TWISTED γ-FILTRATION OF A LINEAR ALGEBRAIC GROUP.

KIRILL ZAINOULLINE

Abstract. In the present notes we introduce and study the twisted γ-filtration on K_{0}(G_{s}), where G_{s} is a split simple linear algebraic group over a field k of characteristic prime to the order of the center of G_{s}. We apply this filtration to construct torsion elements in the γ-ring of some twisted flag varieties.

1. Introduction

Let X be a smooth projective variety over a field k. Consider the Grothendieck γ-filtration on K_{0}(X). It is given by the ideals [6, §2.3] (see also [8, §2])

γ^{i}K_{0}(X) = ⟨c_{n_{1}}(b_{1}) · . . . · c_{n_{m}}(b_{m}) | n_{1} + … + n_{m} ≥ i, b_{1}, …, b_{m} ∈ K_{0}(X)⟩, i ≥ 0

generated by products of Chern classes in K_{0}. Let γ^{i}(X) denote the i-th subsequent quotient and let γ^{∗}(X) = ⊕_{i≥0}γ^{i}(X) denote the associated graded commutative ring called the γ-ring of X.

The ring γ^{∗}(X) was invented by Grothendieck to approximate the topological filtration on K_{0} and, hence, the Chow ring CH^{∗}(X) of algebraic cycles modulo rational equivalence. Indeed, by the Riemann-Roch theorem (see [6, §2]) the i-th Chern class c_{i} induces an isomorphism with ℚ-coefficients, i.e. c_{i}: γ^{i}(X; ℚ) →^{≃} CH^{i}(X; ℚ). Moreover, in some cases the ring γ^{∗}(X) can be used to compute CH^{∗}(X), e.g. γ^{1}(X) = CH^{1}(X) and there is a surjection γ^{2}(X) ↠ CH^{2}(X) (see [7, Ex. 15.3.6]).

In the present notes we provide a uniform lower bound for the torsion part of γ^{∗}(X), where X = _{ξ}𝔅_{s} is a twisted form of the variety of Borel subgoups 𝔅_{s} of a split simple linear algebraic group G_{s} by means of a cocycle ξ ∈ H^{1}(k, G_{s}). Note that the groups γ^{2}(X) and γ^{3}(X) have been studied for G_{s} = P GL_{n} in [8] and for strongly inner forms in [4]. In particular, it was shown in [4, §3,7] that in the strongly inner case the torsion part of γ^{2}(X) determines the Rost invariant.

Our main tool is the twisted γ-filtration on K_{0}(G_{s}), where G_{s} is a split simple linear algebraic group. Roughly speaking, it is defined to be the image (see Definition 4.4) of the γ-filtration on K_{0} of the twisted form X under the composition K_{0}(X) → K_{0}(𝔅_{s}) → K_{0}(G_{s}), where the first map is given by the restriction and the second map is induced by taking the quotient.

Let γ_{ξ}^{∗} denotes the associated graded ring of the twisted γ-filtration. It has the following important properties:

(i) The ring γ_{ξ}^{∗} can be explicitly computed (see Theorem 4.5). Observe that γ_{ξ}^{0} = ℤ, γ_{ξ}^{1} = 0 and γ_{ξ}^{i} is torsion and finitely generated for i > 1.

(ii) There is a surjective ring homomorphism γ^{∗}(X) ↠ γ_{ξ}^{∗}. Hence, γ_{ξ}^{∗} gives a

lower bound for the γ-ring of the twisted form X = _{ξ}𝔅_{s}. (iii) The assignment ξ ↦ γ_{ξ}^{∗} respects the base change and, therefore, is an in-

variant of a G_{s}-torsor ξ, moreover, the ring γ_{ξ}^{∗} can be viewed as a substitute for the γ-ring of the inner group _{ξ}G_{s}.

In the last section we use these properties to construct nontrivial torsion elements in γ^{2}(X) for some twisted flag varieties X (see 5.3 and 5.5). In particular, we establish the connection between the indexes of the Tits algebras of ξ and the order of the special cycle θ ∈ γ^{2}(X) constructed in [4].

2. Preliminaries.

In the present section we recall several basic facts concerning linear algebraic groups, characters and the Grothendieck K_{0} (see [9, §24], [4, §1B,§6]).

2.1. Let G_{s} be a split simple linear algebraic group of rank n over a field k. We assume that characteristic of k is prime to the order of the center of G_{s}. We fix a split maximal torus T and a Borel subgroup B such that T ⊂ B ⊂ G_{s}.

Let Λ_{r} and Λ be the root and the weight lattices of the root system of G_{s} with respect to T ⊂ B. Let {α_{1}, …, α_{n}} be a set of simple roots (a basis of Λ_{r}) and let {ω_{1}, …, ω_{n}} be the respective set of fundamental weights (a basis of Λ), i.e. α_{i}^{∨}(ω_{j}) = δ_{ij}. The group of characters T^{∗} of T is an intermediate lattice Λ_{r} ⊂ T^{∗} ⊂ Λ that determines the isogeny class of G_{s}. If T^{∗} = Λ, then the group G_{s} is simply connected and if T^{∗} = Λ_{r} it is adjoint.

