UPPER MOTIVES OF OUTER ALGEBRAIC GROUPS

NIKITA A. KARPENKO

Abstract. Let G be a semisimple affine algebraic group over a field F. Let E/F be a minimal field extension such that the group G_{E} is of inner type. Assuming that the degree of E/F is a power of a prime p, we determine the structure of the Chow motives with coefficients in a finite field of characteristic p of the projective G-homogeneous varieties. More precisely, it is known that the motive of any such variety decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. This description is already known for the groups G of inner type and is new for G of outer type.

1. Introduction

We fix an arbitrary base field F. Besides of that, we fix a finite field 𝔽 and we consider the Grothendieck Chow motives over F with coefficients in 𝔽. These are the objects of the category CM(F, 𝔽), defined as in [4].

Let G be a semisimple affine algebraic group over F. According to [3, Corollary 35(4)] (see also Corollary 2.2 here), the motive of any projective G-homogeneous variety decomposes (and in a unique way) into a finite direct sum of indecomposable motives. One would like to describe the indecomposable motives which appear this way. In this paper we do it under certain assumption on G formulated in terms of a minimal field extension E/F such that the group G_{E} is of inner type: the degree of E/F is assumed to be a power of p, where p = char 𝔽. (Note that in the case when E = F, that is, when G is of inner type, the problem has been solved in [5].)

Let us introduce a minimum of terminology and notation needed to formulate the answer. For any intermediate field L of the extension E/F and any projective G_{L}- homogeneous variety Y, we consider the upper (see [5, Definition 2.10]) indecomposable summand M_{Y} of the motive M(Y) ∈ CM(F, 𝔽) of Y (considered as an F -variety at this point). Note that L is the constant field (the algebraic closure of the base field in the function field) of the F -variety Y. The set of the isomorphism classes of the motives M_{Y} for all L and Y is called the set of upper motives of the algebraic group G.

We are going to claim that the complete motivic decomposition of any projective Ghomogeneous variety X consists of shifts of upper motives of G. In fact, the information we have is a bit more precise:

Date: June 2009.

Key words and phrases. Algebraic groups, projective homogeneous varieties, Chow groups.

2000 Mathematical Subject Classifications: 14L17; 14C25.

Supported by the Collaborative Research Centre 701 of the Bielefeld University.

Theorem 1.1. For F, G, E, and X as above, the complete motivic decomposition of X consists of shifts of upper motives of the algebraic group G. More precisely, any indecomposable summand of the motive of X is isomorphic the upper motive M_{Y} of a variety Y such that the Tits index of G over the function field of Y contains the Tits index of G over the function field of X_{L}, where L is the constant field of Y.

The proof of Theorem 1.1 is given in §4. Before this, we get some preparation results which are also of independent interest. In §2, we prove the nilpotence principle for the quasi-homogeneous varieties. In §3, we establish some properties of a motivic corestriction functor.

By sum of motives we always mean the direct sum; a summand is a direct summand.

2. Nilpotence principle for quasi-homogeneous varieties

Let us consider the category CM(F, Λ) of Grothendieck Chow motives over a field F with coefficients in an arbitrary associative commutative unital ring Λ.

We say that a smooth complete F -variety X satisfies the nilpotence principle, if for any field extension K/F the kernel of the change of field homomorphism

End (M(X)) → End (M(X_{K}))

consists of nilpotents, where M(X) stands for the motive of X in CM(F, Λ).

We say that an F -variety X is quasi-homogeneous, if each connected component X^{0} of X has the following property: there exists a finite separable field extension L/F, a semisimple affine algebraic group G over L, and a projective G_{L}-homogeneous variety Y such that Y, considered as an F -variety via the composition Y → Spec L → Spec F, is isomorphic to X^{0}. (Note that the algebraic group G needs not to be defined over F in this definition.)

We note that any variety which is projective quasi-homogeneous in the sense of [1, §4] is also quasi-homogeneous in the above sense. The following statement generalizes [2, Theorem 8.2] (see also [1, Theorem 5.1]) and [3, Theorem 25]:

Theorem 2.1. Any quasi-homogeneous variety satisfies the nilpotence principle.

