RATIONALITY OF CYCLES ON FUNCTION FIELD OF EXCEPTIONAL PROJECTIVE HOMOGENEOUS VARIETIES

RAPHAEL FINO

Abstract. In this article we prove a result comparing rationality of algebraic cycles over the function field of a projective homogeneous variety under a linear algebraic group of type F_{4} or E_{8} and over the base field, which can be of any characteristic. Keywords: Chow groups and motives, exceptional algebraic groups, projective homogeneous varieties.

1. Introduction

Let G be a linear algebraic group of type F_{4} or E_{8} over a field F and let X be a projective homogeneous G-variety. We write Ch for the Chow group with coefficient in ℤ/pℤ, with p = 3 when G is of type F_{4} and p = 5 when G is of type E_{8}. The purpose of this note is to prove the following theorem dealing with rationality of algebraic cycles on function field of such a projective homogeneous G-variety.

Theorem 1.1. For any equidimensional variety Y, the change of field homomorphism

Ch(Y) → Ch(Y_{F (X)})

is surjective in codimension < p + 1. It is also surjective in codimension p + 1 for a given Y provided that 1 ∉ deg Ch_{0}(X_{F (ζ)}) for each generic point ζ ∈ Y.

The proof is given in section 3.

In previous papers ([2], [3], after the so-called Main Tool Lemma by A. Vishik, cf [16], [17]), similar issues about rationality of cycles, with quadrics instead of exceptional projective homogeneous varieties, have been treated. The above statement is to put in relation with [10, Theorem 4.3], where generic splitting varieties have been considered. Also, Theorem 1.1 is contained in [10, Theorem 4.3] if char(F) = 0.

On the one hand, our method of proof is basically the method used to prove [10, Theorem 4.3]. On the other hand, our method mainly relies on a motivic decomposition result for projective homogeneous varieties due to V. Petrov, N. Semenov and K. Zainoulline (cf [14, Theorem 5.17]). It also relies on a linkage between the γ-filtration and Chow groups, in the spirit of [5]. Our method works in any characteristic and is particularly suitable for groups of type F_{4} and E_{8} mainly because the latter have an opportune J-invariant.

In the aftermath of Theorem 1.1, we get the following statement dealing with integral Chow groups (see [10, Theorem 4.5]).

Date: 2 June 2013.

2010 Mathematics Subject Classification. 14C25; 20G41.

Corollary 1.2. If p ∈ deg CH_{0}(X) then for any equidimensional variety Y, the change of field homomorphism

CH(Y) → CH(Y_{F (X)})

is surjective in codimension < p + 1. It is also surjective in codimension p + 1 for a given Y provided that 1 ∉ deg Ch_{0}(X_{F (ζ)}) for each generic point ζ ∈ Y.

Remark 1.3. Our method of proof for Theorem 1.1 works for groups of type G_{2} as well (with p=2). However, the case of G_{2} can be treated in a more elementary way if char(F) = 0.

Indeed, it is known that to each group G of type G_{2} one can associate a 3-fold Pfister quadratic form ρ such that, by denoting X_{ρ} the Pfister quadric associated with ρ, the variety X has a rational point over F (X_{ρ}) and vice-versa. Thus, for any equidimensional variety Y, one has the commutative diagram

\xymatrix{Ch(Y) \ar[r] \ar[d] & Ch(Y_{F (X)}) \ar[d] \\ Ch(Y_{F (X_{ρ})}) \ar[r] & Ch(Y_{F (X_{ρ}×X)})}

where the right and the bottom maps are isomorphisms. Furthermore, as suggested in [17, Remark on Page 665] (where the assumption char(F) = 0 is required), the change of field homomorphism Ch(Y) → Ch(Y_{F (Q)}) is surjective in codimension < 3.

Acknowledgements. I gracefully thank Nikita Karpenko for sharing his great knowledge and his valuable advice.

2. Filtrations on projective homogeneous varieties

In this section, we prove two propositions which play a crucial role in the proof of Theorem 1.1.

