-dimensional quadric we know from the previous paragraph that

[L_{i}] = \frac{1}{2^{i+1}} (h^{d−1} + 2h^{d−2} + ⋯ + 2^{i}h^{d−1−i}) ∈ K(Y) .

Applying the push-forward with respect to the embedding Y ↪ X_{φ} and using once again the projection formula for the right-hand side we get

[L_{i}] = \frac{1}{2^{i+1}} (h^{d−1} + 2h^{d−2} + ⋯ + 2^{i}h^{d−1−i}) · [Y] ∈ K(X_{φ}) .

Since [Y] = h we are done.

Lemma 3.6. For any 0 ≤ i < s one has:

• 2l_{i} = h^{d−i} + l_{i−1};

• hl_{i} = l_{i−1};

• dim l_{i} > dim l_{i−1};

• if φ is anisotropic then dim l_{i} > i.

Proof. The first two properties are obvious from the formula (3.4) defining l_{i}. Since the multiplication in K respects the filtration and h ∈ K^{(1)} the second property implies the third one. If φ is anisotropic, the degree of any closed point on X_{φ} is even whence l_{0} ∉ K_{(0)}, i.e. dim l_{0} > 0; thus dim l_{i} ≥ i + dim l_{0} > i.

Corollary 3.7. If φ is anisotropic every element l̄_{i} ∈ G^{∗}K, 0 ≤ i < s has order 2.

Proof. By an agreement in the beginning of § we denote by l̄_{i} the class of l_{i} ∈ K in G_{dim l_{i}}K. By (3.6) 2l_{i} = h^{d−i} + l_{i−1}, dim l_{i} > dim l_{i−1} and dim l_{i} > i = dim h^{d−i}. Thus dim l_{i} > dim 2l_{i}, i.e. 2l̄_{i} = 0.

Definition 3.8. Let us denote by I^{∗} ⊂ Tors G^{∗}K the subgroup generated by all l̄_{i}, 0 ≤ i < s. The quotient Tors G^{∗}K/I^{∗} will be denoted by II^{∗}.

Theorem 3.9. Assume that the quadratic form φ is anisotropic. There exits an exact sequence of graded groups

0 → I^{∗} → Tors G^{∗}K → II^{∗} → 0

where I^{∗} and II^{∗} have the following properties:

• #I^{p} ≤ 2 for any p;

• #I^{∗} = 2^{s} where s = s(φ) is defined in (3.1);

• if φ ∉ I^{2}(F) then II^{∗} = 0; moreover, in the case φ ∈ I^{2}(F) it holds:

• for every p the group II^{p} is cyclic;

• II^{p} = 0 for p ≥ d/2;

• if there exists a field extension of degree 2^{n} which completely splits φ then #II^{∗} divides 2^{n+s−d/2};

• if II^{0} = II^{1} = ⋯ = II^{p} = 0 for some p < d/2 then I^{0} = I^{1} = ⋯ = I^{p} = I^{p+1} = 0.

Proof. The graded groups I^{∗} and II^{∗} are defined in (3.8). The group I^{∗} has exactly s non-trivial components: these are components of dimensions dim l_{i}, i = 0, 1, …, s− 1 (by (3.6) all the numbers dim l_{i} are distinct). Every non-trivial component has order 2 because it is generated by an element l̄_{i} (3.7). So, two first statements of the theorem hold by the very definition of I^{∗}.

Suppose that φ ∉ I^{2}(F). If we consider on H and K/H the filtrations induced from K the exact sequence 0 → H → K → K/H → 0 will give an exact sequence of the adjoint graded groups:

0 → G^{∗}H → G^{∗}K → G^{∗}(K/H) → 0 .

Since G^{∗}H is torsion-free we obtain an injection Tors G^{∗}K ↪ G^{∗}(K/H). Note that [13, theorem 1], (3.2) and (3.3) imply #K/H = 2^{s}. Since Tors G^{∗}K ⊃ I^{∗}, #I^{∗} = 2^{s} and #G^{∗}(K/H) = #K/H = 2^{s} we obtain that Tors G^{∗}K = I^{∗}, i.e. II^{∗} = 0.

Now suppose that φ ∈ I^{2}(F). Denote by N the subgroup of K generated by H and 2^{−s}[U]. Considering on N and K/N the induced filtrations we get an exact sequence of the adjoint graded groups

0 → G^{∗}N → G^{∗}K → G^{∗}(K/N) → 0 .

So, the torsion subgroups are connected by the exact sequence:

0 → Tors G^{∗}N → Tors G^{∗}K → Tors G^{∗}(K/N) .

The same arguments as above show that Tors G^{∗}N = I^{∗}. Thus the latter exact sequence produces an embedding II^{∗} ↪ G^{∗}(K/N). Since the quotient K/N is a cyclic group every component G^{p}(K/N) is cyclic too; whence the fourth statement of the theorem.

