TWISTED GAMMA FILTRATION AND ALGEBRAS WITH ORTHOGONAL INVOLUTION

CAROLINE JUNKINS

Abstract. For the Grothendieck group of a split simple linear algebraic group, the twisted γ-filtration provides a useful tool for constructing torsion elements in γ-rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the γ-ring of a complete flag variety twisted by means of a P GO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline. We use this torsion element to study algebras with orthogonal involutions.

Introduction

Let X be a variety of Borel subgroups of a simple linear algebraic group G over a field k. Consider the Grothendieck group K_{0}(X) and the associated graded ring γ^{∗} K_{0}(X) whose degree i components are determined by the Grothendieck γfiltration. The ring γ^{∗} K_{0}(X) was introduced by Grothendieck as an approximation of the topological filtration on K_{0}, which in turn is related to the Chow groups CH^{∗}(X) of algebraic cycles modulo rational equivalence. By the Riemann-Roch theoreom (cf. [8]), there exists a surjection γ^{2/3} K_{0}(X) ↠ CH^{2}(X) (cf. [4, Ex. 15.3.6]), and so the torsion in γ^{2} K_{0} may be viewed as an upper bound for the torsion in CH^{2}.

Determining torsion in CH^{d}(X) is a non-trivial problem, and only partial results are known. For d = 2, 3 and G stronly inner, we refer to [15] and [6]. The case of quadrics was considered in [12] for d = 2, 3, 4. The γ-filtration was used in [10], and in [11] it was found that T ors CH^{4} can in fact be infinitely generated. Results for arbitrary d have been obtained recently in [2] and [1], by providing upper bounds for the annihilators of T ors CH^{d}.

The γ-filtration has also proved useful for studying the motivic J-invariant in [16] and [9]. In the present paper we use the twisted γ-filtration, introduced in [19], to provide a non-trivial torsion element in γ^{2} K_{0}(X), where X is a complete flag variety twisted by means of a P GO-torsor. Namely, we prove the following

0.1. Theorem. Let X be the variety of Borel subgroups for a linear algebraic group G of type D_{n}, for n ≥ 4 even. Suppose that the indicies 2^{i_{a}} ≤ 2^{i_{b}} ≤ 2^{i_{c}} of the Tits algebras of G are all non-trivial, are bounded above by n, and that they satisfy the additional condition that i_{a}+i_{b} > i_{c} ≥ 2. Under these conditions, there exists a nontrivial torsion element in γ^{2} K_{0}(X) which is nontrivial in the twisted γ-filtration of K_{0}(X) and which vanishes over a splitting field of G.

Furthermore, we may relate this element to a torsion element provided in [19] for orthogonal groups, by considering a finite field extension of k over which one of the Tits algebras of G splits.

This paper is organized as follows. In the first section we recall the definition of the γ-filtration and provide a description through use of the Steinberg basis and Tits algebras of G. In the second and third sections we introduce the twisted γfiltration γ_{ξ}^{∗}𝔊_{s}, and compute γ_{ξ}^{2/3}𝔊_{s} in the case that G = P GO_{2n} for n ≥ 4 even. In the final section, we construct an explicit torsion element in γ^{2} K_{0}(X), provide an example where such an element exists, and finally consider the behaviour of such an element over a finite field extension.

1. Preliminaries

In this section we recall the constructions and results which will be neccessary for the main theorem. We recall the definition of Grothendieck’s γ-filtration, introduce the notion of the Steinberg basis for K_{0} of a complete flag variety and recall basic properties of K_{0} of twisted flag varieties. The reader is advised to consult [4, Chap. 15], [5, Chap. 3] or [6] for further details.

Consider a smooth projective variety X over a field k and its Grothendieck group K_{0}(X). For an element x ∈ K_{0}(X), let c(x) = ∑_{i≥0} c_{i}(x) be the total Chern class of x with values in K_{0}(X) (cf. [4]). We follow the convention that for a line bundle L over X, c_{1}([L]) = 1 − [L^{∨}].

The tensor product of vector bundles endows K_{0}(X) with a ring structure, and while it does not have a canonical grading, we may impose one through the construction of a filtration. The γ-filtration is defined using the above Chern classes. We define

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

and let γ^{i/i+1}K_{0}(X) = γ^{i} K_{0}(X)/γ^{i+1} K_{0}(X) denote the degree i component of the corresponding graded ring (cf. [8, §2.3]).