First of all, we recall that for any smooth projective variety X over a field E, one can consider two particular filtrations on the Grothendieck ring K(X) (see [5, §1.A]), i.e the γ-filtration and the topological filtration, whose respective terms of codimension i are given by

γ^{i}(X) = ⟨c_{n_{1}}(a_{1}) ⋯ c_{n_{m}}(a_{m}) | n_{1} + ⋯ + n_{m} ≥ i and a_{1}, …, a_{m} ∈ K(X)⟩

and

τ^{i}(X) = ⟨[O_{Z}] | Z ↪ X and codim(Z) ≥ i⟩,

where c_{n} is the n-th Chern Class with values in K(X) and [O_{Z}] is the class of the structure sheaf of a closed subvariety Z. We write γ^{i/i+1}(X) and τ^{i/i+1}(X) for the respective quotients. For any i, one has γ^{i}(X) ⊂ τ^{i}(X) and one even has γ^{i}(X) = τ^{i}(X) for i ≤ 2. We denote by pr the canonical surjection

\begin{matrix}CH^{i}(X) & −↠ & τ^{i/i+1}(X) \\ [Z] & ⟼ & [O_{Z}] \end{matrix} ,

where CH stands for the integral Chow group.

The method of proof of the following proposition is largely inspired by the proof of [9, Theorem 6.4 (2)].

Proposition 2.1. Let G_{0} be a split semisimple linear algebraic group over a field F and let B be a Borel subgroup of G_{0}. There exist an extension E/F and a cocycle ξ ∈ H^{1}(E, G_{0}) such that the topological filtration and the γ-filtration coincide on K(_{ξ}(G_{0}/B)).

Proof. Let n be an integer such that G_{0} ⊂ GL_{n} and let us set S := GL_{n} and E := F (S/G_{0}). We denote by T the E-variety S ×_{S/G_{0}} Spec(E) given by the generic fiber of the projection S → S/G_{0}. Note that since T is clearly a G_{0}-torsor over E, there exists a cocycle ξ ∈ H^{1}(E, G_{0}) such that the smooth projective variety X := T/B_{E} is isomorphic to _{ξ}(G_{0}/B). We claim that the Chow ring CH(X) is generated by Chern classes.

Indeed, the morphism h : X → S/B induced by the canonical G_{0}-equivariant morphism T → S being a localisation, the associated pull-back

h^{∗} : CH(S/B) ⟶ CH(X)

is surjective. Furthermore, the ring CH(S/B) itself is generated by Chern classes: by [9, §6,7] there exist a morphism

(2.2)  𝕊(T^{∗}) ⟶ CH(S/B),

(where 𝕊(T^{∗}) is the symmetric algebra of the group of characters T^{∗} of a split maximal torus T ⊂ B) with its image generated by Chern classes. Moreover, the morphism (2.2) is surjective by [9, Proposition 6.2]. Since h^{∗} is surjective and Chern classes commute with pull-backs, the claim is proved.

We show now that the two filtrations coincide on K(X) by induction on dimension. Let i ≥ 0 and assume that τ^{i+1}(X) = γ^{i+1}(X). Since for any j ≥ 0, one has γ^{j}(X) ⊂ τ^{j}(X), the induction hypothesis implies that

γ^{i/i+1}(X) ⊂ τ^{i/i+1}(X).

Thus, the ring CH(X) being generated by Chern classes, one has γ^{i/i+1}(X) = τ^{i/i+1}(X) by [8, Lemma 2.16]. Therefore one has τ^{i}(X) = γ^{i}(X) and the proposition is proved. □

Note that this result remains true when one consider a special parabolique subgroup P instead of B.

Now, we prove a result which will be used in section 3 to get the second conclusion of Theorem 1.1.

We recall that for any smooth projective variety X over a field and for any i < p+1, the canonical surjection pr : Ch^{i}(X) ↠ τ^{i/i+1}(X) with ℤ/pℤ-coefficient is an isomorphism (cf [5, §1.A] for example). The following proposition extends this fact to i = p + 1 provided that X is a projective homogeneous variety under a linear algebraic group G of type F_{4} or E_{8}.

Proposition 2.3. Let X be a projective homogeneous variety under a group G of type F_{4} or E_{8}, then the canonical surjection

pr : Ch^{p+1}(X) ↠ τ^{p+1/p+2}(X)

is injective.

Proof. The epimorphism pr : Ch^{p+1}(X) ↠ τ^{p+1/p+2}(X) coincides with the edge homomorphism of the spectral Brown-Gersten-Quillen structure E_{2}^{p+1,−p−1}(X) ⇒ K(X), i.e E_{r}^{p+1,−p−1}(X) stabilizes for r >> 0 with E_{∞}^{p+1,−p−1}(X) = τ^{p+1/p+2}(X), and for any r ≥ 2 the differential E_{r}^{p+1,−p−1}(X) → E_{r}^{p+1+r,−p−r}(X) is zero, so that the epimorphism pr coincides with the composition

Ch^{p+1}(X) ≃ E_{2}^{p+1,−p−1}(X) ↠ E_{3}^{p+1,−p−1}(X) ↠ ⋯ ↠ E_{∞}^{p+1,−p−1}(X) = τ^{p+1/p+2}(X).

