CODIMENSION 2 CYCLES ON PRODUCTS OF PROJECTIVE HOMOGENEOUS SURFACES

SANGHOON BAEK

Abstract. In the present paper, we provide general bounds for the torsion in the codimension 2 Chow groups of the products of projective homogeneous surfaces. In particular, we determine the torsion for the product of four Pfister quadric surfaces and the maximal torsion for the product of three Severi-Brauer surfaces. We also find an upper bound for the torsion of the product of three quadric surfaces with the same discriminant.

Contents

1. Introduction 1 2. Two filtrations on the Grothendieck ring 3 3. Grothendieck group of the product of Severi-Brauer varieties 4 4. Product of Conics 5 4.1. Four Conics 7 4.2. Galois cohomology and torsion groups 13 5. Product of Severi-Brauer surfaces 13 5.1. Three Severi-Brauer surfaces 16 6. Product of Quadric surfaces 20 6.1. Three quadric surfaces with the same discriminant. 24 References 28

1. Introduction

Let X be a projective homogenous variety under the action of a semisimple group G over an algebraically closed field F. The Chow group CH(X) of algebraic cycles modulo the rational equivalence relation is well-understood as well as its ring structure. Namely, the Chow group of X is a free abelian group with the basis of Schubert cycles. For an arbitrary base field F, this is no longer true: the Chow group CH(X) can have torsion. Indeed, by a transfer argument the problem of determining the Chow group of X over an arbitrary field F reduces to computing its torsion subgroup.

For codimension d ≤ 1, the Chow group CH^{d}(X) is torsion-free. A nontrivial torsion first appears in codimension 2 cycles on X and the exact structure of the torsion subgroup is known in many cases. For a projective quadric X, the torsion subgroup of CH^{2}(X) is either 0 or ℤ/2ℤ [6]. For a Severi-Brauer variety X, it is shown that the torsion subgroup of CH^{2}(X) is either 0 or a cyclic group if the corresponding simple algebra satisfies certain conditions [8]. For a simple simply connected group G splits over F (X), it is known that the torsion subgroup of CH^{2}(X) is a cyclic group generated by the Rost invariant [2]. However, the only partial results are known for their products. In this case, the structure of the torsion subgroup is more complicated.

As a first step in determining torsion in codimension 2 cycles on the product of flag varieties, we consider in this paper the product of two dimensional flag varieties of the same type. For an integer n ≥ 1, this can be divided into two classes: the set of all products of n Severi-Brauer surfaces, denoted by SB_{n} and the set of all products of n quadric surfaces, denoted by Q_{n}. The latter has a special subclass PQ_{n} consisting of all product of n Pfister quadric surfaces. Here, we view the product of two conics as the product of two Pfister quadric surfaces since they have the same torsion subgroup in codimension 2 cycles by [3, Corollary 2.5]. To measure the size of the torsion subgroup of codimension 2 cycles, we introduce the following notation: M(A) = max_{X∈A} | CH^{2}(X)_{tors}|, where A is any of SB_{n}, Q_{n}, and PQ_{n}. Hence, the torsion subgroup of any element of A is an elementary abelian group whose order is a divisor of M(A). We denote by C_{n} the set of all products of n conics, thus we have M(PQ_{n}) = M(C_{n}).

It is well-known that M(SB_{1}) = M(Q_{1}) = M(C_{2}) = 1. In [10] Peyre proved that M(C_{3}) = 2, thus M(PQ_{3}) = 2. In [4], [3] and [5] Izhboldin and Karpenko proved that M(SB_{2}) = 3 and M(Q_{2}) = 2. As it is showed in [11, Theorem 2.1] and [10, Theorem 4.1], the torsion subgroup of codimension 2 cycles is closely related to a relative Galois cohomology group in degree 3 and the above results were used to describe the cohomology groups. Moreover, Izhboldin and Karpenko’s result was the key part of their results on isotropy of quadratic forms.

The main goal of this paper is to extend their results to arbitrary n. First, we determine the maximal torsion M(C_{n}) (= M(PQ_{n})) in Corollary 4.4, which gives a general lower bound of M(Q_{n}). In particular, we determine the torsion in the gamma filtration of the product of four conics in Theorem 4.6 as well as its torsion in Chow group. Secondly, we find a general lower bound of M(SB_{n}) in Proposition 5.2. Especially, we show that the bound is sharp for the product of three Severi-Brauer surfaces in Theorem 5.4. The results M(PQ_{4}) = 2^{5} and M(SB_{3}) = 3^{8} give the first examples such that the torsion subgroup CH^{2}(X)_{tors} is not cyclic in their classes. In the last part, we provide an elementary proof of Izhboldin and Karpenko’s result for the product of two quadric surfaces and find an upper bound for the torsion of the product of three quadric surfaces with the same discriminant in Theorem 6.1, Proposition 6.3, and Theorem 6.5.

As Karpenko showed in [7] and [8], the topological filtration and the gamma filtration on the Grothendieck ring can be used to find the torsion in Chow groups of codimension 2 of projective homogeneous varieties. Moreover, the torsion in the filtrations can be computed by studying the divisibility of certain polynomials produced by the characteristic classes on the Grothendieck ring. We use this general approach to find the torsion in codimension 2 cycles on the product of projective homogeneous surfaces together with some additional combinatorial arguments.

This paper is organized as follows. In Sections 2 and 3, we recall basics of the topological filtration and the gamma filtration on the Grothendieck ring as well as their torsion subgroups of the product of Severi-Brauer varieties. In Section 4, we determine the bound of the torsion in codimension 2 cycles on the product of n conics or the product of n Pfister quadric surfaces. In particular, we determine the torsion of the product of four conics or the product of four Pfister quadric surfaces in terms of the indexes of the corresponding algebras. In the last part of this section, we also present its application to a Galois cohomology group in degree 3. In Section 5, we find a general lower bound of the torsion in codimension 2 cycles on the product of n Severi-Brauer surfaces. Using this bound, we determine the maximal torsion subgroup of the Chow group of codimension 2 cycles on the product of three Severi-Brauer surfaces. In the last section, we recover a result of Izhboldin and Karpenko and extend it to the product of three quadric surfaces with the same discriminant.

In the present paper, A_{tors} denotes the torsion subgroup of an abelian group A and I_{n} = {1, …, n} for any integer n ≥ 1. We denote by min{·} and max{·} the minimum integer of a set and the maximum of a set, respectively.

2. Two filtrations on the Grothendieck ring

In this section, we briefly recall definitions and properties of the topological filtration and the gamma filtration on the Grothendieck ring K of a smooth projective variety (see [1] and [8] for details). We also provide a useful fact concerning the torsion part of these two filtrations on a smooth projective homogeneous variety.

Let X be a smooth projective variety and let K(X) be the Grothendieck ring of X. The topological filtration

K(X) = T^{0}(X) ⊃ T^{1}(X) ⊃ …

is given by the ideal T^{d}(X) generated by the class [O_{Y}] of the structure sheaf of a closed subvariety Y of codimension at least d. We write T^{d/d+1}(X) for the quotient T^{d}(X)/T^{d+1}(X).

Let Γ^{0}(X) = K(X) and let Γ^{1}(X) be the kernel of the rank map K(X) → ℤ. The gamma filtration

K(X) = Γ^{0}(X) ⊃ Γ^{1}(X) ⊃ …

is given by the ideals Γ^{d}(X) generated by the product γ_{d_{1}}(x_{1}) ⋯ γ_{d_{i}}(x_{i}) with x_{j} ∈ Γ^{1}(X) and d_{1} + ⋯ + d_{i} ≥ d, where γ_{d_{j}} is the gamma operation on K(X). For instance, we have γ_{0}(x) = 1 and γ_{1} = id, where x ∈ K(X). Indeed, the gamma operation defines the Chern class c_{j}(x) := γ_{j}(x − rank(x)) with values in K.

For any d ≥ 0, the gamma filtration Γ^{d}(X) is contained in the topological filtration T^{d}(X). For small degree d = 1, 2, two filtrations coincide. Moreover, the second quotient of the topological filtration can be identified with the codimension 2 cycles so that we have:

(1)  Γ^{2/3}(X) ↠ T^{2/3}(X) = CH^{2}(X).

Now we assume that X is a smooth projective homogeneous variety over a field F. Let E be a splitting field of X. Then, by [11, Proposition 3.4] we have

(2)  T^{d}(X) = Γ^{d}(X) = Γ^{d}(X_{E}) ∩ K(X)

for d = 1, 2.

Let F be either the gamma-filtration Γ or the topological filtration T on K(X). Applying the Snake lemma to the commutative diagram involving the exact sequences 0 → F^{d+1}(X) → F^{d}(X) → F^{d/d+1}(X) → 0 and the one over a splitting field E, we have the following useful formula [7, Proposition 2]:

(3)  |⊕ F^{d/d+1}(X)_{tors}| · |K(X_{E})/K(X)| = ∏_{d=1}^{dim(X)} |F^{d/d+1}(X_{E})/ Im(res^{d/d+1})|,

where res^{d/d+1} : F^{d/d+1}(X) → F^{d/d+1}(X_{E}) is the restriction map.

3. Grothendieck group of the product of Severi-Brauer varieties

We now recall the Grothendieck group of a product of Severi-Brauer varieties. In addition, we state some basic facts about codimension 2 cycles of a product of Severi-Brauer varieties.

Let A_{i} be a central simple F -algebra of degree d_{i} for 1 ≤ i ≤ n. Consider the restriction map

(4)  K(∏_{i=1}^{n} SB(A_{i})) → K(∏_{i=1}^{n} ℙ_{E}^{d_{i}−1}),

where the corresponding Severi-Brauer variety SB(A_{i}) over a splitting field E is identified with the projective space ℙ_{E}^{d_{i}−1}. The latter ring in (4) is isomorphic to the quotient ring ℤ[x_{1}, ⋯, x_{n}]/((x_{1} − 1)^{d_{1}}, ⋯, (x_{n} − 1)^{d_{n}}), where x_{i} is the pullback of the class of the tautological line bundle on ℙ_{E}^{d_{i}−1}. Then by [12, §8 Theorem 4.1] (see also [10, Proposition 3.1]) the image of the map (4) coincides with the sublattice with basis

(5)  {ind(A_{1}^{⊗i_{1}} ⊗ ⋯ ⊗ A_{n}^{⊗i_{n}}) · x_{1}^{i_{1}} ⋯ x_{n}^{i_{n}} | 0 ≤ i_{j} ≤ d_{j} − 1, 1 ≤ j ≤ n}.

Let X be the product of Severi-Brauer varieties SB(A_{i}) as above. For codimension 2 cycles, one can simplify the computation of torsion subgroup by using [4, Proposition 4.7]: if ⟨A′_{1}, …, A′_{n′}⟩ = ⟨A_{1}, …, A_{n}⟩ in the Brauer group Br(F), then

(6)  CH^{2}(X)_{tors} ≃ CH^{2}(X′)_{tors},

where X′ = ∏_{i=1}^{n′} SB(A′_{i}).

Now we restrict our attention to p-primary algebras. Let A_{1}, …, A_{n} be central simple algebras of p-power degree with given indices ind(A_{1}^{⊗i_{1}} ⊗ ⋯ ⊗ A_{n}^{⊗i_{n}}) for all nonnegative integers i_{1}, …, i_{n}. Let X be the product of SB(A_{1}), …, SB(A_{n}). Then, by [8, Corollary 2.15] the map (1) induces a surjection on torsion subgroups Γ^{2/3}(X)_{tors} ↠ CH^{2}(X)_{tors}. Moreover, by [4, Theorem 4.5] and [9, Proposition III.1] there is a product X̄ of Severi-Brauer varieties SB(Ā_{i}) of algebras Ā_{i} with ind(A_{1}^{⊗i_{1}} ⊗⋯⊗A_{n}^{⊗i_{n}}) = ind(Ā_{1}^{⊗i_{1}} ⊗⋯⊗Ā_{n}^{⊗i_{n}}) such that the above surjective map becomes bijective map

(7)  Γ^{2/3}(X̄)_{tors} →^{∼} CH^{2}(X̄)_{tors}.

This variety X̄ will be called a generic variety corresponding to X.

4. Product of Conics

In the present section, we provide a general bound for the torsion in codimension 2 cycles of the product of n conics in Corollary 4.4. In case of the product of four conics, we determine its torsion in terms of the indexes of the corresponding algebras in Theorem 4.6. In the last subsection, we present its application to a Galois cohomology group.

The following example was proved in a different way [10, Corollary 3.9].

Example 4.1. Consider the product of two conics X = SB(Q_{1}) × SB(Q_{2}), where Q_{1} and Q_{2} are quaternions. Let a = ind(Q_{1}), b = ind(Q_{2}), and ind(Q_{1} ⊗ Q_{2}) = c. If ab = 1, then the Chow group of X is torsion free, thus we may assume that ab ≥ 1. By (5), we have a basis {1, ax_{1}, bx_{2}, cx_{1}x_{2}} of K(X), where x_{1} and x_{2} are the pullbacks of the classes of the line bundles on the projective line over a splitting field E. Therefore, we have |K(X_{E})/K(X)| = abc. By substitution y_{n} = x_{n} − 1 for n = 1, 2, we have a different basis {1, ay_{1}, by_{2}, c(y_{1}y_{2} + y_{1} + y_{2})} of K(X). Let α_{n} = |T^{n/n+1}(X_{E})/ Im(res^{n/n+1})|. If c ≥ 2, then by (2) we have ay_{1}, by_{2} ∈ T^{1}(X) and cy_{1}y_{2} ∈ T^{2}(X). Therefore, |⊕ T^{n/n+1}(X)_{tors}| is trivial. Otherwise, by the diagonal embedding SB(Q_{1}) ↪ X we have y_{1} + y_{2} ∈ Im(res^{1/2}), thus α_{1} ≤ min{a, b}. As 2y_{1}y_{2} ∈ T^{2}(X), we get α_{2} ≤ 2. Hence, |⊕ T^{n/n+1}(X)_{tors}| is trivial as well. In any case, CH^{2}(X) is trivial. In particular, M(C_{2}) = M(PQ_{2}) = 0.

We determine the Chow group of codimension 2 of the product of three conics.

Proposition 4.2. (cf. [10, Proposition 6.1, Proposition 6.3]) Let Q_{1}, Q_{2}, Q_{3} be quaternions and X = SB(Q_{1}) × SB(Q_{2}) × SB(Q_{3}). Then, we have M(C_{3}) = M(PQ_{3}) = 2. Moreover, CH^{2}(X)_{tors} is trivial except the cases where the division algebras Q_{i} satisfying ind(Q_{i}⊗Q_{j}) = ind(Q_{1} ⊗ Q_{2} ⊗ Q_{3}) = 2 or 4 for all 1 ≤ i ≠ j ≤ 3 and CH^{2}(X̄)_{tors} = ℤ/2ℤ in these cases, where X̄ is the corresponding generic variety.