2.2. Let ℤ[T^{∗}] be the integral group ring of T^{∗}. Its elements are finite linear combinations ∑_{i} a_{i}e^{λ_{i}}, λ_{i} ∈ T^{∗}. Let 𝔅_{s} denote the variety of Borel subgroups G_{s}/B of G_{s}. Consider the characteristic map for K_{0} (see [3, §2.8])

𝔠: ℤ[T^{∗}] → K_{0}(𝔅_{s})

defined by sending e^{λ}, λ ∈ T^{∗}, to the class of the associated line bundle [L(λ)]. Observe that the ring K_{0}(𝔅_{s}) does not depend on the isogeny class of G_{s} while the group of characters T^{∗} and, hence, the image of 𝔠 does.

Since K_{0}(𝔅_{s}) is generated by the classes [L(ω_{i})], i = 1 … n, the characteristic map 𝔠 is surjective if G_{s} is simply connected. If G_{s} is adjoint, then the image of 𝔠 is generated by the classes [L(α_{i})], where

α_{i} = ∑_{j} c_{ij}ω_{j} and, therefore, L(α_{i}) = ⊗_{j}L(ω_{j})^{⊗c_{ij}},

and c_{ij} = α_{i}^{∨}(α_{j}) are the coefficients of the Cartan matrix of G_{s}.

2.3. The Weyl group W of G_{s} acts on weights via simple reflections s_{α_{i}} as

s_{α_{i}}(λ) = λ − α_{i}^{∨}(λ)α_{i}, λ ∈ Λ.

For each element w ∈ W we define (cf. [13, §2.1]) the weight ρ_{w} ∈ Λ as

ρ_{w} = ∑_{{i∈1...n|w^{−1}(α_{i})<0}} w^{−1}(ω_{i}).

In particular, for a simple reflection w = s_{α_{j}} we have

ρ_{w} = ∑_{{i∈1...n|s_{α_{j}} (α_{i})<0}} s_{α_{j}}(ω_{i}) = s_{α_{j}}(ω_{j}) = ω_{j} − α_{j}.

Observe that the quotient Λ/Λ_{r} coincides with the group of characters of the center of the simply connected cover of G_{s}. Since W acts trivially on Λ/Λ_{r}, we have

ρ̄_{w} = ∑_{{i∈1...n|w^{−1}(α_{i})<0}} ω̄_{i} ∈ Λ/T^{∗},

where ρ̄_{w} denotes the class of ρ_{w} ∈ Λ modulo T^{∗}. In particular, ω̄_{i} = ρ̄_{s_{α_{i}}}.

2.4. Let ℤ[Λ]^{W} denote the subring of W -invariant elements. Then the integral group ring ℤ[Λ] is a free ℤ[Λ]^{W} -module with the basis {e^{ρ_{w}}}_{w∈W} (see [13, Thm.2.2]). Now let ϵ: ℤ[Λ] → ℤ, e^{λ} ↦ 1 be the augmentation map. By the Chevalley Theorem the kernel of the surjection 𝔠 is generated by elements x ∈ ℤ[Λ]^{W} such that ϵ(x) = 0. Hence, there is an isomorphism

ℤ[Λ] ⊗_{ℤ[Λ]^{W}} ℤ ≃ ℤ[Λ]/ ker(𝔠) ≃ K_{0}(𝔅_{s}).

So the elements

{g_{w} = 𝔠(e^{ρ_{w}}) = [L(ρ_{w})]}_{w∈W}

form a ℤ-basis of K_{0}(𝔅_{s}) called the Steinberg basis.

2.5. Following [14] we associate with each χ ∈ Λ/T^{∗} and each cocycle ξ ∈ Z^{1}(k, G_{s}) the central simple algebra A_{χ,ξ} over k called the Tits algebra. This defines a group homomorphism

β_{ξ}: Λ/T^{∗} → Br(k) with β_{ξ}(χ) = [A_{χ,ξ}].

Let 𝔅 = _{ξ}𝔅_{s} denote the twisted form of the variety of Borel subgroups 𝔅_{s} by means of ξ. Consider the restriction map on K_{0} over the separable closure k_{sep}

res: K_{0}(𝔅) → K_{0}(𝔅 ×_{k} k_{sep}) = K_{0}(𝔅_{s}),

where we identify K_{0}(𝔅 ×_{k} k_{sep}) with K_{0}(𝔅_{s}). By [11, Thm.4.2] the image of the restriction can be identified with the sublattice

⟨ı_{w} · g_{w}⟩_{w∈W} ,

where g_{w} = [L(ρ_{w})] is an element of the Steinberg basis and ı_{w} = ind(β_{ξ}(ρ̄_{w})) is the index of the respective Tits algebra. Observe that if G_{s} is simply connected, then all indexes ı_{w} are trivial and the restriction map becomes an isomorphism.

3. The K_{0} of a split simple (adjoint) group

In the present section we provide an explicit description of the ring K_{0}(G_{s}) in terms of generators and relations for every simple split linear algebraic group G_{s}. The method to compute K_{0}(G_{s}) was known before, however, due to the lack of precise references we provide the computations here.