Consider a split simple linear algebraic group G_{s} of rank n over k. We fix a split maximal torus T and Borel subgroup B such that T ⊂ B ⊂ G_{s}. Let X_{s} = G_{s}/B be the variety of Borel subgroups. Let K_{0}(X_{s}) be the Grothendieck group of X_{s} and recall that since G_{s} is split, K_{0}(X_{s}) is generated by line bundles (see [8]).

Let Λ_{r} and Λ be the root and weight lattices of G_{s} respectively. Let {α_{1}, …, α_{n}} be a set of simple roots and {ω_{1}, …, ω_{n}} the respective set of fundamental weights, that is α_{i}^{∨}(ω_{j}) = δ_{ij}. The character group T^{∗} of T is an intermediate lattice Λ_{r} ⊆ T^{∗} ⊆ Λ which determines the isogeny class of G_{s}. For instance if T^{∗} = Λ_{r} then G_{s} is adjoint, and if T^{∗} = Λ then G_{s} is simply connected.

With this, we may define the integral group ring ℤ[T^{∗}] whose elements are linear combinations ∑_{i} a_{i}e^{λ_{i}}, for a_{i} ∈ ℤ and λ_{i} ∈ T^{∗}. Let

𝔠 : ℤ[T^{∗}] → K_{0}(X_{s})

be the characteristic map, defined by sending e^{λ} to the associated line bundle [L(λ)] for λ ∈ Λ. Note that while K_{0}(X_{s}) does not depend on the isogeny class of G_{s}, the image of this map does. In particular, if G_{s} is simply connected then the characteristic map is surjective [8].

The Weyl group W of G_{s} acts on weights via simple reflections s_{α_{i}}, i = 1, …, n. That is s_{α_{i}}(λ) = λ − α_{i}^{∨}(λ)α_{i}, for λ ∈ Λ. So, for each element w ∈ W we may define the weight ρ_{w} ∈ Λ (cf. [17, §2.1]) as

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

In particular if w = s_{α_{i}}, then ρ_{w} = s_{α_{i}}(ω_{i}) = ω_{i} − α_{i}. Since W acts trivially on Λ/Λ_{r}, we have

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

where \overline{ρ}_{w} denotes the class of ρ_{w} ∈ Λ modulo T^{∗}. By the characteristic map of the simply connected cover of G_{s}, we may associate to each w ∈ W the class of the associated line bundle g_{w} := 𝔠(e^{ρ_{w}}) = [L(ρ_{w})]. These elements form a ℤ-basis of K_{0}(X_{s}) called the Steinberg basis. The Steinberg basis allows a nice description of the γ-filtration of X_{s}, since we may express generators in terms of only first Chern classes:

γ^{i} K_{0}(X_{s}) = ⟨c_{1}(g_{w_{1}}) · . . . · c_{1}(g_{w_{m}}) | m ≥ i, w_{1}, …, w_{m} ∈ W ⟩.

Unfortunately, for a cocyle ξ ∈ Z^{1}(k, G_{s}) the twisted flag variety X = _{ξ}X_{s} is not in general generated by line bundles, and so a description of γ^{i} K_{0}(X) is not as straightforward.

For a fixed ξ ∈ Z^{1}(k, G_{s}), we can associate to each weight \overline{λ} ∈ Λ/T^{∗} a central simple k-algebra A_{ξ,\overline{λ}}, called a Tits algebra of G_{s} with index ind(A_{ξ,\overline{λ}}) (cf. [18]). We define the Tits map

β_{ξ} : Λ/T^{∗} → Br(k)

by sending \overline{λ} to the class [A_{ξ,\overline{λ}}] in the Brauer group. This map is a group homomorphism for a fixed ξ, with \overline{λ_{1}} + \overline{λ_{2}} ↦ [A_{ξ,\overline{λ_{1}}}] ⊗ [A_{ξ,\overline{λ_{2}}}].

Let X = _{ξ}X_{s}, let k_{sep} be the separable closure of X, and consider the restriction map

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

where we identify K_{0}(X ×_{k} k_{sep}) with K_{0}(X_{s}). The main result of [14] says that the image of this restriction map coincides with the sublattice ⟨ind(A_{w})g_{w}⟩_{w∈W} , where g_{w} is an element of the Steinberg basis and ind(A_{w}) ≥ 1 is the index of the respective Tits algebra A_{ξ,\overline{ρ_{w}}}. Note that if G_{s} is simply connected the restriction map is an isomorphism.

