J-invariant of linear algebraic groups

V. Petrov, N. Semenov, K. Zainoulline ^{∗}

January 30, 2007

Abstract

Let G be a linear algebraic group over a field F and X be a projective homogeneous G-variety such that G splits over the function field of X. In the present paper we introduce an invariant of G called J-invariant which characterizes the motivic behaviour of X. This generalizes the respective notion invented by A. Vishik in the context of quadratic forms. As a main application we obtain a uniform proof of all known motivic decompositions of generically split projective homogeneous varieties (Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, G_{2}- and F_{4}-varieties) as well as provide new examples (exceptional varieties of types E_{6}, E_{7} and E_{8}). We also discuss relations with torsion indices, canonical dimensions and cohomological invariants of the group G.

Introduction

Let G be a simple linear algebraic group over a field F and X be a projective homogeneous G-variety. In the present paper we address the problem of computing the Grothendieck-Chow motive M(X) of X or, in other words, providing a direct sum decomposition of M(X).

When the group G is split, i.e., contains a split maximal torus, the motive of X has the simplest decomposition – it is isomorphic to a direct sum of twisted Tate motives. This was first observed by B. Köck in [Kö91]. The next step was done by V. Chernousov, S. Gille and A. Merkurjev (see [CGM]). They provided an algorithm of expressing the motive of any rational projective homogeneous G-variety in terms of motives of anisotropic G-varieties, i.e., those which have no rational points. Finally, P. Brosnan (see [Br05]) generalized this result to the case of an isotropic group G and possibly anisotropic X. In all these proofs one constructs a (relative) cellular filtration on X, which allows to express the motive of the total space X in terms of motives of the base. Since the latter consists of homogeneous varieties of anisotropic groups, it reduces the problem to the case of anisotropic group G.

∗The paper is based on the PhD Thesis of the first author and was partially supported by CNRS, DAAD A/04/00348, INTAS 03-51-3251, RTN HPRN-CT-2002-00287, SFB701.

For anisotropic groups the situation becomes more complicated: only very few partial results are known. Observe that in this case the components of a motivic decomposition are expected to have a non-geometric nature, i.e., can not be identified with (twisted) motives of some other varieties. The first examples of such decompositions were provided by M. Rost [Ro98]. He proved that the motive of a Pfister quadric decomposes as a direct sum of twisted copies of a certain non-geometric motive R known in the literature as Rost motive. Motives of Severi-Brauer varieties were described by N. Karpenko [Ka96]. For exceptional varieties examples of motivic decompositions were provided by J.-P. Bonnet [Bo03] (varieties of type G_{2}) and by S. Nikolenko, N. Semenov, K. Zainoulline [NSZ] (varieties of type F_{4}).

In the present paper we provide a uniform proof of all these results. Observe that in all examples the variety X has the following property: Over the generic point of X the motive M(X) splits as a direct sum of twisted Tate motives. Varieties which satisfy this condition will be called generically split. The main result of the paper says that (see Theorem 5.1)

Theorem. Let G be a simple linear algebraic group of inner type over a field F and p be a prime integer. Let X be a generically split projective homogeneous G-variety. Then the Chow motive of X with ℤ/p-coefficients is isomorphic to a direct sum

M(X; ℤ/p) ≃ ⨁_{i≥0} R_{p}(G)(i)^{⊕a_{i}}

of twisted copies of a certain indecomposable motive R_{p}(G), where the integers a_{i} are coefficients of the quotient of the respective Poincaré polynomials.

Observe that the motive R_{p}(G) depends only on G and p but not on the type of a parabolic subgroup defining X. Moreover, considered with ℚ-coefficients it always splits as a direct sum of Tate motives.

Our proof is based on two different observations. The first comes from the topology of compact Lie groups. Namely, to compute Chow rings of compact Lie groups V. Kac [Kc85] invented the notion of p-exceptional degrees – numbers which encode the information about the Chow ring of a group modulo p. These numbers can be expressed in terms of degrees of basic polynomial invariants and, therefore, deal with the combinatorics of the Weyl group. From the other hand the results of N. Karpenko, A. Merkurjev [KM06] and K. Zainoulline [Za06] concerning canonical p-dimensions of algebraic groups tell us that there is a strong interrelation between p-exceptional degrees and the ‘size’ of the image of the restriction map res: CH^{∗}(X×X) → CH^{∗}(\overline{X}×\overline{X}) to the separable closure of F. To relate this image with a motivic decomposition of X we use Rost Nilpotence Theorem. It was first proved for quadrics by M. Rost and lately generalized to arbitrary projective homogeneous varieties by V. Chernousov, S. Gille and A. Merkurjev (see [CGM, Theorem 8.2]). Roughly speaking, it says that any decomposition which is given by idempotent cycles from this image, indeed, comes from F.

All this together lead to the notion of J-invariant J_{p}(G) of a group G modulo p (see Definition 4.5) which generalizes the respective notion introduced by A. Vishik [Vi05] in the case of quadrics. Our main observation is that J_{p}(G) completely characterizes the motive R_{p}(G) and, hence, the motivic decomposition of X. Observe that if the J-invariant takes its minimal possible non-trivial value J_{p}(G) = (1), then the motive R_{p}(G) ⊗ ℚ has the following recognizable decomposition (cf. [Vo03, §5])

R_{p}(G) ⊗ ℚ ≃ ⨁_{i=0}^{p−1} ℚ(i · \frac{p^{n−1}−1}{p−1})

where n = 2 or 3 (see the last section).

Apart from the notion of J-invariant we generalize some of the results of paper [CPSZ]: Using the motivic version of a result of D. Edidin and W. Graham [EG97] on cellular fibrations we provide a general formulae which expresses the motive of the total space of a cellular fibration in terms of the motives of its base (see Theorem 3.8). We also provide several criteria for the existence of liftings of motivic decompositions via the reduction map ℤ → ℤ/m. We prove that such a lifting always exists if m = 2, 3, 4 or 6 (see Theorem 2.16).

The paper is organized as follows. In the first section we recall the definition of Chow motives and some properties of generically split varieties.

Rather technical section 2 is devoted to lifting of idempotents. In section 3 we discuss motives of cellular fibrations. The proof of the main result actually starts with section 4, where we introduce the notion of J-invariant and provide a motivic decomposition for the variety of complete flags. In section 5 we finish the proof and give some properties of the motive R_{p}(G). The last two sections are devoted to various applications of J-invariant and examples of motivic decompositions.

1 Motives of generically split varieties

1.1. In the present paper we will work with Chow motives of smooth projective varieties over a field F. We will use the following notation (cf. [Ma68], [CGM, §7] or [EKM, XII])

Given a smooth projective variety X over a field F we denote by M(X) its Chow motive, and by M(X)(n) = M(X) ⊗ ℤ(n) the respective twist by a Tate object. A morphism of motives between M(X)(n) and M(Y)(m), where X is irreducible, is given by a class ϕ of rationally equivalent cycles of dimension dim X + n − m on X × Y. Hence, the group of endomorphisms End(M(X)) coincides with the Chow group CH_{dim X}(X × X). The element ϕ is called a correspondence between X and Y of degree n − m.

Given a correspondence ϕ of degree d and k ∈ ℤ the composite

\xymatrix{CH_{k}(X) \ar[r]^{(pr_{X})^{∗}} & CH_{k+dim Y} (X × Y) \ar[r]^{−∩ϕ} & CH_{k+d}(X × Y) \ar[r]^{(pr_{Y} )_{∗}} & CH_{k+d}(Y)}

of the pull-back (pr_{X})^{∗}, intersection product with ϕ and the push-forward (pr_{Y})_{∗} is called the realization of ϕ and is denoted by ϕ_{⋆}. Given correspondences ϕ ∈ CH_{dim X+d}(X × Y) and ψ ∈ CH_{dim Y +e}(Y × Z) of degrees d and e respectively the correspondence of degree d + e

(pr_{X×Z})_{∗}((pr_{Y ×Z})^{∗} ∩ (pr_{X×Y})^{∗}) ∈ CH_{dim X+d+e}(X × Z)

is called the correspondence product of ϕ and ψ and is denoted by ψ ∘ ϕ. By definition (ψ ∘ ϕ)_{⋆} = ψ_{⋆} ∘ ϕ_{⋆}. Given a correspondence ϕ we denote by ϕ^{t} its transpose.

The correspondence product endows the group End(M(X)) with the ring structure. The identity element of this ring is the class of the diagonal Δ_{X}.

Finally, by M(X; Λ), where Λ is a commutative ring, we denote an object of the category of motives with Λ-coefficients obtained by taking correspondences ϕ with Λ-coefficients, i.e., replacing CH(X × Y) by CH(X × Y) ⊗_{ℤ} Λ.

1.2 Definition. Let L/F be a field extension. We say L is a splitting field of a smooth projective variety X or, equivalently, a variety X splits over L if the motive M(X) splits over L as a finite direct sum of twisted Tate motives, i.e.,

M(X)_{L} ≃ ⨁_{n} ℤ_{L}(n)

We say a variety X is generically split if it splits over the field of rational functions F (X).

1.3 Example. A variety X is called generically cellular if it is cellular over its generic point, i.e., X_{F (X)} = X ×_{F} F (X) has a proper descending filtration by closed subvarieties X_{i} such that each complement X_{i} \ X_{i+1} is a disjoint union of affine spaces defined over F (X). According to [EKM, Corollary 67.2] if X is generically cellular, then X is generically split.

1.4 Example. Let G be a semisimple linear algebraic group over a field F and X be a projective homogeneous G-variety. Let L/F be a field extension. If the group G splits over L, i.e., G_{L} = G×_{F} L contains a split maximal torus defined over L, then X_{L} is cellular and, hence, L is a splitting field of X. In particular, if G splits over the function field of X, then X is generically split. Many examples of such varieties can be provided investigating Tits indices of G (see Example 3.7).

1.5. Assume X has a splitting field L. We will write CH(\overline{X}; Λ) for CH(X_{L}; Λ) and \overline{CH}(X; Λ) for the image of the restriction map CH(X; Λ) → CH(\overline{X}; Λ) (cf. [KM06, 1.2]). Similarly, by M(\overline{X}) we denote the motive of X considered over L. If M is a direct summand of M(X)(n), by \overline{M} we denote the motive M_{L}. Elements of \overline{CH}(X) will be called rational cycles on X_{L} with respect to the field extension L/F. If L′ is another splitting field of X, then there is a chain of canonical isomorphisms CH(X_{L}) ≃ CH(X_{LL′}) ≃ CH(X_{L′}), where LL′ is the composite of L and L′. Hence, the groups CH(\overline{X}) and \overline{CH}(X) do not depend on the choice of L.