3.1. Definition. Let 𝔠: ℤ[Λ] → K_{0}(𝔅_{s}) be the characteristic map for the simply connected cover of G_{s}. We define the ring 𝔊_{s} to be the quotient

𝔊_{s} := ℤ[Λ/T^{∗}]/\overline{(ker 𝔠)}

and the surjective ring homomorphism q to be the composite

\xymatrix{q: K_{0}(𝔅_{s}) \ar[r]^{𝔠^{−1}}_{≃} & ℤ[Λ]/(ker 𝔠) \ar@{->>}[r] & ℤ[Λ/T^{∗}]/\overline{(ker c)} = 𝔊_{s}.}

Observe that if G_{s} is simply connected, then 𝔊_{s} = ℤ.

3.2. Remark. By [10, Cor.33] applied to X = G_{s} and to the simply-connected cover G = Ĝ_{s} of G_{s}, there is an isomorphism

K_{0}(G_{s}) ≃ ℤ ⊗_{R(Ĝ_{s})} K_{0}(Ĝ_{s}, G_{s}),

where R(Ĝ_{s}) ≃ ℤ[Λ]^{W} is the representation ring. By [10, Cor.5] applied to G = Ĝ_{s}, X = Spec k and G/H = G_{s} there is an isomorphism

K_{0}(Ĝ_{s}, G_{s}) ≃ R(H),

where R(H) ≃ ℤ[Λ/T^{∗}] is the representation ring. Therefore,

K_{0}(G_{s}) ≃ ℤ ⊗_{ℤ[Λ]^{W}} ℤ[Λ/T^{∗}] ≃ 𝔊_{s}.

3.3. Lemma. The ideal \overline{(ker 𝔠)} ⊂ ℤ[Λ/T^{∗}] is generated by the elements

d_{i}(1 − eω̄_{i}), i = 1 … n,

where d_{i} is the dimension of the i-th fundamental representation.

Proof. By the Chevalley Theorem the subring of invariants ℤ[Λ]^{W} can be identified with the polynomial ring ℤ[ρ_{1}, …, ρ_{n}], where ρ_{i} is the i-th fundamental representation, i.e.

ρ_{i} = ∑_{λ∈W (ω_{i})} e^{λ}

(here W (ω_{i}) denotes the W -orbit of the fundamental weight ω_{i}).

Since d_{i} = ϵ(ρ_{i}), ker 𝔠 = (d_{1} − ρ_{1}, …, d_{n} − ρ_{n}). To finish the proof observe that \overline{(d_{i} − ρ_{i})} = d_{i}(1 − eω̄_{i}). □

3.4. Remark. Observe that by definition and 3.3 we have 𝔊_{s} ⊗ ℚ ≃ ℚ.

3.5. In the following examples we compute the ring 𝔊_{s} ≃ K_{0}(G_{s}) for every simple split linear algebraic group G_{s} (we refer to [9, §24] for the description of Λ/T^{∗} and to [1, Ch.8, Table 2] for the dimensions of fundamental representations).

Λ/T^{∗}  G_{s}, m ≥ 1  Example
ℤ/mℤ, m ≥ 2 SL_{n+1}/μ_{m}  (3.6)
ℤ/2ℤ  O_{m+4}^{+}, P Sp_{2m+2}, HSpin_{4m+4}, E_{7}^{ad} (3.7)
ℤ/2ℤ ⊕ ℤ/2ℤ P GO_{4m+4}^{+}  (3.8)
ℤ/3ℤ  E_{6}^{ad}  (3.9)
ℤ/4ℤ  P GO_{4m+2}^{+}  (3.10)

3.6. Example. Consider the case G_{s} = SL_{n+1}/μ_{m}, m ≥ 2. The group G_{s} has type A_{n} and Λ/T^{∗} = ⟨σ⟩ is cyclic of order m. The quotient map Λ/Λ_{r} → Λ/T^{∗} sends ω̄_{i} ∈ Λ/Λ_{r}, i = 1 … n to (i mod m)σ ∈ Λ/T^{∗}.

By Definition 3.1 and Lemma 3.3 we have

𝔊_{s} ≃ ℤ[y]/(1 − (1 − y)^{m}, d_{1}y, …, d_{m−1}y^{m−1}),

where y = (1 − e^{σ}) and d_{j} = gcd\{\binom{n+1}{i} | i ≡ j mod m, i = 1 … n\}.

In particular, for G_{s} = SL_{p}/μ_{p} = P GL_{p}, where p is a prime, we obtain

𝔊_{s} ≃ ℤ[y]/(\binom{p}{1}y, \binom{p}{2}y^{2}, …, \binom{p}{p−1}y^{p−1}, y^{p}).

3.7. Example. Assume that Λ/T^{∗} = ⟨σ⟩ has order 2. Then

𝔊_{s} ≃ ℤ[y]/(y^{2} − 2y, dy),

where y = (1−e^{σ}) and d is the g.c.d. of dimensions of representations corresponding to ω_{i} with ω̄_{i} = σ. The integer d can be determined as follows:

B_{n}: We have Λ/Λ_{r} = {0, ω̄_{n}} ≃ ℤ/2ℤ which corresponds to the adjoint group G_{s} = O_{2n+1}^{+}. Since ω̄_{i} = 0 for each i ≠ n, d coincides with the dimension of ω_{n} that is 2^{n}.