Since characteristic classes commute with restrictions, the restriction map res: γ^{i} K_{0}(X) → γ^{i} K_{0}(X_{s}) is well defined, and

res(γ^{i} K_{0}(X)) = ⟨∏_{j=1}^{m} \binom{ind(A_{w_{j}})}{n_{j}}c_{1}(g_{w_{j}})^{n_{j}} | n_{1} + ⋯ + n_{m} ≥ i, w_{j} ∈ W ⟩ .

2. The twisted γ-filtration

While we have an explicit description of the image res(γ^{2/3} K_{0}(X)) in terms of the Steinberg basis and the indices of the Tits algebras, it is not as practical to work with as one would hope. One problem is that expressing the tensor product of two Steinberg elements as a linear combination of Steinberg elements is a non-trivial task (especially when W is large). The twisted γ-filtration was introduced in [19] as a tool for getting identifying elements in γ^{2/3} K_{0}(X) more easily.

Returning to the characteristic map and the canonical surjection Λ ↦ Λ/T^{∗}, we have the following diagram,

\xymatrix{ℤ[Λ] \ar@{->>}[r]^{𝔠} \ar@{->>}[d] & K_{0}(X_{s}) \ar[dr]^{q} \ar[r]^{≃} & ℤ[Λ]/ ker(𝔠) \ar@{->>}[d] \\ ℤ[Λ/T^{∗}] \ar@{->>}[rr] &  & ℤ[Λ/T^{∗}]/\overline{ker(𝔠)}}

which allows us to define the quotient ring

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

and the composite map q : K_{0}(X_{s}) → 𝔊_{s}, which is a surjective ring homomorphism. Observe that if G_{s} is simply connected then 𝔊_{s} ≃ ℤ.

By Lemma 3.3 in [19] 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 number of elements in the W -orbit of the fundamental weight ω_{i}.

Consider next the γ-adic filtration on ℤ[Λ], where the i-th power γ^{i} is generated by products of at least i differences. That is,

γ^{i} = ⟨(1 − e^{λ_{1}}) … (1 − e^{λ_{k}}) | k ≥ i⟩.

With this we have γ^{i} K_{0}(X_{s}) = 𝔠(γ^{i}) and we define γ^{i}𝔊_{s} := q(γ^{i} K_{0}(X_{s})), i ≥ 0.

Given a fixed G_{s}-torsor ξ and the respective twisted form X = _{ξ}X_{s}, define the twisted γ-filtration on 𝔊_{s} to be:

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

and let γ_{ξ}^{i/i+1} 𝔊_{s} = γ_{ξ}^{i}𝔊_{s}/γ_{ξ}^{i+1} 𝔊_{s}. The associated graded ring γ_{ξ}^{∗} := ⨁_{i≥0} γ_{ξ}^{i/i+1}𝔊_{s} is called the γ-invariant of ξ.

Futhermore, there exists a surjective ring homomorphism γ^{∗}(X) ↠ γ_{ξ}^{∗}, so we may provide a description of the twisted γ-filtration similar to that in the previous section. By Theorem 4.5 in [19],

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

3. Algebras with orthogonal involution

Let G_{s} be the adjoint group P GO_{2n}^{+}, for n ≥ 4 even; ie. G is of type D_{n}, n even. In this case, Λ/T^{∗} = Λ/Λ_{r} = \{0, \overline{ω}_{1}, \overline{ω}_{n−1}, \overline{ω}_{n}\} ≃ ℤ/2ℤ ⊕ ℤ/2ℤ. That is, 2ω̄_{1} = 0, ω̄_{1} = ω̄_{n−1} + ω̄_{n}, and ω̄_{s} = sω̄_{1} for 2 ≤ s ≤ n − 2. Let C_{+} and C_{−} denote the Tits algebras corresponding to \overline{ω}_{n−1} and \overline{ω}_{n}. Let A denote the Tits algebra corresponding to \overline{ω}_{1}. We note that C_{+} × C_{−} is the even part of the Clifford algebra of the algebra with involution A, and [A] + [C_{+}] + [C_{−}] = 0 in Br(k) (cf.[13]).