There is the Künneth decomposition End(M(\overline{X})) = CH(\overline{X} × \overline{X}) = CH(\overline{X}) ⊗ CH(\overline{X}) and Poincaré duality. The latter means that given a basis of CH(\overline{X}) there is a dual one with respect to the pairing (α, β) ↦ deg(α · β), where deg is the degree map. In view of the Künneth decomposition the correspondence product of cycles in on CH(\overline{X} × \overline{X}) is given by the formula (α_{1} × β_{1}) ∘ (α_{2} × β_{2}) = deg(α_{1}β_{2})(α_{2} × β_{1}), the realization by (α × β)_{⋆}(γ) = deg(αγ)β and the transpose by (α × β)^{t} = β × α. Since CH_{∗}(\overline{X}) is a free graded ℤ-module, we may define its Poincaré polynomial as P (CH_{∗}(\overline{X}), t) = ∑_{i≥0} rk_{ℤ} CH_{i}(\overline{X}) · t^{i}.

Sometimes we will use contravariant notation CH^{∗} for Chow groups, where CH^{k}(X) = CH_{dim X−k}(X) for irreducible X.

1.6 Lemma. Let X and Y be two smooth projective varieties such that F (Y) is a splitting field of X and Y has a splitting field. Consider the projection in the Künneth decomposition

pr_{0} : CH^{r}(\overline{X} × \overline{Y}) = ⨁_{i=0}^{r} CH^{r−i}(\overline{X}) ⊗ CH^{i}(\overline{Y}) → CH^{r}(\overline{X}).

Then for any ρ ∈ CH^{r}(\overline{X}) we have pr_{0}^{−1}(ρ) ∩ \overline{CH^{r}}(X × Y) ≠ ∅.

Proof. Let L be a common splitting field of X and Y. Lemma follows from the commutative diagram

\xymatrix{CH^{r}(X ×_{F} Y) \ar[r]^{^{res^{L/F}}} \ar@{->>}[d] & CH^{r}(X_{L} ×_{L} Y_{L}) \ar@{->>}[d] \ar@{-->}[dr]^{_{pr_{0}}} \\ CH^{r}(X_{F (Y)}) \ar[r]^{≃} & CH^{r}((X_{L})_{L(Y_{L})}) \ar[r]^{≃} & CH^{r}(X_{L})}

where the left square is obtained by taking the generic fiber of the base change morphism X_{L} → X; the vertical arrows are taken from the localization sequence for Chow groups and, hence, are surjective; and the bottom horizontal maps are isomorphisms since L is a splitting field.

We will extensively use the following version of Rost Nilpotence Theorem.

1.7 Lemma. Let X be a smooth projective variety such that it splits over any field K over which it has a rational point. Then for any α in the kernel of the natural map End(M(X)) → End(M(\overline{X})) we have α^{∘(dim X+1)} = 0.

Proof. See [EKM, Theorem 68.1].

2 Lifting of idempotents

2.1 Definition. Given a ℤ-graded ring A^{∗} and two idempotents ϕ_{1}, ϕ_{2} ∈ A^{0} we say ϕ_{1} and ϕ_{2} are orthogonal if ϕ_{1}ϕ_{2} = ϕ_{2}ϕ_{1} = 0. We say an element θ_{12} provides an isomorphism of degree d between idempotents ϕ_{1} and ϕ_{2} if θ_{12} ∈ ϕ_{2}A^{−d}ϕ_{1} and there exists θ_{21} ∈ ϕ_{1}A^{d}ϕ_{2} such that θ_{12}θ_{21} = ϕ_{2} and θ_{21}θ_{12} = ϕ_{1}.

2.2 Example. Let Λ be a commutative ring. Set A^{∗} = End^{∗}(M(X; Λ)), where

End^{k}(M(X; Λ)) = CH^{k}(X × X; Λ), k ∈ ℤ

and the multiplication is given by the correspondence product. By definition End^{0}(M(X; Λ)) is the ring of endomorphisms of the motive M(X; Λ). Note that a direct summand of M(X; Λ) can be identified with a pair (X, ϕ), where ϕ is an idempotent, i.e., ϕ ∘ ϕ = ϕ (see [EKM, ch. XII]). Then an isomorphism θ_{12} of degree d between ϕ_{1} and ϕ_{2} can be identified with an isomorphism between the motives (X, ϕ_{1}) and (X, ϕ_{2})(d).

2.3 Definition. Let f: A^{∗} → B^{∗} be a homomorphism of ℤ-graded rings. We say that f is decomposition preserving if given a family ϕ_{i} ∈ B^{0} of pairwise orthogonal idempotents such that ∑_{i} ϕ_{i} = 1_{B}, there exists a family of pair-wise orthogonal idempotents φ_{i} ∈ A^{0} such that ∑_{i} φ_{i} = 1_{A} and each f(φ_{i}) is isomorphic to ϕ_{i} by means of an isomorphism of degree 0. We say f is strictly decomposition preserving if, moreover, one can choose φ_{i} such that f(φ_{i}) = ϕ_{i}.

We say f is isomorphism preserving if for any idempotents φ_{1} and φ_{2} in A^{0} and any isomorphism θ_{12} of degree d between idempotents f(φ_{1}) and f(φ_{2}) in B^{0} there exists an isomorphism ϑ_{12} of degree d between φ_{1} and φ_{2}. We say f is strictly isomorphism preserving if, moreover, one can choose ϑ_{12} such that f(ϑ_{12}) = θ_{12}.

2.4. By definition we have the following properties of (strictly) decomposition and isomorphism preserving morphisms:

(i) Let f: A^{∗} → B^{∗} and g: B^{∗} → C^{∗} be homomorphisms such that g ∘ f is decomposition (resp. isomorphism) preserving and g is isomorphism preserving. Then f is decomposition (resp. isomorphism) preserving.

(ii) Assume we are given a commutative diagram with ker f′ ⊂ im i

\xymatrix{A^{∗} \ar@{->>}[r]^{f} \ar@{^{(}->}[d]_{i} & B^{∗} \ar@{^{(}->}[d]^{i′} \\ A^{′∗} \ar@{->>}[r]^{f′} & B^{′∗}.}

If f′ is strictly decomposition (resp. strictly isomorphism) preserving, then so is f.

2.5 Proposition. Let f: A^{∗} → B^{∗} be a surjective homomorphism such that the kernel of the restriction of f to A^{0} consists of nilpotent elements. Then f is strictly decomposition (cf. [EKM, Proposition 95.1]) and strictly isomorphism preserving.

Proof. The fact that f is strictly decomposition preserving follows from [AF92, Proposition 27.4]. The fact that f is strictly isomorphism preserving follows from Lemma 2.6 below.

2.6 Lemma. Let A, B be two rings, A^{0}, B^{0} be their subrings, f^{0} : A^{0} → B^{0} be a ring homomorphism, f: A → B be a map of sets satisfying the following conditions:

• f(α)f(β) equals either f(αβ) or 0 for all α, β ∈ A;

• f^{0}(α) equals f(α) if f(α) ∈ B^{0} or 0 otherwise;

• ker f^{0} consists of nilpotent elements.

Let φ_{1} and φ_{2} be two idempotents in A^{0}, ψ_{12} and ψ_{21} be elements in A such that ψ_{12}A^{0}ψ_{21} ⊂ A^{0}, ψ_{21}A^{0}ψ_{12} ⊂ A^{0}, f(ψ_{21})f(ψ_{12}) = f(φ_{1}), f(ψ_{12})f(ψ_{21}) = f(φ_{2}). Then there exist elements ϑ_{12} ∈ φ_{2}A^{0}ψ_{12}A^{0}φ_{1} and ϑ_{21} ∈ φ_{1}A^{0}ψ_{21}A^{0}φ_{2} such that ϑ_{21}ϑ_{12} = φ_{1}, ϑ_{12}ϑ_{21} = φ_{2}, f(ϑ_{12}) = f(φ_{2})f(ψ_{12}) = f(ψ_{12})f(φ_{1}), f(ϑ_{21}) = f(φ_{1})f(ψ_{21}) = f(ψ_{21})f(φ_{2}).

Proof. Since ker f^{0} consists of nilpotents, f^{0} sends non-zero idempotents in A^{0} to non-zero idempotents in B^{0}; in particular, f(φ_{1}) = f^{0}(φ_{1}) ≠ 0, f(φ_{2}) = f^{0}(φ_{2}) ≠ 0. Observe that

f(ψ_{12})f(φ_{1}) = f(ψ_{12})f(ψ_{21})f(ψ_{12}) = f(φ_{2})f(ψ_{12})

and, similarly, f(ψ_{21})f(φ_{2}) = f(φ_{1})f(ψ_{21}). Changing ψ_{12} to φ_{2}ψ_{12}φ_{1} and ψ_{21} to φ_{1}ψ_{21}φ_{2} we may assume that ψ_{12} ∈ φ_{2}Aφ_{1} and ψ_{21} ∈ φ_{1}Aφ_{2}. We have

f^{0}(φ_{2}) = f(φ_{2}) = f(ψ_{12})f(ψ_{21}) = f(ψ_{12}ψ_{21}) = f^{0}(ψ_{12}ψ_{21});

therefore α = ψ_{12}ψ_{21} − φ_{2} ∈ A^{0} is nilpotent, say α^{n} = 0. Note that φ_{2}α = α = αφ_{2}. Set α^{∨} = φ_{2} − α + … + (−1)^{n−1}α^{n−1} ∈ A^{0}; then αα^{∨} = φ_{2} − α^{∨}, φ_{2}α^{∨} = α^{∨} = α^{∨}φ_{2} and f(φ_{2}) = f^{0}(φ_{2}) = f^{0}(α^{∨}) = f(α^{∨}). Therefore setting ϑ_{21} = ψ_{21}α^{∨} we have ϑ_{21} ∈ φ_{1}Aφ_{2}, ψ_{12}ϑ_{21} = φ_{2} and f(ϑ_{21}) = f(ψ_{21}).