C_{n}: We have Λ/Λ_{r} = {0, σ = ω̄_{1} = ω̄_{3} = …} ≃ ℤ/2ℤ that is G_{s} = P Sp_{2n}. Since ω̄_{i} = 0 for even i, d is the g.c.d. of dimensions of ω_{1}, ω_{3}, …, i.e.

d = gcd(2n, \binom{2n}{3} − \binom{2n}{1}, \binom{2n}{5} − \binom{2n}{3}, …).

D_{n}: If n is odd, then Λ/Λ_{r} = {0, ω̄_{n−1}, ω̄_{1}, ω̄_{n}} ≃ ℤ/4ℤ, where ω̄_{1} = 2ω̄_{n−1} = 2ω̄_{n}. Therefore, Λ/T^{∗} ≃ ℤ/2ℤ if it is a quotient of Λ/Λ_{r} modulo the subgroup {0, ω̄_{1}}. In this case Λ/T^{∗} = {0, σ = ω̄_{n−1} = ω̄_{n}} which corresponds to the special orthogonal group G_{s} = O_{2n}^{+}. Since ω̄_{s} = sω̄_{1} for 2 ≤ s ≤ n − 2 and ω̄_{1} = 0 in Λ/T^{∗}, d is the g.c.d. of the dimensions of ω_{n−1} and ω_{n} that is 2^{n−1}.

If n is even, then Λ/Λ_{r} = {0, ω̄_{n−1}}⊕{0, ω̄_{n}} ≃ ℤ/2ℤ⊕ℤ/2ℤ, where ω̄_{1} = ω̄_{n−1}+ω̄_{n}. In this case, we have two cases for Λ/T^{∗}:

(1) It is the quotient of Λ/Λ_{r} modulo the diagonal subgroup {0, ω̄_{n−1} + ω̄_{n}}. Then Λ/T^{∗} = {0, σ = ω̄_{n−1} = ω̄_{n}}, G_{s} = O_{2n}^{+} and d is the same as in the odd case, i.e. d = 2^{n−1}.

(2) It is the quotient modulo one of the factors, e.g. Λ/T^{∗} = {0, σ = ω̄_{n−1}}, where ω̄_{n} = 0. This corresponds to the half-spin group G_{s} = HSpin_{2n}. We have ω̄_{1} = ω̄_{3} = … = ω̄_{n−1} and ω̄_{i} = 0 if i is even. Therefore, d = gcd(2n, \binom{2n}{3}, …, \binom{2n}{n−3}, 2^{n−1}) which implies that d = 2^{v_{2}(n)+1}, where v_{2}(n) denotes the 2-adic valuation of n.

E_{7}: We have Λ/Λ_{r} = {0, σ = ω̄_{7} = ω̄_{5} = ω̄_{2}} ≃ ℤ/2ℤ with ω̄_{1} = ω̄_{3} = ω̄_{4} = ω̄_{6} = 0. Therefore, d = gcd(56, \binom{56}{3}, 912) = 8.

3.8. Example. Assume that Λ/T^{∗} = ⟨σ_{1}⟩ ⊕ ⟨σ_{2}⟩, where σ_{1} and σ_{2} are of order 2. In this case G_{s} = P GO_{2n}^{+} is an adjoint group (T^{∗} = Λ_{r}) of type D_{n} with n even. We have σ_{1} = ω̄_{n−1} and σ_{2} = ω̄_{n}, ω̄_{s} = sω̄_{1}, 2 ≤ s ≤ n − 2, 2ω̄_{1} = 0 and ω̄_{1} = ω̄_{n−1} + ω̄_{n}. Then

𝔊_{s} ≃ ℤ[y_{1}, y_{2}]/(y_{1}^{2} − 2y_{1}, y_{2}^{2} − 2y_{2}, d_{1}y_{1}, d_{2}y_{2}, d(y_{1} + y_{2} − y_{1}y_{2})),

where y_{1} = (1 − e^{σ_{1}}), y_{2} = (1 − e^{σ_{2}}); d_{1} (resp. d_{2}) is the g.c.d. of dimensions of ω_{i} with ω̄_{i} = ω̄_{n−1} (resp. ω̄_{i} = ω̄_{n}) that is d_{1} = d_{2} = 2^{n−1}; and d is the g.c.d. of dimensions of ω_{1}, ω_{3}, …, ω_{n−3} that is d = gcd(2n, \binom{2n}{3}, … \binom{2n}{n−3}).

In particular, for G_{s} = P GO_{8}^{+} we obtain

𝔊_{s} ≃ ℤ[y_{1}, y_{2}]/(y_{1}^{2} − 2y_{1}, y_{2}^{2} − 2y_{2}, 8y_{1}, 8y_{2}, 8y_{1}y_{2}).