In order to simplify some calculations, we choose the generators σ_{1} and σ_{2} from \{\overline{ω}_{1}, \overline{ω}_{n−1}, \overline{ω}_{n}\} such that ind(β_{ξ}(σ_{1})) ≤ ind(β_{ξ}(σ_{2})) ≤ ind(β_{ξ}(σ_{1} + σ_{2})). For ease of notation we will define the following non-negative integers:

i_{a} = v_{2}(ind(β_{ξ}(σ_{1}))), i_{b} = v_{2}(ind(β_{ξ}(σ_{2}))), i_{c} = v_{2}(ind(β_{ξ}(σ_{1} + σ_{2})))

Thus i_{a} ≤ i_{b} ≤ i_{c}, and by the relation in the Brauer group, we have i_{c} ≤ i_{a} + i_{b}. Then letting y_{1} = 1 − e^{σ_{1}} and y_{2} = 1 − e^{σ_{2}}, we have

𝔊_{s} ≅ ℤ[y_{1}, y_{2}]/(y_{1}^{2} − 2y_{1}, y_{2}^{2} − 2y_{2}, d_{a}y_{1}, d_{b}y_{2}, d_{c}(y_{1} + y_{2} − y_{1}y_{2})),

where d_{a}, d_{b}, d_{c} are determined by the number of elements in the W -orbit of the three corresponding fundamental weights.

3.1. Example. If n = 4 (ie. G_{s} = P GO_{8}^{+}), then d_{a} = d_{b} = d_{c} = 2^{3}, and so

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

To compute γ_{ξ}^{2/3}𝔊_{s} we first make note of 4 cases, dependent on the indices of the Tits algebras. The arguments used are independent of the choice of generator, so without loss of generality we consider the generator y_{1} and its associated index 2^{i_{a}}.

Consider first the case that the associated Tits algebra is split, ie. i_{a} = 0. Then, y_{1}^{2} ≡ 2y_{1} ∈ γ_{ξ}^{2}𝔊_{s}, and y_{1}^{3} ≡ 4y_{1} ∈ γ_{ξ}^{3}𝔊_{s}. Thus γ_{ξ}^{2/3}𝔊_{s} has a summand isomorphic to ℤ/2ℤ.

Next, suppose i_{a} = 1. Then \binom{2}{2}y_{1}^{2} ≡ 2y_{1} ∈ γ_{ξ}^{2}𝔊_{s}, and \binom{2}{2}\binom{2}{1}y_{1}^{3} ≡ 8y_{1} ∈ γ_{ξ}^{3}𝔊_{s}.

This time, γ_{ξ}^{2/3}𝔊_{s} has a summand isomorphic to ℤ/4ℤ.

Thirdly, consider the case that 2 ≤ i_{a} < v_{2}(d_{a}). Then \binom{2^{ia}}{2}y_{1}^{2} ≡ 2^{i_{a}}y_{1} ∈ γ_{ξ}^{2}(G)_{s},

and \binom{2^{ia}}{4}y_{1}^{4} ≡ 2^{i_{a}+1}y_{1} ∈ γ_{ξ}^{4}𝔊_{s} ⊆ γ_{ξ}^{3}𝔊_{s}. So, γ_{ξ}^{2/3}𝔊_{s} has a summand isomorphic to ℤ/2ℤ

Finally, suppose i_{a} ≥ v_{2}(d_{a})). In this situation we have \binom{2^{ia}}{r}y_{1}^{r} | 2^{i_{a}}y_{1} ≡ 0 for all r ≥ 2, and so the summand is trivial.

For y_{1} + y_{2} − y_{1}y_{2}, the results are slightly different for i_{c} ≥ 2. We have 2^{i_{c}}(y_{1} + y_{2} − y_{1}y_{2}) ∈ γ^{2}𝔊_{s} as before; however, in γ^{4}𝔊_{s} we have the additional element 2^{i_{a}}2\binom{2^{i_{b}}}{2}y_{1}^{2}y_{2}^{2} ≡ 2^{i_{a}+i_{b}}y_{1}y_{2}. By the relations in the Brauer group, i_{a} + i_{b} ≥ i_{c}, and so we have a non-trivial multiple of y_{1} + y_{2} − y_{1}y_{2} only if this inequality is strict, or if i_{a} = 0. In the specific case that i_{a} = i_{b} = 1, i_{c} = 2, we have y_{1}^{2}y_{2}^{2} ≡ 4y_{1}y_{2} ∈ γ^{3}𝔊_{s}, but 4y_{1}, 4y_{2} ∉ γ^{3}𝔊_{s}.

