THE γ-FILTRATION AND THE ROST INVARIANT

SKIP GARIBALDI AND KIRILL ZAINOULLINE

Abstract. Let X be the variety of Borel subgroups of a simple and strongly inner linear algebraic group G over a field k. We prove that the torsion part of the second quotient of Grothendieck’s γ-filtration on X is a cyclic group of order the Dynkin index of G.

As a byproduct of the proof we obtain an explicit cycle θ that generates this cyclic group; we provide an upper bound for the torsion of the Chow group of codimension-3 cycles on X; we relate the cycle θ with the Rost invariant and the torsion of the respective generalized Rost motives; we use θ to obtain a uniform lower bound for the essential dimension of (almost) all simple linear algebraic groups.

Grothendieck’s celebrated γ-filtration of the ring K_{0}(X) gives a way to estimate the Chow group CH^{∗}(X) of algebraic cycles on X modulo rational equivalence when X is a smooth projective variety over a field k. Namely, by the Riemann-Roch theorem without denominators [8, §15.3] the i-th Chern class provides a well-defined group homomorphism

c_{i}: γ^{i/i+1}(X) → CH^{i}(X), i ≥ 0

from the i-th quotient of the γ-filtration to the Chow group of codimension-i cycles on X. Observe that for i = 0 it is the identity map and for i = 1 it is an isomorphism identifying CH^{1}(X) with the Picard group of X.

In the present paper we study these homomorphisms in the cases i = 2, 3 and X is a generically split projective homogeneous variety under a semisimple linear algebraic group G. Our core determines and bounds respectively the torsion subgroup of γ^{2/3}(𝔅) and γ^{3/4}(𝔅) for the variety of Borel subgroups 𝔅 of strongly inner G (Theorem 3.1). For instance, we show that the torsion subgroup Tors γ^{2/3}(𝔅) is cyclic of order the Dynkin index of G and exhibit a generator θ for it (Definition 3.3).

This fact together with the Riemann-Roch theorem imply (see §4) that the surjection

\xymatrix{⟨θ⟩ = Tors γ^{2/3}(𝔅) \ar@{->>}[r]^{c_{2}} & Tors CH^{2}(𝔅)}

can be viewed as a substitute of the key map Q(V) → H^{3}(k, ℚ/ℤ(2)) in the definition of the Rost invariant [10, pp. 126-127]. Indeed, a theorem of Peyre-Merkurjev [27] shows that Tors CH^{2}(𝔅) can be identified with the kernel of the restriction H^{3}(k, ℚ/ℤ(2)) → H^{3}(k(𝔅), ℚ/ℤ(2)). Furthermore, the order of c_{2}(θ) in Tors CH^{2}(𝔅) equals to the order of the Rost invariant of G (see Prop. 3.2).

Our result gives bounds for the torsion in CH^{3} for generically split X (see §5) and provides explicit generators of torsion subgroups of CH^{2} of certain generalized Rost-Voevodsky motives. Note that typically, one does not even know a priori if the torsion subgroup of CH^{i}(X), i ≥ 3, is finitely generated. However, determining the torsion subgroup determines CH^{i}(X) as an abelian group, since the dimension of its free part CH^{i}(X) ⊗ ℚ can be easily computed.

In section 6, we study the behaviour of the image of c_{2} under field extensions. In particular, we show that this image is non-trivial if G has Tits algebras of index 2 (Prop. 6.5). Using this fact we prove (Prop. 7.1) that the essential dimension ed(G) of any absolutely almost simple linear algebraic group not of type A nor isomorphic to Sp_{2n} is greater or equal than 3.

1. Preliminaries

We now provide several facts and observations concerning Chow groups, characteristic maps, invariants, Dynkin indices and filtrations on K_{0} for varieties of Borel subgroups of split simple linear algebraic groups.

§1A. Two filtrations on K_{0}. All facts provided here can be found in [17, §2], [8, §15] and [9, Ch. 3,5]. Let X be a smooth projective variety over a field k. Consider the γ-filtration on K_{0}(X). It is given by the subgroups

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

where c_{n}^{K_{0}} denote the n-th Chern class with values in K_{0}. For example, for the class of a line bundle we have c_{1}^{K_{0}}([L]) = 1 − [L^{∗}]. Let γ^{i/i+1}(X) = γ^{i}(X)/γ^{i+1}(X) denote the respective quotient. Consider the topological filtration on K_{0}(X) given by the subgroups

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

where [O_{V}] is the class of the structure sheaf of a closed subvariety V. Let τ^{i/i+1}(X) = τ^{i}(X)/τ^{i+1}(X) denote the corresponding quotient.

There is an obvious surjection pr: CH^{i}(X) ↠ τ^{i/i+1}(X) from the Chow group of codimension i cycles given by V ↦ [O_{V}]. By the Riemann-Roch Theorem without denominators the i-th Chern class induces the map in the opposite direction

c_{i}: τ^{i/i+1}(X) → CH^{i}(X)

and the composite c_{i} ∘ pr is the multiplication by (−1)^{i−1}(i − 1)! which is an isomorphism for i ≤ 2 [8, Ex.15.3.6]. For example, by the very definition we have

c_{i}(∏_{j=1}^{i} c_{1}^{K_{0}}([L_{j}])) = (−1)^{i−1}(i − 1)! ∏_{j=1}^{i} c_{1}^{CH}(L_{j}),

where L_{j} is a line bundle. Observe also that c_{i} becomes an isomorphism after tensoring with ℚ.

There is an embedding γ^{i}(X) ⊂ τ^{i}(X) for all i. Moreover, γ^{i}(X) = τ^{i}(X) for i ≤ 2. Observe that γ^{1/2}(X) = τ^{1/2}(X) = CH^{1}(X) is the Picard group and by [17, Cor. 2.15] there is an exact sequence

(1.1)  0 → τ^{3}(X)/γ^{3}(X) → Tors γ^{2/3}(X) ⟶^{c_{2}} Tors CH^{2}(X) → 0,

where we have written c_{2} for the composition γ^{2/3}(X) → τ^{2/3}(X) ⟶^{c_{2}} CH^{2}(X).

§1B. Characteristic maps and invariants. Let G_{s} be a split simply connected simple linear algebraic group of rank n over a field k. We fix a split maximal torus T and a Borel subgroup B such that T ⊂ B ⊂ G_{s}. Let 𝔅_{s} denote the variety of Borel subgroups of G_{s} and let T^{∗} denote the group of characters of T. We fix a basis of T^{∗} given by the fundamental weights ω_{1}, …, ω_{n}.

Let 𝕊(T^{∗}) be the symmetric algebra of T^{∗}. Its elements are polynomials in the fundamental weights ω_{i} with coefficients in ℤ. Let ℤ[T^{∗}] be the integral group ring of T^{∗}. Its elements are integral linear combinations ∑_{i} a_{i}e^{λ_{i}}, λ_{i} ∈ T^{∗}. Consider the characteristic maps for CH and K_{0} (see [5, §8, 9] and [6, §1.5, 1.6])

𝔠: 𝕊(T^{∗}) → CH(𝔅_{s}) and 𝔠′: ℤ[T^{∗}] → K_{0}(𝔅_{s})

given by

𝔠: ω_{i} ↦ c_{1}^{CH}(L(ω_{i})) and 𝔠′: e^{λ} ↦ [L(λ)]

where L(λ) is the line bundle over 𝔅_{s} associated to the character λ.

There are obvious augmentation maps 𝕊(T^{∗}) → ℤ and aug: ℤ[T^{∗}] → ℤ given by ω_{i} ↦ 0 and e^{λ} ↦ 1 respectively. The Weyl group acts naturally on T^{∗}, hence also on 𝕊(T^{∗}) and ℤ[T^{∗}]. Consider the subrings of invariants 𝕊(T^{∗})^{W} and ℤ[T^{∗}]^{W}. Let I (resp. I′) be the ideal generated by the elements of 𝕊(T^{∗})^{W} (resp. ℤ[T^{∗}]^{W}) from the kernel of the augmentation map. Then we have

ker 𝔠 = I and ker 𝔠′ = I′.

Therefore we have embeddings

𝔠: 𝕊(T^{∗})/I ↪ CH(𝔅_{s}) and 𝔠′: ℤ[T^{∗}]/I′ ⟶^{≃} K_{0}(𝔅_{s}),

where the second map is surjective since G_{s} is simply connected (see [28]). By [5, §2 and Cor.2] the kernel I of 𝔠 consists of elements g such that

(1.2)  m · g = ∑_{i} g_{i} · f_{i},

for m ∈ ℤ, f_{i} the basic polynomial invariants, and g_{i} ∈ 𝕊(T^{∗}).

There is a W -invariant quadratic form q on T^{∗} ⊗ ℚ that is uniquely determined up to a scalar multiple [2, §VI.1.1–2]. We normalize q so that it takes the value 1 on every short coroot; as q is indivisible, it can be taken as the generator of I of degree 2. To say it differently, each element of I of degree 2 is a multiple of q by an integer.