Proof. Let d = ind(Q_{1} ⊗ Q_{2} ⊗ Q_{3}) and e_{ij} = ind(Q_{i} ⊗ Q_{j}) for 1 ≤ i ≠ j ≤ 3. If one of ind(Q_{i}), e_{ij}, and d is 1, then by (6) and Example 4.1 CH^{2}(X)_{tors} is trivial. Therefore, by (5) we have the following basis

{1, 2y_{i}, e_{ij}y_{i}y_{j}, d(y_{1}y_{2}y_{3} + y_{1}y_{2} + y_{1}y_{3} + y_{2}y_{3})}

of K(X), where y_{i} = x_{i} − 1, x_{i} is the pullback of the tautological line bundle on the projective line over a splitting field E, and e_{ij}, d ≥ 2.

If either d = 8 or d = 4 and e_{ij} = 2 for some i, j, then the set K(X) ∩ T^{3}(X_{E}) has only one element dy_{1}y_{2}y_{3} of the basis. As e_{ij}y_{i}y_{j} ∈ T^{2}(X) and 2y_{k} ∈ T^{1}(X) for k ≠ i, j, we have dy_{1}y_{2}y_{3} = 2e_{ij}y_{1}y_{2}y_{3} = 2y_{k}(e_{ij}y_{i}y_{j}) ∈ T^{3}(X), which implies that T^{2/3}(X)_{tors} = CH^{2}(X)_{tors} is trivial.

Now we assume that d = e_{ij} = 2 or 4 for all 1 ≤ i ≠ j ≤ 3. Then, we have |K(X_{E})/K(X)| = 2^{3}e_{ij}^{3}d. Let α_{n} = |Γ^{n/n+1}(X_{E})/ Im(res^{n/n+1})|. Then, we obtain α_{1} ≤ 2^{3}, α_{2} ≤ e_{ij}^{3}, and α_{3} ≤ 2d as 2dy_{1}y_{2}y_{3} = 2y_{1}(dy_{2}y_{3}) ∈ Γ^{3}(X). Hence, |⊕ Γ^{n/n+1}(X)_{tors}| ≤ 2. As c_{2}(dx_{1}x_{2}x_{3}) = \binom{d}{2}(6y_{1}y_{2}y_{3} + 2y_{1}y_{2} + 2y_{1}y_{3} + 2y_{2}y_{3}) ∈ Γ^{2}(X), we have dy_{1}y_{2}y_{3} ∈ Γ^{2}(X). Observe that Γ^{3}(X) is generated by Γ^{1}(X) · Γ^{2}(X) and any element of Γ^{1}(X) · Γ^{2}(X) is divisible by 2d. Therefore, the class of dy_{1}y_{2}y_{3} gives a torsion of order 2 in Γ^{2/3}(X). Hence, we have Γ^{2/3}(X)_{tors} = ℤ/2ℤ and CH^{2}(X̄)_{tors} = ℤ/2ℤ by (7). □

We determine the maximal torsion in Chow group of codimension 2 of the product of n conics or the product of n Pfister quadric surfaces.

Proposition 4.3. Let n ≥ 2 and 1 ≤ i ≤ n be integers, Q_{i} quaternion algebras satisfying ind(⊗_{k=1}^{m}Q_{i_{k}}) = 2 for any 1 ≤ m ≤ n and all distinct i_{k}, and X = ∏_{i=1}^{n} SB(Q_{i}). Then, the torsion subgroup CH^{2}(X̄)_{tors} of the corresponding generic variety X̄ is

(ℤ/2ℤ)^{⊕N}, where N = 2^{n} − (\binom{n}{2} + n + 1).

In particular, M(C_{n}) = M(PQ_{n}) ≥ 2^{N}.

Proof. Let 1 ≤ i ≤ n and let Q_{i} be a quaternion such that ind(⊗_{k=1}^{m}Q_{i_{k}}) = 2 for any 1 ≤ m ≤ n and all distinct i_{k}, and X = ∏_{i=1}^{n} SB(Q_{i}). Then, by (5) we have a basis {1, 2x_{i_{1}} ⋯ x_{i_{m}}} of K(X), where x_{i} is the pullback of the class of the tautological line bundle on the projective line over a splitting field E. Let y_{i} = x_{i} − 1. Consider another basis {1, 2y_{i_{1}} ⋯ y_{i_{m}}} of K(X).

Let j ≥ 1. As any element of Γ^{2j+1}(X) is divisible by 2^{j+1}, we have

(8)  2^{j}y_{i_{1}} ⋯ y_{i_{k}} ∈ Γ^{2j}(X)\Γ^{2j+1}(X)

for any 2j + 1 ≤ k ≤ n. Moreover, if 2j + 1 ≤ k ≤ 2j + 2, then we obtain

(9)  2^{j+1}y_{i_{1}} ⋯ y_{i_{k}} ∈ Γ^{2j+2}(X).

Then, it follows from (8) and (9) that for any j ≥ 1 the class of 2^{j}y_{i_{1}} ⋯ y_{i_{k}} generates a subgroup of Γ^{2j/2j+1}(X)_{tors} of order 2. By the divisibility of an element of Γ^{2j+1}(X), any two subgroups generated by different classes of the elements 2^{j}y_{i_{1}} ⋯ y_{i_{k}} and 2^{j}y_{i_{1}} ⋯ y_{i_{k′}} have trivial intersection. Hence, we have

(10)  (ℤ/2ℤ)^{⊕N_{j}} ⊆ Γ^{2j/2j+1}(X)_{tors},

where N_{j} = ∑_{k=2j+1}^{n} \binom{n}{k}.

Let β_{i} = |Γ^{i/i+1}(X_{E})/ Im(res^{i/i+1})|/|K^{i}(X_{E})/K^{i}(X)|, where K^{i}(X_{E}) (resp. K^{i}(X)) is the codimension i part of K(X_{E}) (resp. K(X)). Then, It follows from the base of K(X) that β_{1}, β_{2} ≤ 1 and

β_{i} ≤ 2^{([i+1/2]−1)\binom{n}{i}} = 2^{[i+1/2]\binom{n}{i}}/2^{\binom{n}{i}}

for any 3 ≤ i ≤ n. Hence, by (3) we have

(11)  |⊕ Γ^{i/i+1}(X)_{tors}| ≤ 2^{∑_{i=3}^{n}([i+1/2]−1)\binom{n}{i}}.

Therefore, it follows from (10) and (11) that for any j ≥ 1

Γ^{2j/2j+1}(X)_{tors} = (ℤ/2ℤ)^{N_{j}}.

As N_{1} = N, the result follows from (7). □

Corollary 4.4. For any n ≥ 2, we have M(C_{n}) = M(PQ_{n}) = 2^{N}.

Proof. Let X = ∏_{i=1}^{n} SB(Q_{i}) ∈ C_{n} such that ind(Q_{i}) = 2 for all i. For 1 ≤ m ≤ n, set e_{i_{1}...i_{m}} = max{ind(Q_{i_{1}}^{⊗j_{1}} ⊗⋯⊗Q_{i_{m}}^{⊗j_{m}})}, where the maximum ranges over 0 ≤ j_{1}, …, j_{m} ≤ 1. If m ≥ 3, then e_{i_{1}...i_{m}}y_{i_{1}} ⋯ y_{i_{m−1}} ∈ T^{2}(X). As 2y_{i_{m}} ∈ T^{1}(X), we have

(12)  2e_{i_{1}...i_{m}}y_{i_{1}} ⋯ y_{i_{m}} ∈ T^{3}(X).

Since the subgroup T^{3}(X_{E}) ∩ K(X) is generated by e_{i_{1}...i_{m}}y_{i_{1}} ⋯ y_{i_{m}} for m ≥ 3, it follows from (1), (2), and (12) that | CH^{2}(X)_{tors}| ≤ 2^{N}. If ind(Q_{i}) = 1 for some i, then by (6) the computation of the upper bound for the torsion subgroup reduces to the case of product of less than n conics. Hence, we have M(C_{n}) ≤ 2^{N}, thus by Proposition 4.3 we obtain the result. □

4.1. Four Conics. Let Q_{i} be a quaternion algebra over a field F for 1 ≤ i ≤ 4. Consider the product X of the corresponding conics SB(Q_{i}). If ind(Q_{i}) = 1 for some i, then by (6) the problem to find torsion in CH^{2}(X) is reduced to the case of product of three conics (Proposition 4.2). Hence, we may assume that ind(Q_{i}) = 2 for all i. Let g_{ij} = ind(Q_{i} ⊗Q_{j}), h_{i} = ind(Q_{j} ⊗ Q_{k} ⊗ Q_{l}), and d = ind(Q_{1} ⊗ Q_{2} ⊗ Q_{3} ⊗ Q_{4}) for all integers i, j, k, l such that {i, j, k, l} = I_{4}. For the same reason, we may assume that g_{ij}, h_{i}, d ≥ 2. By (5), we have the following basis of K(X)

(13)  {1, 2x_{i}, g_{ij}x_{i}x_{j}, h_{i}x_{j}x_{k}x_{l}, dx_{1}x_{2}x_{3}x_{4}},

where x_{i}, x_{j}, x_{k}, x_{l} are the pullbacks of the classes of the tautological line bundles on the projective lines. By substitution y_{i} = x_{i} − 1 for all 1 ≤ i ≤ 4, it follows from (13) that we have another basis K(X)

(14) {1, 2y_{i}, g_{ij}y_{i}y_{j}, h_{i}(y_{j}y_{k}y_{l} + y_{j}y_{k} + y_{j}y_{l} + y_{k}y_{l}), d(y_{1}y_{2}y_{3}y_{4} + ∑ y_{i}y_{j}y_{k} + ∑ y_{i}y_{j}}.

Now we will set up some notations, which will be used in this subsection. For any integer m ∈ {2, 4, 8}, we set

H_{m} = {1 ≤ i ≤ 4 | h_{i} = m}.

Let J be the set of all indices {12, 13, 14, 23, 24, 34} of g_{ij}. We consider the decompositions of J:

J = J_{1} ∪ J_{2} ∪ J_{3} = K_{i} ∪ L_{i},

where J_{1} = {12, 34}, J_{2} = {13, 24}, J_{3} = {14, 23}, K_{i} = {jk, jl, kl}, and L_{i} = J\K_{i} for j < k < l. We set

G = {ij ∈ J | g_{ij} = 2}.

We will use the following lemma to find bounds for the torsion in CH^{2}(X).

Lemma 4.5. Let 1 ≤ m ≤ 3 and 1 ≤ p < q < r ≤ 4 be integers and let i, j, k, l be integers such that {i, j, k, l} = I_{4}. Then, in codimension 2, 3, and 4 respectively, we have

(1) Im(res^{2/3}) ∋ \begin{cases}2 ∑ y_{p}y_{q} & if d = 2, \\ 2(y_{i}y_{j} + y_{i}y_{k} + y_{j}y_{k}) & if h_{l} = 2, \end{cases}

(2) Im(res^{3/4}) ∋\begin{cases}4y_{i}y_{j}y_{k}, 4y_{i}y_{j}y_{l} & if g_{ij} = 2, \\ 4y_{i}y_{j}y_{k}, −4y_{i}y_{j}y_{k}+4 ∑ y_{p}y_{q}y_{r} & if h_{l} = 2, \\ 4y_{i}y_{j}y_{k}, 4y_{i}y_{j}y_{l}, 4y_{i}y_{k}y_{l} & if g_{ij} = g_{ik} = 2, \\ ∀ 4y_{p}y_{q}y_{r} & if d = 2 or |G| ≥ 4 or |G ∩ J_{m}|=2 \\ & or |G ∩ K_{i}|=3, \end{cases}

(3) Γ^{4}(X) ∋ \begin{cases}8y_{1}y_{2}y_{3}y_{4} & if d ∈ \{2, 4\} or |G| ≥ 1 or |H_{2}| ≥ 1, \\ 4y_{1}y_{2}y_{3}y_{4} & if |G ∩ J_{m}| = 2. \end{cases}

Proof. If d = 2, then we have

c_{2}(2x_{1}x_{2}x_{3}x_{4}) = 2(y_{1}y_{2}y_{3}y_{4} + ∑ y_{p}y_{q}y_{r} + ∑ y_{p}y_{q}) ∈ Γ^{2}(X),

thus, we obtain c_{1}(2x_{i})c_{1}(2x_{j})c_{2}(2x_{1}x_{2}x_{3}x_{4}) = 8y_{1}y_{2}y_{3}y_{4} ∈ Γ^{4}(X), which imply the results in Lemma 4.5 (1), (3), respectively. Since 8 ∑ y_{p}y_{q}y_{r} ∈ Γ^{3}(X), the result in Lemma 4.5 (2) follows from

c_{1}(2x_{i})c_{2}(2x_{1}x_{2}x_{3}x_{4}) = 12y_{1}y_{2}y_{3}y_{4} + 4y_{i}(y_{j}y_{k} + y_{j}y_{l} + y_{k}y_{l}) ∈ Γ^{3}(X) and
c_{1}(2x_{1}x_{2}x_{3}x_{4})c_{2}(2x_{1}x_{2}x_{3}x_{4}) = 72y_{1}y_{2}y_{3}y_{4} + 12 ∑ y_{p}y_{q}y_{r} ∈ Γ^{3}(X).

If h_{l} = 2, then it follows from (2) and (14) that

(15)  2(y_{i}y_{j}y_{k} + y_{i}y_{j} + y_{i}y_{k} + y_{j}y_{k}) ∈ Γ^{2}(X),

which completes the proof of Lemma 4.5 (1). Multiplying (15) by c_{1}(2x_{k}) and c_{1}(2x_{l}), respectively, we obtain the results in Lemma 4.5 (2). Multiplying 4y_{i}y_{j}y_{k} by c_{1}(2x_{l}), we have the result in Lemma 4.5 (3).

If |G ∩ J_{m}| = 2 for some m, then we have g_{ij} = g_{kl} = 2 for some i, j, k, l, thus the results in Lemma 4.5 (2)(3) follow from (14) and (2).

If g_{ij} = 2, then it follows from (14) and (2) that

(16)  2y_{i}y_{j}c_{1}(2y_{k}), 2y_{i}y_{j}c_{1}(2y_{l}) ∈ Γ^{3}(X).

Multiplying the first element in (16) by c_{1}(2x_{l}), we obtain the result in Lemma 4.5 (3). If d = 4, then we have c_{4}(4x_{1}x_{2}x_{3}x_{4}) = 24y_{1}y_{2}y_{3}y_{4} ∈ Γ^{4}(X). As 16y_{1}y_{2}y_{3}y_{4} ∈ Γ^{4}(X), we complete the proof of Lemma 4.5 (3).

If |G| ≥ 4, then we have |G ∩ J_{m}| = 2 for some m, hence the result in Lemma 4.5 (2) follows from the case of |G ∩ J_{m}| = 2. If g_{ij} = g_{ik} = 2, then it follows from (14) and (2) that

2y_{i}y_{j}c_{1}(2x_{k}), 2y_{i}y_{j}c_{1}(2x_{l}), 2y_{i}y_{k}c_{1}(2x_{l}) ∈ Γ^{3}(X).

If |G ∩ K_{i}| = 3, then we get g_{jk} = g_{jl} = g_{kl} = 2, thus, by the same argument we complete the proof of Lemma 4.5 (2). □

We determine the codimension 2 cycles of the product of four conics as in Proposition 4.2.

Theorem 4.6. The torsion in Chow group of codimension 2 of the product X of four conics is trivial if it satisfies one of the conditions (17), (22), (23), (25), (31), (32), (33), d = 16. Otherwise, the torsion subgroup admits all elementary abelian group whose order is a divisor of 2^{5}.