Now ϑ_{21}ψ_{12} is an idempotent. We have

f^{0}(φ_{1}) = f(φ_{1}) = f(ϑ_{21})f(ψ_{12}) = f(ϑ_{21}ψ_{12}) = f^{0}(ϑ_{21}ψ_{12});

therefore β = ϑ_{21}ψ_{12} − φ_{1} ∈ A^{0} is nilpotent. Note that βφ_{1} = β = φ_{1}β. Now φ_{1} + β = (φ_{1} + β)^{2} = φ_{1} + 2β + β^{2} and therefore β(1 + β) = 0. But 1 + β is invertible and hence we have β = 0. It means that ϑ_{21}ψ_{12} = φ_{1} and we can set ϑ_{12} = ψ_{12}.

2.7 Corollary. The map End^{∗}(M(X; ℤ/p^{n})) → End^{∗}(M(X; ℤ/p)) is strictly decomposition (cf. [EKM, Corollary 95.3]) and strictly isomorphism preserving.

Proof. Apply Proposition 2.5 to the case A^{∗} = End^{∗}(M(X; ℤ/p^{n})), B^{∗} = End^{∗}(M(X; ℤ/p)) and the reduction map f: A^{∗} → B^{∗}.

2.8 Lemma. Let m = m_{1}m_{2} be a product of two coprime integers. Then the map End^{∗}(M(X; ℤ/m)) → End^{∗}(M(X; ℤ/m_{1}))×End^{∗}(M(X; ℤ/m_{2})) is an isomorphism.

Proof. Apply Chinese Remainder Theorem.

2.9. From now on assume that X is a smooth projective variety which has a splitting field such that the kernel of the restriction map

res_{E}: End^{∗}(M(X_{E}; Λ)) → End^{∗}(M(\overline{X}; Λ))

consists of nilpotent elements for all extensions E/F and all rings of coefficients Λ. By Lemma 1.7 the latter holds if X splits over any field extension over which it has a rational point. We denote by \overline{End^{∗}}(M(X; Λ)) the image of res_{F}.

2.10 Corollary. The map End^{∗}(M(X_{E}; Λ)) → \overline{End^{∗}}(M(X_{E}; Λ)) is strictly decomposition and strictly isomorphism preserving for any field extension E/F.

Proof. Apply Proposition 2.5 to the homomorphism res_{E}: A^{∗} → B^{∗} between the graded rings A^{∗} = End^{∗}(M(X_{E}; Λ)) and B^{∗} = \overline{End^{∗}}(M(X_{E}; Λ)).

2.11 Definition. We say that a field extension E/F is rank preserving with respect to X if the restriction map res_{E/F} : CH(X) → CH(X_{E}) becomes an isomorphism after tensoring with ℚ.

2.12 Lemma. Assume X has a splitting field. Then for any rank preserving finite field extension E/F we have [E : F] · \overline{CH}(X_{E}) ⊂ \overline{CH}(X).

Proof. Let L be a splitting field containing E. Let γ be any element in \overline{CH}(X_{E}). By definition there exists α ∈ CH(X_{E}) such that γ = res_{L/E}(α). Since res_{E/F} ⊗ ℚ is an isomorphism, there exists an element β ∈ CH(X) and a non-zero integer n such that res_{E/F} (β) = nα. By projection formula

n · cores_{E/F} (α) = cores_{E/F} (res_{E/F} (β)) = [E : F] · β.

Applying res_{L/E} to both sides we obtain n(res_{L/E}(cores_{E/F} (α))) = n[E : F]·γ. Therefore, res_{L/E}(cores_{E/F} (α)) = [E : F] · α.

2.13 Corollary. Assume X has a splitting field, E/F is a field extension of degree coprime with m, which is rank preserving with respect to X × X. Then the map End^{∗}(M(X; ℤ/m)) → End^{∗}(M(X_{E}; ℤ/m)) is decomposition and isomorphism preserving.

Proof. By Lemma 2.12 we have \overline{End^{∗}}(M(X_{E}; ℤ/m)) = \overline{End^{∗}}(M(X; ℤ/m)). Now apply Corollary 2.10 and 2.4.(i) with A^{∗} = End^{∗}(M(X; ℤ/m)), B^{∗} = End^{∗}(M(X_{E}; ℤ/m)) and C^{∗} = \overline{End^{∗}}(M(X_{E}; ℤ/m)).

2.14 Lemma. The map SL_{l}(ℤ) → SL_{l}(ℤ/m) induced by the reduction modulo m is surjective.

Proof. Since ℤ/m is a semi-local ring, the group SL_{l}(ℤ/m) is generated by elementary matrices (see [HOM, Theorem 4.3.9]).

Given a free graded ℤ-module V^{∗} set End^{−d}(V^{∗}), d ∈ ℤ, to be the group of endomorphisms of V^{∗} decreasing the degree by d.

2.15 Proposition. (cf. [EKM, §96]) Consider a free graded ℤ-module V^{∗} of finite rank and the reduction map f: End^{∗}(V^{∗}) → End^{∗}(V^{∗} ⊗_{ℤ}ℤ/m). Assume that the graded components of the respective im ϕ_{i} (see Definition 2.3) are free ℤ/m-modules. Then f is strictly decomposition preserving. Moreover, if (ℤ/m)^{×} = {±1}, then f is strictly isomorphism preserving.

Proof. We are given a decomposition V^{k} ⊗_{ℤ} ℤ/m = ⊕_{i}W_{i}^{k}, where W_{i}^{k} is the k-graded component of im ϕ_{i}. Present V^{k} as a direct sum V^{k} = ⨁_{i} V_{i}^{k} of free ℤ-modules such that rk_{ℤ} V_{i}^{k} = rk_{ℤ/m} W_{i}^{k}. Fix a ℤ-basis {v_{ij}^{k}}_{j} of V_{i}^{k}. For each W_{i}^{k} choose a basis {w_{ij}^{k}}_{j} such that the linear transformation D^{k} of V^{k} ⊗_{ℤ} ℤ/m sending each v_{ij}^{k} ⊗ 1 to w_{ij}^{k} has determinant 1. By Lemma 2.14 there is a lifting D̃^{k} of D^{k} over ℤ. So we obtain V^{k} = ⨁_{i} W̃_{i}^{k}, where W̃_{i}^{k} = D̃^{k}(V_{i}^{k}) satisfies W̃_{i}^{k} ⊗_{ℤ} ℤ/m = W_{i}^{k}. Define φ_{i} on each V^{k} to be the projection onto W̃_{i}^{k}.

Now let φ_{1}, φ_{2} be two idempotents in End^{∗}(V^{∗}). Denote by V_{i}^{k} the kgraded component of im φ_{i}. An isomorphism θ_{12} between φ_{1} ⊗ 1 and φ_{2} ⊗ 1 of degree d can be identified with a family of isomorphisms θ_{12}^{k}: V_{1}^{k} ⊗ℤ/m → V_{2}^{k−d}⊗ℤ/m. In the case (ℤ/m)^{×} = {±1} all these isomorphisms are given by matrices with determinants {±1} and, hence, can be lifted to isomorphisms ϑ_{12}^{k}: V_{1}^{k} → V_{2}^{k−d} by Lemma 2.14.

Now we are ready to formulate and prove the main result of this section

2.16 Theorem. Assume X has a splitting field of degree m which is rank preserving with respect to X × X. Then the map

End^{∗}(M(X)) → End^{∗}(M(X; ℤ/m))

preserves decompositions with the property that im res(ϕ_{i}) are free ℤ/m-modules, where

res: End^{∗}(M(X; ℤ/m)) → End^{∗}(M(\overline{X}; ℤ/m))

is the restriction. If additionally (ℤ/m)^{×} = {±1} then this map is isomorphism preserving.

Proof. Consider the diagram

\xymatrix{End^{∗}(M(X)) \ar@{->>}[r]^{f} \ar@{->>}[d] & End^{∗}(M(X; ℤ/m)) \ar@{->>}[d] \\ \overline{End^{∗}}(M(X)) \ar@{->>}[r]^{f̄} \ar@{^{(}->}[d]_{i} & \overline{End^{∗}}(M(X; ℤ/m)) \ar@{^{(}->}[d] \\ End^{∗}(M(\overline{X})) \ar@{->>}[r]^{f′} & End^{∗}(M(\overline{X}; ℤ/m)).}

Note that using Poincaré duality (see 1.5) we can identify End^{−d}(M(\overline{X})) with the group of endomorphisms of CH^{∗}(\overline{X}) which decrease the grading by d. Applying Proposition 2.15 to the case V^{∗} = CH^{∗}(\overline{X}) we obtain that the map f′ is strictly decomposition preserving. Moreover, if (ℤ/m)^{×} = {±1} then f′ is strictly isomorphism preserving. By Lemma 2.12 ker f′ ⊂ im i and, therefore, applying 2.4.(ii) we obtain that f̄ is strictly decomposition preserving and, moreover, f̄ is strictly isomorphism preserving if (ℤ/m)^{×} = {±1}. Now by Corollary 2.10 the vertical arrows of the top square are strictly decomposition and strictly isomorphism preserving. It remains to apply 2.4.(i).

3 Motives of fibered spaces

3.1 Definition. Let X be a smooth projective variety over a field F. We say a smooth projective morphism f: Y → X is a cellular fibration if it is a locally trivial fibration whose fiber F is cellular, i.e., has a decomposition into affine cells (see [EKM, §67]).

3.2 Lemma. Let f: Y → X be a cellular fibration. Then M(Y) is isomorphic to M(X) ⊗ M(F).

Proof. We follow the proof of [EG97, Proposition 1]. Define the morphism

φ: ⨁_{i∈I} M(X)(codim B_{i}) → M(Y)

to be the direct sum φ = ⨁_{i∈I} φ_{i}, where each φ_{i} is given by the cycle [pr_{Y}^{∗} (B_{i})·Γ_{f}] ∈ CH(X ×Y) produced from the graph cycle Γ_{f} and the chosen (non-canonical) basis {B_{i}}_{i∈I} of CH(Y) over CH(X). The realization of φ coincides exactly with an isomorphism of abelian groups CH(X)⊗CH(F) → CH(Y) constructed in [EG97, Proposition 1]. Then, by Manin’s identity principle (see [Ma68, §3]) φ is an isomorphism.

