THE J-INVARIANT AND TITS INDICES FOR GROUPS OF INNER TYPE E_{6}

CAROLINE JUNKINS

Abstract. A connection between the indices of the Tits algebras of a split linear algebraic group G and the degree one parameters of its motivic J-invariant was introduced by Quéguiner-Mathieu, Semenov and Zainoulline through use of the second Chern class map in the Riemann-Roch theorem without denominators. In this paper we extend their result to higher Chern class maps and provide applications to groups of inner type E_{6}.

Introduction

For a linear algebraic group G, an invariant known as the Tits algebra was introduced by J. Tits in [18] and has proven to be an invaluable tool for the computation of the K-theory of twisted flag varieties by Panin [11] and for the index reduction formulas by Merkurjev, Panin and Wadsworth [10]. Furthermore, the Tits algebra has applications to both the classification of linear algebraic groups and the study of the associated homogeneous varieties.

The J-invariant, as defined by Petrov, Semenov and Zainoulline in [13], is an invariant of G which describes the motivic behaviour of the variety of Borel subgroups of G. For a prime p, the J-invariant of G modulo p is given by an r-tuple of integers J_{p}(G) = (j_{1}, …, j_{r}). We consider also a (possibly empty) subtuple J_{p}^{(1)}(G) = (j_{1}, …, j_{s}), s ≤ r, consisting of the parameters of the J-invariant of degree 1.

Motivated by the work [5], Quéguiner-Mathieu, Semenov and Zainoulline discovered a connection between these degree 1 parameters and the indices of the Tits algebras of G. This connection is developed in [14], through use of the second Chern class map in the Riemann-Roch theorem without denominators. The goal of this paper is to extend their result through use of higher Chern class maps (see Theorem 4.1). We then apply this result to a group G of inner type E_{6}. We provide an explicit connection between the values of J_{3}^{(1)}(G) and the index of the Tits algebra of G (see Proposition 5.2).

This paper is organized as follows. In the first section, we review the notion of characteristic classes and introduce the topological and γ-filtrations on the Grothendieck group K_{0} of a smooth projective variety. In Section 2, we look more closely at the γ-filtration of the variety of Borel subgoups X, and give a simplified definition through use of the Steinberg basis. We then consider a twisted form _{ξ}X of X by means of a G-torsor ξ ∈ H^{1}(k, G) and the γ-filtration of K_{0}(_{ξ}X). In Section 3, we recall the definition of the Tits map and its relation to our primary objects of concern, the common index i_{c} and the J-invariant of ξ. Section 4 provides the main result of the paper, a relationship between the common index and the possible values of the indices of the J-invariant of degree 1. In the final section, we give an application of this result to a group of inner type E_{6}.

Acknowledgments. I am very 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. The last part of these notes has been written during my research visit to the Johannes Gutenberg University of Mainz. My special gratitude is due to N. Semenov and the University of Mainz for hospitality and support.

1. Gamma filtration on K_{0} and the Chern class map

In the present section we recall several useful properties of the topological filtration and the γ-filtration on Grothendieck’s K_{0} of a smooth projective variety. The reader is advised to look at [16], [3] and [4, ch.15] for more details.

Let X be a smooth projective variety over a field k. Consider the topological filtration on K_{0}(X) (see [4, Ex. 15.1.5]) given by

τ^{i} K_{0}(X) = ⟨[O_{V}] | codim V ≥ i⟩,

where O_{V} is the structure sheaf of a closed subvariety V in X. We denote by τ^{i/i+1}(X), i ≥ 0 the i-th subsequent quotient τ^{i} K_{0}(X)/τ^{i+1} K_{0}(X). There is a surjection

p: CH^{i}(X) ↠ τ^{i/i+1}K_{0}(X), V ↦ [O_{V}],

from the Chow group of codimension i cycles.

For a vector bundle E on X, the total Chern class c(E) = 1+c_{1}(E)t+c_{2}(E)t^{2}+… is an element of CH(X), and by the Whitney sum formula, it defines a group homomorphism

c : K_{0}(X) → CH(X)

For α ∈ K_{0}(X), c_{i}(α) is the component of c(α) in CH^{i}(X), giving a group homomorphism

c_{i}: τ^{i} K_{0}(X) → CH^{i}(X)

defined by taking the i-th Chern class.

1.1. Lemma. c_{i}(τ^{j} K_{0}(X)) = 0 for all 0 < i < j.

Proof. From [4, Ex 15.3.6] it can be seen that c_{i}(τ^{i+1} K_{0}(X)) = 0 for all i ∈ ℤ_{>0}. By the definiton of the topological filtration, we have τ^{j} K_{0}(X) ⊆ τ^{i} K_{0}(X) for all i ≤ j, and so i ⪇ j implies that in particular we have τ^{j} ⊆ τ^{i+1} K_{0}(X). The result follows immediately. □

Thus we have an induced homomorphism

c_{i}: τ^{i/i+1}(X) → CH^{i}(X)

such that the composite c_{i} ∘ p is the multiplication by (−1)^{i−1}(i − 1)!. This result implies that c_{i} is an isomorphism for i ≤ 2 (see [4, Ex. 15.3.6 ]) and, moreover,

1.2. Lemma. The map c_{i} is an isomorphism over the coefficient ring ℤ[\frac{1}{(i−1)!}].

Let γ_{i} denote the i-th characteristic class with values in K_{0} (see [4, Ex. 3.2.7(b)]. We follow the convention γ_{1}([L]) = 1 − [L^{∨}] for any line bundle L over X, where L^{∨} denotes the dual of L.