The form q should be familiar. Its polar bilinear form b_{q} amounts to the restriction of the “reduced Killing form” to the Cartan subalgebra of the Lie algebra of G_{s} as described in [15, §5]. In the case where the roots are all one length, an explicit formula for b_{q} is well known: its Gram matrix is the Cartan matrix of the root system.

§1C. Degree 3 elements of I. If G_{s} is not of type A_{n} (n ≥ 2), then there is no basic invariant of degree 3 [16, p. 59]. Then by (1.2) and the indivisibility of q, every g ∈ I of degree 3 can be written as g = (∑ a_{i}ω_{i})q for some a_{i} ∈ ℤ.

If G_{s} is of type A_{n} for some n ≥ 2, then 𝕊(T^{∗})^{W} has a basic generator f_{3} of degree 3. We view the weight lattice as in [2], meaning that it is contained in a lattice with basis ε_{1}, …, ε_{n+1} so that the embedding is defined by ε_{1} = ω_{1}, ε_{n+1} = −ω_{n}, and ε_{i} = ω_{i} − ω_{i−1} for 2 ≤ i ≤ n, and W is the permutation group of the ε’s. Then q and

f_{3} := (ε_{1}^{3} + ⋯ + ε_{n+1}^{3})/3 = ∑_{i=2}^{n} ω_{i−1}^{2}ω_{i} − ω_{i−1}ω_{i}^{2}

are members of a set of basic invariants [16, §3.12].

For g ∈ I of degree 3, we have mg = g_{2}q + g_{3}f_{3} for some g_{3}, m ∈ ℤ and g_{2} = ∑ a_{i}ω_{i} with a_{i} ∈ ℤ. On the right side, the monomial ω_{i}^{3} occurs only in g_{2}q and has coefficient a_{i}, hence m divides a_{i} for all i, hence m divides g_{2} and also g_{3}. In summary, g = (g_{2}/m)q + (g_{3}/m)f_{3} for some g_{2}/m, g_{3}/m ∈ 𝕊(T^{∗}).

§1D. The γ-filtration on the variety of Borel subgroups. Consider the γfiltration on the variety 𝔅_{s} of Borel subgroups of G_{s}. Let γ^{m} denote the subgroup of ℤ[T^{∗}] generated by products of at least m elements of the form (1−e^{−ω_{i}}), where ω_{i} is a fundamental weight. Then the isomorphism 𝔠′ induces an isomorphism

γ^{m/m+1}(𝔅_{s}) ≃ γ^{m}/(γ^{m+1} + I′) for each i.

For example γ^{1/2}(X) ≃ γ^{1}/(γ^{2} + I′) is a free abelian group with a basis given by the classes of the elements

(1 − e^{−ω_{i}}) ∈ γ^{1}, i = 1, …, n.

Indeed, c_{1}^{K_{0}}([L(ω_{i})]) = 1 − [L(−ω_{i})], the map c_{1} : γ^{1/2}(𝔅_{s}) → CH^{1}(𝔅_{s}) is an isomorphism and the elements c_{1}(L(ω_{i})) for i = 1, …, n form a basis of the Picard group CH^{1}(𝔅_{s}).

Since K_{0}(𝔅_{s}) is generated by classes of line bundles (see [28]), so is γ^{i}(𝔅_{s}). Therefore, we have

γ^{i}(𝔅_{s}) = ⟨c_{1}^{K_{0}}([L_{1}]) ⋯ c_{1}^{K_{0}}([L_{m}]) | m ≥ i and L_{j} is a line bundle over 𝔅_{s}⟩.

Let λ = ∑_{i} a_{i}ω_{i} be a presentation of a character λ in terms of the fundamental weights. Then L(λ) = ⊗_{i}L(ω_{i})^{⊗a_{i}}. Since for any two line bundles L_{1} and L_{2} we have

c_{1}^{K_{0}}([L_{1} ⊗ L_{2}]) = c_{1}^{K_{0}}([L_{1}]) + c_{1}^{K_{0}}([L_{2}]) − c_{1}^{K_{0}}([L_{1}])c_{1}^{K_{0}}([L_{2}])

applying this formula recursively we can express any element of γ^{i/i+1}(𝔅_{s}) as a linear combination of the products of the first Chern classes of the bundles L(ω_{i}), i = 1 … n. For instance, any element of γ^{2/3}(𝔅_{s}) can be written as a class of

∑_{i=1}^{n} ∑_{j=1}^{n} a_{ij}(1 − e^{−ω_{i}})(1 − e^{−ω_{j}}) ∈ γ^{2} mod γ^{3} + I′, where a_{ij} ∈ ℤ.

§1E. The Dynkin index. Let N denote the map ℤ[T^{∗}]^{W} → ℤ defined by fixing a long root α and setting

N (∑_{i} a_{i}e^{λ_{i}}) := \frac{1}{2} ∑_{i} a_{i}⟨λ_{i}, α^{∨}⟩^{2}.

This does not depend on the choice of α and takes values in ℤ (and not merely in \frac{1}{2}ℤ), cf. Lemma 2.5 below. The number N(χ) is called the Dynkin index of χ. Note that for m ∈ ℤ, we have N(m) = N(me^{0}) = 0, so N(χ) only depends on the image of χ in the kernel of the augmentation map.

In case G_{s} has two root lengths, it is natural to wonder what one would find if one used a short root, say, δ in the definition of N instead of the long root α. We claim that

(1.3)  \frac{1}{2} ∑ a_{i}⟨λ_{i}, δ^{∨}⟩^{2} = q(δ^{∨}) [\frac{1}{2} ∑ a_{i}⟨λ_{i}, α^{∨}⟩^{2}] ,

where q is the form introduced in §1B. In other words, one obtains something that differs by a factor of q(δ^{∨}). (We will use this observation later.) To prove it, define quadratic forms n_{α} and n_{δ} on T^{∗} via n_{α}(λ) = ∑_{w∈W} ⟨wλ, α^{∨}⟩^{2} and similarly for δ^{∨}. For example, n_{δ}(α) = q(δ^{∨})^{2}n_{α}(δ). But n_{α} is a W -invariant quadratic form on T^{∗}, hence it is a scalar multiple of q. As q(α) = q(δ^{∨})q(δ), we have n_{α}(α) = q(δ^{∨})n_{α}(δ). But n_{δ} is also a scalar multiple of q, so we conclude that n_{δ} = q(δ^{∨})n_{α}, proving the claim.

The Dynkin index N(G_{s}) is defined to be the gcd of N(χ) as χ varies over the characters of finite-dimensional representations of G_{s}. The number N(G_{s}) is calculated in [10], [18], or [19]:

type of G_{s} A or C B_{n} (n ≥ 3), D_{n} (n ≥ 4), G_{2} F_{4} or E_{6} E_{7} E_{8}
N(G_{s})  1  2  6  12 60

If G is a simple and strongly inner group, then, for the purposes of this paper, we define the Dynkin index N(G) of G to be the Dynkin index N(G_{s}) of the split simply connected group of the same Killing-Cartan type.

2. Dynkin indices and the map ϕ

Let G_{s} denote a split simply connected simple linear algebraic group of rank n over a field k. We fix a pinning for G_{s} and in particular a split maximal torus T and fundamental weights ω_{1}, …, ω_{n}. As G_{s} is simply connected, T_{∗} (= Hom(𝔾_{m}, T)) and T^{∗} are canonically identified with the coroot and weight lattices respectively.

2.1. Definition. Put ℤ[T^{∗}] := ℤ[e^{ω_{1}}, …, e^{ω_{n}}], the integral group ring, and 𝕊(T^{∗}) := ℤ[ω_{1}, …, ω_{n}], the symmetric algebra of T^{∗}. We define a ring homomorphism

ϕ_{m}: ℤ[T^{∗}]/γ^{m+1} → 𝕊(T^{∗})/(𝕊^{m+1}(T^{∗})), m ≥ 2,

via ϕ_{m} (e^{∑_{i=1}^{n} a_{i}ω_{i}}) = ∏_{i=1}^{n}(1 − ω_{i})^{−a_{i}}.

In particular, ϕ_{m}(e^{ω_{i}}) = 1+ω_{i} +⋯+ω_{i}^{m} and ϕ_{m}(e^{−ω_{i}}) = 1−ω_{i}. (Note that ℤ[T^{∗}] can be viewed as Laurent polynomials in the variables ω_{1}, …, ω_{n}, and from this perspective it is clear that the formula for ϕ gives a well-defined ring homomorphism on ℤ[T^{∗}] and ϕ_{m}(γ^{m+1}) is zero in 𝕊(T^{∗})/(𝕊^{m+1}(T^{∗})).)

