CHOW GROUPS OF PRODUCTS OF SEVERI-BRAUER VARIETIES AND INVARIANTS OF DEGREE 3

SANGHOON BAEK

Abstract. We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension 2 Chow groups of a product of Severi-Brauer varieties. In particular, for any n ≥ 2 we completely determine the degree 3 invariants of a split semisimple group, the quotient of (SL_{2})^{n} by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants of a split semisimple group of type A.

1. Introduction

Let G be a linear algebraic group over a field F. The notion of a cohomological invariant of G was introduced by Serre [6] as follows. Consider the functor G -torsors taking a field K/F to the set G -torsors(K) of isomorphism classes of G-torsors over Spec(K). A degree d (cohomological) invariant of G with values in a cohomology group H^{d}(K) := H^{d}(K, C) for some Galois module C is a morphism of functors G -torsors → H^{d}. All the invariants of degree d of G form a group Inv^{d}(G). We refer to [6] and [5] for various examples and general discussion of invariants.

In this paper, we work with the Galois module C = ℚ/ℤ(d − 1), i.e., the Galois cohomology functor H^{d}(−) := H^{d}(−, ℚ/ℤ(d − 1)), where ℚ/ℤ(d − 1) := ⨁_{p} \underrightarrow{lim}_{n} μ_{p^{n}}^{⊗d−1} if char(F) ≠ p and ⨁_{p} \underrightarrow{lim}_{n} W_{n}Ω_{log}^{d−1}[1 − d] otherwise (W_{n}Ω_{log}^{d−1} is the logarithmic de Rham-Witt sheaf). In particular, one has H^{2}(K) = Br(K).

From now on we focus on degree 3 invariants. The group Inv^{3}(G) was determined by Rost in the case where G is simply connected quasi-simple [6]. In [7] Mekurjev constructed an exact sequence which generalize Rost’s result to an arbitrary semisimple group. In particular, when G is an adjoint group of inner type the group Inv^{3}(G) was computed by means of decomposable and indecomposable invariants.

Recently, a new type of degree 3 invariant of a split semisimple group, called a semidecomposable invariant was introduced by Merkurjev-Neshitov-Zainoulline [9]. This invariant is locally decomposable, so any decomposable invariant is semi-decomposable. In their paper, authors established a relation between nontrivial semi-decomposable invariants and the torsion in the codimension 2 Chow groups of a generic flag variety and showed that there is no nontrivial semi-decomposable invariant of a split simple group in [9, Theorem].

The first and, so far, the only example of nontrivial semi-decomposable invariants comes from a semisimple group SO_{4} ≃ (SL_{2} × SL_{2})/μ, where μ = {(λ_{1}, λ_{2}) ∈ μ_{2} × μ_{2} | λ_{1}λ_{2} = 1}; see [9, Example 3.1]. Indeed, this invariant is given by ϕ := a⟨⟨b, c⟩⟩ ↦ (a) ∪ [b, c], where ϕ is a 4-dimensional quadratic form with trivial discriminant over a field extension K/F and [b, c] is the class of a quaternion algebra in the Brauer group Br(K); see [6, Example 20.3]. Since the class [b, c] is the image of ϕ under the Hasse-Witt invariant, the invariant is semi-decomposable. On the other hand, it is not decomposable as it is nontrivial over an algebraic closure of F.

In the present paper, we completely determine the degree 3 invariants of split semisimple groups (SL_{2})^{n}/μ for all n ≥ 2 (see Theorem below), which generalize the above example (observe that there is no indecomposable degree 3 invariant of SO_{n} for n ≥ 5 [6, §19, 20]). Furthermore, we also determine the degree 3 invariants of split reductive groups (GL_{2})^{n}/μ for any n ≥ 2 and provide an explicit description of all semi-decomposable invariants. Namely, our main result reads as follows.

Theorem. Let μ = {(λ_{1}, …, λ_{n}) ∈ (μ_{2})^{n} | λ_{1} ⋯ λ_{n} = 1} be a central subgroup of (GL_{2})^{n} over a field F and let G = (GL_{2})^{n}/μ and H = (SL_{2})^{n}/μ be factor groups. Then, for any n ≥ 2 the group Inv^{3}(H)_{ind} of indecomposable invariants of H is cyclic of order 2 containing the subgroups Inv^{3}(G)_{ind} = ℤ/2ℤ if n ≥ 3 and 0 otherwise,

\frac{Sdec(H)}{Dec(H)} = \begin{cases}ℤ/2ℤ & if 2 ≤ n ≤ 4, \\ 0 & otherwise, \end{cases} and \frac{Sdec(G)}{Dec(G)} = \begin{cases}ℤ/2ℤ & if 3 ≤ n ≤ 4, \\ 0 & otherwise, \end{cases}

i.e., the semi-decomposable invariants of H (respectively, G) coincide with the decomposable invariants if n ≥ 5 (respectively, n = 2 or n ≥ 5). Moreover, if char(F) ≠ 2, then the semi-decomposable invariant of G (but not decomposable) is given by

\begin{cases}(Q_{1}, Q_{2}, Q_{3}) ↦ [Q_{1}] ∪ (a) + [Q_{2}] ∪ (−1) & if n = 3, \\ (Q_{1}, Q_{2}, Q_{3}, Q_{4}) ↦ [Q_{1}] ∪ (b) + [Q_{2}] ∪ (−b) + [Q_{3}] ∪ (−1) & if n = 4, \end{cases}

where Q_{1}, …, Q_{4} are quaternions over a field extension K/F and a, b ∈ K^{×}.

In the proof of the Theorem we compute the torsion in codimension 2 cycles of a product of Severi-Brauer varieties associated to quaternion algebras and relate it to the torsion of the generic variety of (SL_{2})^{n}/μ. We then combine it with the group of indecomposable invariants of (SL_{2})^{n}/μ and (GL_{2})^{n}/μ using the short exact sequence in [9, Theorem] and its extension to reductive groups (Proposition 2.1).

We would like to emphasize that our method developed in the proof of the theorem can be applied to find nontrivial semi-decomposable invariants of an arbitrary split semisimple group of type A: In the last section, using the same methods developed so far, we present an example (Proposition 7.4) of a semisimple group which has a nontrivial semi-decomposable invariant.

CHOW GROUPS OF PRODUCTS OF SEVERI-BRAUER VARIETIES AND INVARIANTS 3

The paper is organized as follows. In Section 2, we extend the notion of semidecomposable invariants of split semisimple groups to the class of split reductive groups and provide its connection to the invariants of semisimple groups. In Sections 3 and 4, we determine the indecomposable invariants of our groups in Theorem and find all their semi-decomposable invariants. In Section 5, we compute the torsion in CH^{2} on the product of Severi-Brauer varieties associated to the generic variety. In Section 6, we combine all the results from the previous sections and prove our main result. In the last section, we discuss applications of our method.

2. Semi-decomposable invariants of a split reductive group

Let G be a linear algebraic group over a field F. An invariant α ∈ Inv^{d}(G) is called normalized if α(Z) = 0 for the trivial G-torsor Z, and we write Inv^{d}(G)_{norm} for the subgroup of normalized invariants in Inv^{d}(G). Then, we have an isomorphism Inv^{d}(G) ≃ Inv^{d}(G)_{norm} ⊕ H^{d}(F).