We may summarize this as follows.

3.2. Lemma. Let γ_{ξ}^{2/3}𝔊_{s} = G_{a} ⊕ G_{b} ⊕ G_{c}. Then we have

G_{a} ≃ \begin{cases}ℤ/2ℤ & if i_{a} = 0 \\ ℤ/4ℤ & if i_{a} = 1 \\ ℤ/2ℤ & if 2 ≤ i_{a} < v_{2}(d_{a}) \\ 0 & if i_{a} ≥ v_{2}(d_{a}) \end{cases} G_{b} ≃ \begin{cases}ℤ/2ℤ & if i_{b} = 0 \\ ℤ/4ℤ & if i_{b} = 1 \\ ℤ/2ℤ & if 2 ≤ i_{b} < v_{2}(d_{b}) \\ 0 & if i_{b} ≥ v_{2}(d_{b}) \end{cases}

G_{c} ≃ \begin{cases}ℤ/2ℤ & if i_{c} = 0 \\ ℤ/4ℤ & if i_{c} = 1 \\ ℤ/2ℤ & if i_{c} = 2, i_{a} = i_{b} = 1 \\ ℤ/2ℤ & if 2 ≤ i_{c} < v_{2}(d_{c}), i_{a} + i_{b} < i_{c} \\ 0 & if i_{c} ≥ 2, i_{a} + i_{b} = i_{c} \\ 0 & if i_{c} ≥ v_{2}(d_{c}) \end{cases}

3.3. Example. Let G_{s} = HSpin_{2n} be a half-spin group of rank n ≥ 4. Then, G_{s} is of type D_{n} for n even, and we have Λ/T^{∗} = ⟨σ⟩ where σ is of order 2 [13].

This corresponds to taking the quotient of Λ/Λ_{r} ≃ ⟨σ_{1}⟩ ⊕ ⟨σ_{2}⟩ modulo one of the generators, eg. σ_{2} ≡ 0. In this situation, we have

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

where d = 2^{v_{2}(n)+1}. Thus we may describe the twisted γ-filtration of G_{s} by taking the quotient of our previous description of γ^{2/3}𝔊_{s} for G_{s} = P GO_{2n}^{+} by ⟨2^{v_{2}(n)+1}y_{1}, y_{2}⟩. So,

γ^{2/3}𝔊_{s} ≃ \begin{cases}ℤ/2ℤ & if i_{a} = 0 \\ ℤ/4ℤ & if i_{a} = 1 \\ ℤ/2ℤ & if 2 ≤ i_{a} ≤ v_{2}(n) \\ 0 & if ]ia > v_{2}(n) \end{cases}

Note that this corresponds to the result given in [19, Example 4.8].

4. Construction of a torsion element

In this section, we provide an explicit torsion element in γ^{2/3} K_{0}(X) using the twisted γ-filtration. Namely, we prove the following

4.1. Theorem. Assume that 0 < i_{a} ≤ i_{b} ≤ i_{c} ≤ v_{2}(n), and that i_{a} + i_{b} > i_{c} ≥ 2. Then there exists a non-trivial torsion element of order dividing 2^{i_{a}+i_{b}−i_{c}} in γ^{2/3} K_{0}(X). Furthermore, its image in γ_{ξ}^{2/3}𝔊_{s} via q is non-trivial, and its image in γ^{2/3} K_{0}(X_{s}) via res is trivial.

Proof. By [7, Cor. 5.5], 𝔠(ℤ[T^{∗}]) ⊂ K_{0}(X), so we begin by considering classes of bundles lying in this image.

Denote by g_{a}, g_{b}, g_{c} classes of line bundles L(λ), where λ̄ = σ_{1}, σ_{2} and σ_{1} + σ_{2} respectively. Then, 2λ ∈ T^{∗} in each case and so c_{1}(g^{2}) = c_{1}([L(2λ)]) ∈ γ^{1} K_{0}(X) for g = g_{a}, g_{b} and g_{c}. In particular, the following element then lies in γ^{2} K_{0}(X):

c_{1}(g_{a}^{2})c_{1}(g_{b}^{2}) = 4c_{1}(g_{a})c_{1}(g_{b}) − 2c_{1}(g_{a})c_{1}(g_{b})^{2} − 2c_{1}(g_{a})^{2}c_{1}(g_{b}) + c_{1}(g_{a})^{2}c_{1}(g_{b})^{2}

= c_{1}(g_{a})^{2}c_{1}(g_{b})^{2} + 2c_{1}(g_{a})c_{1}(g_{b})c_{1}(g_{c}) − 2c_{1}(g_{a})^{2} − 2c_{1}(g_{b})^{2} + 2c_{1}(g_{c})^{2}.