The homomorphism ϕ_{m} is an isomorphism. To see this, define a homomorphism 𝕊(T^{∗}) → ℤ[T^{∗}]/γ^{m+1} via ψ_{m}(ω_{i}) = 1 − e^{−ω_{i}} for all i; it induces a homomorphism 𝕊(T^{∗})/(𝕊^{m+1}(T^{∗})) → ℤ[T^{∗}]/γ^{m+1} that we also denote by ψ_{m}. As the compositions ϕ_{m}ψ_{m} and ψ_{m}ϕ_{m} are both the identity on generators, the claim is proved.

The goal of this section is to give a formula for ϕ_{2} on I′.

2.2. Proposition. If G_{s} is simple, then for χ ∈ ℤ[T^{∗}]^{W}, we have:

ϕ_{2}(χ) = aug(χ) + N(χ) · q ∈ (𝕊(T^{∗})/(𝕊^{3}(T^{∗})))^{W} ,

where q is the invariant form introduced in §§1B.

The proof would be much easier if we already knew that ϕ_{2} takes W -invariant elements to W -invariant elements, but this only comes as a consequence of the proof of the proposition. We give some preliminary material before the proof.

2.3. Example (SL_{2}). In case G_{s} = SL_{2}, write ω for the unique fundamental weight. For n > 0, we have:

ϕ_{2}(e^{nω} + e^{−nω}) = (1 + ω + ω^{2})^{n} + (1 − ω)^{n} = 2 + n^{2}ω^{2},

which verifies Prop. 2.2 for this group.

2.4. Example (SL_{2} × SL_{2}). In case G_{s} = SL_{2} × SL_{2} there are two fundamental weights ω_{1}, ω_{2} and the Weyl group W is the Klein four-group; it acts by flipping the signs of ω_{1} and ω_{2}. The definition of ϕ_{2} above makes sense here even though G_{s} is not simple. We find:

ϕ_{2}(W e^{a_{1}ω_{1}+a_{2}ω_{2}}) = 4 + 2 [a_{1}^{2}ω_{1}^{2} + a_{2}^{2}ω_{2}^{2}] .

One final observation about Weyl group actions. We write W λ for the W -orbit of λ ∈ T^{∗}.

2.5. Lemma. For every root α and weight λ ∈ T^{∗}, the map W λ → ℤ defined by π ↦ ⟨π, α^{∨}⟩ hits x and −x the same number of times, for every x ∈ ℤ. If α, β are orthogonal roots, then for every weight λ ∈ T^{∗}, the map W λ → ℤ × ℤ defined by π ↦ (⟨π, α^{∨}⟩, ⟨π, β^{∨}⟩) hits (x, y), (−x, y), (x, −y), and (−x, −y) the same number of times, for every x, y ∈ ℤ.

Sketch of proof. It is an exercise to show the analogous statements for the map W → ℤ defined by w ↦ ⟨wλ, α^{∨}⟩ and similarly for the second claim. The lemma follows. □

Proof of Prop. 2.2. We may assume that χ = ∑ e^{λ_{j}} where λ_{1}, …, λ_{r} is the Weyl orbit of some λ ∈ T^{∗}. Put λ_{j} = ∑_{i=1}^{n} a_{ij}ω_{i}, so ϕ(χ) = ∑_{j=1}^{r} ∏_{i=1}^{n}(1 − ω_{i})^{−a_{ij}}. Obviously, the degree 0 component of ϕ(χ) is r = aug(χ).

The degree 1 component of ϕ(χ) is ∑_{j} ∑_{i} a_{ij}ω_{i} = ∑_{i} (∑_{j} a_{ij}) ω_{i}. Here the claim is that ∑_{j} a_{ij} = 0 for each i. The a_{ij}’s are the images of W λ in ℤ under the map λ_{j} ↦ ⟨λ_{j}, α_{i}^{∨}⟩ where α_{i} denotes the simple root corresponding to the fundamental weight ω_{i}, hence the claim follows from Lemma 2.5.

The crux is to check the claim on the degree 2 component q_{1} of ϕ(χ); it is an integer-valued quadratic form on the coroot lattice T_{∗} and we check that it equals q_{2} := N(χ)q. We write out for ℓ = 1, 2:

(2.6)  q_{ℓ}(∑ d_{i}α_{i}^{∨}) = ∑_{i} d_{i}^{2}q_{ℓ}(α_{i}^{∨}) + ∑_{i<j} d_{i}d_{j}b_{q_{ℓ}}(α_{i}^{∨}, α_{j}^{∨}),

where b_{q_{ℓ}} is the polar bilinear form of q_{ℓ}. We will check that the value of this expression is the same for ℓ = 1, 2.

First suppose that δ^{∨} := ∑ d_{i}α_{i}^{∨} is a coroot and every d_{i} is 0 or 1. Then it defines a homomorphism η: SL_{2} → G_{s} so that, roughly speaking, the simple coroot α^{∨} of SL_{2} (viewed as a map 𝔾_{m} → T_{1} := η^{−1}(T)) satisfies η(α^{∨}) = δ^{∨}. We check that the diagram

(2.7)  \begin{matrix}ℤ[T^{∗}] & ⟶^{ϕ_{2}} & 𝕊(T^{∗})/(𝕊^{3}(T^{∗})) \\ ^{η^{∗}}↓ & & ↓^{η^{∗}} \\ ℤ[T_{1}^{∗}] & ⟶^{ϕ_{2}} & 𝕊(T_{1}^{∗})/(𝕊^{3}(T_{1}^{∗})) \end{matrix}

commutes. Since ω_{j}(δ^{∨}) = d_{j}, we have η^{∗}(ω_{j}) = d_{j}ω for ω the fundamental weight of SL_{2} dual to α^{∨}. We find:

η^{∗}ϕ_{2}(e^{∑ c_{j}ω_{j}}) = ∏_{j}(1 − d_{j}ω)^{−c_{j}} = (1 − ω)^{− ∑ c_{j}d_{j}},

because the d_{j} are all 0 or 1. As this is ϕ_{2}(e^{(∑ d_{j}c_{j})ω}) = ϕ_{2}η^{∗}(e^{∑ c_{j}ω_{j}}), we have confirmed the commutativity of (2.7).

Put ϕ^{2} for the composition of ϕ_{2} with the projection onto the degree 2 component 𝕊^{2}, so q_{1} = ϕ^{2}(χ). Then q_{1}(δ^{∨}) = (η^{∗}ϕ^{2}(χ))(α^{∨}) obviously, which is (ϕ^{2}η^{∗}(χ))(δ^{∨}) by commutativity of (2.7). We have η^{∗}(χ) = ∑_{j} e^{∑_{i} a_{ij}ω} and by Lemma 2.5, the multiset of the j integers ∑_{i} a_{ij}d_{i} is symmetric under multiplication by −1, hence by Example 2.3 we find:

q_{1}(δ^{∨}) = \frac{1}{2} (∑_{j} (∑_{i} a_{ij}d_{i})^{2}) = \frac{1}{2} ∑_{j} ⟨λ_{j}, δ^{∨}⟩^{2}.

By (1.3) this equals q(δ^{∨})N(χ) = q_{2}(δ^{∨}).

Returning to equation (2.6), this shows that the term q_{ℓ}(α_{i}^{∨}) does not depend on ℓ. Similarly, if α_{i}^{∨} and α_{j}^{∨} are not orthogonal coroots, then α_{i}^{∨} and α_{j}^{∨} are adjacent in the Dynkin diagram and α_{i}^{∨} + α_{j}^{∨} is a coroot [2, VI.1.6, Cor. 3b]. The preceding two paragraphs show that the value of

b_{q_{ℓ}}(α_{i}^{∨}, α_{j}^{∨}) = q_{ℓ}(α_{i}^{∨} + α_{j}^{∨}) − q_{ℓ}(α_{i}^{∨}) − q_{ℓ}(α_{j}^{∨})

does not depend on ℓ.

It remains to consider b_{q_{ℓ}}(α_{i}^{∨}, α_{j}^{∨}) where α_{i}^{∨} and α_{j}^{∨} are orthogonal (relative to the polar form of q – it follows that they are orthogonal relative to b_{q_{2}}. We use α_{i}^{∨} and α_{j}^{∨} to define a homomorphism τ: SL_{2} × SL_{2} → G_{s} and – as we did for SL_{2} above – we fix a torus T_{2} = T_{1}×T_{1} ⊂ SL_{2}×SL_{2} such that τ(T_{2}) = im(α_{i}^{∨}×α_{j}^{∨}) ⊂ T. Arguing using a commutative diagram analogous to (2.7), it suffices to check that the simple roots of SL_{2} ×SL_{2} are orthogonal relative to τq_{1} = ϕ^{2}τ(χ), which follows from Example 2.4 and Lemma 2.5. □

In view of §§1E, Prop. 2.2 gives:

2.8. Corollary. ϕ_{2}(I′) = ℤ · N(G_{s}) · q. □

2.9. Example. Suppose G has type A_{n} for some n ≥ 2. We continue the notation of §1C, and we compute:

ϕ_{3}(W e^{ω_{1}}) = ϕ_{3}(e^{ω_{1}} + e^{−ω_{n}} + ∑_{i=2}^{n} e^{ω_{i}−ω_{i−1}})

= (n + 1) + q + ∑_{j=1}^{n} ω_{i}^{3} − ∑_{i=2}^{n} ω_{i−1}ω_{i}^{2}.