Proof. Let m be an integer in I_{3}, Q_{i} a quaternion division algebra over F and X the product of the corresponding conics SB(Q_{i}) for 1 ≤ i ≤ 4. Set

β_{i} = |Γ^{i/i+1}(X_{E})/ Im(res^{i/i+1})|/|K^{i}(X_{E})/K^{i}(X)|,

where E is a splitting field of X and K^{i}(X_{E}) (resp. K^{i}(X)) is the codimension i part of K(X_{E}) (resp. K(X)). We find upper bounds of β_{i} using case by case analysis. Note that we have β_{1} ≤ 1.

We begin with some observations on d: if d = 2^{4}, then we have h_{i} = d/2 for all i, which implies that g_{ij} = d/4 for all 1 ≤ i ≠ j ≤ 4. Therefore, we obtain β_{i} ≤ 1 for all i, thus, |⊕ Γ^{i/i+1}(X)_{tors}| ≤ 1, i.e., CH^{2}(X)_{tors} is trivial. Hence, we may assume that 2 ≤ d ≤ 2^{3}. Note also that we have

d ≤ 4 if |H_{2}| ≥ 1, and |H_{8}| = 0 if d = 2,

thus we only consider the cases where d = 4, 8 (resp. d = 2, 4) in the first 3 cases (resp. the last 4 cases) of the following. In case where |H_{2}| = 4, we have d ∈ {2, 4, 8}.

Case: |H_{8}| ≥ 3. In this case, we have g_{ij} = 4 for all i, j. It follows from Lemma 4.5 (1) that β_{2} ≤ (4^{5} · 2^{2−|H_{2}|})/4^{6}. By Lemma 4.5 (2), we obtain

β_{3} ≤ \begin{cases}(8^{2} · 4^{2})/8^{3} · 2 & if |H_{2}| = 1, \\ 2^{|H_{4}|} & otherwise. \end{cases}

It follows by Lemma 4.5 (3) that β_{4} ≤ 2. Hence, by (3) the order of the group ⊕Γ^{i/i+1}(X)_{tors} is nontrivial except the case where

(17)  |H_{2}| = 1, |H_{8}| = 3.

If |H_{8}| = 4, d = 8, then by the divisibility of elements in Γ^{3}(X) the class of 8y_{1}y_{2}y_{3}y_{4} gives a torsion element of Γ^{2/3}(X)_{tors} of order 2. As β_{1}β_{2}β_{3}β_{4} ≤ 2, we have

(18)  Γ^{2/3}(X)_{tors} = ℤ/2ℤ.

Similarly, if |H_{8}| = 3, h_{i} = 4 and d = 4, then we have

(19)  Γ^{2/3}(X)_{tors} = (ℤ/2ℤ)^{⊕2}

generated by the classes of 4y_{j}y_{k}y_{l} and 4(y_{i}y_{j}y_{k} + y_{i}y_{j}y_{l} + y_{i}y_{k}y_{l} + y_{1}y_{2}y_{3}y_{4}).

Case: |H_{8}| = 2. A simple calculation of index implies that 0 ≤ |G| ≤ 1. Hence, it follows from Lemma 4.5 (1) that

(20)  β_{2} ≤ (4^{6−|G|−|H_{2}|} · 2^{|G|+|H_{2}|})/4^{6−|G|} · 2^{|G|}.

If |G| = |H_{2}| = 0, then we have β_{3} ≤ 8^{4}/(8^{2} · 4^{2}). Otherwise, by Lemma 4.5 (2) we obtain

β_{3} ≤ (8^{3−|G|−|H_{2}|} · 4^{1+|G|+|H_{2}|})/8^{2} · 4^{2−|H_{2}|} · 2^{|H_{2}|}.

In summary, we have

β_{3} ≤ \begin{cases}1 & if |G| = 1, |H_{2}| = 0 or |G| = |H_{2}| = 1, \\ 2 & if |G| = 0, |H_{2}| = 1 or |H_{2}| = 2, \\ 2^{2} & otherwise. \end{cases}

It follows from Lemma 4.5 (3) that

(21)  β_{4} ≤ \begin{cases}1 & if |G| ≥ 1, |H_{2}| = 0, d = 8, \\ 2 & otherwise. \end{cases}

Therefore, it follows by the same argument as in the previous case that |⊕ Γ^{i/i+1}(X)_{tors}| is nontrivial except the case where

(22)  |H_{2}| = 2 or |G| = |H_{2}| = 1 or |G| = 1, |H_{2}| = 0, d = 8.

Case: |H_{8}| = 1. By this assumption, we have 0 ≤ |G| ≤ 3. It follows by the same argument as in the previous case that we have the same upper bounds in (20) and (21) for β_{2} and β_{4}, respectively. Let G_{n} = G ∩ (∩_{i∈H_{n}}K_{i}) for n = 2, 4. If |G| = |G_{2}| = |H_{2}| = 1 (resp. |G| = |G_{4}| = 1, |H_{2}| = 2), then we have h_{i} = 2, h_{l} = 8, h_{j} = h_{k} = 4 (resp. h_{j} = 2, h_{k} = 4), and g_{il} = 2 (resp. g_{kl} = 2) for {i, j, k, l} = I_{4}, which implies that β_{3} ≤ 4^{4}/(8 · 4^{2} · 2) (resp. β_{3} ≤ 8·4^{3}/(8·4·2^{2})) by Lemma 4.5 (2). If |H_{2}| = 0, |G| ≥ 2 (resp. |H_{2}| = 0, |G| = 0), then by Lemma 4.5 (2) we obtain β_{3} ≤ 8 · 4^{3}/(8 · 4^{3}) (resp. β_{3} ≤ 8^{4}/(8 · 4^{3})). Otherwise, by Lemma 4.5 (2) we have

β_{3} ≤ min{1, 8^{3−|G|−|H_{2}|}} · min{4^{4}, 4^{1+|G|+|H_{2}|}}/(8 · 4^{3−|H_{2}|} · 2^{|H_{2}|}).

Therefore, we conclude that

β_{3} ≤ \begin{cases}1 & if 0 ≤ |H_{2}| ≤ 1, |G| ≥ 2 or |H_{2}| = |G_{2}| = |G| = 1, \\ 2 & otherwise, \\ 2^{2} & if 1 ≤ |H_{2}| ≤ 2, |G| = 0 or |H_{2}| = 2, |G| = |G_{4}| = 1, \\ 2^{3} & if |H_{2}| = |G| = 0. \end{cases}

By the same argument, the order of the torsion |⊕ Γ^{i/i+1}(X)_{tors}| is nontrivial except the following cases: (23) |G| ≥ 2, |H_{2}| = 0, d = 8 or |G| ≥ |H_{2}||H_{4}| = 2 or |H_{2}||H_{4}| = 2, |G| = |G_{2}| = 1 or |H_{2}| = 3.

If d = 8, |G| = 6, and h_{i} = 8, then by the same argument as above we obtain

(24)  Γ^{2/3}(X)_{tors} = (ℤ/2ℤ)^{⊕3} and Γ^{3/4}(X)_{tors} = ℤ/2ℤ

generated by 4y_{i}y_{k}y_{l}, 4y_{i}y_{j}y_{l}, 4y_{i}y_{j}y_{k}, and 8y_{1}y_{2}y_{3}y_{4}, respectively.

Case: |H_{4}| = 4. By Lemma 4.5 (1), we have

β_{2} ≤ \begin{cases}(4^{5−|G|} · 2^{|G|+1})/4^{6−|G|} · 2^{|G|} & if d = 2, |G| ≠ 6, \\ (4^{6−|G|} · 2^{|G|})/4^{6−|G|} · 2^{|G|} & otherwise. \end{cases}

It follows from Lemma 4.5 (2) that

4^{4} · β_{3} ≤ \begin{cases}4^{4} & otherwise, \\ 8 · 4^{3} & if d ∈ \{4, 8\}, 2 ≤ |G| = |G ∩ L_{i}| ≤ 3 for some i, \\ 8^{2} · 4^{2} & if d ∈ \{4, 8\}, |G| = 1, \\ 8^{4} & if d ∈ \{4, 8\}, |G| = 0. \end{cases}

By Lemma 4.5 (3) and (14), we obtain

β_{4} ≤ \begin{cases}1 & if d = 8, |G| ≠ 0 or d = 4, |G ∩ J_{m}| = 2 for some m, \\ 2 & otherwise, \\ 2^{2} & if d = 2, |G| ≠ 6, |G ∩ J_{m}| ≠ 2 ∀m. \end{cases}

Applying the same argument, we have that the order of ⊕Γ^{i/i+1}(X)_{tors} is nontrivial except the following cases: for some m and i

(25)  d = 2, |G| ≠ 6, |G ∩ J_{m}| = 2 or d = 4, |G ∩ J_{m}| = 2 or d = 8, |G ∩ K_{i}| = 3.

If |G| = 0 and d = 4, then by (14) and (2) one has h_{i}y_{j}y_{k}y_{l}, 4y_{1}y_{2}y_{3}y_{4} ∈ Γ^{2}(X)\Γ^{3}(X) for all {i, j, k, l} = I_{4} since any element of Γ^{3}(X) is divisible by 2^{3}. Therefore, by (14) and Lemma 4.5 (3) the classes of h_{i}y_{j}y_{k}y_{l} and 4y_{1}y_{2}y_{3}y_{4} give torsion elements of Γ^{2/3}(X) of order 2. Moreover, the subgroups generated by these classes have trivial intersection by the divisibility of an element of Γ^{3}(X). Hence, it follows from Corollary 4.4 that

(26)  Γ^{2/3}(X)_{tors} = (ℤ/2ℤ)^{⊕5}, thus CH^{2}(X̄)_{tors} = (ℤ/2ℤ)^{⊕5},

where X̄ is the corresponding generic variety. If |G| = 0 and d = 8, then by the same argument as above the classes of h_{i}y_{j}y_{k}y_{l} generate different subgroups of Γ^{2/3}(X)_{tors} of order 2 and the class of 8y_{1}y_{2}y_{3}y_{4} = 2y_{1}(4y_{2}y_{3}y_{4}) generates a subgroup of Γ^{3/4}(X)_{tors} of order 2. Therefore, it follows Corollary 4.4 that

(27)  Γ^{2/3}(X)_{tors} = (ℤ/2ℤ)^{⊕4} and Γ^{3/4}(X)_{tors} = ℤ/2ℤ.

Case: |H_{2}| = 4. By (14) and Lemma 4.5 (1), one has

(28)  β_{2} ≤ \begin{cases}d/2^{5} & otherwise, \\ d/2^{4} & if |G| = d = 4 or |G| = 3, d = 2, \\ d/2^{3} & if |G| = 4, d = 2 or |G| = 5, d = 4, \\ d/2^{2} & if |G| = 6, d = 4 or |G| = 5, d = 2, \\ d/2 & if |G| = 6, d = 2. \end{cases}

It follows from (14) that β_{3} ≤ 2^{4} = 8^{4}/4^{4}. By Lemma 4.5 (3), we have

(29)  β_{4} ≤ \begin{cases}2^{2}/d & if |G| ≥ 2, |G ∩ J_{m}| = 2 for some m, \\ 2^{3}/d & otherwise. \end{cases}

Applying the same argument together with the upper bounds β_{i}, we have Γ^{2/3}(X)_{tors} ≠ 0 for any case. In particular, if |G| = 6 and d = 2, then by Proposition 4.3 we have

(30)  Γ^{2/3}(X)_{tors} = (ℤ/2ℤ)^{⊕5}, thus CH^{2}(X̄)_{tors} = (ℤ/2ℤ)^{⊕5}.

Case: |H_{2}| = 3, |H_{4}| = 1. Let H′_{2} = {i ∈ H_{2} | |K_{i} ∩G| = 3}. Then, by the same argument as in the previous case we have the same upper bound (28) for β_{2} if we add one case |G| = 3, d = 4, |H′_{2}| = 1 to the second line of (28). It follows by Lemma 4.5 (2) that β_{3} ≤ 2^{3}. By applying the same argument as in the previous case we have the same upper bound (29) for β_{4}. it follows by the same argument as in the previous case that |⊕ Γ^{i/i+1}(X)_{tors}| is nontrivial except the case:

(31)  |G| = 2, |G ∩ J_{m}| = 2 or |G| = 3, |G ∩ J_{m}| = 2, d = 4 for some m.

Case: |H_{2}| = 2, |H_{4}| = 2. By Lemma 4.5 (1), we have

β_{2} ≤ \begin{cases}d/2^{4} & otherwise, \\ d/2^{3} & if 3 ≤ |G| ≤ 4, |H′_{2}|=1 or |G| =4, |H′_{2}|=0, d=2 or |G|= 5, |H′_{2}| ≤ 1, d=4, \\ d/2^{2} & if |G| = 5, |H′_{2}| = 2 or |G| = 5, |H′_{2}| ≤ 1, d = 2 or |G| = 6, d = 4, \\ d/2 & if |G| = 6, d = 2. \end{cases}

It follows from Lemma 4.5 (2) that

β_{3} ≤ \begin{cases}2^{2} & otherwise, \\ 2^{3} & if |G| = 0, d = 4 or |G| = |G_{2}| = 1, d = 4. \end{cases}

Applying the same argument as in the previous case, one has the same upper bound (29) for β_{4}. Therefore, by the same argument, the order of the torsion |⊕ Γ^{i/i+1}(X)_{tors}| is nontrivial except the following cases: for some m and i

(32)  |G| = |G ∩ J_{m}| = 2 or |G| = |G ∩ K_{i}| = 3 or |G| = 4, |H′_{2}| = 0, d = 4.

Case: |H_{2}| = 1, |H_{4}| = 3. By Lemma 4.5 (1), we obtain

β_{2} ≤ \begin{cases}d/2^{3} & otherwise, \\ d/2^{2} & if 3 ≤ |G| ≤ 5, |H′_{2}| = 1 or |G| = 5, |H′_{2}| = 0, d = 2 or |G| = 6, d = 4, \\ d/2 & if |G| = 6, d = 2. \end{cases}

In codimension 3, it follows from Lemma 4.5 (2) that

β_{3} ≤ \begin{cases}2 & otherwise, \\ 2^{2} & if |G| = |G_{2}| = 1, d = 4, \\ 2^{3} & if |G| = 0, d = 4. \end{cases}

In codimension 4, it follows by Lemma 4.5 (2) that

β_{4} ≤ \begin{cases}1 & if 2 ≤ |G| ≤ 3, |G ∩ J_{m}| = 2 for some m, d = 4 or |G| ≥ 4, d = 4, \\ 2 & otherwise, \\ 2^{2} & if 0 ≤ |G| ≤ 3, |G ∩ J_{m}| ≠ 2 ∀m, d = 2. \end{cases}

Hence, by the same argument as above the order of ⊕Γ^{i/i+1}(X)_{tors} is nontrivial except the following cases:

(33) 2 ≤ |G| ≤ 3, |G∩J_{m}| = 2 for some m or |G| = 4, |H′_{2}| = 0 or |G| = 5, |H′_{2}| = 0, d = 4.