Now, it is equivalent in order to prove the proposition to prove that for any r ≥ 2, the differential E_{r}^{p+1−r,−p−2+r}(X) → E_{r}^{p+1,−p−1}(X) is zero.

First of all, since we work with ℤ/pℤ-coefficient, by [12,Theorem 3.6], the differential E_{r}^{p+1−r,−p−2+r}(X) → E_{r}^{p+1,−p−1}(X) is zero for any r ≥ 2 with r ≠ p. Hence, one only has to show that the differential E_{p}^{1,−2}(X) → E_{p}^{p+1,−p−1}(X) is zero.

Let us consider the following composition given by the BGQ-structure

E_{∞}^{1,−2}(X) ↪ ⋯ ↪ E_{3}^{1,−2}(X) ↪ E_{2}^{1,−2}(X).

Note that one has E_{∞}^{1,−2}(X) ≃ E_{2}^{1,−2}(X) if and only if for any r ≥ 2 the differential E_{r}^{1,−2}(X) → E_{r}^{1+r,−2−r+1}(X) is zero. Therefore it is sufficient to prove that E_{∞}^{1,−2}(X) ≃ E_{2}^{1,−2}(X) to get that the differential E_{p}^{1,−2}(X) → E_{p}^{p+1,−p−1}(X) is zero.

On the one hand, by the very defintion, the group E_{∞}^{1,−2}(X) is the first quotient K_{1}^{(1/2)}(X) of the topological filtration on K_{1}(X). On the other hand, one has E_{2}^{1,−2}(X) ≃ H^{1}(X, K_{2}) (for any integers p and q, one has E_{2}^{p,q}(X) ≃ H^{p}(X, K_{−q})).

Let us now consider the commutative diagram (cf [7,§4])

\xymatrix{K_{1}^{(1/2)}(X) \ar@{^{(}->}[rr] &  & H^{1}(X, K_{2}) \\  & H^{0}(X, K_{1}) ⊗ Ch^{1}(X) \ar@{-->}[ul] \ar@{-->}[ur]}

We claim that the natural map H^{0}(X, K_{1}) ⊗ Ch^{1}(X) → H^{1}(X, K_{2}) is an isomorphism. Indeed since G is of type F_{4} or E_{8}, it has only trivial Tits algebras, and therefore, by [11, Theorem], one has

H^{1}(X, K_{2}) ≃ H^{1}(X_{sep}, K_{2})^{Γ},

where Γ is the absolute Galois group of F. Moreover, since the variety X_{sep} is cellular, by [11, Proposition 1], one has

H^{1}(X_{sep}, K_{2}) ≃ K_{1}F_{sep} ⊗ Ch^{1}(X_{sep}).

Thus, since the Picard group of any homogeneous projective variety under a group of type F_{4} or E_{8} is rational (cf [15, Example 4.1.1]) and since (K_{1}F_{sep})^{Γ} = K_{1}F = H^{0}(X, K_{1}), one has

H^{1}(X, K_{2}) ≃ K_{1}F ⊗ Ch^{1}(X) ≃ H^{0}(X, K_{1}) ⊗ Ch^{1}(X),

and the claim is proved. Therefore, one has E_{∞}^{1,−2}(X) ≃ E_{2}^{1,−2}(X) and the proposition is proved. □

Remark 2.4. Assume that G_{0} of strongly inner type (e.g F_{4} and E_{8}) and consider an extension E/F and a cocycle ξ ∈ H^{1}(E, G_{0}). By [13, Theorem 2.2.(2)], the change of field homomorphism

K(_{ξ}(G_{0}/B)_{E}) → K(_{ξ}(G_{0}/B)\overline{E}) ≃ K(G_{0}/B)

is an isomorphism, where \overline{E} denotes an algebraic closure of E. Therefore, since the γfiltration is defined in terms of Chern classes and the latter commute with pull-backs, the quotients of the γ-filtration on K(_{ξ}(G_{0}/B)_{E}) do not depend nor on the extension E/F neither on the choice of ξ ∈ H^{1}(E, G_{0}).

3. Proof of Theorem 1.1

In this section, we prove Theorem 1.1.