3.3 Lemma. Let G be a linear algebraic group over a field F, X be a projective homogeneous G-variety and Y be a G-variety. Let f: Y → X be a G-equivariant projective morphism. Assume that the fiber of f over F (X) is isomorphic to F_{F (X)} for some variety F over F. Then f is a locally trivial fibration with the fiber F.

Proof. By the assumptions, we have Y ×_{X}Spec F (X) ≃ (F×X)×_{X}Spec F (X) as schemes over F (X). Since F (X) is a direct limit of O(U) taken over all non-empty affine open subsets U of X, by [EGA IV, Corollaire 8.8.2.5] there exists U such that f^{−1}(U) = Y ×_{X} U is isomorphic to (F ×X)×_{X} U ≃ F ×U as a scheme over U. Since G acts transitively on X and f is G-equivariant, the map f is a locally trivial fibration.

3.4 Corollary. Let X be a projective G-homogeneous variety, Y be a projective variety such that Y_{F (X)} ≃ F_{F (X)} for some variety F. Then the projection map X × Y → X is a locally trivial fibration with the fiber F. Moreover, if F is cellular, then M(X × Y) ≃ M(X) ⊗ M(F).

Proof. Apply Lemma 3.3 to the projection map X × Y → X and use Lemma 3.2.

3.5. Let G be a simple (connected) linear algebraic group over a field F, X be a projective homogeneous G-variety. Denote by D the Dynkin diagram of G. According to [Ti66] one can always choose a quasi-split group G_{0} over F with the same Dynkin diagram, a parabolic subgroup P of G_{0} and a cocycle ξ ∈ H^{1}(F, G_{0}) such that G is isogenic to_{ξ}G_{0} and X is isomorphic to_{ξ}(G_{0}/P). We will use the following standard notation: If G_{0} is split, then G is called to be of inner type over F.

3.6 Lemma. Let G be a semisimple linear algebraic group over F, X and Y be projective homogeneous G-varieties corresponding to parabolic subgroups P and Q of G_{0}, Q ⊆ P. Denote by f: Y → X the map induced by the quotient map G_{0}/Q → G_{0}/P. If G becomes split over F (X) then f is a cellular fibration with the fiber F = P/Q.

Proof. Since G splits over F (X), the fiber of f over F (X) is isomorphic to (P/Q)_{F (X)} = F_{F (X)}. Now apply Lemma 3.3 and note that F is cellular.

3.7 Example. Let P = P_{Θ} be the standard parabolic subgroup of a split group G_{0}, corresponding to a subset Θ of the respective Dynkin diagram D (enumeration of roots follows Bourbaki). In this notation the Borel subgroup corresponds to the empty set. Let ξ be a cocycle in H^{1}(F, G_{0}). Set G = _{ξ}G_{0} and X = _{ξ}(G_{0}/P). Denote by q the degree of a splitting field of G_{0} and by d the index of associated Tits algebra (for groups of classical types d is given in [Ti66, Table II]). Analyzing Tits indices of G we see that G becomes split over F (X) and, therefore, X becomes generically split over F if the subset D \ Θ contains one of the following vertices k (cf. [KR94, §7]):

G_{0} ^{1}A_{n}  B_{n}  C_{n}  ^{1}D_{n}
k gcd(k, d) = 1 k = n;  k is odd; k = n − 1;
any k in the  k = n if 2 ∤ n or d = 1;
Pfister case  any k in the Pfister case

G_{0} G_{2} F_{4}  ^{1}E_{6}  E_{7}  E_{8}
k any k = 1, 2, 3; k = 3, 5;  k = 2, 5;  k = 2, 3, 4, 5;
any k if  k = 2, 4 if d = 1; k = 3, 4 if d = 1; any k if
q = 3  k = 1, 6 if q is odd k ≠ 7 if q = 3  q = 5

(here by the Pfister case we mean the case when the cocycle ξ corresponds to a Pfister form or its maximal neighbor)

Case-by-case arguments of paper [CPSZ] show that under certain conditions the Chow motive of a twisted flag variety X can be expressed in terms of the motive of a minimal flag. These conditions cover almost all twisted flag varieties corresponding to groups of types A_{n} and B_{n} together with some examples of types C_{n}, G_{2} and F_{4}. Using the following theorem we provide a uniform proof of these results as well as extend it to some other types.

3.8 Theorem. Let Y and X be taken as in Lemma 3.6. Then the Chow motive M(Y) of Y is isomorphic to a direct sum of twisted copies of the motive M(X), i.e.,

M(Y) ≃ ⨁_{i≥0} M(X)(i)^{⊕c_{i}},

where ∑ c_{i}t^{i} = P (CH_{∗}(\overline{Y}), t)/P (CH_{∗}(\overline{X}), t).

Proof. Apply Lemmas 3.6 and 3.2.

3.9 Remark. The explicit formula for P (CH_{∗}(\overline{X}), t) involves the degrees of basic polynomial invariants of G_{0} and is provided in [Hi82, Ch. IV, Cor. 4.5].

4 Varieties of complete flags

4.1. Let G_{0} be a split simple linear algebraic group with a maximal split torus T and a Borel subgroup B containing T. Let G = _{ξ}G_{0} be a twisted form of G_{0} given by a cocycle ξ ∈ H^{1}(F, G_{0}) and X = _{ξ}(G_{0}/B) be the corresponding variety of complete flags. Observe that the group G splits over any field K over which X has a rational point, in particular, over the function field F (X). According to [De74] the Chow ring CH(\overline{X}) can be expressed in purely combinatorial terms and, therefore, depends only on the type of G but not on the base field F.

4.2. Let p be a prime integer. To simplify the notation we denote by Ch(X) the Chow ring of X = _{ξ}(G_{0}/B) with ℤ/p-coefficients and by \overline{Ch}(X) the image of the restriction map CH(X; ℤ/p) → CH(\overline{X}; ℤ/p). Let T̂ denote the group of characters of T and S(T̂) be the symmetric algebra. By R we denote the image of the characteristic map c: S(T̂) → Ch(\overline{X}) (see [Gr58, (4.1)]). According to [KM06, Thm.6.4] there is an embedding

R ⊆ \overline{Ch}(X),  (1)

where the equality holds if the cocycle ξ corresponds to a generic torsor.

4.3. Let Ch(\overline{G}) denote the Chow ring with ℤ/p-coefficients of the group G_{0}. Consider the pull-back induced by the quotient map

π: Ch(\overline{X}) → Ch(\overline{G})

According to [Gr58, Rem. 2^{∘}] π is surjective with the kernel generated by R^{+}, where R^{+} stands for the subgroup of non-constant elements of R.

The explicit presentation of Ch(\overline{G}) is known for all types of G and all torsion primes p of G (see [Gr58, Definition 3]). Namely, by [Kc85, Theorem 3] it is a quotient of the polynomial ring in r variables x_{1}, …, x_{r} of codimensions d_{1} ≤ d_{2} ≤ … ≤ d_{r} coprime to p, modulo an ideal generated by certain p-powers x_{1}^{p^{k_{1}}}, …, x_{r}^{p^{kr}} (k_{i} ≥ 0, i = 1, …, r)

Ch^{∗}(\overline{G}) = (ℤ/p)[x_{1}, …, x_{r}]/(x_{1}^{p^{k_{1}}}, …, x_{r}^{p^{kr}}).  (2)

In the case p is not a torsion prime of G we have Ch^{∗}(\overline{G}) = ℤ/p, i.e., r = 0.

Note that the complete list of numbers {d_{i}p^{k_{i}}}_{i=1...r} called p-exceptional degrees of G_{0} was provided in [Kc85, Table II]. Taking the p-primary and p-coprimary parts of each p-exceptional degree one immediately restores the respective k_{i} and d_{i}.

4.4. We introduce two orders on the set of additive generators of Ch(\overline{G}), i.e., monomials x_{1}^{m_{1}} … x_{r}^{m_{r}}. To simplify the notation, we will denote the monomial x_{1}^{m_{1}} … x_{r}^{m_{r}} by x^{M}, where M is an r-tuple of integers (m_{1}, …, m_{r}). The codimension of x^{M} will be denoted by |M|. Observe that |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 x^{M} ≼ x^{N} (or equivalently M ≼ N) if m_{i} ≤ n_{i} for all i. This gives a partial ordering on the set of all monomials (r-tuples).

• Given two r-tuples M = (m_{1}, …, m_{r}) and N = (n_{1}, …, n_{r}) we say 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}. This gives a wellordering on the set of all monomials (r-tuples) known also as DegLex order.

4.5 Definition. Let X = _{ξ}(G_{0}/B) be the twisted form of the variety of complete flags by means of a cocycle ξ ∈ H^{1}(F, G_{0}). Let \overline{Ch}(G) denote the image of the composite

Ch(X) ⟶^{res} Ch(\overline{X}) ⟶^{π} Ch(\overline{G})

Since both maps are ring homomorphisms, \overline{Ch}(G) is a subring of Ch(\overline{G}).

For each 1 ≤ i ≤ r set j_{i} to be the smallest non-negative integer such that the subring \overline{Ch}(G) contains an element a with the greatest monomial x_{i}^{p^{j_{i}}} with respect to the DegLex order on Ch(\overline{G}), i.e., of the form

a = x_{i}^{p^{j_{i}}} + ∑_{x^{M} ⪇x_{i}^{p^{j_{i}}}} c_{M}x^{M}.

The r-tuple of integers (j_{1}, …, j_{r}) will be called J-invariant of G modulo p and will be denoted by J_{p}(G).

4.6 Example. From presentation (2) we have j_{i} ≤ k_{i} for all i = 1, …, r. According to (1) the J-invariant takes its maximal value J_{p}(G) = (k_{1}, …, k_{r}) if the cocycle ξ corresponds to a generic torsor. Later on (see Corollary 6.10) it will be shown that the J-invariant takes its minimal possible value J_{p}(G) = (0, …, 0) if and only if the group G splits by a finite field extension of degree coprime to p.

4.7 Example. If the Chow ring Ch(\overline{G}) has only one generator, i.e., r = 1, then the J-invariant is equal to the smallest non-negative integer j_{1} such that x_{1}^{p^{j_{1}}} ∈ \overline{Ch}(G).