1.3. Example. Using the Whitney sum formula we obtain the following results for i = 1, 2 by computing the total Chern classes,

c(γ_{1}([L])) = c(1 − [L^{∨}])

= \frac{1}{1 − c_{1}(L)t}

= 1 + c_{1}(L)t + c_{1}(L)^{2}t^{2} + …

This gives c_{1}(γ_{1}([L])) = c_{1}(L). Similarly,

c(γ_{1}([L_{1}])γ_{1}([L_{2}])) = c((1 − [L_{1}^{∨}])(1 − [L_{2}^{∨}])) = c(1 − [L_{1}^{∨}] − [L_{2}^{∨}] + [(L_{1} ⊗ L_{2})^{∨}])

= \frac{c(1) · c([(L_{1} ⊗ L_{2})^{∨}])}{c([L_{1}^{∨}]) · c([L_{2}^{∨}])} = \frac{1 − (c_{1}(L_{1}) + c_{1}(L_{2}))t}{(1 − c_{1}(L_{1})t)(1 − c_{1}(L_{2})t)}

= (1−(c_{1}(L_{1})+c_{1}(L_{2}))t)(1+c_{1}(L_{1})t+c_{1}(L_{1})^{2}t^{2}+…)(1+c_{1}(L_{2})t+c_{1}(L_{2})^{2}t^{2}+…)

= 1 − c_{1}(L_{1})c_{1}(L_{2})t^{2} + …

Hence c_{2}(γ_{1}([L_{1}])γ_{1}([L_{2}])) = −c_{1}(L_{1})c_{1}(L_{2}).

By the definition of these characteristic classes (see [4, Ex 15.3.6]), we have in general

(1)  c_{i}(γ_{1}([L_{1}]) … γ_{1}([L_{i}])) = (−1)^{i−1}(i − 1)! · c_{1}(L_{1}) · . . . · c_{1}(L_{i}).

The Grothendieck γ-filtration on K_{0}(X) is defined by

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

(see [4, Ex.15.3.6], [3, Ch.3 and 5]). Let γ^{i/i+1}(X) denote the i-th subsequent quotient γ^{i} K_{0}(X)/γ^{i+1} K_{0}(X).

It is known that γ^{i} K_{0}(X) is contained in τ^{i} K_{0}(X) for every i ≥ 0, and they coincide for i ≤ 2 (see [8, Prop.2.14]). Therefore, by Lemma 1.1 the Chern class map c_{i} restricted to γ^{i} K_{0}(X) vanishes on γ^{i+1} K_{0}(X), and hence induces a map

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

1.4. Example. For i = 1 we have γ^{1/2} K_{0}(X) = τ^{1/2} K_{0}(X) and c_{1} ∘ p = id_{CH^{1}(X)}, giving an isomorphism

c_{1} : γ^{1/2} K_{0}(X) → CH^{1}(X).

In the previous example, we saw that the map c_{1} sends γ_{1}([L]) to c_{1}(L). For i = 2 we again have γ^{2} K_{0}(X) = τ^{2} K_{0}(X), but this time γ^{3} K_{0}(X) does not necessarily coincide with τ^{3} K_{0}(X). We may form an exact sequence,

0 → τ^{3} K_{0}(X)/γ^{3} K_{0}(X) → γ^{2/3} K_{0}(X) → τ^{2/3} K_{0}(X) → 0.

Replacing τ^{2/3} K_{0}(X) with CH^{2}(X), we see that the map c_{2} : γ^{2/3} K_{0}(X) → CH^{2}(X) is surjective. In addition, ker(c_{2}) ≅ τ^{3} K_{0}(X)/γ^{3} K_{0}(X). Note that for all i ≥ 0, (see [8, Prop. 2.14])

τ^{i} K_{0}(X) ⊗ ℚ ≅ γ^{i} K_{0}(X) ⊗ ℚ,

thus ker(c_{2}) is torsion.

1.5. Proposition. The Chern class map c_{i}: γ^{i/i+1}(X) → CH^{i}(X) is surjective over the coefficient ring ℤ[\frac{1}{(i−1)!}].

Proof. By Lemma 1.2, the map c_{τ,i} : τ^{i/i+1}(X) → CH^{i}(X) is surjective over the coefficient ring ℤ[\frac{1}{(i−1)!}]. Since γ^{i} K_{0}(X) ⊆ τ^{i} K_{0}(X) for all i, we have an obvious map γ^{i/i+1} K_{0}(X) → τ^{i/i+1} K_{0}(X) defined by sending x + γ^{i+1} K_{0}(X) ↦ x + τ^{i+1} K_{0}(X). By definition, c_{γ,i} is the composition of these two maps.

\xymatrix{γ^{i+1} K_{0}(X) \ar[r] \ar[rr]_{c_{γ,i}} & τ^{i/i+1} K_{0}(X) \ar@{->>}[r]^{c_{τ,i}} & CH^{i}(X)}

Over ℤ[\frac{1}{(i−1)!}] we have im(c_{γ,i}) ⊆ im(c_{τ,i}) = CH^{i}(X), and so it remains to show that CH^{i}(X) ⊆ im(c_{γ,i}) for all i. Consider an arbitrary element x ∈ K_{0}(X). By the splitting principle we may write x = L_{1} ⊕ ⋯ ⊕ L_{n} where L_{1}, …, L_{n} are line bundles over X. By the properties of characteristic classes, γ_{i}(L_{1} ⊕ ⋯ ⊕ L_{n}) = 0 for all i > n and γ_{i}(L_{1} ⊕ ⋯ ⊕ L_{n}) = s_{i}(γ_{1}(L_{1}), …, γ_{1}(L_{n})) for all 0 < i ≤ n, where s_{i}(γ_{1}(L_{1}), …, γ_{1}(L_{n})) is the i-th elementary symmetric polynomial in variables γ_{1}(L_{1}), …, γ_{1}(L_{n}). Thus, taking the total Chern class, we have by (1) and Lemma 1.1

c(γ_{i}(L_{1} ⊕ ⋯ ⊕ L_{n})) = c(s_{i}(γ_{1}(L_{1}), …, γ_{1}(L_{n})))

= ∏_{1≤j_{1}<···<j_{i}≤n}(1 + (−1)^{i−1}(i − 1)!c_{1}(L_{j_{i}}) ⋯ c_{1}(L_{j_{i}})t^{i} + …)

= 1 + (−1)^{i−1}(i − 1)! · s_{i}(c_{1}(L_{1}), …, c_{1}(L_{n}))t^{i} + …

Thus, c_{i}(γ_{i}(L_{1} ⊕ ⋯ ⊕ L_{n})) = (−1)^{i−1}(i − 1)!s_{i}(c_{1}(L_{1}), …, c_{1}(L_{n})). In general, we have

(2)  c_{i}(γ_{i}(x)) = (−1)^{i−1}(i − 1)!c_{i}(x) for all x ∈ K_{0}(X).