First of all, note that the F -variety X is A-trivial in the sense of [10, Definition 2.3] (see [10, Example 2.5]), i.e for any extension L/F with X(L) ≠ ∅, the degree homomorphism deg : Ch_{0}(X_{L}) → ℤ/pℤ is an isomorphism. Therefore, by [10, Lemma 2.9], the change of field homomorphism Ch(Y) → Ch(Y_{F (X)}) is an isomorphism (in any codimension) if 1 ∈ deg Ch_{0}(X). Hence, one can assume that 1 ∉ deg Ch_{0}(X).

Now, we know from [14, Table 4.13] that the J-invariant J_{p}(G) of G is equal to (1) or (0). However, the assumption J_{p}(G) = (0) implies that there exists a splitting field K/F of degree coprime to p (see [14, Corollary 6.7]), and in that case one has Ch_{0}(X) ≃ Ch_{0}(X_{K}) and 1 ∈ deg Ch_{0}(X_{K}) by A-triviality of X. Thus, under the assumption 1 ∉ deg Ch_{0}(X), one necessarly has J_{p}(G) = (1) and that is why we can assume J_{p}(G) = (1) in the sequel.

Since X is A-trivial, one can use the following proposition (cf [10, Proposition 2.8]).

Proposition 3.1 (Karpenko, Merkurjev). Given an equidimensional F -variety Y and an integer m such that for any i and any point y ∈ Y of codimension i the change of field homomorphism

Ch^{m−i}(X) → Ch^{m−i}(X_{F (y)})

is surjective, the change of field homomorphism

Ch^{m}(Y) → Ch^{m}(Y_{F (X)})

is also surjective.

Consequently, it is sufficient in order to prove the first conclusion of Theorem 1.1 to show that for any extension L/F, the change of field homomorphism

(3.2)  Ch(X) ⟶ Ch(X_{L})

is surjective in codimension < p + 1.

Moreover, the F -variety being generically split (see [14, Example 3.6]), one can apply the motivic decomposition result [14, Theorem 5.17] to X and get that the motive M(X, ℤ/pℤ) decomposes as a sum of twists of an indecomposable motive R_{p}(G) (in the same way as (3.5)). Note that the quantity and the value of those twists do not depend on the base field. In particular, we get that for any extension L/F and any integer k, the group Ch^{k}(X_{L}) is isomorphic to a direct sum of groups Ch^{k−i}(R_{p}(G)_{L}) with 0 ≤ i ≤ k.

Therefore, the surjectivity of (3.2) in codimension < p+1 is a consequence of the following proposition.

Proposition 3.3. For any extension L/F, the change of field

(3.4)  Ch(R_{p}(G)) ⟶ Ch(R_{p}(G)_{L})

is surjective in codimension < p + 1.

Proof. Let G_{0} be a split linear algebraic group of the same type of the type of G and let ξ ∈ H^{1}(F, G_{0}) be a cocycle such that G is isogenic to the twisted form _{ξ}G_{0}. We write 𝔅 for the Borel variety of G (i.e 𝔅 = _{ξ}(G_{0}/B), where B is a Borel subgroup of G_{0}).

By [14, Theorem 5.17], one has the motivic decomposition

(3.5)  M(𝔅, ℤ/pℤ) ≃ ⨁_{i≥0} R_{p}(G)(i)^{⊕a_{i}},

where Σ_{i≥0}a_{i}t^{i} = P (CH(\overline{𝔅}), t)/P (CH(\overline{R_{p}(G)}), t), with P (−, t) the Poincaré polynomial. Thus, for any integer k,we get the following decomposition concerning Chow groups

(3.6)  Ch^{k}(𝔅_{L}) ≃ ⨁_{i≥0} Ch^{k−i}(R_{p}(G)_{L})^{⊕a_{i}}

First of all, the homomorphism (3.4) is clearly surjective in codimension 0 since one has Ch^{0}(R_{p}(G)_{L}) = ℤ/pℤ for any extension L/F. Then, Ch^{1}(\overline{𝔅}) is identified with the Picard group Pic(\overline{𝔅}) and is rational (see [15, Example 4.1.1]). Furthermore, thanks to the Solomon Theorem for example (see [15, §2.5]), one can compute the coefficients a_{i}’s: we get a_{0} = 1 and a_{1} = rank(G) = rank(Ch^{1}(\overline{𝔅})). Thus, the isomorphism (3.6) implies that Ch^{1}(R_{p}(G)_{L}) = 0 for any extension L/F.