Proof. By [4, Theorem 92.4] it suffices to show that the quasi-homogeneous varieties form a tractable class. We first recall the definition of a tractable class C (over F). This is a disjoint union of classes C_{K} of smooth complete K-varieties, where K runs over all field extensions of F, having the following properties:

(1) if Y_{1} and Y_{2} are in C_{K} for some K, then the disjoint union Y_{1} ∐ Y_{2} is also in C_{K};

(2) if Y is in C_{K} for some K, then each component of Y is also in C_{K};

(3) if Y is in C_{K} for some K, then for any field extension K′/K the K′-variety Y_{K′} is in C_{K′};

(4) if Y is in C_{K} for some K, Y is irreducible, dim Y > 0, and Y (K) ≠ ∅, then C_{K} contains a (not necessarily connected) variety Y_{0} such that dim Y_{0} < dim Y and M(Y) ≃ M(Y_{0}) in CM(K, Λ).

Let us define a class C as follows. For any field extension K/F, C_{K} is the class of all quasi-homogeneous K-varieties.

We claim that the class C is tractable. Indeed, the properties (1)–(3) are trivial and the property (4) is [2, Theorem 7.2]. □

We turn back to the case where the coefficient ring Λ is a finite field 𝔽.

Corollary 2.2. Let M ∈ CM(F, 𝔽) be a summand of the motive of a quasi-homogeneous variety. Then M decomposes in a finite direct sum of indecomposable motives; moreover, such a decomposition is unique (up to a permutation of the summands).

Proof. Any quasi-homogeneous variety is geometrically cellular. In particular, it is geometrically split in the sense of [5, §2a]. Finally, by Theorem 2.1, it satisfies the nilpotence principle. The statement under proof follows now by [5, Corollary 2.6]. □

3. Corestriction of scalars for motives

As in the previous section, we fix an arbitrary coefficient ring Λ. We write Ch for the Chow group with coefficients in Λ. Let C(F, Λ) be the category whose objects are pairs (X, i), where X is a smooth complete equidimensional F -variety and i is an integer. A morphism (X, i) → (Y, j) in this category is an element of the Chow group Ch_{dim X+i−j}(X × Y) (and the composition is the usual composition of correspondences). The category C(F, Λ) is preadditive. Taking first the additive completion of it, and taking then the idempotent completion of the resulting category, one gets the category of motives CM(F, Λ), cf. [4, §63 and §64].

Let L/F be a finite separable field extension. We define a functor

cor_{L/F} : C(L, Λ) → C(F, Λ)

as follows: on the objects cor_{L/F} (X, i) = (X, i), where on the right-hand side X is considered as an F -variety via the composition X → Spec L → Spec F; on the morphisms, the map

Hom_{C(L,Λ)} ((X, i), (Y, j)) → Hom_{C(F,Λ)} ((X, i), (Y, j))

is the push-forward homomorphism Ch_{dim X+i−j}(X ×_{L} Y) → Ch_{dim X+i−j}(X ×_{F} Y) with respect to the closed imbedding X ×_{L} Y ↪ X ×_{F} Y. Passing to the additive completion and then to the idempotent completion, we get an additive and commuting with the Tate shift functor CM(L, Λ) → CM(F, Λ), which we also denote by cor_{L/F}.

The functor cor_{L/F} : C(L, Λ) → C(F, Λ) is left-adjoint and right-adjoint to the change of field functor res_{L/F} : C(F, Λ) → C(L, Λ), associating to (X, i) the object (X_{L}, i). Therefore the functor cor_{L/F} : CM(L, Λ) → CM(F, Λ) is also left-adjoint and right-adjoint to the change of field functor res_{L/F} : CM(F, Λ) → CM(L, Λ). (This makes a funny difference with the category of varieties, where the functor cor_{L/F} is only left-adjoint to res_{L/F}, while the right-adjoint to res_{L/F} functor is the Weil restriction.) It follows that for any M ∈ CM(L, Λ) and any i ∈ ℤ, the Chow groups Ch^{i}(M) and Ch^{i}(cor_{L/F} M) are canonically isomorphic as well as the Chow groups Ch_{i}(M) and Ch_{i}(cor_{L/F} M) are. Indeed, since res_{L/F} Λ = Λ ∈ CM(L, Λ), we have

Ch^{i}(M) := Hom (M, Λ(i)) = Hom ( cor_{L/F} M, Λ(i)) =: Ch^{i}(cor_{L/F} M)  and
Ch_{i}(M) := Hom (Λ(i), M) = Hom (Λ(i), cor_{L/F} M) =: Ch_{i}(cor_{L/F} M).