The element Δ := W e^{ω_{1}} −W e^{ω_{n}} is in I′, and by symmetry we see that ϕ_{3}(Δ) = f_{3}.

3. Torsion in the γ-filtration

Let 𝔅 denote the variety of Borel subgroups of a strongly inner simple linear algebraic group G over k. Recall that G is strongly inner if the simply connected cover of G is isomorphic to G_{s} twisted by a cocycle ξ ∈ H_{ét}^{1}(k, G_{s}), where G_{s} denotes the simply connected split group of the same Killing-Cartan type as G. Observe that the variety 𝔅 is always defined over k by [7, Cor. XXVI.3.6]; it is a twisted form of the variety of Borel subgroups 𝔅_{s} of G_{s}, i.e., 𝔅 and 𝔅_{s} become isomorphic over the algebraic closure of k.

In the present section we determine and bound respectively the torsion parts of the second and the third quotients of the γ-filtration on the variety 𝔅. The main result is the following.

3.1. Theorem. Let 𝔅 be the variety of Borel subgroups of a strongly inner simple linear algebraic group G over a field k. Then

(i) Tors γ^{2/3}(𝔅) is a cyclic group of order the Dynkin index N(G) and generated by 𝔠′(θ) for θ as in Def. 3.3.

(ii) The subgroup τ^{3}(𝔅)/γ^{3}(𝔅) of Tors γ^{2/3}(𝔅) is generated by o(r(G)) 𝔠′(θ);

it is cyclic of order N(G)/o(r(G)). (iii) 2 Tors γ^{3/4}(𝔅) is a quotient of (ℤ/N(G))^{⊕(rank G)}.

There is some new notation in the statement of the theorem. The Rost invariant r is a map H^{1}(k, G_{s}) → H^{3}(k, ℚ/ℤ(2)), and for our G, the element r(ξ) depends only on G and not on the choice of ξ by [11, Lemma 2.1]; we write simply r(G) for this element and o(r(G)) for its order in the abelian group H^{3}(k, ℚ/ℤ(2)).

Philippe Gille pointed out to us that pasting together two results in the literature gives a description of Tors CH^{2}(X) for some X.

3.2. Proposition. Let X be a projective homogeneous variety under G. If G is split by k(X), then Tors CH^{2}(X) is a cyclic group whose order is the same as the order of r(G) in H^{3}(k, ℚ/ℤ(2)); in particular its order divides N(G).

Proof. We view ξ as a principal homogeneous G_{s}-variety. The kernel of the scalar extension map H^{3}(k, ℚ/ℤ(2)) → H^{3}(k(ξ), ℚ/ℤ(2)) is the cyclic group generated by r(G) by [10, p. 129]. For every extension L/k, ξ has an L-point if and only if G is split, if and only if X has an L-point. Therefore, this kernel is the same as the kernel of H^{3}(k, ℚ/ℤ(2)) → H^{3}(k(X), ℚ/ℤ(2)). A theorem of Peyre-Merkurjev [27] shows that this kernel is isomorphic to Tors CH^{2}(X). □

Obviously, one can take X = 𝔅 in the proposition. Furthermore, the same proof shows that the proposition still holds if one replaces “G is strongly inner” and “G is split by k(X)” with “G has trivial Tits algebras” and “G becomes quasi-split over k(X)”.

Also, statement 3.1(i) makes use of the following definition.

3.3. Definition. Write the form q from subsection §1B (relative to the split group G_{s}) as q = ∑_{i≤j} c_{ij}ω_{i}ω_{j} ∈ 𝕊(T^{∗}). We call the element

θ := ∑_{i≤j} c_{ij}(1 − e^{−ω_{i}})(1 − e^{−ω_{j}}) ∈ ℤ[T^{∗}]

the special cycle. Its image in ℤ[T^{∗}]/γ^{m+1} is ψ_{m}(q) for all m ≥ 2.

Proof of Theorem 3.1. By the result of Panin [25, Thm. 2.2.(2)] since G is strongly inner, the restriction map

(3.4)  res: K_{0}(𝔅) → K_{0}(𝔅 × k_{alg}) ≃ K_{0}(𝔅_{s} × k_{alg}) ≃ K_{0}(𝔅_{s})

is an isomorphism, where k_{alg} denotes an algebraic closure of k. Since the γfiltration is defined in terms of Chern classes and the latter commute with restrictions, it induces an isomorphism between the γ-quotients, i.e.,

res: γ^{i/i+1}(𝔅) ⟶^{≃} γ^{i/i+1}(𝔅_{s}).

Therefore, we may reduce to the split case G = G_{s}. Let T, T^{∗}, etc., be as in subsection §1B.

There is a commutative diagram

(3.5)  \xymatrix{γ^{m/m+1}(𝔅_{s}) \ar[r]^{c_{m}} & CH^{m}(𝔅_{s}) \\ γ^{m}/γ^{m+1} \ar@{->>}[u]^{𝔠′} \ar[r]^{(−1)^{m−1}(m−1)!·ϕ_{m}} & 𝕊^{m}(T^{∗})/(𝕊^{m+1}(T^{∗})) \ar[u]_{𝔠}}

First take m = 2 and suppose that x ∈ γ^{2/3} maps to a torsion element in γ^{2/3}(𝔅_{s}). As CH^{2}(𝔅_{s}) has zero torsion, the commutativity of (3.5) shows that ϕ_{2}(x) is in the kernel I of 𝔠. Writing x = ∑_{i,j} a_{ij}(1 − e^{ω_{i}})(1 − e^{ω_{j}}) mod γ^{3}, we have ϕ_{2}(x) = ∑ a_{ij}ω_{i}ω_{j} of degree 2 in I, hence ϕ_{2}(x) = aq for some a ∈ ℤ. Then modulo γ^{3}, we have x ≡ ψ_{2}ϕ_{2}(x) ≡ aθ, so Tors γ^{2/3}(𝔅_{s}) is a cyclic group generated by the class of the special cycle θ modulo γ^{3} + I′.

By Corollary 2.8 there exists χ ∈ I′ such that ϕ_{2}(χ) = N(G_{s}) · q. Applying ψ_{2} we obtain that

0 ≡ χ ≡ N(G_{s}) · θ mod γ^{3} + I′,

hence, the order of θ modulo γ^{3} + I′ divides the Dynkin index N(G_{s}). This shows that Tors γ^{2/3}(𝔅) is a cyclic group of order dividing N(G) with generator 𝔠′(θ).

Let ξ′ ∈ H^{1}(k′, G_{s}) be a versal G_{s}-torsor for some extension k′ of k, and write 𝔅′ for the Borel variety (over k′) of the group G_{s} twisted by ξ′. The element r(ξ′) has order N(G_{s}) in H^{3}(k′, ℚ/ℤ(2)) by [10, pp. 31, 133]. But Tors γ^{2/3}(𝔅′) is cyclic of order dividing N(G_{s}), hence Prop. 3.2 and the exactness of (1.1) give that Tors γ^{2/3}(𝔅′) also has order N(G_{s}). Now take K to be an algebraically closed field containing k′. The restriction maps for k → K and k′ → K give isomorphisms Tors γ^{2/3}(𝔅_{/k}) ≃ Tors γ^{2/3}((𝔅_{s})_{/K}) ≃ Tors γ^{2/3}(𝔅′_{/k′}), which is itself ℤ/N(G), completing the proof of (i). Claim (ii) follows from the exactness of (1.1).

Now take m = 3 and suppose that x ∈ γ^{3/4} maps to a torsion element in γ^{3/4}(𝔅_{s}). As CH^{3}(𝔅_{s}) has zero torsion, diagram (3.5) shows that 2ϕ_{3}(x) is in the kernel I of 𝔠. As in the m = 2 case, 2ϕ_{3}(x) has degree 3.

Suppose G_{s} is not of type A_{n} for n ≥ 2. Then by §§1C, 2ϕ_{3}(x) = q · f, where f = ∑_{i=1}^{n} a_{i}ω_{i}. Applying ψ_{3} we obtain that 2x = θ · f′, where f′ = ∑_{i=1}^{n} a_{i}(1 − e^{−ω_{i}}). In other words, the torsion part of 2γ^{3/4}(𝔅_{s}) is generated by the elements 𝔠′(θ · (1 − e^{−ω_{i}})) for i = 1, …, n.

By Corollary 2.8 there exists χ ∈ I′ such that ϕ_{3}(χ · (1 − e^{−ω_{i}})) ≡ N(G_{s}) · q · ω_{i} mod (𝕊^{4}(T^{∗})). Applying ψ_{3} we obtain that

0 ≡ χ · (1 − e^{−ω_{i}}) ≡ N(G_{s}) · θ · (1 − e^{−ω_{i}}) mod γ^{4} + I′,

hence, the torsion part of 2γ^{3/4}(𝔅_{s}) is a product of n cyclic groups of orders dividing the N(G_{s}).