Finally, the second statement of the theorem follows from (18), (19), (24), (26), and (27). □

Remark 4.7. Indeed, one can easily show that the upper bounds β_{1}β_{2}β_{3}β_{4} for each case of the proof of Theorem 4.6 are sharp.

4.2. Galois cohomology and torsion groups. As mentioned in Section 1, the torsion subgroup of the Chow group of codimension 2 cycles can be used to measure how far is a relative Galois cohomology group from being a decomposable subgroup (generated by the class of A_{i} below) [10, Theorem 4.1]. Namely, for an F -variety X = ∏_{i} SB(A_{i}) ∈ C_{n} or SB_{n} we have

(34)  CH^{2}(X)_{tors} ≃ H^{3}(F (X)/F, ℚ/ℤ(2))/ ⊕_{i} H^{1}(F, ℚ/ℤ(1)) ∪ [A_{i}],

where H^{3}(F (X)/F, ℚ/ℤ(2)) denotes the kernel of H^{3}(F, ℚ/ℤ(2)) → H^{3}(F (X), ℚ/ℤ) of Galois cohomology groups with coefficient in ℚ/ℤ(2) and [A_{i}] denotes the class in the Brauer group Br(F) = H^{2}(F, ℚ/ℤ(1)). Therefore, our main results (Corollary 4.4, Proposition 5.2, and Theorem 5.4) tell us how large indecomposable subgroups we can have.

Moreover, by [10, Remark 4.1] there is a canonical injection from a Galois cohomology group with the finite coefficient μ_{n}^{⊗2} to the torsion subgroup:

(35)  H^{3}(F (X)/F, μ_{n}^{⊗2})/ ⊕_{i} H^{1}(F, μ_{n}) ∪ [A_{i}] ↪ CH^{2}(X)_{tors}.

Therefore, if the torsion subgroup CH^{2}(X)_{tors} is trivial, then one can write the relative Galois cohomology group in terms of decomposable subgroups with the finite coefficient μ_{n}. For instance, if X ∈ C_{4} satisfying one of the conditions (17), (22), (23), (25), (31), (32), (33), d = 16 in Theorem 4.6, then we obtain | CH^{2}(X)_{tors}| = 1, thus by (35) we have

H^{3}(F (X)/F, ℤ/2ℤ) = ⊕_{i=1}^{4}H^{1}(F, ℤ/2ℤ) ∪ [Q_{i}].

5. Product of Severi-Brauer surfaces

In this section, we find a general lower bound of the torsion in Chow group of codimension 2 of the product of n Severi-Brauer surfaces in Proposition 5.2 and prove that the lower bound is sharp for n = 3 in Theorem 5.4.

To prove Proposition 5.2 we shall need the following lemma.

Lemma 5.1. Let p be an odd prime and let n ≥ 2 and m_{i} be integers such that 1 ≤ m_{i} ≤ p−1 for all 1 ≤ i ≤ n. Let Φ be the polynomial (∏_{i=1}^{n}(s_{i} + 1)^{m_{i}} − 1)^{p} in ℤ[s_{1}, …, s_{n}]. Then, the alternating sum ∑_{j_{1}=···=j_{n}=1}^{p−1}(−1)^{j_{1}+···+j_{n}}C_{j_{1}···j_{n}} in the quotient ℤ[s_{1}, …, s_{n}]/(s_{1}^{p}, …, s_{n}^{p}) is divisible by p^{2}, where C_{j_{1}···j_{n}} is the coefficient of the monomial s_{1}^{j_{1}} ⋯ s_{n}^{j_{n}} in Φ.

Proof. Let t_{n} = (s_{n} + 1)^{m_{n}}. For any 1 ≤ j_{1}, …, j_{n−1} ≤ p − 1, we write ∑_{k=1}^{p} d_{j_{1}···j_{n−1}k}t^{k} for the coefficient of s_{1}^{j_{1}} ⋯ s_{n−1}^{j_{n−1}} in Φ. Let e_{j_{1}···j_{n−1}} be the coefficient of s_{1}^{j_{1}} ⋯ s_{n−1}^{j_{n−1}} in Ψ_{n} := (s_{1} + 1)^{pm_{1}} ⋯ (s_{n−1} + 1)^{pm_{n−1}}. By expanding each factor (s_{i} + 1)^{pm_{i}} = ((s_{i} + 1)^{p})^{m_{i}} of Ψ_{n}, we have

(36)  e_{j_{1}···j_{n−1}} = (\binom{p}{j_{1}}m_{1} + α_{1}) ⋯ (\binom{p}{j_{n−1}}m_{n−1} + α_{n−1}),

where p^{2} | α_{i} for all 1 ≤ i ≤ n − 1.

We prove by induction on n. Assume n = 2. First, observe that C_{j_{1}0} = 0 in the quotient ℤ[s_{1}, s_{2}]/(s_{1}^{p}, s_{2}^{p}). Hence, we have

∑_{j_{2}=1}^{p−1}(−1)^{j_{2}}C_{j_{1}j_{2}} =∑_{j_{2}=0}^{p−1}(−1)^{j_{2}}C_{j_{1}j_{2}} =∑_{k=1}^{p} d_{j_{1}k}∑_{i=0}^{p−1}(−1)^{i}\binom{m_{2}k}{i}=∑_{k=1}^{p} d_{j_{1}k}\binom{m_{2}k − 1}{p − 1},

which implies that

(37)  ∑_{j_{1}=j_{2}=1}^{p−1}(−1)^{j_{1}+j_{2}}C_{j_{1}j_{2}} = ∑_{j_{1}=1}^{p−1} ∑_{k=1}^{p} d_{j_{1}k}\binom{m_{2}k − 1}{p − 1}(−1)^{j_{1}}.

For each 1 ≤ k ≤ p − 1, we have

(38)  p | \binom{m_{2}k − 1}{p − 1}.

As

Φ = ∑_{i=0}^{p} \binom{p}{i}(s_{1} + 1)^{m_{1}i}t_{2}^{i}(−1)^{p−i},

for any 1 ≤ k ≤ p we obtain

(39)  p | d_{j_{1}k}.

From (38) and (39), it suffices to show that

(40)  p^{2} | ∑_{j_{1}=1}^{p−1} d_{j_{1}p}\binom{m_{2}p − 1}{p − 1}(−1)^{j_{1}}.

Since d_{j_{1}p} = e_{j_{1}} and ∑_{j_{1}=1}^{p−1} \binom{p}{j_{1}}(−1)^{j_{1}} = 0, the divisibility in (40) follows from (36).

Now we assume that the result holds for n − 1. By the induction hypothesis, it is enough to show that ∑_{j_{1}=···=j_{n−1}=1,j_{n}=0}^{p−1}(−1)^{j_{1}+···+j_{n}}C_{j_{1}···j_{n}} is divisible by p^{2}. Applying the same argument as in the case n = 2, we have

(41) ∑_{j_{1}=···=j_{n−1}=1,j_{n}=0}^{p−1} (−1)^{j_{1}+···+j_{n}}C_{j_{1}···j_{n}} = ∑_{j_{1}=···=j_{n−1}=1}^{p−1} ∑_{k=1}^{p} d_{j_{1}···j_{n−1}k}\binom{m_{n}k − 1}{p − 1}(−1)^{j_{1}+···+j_{n−1}}.

We have p | \binom{m_{n}k−1}{p−1} for each 1 ≤ k ≤ p − 1. By the same argument with t_{n}, we also have p | d_{j_{1}···j_{n−1}k} for each 1 ≤ k ≤ p. For k = p, we have p^{2} | d_{j_{1}···j_{n−1}p} by (36). Therefore, the result immediately follows. □

We will provide lower bounds of the torsion in Chow group of codimension 2 of the product of n Severi-Brauer surfaces, which generalize the case of n = 2 in [4, Proposition 6.3].

Proposition 5.2. Let n ≥ 2 and 1 ≤ i ≤ n be integers, A_{i} a central simple algebra satisfying ind(A_{1}^{⊗j_{1}} ⊗ ⋯ ⊗ A_{n}^{⊗j_{n}}) = 3 for any integers 0 ≤ j_{1}, …, j_{n} ≤ 2, not all equal to 0, and X = ∏_{i=1}^{n} SB(A_{i}). Then, the torsion subgroup CH^{2}(X̄)_{tors} of the corresponding generic variety X̄ contains

(ℤ/3ℤ)^{⊕N}, where N = 2^{n} + 4\binom{n}{3} − (n + 1).

In particular, M(SB_{n}) ≥ 3^{N}.

Proof. Let 1 ≤ i ≤ n and let A_{i} be a central simple algebra such that ind(⊗_{k=1}^{n}A_{k}^{⊗j_{k}}) = 3 for any integers 0 ≤ j_{1}, …, j_{n} ≤ 2, and X = ∏_{i=1}^{n} SB(A_{i}). Then, by (5) we have a basis {1, 3x_{i_{1}}^{j_{1}} ⋯ x_{i_{m}}^{j_{m}}} of K(X), where x_{i} is the pullback of the class of the tautological line bundle on the projective plane and 1 ≤ m ≤ n.

Let y_{i} = x_{i} − 1. Consider another basis {1, 3y_{i_{1}}^{j_{1}} ⋯ y_{i_{m}}^{j_{m}}} of K(X). Let B_{1} = {b_{pq} := 3y_{p}^{2}y_{q}^{2}}, B_{2} = {b_{pqr} := 3y_{p}y_{q}y_{r}}, B_{3} = {b′_{pqr} := 3y_{p}^{2}y_{q}y_{r}}, and D_{s} = {d_{i_{1}···i_{s}} := 3(y_{i_{1}} ⋯ y_{i_{s}})^{2}} for all distinct 1 ≤ p, q, r ≤ n and 3 ≤ s ≤ n. Then we have y_{i}^{3} = 0, |B_{1}| = \binom{n}{2}, |B_{2}| = \binom{n}{3}, |B_{3}| = 3\binom{n}{3}, and |D_{s}| = \binom{n}{m}. Set

N = |B_{1}| + |B_{2}| + |B_{3}| + ∑_{s=3}^{n} |D_{s}| = 2^{n} + 4\binom{n}{3} − (n + 1).

It follows from (2) that each element of B_{1}, B_{2}, B_{3}, D_{s} is in Γ^{2}(X). If any element of B_{1}, B_{2}, B_{3}, D_{s} is in Γ^{3}(X), then by applying Lemma 5.1 with y_{i} = −1 for all 1 ≤ i ≤ n we have 3 is divisible by 9, which is impossible. Therefore, any element of B_{1}, B_{2}, B_{3}, D_{s} is in Γ^{2}(X)\Γ^{3}(X). Since 3b_{pq} = 3y_{p}^{2}(3y_{q}^{2}) ∈ Γ^{4}(X), 3b_{pqr} = 3y_{p}y_{q}(3y_{r}) ∈ Γ^{3}(X), 3b′_{par} = 3y_{p}^{2}(3y_{q}y_{r}) ∈ Γ^{3}(X), 3d_{i_{1}···i_{s}} = 3y_{i_{1}}^{2}(3y_{i_{2}}^{2} ⋯ y_{i_{m}}^{2}) ∈ Γ^{4}(X), any element of B_{1}, B_{2}, B_{3}, D_{s} gives a torsion of Γ^{2/3}(X) of order 3.

We show that any two subgroups generated by any two elements of B_{1}, B_{2}, B_{3}, D_{s} have trivial intersection. Let b_{pq} and b_{tu} be two different elements of B_{1}. If b_{pq} ± b_{tu} ∈ Γ^{3}(X), then by applying Lemma 5.1 with y_{p} = y_{q} = −1, y_{l} = 0 for all 1 ≤ l ≠ p, q ≤ n we obtain 3 is divisible by 9, which is a contradiction. Hence, the subgroups generated by b_{pq} and b_{tu} have trivial intersection. Moreover, by the same argument the subgroup generated by b_{pq} has trivial intersection with any subgroup generated by any element of B_{2}, B_{3}, and D_{s}.

Let z be either 1 or a product of x_{1}, ⋯, x_{n} which does not contain any of x_{p}, x_{q}, and x_{r}. Consider the sequence β′_{pqr} consisting of the coefficient of b′_{pqr}/3 in

c_{3}(3x_{p}^{2}x_{q}x_{r}z), c_{3}(3x_{p}x_{q}^{2}x_{r}z), c_{3}(3x_{p}x_{q}x_{r}^{2}z), c_{3}(3x_{p}^{2}x_{q}^{2}x_{r}z), c_{3}(3x_{p}^{2}x_{q}x_{r}^{2}), (42)

c_{3}(3x_{p}x_{q}^{2}x_{r}^{2}z), c_{3}(3x_{p}^{2}x_{q}^{2}x_{r}^{2}z), c_{3}(3x_{p}x_{q}x_{r}z), respectively.

Then, by a direct calculation, we have β′_{pqr} = (66, 30, 30, 132, 132, 60, 264, 15). Hence, each element of β′_{pqr} − β′_{qpr}, β′_{pqr} − β′_{rpq}, and β′_{qpr} − β′_{rpq} is divisible by 9, i.e.,

(43)  9 | β′_{pqr} − β′_{qpr}, β′_{pqr} − β′_{rpq}, β′_{qpr} − β′_{rpq}.

Consider another sequence β_{pqr} consisting of the coefficient of b_{pqr}/3 in (42). Then, we have β_{pqr} = (12, 12, 12, 24, 24, 24, 48, 6). Therefore, we have

(44)  9 | β_{pqr} − β′_{pqr}, β_{pqr} − β′_{qpr}, β_{pqr} − β′_{rpq}.

Let b′_{pqr} and b′_{tuv} be two different elements of B_{3}. If b′_{pqr} ±b′_{tuv} ∈ Γ^{3}(X), then by applying Lemma 5.1 with y_{p} = y_{q} = y_{r} = −1, y_{l} = 0 for all 1 ≤ l ≠ p, q, r ≤ n we obtain 3 is divisible by 9, which is impossible. Therefore, the subgroups generated by b′_{pqr} and b′_{tuv} have trivial intersection. Assume that b′_{pqr} ± b_{tuv} ∈ Γ^{3}(X) or b′_{pqr} ± d_{i_{1}···i_{s}} ∈ Γ^{3}(X). Then, this contradicts the divisibility in (43). Hence, the subgroup generated by b′_{pqr} has trivial intersection with any subgroup generated by any element of B_{2} and D_{s}.

Let b_{pqr} and b_{tuv} be two different elements of B_{2}. Suppose that b_{pqr} ±b_{tuv} ∈ Γ^{3}(X). Then, by applying Lemma 5.1 with y_{p} = y_{q} = y_{r} = −1, y_{l} = 0 for all 1 ≤ l ≠ p, q, r ≤ n we obtain −3 is divisible by 9, which is a contradiction. Therefore, the subgroups generated by b_{pqr} and b_{tuv} have trivial intersection. If b_{pqr} ± d_{i_{1}···i_{s}} ∈ Γ^{3}(X), then we obtain a contradiction by the divisibility in (44). Hence, the subgroup generated by b_{pqr} has trivial intersection with any subgroup generated by any element of D_{s}.