By the retriction map, c_{2}(i_{w}g_{w}) = \binom{i_{w}}{2}c_{1}(g_{w})^{2} ∈ γ^{2} K_{0}(X) for all w ∈ W. So, under the hypothesis that i_{c} ≥ 2, we may consider the following element in γ^{2} K_{0}(X):

μ = 2^{i_{c}−2}c_{1}(g_{a}^{2})c_{1}(g_{b}^{2}) + c_{2}(2^{i_{c}}g_{a}) + c_{2}(2^{i_{c}}g_{b}) − c_{2}(2^{i_{c}}g_{c}) ∈ γ^{2} K_{0}(X).

Note that we may permute the line bundles g_{a}, g_{b}, g_{c} in the definition of μ (but keep the coefficients as 2^{i_{c}−2} and 2^{i_{c}} respectively), and we will still have an element of γ^{2} K_{0}(X). In fact, all subsequent arguments are valid for such permutations as well.

By Chern class operations and the relation c_{1}(g_{c}) = c_{1}(g_{a})+c_{1}(g_{b})−c_{1}(g_{a})c_{1}(g_{b}),

μ = 2^{i_{c}−2}c_{1}(g_{a}^{2})c_{1}(g_{b}^{2}) + 2^{i_{c}−1}c_{1}(g_{a})^{2} + 2^{i_{c}−1}c_{1}(g_{b})^{2} − 2^{i_{c}−1}c_{1}(g_{c})^{2}
= 2^{i_{c}−1}c_{1}(g_{a})^{2}c_{1}(g_{b}) + 2^{i_{c}−1}c_{1}(g_{a})c_{1}(g_{b})^{2} − 2^{i_{c}−2}c_{1}(g_{a})^{2}c_{1}(g_{b})^{2}.

From this, it is clear that res(μ) ∈ γ^{3} K_{0}(X_{s}). Next we apply the map q,

q(res(μ)) = 2^{i_{c}−2}y_{1}^{2}y_{2}^{2} + 2^{i_{c}−1}y_{1}y_{2}(y_{1} + y_{2} − y_{1}y_{2}) = 2^{i_{c}}y_{1}y_{2}.

By the hypothesis that i_{c} ≤ v_{2}(n), we have 2^{i_{c}}y_{1}y_{2} ∉ γ_{ξ}^{3}𝔊_{s}, and hence μ ∉ γ^{3} K_{0}(X).

Finally, we must check that μ is torsion. Consider the following elements, which by definition lie in res(γ^{3} K_{0}(X)):

\binom{2^{i_{a}}}{2}\binom{2^{i_{b}}}{1}c_{1}(g_{a})^{2}c_{1}(g_{b}) = 2^{i_{a}+i_{b}−1}c_{1}(g_{a})^{2}c_{1}(g_{b})
\binom{2^{i_{a}}}{1}\binom{2^{i_{b}}}{2}c_{1}(g_{a})c_{1}(g_{b})^{2} = 2^{i_{a}+i_{b}−1}c_{1}(g_{a})c_{1}(g_{b})^{2}
\binom{2^{i_{a}}}{2}\binom{2^{i_{b}}}{2}c_{1}(g_{a})^{2}c_{1}(g_{b})^{2} = 2^{i_{a}+i_{b}−2}c_{1}(g_{a})^{2}c_{1}(g_{b})^{2}.

Thus,

2^{i_{a}+i_{b}−i_{c}}μ = 2^{i_{a}+i_{b}−1}c_{1}(g_{a})^{2}c_{1}(g_{b}) + 2^{i_{a}+i_{b}−1}c_{1}(g_{a})c_{1}(g_{b})^{2} − 2^{i_{a}+i_{b}−2}c_{1}(g_{a})^{2}c_{1}(g_{b})^{2}.