We have already shown that the homomorphism (3.4) is surjective in codimension 0 and 1. The following lemma implies the surjectivity in codimension 2 and 3 (and therefore proves the first conclusion of Theorem 1.1 if G is of type F_{4}).

Lemma 3.7. Under the assumption J_{p}(G) = (1), one has

Ch^{2}(R_{p}(G)) = ℤ/pℤ and Ch^{3}(R_{p}(G)) = 0

Proof. Since J_{p}(G) = (1), by [6, Example 5.3], the cocycle ξ ∈ H^{1}(F, G_{0}) match with a generic G_{0}-torsor in the sense of [6]. Thus, by [5, Proposition 3.2] and [4, pp. 31, 133], one has Tors_{p}CH^{2}(𝔅) ≠ 0 (note that since an algebraic group of type F_{4} or E_{8} is simply connected, it is of strictly inner type, and we can use material from [5, §3]). The conclusion is given by [5, Proposition 5.4]. □

Let us fix an extension L/F. We now prove the surjectivity of (3.4) in codimension 2 and 3. By [14, Example 4.7], one has J_{p}(G_{L}) = (0) or J_{p}(G_{L}) = (1).

If J_{p}(G_{L}) = (0) then one has R_{p}(G_{L}) = ℤ/pℤ by [14, Corollary 6.7], and on the other hand the motivic decomposition given in [14, Proposition 5.18 (i)] implies the following decomposition on Chow groups for any integer k

(3.8)  Ch^{k}(R_{p}(G)_{L}) ≃ ⨁_{i=0}^{p−1} Ch^{k−i(p+1)}(R_{p}(G_{L})).

In particular, one has Ch^{k}(R_{p}(G)_{L}) = 0 for k = 2 or 3 and the conclusion follows.

If J_{p}(G_{L}) = (1) then by Lemma 3.7 one has Ch^{2}(R_{p}(G_{L})) = ℤ/pℤ and Ch^{3}(R_{p}(G_{L})) = 0. Moreover, since J_{p}(G_{L}) = J_{p}(G), one has R_{p}(G_{L}) ≃ R_{p}(G)_{L} (see [14, Proposition 5.18 (i)]). Therefore, the homomorphism (3.4) is clearly surjective in codimension 3.

We claim that it is also surjective in codimension 2. By (3.6) it suffices to show that the change of field Ch^{2}(𝔅) → Ch^{2}(𝔅_{L}) is an isomorphism. We use material and notation introduced in section 2. Since J_{p}(G) = J_{p}(G_{L}) = (1), the cocycles ξ and ξ_{L} match with generic G_{0}-torsors and one consequently has γ^{3}(𝔅) = τ^{3}(𝔅) and γ^{3}(𝔅_{L}) = τ^{3}(𝔅_{L}) (see [5, Theorem 3.1(ii)]). It follows that

γ^{2/3}(𝔅) = τ^{2/3}(𝔅) and γ^{2/3}(𝔅_{L}) = τ^{2/3}(𝔅_{L}).

Therefore, since 2 < p + 1, the homomorphsim Ch^{2}(𝔅) → Ch^{2}(𝔅_{L}) coincides with

Ch^{2}(𝔅) ≃ γ^{2/3}(𝔅) → γ^{2/3}(𝔅_{L}) ≃ Ch^{2}(𝔅_{L})

and the center arrow is an isomorphism by Remark 2.4.

The surjectivity of (3.4) in codimension 4 and 5 is a direct consequence of the following statement, where G is of type E_{8} and p = 5. Consequently, Lemma 3.9 completes the proof of the first conclusion of Theorem 1.1 for G of type E_{8}.

Lemma 3.9. For any extension L/F, one has

Ch^{4}(R_{5}(G)_{L}) = 0 and Ch^{5}(R_{5}(G)_{L}) = 0

Proof. Since J_{5}(G) = (1), we know that J_{5}(G_{L}) = (1) or (0). If J_{5}(G_{L}) = (0) then one has R_{5}(G_{L}) = ℤ/5ℤ and the isomorphism (3.8) implies that Ch^{4}(R_{5}(G)_{L}) = Ch^{5}(R_{5}(G)_{L}) = 0. Thus, one can assume L = F and we have to prove that Ch^{4}(R_{5}(G)) = Ch^{5}(R_{5}(G)) = 0.

By Proposition 2.1 there exist an extension E/F and a cocycle ξ′ ∈ H^{1}(E, G_{0}) such that the topological filtration and the γ-filtration coincide on K(𝔅′), with 𝔅′ = _{ξ′}(G_{0}/B). Let us denote G′ the variety _{ξ′}G_{0}.