It remains to consider the case m = 3 and G_{s} of type A_{n} for n ≥ 2; the claim is that 2x is in I′. As in the preceding paragraph, §§1C and Example 2.9 show that 2ϕ_{3}(x) = q · f + bϕ_{3}(Δ) for some b ∈ ℤ. Applying ψ_{3}, we find 2x = θ · f′ + Δ · b. As θ and Δ belong to I′, the proof of (iii) is complete. □

3.6. Remark. Observe that in the proof we used the structure of the ideal of invariants I in degrees 2 and 3. In principle it is possible to extend the proof to higher degrees, but then one must extend the arguments of §§1C and Example 2.9.

The next lemma can be used to extend the bounds obtained in Th. 3.1 to the case of a semisimple group.

3.7. Lemma. Let G_{1}, …, G_{m} be simple and strongly inner groups and write 𝔅_{j} for the Borel variety of G_{j}. The Borel variety for ∏ G_{j} is ∏ 𝔅_{j}, and we have:

Tors γ^{2/3}(∏ 𝔅_{j}) ≃ ⨁ Tors γ^{2/3}(𝔅_{j}),

and

Tors γ^{3/4}(∏ 𝔅_{j}) ≃ ⨁_{j=1}^{m} ( Tors γ^{3/4}(𝔅_{j}) ⊕ Tors γ^{2/3}(𝔅_{j})).

Proof. Apply the Künneth decomposition and the fact that γ^{i/i+1}(𝔅_{j}) has no torsion for i = 0 and 1. □

4. Examples of torsion in CH^{2}

We now make a few remarks regarding torsion in CH^{2}. We maintain the notation of the previous section.

Using Prop. 3.2, one can view the the assignment G ↦ Tors CH^{2}(𝔅) as a replacement for the Rost invariant of G. Furthermore, the group Tors CH^{2}(𝔅) is generated by the image of the special cycle θ ∈ ℤ[T^{∗}] which is defined in purely combinatorial terms.

The next two examples show that the image of θ generates the torsion in CH^{2} for certain (generalized) Rost motives.

4.1. Example. Assume that the motive of 𝔅 splits as a direct sum of twisted copies of the Rost motive R_{2} corresponding to a 3-fold Pfister form. According to [26, §7] this can happen for G of type B_{n}, D_{n}, G_{2}, F_{4}, or E_{6}. Then we have

(Tors CH^{2}(𝔅)) ⊗ ℤ/2ℤ ≃ Ch^{2}(R_{2}),

where Ch denotes the Chow group with ℤ/2ℤ-coefficients and the image of the special cycle θ generates (Tors CH^{2}(𝔅)) ⊗ ℤ/2ℤ and, hence, the group Ch(R_{2}).

4.2. Example. Suppose G is an anisotropic group of type F_{4} over k that is split by a cubic extension of k. (Such a group exists if and only if H^{3}(k, ℤ/3ℤ(2)) is not zero.) By [24, Rem. 4.5] and the main theorem of [26] the motive of the Borel variety 𝔅 splits as a direct sum of generalized Rost motives R_{3} corresponding to the Rost-Serre invariant g_{3} (the mod-3 portion of the Rost invariant) of G. Therefore, we have

(Tors CH^{2}(𝔅)) ⊗ ℤ/3ℤ ≃ Ch^{2}(R_{3}),

where Ch denotes the Chow group with ℤ/3ℤ-coefficients.

By the recent results of Merkurjev-Suslin [23] and Yagita [34, Thm. 10.5, Cor. 10.8] we have Ch^{2}(R_{3}) ≃ ℤ/3ℤ. On the other hand, the image of the special cycle θ generates (Tors CH^{2}(𝔅)) ⊗ ℤ/3ℤ and, hence, the group Ch^{2}(R_{3}).

5. Torsion in CH^{3}

Let X be a projective homogeneous G-variety such that G is split over k(X). (“X is generically split.”) Thanks to Proposition 3.2, we may view Tors CH^{2}(X) as known, so we now investigate Tors CH^{3}(X). We retain the meaning of G and 𝔅 from the previous section. Let n denote the rank of G and let r denote the rank of the Picard group of X over an algebraic closure of k.

We remark that the results in this section only use the fact that Tors CH^{2}(X) is cyclic of order dividing N(G), which follows from our Theorem 3.1(i) and Eq. (1.1). They do not need the finer result of Prop. 3.2, hence also do not need material from [10] and [27].

For an abelian group A and a prime p, write Tors_{p} A for the subgroup of A consisting of elements of order a power of p.

5.1. Lemma. The restriction of the m-th Chern class gives a surjection

Tors τ^{m/m+1}(X) ↠ (m − 1)! Tors CH^{m}(X)

and for each prime p not dividing (m − 1)!, c_{m} is an isomorphism

Tors_{p} τ^{m/m+1}(X) ⟶^{≃} Tors_{p} CH^{m}(X).

Proof. By Riemann-Roch (see subsection §1A), the composition

\xymatrix{CH^{m}(X) \ar@{->>}[r]^{pr} & τ^{m/m+1}(X) \ar[r]^{c_{m}} & CH^{m}(X)}

is multiplication by (−1)^{m−1}(m − 1)!, hence c_{m}(τ^{m/m+1}(X)) is (m − 1)! CH^{m}(X). For x ∈ Tors CH^{m}(X), we have (m − 1)! · x = c_{m}(pr(x)), where pr(x) is in Tors τ^{m/m+1}(X). This proves the first claim, from which the second claim follows immediately. □

5.2. Proposition. If τ^{3}(𝔅) = γ^{3}(𝔅), then Tors 4 · CH^{3}(𝔅) is a quotient of the direct sum (ℤ/N(G)ℤ)^{⊕n}. In particular, the torsion part of CH^{3}(𝔅) can consist only of subgroups ℤ/2^{s}ℤ for s ≤ 4, ℤ/3ℤ, or ℤ/5ℤ.

Theorem 3.1(ii) gives a way to check the hypothesis on the filtration.

Proof. By the hypothesis, the map γ^{3/4}(𝔅) → τ^{3/4}(𝔅) is surjective. Now combine Lemma 5.1 and Theorem 3.1(iii). □

As an alternative to making a hypothesis on the filtrations, we may control the torsion on CH^{3}(X) based on information about the torsion in CH^{2}(X) and the motivic decomposition of X, as we now illustrate.

Fix a prime p. In the category of Chow motives with ℤ/pℤ-coefficients, the motive of X is a direct sum of shifts of an indecomposable motive R, see [26, Th. 5.17], where R depends on G but not the choice of X (ibid., Th. 3.7). We write Ch^{m}(R) for the m-th Chow group of R with ℤ/pℤ coefficients.

5.3. Lemma. We have:

(i) (Tors_{p} CH^{2}(X)) ⊗ ℤ/pℤ ≃ Ch^{2}(R); (ii) (Tors_{p} CH^{3}(X)) ⊗ ℤ/pℤ ≃ (Ch^{2}(R))^{⊕r} ⊕ Ch^{3}(R). (iii) Tors CH^{3}(𝔅) ≃ (Tors CH^{2}(X))^{⊕(n−r)} ⊕ Tors CH^{3}(X).

Proof. The expression of the motive of X from [26] gives:

(Tors_{p} CH^{m}(X)) ⊗ ℤ/pℤ ≃ \overline{Ch}^{m}(R) ⊕ (\overline{Ch}^{m−1}(R))^{⊕r} ⊕ (\overline{Ch}^{m−2}(R))^{⊕...} ⊕ …

where \overline{Ch}^{m}(R) denotes the kernel of the restriction Ch^{m}(R) → Ch^{m}(R×_{k} k̄) to the algebraic closure k̄. By the formula for the generating function [26, Thm. 5.13(3)] and table 4.13 in ibid., we have \overline{Ch}^{0}(R) = \overline{Ch}^{1}(R) = 0 and \overline{Ch}^{i}(R) = Ch^{i}(R) for i = 2, 3. This implies claims (i) and (ii).

Claim (iii) is proved similarly, but using the integral motivic decomposition from [26, Th. 3.7] with Y = 𝔅. □

5.4. Proposition. Fix an odd prime p. If Tors_{p} CH^{2}(X) ≠ 0, then (1) p = 3 or 5;

(2) Ch^{2}(R) ≃ ℤ/pℤ and Ch^{3}(R) = 0.

(3) Tors_{p} CH^{2}(X) ≃ ℤ/pℤ and Tors_{p} CH^{3}(X) ≃ (ℤ/pℤ)^{⊕r}.

Proof. By Prop. 3.2 (or [26, Th. 3.7]), CH^{2}(X) and CH^{2}(𝔅) have the same ptorsion. As Tors CH^{2}(𝔅) has order dividing N(G) by Theorem 3.1(i), the list of Dynkin indexes in §1E gives that p = 3 or 5 and Tors_{p} CH^{2}(𝔅) ≃ ℤ/pℤ. Combining this with Lemma 5.3(i), it only remains to prove the claims about Ch^{3}(R) and CH^{3}(X).