Let y ∈ CH^{i}(X). Then by the sujectivity of the Chern class map, y = c_{i}(x) for some x ∈ τ^{i/i+1} K_{0}(X). By (2), c_{i}(x) = \frac{(−1)^{i−1}}{(i−1)!} · c_{i}(γ_{i}(x)), where γ_{i}(x) ∈ γ^{i/i+1} K_{0}(X) by definiton of the γ-filtration. Thus y ∈ im(c_{γ,i}) over the coefficient ring ℤ[\frac{1}{(i−1)!}], as required. □

2. Gamma filtration on twisted flag varieties

In the present section we discuss the γ-filtration on the variety of Borel subgroups. A reader is encouraged to look at [14] and [5] for further details.

Let G be a split simple linear algebraic group over a field k. We fix a split maximal torus T and a Borel subgroup B ⊃ T of G. Let W denote the Weyl group of G. Let X denote the variety of Borel subgroups of G.

Let {g_{w}}_{w∈W} be the Steinberg basis of K_{0}(X) (cf. [17]). For each w ∈ W, g_{w} = [L_{w}] is the class of a line bundle over X, and together they form a ℤ-basis of K_{0}(X).

2.1. Lemma. The i-th term of the γ-filtration on X is generated by products

γ^{i/i+1} K_{0}(X) = {γ_{1}(g_{w_{1}}) · . . . · γ_{1}(g_{w_{i}}), w_{1}, …, w_{i} ∈ W }.

Proof. As a temporary notation, let

Γ^{i} = ⟨γ_{1}(g_{w_{1}}) · . . . · γ_{1}(g_{w_{i}}), w_{1}, …, w_{i} ∈ W ⟩.

By the definition of γ^{i} K_{0}(X), it is clear that Γ^{i} ⊆ γ^{i} K_{0}(X) for all i. It remains to show the other inclusion. Consider first the case i = 1. Let x ∈ K_{0}(X) such that x = ∑_{w∈W} a_{w}g_{w} for some a_{w} ∈ ℤ. Computing the total characteristic class, we have

γ(x) = γ( ∑_{w∈W} a_{w}g_{w}) = ∏_{w∈W} γ(g_{w})^{a_{w}} = ∏_{w∈W}(1 + γ_{1}(g_{w})t)^{a_{w}}

= ∏_{w∈W}(∑_{k=0}^{a_{w}} \binom{a_{w}}{k}γ_{1}(g_{w})^{k}t^{k}) = ∏_{w∈W}(1 + a_{w}γ_{1}(g_{w})t + \binom{a_{w}}{2}γ_{1}(g_{w})^{2}t^{2} + …)

(3) = 1+( ∑_{w∈W} a_{w}γ_{1}(g_{w}))t+( ∑_{w∈W} \binom{a_{w}}{2}γ_{1}(g_{w})^{2}+∑_{v≠w} a_{v}a_{w}γ_{1}(g_{v})γ_{1}(g_{w}))t^{2}+…

Thus we have γ_{1}(x) = ∑_{w∈W} a_{w}γ_{1}(g_{w}) ∈ Γ^{1}, as required.

In general, it is clear that Γ defines a graded ring. That is, if x ∈ Γ^{i} and y ∈ Γ^{j}, then xy ∈ Γ^{i+j}. Suppose for induction that γ^{j} K_{0}(X) = Γ^{j} for all j < i and consider an element y = γ_{i_{1}}(x_{1}) ⋯ γ_{i_{m}}(x_{m}) ∈ γ^{i} K_{0}(X) such that i_{1} + ⋯ + i_{m} = i. In the case that m = 1 (i.e. y = γ_{i}(x)) the expansion (3) shows that y ∈ Γ^{i}. If instead m > 1, then we have i_{l} < i for all l = 1, …, m. By induction, γ_{i_{l}}(x_{l}) ∈ Γ^{i_{l}} for each l = 1, …, m, and so

y = γ_{i_{1}}(x_{1}) ⋯ γ_{i_{m}}(x_{m}) ∈ Γ^{i_{1}} ⋯ Γ^{i_{m}} ⊆ Γ^{i}.
□

Consider the twisted form _{ξ}X of X by means of a G-torsor ξ ∈ H^{1}(k, G). In general the group K_{0}(_{ξ}X) is not generated by classes of line bundes. Hence, in the definition of γ^{i} K_{0}(_{ξ}X) some higher characteristic classes γ_{j>1}(−) may appear.

Note that characteristic classes commute with restrictions, i.e. for every field extension k′/k and variety Y over k we have

res_{k′/k} ∘ γ_{j} = γ_{j} ∘ res_{k′/k} and c_{j} ∘ res_{k′/k} = res_{k′/k} ∘ c_{j}

where res_{k′/k}: K_{0}(Y) → K_{0}(Y ×_{k} k′). Moreover, for every k′/k the restriction map res: K_{0}(X) → K_{0}(X ×_{k} k′) is an isomorphism.