If d = 1, then Inv^{1}(G)_{norm} is trivial for a connected group G. For d = 2, it was shown in [4, Theorem 2.4] that the group Inv^{2}(G)_{norm} is isomorphic to the Picard group Pic(G) for a smooth connected group G. In particular, for a split reductive group G and its derived subgroup H one has

(1)  Inv^{2}(G)_{norm} ≃ Inv^{2}(H)_{norm} ≃ M^{∗}, thus Inv^{2}(G) ≃ Inv^{2}(H),

where M^{∗} is the character group of the kernel of the universal covering morphism of H (see [8, Example 2.1]). Indeed, the isomorphism M^{∗} →^{∼} Inv^{2}(H)_{norm} is given by the composition of the connecting morphism δ with the induced morphism χ_{∗} : H^{2}(K, M) → H^{2}(K, 𝔾_{m}) = Br(K):

(2)  H -torsors(K) →^{δ} H^{2}(K, M) →^{χ_{∗}} Br(K)

for each character χ ∈ M^{∗}(K) over a field extension K/F.

For d = 3, the group Inv^{3}(G)_{norm} is finite cyclic with canonical generator (called the Rost invariant) if G is simply connected quasi-simple [6, Theorem 9.11]. For an arbitrary semisimple group G, the group Inv^{3}(G)_{norm} is determined by means of an exact sequence of five terms [7, Theorem]. In particular, it was calculated for all adjoint groups of inner type [7] and for the remaining split simple groups [3].

Let G be a split reductive group over F. A normalized invariant α ∈ Inv^{3}(G)_{norm} is called decomposable if it is given by a cup product of an invariant in degree 2 with a constant invariant in degree 1 [7, §3]. The subgroup of decomposable invariants will be denoted by Dec(G). Observe that α ∈ Dec(G) if and only if αF̄ is trivial. We shall write Inv^{3}(G)_{ind} = Inv^{3}(G)_{norm}/ Dec(G) and call Inv^{3}(G)_{ind} the indecomposable invariants [7, §3].

Let H be a split semisimple group over F. In [9] a semi-decomposable invariant of H was introduced. We write Sdec(H) for the subgroup of semi-decomposable invaraints. This invariant can be viewed as a locally decomposable invariant so that Dec(H) ⊆ Sdec(H) ⊆ Inv^{3}(H)_{norm}.

Consider a generically free representation V of H, i.e., there exists a nonempty H-invariant open subset U ⊆ V such that U → U/H is a H-torsor. Then, the generic fiber Ū → Spec F (U/H) of U → U/H is a versal H-torsor [6]. Let B be a Borel subgroup of H and let X̄ be the associated flag variety Ū/B over Spec F (U/H), which is called the generic variety of H [9]. Then, it was shown in [9, Theorem] that there is an exact sequence

(3)  0 → \frac{Sdec(H)}{Dec(H)} → Inv^{3}(H)_{ind} → CH^{2}(X̄)_{tors} → 0.

In particular, it was proved that Dec(H) = Sdec(H) if H is simple.

We extend the definition of semi-decomposable invariants of split semisimple groups [9, Definition 2.9] to the class of split reductive groups: let G be a split reductive group. A normalized invariant α of degree 3 of G is semi-decomposable if for any field extension K/F and any Z ∈ G -torsors(K) there exist β_{i} ∈ Inv^{2}(G)_{norm} and x_{i} ∈ K^{×} such that

α(Z) = ∑_{i} β_{i}(Z) ∪ (x_{i}).

We extend the exact sequence (3) to the class of split reductive groups:

Proposition 2.1. Let G be a split reductive group over a field F and let H be its (semisimple) derived subgroup. Then, there is a commutative diagram

[figure: 0; \frac{Sdec(H)}{Dec(H)}; Inv^{3}(H)_{ind}; CH^{2}(X̄)_{tors}; 0; 0; \frac{Sdec(G)}{Dec(G)}; Inv^{3}(G)_{ind}; CH^{2}(X̄)_{tors}]

with the exact rows, where X̄ is the generic variety associated to H.

Proof. Consider the exact sequence 1 → H ⟶^{ϕ} G → T → 1, where T is a split torus induced by the embedding ϕ. Then, for a field extension K/F this yields a surjective morphism

(4) ϕ_{∗} : H -torsors(K) → G -torsors(K) given by Z ↦ ϕ_{∗}(Z) := (Z × G)/H,

which yields an injective morphism

ϕ̄: Inv^{d}(G) → Inv^{d}(H) given by ϕ̄(α)(Z) := α(ϕ_{∗}(Z))

for any degree d. Hence, we have the injectivity of the middle column (see [8, Proposition 6.1]).

It remains to show that Sdec(G) = Sdec(H) ∩ Inv^{3}(G). Let α ∈ Sdec(G). Then, for Z ∈ H -torsors(K) there exist β_{i} ∈ Inv^{2}(G)_{norm} and x_{i} ∈ K^{×} such that

ϕ̄(α)(Z) = α(ϕ_{∗}(Z)) = ∑_{i} β_{i}(ϕ_{∗}(Z)) ∪ (x_{i}) = ∑_{i} ϕ̄(β_{i})(Z) ∪ (x_{i}),

thus ϕ̄(α) ∈ Sdec(H). Conversely, let α ∈ Inv^{3}(G) such that ϕ̄(α) ∈ Sdec(H). It follows from (4) that for Y ∈ G -torsors(K) there exists Z ∈ H -torsors(K) such that ϕ_{∗}(Z) = Y. Therefore, we have

(5)  α(Y) = ϕ̄(α)(Z) = ∑_{i} β_{i}(Z) ∪ (x_{i})

for some β_{i} ∈ Inv^{2}(G)_{norm} and x_{i} ∈ K^{×}. Since ϕ̄ is an isomorphism by (1), there exists γ_{i} ∈ Inv^{2}(G) such that ϕ̄(γ_{i}) = β_{i}, thus the rightmost term in (5) equals to ∑_{i} γ_{i}(Y) ∪ (x_{i}). Hence, α ∈ Sdec(G). □

Remark 2.2. In general, the map Inv^{3}(G)_{ind} → CH^{2}(X̄)_{tors} in the bottom row of Proposition 2.1 is not necessarily surjective. Consder the reductive group G = GL_{8} /μ_{2} and its derived group H = SL_{8} /μ_{2}. It follows from [2, Theorem 2.8] that Inv^{3}(G)_{ind} = 0. On the other hand, by [12, Example 4.15] we have CH^{2}(X)_{tors} = ℤ/2ℤ. Note that by [3, Theorem 4.1] we also have Inv^{3}(H)_{ind} = ℤ/2ℤ.

3. Degree 3 invariants of (GL_{2})^{n}/μ

In the present section, we give an explicit description of semi-decomposable invariants of (GL_{2})^{n}/μ for n = 3, 4, which in turn are all nontrivial semi-decomposable invariants of (GL_{2})^{n}/μ for any n ≥ 2.