Tensoring sequence (1.1) with ℤ/pℤ, we find that

γ^{3}(𝔅) ⊗ ℤ/pℤ = τ^{3}(𝔅) ⊗ ℤ/pℤ.

Combining Lemma 5.1 and Theorem 3.1(iii) gives that Tors_{p} CH^{3}(𝔅) is a product of at most n copies of ℤ/pℤ. By Lemma 5.3(ii) applied to X = 𝔅 we obtain

(Tors_{p} CH^{3}(𝔅)) ⊗ ℤ/pℤ ≃ (ℤ/pℤ)^{⊕n} ⊕ Ch^{3}(R).

Since the right hand side already contains n copies of ℤ/pℤ, Tors_{p} CH^{3}(𝔅) = (ℤ/pℤ)^{⊕n} and Ch^{3}(R) is zero. The second part of (3) now follows by Lemma 5.3(iii). □

6. Cohomological invariants and the Tits algebras

So far, we have studied the case where G is strongly inner and we constructed the special cocycle 𝔠′(θ) in K_{0}(𝔅), cf. Example 6.3 below. We now relax our hypothesis on G and ask if 𝔠′(θ) is still defined over k.

In the present section G_{s} denotes an adjoint split simple linear algebraic group over a field k. As it is adjoint, the character group T^{∗} of a split maximal torus of G_{s} is naturally identified with the root lattice Λ_{r}.

We fix a pinning for G_{s}, which includes a set of simple roots Δ = {α_{1}, …, α_{n}} in Λ_{r}. Write ω_{i} for the fundamental weight corresponding to α_{i} and s_{i} for the reflection of the weight lattice Λ in the hyperplane orthogonal to α.

The Steinberg basis. For each element w of the Weyl group W (of T) we define

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

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 } by [32, Th. 2.2].

Let 𝔅_{s} denote the variety of Borel subgroups of G_{s}. Consider the characteristic map for the simply connected cover of G_{s}

𝔠′: ℤ[Λ] ↠ K_{0}(𝔅_{s}).

Since the kernel of the surjection 𝔠′ is generated by elements x ∈ ℤ[Λ]^{W} in the kernel of the augmentation map, there is an isomorphism

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

The elements

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

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

Observe that the quotient group Λ/Λ_{r} coincides with the group of characters of the center of the simply connected cover of G_{s}. Consider the surjective ring homomorphism induced by the restriction ℤ[Λ] → ℤ[Λ/Λ_{r}]. Since W acts trivially on Λ/Λ_{r}, we obtain that

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

where ρ̄ means the restriction to Λ/Λ_{r}.

6.1. Example. (a) For a simple reflection s_{j} we have

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

(b) More generally, let w := s_{i_{1}}s_{i_{2}} … s_{i_{m}} be a product of m distinct simple reflections such that the simple roots α_{i_{j}}, α_{i_{ℓ}} are orthogonal for all j ≠ ℓ. Then

ρ_{s_{i_{1}} s_{i_{2}} ...s_{im}} = ρ_{s_{i_{1}}} + ρ_{s_{i_{2}}} + … + ρ_{s_{im}} .

because w^{−1}(α_{i}) is negative if and only if i = i_{j} for some j.

(c) For a product of two simple reflections s_{i}s_{j} such that c_{ij} = α_{i}^{∨}(α_{j}) < 0 we

obtain

ρ_{s_{i}s_{j}} = ρ_{s_{i}} + c_{ij}α_{j}.

The Tits algebras and the base change. Let G be a twisted form of G_{s}, i.e. G is obtained by twisting G_{s} by a cocycle ξ ∈ Z^{1}(k, Aut(G_{s})). More specifically, our choice of pinning for G_{s} defines a section s of the quotient map π: Aut(G_{s}) → Aut(Δ). Twisting G_{s} by ξ′ := sπ(ξ) gives a quasi-split group G_{q} and we pick ξ′′ ∈ Z^{1}(k, G_{q}) that maps via twisting to ξ—i.e., we pick ξ′′ so that G is isomorphic to _{ξ′′}G_{q}.

Let 𝔅 = _{ξ}𝔅_{s} be the variety of Borel subgroups of G. Let Γ denote the absolute Galois group of k; it acts on the weight lattice Λ via the cocycle ξ′.

Following [33] (see also [25, §3.1, §11.7] and [22, §2]) we associate with each χ ∈ Λ/Λ_{r} the field of definition k_{χ} of χ and the central simple algebra A_{χ,ξ} over k_{χ} called the Tits algebra. Here k_{χ} is a fixed subfield for the stabilizer

Γ_{χ} = {τ ∈ Γ | τ(χ) = χ}.

There is a group homomorphism

β: (Λ/Λ_{r})^{Γ_{χ}} → Br(k_{χ}) with β(χ′) = [A_{χ′,ξ}].

By [27, Thm. 2.1] there is an isomorphism

Tors CH^{2}(𝔅) ≃ \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(χ))},

where the numerator is the kernel of the restriction map to the field of fractions k(𝔅) of 𝔅 and N_{k_{χ}/k} is the norm map. Let H_{β}^{3}(k, ℚ/ℤ(2)) denote the cohomology quotient

H_{β}^{3}(k, ℚ/ℤ(2)) = H^{3}(k, ℚ/ℤ(2))/ ⨁_{χ∈Λ/Λ_{r}} N_{k_{χ}/k}(k_{χ}^{∗} ∪ β(χ))

so that Tors CH^{2}(𝔅) ⊆ H_{β}^{3}(k, ℚ/ℤ(2)).

Let l/k be a field extension. Since the Chern classes commute with restrictions, there is the induced map

res_{l/k}: γ^{i/i+1}(𝔅) → γ^{i/i+1}(𝔅_{l}),

where 𝔅_{l} = 𝔅 ×_{k} l, with the image generated by the products

⟨c_{n_{1}}^{K_{0}}(x_{1}) ⋯ c_{n_{m}}^{K_{0}}(x_{m}) | n_{1} + ⋯ + n_{m} = i, x_{1}, …, x_{m} ∈ res_{l/k}(K_{0}(𝔅))⟩

and there is a commutative diagram

(6.2)  \xymatrix{Tors γ^{2/3}(𝔅) \ar@{->>}[r]^{c_{2}} \ar[d]_{_{res_{l/k}}} & Tors CH^{2}(𝔅) ⊆ H_{β}^{3}(k, ℚ/ℤ(2)) \ar[d]^{_{res_{l/k}}} \\ Tors γ^{2/3}(𝔅_{l}) \ar@{->>}[r]^{c_{2}} & Tors CH^{2}(𝔅_{l}) ⊆ H_{β}^{3}(l, ℚ/ℤ(2))}

Observe that the image res_{l/k}(K_{0}(𝔅)) can be computed using [25]. For instance, if G is an inner group, i.e., ξ′ = 0, then Γ acts trivially on Λ/Λ_{r}, i.e. k_{χ} = k for all χ and by [25, Thm. 4.2] the image of the restriction map K_{0}(𝔅) → K_{0}(𝔅_{s}) from (3.4) coincides with the sublattice

⟨ind(A_{ρ̄_{w},ξ}) · g_{w} | w ∈ W ⟩.

Using (6.2) one can provide a non-trivial element in H_{β}^{3}(k, ℚ/ℤ(2)) as follows:

• Assume that we are given non-trivial elements over l, i.e. that there is a non-trivial element θ ∈ Tors γ^{2/3}(𝔅_{l}) such that c_{2}(θ) ∈ H_{β}^{3}(l, ℚ/ℤ(2)) is non-trivial.

• Assume that we know that θ is defined over k, i.e. that θ = res_{l/k}(θ′) for

some θ′ ∈ Tors γ^{2/3}(𝔅). Then the image c_{2}(θ′) provides a non-trivial element in H_{β}^{3}(k, ℚ/ℤ(2)).

6.3. Example (strongly inner case). If G is strongly inner—i.e., if G is inner and β is the trivial homomorphism—then for any field extension l/k the left vertical arrow in (6.2) is an isomorphism, hence, identifying Tors γ^{2/3}(𝔅) with the cyclic group generated by the special cycle θ. As in Prop. 3.2 and its proof Tors CH^{2}(𝔅) coincides with the usual unramified cohomology generated by the Rost invariant r(G) of G and ⟨c_{2}(θ)⟩ = ⟨r(G)⟩ in H^{3}(k, ℚ/ℤ(2)).

6.4. Lemma. Assume that G is inner.

(a) If a weight ω is such that β(ω) = 0, then [L(ω)] is in the image of

res: K_{0}(𝔅) → K_{0}(𝔅_{s}).