We will use the following commutative diagram

\xymatrix{γ^{i/i+1}(_{ξ}X) \ar[r]^{res_{γ}} \ar[d]_{c_{i}} & γ^{i/i+1}(X) \ar[d]^{c_{i}} \\ CH^{i}(_{ξ}X) \ar[r]^{res_{k′/k}} & CH^{i}(X).}

Let l be a splitting field of ξ. The main result of [11] says that the image of the restriction map

res_{l/k}: K_{0}(_{ξ}X) → K_{0}(_{ξ}X ×_{k} l) ≃ K_{0}(X ×_{k} l) ≃ K_{0}(X)

coincides with the sublattice

⟨i_{w,ξ}g_{w}⟩_{w∈W} ,

where i_{w,ξ} ≥ 1 are indices of the respective Tits algebras, which will be introduced in the next section.

2.2. Proposition. Consider the composite

π_{i}: γ^{i/i+1}(_{ξ}X) ⟶^{res_{γ}} γ^{i/i+1}(X) ⟶^{c_{i}} CH^{i}(X).

The image of π_{1} is generated by i_{w,ξ}c_{1}(g_{w}) for all w ∈ W. The image of π_{2} is generated by the elements i_{w_{1},ξ}i_{w_{2},ξ}c_{1}(g_{w_{1}})c_{1}(g_{w_{2}}) and \binom{i_{w,ξ}}{2}c_{1}(g_{w})^{2} for all w_{1}, w_{2}, w ∈ W. In general, the image of π_{i} is generated by the elements

(i − 1)!\binom{i_{w_{1}}}{i_{1}} ⋯ \binom{i_{w_{m}}}{i_{m}}c_{1}(g_{w_{1}})^{i_{1}} ⋯ c_{1}(g_{w_{m}})^{i_{m}}

where i_{1} + ⋯ + i_{m} = i for all w_{1}, …, w_{m} ∈ W.

Proof. By the defintions of the restriction map and the γ-filtration on K_{0}(X), we can see that the image of res_{γ}^{(i)} is generated by products

res_{γ}(γ^{i/i+1} K_{0}(_{ξ}X)) = {γ_{i_{1}}(i_{w_{1}}g_{w_{1}}) ⋯ γ_{i_{m}}(i_{w_{m}}g_{w_{m}})|i_{1} + ⋯ + i_{m} = i},

where w_{1}, …, w_{m} ∈ W.

We also note that since the g_{w}’s are line bundles, we have

γ(i_{w}g_{w}) = γ(g_{w})^{i_{w}}

= (1 + γ_{1}(g_{w})t)^{i_{w}}

= ∑_{k=1}^{i_{w}} \binom{i_{w}}{k}γ_{1}(g_{w})^{k}t^{k}

and hence for all 0 ≤ j ≤ i_{w} we have

(4)  γ_{j}(i_{w}g_{w}) = \binom{i_{w}}{j}γ_{1}(g_{w})^{j}.

This property holds for all characteristic classes, and hence

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

We consider first the case i = 1. Then im(res_{γ}^{(1)}) is generated by the elements γ_{1}(i_{w}g_{w}) = i_{w}γ_{1}(g_{w}) for any w ∈ W. Thus, by (2) and (5), the image of π_{1} is generated by elements of the form

c_{1}(γ_{1}(i_{w}g_{w})) = c_{1}(i_{w}g_{w})
= i_{w}c_{1}(g_{w})

for all w ∈ W.

In general, consider an element of the form x = γ_{i_{1}}(i_{w_{1}}g_{w_{1}}) ⋯ γ_{i_{m}}(i_{w_{m}}g_{w_{m}}) such that i_{1} + ⋯ + i_{m} = i for some w_{1}, …, w_{m} ∈ W. By (4)

γ_{i_{1}}(i_{w_{1}}g_{w_{1}}) ⋯ γ_{i_{m}}(i_{w_{m}}g_{w_{m}}) = \binom{i_{w_{1}}}{i_{1}} ⋯ \binom{i_{w_{m}}}{i_{m}}γ_{1}(g_{w_{1}})^{i_{1}} ⋯ γ_{1}(g_{w_{m}})^{i_{m}}.

Taking total Chern classes then gives

c(x) = c(\binom{i_{w_{1}}}{i_{1}} ⋯ \binom{i_{w_{m}}}{i_{m}}γ_{1}(g_{w_{1}})^{i_{1}} ⋯ γ_{1}(g_{w_{m}})^{i_{m}})
= c(γ_{1}(g_{w_{1}})^{i_{1}} ⋯ γ_{1}(g_{w_{m}})^{i_{m}})^{\binom{iw_{1}}{i_{1}}}^{···}^{\binom{iw_{m}}{im}}
= (1 + (−1)^{i−1}(i − 1)!c_{1}(g_{w_{1}})^{i_{1}} ⋯ c_{1}(g_{w_{m}})^{i_{m}}t^{i})^{\binom{iw_{1}}{i_{1}}}^{···}^{\binom{iw_{m}}{im}},

and so,

c_{i}(x) = (−1)^{i−1}(i − 1)!\binom{i_{w_{1}}}{i_{1}} ⋯ \binom{i_{w_{m}}}{i_{m}}c_{1}(g_{w_{1}})^{i_{1}} ⋯ c_{1}(g_{w_{m}})^{i_{m}}.