The next example explains the terminology ‘J-invariant’

4.8 Example. Let ϕ be a quadratic form with trivial discriminant. In [Vi05, Definition 5.11] A. Vishik introduced the notion of J-invariant of ϕ, the tuple of integers, which describes the subgroup of rational cycles on the respective maximal orthogonal Grassmannian. This invariant is an important tool in studying algebraic cycles on quadrics. An equivalent but ‘dual’ (in terms of non-rationality of cycles) definition of J(ϕ) was provided in [EKM, § 88]. Using Theorem 3.8 one can show that J(ϕ) introduced in [EKM] can be expressed in terms of J_{2}(O^{+}(ϕ)) = (j_{1}, …, j_{r}) as follows:

J(ϕ) = {2^{l}d_{i} | i = 1, …, r, 0 ≤ l ≤ j_{i} − 1}.

Since all d_{i} are odd, J_{2}(O^{+}(ϕ)) is uniquely determined by J(ϕ) as well.

Now we are ready to formulate and prove the main result of this section

4.9 Theorem. Given G and p with J_{p}(G) = (j_{1}, …, j_{r}) the motive of X is isomorphic to the direct sum

M(X; ℤ/p) ≃ ⨁_{i≥0} R_{p}(G)(i)^{⊕c_{i}},

where the motive R_{p}(G) is indecomposable, its Poincaré polynomial over a splitting field is equal to

P (\overline{R_{p}(G)}, t) = ∏_{i=1}^{r} \frac{1 − t^{d_{i}p^{j_{i}}}}{1 − t^{d_{i}}} ,  (3)

and the integers c_{i} are the coefficients of the polynomial

∑_{i≥0} c_{i}t^{i} = P (Ch^{∗}(\overline{X}), t)/P (\overline{R_{p}(G)}, t).

Fix preimages e_{i} of x_{i} in Ch(\overline{X}). For an r-tuple M = (m_{1}, …, m_{r}) set e^{M} = ∏_{i=1}^{r} e_{i}^{m_{i}}. Set K = (k_{1}, …, k_{r}) and N = p^{K} −1 = (p^{k_{1}} −1, …, p^{k_{r}} −1). Set d = dim X − |N| = deg(P (R^{∗}, t)).

4.10 Lemma. The Chow ring Ch(\overline{X}) is a free R-module with a basis \{e^{M}\}, M ≼ N.

Proof. Note that R^{+} is a nilpotent ideal in R. Applying Nakayama Lemma we obtain that \{e^{M}\} generate Ch(\overline{X}). By [Kc85, (2)] Ch(\overline{X}) is a free Rmodule, hence, for the Poincaré polynomials we have

P (Ch^{∗}(\overline{X}), t) = P (Ch^{∗}(\overline{G}), t) · P (R^{∗}, t).

Substituting t = 1 we obtain that

rk Ch(\overline{X}) = rk Ch(\overline{G}) · rk R.

To finish the proof observe that rk Ch(\overline{G}) coincides with the number of generators {e^{M}}.

4.11 Proposition. The pairing R × R → ℤ/p given by (α, β) ↦ deg(e^{N}αβ) is non-degenerated, i.e., for any element α ∈ R there exists β such that deg(e^{N}αβ) ≠ 0.

Proof. Choose a homogeneous basis of Ch(\overline{X}). Let α^{∨} be the Poincaré dual of α with respect to this basis. By Lemma 4.10 Ch(\overline{X}) is a free R-module with a basis {e^{M}}, hence, expanding α^{∨} we obtain

α^{∨} = ∑_{M≼N} e^{M}β_{M}, where β_{M} ∈ R.

Note that if M ≠ N then codim αβ_{M} > d, therefore, αβ_{M} = 0. So we can set β = β_{N}.

From now on we fix a homogeneous ℤ/p-basis {α_{i}} of R and the dual basis {α_{i}^{#}} with respect to the pairing introduced in Proposition 4.11.

4.12 Corollary. For |M| ≤ |N| we have

deg(e^{M}α_{i}α_{j}^{#}) = \begin{cases}1, & M = N and i = j \\ 0, & otherwise \end{cases}

Proof. If M = N, then it follows from the definition of the dual basis. Assume |M| < |N|. If deg(e^{M}α_{i}β_{j}) ≠ 0, then codim(α_{i}β_{j}) > d, a contradiction with the fact that α_{i}β_{j} ∈ R. Hence, we reduced to the case M ≠ N and |M| = |N|. Since |M| = |N|, codim(α_{i}β_{j}) = d and, hence, R^{+}α_{i}β_{j} = 0. From the other side there exists i such that m_{i} ≥ p^{k_{i}} and e^{p^{k_{i}}} ∈ Ch(\overline{X}) · R^{+}. Hence, e^{M}α_{i}β_{j} = 0.

4.13 Definition. Given two pairs (L, l) and (M, m), where L, M are r-tuples and l, m are integers, we say (L, l) ≤ (M, m) if either L ⪇ M, or in the case L = M we have l ≤ m. We introduce the filtration on the ring Ch(\overline{X}) as follows:

The (M, m)-th term Ch(\overline{X})_{M,m} is the subring generated by the
elements e^{I}α with I ≤ M, α ∈ R, codim α ≤ m.

Define the associated graded ring as follows:

A^{∗,∗} = ⨁_{(M,m)} A^{M,m}, where A^{M,m} = Ch(\overline{X})_{M,m}/ ⋃_{(L,l)⪇(M,m)}Ch(\overline{X})_{L,l}.

By Lemma 4.10 if M ≼ N the graded component A^{M,m} consists of the classes of elements e^{M}α with α ∈ R and codim α = m. In particular, rk A^{M,m} = rk R^{m}. Comparing the ranks we see that A^{M,m} is trivial when M ⋠ N.

Consider the subring \overline{Ch}(X) of rational cycles with the induced filtration. The associated graded subring will be denoted by A_{rat}^{∗,∗}. From the definition of J-invariant it follows that the elements e_{i}^{p^{j_{i}}}, i = 1, …, r, belong to A_{rat}^{∗,∗}.

Similarly, we introduce the filtration on the ring Ch(\overline{X} × \overline{X}) as follows:

The (M, m)-th term is the subring generated by the elements e^{I}α×e^{L}β with I +L ≤ M, α, β ∈ R and codim α+codim β ≤ m.

The associated graded ring will be denoted by B^{∗,∗}. By definition B^{∗,∗} is isomorphic to the product of graded rings A^{∗,∗} ⊗_{ℤ/p}A^{∗,∗}. The graded subring associated to \overline{Ch}(X × X) will be denoted by B_{rat}^{∗,∗}.

4.14. The key observation is that due to Corollary 4.12 we have

Ch(\overline{X} × \overline{X})_{M,m} ∘ Ch(\overline{X} × \overline{X})_{L,l} ⊂ Ch(\overline{X} × \overline{X})_{M+L−N,m+l−d} and

(Ch(\overline{X} × \overline{X})_{M,m})_{⋆}(Ch(\overline{X})_{L,l}) ⊂ Ch(\overline{X})_{M+L−N,m+l−d}

and, therefore, we have the correctly defined composition law

∘: B^{M,m} × B^{L,l} → B^{M+L−N,m+l−d}

and realization map (see 1.5)

⋆: B^{M,m} × A^{L,l} → A^{M+L−N,m+l−d}

In particular, B^{N+∗,d+∗} can be viewed as a graded ring with respect to the composition such that (α ∘ β)_{⋆} = α_{⋆} ∘ β_{⋆}. Note also that both operations preserve rationality of cycles.

The proof of the following result is based on the fact that the variety X is generically split

4.15 Lemma. The elements e_{i} × 1 − 1 × e_{i}, i = 1, …, r, belong to B_{rat}^{∗,∗}.

Proof. The proof is based on the fact that the variety X is generically split. Fix an i. Since X splits over F (X), by Lemma 1.6 there exists a cycle in \overline{Ch^{d_{i}}}(X × X) of the form

ξ = e_{i} × 1 + ∑_{s} μ_{s} × ν_{s} + 1 × μ,

where codim μ_{s}, codim ν_{s} < d_{i}. Then the cycle

pr_{13}^{∗}(ξ) − pr_{23}^{∗}(ξ) = (e_{i} × 1 − 1 × e_{i}) × 1 + ∑_{s}(μ_{s} × 1 − 1 × μ_{s}) × ν_{s}

belongs to \overline{Ch}(X×X×X). Applying Corollary 3.4 to the projection pr_{3} : X× X ×X → X ×X we conclude that the pull-back pr_{3}^{∗}: Ch(X ×X) → Ch(X × X × X) has a (non-canonical) section, say, δ_{3}. Since the construction of this section preserves base change it preserves rationality of cycles. Hence, passing to a splitting field we obtain a rational cycle

δ_{3}(pr_{13}^{∗}(ξ) − pr_{23}^{∗}(ξ)) = e_{i} × 1 − 1 × e_{i} + ∑_{s}(μ_{s} × 1 − 1 × μ_{s})δ_{3}(1 × 1 × ν_{s})

whose image in B_{rat}^{∗,∗} is e_{i} ⊗ 1 − 1 ⊗ e_{i}.

We will write (e×1−1×e)^{M} for the product ∏_{i=1}^{r}(e_{i} ×1−1×e_{i})^{m_{i}}. and \binom{M}{L} for the product of binomial coefficients ∏_{i=1}^{r} \binom{m_{i}}{l_{i}}. In the computations we will extensively use the following two formulas (the first follows directly from Corollary 4.12 and the second is a well-known binomial identity).

4.16. Let α be an element of R^{∗} and α^{#} be its dual with respect to the non-degenerate pairing from 4.11, i.e., deg(e^{N}αα^{#}) = 1. Then we have

((e × 1 − 1 × e)^{M}(α^{#} × 1))_{⋆}(e^{L}α) = \binom{M}{M + L − N}(−1)^{M+L−N}e^{M+L−N}

4.17 (Lucas’ Theorem). The following identity holds

\binom{n}{k} ≡ ∏_{i≥0} \binom{n_{i}}{k_{i}} mod p,

where k = ∑_{i≥0} k_{i}p^{i} and n = ∑_{i≥0} n_{i}p^{i} are base p presentations of k and n.

Set for brevity J = J_{p}(G) = (j_{1}, …, j_{r}) and recall that K = (k_{1}, …, k_{r}).