For n ≥ 2, consider a central subgroup μ = {(λ_{1}, …, λ_{n}) ∈ (μ_{2})^{n} | λ_{1} ⋯ λ_{n} = 1} of (GL_{2})^{n}. An element of (GL_{2})^{n}/μ -torsors(K), where K/F is a field extension, is the class of n−tuples (Q_{1}, …, Q_{n}) of quaternions over K satisfying the relation [Q_{1}] + ⋯ + [Q_{n}] = 0 in Br(K).

The reduced norm form N_{Q} = ⟨⟨a, b⟩⟩ of a quaternion algebra Q = (a, b) over K is in I^{2}(K) and the image of N_{Q} under e_{2} : I^{2}(K) → Br_{2}(K) is the class [Q]. Therefore, the sum of norm forms ∑_{i=1}^{n} N_{Q_{i}} of an element (Q_{1}, …, Q_{n}) of (GL_{2})^{n}/μ -torsors(K) is contained in I^{3}(K) = Ker(e_{2}).

Let α_{n} be a degree 3 invariant of (GL_{2})^{n}/μ defined by

(6)  (Q_{1}, …, Q_{n}) ↦ e_{3}(∑_{i=1}^{n} N_{Q_{i}}),

where e_{3} : I^{3}(K) → H^{3}(K) is the Arason invariant for a field extension K/F (see [8, Example 11.2]). If n = 2, then any element (Q_{1}, Q_{2}) ∈ (GL_{2})^{2}/μ -torsors(K) satisfies Q_{1} ≃ Q_{2}, thus the invariant α_{2}(Q_{1}, Q_{2}) = [Q_{1}] ∪ (−1) in H^{3}(K) is decomposable as it is trivial over an algebraic closure K̄ of K.

Lemma 3.1. Let (Q_{1}, Q_{2}, Q_{3}) ∈ (GL_{2})^{3}/μ -torsors(K). Then, the invariant α_{3}, which can be written explicitly as (Q_{1}, Q_{2}, Q_{3}) ↦ [Q_{1}] ∪ (q) + [Q_{2}] ∪ (−1) for some q ∈ K^{×}, is semi-decomposable but not decomposable.

Proof. First, observe that in the Witt ring W (K) one has

(7)  ⟨⟨x, y⟩⟩ + ⟨⟨x, z⟩⟩ = ⟨⟨x, y, z⟩⟩ + ⟨⟨x, yz⟩⟩

for x, y, z ∈ K^{×}.

Since Q_{1} ⊗ Q_{2} is not a division algebra, they have a common splitting quadratic extension. Therefore, Q_{1} = (a, b) and Q_{2} = (a, c) for some a, b, c ∈ K^{×}. It follows from the relation [Q_{1}] + [Q_{2}] + [Q_{3}] = 0 that Q_{3} = (a, bc).

Applying (7) to the sum of norm forms, one obtain

∑_{i=1}^{3} N_{Q_{i}} = ⟨⟨a, b⟩⟩ + ⟨⟨a, c⟩⟩ + ⟨⟨a, bc⟩⟩ = ⟨⟨a, b, c⟩⟩ + ⟨⟨a, bc⟩⟩ + ⟨⟨a, bc⟩⟩

in W (K). As 2 · ⟨⟨a, bc⟩⟩ = ⟨⟨a, bc, −1⟩⟩, we have

α_{3}(Q_{1}, Q_{2}, Q_{3}) = [Q_{1}] ∪ (−c) + [Q_{2}] ∪ (−1) in H^{3}(K),

which is not decomposable as it is nontrivial over K̄. Obviously, this invariant is semi-decomposable. □

Remark 3.2. Lemma 3.1 shows that the invariant α_{n} is nontrivial for n ≥ 3.

Proposition 3.3. Let char(F) ≠ 2. The invariant α_{4} of (GL_{2})^{4}/μ, which can be written explicitly as (Q_{1}, Q_{2}, Q_{3}, Q_{4}) ↦ [Q_{1}] ∪ (q) + [Q_{2}] ∪ (−q) + [Q_{3}] ∪ (−1) for some q ∈ K^{×}, is semi-decomposable but not decomposable.

Proof. Let Q_{1} = (a, b), Q_{2} = (c, d), Q_{3} = (e, f), Q_{4} = (g, h) be quaternions over K such that [Q_{1}] + [Q_{2}] = [Q_{3}] + [Q_{4}] in Br(K). Then, by the chain lemma for biquaternion algebras [14, Proposition 1] one can find x, y, z ∈ K such that

(8)  [ac, x] = [eg, y] = [ac, z] = 0

and

[Q′_{3}] = [Q′_{1}] + [a, z], [Q′_{4}] = [Q′_{2}] + [c, z]

in Br(K), where Q′_{1} = (a, bx), Q′_{2} = (c, dx), Q′_{3} = (e, fy), Q′_{4} = (g, hy) are quaternion algebras over K. Hence,

(9)  [Q_{1}] + [Q_{2}] = [Q′_{1}] + [Q′_{2}] and [Q_{3}] + [Q_{4}] = [Q′_{3}] + [Q′_{4}] in Br(K).

Applying (7), we have

N_{Q_{1}} + N_{Q_{2}} + N_{Q′_{1}} + N_{Q′_{2}} = ⟨⟨a, b⟩⟩ + ⟨⟨c, d⟩⟩ + ⟨⟨a, bx⟩⟩ + ⟨⟨c, dx⟩⟩
= ⟨⟨a, x⟩⟩ + ⟨⟨a, b, bx⟩⟩ + ⟨⟨c, x⟩⟩ + ⟨⟨c, d, dx⟩⟩

= ⟨⟨x, ac⟩⟩ + ⟨⟨a, c, x⟩⟩ + ⟨⟨a, b, bx⟩⟩ + ⟨⟨c, d, dx⟩⟩ (10) in W (K). By using the relations (8),

⟨⟨a, b, bx⟩⟩ = ⟨⟨a, b, x⟩⟩ + ⟨⟨a, b, −1⟩⟩, and ⟨⟨c, d, dx⟩⟩ = ⟨⟨c, d, −x⟩⟩,

we see that the sum (10) equals

(11)  ⟨⟨a, b, x⟩⟩ + ⟨⟨c, d, −x⟩⟩ + ⟨⟨a, c, x⟩⟩ + ⟨⟨a, b, −1⟩⟩

in W (K). By (8) and ⟨⟨a, a⟩⟩ = ⟨⟨a, −1⟩⟩, one has ⟨⟨a, c, x⟩⟩ = ⟨⟨a, x, −1⟩⟩ thus by applying (7) one obtain

(12)  ⟨⟨a, c, x⟩⟩ + ⟨⟨a, b, −1⟩⟩ = ⟨⟨a, bx, −1⟩⟩ + ⟨⟨a, b, x, −1⟩⟩.

in W (K). It follows from (11) and (12) that

(13)  α_{4}(Q_{1}, Q_{2}, Q′_{1}, Q′_{2}) = [Q_{1}] ∪ (x) + [Q_{2}] ∪ (−x) + [Q′_{1}] ∪ (−1)

in H^{3}(K) as a 4-fold Pfister form vanishes under e_{3}.