Let 3 ≤ s, s′ ≤ n and let d_{i_{1}···i_{s}} and d_{i′_{1}···i′_{s′}} be two different elements of D_{s} and D_{s′}, respectively. If d_{i_{1}···i_{s}} ± d_{i′_{1}···i′_{s′}} ∈ Γ^{3}(X), then by applying Lemma 5.1 with y_{i_{1}} = ⋯ = y_{i_{s}} = −1 and y_{l} = 0 for all 1 ≤ l ≠ i_{1}, …, i_{s} ≤ n we have 3 is divisible by 9, which is a contradiction. It follows that the torsion subgroup Γ^{2/3}(X)_{tors} contains (ℤ/3ℤ)^{⊕N}, so does CH^{2}(X̄)_{tors}. □

5.1. Three Severi-Brauer surfaces. We consider the product of three Severi-Brauer surfaces. Let A_{1}, A_{2}, A_{3} be a central simple algebras of degree 3 over F and let X be the product of the corresponding Severi-Brauer surfaces SB(A_{1}), SB(A_{2}), SB(A_{3}). If one of A_{1}, A_{2}, A_{3} is split, then by (6) the problem to compute torsion in Chow group of codimension 2 is reduced to the case of product of two Severi-Brauer varieties, which was done in [4, Theorem 5.1]. Therefore, we may assume that ind(A_{m}) = 3 for all 1 ≤ m ≤ 3. Let e_{i} = ind(A_{j} ⊗A_{k}), f_{i} = ind(A_{j}^{⊗2} ⊗A_{k}), d = ind(A_{1} ⊗A_{2} ⊗A_{3}), and g_{i} = ind(A_{i}^{⊗2} ⊗A_{j} ⊗A_{k}) for all i, j, k such that {i, j, k} = I_{3}. For the same reason, we may assume that e_{i}, f_{i}, d, g_{i} ≥ 3. By (5), we have the following basis of K(X)

(45) {1, 3x_{m}, 3x_{m}^{2}, e_{i}x_{j}x_{k}, f_{i}x_{j}^{2}x_{k}, dx_{1}x_{2}x_{3}, e_{i}x_{j}^{2}x_{k}^{2}, g_{i}x_{i}^{2}x_{j}x_{k}, g_{i}x_{i}x_{j}^{2}x_{k}^{2}, dx_{1}^{2}x_{2}^{2}x_{3}^{2}},

where x_{m} is the pullback of the class of the tautological line bundle on the projective plane. We will need the following lemma to find upper bounds of the torsion.

Lemma 5.3. Let i, j, k be integers such that {i, j, k} = I_{3} and let y_{m} = x_{m} − 1 for all 1 ≤ m ≤ 3. Then, we have

(1) 3y_{m}^{2} ∈ Γ^{2}(X) in any case and 3y_{j}y_{k} ∈ Γ^{2}(X), 3y_{j}^{2}y_{k} + 3y_{j}y_{k}^{2} ∈ Γ^{3/4}(X) if e_{i} = 3,

(2) 3y_{j}^{2}y_{k} − 3y_{j}y_{k}^{2} ∈ Γ^{3/4}(X) if f_{i} = 3,

(3) 3 ∑_{m,l=1}^{3} y_{m}y_{l} ∈ Im(res^{2/3}), 6y_{1}y_{2}y_{3} +3 ∑_{m,l=1}^{3} y_{m}^{2}y_{l} ∈ Γ^{3/4}(X) and 9y_{1}^{2}y_{2}^{2}y_{3}^{2} ∈ Γ^{6}(X) if d = 3,

(4) 3 ∑_{m,l=1}^{3} y_{m}^{2}y_{l}+3(y_{i}y_{j}^{2}+y_{i}^{2}y_{j})+12y_{1}y_{2}y_{3} ∈ Γ^{3/4}(X), 3(y_{j}y_{k} −y_{i}y_{k} −y_{i}y_{j}) ∈ Im(res^{2/3}) if g_{i} = 3,

(5) 3y_{1}^{2}y_{2}^{2}y_{3} − 3y_{1}^{2}y_{2}y_{3}^{2} ∈ Γ^{3}(X) if f_{m} = g_{m} = e_{m} = 3 for all m.

(6) 3y_{1}^{2}y_{2}^{2}y_{3}^{2} − 6y_{1}^{2}y_{2}^{2}y_{3} ∈ Γ^{3}(X) if f_{m} = g_{m} = e_{m} = d = 3 for all m.

Proof. (1) By (45), we have γ_{2}(3x_{m} − 3) = 3y_{m}^{2} ∈ Γ^{2}(X). Since 3y_{j}y_{k} ∈ K(X), this element is in Γ^{2}(X). The rest of them follow from direct computation of c_{3}(3x_{j}x_{k}).

(2) By (45) and Lemma 5.3 (1), the elements 9y_{j}^{2}y_{k} = 3y_{j}^{2}(3y_{k}) and 9y_{j}y_{k}^{2} = 3y_{j}(3y_{k}^{2}) are in Γ^{3}(X), the result follows from the calculation of c_{3}(3x_{j}^{2}x_{k}).

(3) By (45), we obtain 3(y_{1}y_{2}y_{3} + ∑_{m,l=1}^{3} y_{m}y_{l}) ∈ Γ^{2}(X). Thus, the first inclusion immediately follows. As 27(y_{1}y_{2}y_{3})^{2} ∈ Γ^{6}(X), the rest of them follow from the computations of c_{3}(3x_{1}x_{2}x_{3}) and c_{6}(6x_{1}x_{2}x_{3}).

(4) By the calculation of c_{3}(3x_{i}^{2}x_{j}x_{k}), the first inclusion follows. By (45) we get 3x_{i}^{2}x_{j}x_{k} ∈ Γ^{2}(X). Hence, the second inclusion follows by expanding the element 3(y_{i} +1)^{2}(y_{j} +1)(y_{k} + 1).

(5) By direct calculation, we have

150(y_{1}^{2}y_{2}^{2}y_{3}−y_{1}^{2}y_{2}y_{3}^{2})=−6(c_{3}(3x_{1}x_{2}^{2}x_{3})+ c_{3}(3x_{1}x_{2}^{2}))+6(c_{3}(3x_{1}x_{2}x_{3}^{2})+ c_{3}(3x_{1}x_{3}^{2}))
+3(c_{3}(3x_{1}^{2}x_{2}^{2}x_{3})+ c_{3}(3x_{2}^{2}x_{3}))−3(c_{3}(3x_{1}^{2}x_{2}x_{3}^{2}) + c_{3}(3x_{2}x_{3}^{2}))
− 9c_{3}(3x_{1}^{2}x_{2}) + 9c_{3}(3x_{1}^{2}x_{3}) + 42c_{3}(3x_{1}x_{2}) − 42c_{3}(3x_{1}x_{3}).

Since 3y_{1}^{2}(3y_{2}^{2}y_{3}) − 3y_{1}^{2}(3y_{2}y_{3}^{2}) ∈ Γ^{4}(X), the result follows.

(6) It follows by a direct computation that

150y_{1}^{2}y_{2}^{2}y_{3}^{2}−300y_{1}^{2}y_{2}^{2}y_{3} = 4(c_{3}(3x_{1}x_{2}x_{3}^{2})−c_{3}(3x_{1}^{2}x_{2}^{2}x_{3}))+6(c_{3}(3x_{1}^{2}x_{2}) + c_{3}(3x_{1}x_{2}^{2}))
+2(c_{3}(3x_{1}x_{3}^{2}) + c_{3}(3x_{2}x_{3}^{2}))−2(c_{3}(3x_{1}^{2}x_{2}x_{3}^{2}) + c_{3}(3x_{1}x_{2}^{2}x_{3}^{2}))
+ 8(c_{3}(3x_{1}^{2}x_{2}x_{3})+ c_{3}(3x_{1}x_{2}^{2}x_{3}))−8(c_{3}(3x_{1}x_{3}) + c_{3}(3x_{2}x_{3}))
+ c_{3}(3x_{1}^{2}x_{2}^{2}x_{3}^{2}) − 16c_{3}(3x_{1}x_{2}x_{3}) − 36c_{3}(3x_{1}x_{2}).

As 9y_{1}^{2}y_{2}^{2}y_{3}^{2} ∈ Γ^{4}(X) by Lemma 5.3 (3) and 3y_{1}^{2}(3y_{2}^{2}y_{3}) ∈ Γ^{4}(X), the result follows. □

Applying Proposition 5.2, we prove the main result of this section.

Theorem 5.4. The maximal torsion in Chow group of codimension 2 of the product of three Severi-Brauer surfaces is (ℤ/3ℤ)^{⊕8}. In other words, M(SB_{3}) = 3^{8}.

Proof. Let A_{1}, A_{2}, A_{3} be division algebras of degree 3 over F and let X be the product of the corresponding Severi-Brauer surfaces SB(A_{1}), SB(A_{2}), SB(A_{3}). Set

β_{n} = |Γ^{n/n+1}(X_{E})/ Im(res^{n/n+1})|/|K^{n}(X_{E})/K^{n}(X)|,

where E is a splitting field of X, 1 ≤ n ≤ 6, and K^{n}(X_{E}) (resp. K^{n}(X)) is the codimension n part of K(X_{E}) (resp. K(X)).

We shall find upper bounds of β_{n} for 1 ≤ n ≤ 6. First of all, by (45) we have β_{1} ≤ 1. For the rest of them, we will find upper bounds using case by case analysis. Let i, j, k be integers such that {i, j, k} = I_{3}, f = f_{1}f_{2}f_{3}, g = g_{1}g_{2}g_{3}, e = e_{1}e_{2}e_{3}, G = {1 ≤ m ≤ 3 | g_{m} = 3} and H = {1 ≤ m ≤ 3 | f_{m} = 9}. Observe that if one of {e_{1}, e_{2}, e_{3}} is 3 (say, e_{i} = 3), then d, g_{m} ≤ 9 for 1 ≤ m ≤ 3 and 9y_{1}y_{2}y_{3} = 3y_{j}y_{k}(3y_{i}) ∈ Γ^{3}(X) by Lemma 5.3 (1).

Case: e_{1} = e_{2} = e_{3} = 3. By Lemma 5.3 (1), we have β_{2} ≤ 1. If d = 3, then by Lemma 5.3 (1) and (3) we get 3y_{1}y_{2}y_{3} ∈ Im(res^{3/4}). Hence, by Lemma 5.3 (1), (2), and (4) we have

f^{2} · β_{3} ≤ \begin{cases}3^{6} & if |H| = 0, \\ 3^{3}f_{i}f_{j} min\{g_{i}, g_{j}\} & if |H| = 1, f_{k} = 9, \\ 3^{3}f_{i} max\{3^{4−|G|}, 3^{2}\} & if |H| = 2, f_{i} = 3, \\ 3^{3} max\{3^{6−|G|}, 3^{4}\} & if |H| = 3. \end{cases}

It follows from 5.3 (1) that 9y_{i}y_{j}y_{k}^{2}, 9y_{i}^{2}y_{j}^{2} ∈ Im(res^{4/5}), thus we obtain β_{4} ≤ 3^{12}/3^{3}g. Again, by Lemma 5.3 (1), we have 3y_{i}^{2}(3y_{j}^{2}y_{k} + 3j_{j}y_{k}^{2}) ∈ Im(res^{4/5}). Therefore, we have β_{5} ≤ 3^{6}/g. It follows from Lemma 5.3 (3) that we have β_{6} ≤ 3^{3}/3^{2}. Finally, by (3) we conclude that

(46)  |⊕ Γ^{n/n+1}(X)_{tors}| ≤ \begin{cases}3^{16}/g^{2} & if |H| = 0, \\ 3^{13} min\{g_{i}, g_{j}\}/g^{2} & if |H| = 1, f_{k} = 9, \\ 3^{10} max\{3^{4−|G|}, 3^{2}\}/g^{2} & if |H| = 2, \\ 3^{7} max\{3^{6−|G|}, 3^{4}\}/g^{2} & if |H| = 3. \end{cases}

The maximum upper bound of (46) is 3^{10} when g_{m} = f_{m} = 3 for all 1 ≤ m ≤ 3 and d ∈ {3, 9}. If g_{m} = f_{m} = d = 3, then by Lemma 5.3 (5) and (6) we have a := 3y_{1}^{2}y_{2}^{2}y_{3} − 3y_{1}^{2}y_{2}y_{3}^{2}, b := 3y_{1}^{2}y_{2}^{2}y_{3}^{2} − 6y_{1}^{2}y_{2}^{2}y_{3} ∈ Γ^{3}(X)\Γ^{4}(X) as any element of Γ^{4}(X) is divisible by 9. Hence, the classes of a and b give torsion elements of Γ^{3/4}(X) of order 3 since 3a = 3y_{1}^{2}(3y_{2}^{2}y_{3}) − 3y_{1}^{2}(3y_{2}y_{3}^{2}), 3b = 9y_{1}^{2}y_{2}^{2}y_{3}^{2} − 3y_{1}^{2}(3y_{2}^{2}y_{3}) ∈ Γ^{4}(X). Moreover, we have a − b, a + b ∉ Γ^{4} as any element of Γ^{4}(X) is divisible by 9, thus the subgroups generated by a and b have trivial intersection. By Proposition 5.2, we have

Γ^{2/3}(X)_{tors} = (ℤ/3ℤ)^{⊕8} and Γ^{3/4}(X)_{tors} = (ℤ/3ℤ)^{⊕2}.

In this case, by (7) we have

(47)  CH^{2}(X̄)_{tors} = (ℤ/3ℤ)^{⊕8},

where X̄ is the corresponding generic variety. If g_{m} = f_{m} = 3 and d = 9, then by the same argument the class a is a torsion of Γ^{3/4}(X). Moreover, we have either 9y_{1}^{2}y_{2}^{2}y_{3}^{2} is a torsion of order 3 in Γ^{4/5}(X) ⊕ Γ^{5/6}(X) or β_{6} ≤ 1. Therefore, we obtain

(48)  |⊕ Γ^{n/n+1}(X)_{tors}| ≤ 3^{8}.

Case: e_{i} = e_{j} = 3 and e_{k} = 9. By Lemma 5.3 (1), (3) and (4) we have

β_{2} ≤ (3^{5} min{d, g_{1}, g_{2}, g_{3}})/3^{3}e.

If d = 9, then by Lemma 5.3 (1), (2), and (4) we have

f^{2} · β_{3} ≤ \begin{cases}3^{2}f min\{9, g_{k}\} & if |H| = 0, \\ 3^{2}f_{t}f_{k}g_{t}g_{k} & if |H| = 1, f_{s} = 9, \\ 3^{2}f_{i}f_{j}g_{k} min\{g_{i}, g_{j}\} & if |H| = 1, f_{k} = 9, \\ 3^{|H|+1}g & if |H| ≥ 2, \end{cases}

where s and t are integers such that {s, t} = {i, j}. Similarly, if d = 3, then it follows by Lemma 5.3 (1), (2), (3), and (4) that

f^{2} · β_{3} ≤ \begin{cases}3^{2}f min\{9, g_{m}\} & if |H| = 0, \\ 3^{2}f_{t}f_{k}g_{s} min\{g_{t}, g_{k}\} & if |H| = 1, f_{s} = 9, \\ 3^{2}f_{i}f_{j} max\{3^{4−|G|}, 3^{2}\} & if |H| = 1, f_{k} = 9, \\ 3^{|H|+1}g & if |H| ≥ 2, \end{cases}

where the minimum of the first inequality ranges over 1 ≤ m ≤ 3. By Lemma 5.3 (1), we have 9y_{i}y_{j}y_{k}^{2}, 9y_{i}^{2}y_{j}^{2} ∈ Im(res^{4/5}). Hence, β_{4} ≤ 3^{12}/(3^{4}g).