4.18 Proposition. Let {α_{i}} be a homogeneous ℤ/p-basis of R. Then the set of elements B = {e^{p^{J} L}α_{i} | L ≼ p^{K−J} − 1} forms a ℤ/p-basis of A_{rat}^{∗,∗}.

Proof. According to Lemma 4.10 the elements from B are linearly independent. Assume B does not generate A_{rat}^{∗,∗}. Choose an element ω ∈ A_{rat}^{M,m} of the smallest index (M, m) which is not in the linear span of B. By definition of A^{M,m} (see Definition 4.13) ω can be written as ω = e^{M}α, where M ≼ N, α ∈ R^{m} and M can not be presented as M = p^{J}L′ for an r-tuple L′. The latter means that in the decomposition of M into p-primary and p-coprimary components M = p^{S}L, where S = (s_{1}, …, s_{r}), L = (l_{1}, …, l_{r}) and p ∤ l_{k} for k = 1, …, r, we have J ⋠ S. Choose an i such that s_{i} < j_{i}. Denote M_{i} = (0, …, 0, m_{i}, 0, …, 0) and S_{i} = (0, …, 0, s_{i}, 0, …, 0), where m_{i} and s_{i} stand at the i-th place.

Set T = N − M + M_{i}. By Lemma 4.15 and 4.16 together with observation 4.14 the element

((e × 1 − 1 × e)^{T} (α^{#} × 1))_{⋆}(e^{M}α) = \binom{p^{k_{i}} − 1}{m_{i}}(−1)^{m_{i}}e^{m_{i}}

belongs to A_{rat}^{M_{i},0}. By 4.17 we have p ∤ \binom{p^{k_{i}}−1}{m_{i}} and, therefore, this element is non trivial. Moreover, since s_{i} < j_{i}, this element is not in the span of B. Since (M, m) was chosen to be the smallest one and (M_{i}, 0) ≤ (M, m) we obtain that (M, m) = (M_{i}, 0). Repeating the same arguments for T = N − M_{i} + p^{S_{i}} we obtain that M_{i} = p^{S_{i}}, i.e., l_{i} = 1.

Now let γ be a representative of ω = e_{i}^{p^{s_{i}}} in \overline{Ch}(X). Then its image π(γ) in \overline{Ch}(G) has the leading term x_{i}^{p^{s_{i}}} with s_{i} < j_{i}. This contradicts to the definition of J-invariant.

4.19 Corollary. The elements

{(e × 1 − 1 × e)^{S}(e^{p^{J} L}α_{i} × e^{p^{J} (p^{K−J} −1−M)}α_{j}^{#}) | L, M ≼ p^{K−J} − 1, S ≼ p^{J} − 1}

form a ℤ/p-basis of B_{rat}^{∗,∗}. In particular, they form a basis of B_{rat}^{N,d} if and only if S = p^{J} − 1 and L = M.

Proof. According to 4.10 these elements are linearly independent and their number is p^{|2K−J|}(rk R)^{2}. They are rational according to Definition 4.13 and Lemma 4.15. Applying Corollary 3.4 we obtain that

rk B_{rat}^{∗,∗} = rk \overline{Ch}(X × X) = rk \overline{Ch}(X) · rk Ch(\overline{X}),

where the latter coincides with rk A_{rat}^{∗,∗}·p^{|K|} rk R = p^{|2K−J|}(rk R)^{2} by Lemma 4.10 and Proposition 4.18.

4.20 Lemma. The elements

θ_{L,M,i,j} = (e × 1 − 1 × e)^{p^{J} −1}(e^{p^{J} L}α_{i} × e^{p^{J} (p^{K−J} −1−M)}α_{j}^{#}), L, M ≼ p^{K−J} − 1,

belong to B_{rat}^{∗,∗} and satisfy the relation θ_{L,M,i,j} ∘ θ_{L′,M′,i′,j′} = δ_{LM′}δ_{ij′}θ_{L′,M,i′,j}.

Proof. Follows from Corollary 4.12.

Proof of Theorem 4.9. Consider the projection map

f^{0} : \overline{Ch}(X × X)_{N,d} → B_{rat}^{N,d}.

By Lemma 4.20 the elements θ_{L,L,i,j} form a family of pairwise-orthogonal idempotents whose sum is the identity. The kernel of f^{0} is nilpotent and, therefore, by Proposition 2.5 there exist pair-wise orthogonal idempotents φ_{L,i} in \overline{Ch}(X ×X) which are mapped to θ_{L,L,i,i} and whose sum is the identity. Their components of codimension |N| + d = dim X have the same properties and, hence, we may assume that φ_{L,i} belong to \overline{Ch^{dim X}}(X × X).

We show that φ_{L,i} are indecomposable. By Corollary 4.19 and Lemma 4.20 the ring (B_{rat}^{N,d}, ∘) can be identified with a product of matrix rings over ℤ/p

B_{rat}^{N,d} ≃ ∏_{s=0}^{d} End((ℤ/p)^{p^{|K−J|} rk R^{s}}).

By means of this identification θ_{L,L,i,i}: e^{p^{J} M}α_{j} ↦ δ_{L,M}δ_{i,j}e^{p^{J} L}α_{i} is an idempotent of rank 1 and, therefore, is indecomposable. Since f^{0} preserves isomorphisms, φ_{L,i} are indecomposable as well.

We show that φ_{L,i} is isomorphic to φ_{M,j}. In the ring B_{rat}^{∗,∗} mutually inverse isomorphisms between them are given by θ_{L,M,i,j} and θ_{M,L,j,i}. Let

f: \overline{Ch}(X × X) → B_{rat}^{∗,∗}

be the leading term map; it means that for any γ ∈ \overline{Ch}(X × X) we find the smallest degree (I, s) such that γ belongs to \overline{Ch}(X ×X)_{I,s} and set f(γ) to be the image of γ in B_{rat}^{I,s}. Note that f is not a homomorphism but satisfies the condition that f(ξ)∘f(η) equals either f(ξ∘η) or 0. Choose preimages ψ_{L,M,i,j} and ψ_{M,L,j,i} of θ_{L,M,i,j} and θ_{M,L,j,i} by means of f. Applying Lemma 2.6 we obtain mutually inverse isomorphisms ϑ_{L,M,i,j} and ϑ_{M,L,j,i} between φ_{L,i} and φ_{M,j}. It remains to take their homogeneous components of the appropriate degrees.

Now applying Lemma 1.7 and Corollary 2.10 to the restriction map

res_{F} : End(M(X; ℤ/p)) = Ch(X × X) → \overline{Ch}(X × X) = \overline{End^{∗}}(M(X; ℤ/p))

and the family of idempotents φ_{L,i} we obtain the family of pair-wise orthogonal idempotents ϕ_{L,i} ∈ End(M(X; ℤ/p)) such that

Δ_{X} = ∑_{L,i} ϕ_{L,i}.

Since res_{F} is isomorphism preserving, for the respective motives we have (X, ϕ_{L,i}) ≃ (X, ϕ_{0,0})(|L| + i) for all L and i (see Example 2.2). Denoting R_{p}(G) = (X, ϕ_{0,0}) we obtain the desired motivic decomposition.

As a direct consequence of the proof we obtain

4.21 Corollary. Any direct summand of M(X; ℤ/p) is isomorphic to a direct sum of twisted copies of R_{p}(G).

Proof. Indeed, in the ring B_{rat}^{N,d} any idempotent is isomorphic to a sum of idempotents θ_{L,L,i,i}, and the map f^{0} preserves isomorphisms.

4.22 Remark. Note that Corollary 4.21 can be viewed as a particular case of the Krull-Schmidt Theorem proven by V. Chernousov and A. Merkurjev (see [CM06, Corollary 9.7]).

5 Motivic decompositions

In the present section we prove the main result of this paper

5.1 Theorem. Let G be a simple linear algebraic group of inner type over a field F and p be a prime integer. Let X be a generically split projective homogeneous G-variety. Then the motive of X with ℤ/p-coefficients is isomorphic to the direct sum

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

where R_{p}(G) is an indecomposable motive, whose Poincaré polynomial P (\overline{R_{p}(G)}, t) is given by (3) and, hence, depends only on the J-invariant of G, and a_{i} are the coefficients of the quotient polynomial

∑_{i≥0} a_{i}t^{i} = P (CH^{∗}(\overline{X}), t)/P (\overline{R_{p}(G)}, t).

Proof. The variety X is generically split means that the group G becomes split over F (X). Let Y be the variety of complete G-flags. According to Theorem 3.8 the motive of Y is isomorphic to a direct sum of twisted copies of the motive of X. To finish apply Theorem 4.9 and Corollary 4.21.

5.2 Lemma. Let G be a group of inner type, X be a projective homogeneous G-variety. Then any field extension E/F is rank preserving with respect to X and X × X.

Proof. By [Pa94, Theorem 2.2 and 4.2] the restriction map K_{0}(X) → K_{0}(X_{E}) becomes an isomorphism after tensoring with ℚ. Now the Chern character ch: K_{0}(X) ⊗ ℚ → CH^{∗}(X) ⊗ ℚ is an isomorphism and respects pull-backs, hence E is rank preserving with respect to X. It remains to note that X ×X is G × G-homogeneous variety.

Now we provide several properties of R_{p}(G) which will be extensively used in the applications:

5.3 Proposition. Let G and G′ be two simple groups of inner type, X and X′ be the corresponding varieties of complete flags.

• (base change) For any field extension E/F we have

R_{p}(G)_{E} ≃ ⨁_{i≥0} R_{p}(G_{E})(i)^{⊕a_{i}},

where ∑ a_{i}t^{i} = P (\overline{R_{p}(G)}, t)/P (\overline{R_{p}(G_{E})}, t).

• (transfer argument) If E/F is a field extension of degree coprime to p then J_{p}(G_{E}) = J_{p}(G) and R_{p}(G_{E}) = R_{p}(G)_{E}. Moreover, if R_{p}(G_{E}) ≃ R_{p}(G′_{E}) then R_{p}(G) ≃ R_{p}(G′).

• (comparison lemma) If G splits over F (X′) and G′ splits over F (X) then R_{p}(G) ≃ R_{p}(G′).