Similarly, one has

(14)  α_{4}(Q′_{1}, Q′_{2}, Q′_{3}, Q′_{4}) = [Q′_{1}] ∪ (z) + [Q′_{2}] ∪ (−z) + [Q′_{3}] ∪ (−1)

and

(15)  α_{4}(Q_{3}, Q_{4}, Q′_{3}, Q′_{4}) = [Q_{3}] ∪ (y) + [Q_{4}] ∪ (−y) + [Q′_{3}] ∪ (−1)

in H^{3}(K).

Combining (13) with (14) and (15) and using the relation (9) we obtain

α_{4}(Q_{1}, Q_{2}, Q_{3}, Q_{4}) = α_{4}(Q_{1}, Q_{2}, Q′_{1}, Q′_{2}) + α_{4}(Q′_{1}, Q′_{2}, Q′_{3}, Q′_{4}) + α_{4}(Q_{3}, Q_{4}, Q′_{3}, Q′_{4})
= [Q_{1}] ∪ (x) + [Q_{2}] ∪ (−x) + ([Q_{1}] + [Q_{2}]) ∪ (−z) + [Q′_{3}] ∪ (1)

+ [Q_{3}] ∪ (y) + [Q_{4}] ∪ (−y) (16)

in H^{3}(K). Since [Q′_{3}] ∪ (1) = 0 in H^{3}(K) and [Q_{4}] = [Q_{1}] + [Q_{2}] + [Q_{3}] in Br(K), it follows from (16) that

α_{4}(Q_{1}, Q_{2}, Q_{3}, Q_{4}) = [Q_{1}] ∪ (xyz) + [Q_{2}] ∪ (−xyz) + [Q_{3}] ∪ (−1) in H^{3}(K),

which is obviously semi-decomposable. This invariant is not decomposable as it is nontrivial over K̄. □

Corollary 3.4. Assume that F is an algebraically closed field. Then, the invariant α_{4} vanishes over a field extension K/F if and only if the element q ∈ K^{×} is contained in the group of similarity factors of an Albert form associated to Q_{1} ⊗ Q_{2}.

Proof. It follows from Proposition 3.3 that α_{4}(Q_{1}, Q_{2}, Q_{3}, Q_{4}) = ([Q_{1}]+[Q_{2}])∪(q). Let ϕ = ⟨a, b, −ab, −c, −d, cd⟩ be the Albert form corresponding to Q_{1}⊗Q_{2} for Q_{1} = (a, b) and Q_{2} = (c, d). By [13, Theorem], α_{4} vanishes if and only if ⟨q⟩ · ϕ ≃ ϕ. □

4. Degree 3 invariants of (SL_{2})^{n}/μ

In the present section, we determine the indecomposable invariants of (SL_{2})^{n}/μ for all n ≥ 2. For n = 2, we have (SL_{2} × SL_{4})/μ ≃ SO_{4}, thus it follows by [6, Example 20.3] that Inv^{3}((SL_{2} × SL_{2})/μ)_{ind} = ℤ/2ℤ.

Proposition 4.1. For any n ≥ 2, we have Inv^{3}((SL_{2})^{n}/μ)_{ind} = ℤ/2ℤ.

Proof. Let G = (GL_{2})^{n}/μ and μ = {(λ_{1}, …, λ_{n}) ∈ (μ_{2})^{n} | λ_{1} ⋯ λ_{n} = 1} ⊂ G. Consider the commutative diagram of exact sequences

(17)  [figure: 1; μ; (μ_{2})^{n}; μ_{2}; 1; 1; μ; T̄_{G}; T_{G}; 1]

where T̄_{G} (respectively, T_{G}) is a split maximal torus of (GL_{2})^{n} (respectively, G). This yields the commutative diagram of the corresponding character groups

(18)  [figure: 0; ℤ/2ℤ; (ℤ/2ℤ)^{n}; μ^{∗}; 0; 0; T_{G}^{∗}; ℤ^{2n}; _{μ_{∗}}; 0]

where the map ℤ^{2n} → (ℤ/2ℤ)^{n} is given by (a_{1}, b_{1}, …, a_{n}, b_{n}) ↦ (\overline{a_{1} + b_{1}}, …, \overline{a_{n} + b_{n}}). It follows from (18) that T_{G}^{∗} = \{∑ a_{i}x_{i} +b_{i}y_{i} | \overline{a_{i} + b_{i}} = \overline{a_{j} + b_{j}} for all 1 ≤ i ≠ j ≤ n\}, where {x_{i}, y_{i}} is a standard basis of ℤ^{2n}. Hence, we have the following ℤ-basis of T_{G}^{∗}

(19)  {x_{i} − y_{i}, 2x_{k}, ∑_{i} x_{i} | 1 ≤ i ≤ n, 1 ≤ k ≤ n − 1}.

Let H = (SL_{2})^{n}/μ. We write T̄_{H} (respectively, T_{H}) for a split maximal torus of (SL_{2})^{n} (respectively, H). Then, we have the following commutative diagram of the character groups

(20)  [figure: 0; T_{H}^{∗}; ℤ^{n}; μ^{∗}; 0; 0; T_{G}^{∗}; ℤ^{2n}; _{μ_{∗}}; 0]

where the middle map ℤ^{2n} → (ℤ)^{n} is defined by x_{i} ↦ x̄_{i} and y_{i} ↦ −x̄_{i}. Hence, by (19) we obtain the following ℤ-basis of T_{H}^{∗}

(21)  {2x̄_{k}, ∑_{i} x̄_{i} | 1 ≤ i ≤ n, 1 ≤ k ≤ n − 1}.

Let W = (S_{2})^{n} be the Weyl group of H, which acts on x̄_{i} by x̄_{i} ↦ −x̄_{i}. Let ϕ be a quadratic form on (21) over ℤ, i.e., ϕ ∈ S^{2}(T_{H}^{∗})^{W}. Since the group S^{2}(T̄_{H}^{∗})^{W} is generated by {−x̄_{1}^{2}, …, −x̄_{n}^{2}}, there exist d_{i} ∈ ℤ for 1 ≤ i ≤ n such that

ϕ(z_{1}, …, z_{n}) = ∑_{k} d_{k}(\frac{z_{k}}{2} )^{2} + d_{n}(z_{n} − ∑_{k} \frac{z_{k}}{2} )^{2},

where z_{k} = 2x̄_{k} for 1 ≤ k ≤ n − 1 and z_{n} = ∑_{i} x̄_{i}. Thus, we have

(22)  \begin{cases}d_{1} + d_{2} ≡ 0 mod 4 & if n = 2, \\ d_{i} ≡ 0 mod 2 and d_{i} ≡ d_{j} mod 4 & if n ≥ 3 \end{cases}

for all 1 ≤ i ≠ j ≤ n. Hence, if n ≥ 3, then the group S^{2}(T_{H}^{∗})^{W} is generated by

(23)  {4x̄_{k}^{2}, 2 ∑_{i} x̄_{i}^{2}}.