By Lemma 5.3 (1), we have 3y_{i}^{2}(3y_{j}^{2}y_{k} + 3y_{j}y_{k}^{2}), 3y_{j}^{2}(3y_{i}^{2}y_{k} + 3y_{i}y_{k}^{2}) ∈ Im(res^{5/6}). If f_{i} = 3 (resp. f_{j} = 3), then by Lemma 5.3 (2) we obtain 3y_{i}^{2}(3y_{j}^{2}y_{k} − 3y_{j}y_{k}^{2}) ∈ Im(res^{5/6}) (resp. 3y_{j}^{2}(3y_{i}^{2}y_{k} − 3y_{i}y_{k}^{2}) ∈ Im(res^{5/6})). Moreover, if d = 3, then by Lemma 5.3 (3) 3y_{k}^{2}(−3y_{1}y_{2}y_{3} + 3 ∑_{m,l=1}^{3} y_{m}^{2}y_{l}) = 9y_{i}^{2}y_{j}y_{k}^{2} + 9y_{i}y_{j}^{2}y_{k}^{2} ∈ Im(res^{5/6}). Therefore, we conclude that β_{5} ≤ 3^{6}/g (resp. 3^{7}/g) if one of {f_{i}, f_{j}, d} is 3 (resp. otherwise). By Lemma 5.3 (3), we have β_{6} ≤ 3. In conclusion, we have

(49)  |⊕ Γ^{n/n+1}(X)_{tors}| ≤ 3^{8} for all cases.

Case: e_{i} = 3 and e_{j} = e_{k} = 9. By Lemma 5.3 (1), (3) and (4) we obtain

β_{2} ≤ (3^{4} min{g_{j}, g_{k}} min{d, g_{i}})/3^{3}e.

In codimension 3, by Lemma 5.3 (1), (2), (3), and (4) we have

f^{2} · β_{3} ≤ \begin{cases}3fg_{j}g_{k} & if |H| = 0, d = 9, \\ 3fg_{i} min\{g_{j}, g_{k}\} & if |H| = 0, d = 3, \\ 3^{3}f_{j}f_{k}g_{j}g_{k} & if |H| = 1, f_{i} = 9, d = 9, \\ 3^{3}f_{j}f_{k} max\{3^{4−|G|}, 9\} & if |H| = 1, f_{i} = 9, d = 3, \\ 3^{3}g & if |H| = 1, f_{i} ≠ 9, \\ 3^{|H|+2}g & if |H| ≥ 2. \end{cases}

By Lemma 5.3 (1), we get 3y_{i}^{2}(3y_{j}y_{k}), 3y_{i}(3y_{j}^{2}y_{k} + 3y_{j}y_{k}^{2}) ∈ Im(res^{4/5}). Hence, it follows from Lemma 5.3 (2) that β_{4} ≤ 3^{8}/g (resp. 3^{7}/g) if |H| = 3 (resp. otherwise).

By Lemma 5.3 (1), we obtain 3y_{i}^{2}(3y_{j}^{2}y_{k} + 3y_{j}y_{k}^{2}) ∈ Im(res^{5/6}). Therefore, by Lemma 5.3 (2) and (3) we have

g · β_{5} ≤ \begin{cases}9^{3} & if |H| ≤ 1 or d = 3, \\ 9^{2} · 27 & if |H| = 2, d ≠ 3, \\ 9 · 27^{2} & if |H| = 3, d ≠ 3. \end{cases}

Finally, by Lemma 5.3 (3) β_{6} ≤ 3. Therefore, for all cases we have

(50)  |⊕ Γ^{n/n+1}(X)_{tors}| ≤ 3^{6}.

Case: e_{1} = e_{2} = e_{3} = 9. By Lemma 5.3 (1), (3) and (4) we have

β_{2} ≤ 3^{3} max{1/3^{3}, 1/3^{|G|+[3/d]}}/3^{3}.

In codimension 3, by Lemma 5.3 (2), (3) and (4) we obtain β_{3} ≤ (3^{3}fg)/df^{2}. It follows from Lemma 5.3 (4) that

eg · β_{4} ≤ \begin{cases}9^{5} · 27 & if |H| ≤ 1, \\ 9^{4} · 27^{2} & if |H| = 2, \\ 9^{3} · 27^{3} & if |H| = 3. \end{cases}

In codimension 5, by Lemma 5.3 (2) and (3) we obtain

g · β_{5} ≤ \begin{cases}9^{3} & if d = 3, \\ 9^{2} · 27 & if d ≠ 3, |H| ≤ 1, \\ 9 · 27^{2} & if d ≠ 3, |H| = 2, \\ 27^{3} & if d ≠ 3, |H| = 3. \end{cases}

Finally, by Lemma 5.3 (3) we have β_{6} ≤ 3^{3}/ max{9, d}. Therefore, for all cases we obtain

(51)  |⊕ Γ^{n/n+1}(X)_{tors}| ≤ 3^{7}.

In conclusion, the result follows from (47), (48), (49), (50), and (51). □

6. Product of Quadric surfaces

In this section, we obtain upper bounds for the torsion in Chow group of codimension 2 of the product of two quadric surfaces in Theorem 6.1 and the product of three quadric surfaces with the same discriminant in Theorem 6.5. In the case of the product of two quadric surfaces, we also provide a sharp lower bound in the gamma filtration in Proposition 6.3.

Let F be a field of characteristic different from 2 and let q = ⟨c, −a, −b, ab⟩ be a nondegenerate quadratic form over F of rank 4 for c, a, b ∈ F^{×}. If the discriminant is trivial, then the quadric surface corresponding to the form q is birational to ℙ^{1} × SB(Q), where Q is the quaternion F -algebra determined by a and b. Otherwise, the quadric is isomorphic to R_{L/F} (SB(Q)), where R_{L/F} is the Weil restriction over a quadratic field L = F (\sqrt{c}). We shall write disc Q for the discriminant c.

Consider two quadric surfaces with the corresponding quaternions Q_{1}, Q_{2} and quadratic extensions L_{1}, L_{2} as above. We set

(52)  X = \begin{cases}SB(Q_{1}) × SB(Q_{2}) & if disc Q_{i} = 1, \\ SB(Q_{1}) × R_{L_{2}/F} (SB(Q_{2})) & if disc Q_{1} = 1 ≠ disc Q_{2}, \\ R_{L_{1}/F} (SB(Q_{1})) × R_{L_{2}/F} (SB(Q_{2})) & if disc Q_{i} ≠ 1. \end{cases}

Then, by [3, Corollary 2.5] the torsion in codimension 2 cycles of X of the first and second cases of (52) is isomorphic to that of the product of two quadric surfaces. Therefore, it suffices to consider X for the torsion in codimension 2 cycles of the product of two quadric surfaces. We call X the variety associated to the product of two quadric surfaces.

Consider the last case of (52). If ind(Q_{1})_{L_{1}} = ind(Q_{1})_{L_{1}} = 1, then the associated variety X has torsion-free Chow groups. Thus, we may assume that ind(Q_{1})_{L_{1}} = 2. We choose a splitting field E of X as follows. If ind(Q_{2})_{L_{2}} = 1, then we take a maximal subfield (≠ L_{2}) of Q_{1} for E. Otherwise, we take for E a common maximal subfield (≠ L_{1}, L_{2}) of Q_{1} and Q_{2} if ind(Q_{1} ⊗ Q_{2}) ≤ 2 or the tensor product of maximal subfields E_{1}(≠ L_{2}) of Q_{1} and E_{2}(≠ L_{1}) of Q_{2} if ind(Q_{1} ⊗ Q_{2}) = 4. Hence, d := [E : F] = 4 if ind(Q_{1} ⊗ Q_{2}) = 4 and d = 2 otherwise. For the second case (52), we choose a splitting field E in the same way.

The theorem below was proven in [3]. Here, we give an elementary proof which does not use any cohomological method and K theory of quadrics. Moreover, we find upper bound of the total torsion in the topological filtration of the product of two quadric surfaces with nontrivial discriminants.

Theorem 6.1. [3, Theorems 5.1, 5.7, 5.8, 5.9] Let X be the variety associated to the product of two quadric surfaces. Then, the torsion subgroup CH^{2}(X)_{tors} is either trivial or ℤ/2ℤ, i.e., M(Q_{2}) ≤ 2.

Proof. Let Q_{i} be a quaternion algebra for i = 1, 2, X the variety associated to the product of two quadric surfaces of Q_{i}, and E the splitting field of X. If disc Q_{i} = 1 for all i, then the variety X is torsion free. From now on we only consider the other cases. To apply (3) we shall find upper bounds of α_{n} := |T^{n/n+1}(X_{E})/ Im(res^{n/n+1})| for each of the following 3 cases.

Case: L_{1}L_{2} := L_{1} ⊗ L_{2} is a biquadratic field extension. Let L = L_{1}L_{2}. Then, we have X_{E} = R_{EL_{1}/E}(ℙ^{1})×R_{EL_{2}/E}(ℙ^{1}) and X_{EL} = ℙ^{1}×ℙ^{1}×ℙ^{1}×ℙ^{1}, which have torsion-free Chow groups. For each 1 ≤ k ≤ 4, let x_{k} be the pullback of the class of tautological line bundle on the projective space in K(X_{EL}). We set x_{i_{1}···i_{k}} = x_{i_{1}} ⋯ x_{i_{k}} for 1 ≤ i_{1} < … < i_{k} ≤ 4. By the action of the Galois group of EL/E, we have the following bases of K(X_{E}) and K(X), respectively:

(53) {1, x_{2i−1} + x_{2i}, x_{12}, x_{34}, (x_{1} + x_{2})(x_{3} + x_{4}), x_{12}(x_{3} + x_{4}), x_{34}(x_{1} + x_{2}), x_{1234}} and
(54) {1, e_{i}(x_{2i−1} + x_{2i}), x_{12}, x_{34}, f(x_{1} + x_{2})(x_{3} + x_{4}), e_{2}x_{12}(x_{3} + x_{4}), e_{1}x_{34}(x_{1} + x_{2}), x_{1234}},

where e_{i} = ind(Q_{i})_{L} and f = ind(Q_{1} ⊗ Q_{2})_{L}. Then, we have |K(X_{E})/K(X)| = e_{1}^{2}e_{2}^{2}f. If e_{1} = e_{2} = 1, then f = 1. Hence, we may assume that e_{1}e_{2} ≥ 2.

Let y_{k} = x_{k} − 1. Set y_{i_{1}···i_{k}} = y_{i_{1}} ⋯ y_{i_{k}} for 1 ≤ i_{1} < … < i_{k} ≤ 4. We will use other bases for K(X_{E}) the basis (53) by replacing x_{k} by y_{k} and for K(X)

(55) {1, e_{i}(y_{2i−1} + y_{2i}), z_{12}, z_{34}, f(y_{1} + y_{2})(y_{3} + y_{4}), e_{2}z_{12}(y_{3} + y_{4}), e_{1}z_{34}(y_{1} + y_{2}), z_{12}z_{34}},

where z_{12} = y_{1} + y_{2} + y_{1}y_{2} and z_{34} = y_{3} + y_{4} + y_{3}y_{4}.

It follows from z_{12}, z_{34} ∈ K(X) that y_{1} + y_{2}, y_{3} + y_{4} ∈ Im(res^{1/2}), thus we have α_{1} = 1. By the same argument, we get (y_{1} + y_{2})(y_{3} + y_{4}) ∈ Im(res^{2/3}). In addition, it follows from the basis and (2) that e_{1}y_{12}, e_{2}y_{34} ∈ T^{2}(X). Hence, α_{2} ≤ e_{1}e_{2}.

If ind(Q_{1} ⊗ Q_{2})= 1, then it follows from closed embeddings SB(Q_{1}) × R_{L_{2}/F} (SB(Q_{1})) ↪ X and R_{L_{1}/F} (SB(Q_{1}))×SB(Q_{1}) ↪ X that we have y_{34}(y_{1} +y_{2}), y_{1}y_{2}(y_{3} +y_{4}) ∈ Im(res^{3/4}), respectively. Otherwise, it follows from e_{1}y_{12} · (y_{34} + y_{3} + y_{4}) and e_{2}y_{34} · (y_{12} + y_{1} + y_{2}) that e_{1}y_{12}(y_{3} + y_{4}), e_{2}y_{34}(y_{1} + y_{2}) ∈ Im(res^{3/4}). Therefore, α_{3} ≤ 1 if ind(Q_{1} ⊗ Q_{2}) = 1 and α_{3} ≤ e_{1}e_{2} otherwise.

By a transfer argument, we have α_{4} ≤ d. Hence, we obtain

|⊕ T^{n/n+1}(X)_{tors}| ≤ \begin{cases}1 & if ind(Q_{1} ⊗ Q_{2}) = 1, \\ d/f & otherwise. \end{cases}

If f = 1, then ind(Q_{1} ⊗ Q_{2}) ≤ 2, thus d = 2. Therefore, |T^{2/3}(X)_{tors}| ≤ 2. Moreover, the order of the group ⊕T^{n/n+1}(X)_{tors} is trivial if f = 4 or f = d = 2 or ind(Q_{1} ⊗ Q_{2}) = 1.

Case: L_{1} = L_{2}. Let L = L_{1} = L_{2}. Then, X_{E} = R_{EL/E}(ℙ^{1}) × R_{EL/E}(ℙ^{1}). Applying the same argument with the previous case, we have the basis (53) (resp. (54)) replacing (x_{1} + x_{2})(x_{3} + x_{4}) (resp. f(x_{1} + x_{2})(x_{3} + x_{4})) with two elements x_{13} + x_{24} and x_{14} + x_{23} (resp. f(x_{13} + x_{24}) and f(x_{14} + x_{23}) for K(X_{E}) (resp. K(X)). As e_{1} = 2, we have |K(X_{E})/K(X)| = 4e_{2}^{2}f^{2}.

Similarly, we use other bases for K(X_{E}) the basis by replacing x_{k} with y_{k} and for K(X) the basis replacing f(y_{1} + y_{2})(y_{3} + y_{4}) with two elements f(y_{13} + y_{24} + ∑_{k=1}^{4} y_{k}) and f(y_{14} + y_{23} + ∑_{k=1}^{4} y_{k}).

By the same argument used in the previous case, we have α_{1} = 1. In codimension 2, we have 2y_{12}, e_{2}y_{34}, (y_{1} + y_{2})(y_{3} + y_{4}) ∈ Im(res^{2/3}). If f = 1, then e_{2} = d = 2, and X = R_{L/F} (SB(Q_{1}))×R_{L/F} (SB(Q_{1})), thus by the diagonal embedding R_{L/F} (SB(Q_{1})) ↪ X the sum of all elements of codimension 2 in the basis is contained in the group Im(res^{2/3}). Moreover, if f ≠ 1, then f(y_{13}+y_{24}), f(y_{14}+y_{23}) ∈ T^{2}(X). Therefore, we obtain α_{2} ≤ 2e_{2}f.