□

3. Tits algebras and the J-invariant

Recall that we have defined G to be a split linear algebraic group of rank n over a field k. Also, we have fixed a split maximal torus T ⊂ G and a Borel subgrup B ⊃ T. Let T^{∗} be the character group of T, {α_{1}, …, α_{n}} a set of simple roots with respect to B and {ω_{1}, …, ω_{n}} the respective set of fundamental weights, so that α_{i}^{∨}(ω_{j}) = δ_{ij}. We have Λ_{r} ⊂ T^{∗} ⊂ Λ, where Λ_{r} is the root lattice and Λ is the weight lattice. Consider the simply connected cover G̃ of G with corresponding Borel subgroup B̃ and maximal split torus T̃. Given any λ ∈ Λ = Hom(T̃, 𝔾_{m}), we can lift λ : T̃ → 𝔾_{m} uniquely to λ : B̃ → 𝔾_{m}. Letting

G̃ ×B̃ V_{1} = G̃ × V_{1}/(g, v) ∼ (g · b, λ(b)^{−1} · v),

the projection map G̃ ×B̃ V_{1} → G̃/B̃ defines a line bundle L(λ) over G̃/B̃ = G/B (see [2, §1.5]), the variety of Borel subgroups of G.

For a fixed ξ ∈ H^{1}(k, G), we can associate to each weight λ a central simple k-algebra A_{ξ,λ}, called a Tits algebra of G (cf. [18]). We define the Tits map

β_{ξ} : Λ/Λ_{r} → Br(k)

by sending λ̄ ↦ [A_{ξ,λ̄}], its class in the Brauer group. We note that β_{ξ} is a group homomorphism for a fixed ξ, with λ̄_{1} + λ̄_{2} ↦ [A_{ξ,λ̄_{1}}] ⊗ [A_{ξ,λ̄_{2}}]. Consider all Tits algebras A_{ξ,λ̄}, λ ∈ Λ of ξ and let H be the subgroup in Br(k) generated by the classes of all such A_{ξ,λ̄}. In other words, H is generated by the non-constant elements in the image of the Tits map β_{ξ}. Since Λ/Λ_{r} is a finite abelian group, H is a finite abelian group as well. If H is non-trivial, we define the common index i_{c} of ξ as

i_{c} := gcd{ind(a) | a ∈ H \ {1}}.

where the gcd is taken over all indices of non-trivial elements of H.

3.1. Example. Let G be a group of inner type E_{6}. G has precisely one non-trivial Tits algebra, with index 3^{d} for d = 0, …, 3. Therefore, by the definition of the common index, we have i_{c} = 3^{d} as well.

Let X be the vairety of Borel subgroups of G. The degree 1 characteristic map in the simply connected case

𝔠_{sc}^{(1)} : Λ ˜→ CH^{1}(X)

defines an isomorphim such that the cycles h_{i} = c_{1}(L(ω_{i})), i = 1, …, n form a ℤ-basis of CH^{1}(X). The degree 1 characteristic map is the restriction of this isomorphism to the character group T^{∗}

𝔠^{(1)} : T^{∗} → CH^{1}(X),

mapping λ = ∑_{i=1}^{n} a_{i}ω_{i} ↦ c_{1}(L(λ)) = ∑_{i=1}^{n} a_{i}h_{i}. In general, this defines the characteristic map 𝔠: S^{∗}(T^{∗}) → CH(X).

We denote by π : CH^{∗}(X) → CH^{∗}(G) the pull-back induced by the natural projection G → X. By [6, Section 4, Rem. 2], π is sujective and its kernel is given by the ideal I ⊂ CH^{∗}(X) generated by the non-constant elements in the image of the characteristic map. In particular, we have I^{(1)} = im(𝔠^{(1)}), and

CH^{1}(G) ≃ CH^{1}(X)/(im(𝔠^{(1)})) ≃ Λ/T^{∗}.

Given a prime p, set Ch(X) = CH(X) ⊗ 𝔽_{p}. Taking 𝔽_{p}-coefficients, we have

Ch^{1}(G) ≃ Ch^{1}(X)/(im(𝔠^{(1)})) ≃ Λ/T^{∗} ⊗_{ℤ} 𝔽_{p}.

It is known (see [7]) that Ch(X)/I is isomorphic (as an 𝔽_{p} algebra, as well as a Hopf algebra) to

Ch(X)/I ≅ 𝔽_{p}[x_{1}, …, x_{r}]/(x_{1}^{p^{k_{1}}} , …, x_{r}^{p^{kr}})

for some integers r and k_{i} ≥ 0 for i = 1, …, r, which are dependent on the group G. For each i, we let d_{i} be the degree of the generator x_{i}. The number of generators of degree 1 is given by the dimension over 𝔽_{p} of the vector space Λ/T^{∗} ⊗_{ℤ} 𝔽_{p}.

Let s = dim_{𝔽_{p}}(Ch^{1}(G)). Then, since ω_{1}, …, ω_{n} generate Λ we may choose a minimal set {i_{1}, …, i_{s}} ⊂ {1, …, n} such that the classes of ω_{i_{1}}, …, ω_{i_{s}} generate Λ/T^{∗} ⊗ 𝔽_{p}. Then, h_{i_{l}} = c_{1}(L(ω_{i_{l}})), l = 1, …, s generate Ch^{1}(X) and so we may take x_{l} = π(h_{i_{l}}), l = 1, …, s to be the generators of Ch^{1}(G).

In fact, this definition of the generators x_{1}, …, x_{s} can be simplified using properties of the Steinberg basis. We recall that for all i = 1, …, n, g_{i} = L(ρ_{s_{i}}), where ρ_{s_{i}} = ω_{i} − α_{i}. Thus, we have L(ω_{i}) = g_{i} + L(α_{i}). But, Λ_{r} ⊂ T^{∗} implies that c_{1}(L(α_{i})) ∈ im(𝔠^{(1)}) and hence c_{1}(L(α_{i})) ∈ ker(π) for all i = 1, …, n. So, for each l = 1, …, s, we have

π(h_{i_{l}}) = π(c_{1}(L(ω_{i_{l}})))
= π(c_{1}(g_{i_{l}}) + c_{1}(L(α_{i_{l}})))
= π(c_{1}(g_{i_{l}})) + π(c_{1}(L(α_{i_{l}})))
= π(c_{1}(g_{i_{l}})).