Let c_{2} : ℤ[T_{H}^{∗}]^{W} → S^{2}(T_{H}^{∗})^{W} be the Chern class map defined in [7, 3c]. Then, it follows from (21) that the image of c_{2} is generated by c_{2}(e^{2x̄_{i}} + e^{−2x̄_{i}}) for 1 ≤ i ≤ n and c_{2}(∏_{i}(ex̄_{i} + e^{−x̄_{i}})), i.e.,

(24)  {4x̄_{i}^{2}, 2^{n−1} ∑_{i} x̄_{i}^{2}}.

Therefore, it follows by [7, Theorem], (22), (23), and (24) that the indecomposable group of H is isomorphic to ℤ/2ℤ for any n ≥ 2. □

5. Torsion in CH^{2} of generic variety associated to (SL_{2})^{4}/μ

In this section, we determine the torsion in codimension 2 cycles of the generic variety of (SL_{2})^{n}/μ for n = 3, 4 by using the filtrations on the Grothendieck ring. It turns out that the result for n = 4 suffices to determine the torsion for all n ≥ 4 (see Remark 6.1).

Let X be a smooth projective homogeneous variety over a field F and consider the Grothendieck ring K_{0}(X) of X. We set Γ^{0}(X) = K_{0}(X) and Γ^{1}(X) = Ker(rank : K_{0}(X) → ℤ). The gamma filtration

K_{0}(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_{i} ∈ Γ^{1}(X) and d_{1} + ⋯ + d_{i} ≥ d, where γ_{d_{j}} is the gamma operation on K_{0}(X). For example, we have γ_{0}(x) = 1 and γ_{1}(x) = x, where x ∈ K_{0}(X). Indeed, the gamma operation defines the Chern class c_{i}(x) := γ_{i}(x − rank(x)) with values in K_{0}.

For any d ≥ 0, we write Γ^{d/d+1}(X) for the subsequent quotient Γ^{d}(X)/Γ^{d+1}(X). Let E be a splitting field of X. By a diagram chasing in the diagram involving 0 → Γ^{d+1}(X) → Γ^{d}(X) → Γ^{d/d+1}(X) → 0 and the one over E, one has the following formula [11, Proposition 2]:

(25) |⊕ Γ^{d/d+1}(X)_{tors}| · |K_{0}(X_{E})/K_{0}(X)| = ∏_{d=1}^{dim(X)} |Γ^{d/d+1}(X_{E})/ Im(res^{d/d+1})|,

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

We shall write T^{d}(X) (respectively, T^{d/d+1}(X)) for the topological filtration of degree d given by the ideal generated by the structure sheaf of a closed subvariety of X of codimension at least d (respectively, the quotient T^{d}(X)/ T^{d+1}(X)). For 1 ≤ d ≤ 2, we have Γ^{d}(X) = T^{d}(X). Moreover, Γ^{2/3}(X) ↠ T^{2/3}(X) = CH^{2}(X) as Γ^{3}(X) ⊆ T^{3}(X).

Now we consider the product of Severi-Brauer varieties X = ∏_{i=1}^{n} SB(A_{i}) of a central simple F -algebra A_{i} of degree d_{i}. Let x_{i} be the pullback of the class of the tautological line bundle on ℙ_{E}^{d_{i}−1} over a splitting field E. Then, by [15, §8 Theorem 4.1] the image of the restriction homomorphism K_{0}(X) → K_{0}(∏_{i=1}^{n} ℙ_{E}^{d_{i}−1}) ≃ ℤ[x_{1}, ⋯, x_{n}]/((x_{1} − 1)^{d_{1}}, ⋯, (x_{n} − 1)^{d_{n}}) is generated by

(26)  {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′ = ∏_{i=1}^{n′} SB(A′_{i}) be another product of Severi-Brauer varieties of simple F - algebras A′_{i} such that ⟨A′_{1}, …, A′_{n′}⟩ = ⟨A_{1}, …, A_{n}⟩ in the Brauer group Br(F). Then, by applying projective bundle theorem one has (see [10, Proposition 4.7])

(27)  CH^{2}(X)_{tors} ≃ CH^{2}(X′)_{tors}.

Let H be a split semisimple group over F and let B be a Borel subgroup of H. We denote by X̄ be the generic variety of H as in Section 2. By the localization sequence, the induced pullback of X̄ ↪ U/B gives

CH(U/T) ≃ CH(U/B) ↠ CH(X̄),

where T is a split maximal torus of H containing B. Since the characteristic map S(T^{∗}) → CH(U/T) is an isomorphism, where S(T^{∗}) denotes the symmetric algebra of the character group T^{∗}, the Chow group CH(U/T) is generated by Chern classes of line bundles, thus CH(X̄) is also generated by Chern classes. Therefore, it follows from the commutativity of the diagram [12, Lemma 2.16]

\xymatrix{K_{0}(X̄) \ar[r]^{c_{d}} \ar[d]^{γ_{d}∘(id − rank)} & CH^{d}(X̄) \ar@{->>}[d] \\ Γ^{d/d+1}(X̄) \ar[r] & T^{d/d+1}(X̄)}

that T^{d/d+1}(X̄)_{tors} = Γ^{d/d+1}(X̄)_{tors}. In particular, one has

(28)  CH^{2}(X̄)_{tors} = T^{2/3}(X̄)_{tors} = Γ^{2/3}(X̄)_{tors}.

Lemma 5.1. (cf. [1, Proposition 3.1]) Let X̄_{4} be the generic variety associated to (SL_{2})^{4}/μ. Then, the torsion subgroup CH^{2}(X̄_{4})_{tors} is trivial.

Proof. The generic variety X̄_{4} is the product of conics ∏_{i=1}^{4} SB(Q_{i}) over the function field K of a classifying space of (SL_{2})^{4}/μ, where Q_{1}, Q_{2}, Q_{3}, Q_{4} are quaternion algebras over K satisfying

ind(Q_{1}⊗Q_{2}⊗Q_{3}⊗Q_{4}) = 1, ind(Q_{i}⊗Q_{j}⊗Q_{k}) = 2, ind(Q_{i}⊗Q_{j}) = 4, ind(Q_{i}) = 2

for all 1 ≤ i ≠ j ≠ k ≤ 4.

Let X = ∏_{i=1}^{3} SB(Q′_{i}), where Q′_{1}, Q′_{2}, Q′_{3} are quaternion K-algebras such that

ind(Q′_{1} ⊗ Q′_{2} ⊗ Q′_{3}) = 2, ind(Q′_{i} ⊗ Q′_{j}) = 4, ind(Q′_{i}) = 2

for all 1 ≤ i ≠ j ≤ 3. By (27), it suffices to show CH^{2}(X)_{tors} = 0.

By (26), we have a basis {1, 2x_{i}, 4x_{i}x_{j}, 2x_{1}x_{2}x_{3}} of K_{0}(X), where x_{i} is the pullback of the tautological line bundle on the projective line over a splitting field E of X. Hence, |K_{0}(X_{E})/K_{0}(X)| = 2^{10}. By substitution y_{i} = x_{i} − 1 for all 1 ≤ i ≤ 3 we have another basis of K_{0}(X)

(29)  {1, 2y_{i}, 4y_{i}y_{j}, 2(y_{1}y_{2}y_{3} + y_{1}y_{2} + y_{1}y_{3} + y_{2}y_{3})}.