3.9. Example. Assume that Λ/T^{∗} = ⟨σ⟩ has order 3. Then

𝔊_{s} ≃ ℤ[y]/(y^{3} − 3y^{2} + 3y, d_{1}y, d_{2}y^{2}),

where y = (1 − e^{σ}) and d_{1} (resp. d_{2}) is the greatest common divisor of dimensions of fundamental representations ω_{i}, i = 1 … n such that ω̄_{i} = σ (resp. ω̄_{i} = 2σ).

For the adjoint group of type E_{6} we have Λ/Λ_{r} = {0, σ = ω̄_{1} = ω̄_{5}, 2σ = ω̄_{2} = ω̄_{6}\} with ω̄_{2} = ω̄_{4} = 0. Therefore, d_{1} = d_{2} = gcd(27, \binom{27}{2}) = 27.

3.10. Example. Assume that Λ/T^{∗} = ⟨σ⟩ has order 4. Then

𝔊_{s} ≃ ℤ[y]/(y^{4} − 4y^{3} + 6y^{2} − 4y, d_{1}y, d_{2}y^{2}, d_{3}y^{3}),

where y = (1−e^{σ}). For the group P GO_{2n}^{+} where n is odd we have σ = ω̄_{n−1}, 2σ = ω̄_{1} and 3σ = ω̄_{n}. Therefore, d_{1} = d_{3} = 2^{n−1} and d_{2} = gcd(\binom{2n}{1}, \binom{2n}{3}, …, \binom{2n}{n−2}).

4. The twisted γ-filtration.

In the present section we introduce and study the twisted γ-filtration.

4.1. Let γ = ker ϵ denote the augmentation ideal in ℤ[Λ]. It is generated by the differences

⟨(1 − e^{−λ}), λ ∈ Λ⟩.

Consider the γ-adic filtration on ℤ[Λ]

ℤ[Λ] = γ^{0} ⊇ γ ⊇ γ^{2} ⊇ …

The i-th power γ^{i} is generated by products of at least i differences.

4.2. Definition. We define the filtration on K_{0}(𝔅_{s}) (resp. on 𝔊_{s}) to be the image of the γ-adic filtration on ℤ[Λ] via 𝔠 (resp. via q), i.e.

γ^{i}K_{0}(𝔅_{s}) := 𝔠(γ^{i}) and γ^{i}𝔊_{s} := q(γ^{i}K_{0}(𝔅_{s})), i ≥ 0.

So that we have a commutative diagram of surjective group homomorphisms

\xymatrix{γ^{i} \ar[dr] \ar[r]^{𝔠} & γ^{i}K_{0}(𝔅_{s}) \ar[d]^{q} \\  & γ^{i}𝔊_{s}}

4.3. Lemma. The γ-filtration on K_{0}(𝔅_{s}) coincides with the filtration introduced in Definition 4.2.

Proof. Since K_{0}(𝔅_{s}) is generated by the classes of line bundles,

γ^{i}K_{0}(𝔅_{s}) = ⟨c_{1}([L_{1}]) · . . . · c_{1}([L_{m}]) | m ≥ i, L_{j} ∈ K_{0}(𝔅_{s})⟩.

Moreover, each line bundle L is the associated bundle L = L(λ) for some character λ ∈ Λ. Therefore, c_{1}([L]) = 1 − [L^{∨}] = 𝔠(1 − e^{−λ}) (see [3, §2.8]). □

4.4. Definition. Given a G_{s}-torsor ξ ∈ H^{1}(k, G_{s}) and the respective twisted form 𝔅 = _{ξ}𝔅_{s} we define the twisted filtration on 𝔊_{s} to be the image of the γ-filtration on K_{0}(𝔅) via the composite res ∘q, i.e.

γ_{ξ}^{i}𝔊_{s} := q(res(γ^{i}K_{0}(𝔅))), i ≥ 0.

Let γ_{ξ}^{i/i+1}𝔊_{s} = γ_{ξ}^{i}𝔊_{s}/γ_{ξ}^{i+1}𝔊_{s}. The associated graded ring ⨁_{i≥0} γ_{ξ}^{i/i+1}𝔊_{s} will be called the γ-invariant of the torsor ξ and will be denoted simply as γ_{ξ}^{∗}.

Note that the Chern classes commute with restrictions, therefore the restriction map res: γ^{i}K_{0}(𝔅) → γ^{i}K_{0}(𝔅_{s}) is well-defined. By definition there is a surjective ring homomorphism

γ^{∗}(𝔅) ↠ γ_{ξ}^{∗}.

4.5. Theorem. The twisted filtration γ_{ξ}^{i}𝔊_{s} can be computed as follows:

γ_{ξ}^{i}𝔊_{s} = ⟨∏_{j=1}^{m} \binom{ind(β_{ξ}(ρ̄_{w_{j}}))}{n_{j}}(1 − eρ̄_{w_{j}})^{n_{j}} | n_{1} + … + n_{m} ≥ i, w_{j} ∈ W ⟩.