Since rk G^{d/2}K = 2 [4, (3.1),(2.2),(2.7)] and rk G^{d/2}N = 1 we have

rk G^{d/2}(K/N) = 1 ;

thereby G^{p}(K/N) = 0 for p ≥ d/2 whence the fifth statement of the theorem.

Suppose that there exists a field extension of degree 2^{n} completely splitting φ, let φ̃ be the form φ over this extension. Let P̃ be a simple C_{0}(φ̃)-module. Put ũ = [U ⊗P̃] ∈ K(Xφ̃). The multiple 2^{d/2−s}ũ of ũ lies in K(X_{φ}) and generates the quotient K(X_{φ})/N(X_{φ}). Considering the element ũ itself in the quotient K(Xφ̃)/N(Xφ̃) one has: ũ ∈ (K(Xφ̃)/N(Xφ̃))^{(d/2)}. Taking the transfer we get:

2^{n}ũ ∈ (K(X_{φ})/N(X_{φ}))^{(d/2)} .

Consequently, # Tors G^{∗}(K(X_{φ})/N(X_{φ})) divides 2^{n+s−d/2} and we have proved the sixth statement.

Let us prove the seventh one. Denote by l_{d/2} ∈ K(Xφ̃) the class of a (d/2)- dimensional linear subspace L_{d/2} lying on Xφ̃. Applying the projection formula to the embedding L_{d/2} ↪ Xφ̃ and using (3.5) one gets: hl_{d/2} = l_{d/2−1}. It follows from [13, theorem 1] that 2^{d/2−s}K(Xφ̃) ⊂ K(X_{φ}). In particular, l := 2^{d/2−s}l_{d/2} ∈ K(X_{φ}).

Lemma 3.10. One has in K(X_{φ}): dim l ≥ m, dim 2^{n}l = m (here 2^{n} is as above the degree of a field extension completely splitting φ) and dim l_{s−1} < dim l.

Proof. Two first properties are evident. The last one holds since

hl = h(2^{d/2−s}l_{d/2}) = 2^{d/2−s}l_{d/2−1} ≡ l_{s−1} mod H_{(d/2−1)} .

Let I_{p} be the non-trivial component of I^{∗} of maximal dimension and suppose that p ≥ d/2. To prove the last statement of the theorem it suffices to find a number q > p with Tors G_{q}K(X_{φ}) ≠ 0. Put q = dim l. Since p = dim l_{s−1} we have by the lemma: q > p. The group G_{q}K(X_{φ}) contains a non-zero element l̄, moreover 2^{n}l̄= 0 by the lemma. Thus Tors G_{q}K(X_{φ}) ≠ 0 and we are done.

Corollary 3.11. If for some p

Tors G^{0}K = Tors G^{1}K = ⋯ = Tors G^{p}K = 0

then the group Tors G^{p+1}K is cyclic.

Proof. According to the theorem a group Tors G^{p+1}K might be non-cyclic only in the case when φ ∈ I^{2}(F) and p < d/2. In this case we can apply the last statement of the theorem.

4. Torsion in CH^{4}

Theorem 4.1. If dim φ > 8 then # Tors CH^{4}(X_{φ}) ≤ 4.

Proof. If φ is isotropic, say φ ≃ ℍ⊥ψ then CH^{4}(X_{φ}) ≃ CH^{3}(X_{ψ}) [11, proposition 1], [4, (2.2)]; by [5, theorem] (see also (5.1)) # Tors CH^{3}(X_{ψ}) ≤ 2 always.

Below in the proof we assume that φ is anisotropic.

Suppose that φ is not a 4-Pfister neighbor. Then by (2.5) we have an isomorphism CH^{4}(X_{φ}) ≃ G^{4}K(X_{φ}). If φ ∉ I^{2}(F) or dim φ ≤ 10 then # Tors G^{4}K(X_{φ}) = #I^{4} ≤ 2 by (3.9). So, only the case φ ∈ I^{2}(F) and dim φ ≥ 12 is left.

If dim φ > 12 all the groups CH^{p}(X_{φ}) with p ≤ 3 are torsion-free. Hence the groups G^{p}K(X_{φ}) with p ≤ 3 are torsion-free too and thereby G^{4}K(X_{φ}) is cyclic (3.11). If dim φ > 14 let us take a quadratic extension L/F such that φ_{L} is isotropic. Then CH^{4}(X_{φ_{L}}) ≃ CH^{3} X_{ψ} for a quadratic form ψ with dim ψ > 12 whence Tors CH^{4}(X_{φ_{L}}) ≃ Tors CH^{3}(X_{ψ}) = 0. Applying the transfer we get 2 Tors CH^{4}(X_{φ}) = 0, i.e. # Tors CH^{4}(X_{φ}) ≤ 2 in this case.