If f = 1, then we have z_{12}(y_{12} + y_{34} + y_{13} + y_{24} + y_{14} + y_{23}) ∈ Im(res^{3/4}). Otherwise, we obtain 2y_{12}(y_{3} + y_{4}), e_{2}y_{34}(y_{1} + y_{2}) ∈ Im(res^{3/4}). Hence, we have α_{3} ≤ 2 if f = 1 and α_{3} ≤ 2e_{2} otherwise. Finally, we get α_{4} ≤ d, thus

(56)  |⊕ T^{n/n+1}(X)_{tors}| ≤ \begin{cases}1 & if f = 1, \\ d/f & otherwise. \end{cases}

Hence, the group ⊕T^{n/n+1}(X)_{tors} is trivial except the case where f = 2 and d = 4. In the latter case, we can further reduce the upper bound of α_{4} to 2 if 2y_{1234} ∈ T^{4}(X). Hence, the group ⊕T^{n/n+1}(X)_{tors} is trivial in this case. If 2y_{1234} ∉ T^{4}(X), then the class of 2y_{1234} = 2y_{12} · z_{34} − z_{12} · 2(y_{13} + y_{24}) ∈ T^{3}(X) gives a torsion element of order 2 in T^{3/4}(X)_{tors}. Therefore, the group T^{2/3}(X)_{tors} is trivial in all cases.

Case: one of disc Q_{i} is trivial. Let disc Q_{1} = 1. Then, we have X_{E} = ℙ^{1} × R_{L_{2}E/E}(ℙ^{1}). We have the following bases of K(X_{E}) and K(X), respectively:

{1, x_{1}, x_{2} + x_{3}, x_{23}, x_{1}(x_{2} + x_{3}), x_{123}} and {1, e_{1}x_{1}, e_{2}(x_{2} + x_{3}), x_{23}, fx_{1}(x_{2} + x_{3}), e_{1}x_{123}},

where e_{1} = ind(Q_{1})_{L_{2}}, e_{2} = ind(Q_{2})_{L_{2}}, f = ind(Q_{1} ⊗ Q_{2})_{L_{2}}. It follows that we obtain |K(X_{E})/K(X)| = e_{1}^{2}e_{2}f. We will use other bases for K(X_{E}) the above basis by replacing x_{k} by y_{k} and for K(X)

(57)  {1, e_{1}y_{1}, e_{2}(y_{2} + y_{3}), z_{23}, f(y_{1} + 1)(y_{2} + y_{3}), e_{1}y_{1}z_{23}},

where z_{23} = y_{2} + y_{3} + y_{23}.

Obviously, we have α_{1} ≤ e_{1}. In codimension 2, we have e_{2}y_{23}, e_{1}y_{1}(y_{2} + y_{3}) ∈ T^{2}(X). If f = 1, then (y_{12} + y_{13} + y_{2} + y_{3}) + z_{23} − 2(y_{2} + y_{3}) = y_{12} + y_{13} + y_{23} ∈ T^{2}(X). Hence, we get α_{2} ≤ min{e_{1}, e_{2}} if f = 1 and α_{3} ≤ e_{1}e_{2} otherwise. Finally, we have α_{3} ≤ d, thus the same upper bound (56) is obtained for the order of the group ⊕T^{n/n+1}(X)_{tors}. Hence, the group ⊕T^{n/n+1}(X)_{tors} is trivial if f = 1 or f = d = 2 or f = 4. Assume that f = 2 and d = 4. Then it follows from (57) that 2y_{123} ∈ T^{2}(X). If 2y_{123} ∈ T^{3}(X), then we have α_{3} ≤ 2. Therefore, the group ⊕T^{n/n+1}(X)_{tors} is trivial in this case. Otherwise, the class of 2y_{123} gives a torsion element of order 2 in T^{2/3}(X). In any case, we have |T^{2/3}(X)_{tors}| ≤ 2. In conclusion, the result follows from (1). □

Remark 6.2. Observe that the proof of Theorem 6.1 indicates when the torsion in Chow group of codimension 2 of X is trivial. Moreover, the proof still works if we replace the topological filtration by the gamma filtration.

Now we provide a nontrivial torsion subgroup in the gamma filtration:

Proposition 6.3. With the above notations, we have Γ^{2/3}(X)_{tors} = ℤ/2ℤ if L is a biquadratic extension, f = 2, and d = 4.

Proof. It follows from the basis (55) that 2y_{1234} ∈ K(X). Hence, by (2) we have 2y_{1234} ∈ Γ^{2}(X). As d = 4, 4y_{1234} ∈ Γ^{3}(X). We show that this element is not contained in Γ^{3}(X) by computing the Chern classes of the elements in the basis (54).

Consider the basis (54) of K(X) with f = e_{i} = 2, where i = 1, 2. It follows from Whitney formula that we have c_{1}(2(x_{2i−1} + x_{2i})) = 2(y_{2i−1} + y_{2i}), c_{1}(x_{2i−1} + x_{2i}) = y_{2i−1} + y_{2i}, and c_{2}(x_{2i−1} + x_{2i}) = y_{2i−1}y_{2i}. Therefore, we have

(58) c_{2}(2(x_{2i−1} + x_{2i})) = 4y_{2i−1}y_{2i}, c_{1}(x_{2i−1}x_{2i})^{2} = 2y_{2i−1}y_{2i} and c_{1}(x_{2i−1}x_{2i})^{3} = 0.

Similarly, we obtain c_{j}(2(x_{2i−1} + x_{2i})) = 0 for 3 ≤ j ≤ 4.

Let z = (x_{1} + x_{2})(x_{3} + x_{4}), z′ = (y_{1} + y_{2})(y_{3} + y_{4}), u = y_{123} + y_{124}, and v = y_{134} + y_{234}. Then, by a direct computation, we have

(59)  c_{j}(z) = \begin{cases}2(∑_{k=1}^{4} y_{k}) + z′ & for j = 1, \\ 2y_{1234} + 3z′ + 4(u + v + y_{12} + y_{34}) & for j = 2, \\ 4(u + v + 3y_{1234}) & for j = 3, \\ 2y_{1234} & for j = 4. \end{cases}

Since c_{2}(2z) = c_{1}(z)^{2} + 2c_{2}(z), it follows from (59) that

(60)  c_{2}(2z) = 8y_{1234} + 14z′ + 16(u + v + y_{12} + y_{34}).

As c_{3}(2z) = 2(c_{1}(z)c_{2}(z) + c_{3}(z)) and c_{4}(2z) = 2(c_{1}(z)c_{3}(z) + c_{4}(z)) + c_{2}(z)^{2}, it follows from (59) that

(61)  c_{j}(2z) ≡ 0 mod 4 for j = 3, 4.

Let w = x_{12}(x_{3} + x_{4}). Then, we obtain c_{1}(w) = 2z_{12} + y_{3} + y_{4} + z′ + u and c_{2}(w)= 4y_{1234} + 3u + 2(y_{12} + v) + y_{13} + y_{14} + y_{23} + y_{24} + y_{34}. Therefore, we have

(62) c_{j}(2w)=\begin{cases}2(8y_{1234} + 9u + 4v + 6y_{12} + 3y_{13} + 3y_{23} + 3y_{14} + 3y_{24} + 2y_{34}) & for j = 2, \\ 4(10y_{1234} + 3u + 2v) & for j = 3, \\ 8y_{1234} & for j = 4. \end{cases}

Let w′ = x_{34}(x_{1} + x_{2}). Then, we have the Chern classes (62) for 2w′ by replacing 1, 2, 3, 4 with 3, 4, 1, 2, respectively.

It follows from (58) that c_{1}(x_{12})^{2}c_{1}(x_{34}) = c_{1}(x_{12})^{2}c_{1}(x_{1234}) = 2(u+y_{1234}), c_{1}(x_{34})^{2}c_{1}(x_{12}) = c_{1}(x_{34})^{2}c_{1}(x_{1234}) = 2(v + y_{1234}). Since c_{1}(x) is divisible by 2 for any element x ∈ {2z, 2w, 2w′, 2(x_{2i−1} + x_{2i})}, one can see easily that the subgroup generated by the products of three of the first Chern classes of any element of the basis (54) is generated by 2(u + y_{1234}) and 2(v + y_{1234}) modulo 4. Similarly, using (58), (60), (62) one sees that the subgroup generated by the products of the first and second Chern classes of any element of the basis (54) is also generated by 2(u + y_{1234}) and 2(v + y_{1234}) modulo 4. It follows from (61) and (62) that the third and fourth Chern classes of 2z, 2w, 2w′ is divisible by 4. Therefore, the subgroup Γ^{3}(X) is generated by 2(u + y_{1234}) and 2(v + y_{1234}) modulo 4. Hence, 2y_{1234} is not contained in Γ^{3}(X) and this element gives a torsion of Γ^{2/3}(X) of order 2. The result immediately follows from Remark 6.2. □

Remark 6.4. If X̄ is a corresponding generic variety to X in Proposition 6.3, then we obtain CH^{2}(X̄) = ℤ/2ℤ, which recovers a theorem of Izhboldin and Karpenko [5, Theorem 14.1]. Indeed, it is possible to find such a variety by showing that the gamma filtration for the variety R_{L_{1}/F} (SB(Q′_{1})) × R_{L_{2}/F} (SB(Q′_{2})) × R_{L/F} (SB(2, Q′_{1} ⊗ Q′_{2})) is torsion free, where L = L_{1}L_{2} is a biquadratic extension and SB(2, Q′_{1} ⊗ Q′_{2}) is the generalized Severi-Brauer variety of rank 2 left ideals in the biquaternion algebra (Q′_{1} ⊗ Q′_{2})_{L}.

6.1. Three quadric surfaces with the same discriminant. In this subsection we consider the product of three quadric surfaces with the same discriminant. Let Q_{1}, Q_{2}, Q_{3} be three quaternion F -algebras and let L be the quadratic extension over F corresponding to three quadratic surfaces with the same discriminant as above.

We set

(63) X = \begin{cases}SB(Q_{1}) × SB(Q_{2}) × SB(Q_{3}) & if disc Q_{i} = 1, \\ R_{L/F} (SB(Q_{1})) × R_{L/F} (SB(Q_{2})) × R_{L/F} (SB(Q_{3})) & otherwise \end{cases}

and call it the variety associated to the product of three quadric surfaces with the same discriminant. Then, by the same argument as in the case of two quadric surfaces, the torsion in codimension 2 cycles of X is isomorphic to that of the product of three quadric surfaces with the same discriminant.

Let h = ind(Q_{1} ⊗ Q_{2} ⊗ Q_{3}), J = {{1, 2}, {3, 4}, {5, 6}}, and f_{pq} = ind(Q_{max{r,s}/2} ⊗ Q_{max{t,u}/2})_{L} for any {{p, q}, {r, s}, {t, u}} = J. Set

F_{m} = {{p, q} ∈ J | f_{pq} = m} for m = 1, 2, 4.

Consider the second case of (63). If ind(Q_{i})_{L} = 1 for all 1 ≤ i ≤ 3, then the variety X has torsion-free Chow groups, thus we may assume that ind(Q_{1})_{L} ≠ 1. We choose a splitting field E of X as follows. If ind(Q_{2})_{L} = ind(Q_{3})_{L} = 1, then we take a maximal subfield for E. If ind(Q_{2})_{L} = 2 and ind(Q_{3})_{L} = 1, then we take for E a common maximal subfield of Q_{1} and Q_{2} if ind(Q_{1} ⊗ Q_{2}) ≤ 2 or the tensor product of maximal subfields of Q_{1} and Q_{2} if ind(Q_{1} ⊗ Q_{2}) = 4.

Now we may assume that ind(Q_{i})_{L} = 2 for all 1 ≤ i ≤ 3. If h = 8, then we take for E the tensor product of maximal subfields of Q_{i}. If h = 1, then |F_{2}| = 3, thus we take the tensor product of a common maximal subfield of Q_{1} and Q_{2} and a maximal subfield of Q_{3} for E. If |F_{1}| ≥ 2, then we have (Q_{1})_{L} ≃ (Q_{2})_{L} ≃ (Q_{3})_{L}, thus we take for E a maximal subfield of Q_{1}. If h ∈ {2, 4}, |F_{1}| ≤ 1, and |F_{4}| ≥ 1, then there exist Q_{i} and Q_{j} such that ind(Q_{i} ⊗ Q_{j}) = 4 for some 1 ≤ i ≠ j ≤ 3, thus we take for E the tensor product of maximal subfields of Q_{i} and Q_{j} which also splits the remaining quaternion algebra. If h ∈ {2, 4}, |F_{1}| = 1, and |F_{4}| = 0, then there exist Q_{i} and Q_{j} such that (Q_{i})_{L} ≃ (Q_{j})_{L} for some 1 ≤ i ≠ j ≤ 3, thus we take for E the tensor product of a maximal subfield of Q_{i} and a maximal subfield of the remaining quaternion algebra. Hence, we have

d := [E : F] = \begin{cases}2 & if |F_{1}| ≥ 2, \\ 4 & if h = 1 or h ∈ \{2, 4\}, |F_{1}| ≤ 1, \\ 8 & if h = 8. \end{cases}

Theorem 6.5. The torsion subgroup in the codimension 2 Chow group of the product of three quadric surfaces with the same discriminant is contained in (ℤ/2ℤ)^{⊕7}.

Proof. Let Q_{i} be a quaternion F -algebra for 1 ≤ i ≤ 3 such that the corresponding qaudrics have the same discriminant. Let X be the associated variety to the product of three quadric surfaces of Q_{i} and E be the splitting field of X as above. If the discriminant is trivial, the result follows from Proposition 4.2. Hence, we may assume that the discriminant is nontrivial, thus we have X_{E} = R_{EL/E}(ℙ^{1}) × R_{EL/E}(ℙ^{1}) × R_{EL/E}(ℙ^{1}) and X_{EL} = ℙ^{1} × ℙ^{1} × ℙ^{1} × ℙ^{1} × ℙ^{1} × ℙ^{1}.

For each 1 ≤ k ≤ 6, let x_{k} be the pullback of the class of tautological bundle on the projective line in K(X_{EL}). Set x_{i_{1}···i_{k}} = x_{i_{1}} ⋯ x_{i_{k}} for 1 ≤ i_{1} < … < i_{k} ≤ 6. It follows from the action of the Galois group ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/2ℤ of LE/E that we have the bases of K(X_{E}) and K(X), repectively:

{1, x_{p} + x_{q}, x_{pq}, x_{pr} + x_{qs}, x_{rs}(x_{p} + x_{q}), x_{prt} + x_{qsu}, x_{pqrs}, x_{pq}(x_{rt} + x_{su}), x_{pqrs}(x_{t} + x_{u}), x_{pqrstu}} and {1, e_{pq}(x_{p} + x_{q}), x_{pq}, f_{tu}(x_{pr} + x_{qs}), e_{pq}x_{rs}(x_{p} + x_{q}), g(x_{prt} + x_{qsu}), x_{pqrs}, f_{pq}x_{pq}(x_{rt} + x_{su}), e_{tu}x_{pqrs}(x_{t} + x_{u}), x_{pqrstu}},

where e_{pq} = ind(Q_{max{p,q}/2})_{L}, g = ind(Q_{1} ⊗ Q_{2} ⊗ Q_{3})_{L}, and p, q, r, s, t, u range over {{p, q}, {r, s}, {t, u}} = J. Then, we have |K(X_{E})/K(X)| = (e_{12}e_{34}e_{56}f_{12}f_{34}f_{56}g)^{4}.