Proof. The first claim follows from Theorem 4.9 and Corollary 4.21. To prove the second claim note that E is rank preserving with respect to X and X × X by Lemma 5.2. Now J_{p}(G_{E}) = J_{p}(G) by Lemma 2.12, and hence R_{p}(G_{E}) = R_{p}(G)_{E} by the first claim. The remaining part of the claim follows from Corollary 2.13 applied to the variety X ∐ X′.

Now we prove the last claim. The variety X×X′ is the variety of complete G×G′-flags. Applying Corollary 3.4 we can express M(X×X′; ℤ/p) in terms of R_{p}(G) and R_{p}(G′). Now apply Corollary 4.21.

5.4 Corollary. We have R_{p}(G) ≃ R_{p}(G_{an}), where G_{an} is the anisotropic kernel of G.

Let m be a positive integer. We say a polynomial g(t) is m-positive, if g ≠ 0, P (\overline{R_{p}(G)}, t) | g(t) and the quotient polynomial g(t)/P (\overline{R_{p}(G)}, t) has non-negative coefficients for all primes p dividing m.

5.5 Proposition. Let G be a simple linear algebraic group of inner type over a field F and X be a generically split projective homogeneous G-variety. Assume that X splits by a field extension of degree m. Let f(t) be an mpositive polynomial dividing P (M(\overline{X}), t) which can not be presented as a sum of two m-positive polynomials. Then the motive of X with integer coefficients splits as a direct sum

M(X) ≃ ⨁_{i} R_{i}(c_{i}), c_{i} ∈ ℤ,

where R_{i} is indecomposable and P (\overline{R}_{i}, t) = f(t) for all i. Moreover, if m = 2, 3, 4 or 6, then all motives R_{i} are isomorphic up to twists.

Proof. First, we apply Corollary 2.7 and Lemma 2.8 to obtain a decomposition with ℤ/m-coefficients. By Lemma 5.2 our field extension is rank preserving so we can apply Theorem 2.16 to lift the decomposition over ℤ.

6 Properties of J-invariant

6.1. Recall (see [Br03]) that if the characteristic of the base field F is different from p then one can construct Steenrod p-th power operations

S^{l}: Ch^{∗}(X) → Ch^{∗+l(p−1)}(X), l ≥ 0

such that S^{0} = id, S^{l} restricted to Ch^{l}(X) coincides with taking to the p-th power, and the total operation S^{•} = ∑_{l≥0} S^{l} is a homomorphism of ℤ/palgebras compatible with pull-backs. In the case of varieties over the field of complex numbers S^{l} is compatible with its topological counterparts: reduced power operation P^{l} if p ≠ 2 and Steenrod square Sq^{2l} if p = 2 (in this case Ch^{∗}(X) can be viewed as a subring in H^{2∗}(X, ℤ/p)).

When X is the variety of complete G-flags the action of Steenrod operations on Ch^{∗}(\overline{X}) can be described in purely combinatorial terms (see [DZ07]) and, hence, doesn’t depend on the choice of a base field. Since Steenrod operations respect pull-back they respect rationality as well.

Over the field of complex numbers Ch^{∗}(\overline{G}) may be identified with the image of the pull-back map H^{2∗}(\overline{X}, ℤ/p) → H^{2∗}(\overline{G}, ℤ/p). An explicit description of this image and formulae describing the action of P^{l} on H^{∗}(\overline{G}, ℤ/p) can be found in [MT91].

The following lemma provides an important technical tool for computing the possible values of J-invariant of G.

6.2 Lemma. Assume that in Ch^{∗}(\overline{G}) we have S^{l}(x_{i}) = x_{m}^{p^{s}} and S^{l}(x_{i′}) < x_{m}^{p^{s}} if i′ < m with respect to the DegLex order. Then j_{m} ≤ j_{i} + s.

Proof. By definition there exists a cycle α ∈ \overline{Ch}(X) such that the leading term of π(α) is x_{i}^{p^{j_{i}}}. For the total operation we have

S(x_{i}^{p^{j_{i}}}) = S(x_{i})^{p^{j_{i}}} = S^{0}(x_{i})^{p^{j_{i}}} + S^{1}(x_{i})^{p^{j_{i}}} + … + S^{d_{i}}(x_{i})^{p^{j_{i}}}.

In particular, S^{lp^{j_{i}}}(x_{i}^{p^{j_{i}}}) = S^{l}(x_{i})^{p^{j_{i}}}. Applying S^{lp^{j_{i}}} to α we obtain a rational cycle whose image under π has the leading term x_{m}^{p^{j_{i}+s}}.

6.3. We summarize information about restrictions on J-invariant which can be obtained using Lemma 6.2 into the following table (numbers d_{i} and k_{i} are taken from [Kc85, Table II]).

G_{0}  p  r  d_{i}  k_{i}  j_{i}
SL_{n} /μ_{m}, m | n p | m 1  1  p^{k_{1}} ∥ n  any
PGSp_{n}, 2 | n 2  1  1  2^{k_{1}} ∥ n  any
SO_{n}  2  [\frac{n+1}{4}] 2i − 1  [log_{2} \frac{n−1}{2i−1}] j_{i} ≥ j_{i+l} if 2 ∤ \binom{i−1}{l},
j_{i} ≤ j_{2i−1} + 1
Spin_{n} 2 [\frac{n−3}{4}] 2i + 1 [log_{2} \frac{n−1}{2i+1}] j_{i} ≥ j_{i+l} if 2 ∤ \binom{i}{l},
j_{i} ≤ j_{2i} + 1
PGO_{2n}, n > 1 2 [\frac{n+2}{2}] 1, i = 1 2^{k_{1}} ∥ n j_{i} ≥ j_{i+l} if 2 ∤ \binom{i−2}{l},
2i − 3, i ≥ 2 [log_{2} \frac{2n−1}{2i−3}] j_{i} ≤ j_{2i−2} + 1
Ss_{2n}, 2 | n 2 \frac{n}{2} 1, i = 1 2^{k_{1}} ∥ n j_{i} ≥ j_{i+l} if 2 ∤ \binom{i−1}{l}
2i − 1, i ≥ 2 [log_{2} \frac{2n−1}{2i−1}] j_{i} ≤ j_{2i−1} + 1
G_{2}, F_{4}, E_{6}  2  1  3  1
F_{4}, E_{6}^{sc}, E_{7}  3  1  4  1
E_{6}^{ad}  3  2  1, 4  2, 1
E_{7}^{sc}  2  3  3, 5, 9  1, 1, 1  j_{1} ≥ j_{2} ≥ j_{3}
E_{7}^{ad}  2  4  1, 3, 5, 9  1, 1, 1, 1 j_{2} ≥ j_{3} ≥ j_{4}
E_{8}  2  4  3, 5, 9, 15  3, 2, 1, 1 j_{1} ≥ j_{2} ≥ j_{3},

j_{1} ≤ j_{2} + 1, j_{2} ≤ j_{3} + 1

E_{8}  3  2  4, 10  1, 1  j_{1} ≥ j_{2}
E_{8}  5  1  6  1

We give some applications of J-invariant. First, as a by-product of the proof of Theorem 4.9 we obtain the following expression for the canonical p-dimension of the variety of complete flags (cf. [EKM, Theorem 90.3] for the case of quadrics).

6.4 Proposition. In the notation of Theorem 4.9 we have

cd_{p}(X) = ∑_{i=1}^{r} d_{i}(p^{j_{i}} − 1).

Proof. Follows from Proposition 4.18 and [KM06, Theorem 5.8].

Let X be a smooth projective variety which has a splitting field.

6.5 Lemma. For any ϕ, ψ ∈ CH^{∗}(\overline{X} × \overline{X}) one has

deg((pr_{2})_{∗}(ϕ · ψ^{t})) = tr((ϕ ∘ ψ)_{∗}).

Proof. Choose a homogeneous basis \{e_{i}\} of CH^{∗}(\overline{X}). Let \{e_{i}^{∨}\} be its Poincaré dual. Since both sides are bilinear, it suffices to check the assertion for ϕ = e_{i} × e_{j}^{∨} and ψ = e_{k} × e_{l}^{∨}. In this case both sides are equal to δ_{il}δ_{jk}.

Denote by d(X) the greatest common divisor of the degrees of all zero cycles on X and by d_{p}(X) its p-primary component.

6.6 Corollary. For any ϕ ∈ \overline{CH}(X × X; ℤ/m) we have

g.c.d.(d(X), m) | tr(ϕ_{∗}).

Proof. Set ψ = ΔX̄ and apply Lemma 6.5.

6.7 Corollary. Assume that M(X; ℤ/p) has a direct summand M. Then

1. d_{p}(X) | P (\overline{M}, 1);

2. if d_{p}(X) = P (\overline{M}, 1) and the kernel of the restriction End(M(X)) → End(M(\overline{X})) consists of nilpotents, then M is indecomposable.

Proof. Set q = d_{p}(X) for brevity. Let M = (X, ϕ). By Corollary 2.7 there exists an idempotent φ ∈ End(M(X); ℤ/q) such that φ mod p = ϕ. Then res(φ) ∈ End(M(\overline{X}); ℤ/q) is a rational idempotent. Since every projective module over ℤ/q is free, we have

tr(res(φ)_{∗}) = rk_{ℤ/q}(res(φ)_{∗}) = rk_{ℤ/p}(res(ϕ)_{∗}) = P (\overline{M}, 1) mod q,

and the first claim follows by Corollary 6.6. The second claim follows from the first, since the second assumption implies that for any nontrivial direct summand M′ of M we have P (\overline{M′}, 1) < P (\overline{M}, 1).

6.8. Let G be a group of inner type. Denote by n(G) the greatest common divisor of degrees of all finite splitting fields of G and by n_{p}(G) its p-primary component. Note that n(G) = d(X) and n_{p}(G) = d_{p}(X), where X is the variety of complete G-flags.

We obtain the following estimate on n_{p}(G) in terms of J-invariant (cf. [EKM, Prop. 88.11] in the case of quadrics).

6.9 Proposition. For a group G of inner type with J_{p}(G) = (j_{1}, …, j_{r}) we have

n_{p}(G) ≤ p^{∑_{i} j_{i}}.

Proof. Follows from Theorem 4.9 and Corollary 6.7.

6.10 Corollary. The following statements are equivalent:

• J_{p}(G) = (0, …, 0);

• n_{p}(G) = 1;

• R_{p}(G) = ℤ/p.