Thus, we may take the generators to be x_{l} = π(c_{1}(g_{i_{l}})) for l = 1, …, s.

We impose an well-ordering on the set of generators, that is, on the monomials x_{1}^{m_{1}} ⋯ x_{r}^{m_{r}} known as the DegLex order [13]. For ease of notation, we denote the monomial x_{1}^{m_{1}} ⋯ x_{r}^{m_{r}} by x^{M}, where M is the r-tuple of integers (m_{1}, …, m_{r}), and set |M| = ∑_{i=1}^{r} d_{i}m_{i}. Given two r-tuples M = (m_{1}, …, m_{r}) and N = (n_{1}, …, n_{r}), we say that x^{M} ≤ x^{N} (or equivalently M ≤ N) if either |M| < |N|, or |M| = |N| and m_{i} ≤ n_{i} for the greatest i such that m_{i} ≠ n_{i}.

Consider the restriction map res_{l/k}: CH^{∗}(_{ξ}X) → CH^{∗}(X). Let I_{ξ} denote the ideal generated by non-constant elements of degrees ≥ 1 from the image of res_{l/k}. Observe that I_{ξ} ⊃ I (see [9]) . Furthermore, it is known that there exists some ξ such that I_{ξ} = I, and such a ξ is called a “generic” torsor.

Now, since I ⊆ I_{ξ}, we have CH(X)/I_{ξ} ⊆ CH(X)/I and hence Ch(X)/I_{ξ} ⊆ Ch(X)/I. Again, there is an isomorphism

Ch(X)/I_{ξ} ≅ 𝔽_{p}[x_{1}, …, x_{r}]/(x_{1}^{p^{j_{1}}} , …, x_{n}^{p^{jr}})

where j_{i} ≤ k_{i} for each 1 ≤ i ≤ r. While r, d_{i} and k_{i} for i = 1, …, r depend only on the group G, the values j_{1}, …, j_{r} depend also on the choice of ξ. Thus given the DegLex ordering defined above, we have a well-defined r-tuple J_{p,ξ}(G) = (j_{1}, …, j_{r}), called the J-invariant of G. We note that (0, …, 0) ≤ (j_{1}, …, j_{r}) ≤ (k_{1}, …, k_{r}) for any choice of ξ.

Let J_{p}^{(1)} = {j_{1}, …, j_{s}} denote the set of indices of the J-invariant of ξ of degree 1. We say that J_{p}^{(1)} > m if for every index j_{l} such that k_{l} > m we have j_{l} > m.

4. The main result

For a fixed prime p, we have defined a minimal subset {ω_{i_{1}}, …, ω_{i_{s}}} ⊂ Λ such that the elements x_{l} = π(c_{1}(g_{i_{l}})), l = 1, …, s generate Ch^{1}(G). By the general definiton of the common index, we may now state that

i_{c} = gcd{ind(A_{ω_{i_{1}}}^{⊗b_{1}} ⊗ ⋯ ⊗ A_{ω_{is}}^{⊗b_{n}}), where at least one of the b_{i} is coprime to p}.

Let I ⊂ CH(X) be the ideal generated by the non-constant elements in the image of the characteristic map 𝔠 : S^{∗}(T^{∗}) → CH(X) and I_{ξ} ⊂ CH(X) be be ideal generated by the non-constant elements in the image of the restriction map res_{l/k} : CH(_{ξ}X) → CH(X). For any integer m, we let I^{(m)} ⊂ CH^{m}(X) and I_{ξ}^{(m)} ⊂ CH^{m}(X) denote the homogeneous parts of these ideals of degree m.

4.1. Theorem. If v_{p}(i_{c}) > 0, then I_{ξ}^{(1)} = I^{(1)}.

If v_{p}(i_{c}) > 1, then I_{ξ}^{(m)} = I^{(m)} for m = 2, …, p.

Proof. Since we know already that I ⊂ I_{ξ}, it suffices to prove that I_{ξ}^{(m)} ⊂ I^{(m)} for all m = 1, …, p under the relevant hypothesis on i_{c}. By Proposition 1.5 and the commutative diagram in Section 2, we have that for any i ≥ 0,

im(res_{l/k}^{(i)}) = c_{i}(im(res_{γ}^{(i)})

over the coefficient ring ℤ[\frac{1}{(i−1)!}].

We begin first with the case m = 1. By the definiton of I_{ξ}, we have I_{ξ}^{(1)} =

im(res_{l/k}^{(1)}). Therefore, to show that I_{ξ}^{(1)} ⊆ I^{(1)}, we must prove that if v_{p}(i_{c}) > 0, then for any w ∈ W, the element i_{w}c_{1}(g_{w}) belongs (after tensoring with 𝔽_{p}) to I^{(1)} = im(𝔠^{(1)}). Recall that g_{w} = L(ρ_{w}), and that we may write ρ_{w} = ∑_{i=1}^{n} a_{i}ω_{i}. Therefore, we have

g_{w} = L(ρ_{w})

= L(∑_{i=1}^{n} a_{i}ω_{i})

= L(ω_{1})^{⊕a_{1}} ⊕ ⋯ ⊕ L(ω_{n})^{⊕a_{n}}.