In particular, if the ring Λ is connected and M ∈ CM(L, Λ) is an upper (see [5, Definition 2.10]) motivic summand of an irreducible smooth complete L-variety X, then cor_{L/F} M is an upper motivic summand of the F -variety X.

Now we turn back to the situation where Λ is a finite field 𝔽:

Proposition 3.1. The following three conditions on a finite galois field extension E/F are equivalent:

(1) for any intermediate field F ⊂ K ⊂ E, the K-motive of SpecE is indecomposable;

(2) for any intermediate fields F ⊂ K ⊂ L ⊂ E and any L-motive M, the K-motive cor_{L/K}(M) is indecomposable;

(3) the degree of E/F is a power of p (where p is the characteristic of the coefficient field 𝔽).

Proof. We start by showing that (3) ⇒ (2). So, we assume that [E : F] is a power of p and we prove (2). The extension L/K decomposes in a finite chain of galois degree p extensions. Therefore we may assume that L/K itself is a galois degree p extension. Let R = End(M). This is an associative, unital, but not necessarily commutative 𝔽algebra. Moreover, since M is indecomposable, the ring R has no non-trivial idempotents. We have End ( cor_{L/K}(M)) = R ⊗_{𝔽} End (M_{K}(Spec L)). According to [3, §7], the ring End (M_{K}(Spec L)) is isomorphic to the group ring 𝔽Γ, where Γ is the Galois group of L/K. Since the group Γ is (cyclic) of order p, we have 𝔽Γ ≃ 𝔽[t]/(t^{p}−1). Since p = char 𝔽, 𝔽[t]/(t^{p} − 1) ≃ 𝔽[t]/(t^{p}). It follows that the ring End ( cor_{L/K}(M)) is isomorphic to the ring R[t]/(t^{p}). We prove (2) by showing that the latter ring does not contain non-trivial idempotents. An arbitrary element of R[t]/(t^{p}) can be (and in a unique way) written in the form a + b, where a ∈ R and b is divisible by t. Note that b is nilpotent. Let us take an idempotent of R[t]/(t^{p}) and write it in the above form a + b. Then a is an idempotent of R. Therefore a = 1 or a = 0. If a = 1, then a + b is invertible and therefore a + b = 1. If a = 0, then a + b is nilpotent and therefore a + b = 0.

We have proved the implication (3) ⇒ (2). The implication (2) ⇒ (1) is trivial. We finish by proving that (1) ⇒ (3).

We assume that [E : F] is divisible by a different from p prime q and we show that (1) does not hold. Indeed, the galois group of E/F contains an element σ of order q. Let K be the subfield of E consisting of the elements of E fixed by σ. We have F ⊂ K ⊂ E and E/K is galois of degree q. The endomorphisms ring of M_{K}(Spec E) is isomorphic to 𝔽[t]/(t^{q} − 1). Since q ≠ char 𝔽, the factors of the decomposition t^{q} − 1 = (t − 1) · (t^{q−1} + t^{q−2} + ⋯ + 1) ∈ 𝔽[t] are coprime. Therefore the ring F [t]/(t^{q} − 1) is the direct product of the rings 𝔽[t]/(t − 1) = 𝔽 and 𝔽[t]/(t^{q−1} + ⋯ + 1), and it follows that the motive M_{K}(Spec E) is not indecomposable. □

Corollary 3.2. Let E/F be a finite p-primary galois field extension and let L be an intermediate field: F ⊂ L ⊂ E. Let M ∈ CM(L, 𝔽) be an upper indecomposable motivic summand of an irreducible smooth complete L-variety X. Then cor_{L/F} M is an upper indecomposable summand of the F -variety X. □

4. Proof of Theorem 1.1

Before starting the proof of Theorem 1.1, let us recall some classical facts and introduce some notation.