If dim φ = 14 we take a biquadratic extension L/F such that the Witt index of φ_{L} is at least 2. Then CH^{4}(X_{φ_{L}}) ≃ CH^{2} X_{ψ} for a quadratic form ψ with dim ψ = 10 whence Tors CH^{4}(X_{φ_{L}}) ≃ Tors CH^{2}(X_{ψ}) = 0 and by the transfer argument 4 Tors CH^{4}(X_{φ}) = 0, i.e. # Tors CH^{4}(X_{φ}) ≤ 4.

For a 12-dimensional quadratic form φ lying in I^{2}(F) let us compute the order of the second kind torsion II^{∗} ⊂ G^{∗}K(X_{φ}). Let L/F be a field extension of degree 2^{d/2−s} (d = 10 now) splitting the Clifford invariant of the form φ. Since φ_{L} is a 12-dimensional from from I^{3}(L) it (completely) splits in a quadratic extension E/L [9, Satz 14]. Putting n = log_{2}[E : F] = d/2 − s + 1 in the formula from (3.9) we get #II^{∗} ≤ 2. Since Tors G^{p}K(X_{φ}) = 0 for p ≤ 2 we have: Tors G^{3}K(X_{φ}) = II^{3} (3.9). Now we can argue as follows: if II^{3} ≠ 0 then II^{3} = II^{∗}, in particular II^{4} = 0, so Tors G^{4}K(X_{φ}) = I^{4} has the order at most 2; otherwise, if II^{3} = 0 the group I^{4} is zero (3.9) and so Tors G^{4}K(X_{φ}) = II^{4} has the order at most 2 again.

We have completed the case when φ is not a 4-Pfister neighbor. Now assume the opposite. Since a Pfister neighbor uniquely determines the Pfister superform the left-hand side term of (2.5) has now the order 2. By this cause we have to show that the right-hand side term, i.e. the group Tors G^{4}K(X_{φ}) is of order at most 2. Looking at the previous part of the current proof we see that it is always the case except when φ ∈ I^{2}(F) and dim φ = 14. But since a 14-dimensional quadratic form of trivial discriminant is evidently not able to be an (anisotropic!) Pfister neighbor this exception does not occurs.

Remark 4.2. The proof of the theorem contains in fact a more precise information on what Tors CH^{4}(X_{φ}) for a particular φ can be. One can also handle the case of dim φ = 7, 8 if φ is not similar to a subform of an anisotropic 4-Pfister form — see (2.2).

5. Torsion in CH^{3}

Theorem 5.1 ([6]). For any φ, one has # Tors CH^{3}(X_{φ}) ≤ 2.

Proof. If φ is isotropic, say φ = ℍ⊥ψ, then CH^{3}(X_{φ}) ≃ CH^{2}(X_{ψ}). Since # Tors CH^{2} ≤ 2 for any quadric [4, theorem (6.1)] we are done in this case. From now on we suppose that φ is anisotropic.

Arguments like (1.1) show that CH^{3}(X_{φ}) ≃ G^{3}K(X_{φ}) [4, corollary (4.5)]. If φ ∉ I^{2}(F) or dim φ ≤ 8 then

# Tors G^{3}K(X_{φ}) ≤ 2

by (3.9). From now on we consider only the case φ ∈ I^{2}(F) and dim φ ≥ 10.

Since dim φ ≥ 10, the groups G^{p}K(X_{φ}) for p ≤ 2 are torsion-free (for p = 2 it holds according to the computation of CH^{2}(X_{φ}) [4, theorem (6.1)]). Hence Tors G^{3}K(X_{φ}) = II^{3} (3.9) which is a cyclic group. The last we need to show is 2 Tors CH^{3}(X_{φ}) = 0. For this it would suffice to find a quadratic extension L/F such that the group CH^{3}(X_{φ_{L}}) = 0 is torsion-free (then one can use the transfer argument).

Take simply an arbitrary quadratic extension L/F which partially splits (i.e. makes isotropic) the form φ, say φ_{L} = ℍ⊥ψ. We have: CH^{3}(X_{φ_{L}}) ≃ CH^{2}(X_{ψ}). If Tors CH^{2}(X_{ψ}) = 0 we are done.

If not then according to the computation of CH^{2} the form ψ is similar to a 3-Pfister form. In this case we can compute the order of the second kind torsion II^{∗} ⊂ G^{∗}K(X_{φ}) by using the formula from (3.9). We have: d = 8, s(φ) = 3 (if s(φ) = 4 then φ should be isotropic as a 10-dimensional form from I^{3}) and since one can split φ by a field extension of degree 4 we can put n = 2. Thus #II^{∗} ≤ 2^{2+3−8/2} = 2. In particular, # Tors G^{3}K(X_{φ}) = #II^{3} ≤ 2.

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Nikita A. Karpenko

Westfälische Wilhelms-Universität Mathematisches Institut Einsteinstr. 62 D-48149 Münster, Germany karpenk@math.uni-muenster.de 66

Documenta Mathematica 1 (1996)