Proof. Since the Chern classes commute with restrictions, the image of the restriction res: γ^{i}K_{0}(𝔅) → γ^{i}K_{0}(𝔅_{s}) is generated by the products

⟨c_{n_{1}}(ı_{w_{1}}g_{w_{1}}) · . . . · c_{n_{m}}(ı_{w_{m}}g_{w_{m}}) | n_{1} + … + n_{m} ≥ i, w_{1}, …, w_{m} ∈ W ⟩,

where {ı_{w_{j}}} are the indexes of the respective Tits algebras from 2.5. Applying the Whitney formula for the Chern classes [7, §3.2] we obtain

c_{j}(ı_{w}g_{w}) = \binom{ı_{w}}{j}c_{1}(g_{w})^{j}.

Therefore, q(\binom{ı_{w}}{j}c_{1}(g_{w})^{j}) = \binom{ı_{w}}{j}(1 − e^{−ρ̄_{w}})^{j}, where ı_{w} = ind(β_{ξ}(ρ̄_{w})). □

4.6. Example. Since γ^{0}(X) ≃ ℤ and γ^{1}(X) = P ic(X) is torsion free for every smooth projective X, we obtain that γ_{ξ}^{0} ≃ ℤ and γ_{ξ}^{1} = 0 for any ξ.

4.7. Example (Strongly-inner case). If β_{ξ} = 0, then \binom{ı_{w_{j}}}{n_{j}} = 1 and γ_{ξ}^{i}𝔊_{s} = γ^{i}𝔊_{s}.

4.8. Example (ℤ/2ℤ-case). As in 3.7 assume that Λ/T^{∗} = ⟨σ⟩ has order 2 and β_{ξ} ≠ 0. Then there is only one non-split Tits algebra A = A_{σ,ξ} and it has exponent 2. Let i_{A} = v_{2}(ind(A)) denote the 2-adic valuation of the index of A. By definition we have

γ_{ξ}^{i}𝔊_{s} = ⟨\binom{2^{i_{A}}}{n_{1}} … \binom{2^{i_{A}}}{n_{m}}2^{n_{1}+...+n_{m}−1}y | n_{1} + … + n_{m} ≥ i⟩

in ℤ[y]/(y^{2} − 2y, dy), where y = 1 − e^{σ} and d is given in 3.7. Observe that modulo the relation y^{2} = 2y these ideals are generated by (for j ≥ 1)

γ_{ξ}^{2j−1}𝔊_{s} = γ_{ξ}^{2j}𝔊_{s} = ⟨2^{2j−1}y⟩  if i_{A} = 1;
γ_{ξ}^{4j−3}𝔊_{s} = γ_{ξ}^{4j−2}𝔊_{s} = ⟨2^{4j−2}y⟩, γ_{ξ}^{4j−1}𝔊_{s} = γ_{ξ}^{4j}𝔊_{s} = ⟨2^{4j−1}y⟩  if i_{A} = 2;
γ_{ξ}^{1}𝔊_{s} = γ_{ξ}^{2}𝔊_{s} = ⟨2^{i_{A}}y⟩, γ_{ξ}^{3}𝔊_{s} = γ_{ξ}^{4}𝔊_{s} = ⟨2^{i_{A}+1}y⟩, γ_{ξ}^{5}𝔊_{s} = ⟨2^{i_{A}+4}y⟩ … if i_{A} > 2.

Taking these generators modulo the relation dy = 0 we obtain the following formulas for the second quotient γ_{ξ}^{2}:

if i_{A} = 1, then γ_{ξ}^{2} = \begin{cases}0 & if v_{2}(d) ≤ 1 \\ ℤ/2ℤ & if v_{2}(d) = 2 \\ ℤ/4ℤ & if v_{2}(d) ≥ 3 \end{cases}

if i_{A} > 1, then γ_{ξ}^{2} = \begin{cases}0 & if v_{2}(d) ≤ i_{A} \\ ℤ/2ℤ & if v_{2}(d) > i_{A} \end{cases}

4.9. Example (ℤ/2ℤ ⊕ ℤ/2ℤ-case). As in 3.8 assume that Λ/T^{∗} = ⟨σ_{1}⟩ ⊕ ⟨σ_{2}⟩, where σ_{1}, σ_{2} have order 2. This is the case for the adjoint group P GO_{2n}^{+} where n is even [9, §25]. Assume that n = 4 which corresponds to the group of type D_{4}, i.e. P GO_{8}^{+}. Let C^{+} and C^{−} denote the Tits algebras corresponding to the generators σ_{1} = ω̄_{3} and σ_{2} = ω̄_{4}. Let A denote the Tits algebra corresponding to the sum σ_{1} + σ_{2}. Note that C^{+} × C^{−} is the even part of the Clifford algebra of the algebra with involution A and [A] = [C^{+} ⊗ C^{−}] in Br(k).

By definition we have in ℤ[y_{1}, y_{2}]

γ_{ξ}^{i}𝔊_{s} = ⟨\binom{ind C_{+}}{n_{1}}y_{1}^{n_{1}} · \binom{ind C_{−}}{n_{2}}y_{2}^{n_{2}} · \binom{ind A}{n_{3}}(y_{1} + y_{2} − y_{1}y_{2})^{n_{3}} | n_{1} + n_{2} + n_{3} ≥ i⟩.