Proof. If J_{p}(G) = (0, …, 0) then n_{p}(G) = 1 by Proposition 6.9. If n_{p}(G) = 1 then there exists a splitting field L of degree m prime to p and, therefore, R_{p}(G) = ℤ/p by the transfer argument (see Proposition 5.3). The remaining implication is obvious.

Finally, we obtain the following reduction formula (cf. [EKM, Cor. 88.7] in the case of quadrics).

6.11 Proposition. Let G be a group of inner type, X be the variety of complete G-flags, Y be a projective variety such that the map CH^{l}(Y) → CH^{l}(Y_{F (x)}) is surjective for all x ∈ X and l ≤ n. Then j_{i}(G) = j_{i}(G_{F (Y)}) for all i such that d_{i}p^{j_{i}(G_{F (Y )})} ≤ n.

Proof. Indeed, by [EKM, Lemma 88.5] the map CH^{l}(X) → CH^{l}(X_{F (Y)}) is surjective for all l ≤ n, and therefore j_{i}(G) ≤ j_{i}(G_{F (Y)}). The converse inequality is obvious.

6.12 Corollary. J_{p}(G) = J_{p}(G_{F (t)}).

Proof. Take Y = ℙ^{1} and apply Proposition 6.11.

7 Examples

In the present section we provide examples of motivic decompositions of projective homogeneous varieties obtained by applying Theorem 5.1.

The case r = d_{1} = 1. According to Table 6.3 this corresponds to the case when G is of type A_{n} or C_{n}. Let A be a central simple algebra corresponding to G. We have A = M_{m}(D), where D is a division algebra of index d ≥ 1 over a field F. Let p be a prime divisor of d (p = 2 in the case of C_{n}). Observe that according to Table 6.3 J_{p}(G) = (j_{1}) for some j_{1} ≥ 0. Let X_{Θ} be the projective homogeneous G-variety given by a subset Θ of vertices of the respective Dynkin diagram such that p ∤ j for some j ∉ Θ (cf. Example 3.7). Then by Theorem 5.1 we obtain that

M(X_{Θ}; ℤ/p) ≃ ⨁_{i≥0} R_{p}(G)(i)^{⊕a_{i}},  (4)

where R_{p}(G) is indecomposable and

\overline{R_{p}(G)} ≃ ⨁_{i=0}^{p^{j_{1}}−1}(ℤ/p)(i).

Now we identify R_{p}(G). Using the comparison lemma (see Proposition 5.3) we conclude that R_{p}(G) depends only on D, so we may assume m = 1. By Table 6.3 we have p^{j_{1}} | d, but on the other hand by Proposition 6.9 we have n_{p}(G) ≤ p^{j_{1}}. Therefore, p^{j_{1}} is a p-primary part of d.

We have D ≃ D_{p} ⊗_{F} D′, where p^{j_{1}} = ind(D_{p}) and p ∤ ind(D′). Passing to a splitting field of D′ of degree prime to p and using Proposition 5.3 we conclude that the motives of X_{Θ} and SB(D_{p}) are direct sums of twisted R_{p}(G). Comparing the Poincaré polynomials we conclude that

7.1 Lemma. M(SB(D_{p}); ℤ/p) ≃ R_{p}(G).

Applying Proposition 5.5 to X = SB(D) and comparing the Poincaré polynomials of M(X) and R_{i} we obtain that

7.2 Corollary. The motive of SB(D) with integer coefficients is indecomposable.

7.3 Remark. Indeed, we provided a uniform proof of the results of paper [Ka96]. Namely, the decomposition of M(SB(A); ℤ/p) (see [Ka96, Cor. 1.3.2]) and indecomposability of M(SB(D); ℤ) (see [Ka96, Thm. 2.2.1]).

The case r = 1 and d_{1} > 1. According to Table 6.3 this holds if

p = 2: G is a group of type Spin_{n}, n = 7, 8, 9, 10, G_{2}, F_{4} or E_{6};

p = 3: G is a group of type F_{4}, E_{7} or strongly inner form of type E_{6};

p = 5: G is a group of type E_{8}.

Observe that in all these cases J_{p}(G) = (0) or (1). Let X be a generically split projective homogeneous G-variety (cf. Example 3.7). By Theorem 5.1 we obtain the decomposition

M(X; ℤ/2) ≃ ⨁_{i≥0} R_{p}(G)(i)^{⊕a_{i}},  (5)

where the motive R_{p}(G) is indecomposable and (cf. [Vo03, (5.4-5.5)])

\overline{R_{p}(G)} ≃ ⨁_{i=1}^{p−1}(ℤ/p)(i · (p + 1)).

Now we identify R_{p}(G). Let 𝔯 be the Rost invariant as defined in [Me03] and 𝔯_{p} denote its p-part.

7.4 Lemma. Let G be a simple linear algebraic group over F satisfying r = 1 and d_{1} > 1 and p be its torsion prime. Then 𝔯_{p}(G) is trivial iff R_{p}(G) ≃ ℤ/p.

Proof. According to [Ga01, Theorem 0.5], [Ch94] and [Gi00, Theoreme 10] the invariant 𝔯_{p}(G) is trivial iff there exists a field extension E/F of degree coprime with p such that the group G splits over E. By Corollary 6.10 the latter is equivalent to the fact that R_{p}(G) ≃ ℤ/p.

7.5 Lemma. Let G and G′ be simple linear algebraic groups over F satisfying r = 1 and d_{1} > 1 and p be its common torsion prime. For groups of type E_{8} we assume in addition that they are twisted forms by means of cocycles taken from the image of the twisted map defined in [Ga06, 13.12] (examples of such groups are provided in [Ga06, 14.7]).

If 𝔯_{p}(G) = 𝔯_{p}(G′)c for some c ∈ (ℤ/p)^{×}, then R_{p}(G) ≃ R_{p}(G′).

Proof. By transfer arguments (see Proposition 5.3) it is enough to prove this over a p-primary closure of F. Let X and X′ be the respective varieties of complete flags. Observe that the invariant 𝔯_{p}(G) becomes trivial over the function field F (X). Since 𝔯_{p}(G) = 𝔯_{p}(G′)c, it becomes trivial over F (X′) as well. By triviality of the kernel of the Rost invariant (see [Ga01, Theorem 0.5] and [Ga06, Theorem 13.14]) this implies that G splits over F (X′). To finish the proof we apply the comparison lemma (see Proposition 5.3) to the groups G and G′.

ℤ-coefficients. Let G be a group of type F_{4} or strongly inner form of type E_{6} which doesn’t split by field extensions of degrees 2 and 3. Observe that such a group splits by an extension of degree 6. Let X be a generically split projective homogeneous G-variety. Then according to Proposition 5.5 the Chow motive of X with integer coefficients splits as a direct sum of twisted copies of an indecomposable motive R(G) such that

R(G) ⊗ ℤ/2 = ⨁_{i=0,1,2,6,7,8} R_{2}(G)(i),  P (\overline{R_{2}(G)}, t) = 1 + t^{3},
R(G) ⊗ ℤ/3 = ⨁_{i=0,1,2,3} R_{3}(G)(i),  P (\overline{R_{3}(G)}, t) = 1 + t^{4} + t^{8},
P (\overline{R(G)}, t) = 1 + t + t^{2} + … + t^{11}.

7.6 Remark. In particular, we provided a uniform proof of the main results of papers [Bo03] and [NSZ], where the cases of some G_{2}- and F_{4}-varieties are considered.

The case r > 1. According to Table 6.3 this holds for groups G of types B_{n} and D_{n} and exceptional types E_{7}, E_{8} for p = 2 and E_{6}^{ad}, E_{8} for p = 3.

Pfister case. Let G = O^{+}(ϕ), where ϕ is a k-fold Pfister form or its maximal neighbor. Assume J_{2}(G) ≠ (0, …, 0). In view of Corollary 6.10 this holds iff n_{2}(G) ≠ 1. By Springer Theorem the latter holds iff ϕ is not split. According to Theorem 5.1 we obtain the decomposition

M(X; ℤ/2) ≃ ⨁_{i≥0} R_{2}(G)(i)^{⊕a_{i}}

where R_{2}(G) is indecomposable. Moreover, according to Theorem 2.16 the similar decomposition exists with ℤ-coefficients.

Now we compute J_{2}(G). Let Y be a projective quadric corresponding to ϕ. Then G splits over F (Y) and Y splits over F (x) for any x ∈ X.

It is known that CH^{l}(\overline{Y}) for l < 2^{k−1} − 1 is generated by CH^{1}(\overline{Y}) and, therefore, is rational. By Proposition 6.11 and Table 6.3 we see that j_{i}(G) = 0 for 0 ≤ i < r, where r = 2^{k−2}. Therefore, J_{2}(G) = (0, …, 0, 1) and P (\overline{R_{2}(G)}, t) = 1 + t^{2^{k−1}−1}. Finally, by Corollary 4.21 the motive R_{2}(G) coincides with the motive introduced in [Ro98] which is called Rost motive.

In this way we obtain Rost decomposition of the motive of a Pfister quadric and its maximal neighbor.

Maximal orthogonal Grassmannian Let G = O^{+}(q), where q: V → F is an arbitrary anisotropic regular quadratic form and X is the respective maximal orthogonal Grassmannian. The variety X is generically split, hence, by Theorem 5.1 we have the decomposition

M(X; ℤ/2) ≃ ⨁_{i≥0} R_{2}(G)(i)^{⊕a_{i}},

where the motive R_{2}(G) is indecomposable. Comparing the Poincaré polynomials of M(X; ℤ/2) and R_{2}(G) we obtain the following particular cases:

• If the group G corresponds to a generic cocycle (see 4.2), the motive M(X; ℤ/2) is isomorphic to R_{2}(G) and, hence, is indecomposable. This corresponds to the maximal value of the J-invariant.

• If q is a Pfister form or its maximal neighbor, by the previous example R_{2}(G) coincides with the Rost motive. This corresponds to the minimal non-trivial value of the J-invariant.

Acknowledgements We are very grateful to Fabien Morel and Ivan Panin for an encouraging attention to our work. We would like to thank Skip Garibaldi, Stefan Gille, Nikita Karpenko and Alexander Vishik for many useful comments on the subject of the paper.

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