In particular, it holds for the classes [L(α_{i})] of simple roots α_{i}. Under the notation of Example 6.1(b) we have

(b) ∑_{j} c_{1}^{K_{0}}([L(ω_{i_{j}})]) − c_{1}^{K_{0}}([L(α_{i_{j}})]) ≡ c_{1}^{K_{0}} (∏_{j} g_{s_{i_{j}}} ) ≡ c_{1}^{K_{0}}(g_{w}) mod γ^{3}(𝔅_{s});

(c) If β(∑_{j} ω_{i_{j}}) = 0, then ∑_{j} c_{1}^{K_{0}}([L(ω_{i_{j}})]) is in the image of

res: γ^{1/2}(𝔅) → γ^{1/2}(𝔅_{s}).

Proof. (a) follows by [14, Cor. 3.1]. (b) follows by the formula for the first Chern class (in K_{0}) of the tensor product of line bundles. According to (a) each c_{1}^{K_{0}}([L(α_{i_{j}})]) is in the image of the restriction map which implies (c). □

6.5. Proposition (quaternionic inner case). Assume that G is inner. If every Tits algebra of _{ξ}G_{s} has index 1 or 2, then the special cycle θ is in the image of the restriction map

res: γ^{2/3}(𝔅) → γ^{2/3}(𝔅_{s}).

In other words, if l/k is an extension that kills im β, then the image of c_{2} over l coincides with the subgroup generated by the respective Rost invariant, i.e. we have

im(c_{2})_{l} = ⟨r(G_{l})⟩ ⊆ H^{3}(l, ℚ/ℤ(2)).

Proof of Prop. 6.5. We may assume that N(G) is not 1 (otherwise θ maps to zero in γ^{2/3}(𝔅_{s}) by Th. 3.1) and Λ/Λ_{r} has even order (otherwise Example 6.3 applies), i.e., we may assume that G has type B, C, D, or E_{7}.

We first make a general observation. Mod γ^{3}(𝔅_{s}), we have:

c_{1}^{K_{0}}([L(ω_{i})])^{2} ≡ (c_{1}^{K_{0}}(g_{s_{i}}) + c_{1}^{K_{0}}([L(α_{i})]))^{2}
≡ c_{1}^{K_{0}}(g_{s_{i}})^{2} + 2c_{1}^{K_{0}}(g_{s_{i}})c_{1}^{K_{0}}([L(α_{i})]) + c_{1}^{K_{0}}([L(α_{i})])^{2}.

The Whitney Sum Formula gives that c_{2}^{K_{0}}(2g_{s_{i}}) = c_{1}^{K_{0}}(g_{s_{i}})^{2} and c_{1}^{K_{0}}(2g_{s_{i}}) ≡ 2c_{1}^{K_{0}}(g_{s_{i}}) mod γ^{2}(𝔅_{s}). Our hypothesis on the Tits algebras gives that 2g_{s_{i}} is in the image of K_{0}(𝔅) → K_{0}(𝔅_{s}), and it follows that c_{1}^{K_{0}}([L(ω_{i})])^{2} is rational – i.e., is in the image of γ^{2/3}(𝔅) → γ^{2/3}(𝔅_{s}) – for all i.

\underline{Type E_{7}:} Suppose that G has type E_{7}. Then

q = (∑_{i=1}^{7} ω_{i}^{2}) − ω_{1}ω_{3} − ω_{3}ω_{4} − ω_{4}ω_{2} − ω_{4}ω_{5} − ω_{5}ω_{6} − ω_{6}ω_{7}

where we have numbered the roots following [2]. Each ω_{i}^{2} contributes a term of the form c_{1}^{K_{0}}([L(ω_{i})])^{2} to the image of 𝔠′(θ) in γ^{2/3}(𝔅_{s}), and such a term is rational by the preceding paragraph. The weights ω_{1}, ω_{3}, ω_{4}, ω_{6} belong to the root lattice and so the term ω_{1}ω_{3} contributes a rational term c_{1}^{K_{0}}([L(ω_{1})])c_{1}^{K_{0}}([L(ω_{3})]) to 𝔠′(θ), and similarly for the term ω_{3}ω_{4}. Next we observe that ω_{4}ω_{2} + ω_{4}ω_{5} contributes

c_{1}^{K_{0}}([L(ω_{4})]) (c_{1}^{K_{0}}([L(ω_{2})]) + c_{1}^{K_{0}}([L(ω_{5})]))

to 𝔠′(θ). But ω_{4} and ω_{2} + ω_{5} both lie in the root lattice, so both terms in the product are rational by Lemma 6.4. The same argument handles ω_{5}ω_{6} + ω_{6}ω_{7}, and we are done with the E_{7} case.

\underline{Type D:} Suppose that G has type D_{n}. Then

q = ∑_{i=1}^{n} ω_{i}^{2} − ∑_{i=1}^{n−2} ω_{i}ω_{i+1} − ω_{n−2}ω_{n}.

The terms ω_{i}^{2} are treated as in the E_{7} case. For the terms in the second sum, we collect around terms with even subscripts: for even i < n − 2, consider ω_{i}(ω_{i−1} + ω_{i+1}). As ω_{i} and ω_{i−1} + ω_{i+1} belong to the root lattice, we see as in the E_{7} case that they contribute rational terms to 𝔠′(θ).

Suppose now that n is even. Then we have not accounted for ω_{n−2}(ω_{n−3} + ω_{n−1} + ω_{n}) from q. As both terms in the product belong to the root lattice, we are finished as in the E_{7} case.

If n is odd, then we have not accounted for ω_{n−2}(ω_{n−1} + ω_{n}) in q. Here Λ/Λ_{r} is isomorphic to ℤ/4 and ω_{n−2}, ω_{n−1}, ω_{n} map to 2, ±1, ±3 respectively. In particular, β(ω_{n−2}) = 2β(ω_{n}), which is zero by our hypothesis on the Tits algebras, so [L(ω_{n−2})] is in the image of res : K_{0}(𝔅) → K_{0}(𝔅_{s}). Similarly, β(ω_{n−1} + ω_{n}) = β(ω_{n−1}) + β(ω_{n}) = 0, and as in the E_{7} case, we see that 𝔠′(θ) is rational.

\underline{Type B or C:} If G has type B_{n} or C_{n}, Λ/Λ_{r} equals ℤ/2. In either case,

q = ∑_{i=1}^{n} c_{ii}ω_{i}^{2} − ∑_{i=1}^{n−1} 2ω_{i}ω_{i+1}.

where the c_{ii} are 1 or 2.

For type C_{n}, the map Λ → Λ/Λ_{r} sends ω_{i} to the class of i. Previous arguments easily handle the n odd case. If n is even, previous arguments leave us to consider the term 2ω_{n−1}ω_{n}. But

2c_{1}^{K_{0}}([L(ω_{n−1})]) ≡ 2(c_{1}^{K_{0}}(g_{s_{n−1}}) + c_{1}^{K_{0}}([L(α_{n−1})])) mod γ^{2}(𝔅_{s})
≡ c_{1}^{K_{0}}(2g_{s_{n−1}}) + 2c_{1}^{K_{0}}([L(α_{n−1})]),

and again we find that 𝔠′(θ) is rational.

For type B_{n}, the map Λ → Λ/Λ_{r} sends ω_{n} to 1 and all other fundamental weights to zero. Consequently, it suffices to consider the term 2ω_{n−1}ω_{n} in q. For this we can apply the argument in the preceding paragraph. □

7. Application to essential dimension

We now apply results from the previous section to give a lower bound on the essential dimension ed(G) for some algebraic groups G. We refer to Reichstein’s 2010 ICM lecture [30] for a definition and survey of this notion. Roughly speaking, it gives the number of parameters required to specify a G-torsor.

7.1. Proposition. Let G be an absolutely almost simple algebraic group. Then ed(G) ≥ 3 unless G is isomorphic to Sp_{2n} for some n ≥ 2 (in which case ed(G) = 0) or G has type A.

The lower bound of 3 is in many cases very weak, but it has the advantage of being uniform and having a proof that is almost as uniform. The existence of the Rost invariant gives the same lower bound on ed(G) when G is simply connected, so our proposition can be viewed as removing the hypothesis “simply connected” from that result.

7.2. Remark. For groups of type A, the lower bound is more complicated and we do not know what the answer is in every case. Of course ed(SL_{n}) = 0. Some other known cases are: For n divisible by the square of a prime, we know that ed(PGL_{n}) ≥ 4 by [29, Th. 16.1(b)]; and for “intermediate” groups of type A in good characteristic, the essential dimension is at least 4 by [31, Th. 8.13]. On the other hand, A.A. Albert conjectured that central simple algebras of prime degree are cyclic, which would imply that ed(PGL_{n}) = 2 for square-free n. However, this is only currently known for n = 2, 3, and 6.