Modulo the relations (y_{1}^{2} − 2y_{1}, y_{2}^{2} − 2y_{2}, 8y_{1}, 8y_{2}, 8y_{1}y_{2}) we obtain that

γ_{ξ}^{2}𝔊_{s} ≃ \frac{(ind C_{+})ℤ}{8ℤ} ⊕ \frac{(ind C_{−})ℤ}{8ℤ} ⊕ \frac{(ind A)ℤ}{8ℤ}

5. Torsion in the γ-filtration.

In the present section we show how the twisted γ-filtration can be used to construct nontrivial torsion elements in the γ-ring of the twisted form 𝔅 of a variety of Borel subgroups.

5.1. For simplicity we consider only the case of G_{s} (see Examples 3.7 and 4.8) with Λ/T^{∗} = ⟨σ⟩ of order 2. Let d denote the g.c.d. of dimensions of fundamental representations corresponding to σ.

Given a G_{s}-torsor ξ ∈ H^{1}(k, G_{s}) let i_{A} denote the 2-adic valuation of the index of the Tits algebra A = A_{σ,ξ}. Let 𝔅 = _{ξ}𝔅_{s} denote the twisted form of the variety of Borel subgroups of G_{s} by means of ξ. Consider the respective twisted filtration γ_{ξ}^{i}𝔊_{s} on 𝔊_{s}.

5.2. Proposition. Assume that v_{2}(d) > i_{A} ≥ 3. Then for each λ ∈ Λ such that λ̄ = σ there exists a non-trivial torsion element of order 2 in γ^{2}(𝔅). Moreover, its image in γ_{ξ}^{2} = ℤ/2 (via q) is non-trivial and in γ^{2}(𝔅_{s}) (via res) is trivial.

Proof. The proof of this result was inspired by the proof of [8, Prop.4.13].

Let g = [L(λ)] denote the class of the associated line bundle. Using the formula for the first Chern class of a tensor product of line bundles for K_{0} we obtain

c_{1}(g)^{2} = 2c_{1}(g) − c_{1}(g^{2}).

Hence,

c_{1}(g)^{4} = (2c_{1}(g) − c_{1}(g^{2}))^{2} = 4c_{1}(g)^{2} − 4c_{1}(g)c_{1}(g^{2}) + c_{1}(g^{2})^{2}.

Therefore,

η = 4c_{1}(g)^{3} − c_{1}(g)^{4} = 4c_{1}(g)^{2} − c_{1}(g^{2})^{2} ∈ γ^{3}K_{0}(𝔅_{s}).

We claim that the class of 2^{i_{A}−3}η gives the desired torsion element.

Indeed, c_{1}(g^{2}) = c_{1}([L(2λ)]). Since 2λ ∈ T^{∗}, [L(2λ)] ∈ 𝔠(T^{∗}) and, therefore, by [5, Cor.3.1] c_{1}(g^{2}) ∈ γ^{1}K_{0}(𝔅). Moreover, we have 2^{i_{A}−1}c_{1}(g)^{2} = c_{2}(2^{i_{A}}g), where 2^{i_{A}}g ∈ K_{0}(𝔅). Hence, 2^{i_{A}−1}c_{1}(g)^{2} ∈ γ^{2}K_{0}(𝔅). Combining these together we obtain that 2^{i_{A}−3}η ∈ γ^{2}K_{0}(𝔅).

Now since 2^{i_{A}−3}η ∈ γ^{2}K_{0}(𝔅) its image in γ_{ξ}^{2}𝔊_{s} can be computed as

q(2^{i_{A}−3}η) = 2^{i_{A}−3}q(η) = 2^{i_{A}−1}q(c_{1}(g)^{2}) = 2^{i_{A}−1}(1 − e^{−σ})^{2} = 2^{i_{A}}y.

But q(2^{i_{A}−3}η) ∉ γ_{ξ}^{3}𝔊_{s} = ⟨2^{i_{A}+1}y⟩. Therefore, 2^{i_{A}−3}η ∉ γ^{3}K_{0}(𝔅).

From the other hand side 2^{i_{A}−2}η = 2^{i_{A}}c_{1}(g)^{3} + 2^{i_{A}−2}c_{1}(g)^{4} is in γ^{3}K_{0}(𝔅). So the class of 2^{i_{A}−3}η gives the desired torsion element of order 2. □

5.3. Example. Let G_{s} = HSpin_{2n} be a half-spin group of rank n ≥ 4. So G_{s} is of type D_{n}, where n is even, Λ/T^{∗} = ⟨σ = ω̄_{1}⟩ ≃ ℤ/2ℤ and according to Example 3.7 we have d = 2^{v_{2}(n)+1}. Let ξ ∈ H^{1}(k, G_{s}) be a non-trivial torsor. Then there is only one Tits algebra A = A_{σ,ξ}; it has exponent 2 and index 2^{i_{A}} such that i_{A} ≤ v_{2}(n)+1.

Recall that each such torsor corresponds to an algebra with orthogonal involution (A, δ) with trivial discriminant and trivial component of the Clifford algebra. The respective twisted form 𝔅 = _{ξ}𝔅_{s} then corresponds to the variety of Borel subgroups of the group P GO^{+}(A, δ).