Taking total chern classes gives

c(g_{w}) = c(L(ω_{1})^{⊕a_{1}} ⊕ ⋯ ⊕ L(ω_{n})^{⊕a_{n}})

= c(L(ω_{1}))^{a_{1}} ⋯ c(L(ω_{n}))^{a_{n}}

= (1 + c_{1}(L(ω_{1}))t)^{a_{1}} ⋯ (1 + c_{1}(L(ω_{n}))t)^{a_{n}}

= (1 + a_{1}c_{1}(L(ω_{1}))t + …) ⋯ (1 + a_{n}c_{1}(L(ω_{n}))t + …)

= 1 + (∑_{i=1}^{n} a_{i}c_{1}(L(ω_{i})))t + …,

and hence c_{1}(g_{w}) = ∑_{i=1}^{n} a_{i}c_{1}(L(ω_{i})). For each i = 1, …, n we have shown in the previous sections that c_{1}(g_{i}) = c_{1}(L(ω_{i})) − c_{1}(L(α_{i})), and c_{1}(L(α_{i})) ∈ im(𝔠^{(1)}). Now, since Ch^{1}(G) is generated by x_{l} = π(c_{1}(g_{i_{l}})) for l = 1, …, s, we may write

c_{1}(g_{w}) = ∑_{l=1}^{s} a_{i_{l}}c_{1}(g_{i_{l}}) mod im(c^{(1)}).

If all a_{i_{l}} ∈ ℤ are divisible by p, we are done. So we assume at least one a_{i_{l}} is coprime to p.

Recall that ρ̄_{w} = ∑_{l=1}^{s} a_{i_{l}} ω̄_{i_{l}} mod T^{∗} for the same coefficients a_{i_{l}}. Applying the Tits map β_{ξ}, we get

β_{ξ}(ρ̄_{w}) = β_{ξ}(∑_{l=1}^{s} a_{i_{l}} ω̄_{i_{l}})

= ⨂_{l=1}^{s} β_{ξ}(ω̄_{i_{l}})^{⊗a_{i_{l}}}

= ⨂_{l=1}^{s}[A_{ξ,ω̄_{i_{l}}}]^{⊗a_{i_{l}}}

By the assumption that at least one of the a_{i_{l}} is coprime to p, we have β_{ξ}(ρ̄_{w}) ∈ H \ {1}. Thus by the hypothesis that v_{p}(i_{c}) > 0, we have p|i_{w} = ind(β_{ξ}(ρ̄_{w})). Therefore i_{w}c_{1}(g_{w}) = 0 in Ch^{1}(X).

For the case m > 1 we work under the hypothesis that v_{p}(i_{c}) > 1, and proceed by induction. We assume that the result I_{ξ}^{(m′)} ⊆ I^{(m′)} holds for all m′ < m. It can be seen that

I_{ξ}^{(m)} = (⨁_{j=1}^{m−1} CH^{m−j}(X) · im(res_{l/k}^{(j)})) ⊕ im(res_{l/k}^{(m)}).

By the inductive hypothesis, im(res_{l/k}^{(j)}) ⊂ I_{ξ}^{(j)} ⊆ I^{(j)} for 1 ≤ j ≤ m − 1, which

implies that CH^{m−j}(X) · im(res_{l/k}^{(j)}) ⊂ I^{(m)} for 1 ≤ j ≤ m − 1. It remains to show

that im(res_{l/k}^{(m)}) ⊂ I^{(m)}.

By Proposition 2.2 we know that im(res_{l/k}^{(m)}) = c_{m}(im(res_{γ}^{(m)})), and is generated by elements of the form

a = (m − 1)!\binom{i_{w_{1}}}{i_{1}} ⋯ \binom{i_{w_{k}}}{i_{k}}c_{1}(g_{w_{1}})^{i_{1}} ⋯ c_{1}(g_{w_{k}})^{i_{k}},

where i_{1} + ⋯ + i_{k} = m, and w_{1}, …, w_{k} ∈ W.

If i_{l} < m for all l = 1, …, k, then \binom{i_{w_{l}}}{i_{l}}c_{1}(g_{w_{l}})^{i_{l}} ∈ I^{(i_{l})} by the inductive hypothesis. Therefore, a ∈ I^{(i_{1})} ⋯ I^{(i_{k})} ⊆ I^{(m)}. If, on the other hand, a is of the form a = \binom{i_{w}}{m}c_{1}(g_{w})^{m}, then we apply the previous argument. Namely, we have

ρ_{w} = ∑_{i=1}^{n} a_{i}ω_{i},

which implies

(c_{1}(g_{w}))^{m} = (∑_{i=1}^{n} a_{i}c_{1}(L(ω_{i})))^{m}

= (∑_{i=1}^{n} a_{i}(c_{1}(g_{i}) + c_{1}(L(α_{i}))))^{m}.

Again, c_{1}(L(α_{i})) ∈ im(𝔠^{(1)}) implies c_{1}(L(α_{i})) ∈ I^{(1)} for all i = 1, …, n, and so all terms in the above expansion that are divisible by some c_{1}(L(α_{i})) are contained in I^{(m)}. As in the previous case, we may write

(c_{1}(g_{w}))^{m} = (∑_{l=1}^{s} a_{i_{l}}c_{1}(g_{i_{l}}))^{m} mod I^{(m)}.

If a_{i_{l}} is divisible by p for all l = 1, …, s, then we are done, so we assume that at least one a_{i_{l}} is coprime to p. As before, this ensures that i_{c} | ind(β_{ξ}(ρ̄_{w}), and so v_{p}(i_{w}) ≥ v_{p}(i_{c}). It is clear that for any b ∈ ℤ_{>0} if v_{p}(b) > 1 then p | \binom{b}{l} for all 1 ≤ l ≤ p. Thus, under the hypothesis that v_{p}(i_{c}) > 1, we have p | \binom{i_{w}}{m}, and so \binom{i_{w}}{m}(c_{1}(g_{w}))^{m} = 0 in Ch^{m}(X). □

4.2. Corollary. Let p be a prime number. If v_{p}(i_{c}) > 0, then J_{p}^{(1)} > 0. If v_{p}(i_{c}) > 1, then J_{p}^{(1)} > 1.

Proof. Consider the diagram

\xymatrix{Ch(_{ξ}X) \ar[r]^{res_{l/k}} & Ch(X) \ar[r]^{π} & Ch(G).}

We begin first with the hypothesis that v_{p}(i_{c}) > 0.