Proof of Prop. 7.1. \underline{Main case:} Suppose first that G is not of type A, C, nor E_{6}. As essential dimension only goes down with field extensions, we may assume that k is algebraically closed and bound ed(G_{s}) where G_{s} is a split simple group not of type A or C. Put G̃_{s} for the simply connected cover of G_{s}. Fix a versal G̃_{s}- torsor ξ̃ ∈ H^{1}(L, G̃_{s}) for some extension L/k. Let K be a field between k and L of minimal transcendence degree such that there is a ξ ∈ H^{1}(K, G_{s}) whose image in H^{1}(L, G_{s}) is the same as the image of ξ̃.

For sake of contradiction, suppose that K has transcendence degree at most 2 over k. By the hypothesis on the type of G, the Tits algebras of _{ξ}G have exponent a power of 2 and so are actually of index 1 or 2 over K by De Jong, see [4] or [20, Th. 4.2.2.3]. By Proposition 6.5, there is a class ψ ∈ γ^{2}(𝔅) whose image under restriction to L is 𝔠′(θ). Now Tors CH^{2}(𝔅_{K}) is zero by Prop. 3.2 because H^{3}(K, ℚ/ℤ(2)) is zero, and it follows that 𝔠′(θ) is zero in Tors CH^{2}(𝔅_{L}). But 𝔠′(θ) has order N(G_{s}) ≠ 1, a contradiction.

\underline{Type C:} If G of type C_{n} (n ≥ 2) is simply connected, by hypothesis it is not Sp_{2n}, so the Rost invariant is not zero on a versal G-torsor and the claim follows. So suppose G is adjoint. (We give an argument that is characteristic-free; if char k ≠ 2, then ed(G) ≥ n + 1 by [3, (1.1)].) If n is odd then we can construct a nonzero normalized cohomological invariant of PSp_{2n} of degree 4 as in [21, Th. 4.1] and that case is settled.

It remains to show that ed(PSp_{2n}) ≥ 3 when k is algebraically closed and n is even. Let k′ be an extension of k that has a quaternion division algebra D and fix a class ζ ∈ H^{1}(k′, PSp_{2n}) with image [D] ∈ H^{2}(k′, μ_{2}) under the natural connecting homomorphism ∂_{k′}. Fix a versal torsor ξ̃ ∈ H^{1}(L,_{ζ} Sp_{2n}) for some extension L/k′. Let K be a field between k and L so that there is a class ξ ∈ H^{1}(K, PSp_{2n}) with the same image as ξ̃ in H^{1}(L, PSp_{2n}). For sake of contradiction, suppose that K has transcendence degree at most 2 over k.

By De Jong, ∂_{K}(ξ) is the class of a quaternion algebra in H^{2}(K, μ_{2}). So we could arrange from the beginning that k′ = K and [D] = ∂_{K}(ξ). We have a well-defined invariant

im [H^{1}(∗,_{ζ} Sp_{2n}) → H^{1}(∗,_{ζ} PSp_{2n})] → H^{3}(∗, ℤ/2ℤ)

defined on extensions of K because the Rost invariant vanishes on H^{1} of the center of every simply connected group of type C_{n} with n even [12]. On the other hand, the class of ξ belongs to the domain of this map and the invariant does not vanish on it; this contradicts the hypothesis that K has transcendence degree at most 2 over k.

The remaining case of type E_{6} is known by arguments as in [13, 9.5, 9.7]. Alternatively, one can repeat the “main case” argument focusing on essential 2-dimension instead of essential dimension. □

Although we stated the proposition for absolutely almost simple groups, it quickly leads to the lower bound ed(G) ≥ 3 for more groups G. We mention: (A) to obtain a lower bound on ed(G) for any linear group G, it suffices to give a lower bound on the essential dimension of the identity component of G [1, 6.19], so one needn’t assume that G is connected. (B) If k/k_{0} is a finite separable extension, one has ed(R_{k/k_{0}}(G)) ≥ ed(G), so the proposition immediately gives a similar (but slightly more complicated to state) result for groups that are simple but not absolutely simple. (C) If G ≃ G_{1} × G_{2}, then ed(G) ≥ max{ed(G_{1}), ed(G_{2})}, and in this way we can weaken the hypothesis “simple” to “semisimple” at the cost of demanding that G be adjoint or simply connected.

Acknowledgments. We thank Zinovy Reichstein and Mark MacDonald for their comments on an earlier version of this paper. The first author’s research was partially supported by the NSF under grant DMS-0653502. The second author was supported by NSERC Discovery 385795-2010 and Accelerator Supplement 396100-2010 grants

References

1. G. Berhuy and G. Favi, Essential dimension: a functorial point of view, Doc. Math. (2003), 279–330.

2. N. Bourbaki, Lie groups and Lie algebras: Chapters 4–6, Springer-Verlag, Berlin, 2002.

3. V. Chernousov and J-P. Serre, Lower bounds for essential dimensions via orthogonal representations, J. Algebra 305 (2006), 1055–1070.

4. A.J. de Jong, The period-index problem for the Brauer group of an algebraic surface, Duke Math. J. 123 (2004), no. 1, 71–94.

5. M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301.

6. , Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88.

7. M. Demazure and A. Grothendieck, Schémas en groupes III: Structure des schemas en groupes reductifs, Lecture Notes in Mathematics, vol. 153, Springer, 1970.

8. W. Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2, Springer-Verlag, Berlin, 1998.

9. W. Fulton and S. Lang, Riemann - roch algebra, Grundlehren der Math. Wiss., vol. 277, Springer-Verlag, 1985.

10. S. Garibaldi, A. Merkurjev, and J-P. Serre, Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, Amer. Math. Soc., 2003.

11. S. Garibaldi and H.P. Petersson, Groups of outer type E_{6} with trivial Tits algebras, Transf. Groups 12 (2007), no. 3, 443–474.

12. S. Garibaldi and A. Quéguiner-Mathieu, Restricting the Rost invariant to the center, St. Petersburg Math. J. 19 (2008), no. 2, 197–213.

13. P. Gille and Z. Reichstein, A lower bound on the essential dimension of a connected linear group, Comment. Math. Helv. 84 (2009), no. 1, 189–212.

14. S. Gille and K. Zainoulline, Equivariant pretheories and invariants of torsors, arxiv:1007.3780v1, 2010.

15. B.H. Gross and G. Nebe, Globally maximal arithmetic groups, J. Algebra 272 (2004), no. 2, 625–642.

16. J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, 1990.

17. N.A. Karpenko, Codimension 2 cycles on Severi-Brauer varieties, K-Theory 13 (1998), no. 4, 305–330.

18. S. Kumar and M.S. Narasimhan, Picard group of the moduli spaces of G-bundles, Math. Ann. 308 (1997), 155–173.

19. Y. Laszlo and C. Sorger, The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Ec. Norm. Sup. (4) 30 (1997), 499–525.

20. M. Lieblich, Twisted sheaves and the period-index problem, Compositio Math. 144 (2008), 1–31.

21. M.L. MacDonald, Cohomological invariants of odd degree Jordan algebras, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 2, 295–303.

22. A. Merkurjev, I. Panin, and A. Wadsworth, Index reduction formula for twisted flag varieties, i., K-Theory 10 (1996), 517–596.

23. A. Merkurjev and A. Suslin, Motivic cohomology of the simplicial motive of a Rost variety, J. Pure Appl. Algebra 214 (2010), 2017–2026.

24. S. Nikolenko, N. Semenov, and K. Zainoulline, Motivic decomposition of anisotropic varieties of type F_{4} into generalized Rost motives., J. K-Theory 3 (2009), 85–102.

25. I.A. Panin, On the algebraic K-theory of twisted flag varieties, K-Theory 8 (1994), no. 6, 541–585.

26. V. Petrov, N. Semenov, and K. Zainoulline, J-invariant of linear algebraic groups, Ann. Ec. Norm. Sup. (4) 41 (2008), 1023–1053.

27. E. Peyre, Galois cohomology in degree three and homogeneous varieties, K-Theory 15 (1998), no. 2, 99–145.

28. H.V. Pittie, Homogeneous vector bundles on homogeneous spaces, Topology 11 (1972), 199– 203.

29. Z. Reichstein, On a theorem of Hermite and Joubert, Canad. J. Math. 51 (1999), 69–95.

30. , Essential dimension, Proceedings of the International Congress of Mathematicians 2010, World Scientific, 2010.

31. Z. Reichstein and B. Youssin, Essential dimensions of algebraic groups and a resolution theorem for G-varieties, Canad. J. Math. 52 (2000), no. 5, 1018–1056, with an appendix by J. Kollár and E. Szabó.

32. R. Steinberg, On a theorem of Pittie, Topology 14 (1975), 173–177.

33. J. Tits, Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196–220.

34. N. Yagita,  Algebraic BP -theory and norm varieties,  available at

    http://hopf.math.purdue.edu/Yagita/abp.pdf, 2006.

(Garibaldi) Department of Mathematics & Computer Science, Emory University, Atlanta, GA 30322, USA

   E-mail address: skip@member.ams.org
   URL: http://www.mathcs.emory.edu/~skip/

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

   E-mail address: kirill@uottawa.ca
   URL: http://www.mathstat.uottawa.ca/~kirill/