Let ϵ_{n} = |Γ^{n/n+1}(X_{E})/ Im(res^{n/n+1})| for 1 ≤ n ≤ 3. Then, we obtain ϵ_{1} ≤ 2^{3}. Since c_{2}(2x_{1}x_{2}x_{3}) = 6y_{1}y_{2}y_{3} + 2(y_{1}y_{2} + y_{1}y_{3} + y_{2}y_{3}) ∈ Γ^{2}(X) and c_{2}(2x_{1}x_{2}x_{3}) · 2y_{1} = 4y_{1}y_{2}y_{3} ∈ Γ^{3}(X), it follows from (29) that ϵ_{2} ≤ 4^{2} · 2 and ϵ_{3} ≤ 4, respectively. Therefore, |⊕ Γ^{n/n+1}(X)_{tors}| ≤ 1, i.e., Γ^{2/3}(X)_{tors} = 0, thus CH^{2}(X)_{tors} = 0. □

Proposition 5.2. Let X̄_{5} be the generic variety associated to (SL_{2})^{5}/μ. Then, we have CH^{2}(X̄_{5})_{tors} = ℤ/2ℤ.

Proof. Let X = ∏_{i=1}^{4} SB(Q′_{i}), where Q′_{1}, Q′_{2}, Q′_{3}, Q′_{4} are quaternion algebras over a field K/F such that

ind(Q′_{1} ⊗ Q′_{2} ⊗ Q′_{3} ⊗ Q′_{4}) = 2, ind(Q′_{i} ⊗ Q′_{j} ⊗ Q′_{k}) = 4, ind(Q′_{i} ⊗ Q′_{j}) = 4

for all 1 ≤ i ≠ j ≠ k ≤ 4. We first show that Γ^{2/3}(X)_{tors} = ℤ/2ℤ.

Let x_{i} be the pullback of the tautological line bundle on ℙ^{1} over a splitting field E of X. By (26), we have a basis {1, 2x_{i}, 4x_{i}x_{j}, 2x_{i}x_{j}x_{k}, 2x_{1}x_{2}x_{3}x_{4}} of K_{0}(X). Therefore, |K_{0}(X_{E})/K_{0}(X)| = 2^{4} · 4^{6} · 4^{4} · 2. Consider another basis

{1, 2y_{i}, 4y_{i}y_{j}, 4y_{i}y_{j}y_{k}, 2(y_{1}y_{2}y_{3}y_{4} + ∑ y_{i}y_{j}y_{k} + ∑ y_{i}y_{j})}

of K_{0}(X), where y_{i} = x_{i} − 1. Let ϵ_{n} = |Γ^{n/n+1}(X_{E})/ Im(res^{n/n+1})| for 1 ≤ n ≤ 4. It is obvious that ϵ_{1} ≤ 2^{4}. As c_{2}(2x_{1}x_{2}x_{3}x_{4}) = 14y_{1}y_{2}y_{3}y_{4} +6 ∑ y_{i}y_{j}y_{k} +2 ∑ y_{i}y_{j} ∈ Γ^{2}(X) and 8y_{1}y_{2}y_{3}y_{4} = 2y_{1} · 2y_{2} · c_{2}(2x_{1}x_{2}x_{3}x_{4}) ∈ Γ^{4}(X), we have ϵ_{2} ≤ 4^{5} · 2 and ϵ_{4} ≤ 2^{3}, respectively. It follows from the computation of 2y_{i} · c_{2}(2x_{1}x_{2}x_{3}x_{4}) that

z_{l} := 4(y_{1}y_{2}y_{3}y_{4} + ∑_{1≤p<q<r≤4} y_{p}y_{q}y_{r}) − 4y_{i}y_{j}y_{k} ∈ Γ^{3}(X)

for all l such that {i, j, k, l} = {1, 2, 3, 4}. Hence, 4(y_{1}y_{2}y_{3}y_{4} + y_{i}y_{j}y_{k}) ∈ Γ^{3}(X), thus ϵ_{3} ≤ 4^{3}. We conclude from (25) that |⊕ Γ^{n/n+1}(X)_{tors}| ≤ 2.

As Γ^{3}(X) is generated by Γ^{1}(X) · Γ^{2}(X), c_{2}(4x_{i}x_{j}) = 12y_{i}y_{j}, and c_{2}(4x_{i}x_{j}x_{k}) = 36y_{i}y_{j}y_{k} + 12(y_{i}y_{j} + y_{i}y_{k} + y_{j}y_{k}), one can easily see that Γ^{3}(X) is generated by

{8y_{1}y_{2}y_{3}y_{4}, 8y_{i}y_{j}y_{k}, z_{l}}

for all i, j, k, l satisfying {i, j, k, l} = {1, 2, 3, 4}. Hence, if 4y_{1}y_{2}y_{3}y_{4} ∈ Γ^{3}(X), then there exist f_{m} in the image of K_{0}(X) (as an element of ℤ[y_{1}, y_{2}, y_{3}, y_{4}]/(y_{1}^{2}, y_{2}^{2}, y_{3}^{2}, y_{4}^{2})) for 1 ≤ m ≤ 9 such that

(30) 4y_{1}y_{2}y_{3}y_{4} = (  ∑_{{i,j,k,l}={1,2,3,4}} f_{l} ·8y_{i}y_{j}y_{k})+f_{5}z_{1} +f_{6}z_{2} +f_{7}z_{3} +f_{8}z_{4} +f_{9} ·8y_{1}y_{2}y_{3}y_{4}

(indeed, we may assume that f_{i} ∈ ℤ). By comparing coefficients of (30), we obtain

8(f_{1} + f_{2} + f_{3} + f_{4} + f_{9}) + 16(f_{5} + f_{6} + f_{7} + f_{8}) = 4,

which is impossible. Therefore, 4y_{1}y_{2}y_{3}y_{4} ∉ Γ^{3}(X).

Since 4y_{1}y_{2}y_{3}y_{4} ∉ Γ^{3}(X) and 4(y_{1}y_{2}y_{3}y_{4} +y_{i}y_{j}y_{k}) ∈ Γ^{3}(X), all the elements 4y_{i}y_{j}y_{k} are not contained in Γ^{3}(X). On the other hand, it follows from the computation of 2c_{2}(4x_{1}x_{2}x_{3}x_{4}) − 3z_{l} that 4y_{i}y_{j}y_{k} ∈ Γ^{2}(X). Therefore, the class of 4y_{i}y_{j}y_{k} generates a subgroup of Γ^{2/3}(X)_{tors} of order 2 as 8y_{i}y_{j}y_{k} ∈ Γ^{3}(X).

As the generic variety X̄_{5} is ∏_{i=1}^{5} SB(Q_{i}), where Q_{i} is a quaternion algebra over the function field K of a classifying space of (SL_{2})^{4}/μ such that

ind(∏_{i=1}^{5} Q_{i}) = 1, ind(Q_{i} ⊗ Q_{j} ⊗ Q_{k} ⊗ Q_{l}) = 2, ind(Q_{i} ⊗ Q_{j} ⊗ Q_{k}) = ind(Q_{i} ⊗ Q_{k}) = 4

for all 1 ≤ i ≠ j ≠ k ≠ l ≤ 5, the result follows from Γ^{2/3}(X)_{tors} = ℤ/2ℤ, (27), and (28). □

Remark 5.3. Note that it was shown that |⊕ Γ^{n/n+1}(X)_{tors}| = 2 in [1, Theorem 3.5].