Let R_{ξ} = im(π ∘ res_{l/k}). Then a ∈ R_{ξ}^{(1)} implies that a ∈ im(res_{l/k}^{(1)}) = I^{(1)}

by Theorem 4.1. Thus, π(a) = 0 ∈ Ch^{1}(X)/I and so R_{ξ}^{(1)} = {0}. Let x_{1}, …, x_{s} be generators of degree 1 in Ch^{1}(G). By the definition of the J-invariant and the DegLex order, we have that for every 1 ≤ i ≤ s, j_{i} is the smallest non-negative integer such that R_{ξ} contains an element of the form

(6)  x_{i}^{p^{j_{i}}} + ∑_{x^{M} <x_{i}^{p^{j_{i}}}} a_{M}x^{M}, a_{M} ∈ 𝔽_{p}.

Thus j_{1} is the smallest non-negative integer m such that x_{1}^{p^{m}} ∈ R_{ξ}. Since x_{1} is non-trivial, we must have x_{1}^{p^{0}} = x_{1} ∉ R_{ξ}^{(1)} by the above argument. Therefore, j_{1} > 0.

The same argument applies for the remaining generators. Let 1 < i ≤ s, then x^{M} < x_{i} implies that x^{M} = x_{j} for some j < i. Since x_{i} + a_{i−1}x_{i−1} + ⋯ + a_{1}x_{1} is non-trivial for any a_{1}, …, a_{i−1} ∈ 𝔽_{p}, it cannot belong to R_{ξ}^{(1)}, and we have by (6)

that j_{i} > 0. Thus, J_{p}^{(1)} > 0.

Under the hypothesis that v_{p}(i_{c}) > 1, suppose again that x_{1}, …, x_{s} are a minimal set of generators of degree 1 in Ch(G). We have the inclusion im(res_{l/k}^{(p)}) ⊂ I_{ξ}^{(p)} and

by Theorem 4.1, I_{ξ}^{(p)} = I^{(p)}. Therefore, we again have R_{ξ}^{(p)} = im(π ∘ res_{l/k}^{(p)}) = {0}.

To show that J_{p}^{(1)} > 1, we begin with the generator x_{1}. If k_{1} ≤ 1 we are done, so suppose k_{1} > 1. Then, x_{1}^{p^{1}} = x_{1}^{p} ∈ Ch^{p}(G) is non-trivial, and so x^{p} ∉ R_{ξ}^{(p)} and we must have j_{1} > 1. Again, we extend the argument for the remaining generators. We suppose that k_{i} > 1 for somel 1 < i ≤ s. Then, the element

x_{i}^{p} + ∑_{(x^{M} <x_{i}^{p})∩(|M|=p)} a_{M}x^{M}

is non-trivial for any a_{M} ∈ 𝔽_{p} and hence cannot belong to R_{ξ}^{(p)}, and so j_{i} > 1.

Thus J_{p}^{(1)} > 1. □

5. Applications

We now apply the results of the previous section to some E_{6} varieties.

Let G be a group of inner type E_{6}. It has one Tits algebra A of index 3^{d} for some d = 0, …, 3. Consider the J-invariant of G modulo p = 3. We note that Ch(G) has precisely two generators, with d_{1} = 1 and d_{2} = 4, where the 3-power relations are defined by k_{1} = 2 and k_{2} = 1 [7, Table II]. Thus

J_{3}(G) = (j_{1}, j_{2}),

where j_{1} = 0, 1, 2, j_{2} = 0, 1. These values are independent on the characteristic of the base field.

We will use the following result concerning the possible values of the J-invariant.

5.1. Lemma. Let G be a semisimple algebraic group of inner type over k, p a prime integer and J_{p}(G) = (j_{1}, …, j_{r}). Assume d_{i} = 1 for some i = 1, …, r. Then j_{i} ≤ max_{A} v_{p}(indA), where A runs through all Tits algebras of G. Conversely, if j_{i} > 0, then there exists a Tits algebra A of G with v_{p}(indA) > 0.

Proof. See Proposition 4.3 part 3 in [15]. □

Observe that J_{3}(G) = (0, 0) iff G splits by a field extension of degree coprime to 3 [13, Corollary 6.7]. Suppose indA = 27, then for any splitting field l/k of A, 27 | [l : k]. By [13, Prop. 6.6], we must then have 3 ≤ j_{1} + j_{2}. Since j_{1} ≤ k_{1} = 2 and j_{2} ≤ k_{2} = 1, the only possible value for the J-invariant is J_{3}(G) = (2, 1).

5.2. Proposition. indA = 1 if and only if j_{1} = 0. indA = 3 if and only if j_{1} = 1.

Proof. Suppose first that indA = 1. Since G has only one Tits algebra, this implies that v_{3}(indB) = 0 for all [B] ∈ H, where H is the subgroup of Br(k) generated by the non-constant elements in the image of the Tits map β_{ξ}. Thus by Lemma 5.1, j_{1} = 0. Suppose conversely that j_{1} = 0. Since there is only one generator of degree 1, we have i_{c} = indA and so v_{3}(i_{c}) = d, where d = 0, …, 3. By Corollary 4.2, j_{1} = 0 implies d = 0 and hence indA = 1.

For the second case, suppose first j_{1} = 1. By Corollary 4.2 this implies that indA = 1 or 3. However, by the first case, j_{1} ≠ 0 implies indA ≠ 1 and so indA = 3. Conversely, suppose indA = 3. Then v_{3}(indB) ≤ 1 for all Tits algebras [B] ∈ Br(k). By Lemma 5.1, j_{1} ≤ max_{B}(indB) = 1 and so j_{1} ≤ 1. Again by the first case, indA ≠ 1 implies j_{1} ≠ 0 and so j_{1} = 1. □

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