□

4.2. Example. Consider again the case that n = 4. Then, the indices which saitsfy the neccessary hypotheses are 1 ≤ i_{a} ≤ 2, i_{b} = i_{c} = 2. So, we have

μ = c_{1}(g_{a}^{2})c_{1}(g_{b}^{2}) + c_{2}(4g_{a})^{2} + c_{2}(4g_{b}) − c_{2}(4g_{c})
= 2c_{1}(g_{a})^{2}c_{1}(g_{b}) + 2c_{1}(g_{a})c_{1}(g_{b})^{2} − c_{1}(g_{a})^{2}c_{1}(g_{b})^{2}.

Now if i_{a} = 1, we have 2^{i_{a}+i_{b}−i_{c}}μ = 2μ ∈ γ^{3} K_{0}(X) by the above argument. If instead i_{a} = 2, then 2^{i_{a}+i_{b}−i_{c}}μ = 4μ ∈ γ^{3} K_{0}(X). In either case μ is a non-trivial torsion element in γ^{2/3} K_{0}(X).

An explict example can be constructed for the case that i_{a} = 1 and i_{b} = i_{c} = 2. We may define G = P GO^{+}(A, σ) of type D_{4}, by the algebra with involution (A, σ), where (B, τ) and (C, γ) are the two components of the Clifford algebra C(A, σ), each endowed with its canonical involution. It follows by the structure theorems [13] that both are degree 8 algebras with orthogonal involutions. The example we present here was contructed in [16], using the notion of direct sums of agebras with involution introduced by Dejaiffe (cf. [3]).

Let F = K(x, y, z, t) be a function field in 4 variables over a field K, and consider the quaternion algebras over F defined by

Q_{1} = (x, zt), Q_{2} = (y, zt), Q_{3} = (xy, t), Q_{4} = (xy, z).

Let (A, σ) be the direct sum of (Q_{1},¯ ) ⊗ (Q_{3},¯ ) and (Q_{2},¯ ) ⊗ (Q_{4},¯ ). Denote by (B, τ) the component of C(A, σ) Brauer equivalent to (Q_{1},¯ )⊗(Q_{3},¯ ) ∼ (x, t)⊗(y, z), and by (C, γ) the component of C(A, σ) Brauer equivalent to (Q_{1},¯ )⊗(Q_{4},¯ ) ∼ (x, z)⊗(y, t). Thus i_{a} = v_{2}(ind(A)) = 1, i_{b} = v_{2}(ind(B)) = 2 and i_{c} = v_{2}(ind(C)) = 2, as required.

We consider the behaviour of the torsion element μ over finite field extensions.

4.3. Example. Consider a finite field extension l/k such that one of the Tits algebras becomes split, that is either C_{+} or C_{−} splits over l. We may look at the image of our element μ under the restriction map res_{l/k} : K_{0}(X) → K_{0}(X × l).

Suppose that σ_{1} + σ_{2} = \overline{ω}_{n−1}, and that C_{+} becomes split over l. Over this field extension we have Λ/T^{∗} = \{0, σ = \overline{ω}_{n}\}, where \overline{ω}_{n−1} = 0 and \overline{ω}_{1} = \overline{ω}_{n}. So, Λ/T^{∗} ≃ ℤ/2ℤ, corresponding to [19, Example 3.6]. Let g = [L(λ)], where \overline{λ} = σ, and let i_{∗} = v_{2}(ind(β_{ξ}(\overline{λ}))), that is, the 2-adic valuation of the index of the Tits algebra of G over l. Since indices of Tits algebras cannot increase over field extensions we have i_{∗} ≤ i_{c}. With this notation, we have

res_{l/k}(c_{1}(g_{a})) = res_{l/k}(c_{1}(g_{b})) = c_{1}(g) and res_{l/k}(c_{1}(g_{c})) = 1.

With this,

res_{l/k}(μ) = 2^{i_{c}}c_{1}(g)^{3} − 2^{i_{c}−2}c_{1}(g)^{4} = 2^{i_{c}−i_{∗}+1}η,

where η = 2^{i_{∗}−3}(4c_{1}(g)^{3} − c_{1}(g)^{4}), the torsion element defined in [19]. Since η is a torsion element of order 2, we observe that μ becomes trivial over this field extension.

Acknowledgments. I am grateful to my PhD supervisor K. Zainoulline for useful discussions on the subject of this paper. This work has been partially supported by his NSERC Discovery grant 385795-2010, by my NSERC Canadian graduate scholarship, and by a graduate scholarship provided by the German Academic Exchange Service. My special gratitude is due to N. Semenov and the University of Mainz for their hospitality and support.

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