6. Proof of theorem

It follows from Proposition 4.1 and Lemma 3.1 that Inv^{3}(H)_{ind} = ℤ/2ℤ for n ≥ 2 and Inv^{3}(G)_{ind} = ℤ/2ℤ for n ≥ 3. To verify the remaining parts of Theorem we apply Propositions 3.3, 4.1, 5.2 and Lemmas 3.1, 5.1 to Proposition 2.1. The proof is divided into four cases.

Case: n = 2. Since G -torsors(K) ≃ PGL_{2} -torsors(K) for any field extension K/F, it follows from [7, §4b] that Inv^{3}(G)_{ind} = 0, thus Sdec(G)/ Dec(G) = 0. On the other hand, it is easy to see that the torsion subgroup CH^{2}(X̄_{2})_{tors} of the corresponding generic variety X̄_{2} is trivial. Hence, it follows from Propositions 2.1 and 4.1 that Sdec(H)/ Dec(H) = ℤ/2ℤ.

Case: n = 3 or 4. By (27), the torsion subgroup CH^{2}(X̄_{3})_{tors} of the generic variety X̄_{3} is isomorphic to the torsion subgroup of a flag variety of dimension 2, thus CH^{2}(X̄_{3}) = 0. Therefore, it follows from Lemma 5.1 and Proposition 4.1 that Sdec(H)/ Dec(H) = ℤ/2Z. On the other hand, the second statement of Theorem follows from Lemma 3.1 and Proposition 3.3, thus Sdec(G)/ Dec(G) = ℤ/2ℤ.

Case: n = 5. It follows from Propositions 4.1 and 5.2 that Sdec(H) = Dec(H). Therefore, by Proposition 2.1 we also have Sdec(G) = Dec(G).

Case: n ≥ 6. Consider the nontrivial invariant α_{n} in (6). We claim that if α_{n} is semi-decomposable but not decomposable, then so is α_{n−1}. Consider the restriction map r : Inv^{3}((GL_{2})^{n}/μ) → Inv^{3}((GL_{2})^{n−1}/μ) defined by

r(α)(Q_{1}, …, Q_{n−1}) = α(Q_{1}, …, Q_{n−1}, 0)

for (Q_{1}, …, Q_{n−1}) ∈ (GL_{2})^{n−1}/μ -torsors(K) and α ∈ Inv^{3}((GL_{2})^{n}/μ). Then, we have r(α_{n})(Q_{1}, …, Q_{n−1}) = e_{3}(∑_{i=1}^{n−1} N_{Q_{i}}) = α_{n−1}(Q_{1}, …, Q_{n−1}), i.e., r(α_{n}) = α_{n−1}. Assume that α_{n} is semi-decomposable. Then, it follows from (2) and (18) that there exist χ_{i} = (b_{1i}, …, b_{ni}) ∈ (ℤ/2ℤ)^{n}/ℤ/2ℤ = μ^{∗} and x_{i} ∈ K^{×} such that

α_{n}(Q_{1}, …, Q_{n}) = ∑_{i}(b_{1i}[Q_{1}] + ⋯ + b_{ni}[Q_{n}]) ∪ (x_{i}).

Hence, α_{n−1} = r(α_{n}) is semi-decomposable. To show that Sdec(G) = Dec(G) we use induction on n together with claim and the previous case. Finally, it follows from Proposition 2.1 that Sdec(H) = Dec(H).

Remark 6.1. The proof of Theorem shows that for any n ≥ 5 the torsion subgroup CH^{2}(X̄_{n})_{tors} of the generic variety associated to SL_{n} /μ is isomorphic to ℤ/2ℤ.

7. Additional example of a nontrivial semi-decomposable invariant

In the present section, we give an example illustrating that a minor modification of our method can be used to determine the semi-decomposable invariants.

Lemma 7.1. Let μ = {(λ_{1}, λ_{2}) ∈ (μ_{4})^{2} | λ_{1}^{2}λ_{2}^{2} = 1} be a central subgroup of SL_{4} × SL_{4}. Then, Inv^{3}((SL_{4} × SL_{4})/μ)_{ind} = ℤ/2ℤ.

Proof. Basically, we follow the proof of Proposition 4.1. Let H = (SL_{4} × SL_{4})/μ and G = (GL_{4} × GL_{4})/μ. The same commutative diagram as (17) with (μ_{2})^{n} replaced by (μ_{4})^{2} yields the same commutative diagram of the corresponding character groups as (18) with (ℤ/2ℤ)^{n} and ℤ^{2n} replaced by ℤ/4ℤ ⊕ ℤ/4ℤ and ℤ^{4} ⊕ ℤ^{4}, where the map ℤ/4ℤ ⊕ ℤ/4ℤ → ℤ^{4} ⊕ ℤ^{4} is given by

(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}) ↦ (\overline{a_{1} + a_{2} + a_{3} + a_{4}}, \overline{b_{1} + b_{2} + b_{3} + b_{4}}).

Hence, T_{G}^{∗} = \{∑ a_{i}x_{i} + b_{i}y_{i} | \overline{∑ a_{i}} = \overline{∑ b_{i}} = ¯0 or \overline{∑ a_{i}} = \overline{∑ b_{i}} = ¯2 for all 1 ≤ i ≤ 4\}, where {x_{i}, y_{i}} is a standard basis of ℤ^{4} ⊕ℤ^{4}. Therefore, we have the following ℤ-basis of T_{G}^{∗}:

{x_{1} − x_{2}, x_{1} − x_{3}, x_{1} − x_{4}, y_{1} − y_{2}, y_{1} − y_{3}, y_{1} − y_{4}, 2x_{1} + 2y_{1}, 2x_{1} − 2y_{1}}.

Consider the same diagram of the character groups as (20) with ℤ^{n} and ℤ^{2n} replaced by ℤ^{3} ⊕ ℤ^{3} and ℤ^{4} ⊕ ℤ^{4}, where the middle map ℤ^{4} ⊕ ℤ^{4} → ℤ^{3} ⊕ ℤ^{3} is defined by x_{i} ↦ x̄_{i}, y_{i} ↦ ȳ_{i}, x_{4} ↦ −x̄_{1} − x̄_{2} − x̄_{3}, and y_{4} ↦ −ȳ_{1} − ȳ_{2} − ȳ_{3} for 1 ≤ i ≤ 3. Then, it follows that we find the following ℤ-basis of T_{H}^{∗}:

(31)  {x̄_{1} − x̄_{2}, x̄_{1} − x̄_{3}, ȳ_{1} − ȳ_{2}, ȳ_{1} − ȳ_{3}, 2x̄_{1} + 2ȳ_{1}, 2x̄_{1} − 2ȳ_{1}}.