Let y_{k} = x_{k} − 1 and y_{i_{1}···i_{k}} = y_{i_{1}} ⋯ y_{i_{k}} for 1 ≤ i_{1} < … < i_{k} ≤ 6. To simplify the computation, we shall use other bases for K(X_{E}) the above basis replacing x_{k} with y_{k} and for K(X)

{1, e_{pq}(y_{p} + y_{q}), z_{pq}, f_{tu}(z_{pr} + z_{qs}), e_{pq}z_{rs}(y_{p} + y_{q}), g(z_{prt} + z_{qsu}), z_{pq}z_{rs}, f_{pq}z_{pq}(z_{rt} + z_{su}), z_{pq}z_{rs}z_{tu}},

where z_{pq} = y_{p}y_{q} + y_{p} + y_{q}, z_{pr} = y_{p}y_{r} + y_{p} + y_{r}, and z_{prt} = y_{prt} + z_{pr} + y_{pt} + y_{rt} + y_{t}.

Let α_{n} = |T^{n/n+1}(X_{E})/ Im(res^{n/n+1})|. We will find upper bounds of α_{n} for 1 ≤ n ≤ 6. Observe that any basis element of K(X_{E}) multiplied by d is contained in the image of the restriction map. Since z_{pq} ∈ K(X), we get y_{p} + y_{q} ∈ Im(res^{1/2}). Hence, we have α_{1} = 1. We divide the proof into three cases.

Case: f_{pq} ≠ 1 for all {p, q} ∈ J, i.e., |F_{1}| = 0. It follows from the basis of K(X) that e_{pq}y_{pq} ∈ T^{2}(X) and (y_{p}+y_{q})(y_{r} +y_{s}) ∈ Im(res^{2/3}) for any {p, q} ≠ {r, s} ∈ J. Since f_{tu} ≠ 1, we obtain f_{tu}(y_{pr} + y_{qs}) ∈ T^{2}(X) for any {t, u} ∈ J. Therefore, we have

(64)  α_{2} ≤ e_{12}e_{34}e_{56}f_{12}f_{34}f_{56}.

Since z_{pq} ∈ T^{1}(X), we have e_{rs}y_{rs} · (y_{p} + y_{q}) ∈ Im(res^{3/4}). Moreover, as z_{12}z_{34}z_{56} ∈ T^{3}(X) we obtain ∑ y_{prt} + y_{qsu} ∈ Im(res^{3/4}), where the sum ranges over all such elements in the basis. As f_{tu} ≠ 1, the element f_{tu}(y_{pr} + y_{qs}) · (y_{t} + y_{u}) is contained in the group Im(res^{3/4}). Hence, we have

(65)  α_{3} ≤ (e_{12}e_{34}e_{56})^{2}f_{12}f_{34}f_{56}d/ max{f_{12}, f_{34}, f_{56}}.

As e_{pq}y_{pq}, f_{pq}(y_{rt}+y_{su}) ∈ T^{2}(X), we obtain e_{pq}e_{rs}y_{pqrs}, e_{pq}y_{pq} ·f_{pq}(y_{rt}+y_{su}) ∈ T^{4}(X). As e_{pq}y_{pq} · z_{rs} · z_{tu} ∈ Im(res^{4/5}), we have α_{4} ≤ ∏_{pq∈J} e_{pq} min{d, e_{12}e_{34}e_{56}/e_{pq}} min{d, f_{pq}e_{pq}}.

It follows from e_{pq}y_{pq} ∈ T^{2}(X) that we have α_{5} ≤ ∏_{pq∈J} min{d, e_{12}e_{34}e_{56}/e_{pq}}. Obviously, α_{6} ≤ d. Hence, we obtain

(66) |⊕ T^{n/n+1}(X)_{tors}| ≤ \frac{d^{2}}{g^{4} max\{f_{12}, f_{34}, f_{56}\}} ∏_{pq∈J} \frac{min\{d, e_{12}e_{34}e_{56}/e_{pq}\} min\{d, f_{pq}e_{pq}\}}{f_{pq}^{2}}.

Let B be the right-hand side of the inequality of (66). We first consider the case where G := g = h. Then, one can compute B for each subcase of g = h. Consider a subcase where G = 4 and |F_{2}| = 1. Then, f_{pq} = 2 for some pq ∈ J. If 2y_{rstu} ∈ T^{4}(X), then 2y_{rstu}(y_{p} + y_{q}) ∈ Im(res^{5/6}), thus we can reduce the upper bound B(= 2^{2}) to 1. Hence, we have | ⊕ T^{n/n+1}(X)_{tors}| ≤ 1. Otherwise, by Lemma 6.6 (1) below we obtain either | ⊕ T^{n/n+1}(X)_{tors}| ≤ 2^{2}, T^{3/4}(X)_{tors} = T^{4/5}(X)_{tors} = ℤ/2ℤ or | ⊕ T^{n/n+1}(X)_{tors}| ≤ 2, T^{3/4}(X)_{tors} = ℤ/2ℤ. Therefore, T^{2/3}(X)_{tors} = 0 in any case.

Let E_{2} = {{p, q} ∈ J | e_{pq} = 2}. Then, we may assume that |E_{2}| ≥ 1. If G = 1, then we have |F_{2}| = |E_{2}| = 3 and d = 4. Therefore, by Lemma 6.6 (2) we can reduce the upper bound 2^{6} (resp. 2^{10}) obtained by (64) (resp. 65) to 2^{5} (resp. 2^{8}).

Applying the same argument, together with Lemma 6.6 to each subcase of g = h, we obtain (67)

|T^{2/3}(X)_{tors}| ≤ \begin{cases}1 & if G = 4, 0 ≤ |F_{2}| ≤ 2 or G = |E_{2}| = d = 2, |F_{2}| = 3, \\ 2 (resp. 2^{5}) & if G = 8 or G = 4 (resp. 2), |F_{2}| = 3, |E_{2}| = 3, \\ 2^{4} & if G = |E_{2}| = 2, |F_{2}| = 3, d = 4 or G = 2, 0 ≤ |F_{2}| ≤ 2, \\ 2^{6} & if G = 1. \end{cases}

Now we consider the case where H := g = h/2. By the same argument, we have the same upper bound (67) for |T^{2/3}(X)_{tors}| if G = H = 1, 2. Similarly, if H = 4, then we obtain

(68)  |T^{2/3}(X)_{tors}| ≤ \begin{cases}1 & if H = 4, |F_{2}| = 2, |E_{2}| = 2, \\ 2^{2} & if H = 4, 2 ≤ |F_{2}| ≤ 3, |E_{2}| = 3, \\ 2^{5} (resp. 2^{3}) & if H = 4, |F_{2}| = 0 (resp. |F_{2}| = 1). \end{cases}

Case: |F_{1}| = 1. Let f_{tu} = 1, e = e_{pq} = e_{rs}, and e′ = e_{tu}. For any number t, we write m(t) for min{d, t}. Then, Lemma 6.6 (3) and the above argument implies that

|⊕ T^{n/n+1}(X)_{tors}| ≤(e^{2}e′f_{pq}f_{rs})(e^{3}e^{′2}m(f_{pq})m(f_{rs})d)(edm(e^{2})m(ee′)^{2}m(f_{pq}e)m(f_{rs}e))·
(m(e^{2})m(ee′)^{2})(d)/(e^{2}e′f_{pq}f_{rs}g)^{4},

where each term inside the parentheses of the numerator is the upper bound of α_{n} for 2 ≤ n ≤ 6.

By the same argument used above, Lemma 6.6 (1) and (2) yield

(69) |T^{2/3}(X)_{tors}| ≤ \begin{cases}2 & if G (or H) = 2, |F_{2}| = 2, |E_{2}| = 2, \\ 2^{2} & if G (or H) = 2, |F_{2}| = 2, |E_{2}| = 1 or G (or H) = 1, \\ 2^{7} & if G (or H) = 2, 0 ≤ |F_{2}| ≤ 2, |E_{2}| = 3. \end{cases}

Case: |F_{1}| ≥ 2. Then, we have |F_{1}| = |E_{2}| = 3, d = 2, and g ∈ {1, 2}. It follows from Lemma 6.6 (3) that y_{pr} + y_{qs} − (y_{pq} + y_{rs}) ∈ T^{2}(X) for all {p, q} ≠ {r, s} ∈ J. Since we have 2y_{pq} ∈ T^{2}(X) and (y_{p} + y_{q})(y_{r} + y_{s}) ∈ Im(res^{2/3}) for all {p, q} ≠ {r, s} ∈ J, we obtain α_{2} ≤ 2^{3}.

As d = 2 and ∑ y_{prt} + y_{qsu} ∈ Im(res^{3/4}), where the sum ranges over all such elements in the basis, it follows from Lemma 6.6 (3) that α_{3} ≤ 2^{6}. Similarly, by Lemma 6.6 (3) we have α_{4} ≤ 2^{5}. As d = 2, we obtain α_{5} ≤ 2^{3} and α_{6} ≤ 2. In conclusion, we have

(70)  |⊕ T^{n/n+1}(X)_{tors}| ≤ 2^{7}/g^{4}

for 1 ≤ g ≤ 2.

The result follows from (67), (68), (69), and (70). □

Lemma 6.6. With the above notation, the followings hold:

(1) If f_{pq} = e_{rs} (resp. f_{pq} = e_{tu}), then \{ \begin{matrix}e_{rs}y_{rstu} (resp. e_{tu}y_{rstu}) ∈ T^{3}(X), \\ e_{rs}y_{rstu}z_{pq} (resp. e_{tu}y_{rstu}z_{pq}) ∈ T^{4}(X), \\ e_{pq}e_{rs} · y_{123456}(resp. e_{pq}e_{tu}y_{123456}) ∈ T^{5}(X). \end{matrix}

Moreover, if in addition e_{rs}y_{rstu} (resp. e_{tu}y_{rstu}) ∉ T^{4}(X), then we have a subgroup ⟨e_{rs}y_{rstu} (resp. e_{tu}y_{rstu})⟩ ⊆ T^{3/4}(X)_{tors} of order e_{tu} (resp. e_{rs}). If in addition e_{rs}y_{rstu}z_{pq} (resp. e_{tu}y_{rstu}z_{pq}) ∉ T^{5}(X) and e_{tu}e_{rs}y_{123456} ∈ T^{5}(X), then we obtain a subgroup ⟨e_{rs}y_{rstu} · (y_{p} + y_{q}) (resp. e_{tu}y_{rstu}(y_{p} + y_{q}))⟩ ⊆ T^{4/5}(X)_{tors} of order e_{tu} (resp. e_{rs}). If in addition e_{pq}e_{rs}y_{123456} (resp. e_{pq}e_{tu}y_{123456}) ∉ T^{6}(X), then we have a subgroup ⟨e_{pq}e_{rs}y_{123456} (resp. e_{pq}e_{tu}y_{123456})⟩ ⊆ T^{5/6}(X)_{tors} of order e_{tu} (resp. e_{rs}).

(2) If g = 1, then \{ \begin{matrix}(y_{pr} + y_{qs}) + (y_{pt} + y_{qu}) + (y_{rt} + y_{su}) − (y_{pq} + y_{rs} + y_{tu}) ∈ Im(res^{2/3}), \\ (y_{r} + y_{s})(y_{pt} + y_{qu} − y_{pq} − y_{tu}) + y_{rs}(y_{p} + y_{q} + y_{t} + y_{u}) ∈ Im(res^{3/4}), \\ (y_{p} + y_{q})(y_{rt} + y_{su} − y_{rs} − y_{tu}) + y_{pq}(y_{r} + y_{s} + y_{t} + y_{u}) ∈ Im(res^{3/4}). \end{matrix}

(3) If f_{tu} = 1, then \{ \begin{matrix}y_{pr} + y_{qs} − (y_{pq} + y_{rs}) ∈ T^{2}(X), \\ y_{pq}(y_{r} + y_{s}) − y_{rs}(y_{p} + y_{q}) ∈ Im(res^{3/4}), \\ y_{pq}(y_{rt} + y_{su} + y_{ru} + y_{st}) − y_{rs}(y_{pt} + y_{qu} + y_{pu} + y_{qt}) ∈ Im(res^{4/5}), \\ y_{pqtu} + y_{rstu} + y_{tu}(y_{pr} + y_{qs}) + y_{tu}(y_{ps} + y_{qr}) ∈ Im(res^{4/5}). \end{matrix}

Proof. (1) For simplicity, we give the proof for the case of f_{pq} = e_{rs}. In this case, we have

e_{rs}y_{rs} · (y_{tu} + y_{t} + y_{u}) − (y_{rs} + y_{r} + y_{s}) · f_{pq}(y_{rt} + y_{su}) = e_{rs}y_{rstu} ∈ T^{3}(X).

It follows from the previous result that e_{rs}y_{rstu} · z_{pq} ∈ T^{4}(X) and e_{pq}y_{pq} · e_{rs}y_{rstu} ∈ T^{5}(X).

Since we have e_{tu}y_{tu} · e_{rs}y_{rs} ∈ T^{4}(X), e_{rs}y_{rs} · e_{tu}y_{tu} · z_{pq} − e_{tu}e_{rs}y_{123456} ∈ T^{5}(X) and e_{pq}y_{pq} · e_{rs}y_{rs} · e_{tu}y_{tu} ∈ T^{6}(X), the second statement immediately follows.

(2) If g = 1, then z_{prt} + z_{qsu} − (z_{pq} + z_{rs} + z_{tu}) ∈ T^{2}(X), which implies the first result. As y_{r} + y_{s}, y_{p} + y_{q} ∈ Im(res^{1/2}), the remaining results follow by multiplication the first result by these elements.

(3) Since f_{tu} = 1, we have y_{pr} + y_{qs} − (y_{pq} + y_{rs}) = z_{pr} + z_{qs} − (z_{pq} + z_{rs}) ∈ T^{2}(X). As y_{p} +y_{q} ∈ Im(res^{1/2}), the second and third results follow from (y_{p} +y_{q})(y_{pr} +y_{qs} −y_{pq} −y_{rs}) ∈ Im(res^{3/4}) and (y_{t} + y_{u})[y_{pq}(y_{r} + y_{s}) − y_{rs}(y_{p} + y_{q})] ∈ Im(res^{4/5}), respectively.

The assumption implies that (Q_{max{p,q}/2})_{L} ≃ (Q_{max{r,s}/2})_{L} =: Q_{L}. Hence, the last one follows from the closed embedding R_{L/F} (SB(Q))×R_{L/F} (SB(Q_{max{t,u}/2})) ↪ R_{L/F} (SB(Q))× R_{L/F} (SB(Q)) × R_{L/F} (SB(Q_{max{t,u}/2})). □

Acknowledgments. The author would like to thank Nikita Karpenko for helpful discussion. This work was partially supported by an internal fund from KAIST, TJ Park Junior Faculty Fellowship of POSCO TJ Park Foundation, and National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2013R1A1A1010171).

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Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea

  E-mail address: sanghoonbaek@kaist.ac.kr