Applying the proposition to this case we obtain that for any such algebra (A, δ) where 8 | ind(A) and A is non-division, there exists a non-trivial torsion element of order 2 in γ^{2}(𝔅) that vanishes over a splitting field of (A, δ).

5.4. Lemma. The γ-filtration on K_{0}(𝔅_{s}) is generated by the first Chern classes c_{1}([L(ω_{i})]), i = 1 … n, i.e.

γ^{i}K_{0}(𝔅_{s}) = ⟨ ∏_{j∈1...n} c_{1}([L(ω_{j})]) | the number of elements in the product ≥ i⟩.

In particular, the second quotient γ^{2}(𝔅_{s}) is additively generated by the products

γ^{2}(𝔅_{s}) = ⟨c_{1}([L(ω_{i})])c_{1}([L(ω_{j})]) | i, j ∈ 1 … n⟩.

Proof. Each b ∈ K_{0}(𝔅_{s}) can be written as a linear combination b = ∑_{w∈W} a_{w}g_{w}. Therefore, any Chern class of b can be expressed in terms of c_{1}(g_{w}).

Each ρ_{w} can be written uniquely as a linear combination of fundamental weights {ω_{1}, …, ω_{n}}. Therefore, by the formula for the Chern class of the tensor product of line bundles [2, 8.2], each c_{1}(g_{w}) can be expressed in terms of c_{1}([L(ω_{i})]). □

5.5. Example. Let G_{s} be an adjoint group of type E_{7} and let ξ ∈ H^{1}(k, G_{s}) be a non-trivial G_{s}-torsor. Then there is only one nonsplit Tits algebra A = A_{σ,ξ} of exponent 2 and i_{A} ≤ 3. Let 𝔅 = _{ξ}𝔅_{s} be the respective twisted flag variety.

By Lemma 5.4 any element of γ^{2}(𝔅) can be written as

x = ∑_{ij} a_{ij}c_{1}([L(ω_{i})])c_{1}([L(ω_{j})]) ∈ γ^{2}(𝔅)

for certain coefficients a_{ij} ∈ ℤ. Since σ = ω̄_{7} = ω̄_{5} = ω̄_{2} and ω̄_{1} = ω̄_{3} = ω̄_{4} = ω̄_{6} = 0, we obtain that

q(x) = C · 2y ∈ γ_{ξ}^{2}, where C = a_{25} + a_{27} + a_{57} + a_{22} + a_{55} + a_{77}.

Therefore, q(x) ≠ 0 in γ_{ξ}^{2} if and only if 4 ∤ C and i_{A} ≤ 2.

Consider the class 𝔠(θ) ∈ γ^{2}K_{0}(𝔅_{s}) of the special cycle θ constructed in [4, Def.3.3]. Note that the image of θ in CH^{2}(𝔅) can be viewed as a generalization of the Rost invariant for split adjoint groups (see Remark 5.7).

If i_{A} = 1, then by [4, Prop.6.5] we know that 𝔠(θ) ∈ γ^{2}(𝔅) is a non-trivial torsion element. If i_{A} = 2, then following the proof of [4, Prop.6.5] we obtain that 2𝔠(θ) ∈ γ^{2}(𝔅).

We claim that if i_{A} ≤ 2, then x = 2𝔠(θ) is non-trivial. Indeed, in this case 4 ∤ C = a_{22} + a_{55} + a_{77} = 6, therefore, we have q(x) ≠ 0, and x ≠ 0 in γ^{2}(𝔅). In particular, this shows that for i_{A} = 1 the order of the special cycle θ in γ^{2}(𝔅) is divisible by 4.

5.6. Example. Let ξ ∈ H^{1}(k, P GO_{8}^{+}). Applying the same arguments as in 5.5 to Example 4.9 we obtain that if ind(A), ind(C_{+}), ind(C_{−}) ≤ 4, then 2𝔠(θ) ∈ γ^{2}(𝔅) is non-trivial.

We finish by the following remark that provides another motivation for the study of the torsion part of γ^{∗}(𝔅)

5.7. Remark. Recall that by the Riemann-Roch theorem the second Chern class induces a surjection c_{2} : T ors γ^{2}(𝔅) ↠ T ors CH^{2}(𝔅) [8, Cor.2.15], where the latter group is isomorphic to the cohomology quotient [12, Thm.2.1]

\frac{ker(H^{3}(k, ℚ/ℤ(2)) → H^{3}(k(𝔅), ℚ/ℤ(2))}{⊕_{χ∈Λ/Λ_{r}}N_{k_{χ}/k}(k_{χ}^{∗} ∪ β_{ξ}(χ))},

where k_{χ} denotes the fixed subfield of χ. Therefore, the group T ors γ^{2}(𝔅) can be viewed as an estimate for the group of cohomological invariants of G_{s}-torsors of degree 3.

Acknowledgments. I am grateful to Alexander Merkurjev for pointing out Remark 3.2. This work was supported by NSERC Discovery 385795-2010 and Accelerator Supplement 396100-2010 grants.

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(Kirill Zainoulline) Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa ON K1N6N5, Canada