As the group S^{2}(T̄_{H}^{∗})^{W} with W = (S_{4})^{2} is generated by q_{1} = ∑_{i} x̄_{i}^{2} + ∑_{i<j} x̄_{i}x̄_{j} and q_{2} = ∑_{i} ȳ_{i}^{2} + ∑_{i<j} ȳ_{i}ȳ_{j} with 1 ≤ i ≠ j ≤ 3, the same argument as in the proof of Proposition 4.1 together with (31) shows that S^{2}(T_{H}^{∗})^{W} is generated by

{4q_{1} + 4q_{2}, 2q_{1} + 6q_{2}}.

Since the image of the 2nd Chern class map is generated by {8q_{1}, 4q_{1} + 4q_{2}}, the indecomposable subgroup of H is isomorphic to ℤ/2ℤ generated by the class of 2q_{1} + 6q_{2}. □

Remark 7.2. Indeed, in S^{2}(T_{H}^{∗})^{W} the element 2q_{1} + 6q_{2} is written as

2(x̄_{1}−x̄_{2}){(x̄_{1}−x̄_{2})+(x̄_{1}−x̄_{3})−2(x̄_{1}+ȳ_{1})−2(x̄_{1}−ȳ_{1})}+12(x̄_{1}+ȳ_{1}){(x̄_{1}+ȳ_{1})−(x̄_{1}−ȳ_{1})}

+2(ȳ_{1}−ȳ_{2}){3(ȳ_{1}−ȳ_{2})+3(ȳ_{1}−ȳ_{3})−6(x̄_{1}+ȳ_{1})+6(x̄_{1}−ȳ_{1})}+12(x̄_{1}−ȳ_{1}){(ȳ_{1}−ȳ_{3})+(x̄_{1}−ȳ_{1})} +2(x̄_{1} − x̄_{3}){(x̄_{1} − x̄_{3}) − 2(x̄_{1} + ȳ_{1}) − 2(x̄_{1} − ȳ_{1})} + 6(ȳ_{1} − ȳ_{3}){(ȳ_{1} − ȳ_{3}) − 2(x̄_{1} + ȳ_{1})}.

Lemma 7.3. Let X̄ be the generic variety associated to (SL_{4} × SL_{4})/μ, where μ = {(λ_{1}, λ_{2}) ∈ (μ_{4})^{2} | λ_{1}^{2}λ_{2}^{2} = 1}. Then, the group CH^{2}(X̄)_{tors} is trivial.

Proof. Let X = SB(A_{1}) × SB(A_{2}), where A_{1} and A_{2} are division algebras of degree 4 over a field K/F such that

ind(A_{1} ⊗ A_{2}) = ind(A_{1} ⊗ A_{2}^{⊗3}) = ind(A_{1}^{⊗3} ⊗ A_{2}) = 16, ind(A_{i}^{⊗2}) = 2,
ind(A_{1} ⊗ A_{2}^{⊗2}) = ind(A_{1}^{⊗2} ⊗ A_{2}) = 4, ind(A_{1}^{2} ⊗ A_{2}^{2}) = 1

for i = 1, 2. By (26), we have the following basis of K_{0}(X)

{1, 4x_{i}, 2x_{i}^{2}, 16x_{1}x_{2}, 4x_{i}^{2}, 4x_{1}^{2}x_{2}, 4x_{1}x_{2}^{2}, x_{1}^{2}x_{2}^{2}, 16x_{1}x_{2}^{3}, 16x_{1}^{3}x_{2}, 4x_{1}^{3}x_{2}^{2}, 4x_{1}^{2}x_{2}^{3}, 16x_{1}^{3}x_{2}^{3}},

where x_{i} is the pullback of the tautological line bundle on ℙ^{3} over a splitting field E of X.

Let δ_{n} = |Γ^{n/n+1}(X_{E})/ Im(res^{n/n+1})|/|K_{0}^{n}(X_{E})/K_{0}^{n}(X)|, where 1 ≤ n ≤ 6 and K^{n}(X_{E}) (respectively, K_{0}^{n}(X)) is the codimension n part of K_{0}(X_{E}) (respectively, K_{0}(X)). We find upper bounds of δ_{n} using case by case analysis. We use another basis for K_{0}(X) the above basis by replacing x_{i} by y_{i} := x_{i}−1. As x_{1}^{2}x_{2}^{2} = (y_{1}+1)^{2}(y_{2}+1)^{2} ∈ Γ^{1}(X), we have 2(y_{1} + y_{2}) ∈ Im(res^{1/2}), thus 2δ_{1} ≤ 1.

In codimension 2, it follows from 2(y_{1}+1)^{2}(y_{2}+1)^{2} ∈ Γ^{2}(X) that 8y_{1}y_{2} ∈ Im(res^{2/3}). Since c_{2}(4x_{i}) = 6y_{i}^{2} ∈ Γ^{2}(X), we obtain 2δ_{2} ≤ 1. In codimension 3, we have c_{3}(4x_{i}) = 4y_{i}^{3} ∈ Γ^{3}(X) and 2(y_{1} + y_{2}) · 2y_{i}^{2} ∈ Im(res^{3/4}), thus δ_{3} ≤ 1. In codimension 4, we see that 2y_{1}^{2} · 2y_{2}^{2} ∈ Γ^{2}(X) · Γ^{2}(X) ⊆ Γ^{4}(X). Furthermore, it follows by a calculation of c_{4}(4x_{1}^{2}x_{2}) ∈ Γ^{4}(X) and c_{4}(4x_{1}x_{2}^{2}) ∈ Γ^{4}(X) that we have 8y_{1}y_{2}^{3}, 8y_{1}^{3}y_{2} ∈ Im(res^{4/5}). Hence, δ_{4} ≤ 1.

In codimension 5, we get 8y_{1}^{3}y_{2}^{2} = 4y_{1}^{3} · 2y_{2}^{2} ∈ Γ^{3}(X) · Γ^{2}(X) ⊆ Γ^{5}(X). Similarly, 8y_{1}^{2}y_{2}^{3} ∈ Γ^{5}(X). Therefore, δ_{5} ≤ 4. Finally, as 4y_{1}^{3} · 4y_{2}^{3} ∈ Γ^{3}(X) · Γ^{3}(X) ⊆ Γ^{6}(X) we have δ_{6} ≤ 1. In conclusion, we obtain |⊕ Γ^{n/n+1}(X)_{tors}| = 1. In particular, the group CH^{2}(X) is torsion-free.

Consider the generic variety X̄ of H := (SL_{4} × SL_{4})/μ over the function field F (U/B), where U is an open subset of a generically free representation of H. Since the canonical morphism X̄ → X is an iterated projective bundle, it follows by the projective bundle theorem that CH^{2}(X̄)_{tors} ≃ CH^{2}(X)_{tors}. Hence, the result follows.

□

This lemma and Lemma 7.1 together with (3) yield the following result:

Proposition 7.4. Let H be a factor group of SL_{4} × SL_{4} by a central subgroup μ = {(λ_{1}, λ_{2}) ∈ (μ_{4})^{2} | λ_{1}^{2}λ_{2}^{2} = 1}. Then, the group Inv^{3}(H)_{ind} is cyclic of order 2 and any degree 3 normalized invariant of H is semi-decomposable.

Acknowledgments. The author would like to express his deep gratitude to Alexander Merkurjev for introducing the problem and giving helpful suggestions to him. 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