We claim that J_{5}(G′) = (1). Indeed, assume that J_{5}(G′) = (0). In that case, one has R_{5}(G′) = ℤ/5ℤ and the isomorphism (3.6) gives that Ch^{2}(𝔅′) = ℤ/5ℤ^{⊕a_{2}}. Since 2 < p+1, it implies that γ^{2/3}(𝔅′) = ℤ/5ℤ^{⊕a_{2}}, and consecutively γ^{2/3}(𝔅) = ℤ/5ℤ^{⊕a_{2}} by Remark 2.4. However, we have γ^{2/3}(𝔅) ≃ τ^{2/3}(𝔅) (because γ^{3}(𝔅) ≃ τ^{3}(𝔅) since ξ ∈ H^{1}(F, G_{0}) is generic). Thus, we have Ch^{2}(𝔅) = ℤ/5ℤ^{⊕a_{2}} which contradicts Ch^{2}(R_{5}(G)) = ℤ/5ℤ and the claim is proved (we recall that for any i < 6 = p + 1, one has τ^{i/i+1}(X) ≃ Ch^{i}(X)).

We now compute the groups γ^{i/i+1}(𝔅′) for i = 3, 4, 5. Note that since K(𝔅′) ≃ K(G_{0}/B) and since the description of the free group K(G_{0}/B) in terms of generators does not depend on the characteristic char(E) of E (see [1, Lemma 13.3(4)]), we can assume that char(E) = 0 in order to compute those groups.

In that case, since J_{5}(G′) ≠ (0), the isomorphism (3.6) combined with the following theorem (adapted from [10, Theorem RM.10] to our situation)

Theorem 3.10 (Karpenko, Merkurjev). Let H be a semisimple linear algebraic group of inner type over a field of characteristic 0 and let p be a torsion prime of H. If J_{p}(H) ≠ (0) then

Ch^{j}(R_{p}(H)) = \{ \begin{matrix}ℤ/pℤ & if j = 0 or j = k(p + 1) − p + 1, 1 ≤ k ≤ p − 1 \\ 0 & otherwise \end{matrix}

gives that

γ^{i/i+1}(𝔅′) ≃ Ch^{i}(𝔅′) = ℤ/5ℤ^{⊕(a_{i−2}+a_{i})} for i = 3, 4, 5

(where the first isomorphism is due to i < p + 1). Therefore, we get

γ^{i/i+1}(𝔅) = ℤ/5ℤ^{⊕(a_{i−2}+a_{i})} for i = 3, 4, 5

(with no particular assumption on char(F)). Thus, since τ^{3/4}(𝔅) ≃ Ch^{3}(𝔅), the isomorphism (3.6) for k = 3 gives that τ^{3/4}(𝔅) ≃ γ^{3/4}(𝔅). Since the γ-filtration is contained in the topological one, we get

τ^{4}(𝔅) = γ^{4}(𝔅),

which implies the existence of an exact sequence

0 → (τ_{5}(𝔅)/γ_{5}(𝔅)) → γ^{4/5}(𝔅) → τ^{4/5}(𝔅) → 0.

Thus, since τ^{4/5}(𝔅) ≃ Ch^{4}(𝔅), by applying the isomorphism (3.6) for k = 4, we get a surjection

ℤ/5ℤ^{⊕(a_{2}+a_{4})} ↠ Ch^{4}(R_{5}(G)) ⊕ ℤ/5ℤ^{⊕(a_{2}+a_{4})},

which implies that Ch^{4}(R_{5}(G)) = 0.

We prove that Ch^{5}(R_{5}(G)) = 0 by proceeding in exactly the same way. □

Consequently, Proposition 3.3 is proved. □

Finally, we want to prove the second conclusion of Theorem 1.1 (p = 3 if G is of type F_{4} and p = 5 if G is of type E_{8}). First of all, since for any generic point ζ of Y, one has

1 ∉ degCh_{0}(X_{F (ζ)}) ⇔ J_{p}(G_{F (ζ)}) = (1),

by Proposition 3.1 and in view of what has already been done, it is sufficient to prove the following lemma to get the second conclusion.

Lemma 3.11. Under the assumption J_{p}(G) = (1), one has Ch^{p+1}(R_{p}(G)) = 0.

Proof. Thanks to Proposition 2.3, one can prove the lemma by proceeding in exactly the same way Lemma 3.9 has been proved. □

This concludes the proof of Theorem 1.1.

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