We write D (or D_{G}) for the set of vertices of the Dynkin diagram of G. The galois group Γ of the field extension E/F acts on D.

Let L be an intermediate field of the extension E/F. Any subset τ in D which is stable under the action of the galois group of E/L, determines a projective G_{L}-homogeneous variety X_{τ,G_{L}} in the way described in [5, §3]. This is the variety corresponding to the set D \ τ in the sense of [6]. For instance, X_{D,G_{L}} is the variety of the Borel subgroups of G_{L}, and X_{∅,G_{L}} = Spec L. Any projective G_{L}-homogeneous variety is G_{L}-isomorphic to X_{τ,G_{L}} for some τ ⊂ D stable under the action of the galois group of E/L.

For any intermediate field L of E/F and any Gal(E/L)-stable subset τ ⊂ D, we write M_{τ,G_{L}} for the upper indecomposable motivic summand of the F -variety X_{τ,G_{L}}.

For any field extension L/F, we consider the Tits index (see [6]) of G over L and we write τ_{L} (or τ_{L,G}) for the subset in D consisting of the circled vertices of the Dynkin diagram.

Proof of Theorem 1.1. This is a recast of [5, proof of Theorem 3.5].

We proof Theorem 1.1 simultaneously for all F, G, X using an induction on n = dim X. The base of the induction is n = 0 where X = Spec F and the statement is trivial.

From now on we are assuming that n ≥ 1 and that Theorem 1.1 is already proven for all varieties of dimension < n.

Let M be an indecomposable summand of M(X). We have to show that M is isomorphic to a shift of M_{τ,G_{L}} for some intermediate field L of E/F and some Gal(E/L)-stable subset τ ⊂ D_{G} containing τ_{L(X)}.

Let G′/F (X) be the semisimple anisotropic kernel of the group G_{F (X)}. The set D_{G′} is identified with D_{G} \ τ_{F (X),G}.

We note that the group G′_{E(X)} is of inner type. The field extension E(X)/F (X) is galois with the galois group Gal(E/F). In particular, its degree is a power of p and any its intermediate field is of the form L(X) for some intermediate field L of the extension E/F.

According to [1, Theorem 4.2], the motive of X_{F (X)} decomposes into a sum of shifts of motives of projective G′_{L(X)}-homogeneous (where L runs over intermediate fields of the extension E/F) varieties Y, satisfying dim Y < dim X = n (we are using the assumption that n > 0 here). It follows by the induction hypothesis and Corollary 3.2, that each summand of the complete motivic decomposition of X_{F (X)} is a shift of M_{τ′,G′_{L(X)}} for some L and some τ′ ⊂ D_{G′}. The complete decomposition of M_{F (X)} is a part of the above decomposition.

Let us choose a summand M_{τ′,G′_{L(X)}}(i) with minimal i in the complete decomposition of M_{F (X)}. We set τ = τ′ ∪ τ_{L(X)} ⊂ D_{G}. We shall show that M ≃ M_{τ,G_{L}}(i) for these L, τ, and i.

We write Y for the F -variety X_{τ,G_{L}} and we write Y′ for the F (X)-variety X_{τ′,G′_{L(X)}}. We write N for the F -motive M_{τ,G_{L}} and we write N′ for the F (X)-motive M_{τ′,G′_{L(X)}}.

By [5, Lemma 2.14] and since M is indecomposable, it suffices to construct morphisms

α : M(Y)(i) → M and β : M → M(Y)(i)

satisfying mult(β ∘ α) = 1.

We construct α first. Since τ′ ⊂ τ, the F (X)(Y)-variety Y′ ×_{L(X)} Spec F (X)(Y) has a rational point. Let α_{1} ∈ Ch_{0} (Y′ ×_{L(X)} Spec F (X)(Y)) be the class of a rational point. Let α_{2} ∈ Ch_{i}(X_{F (X)(Y)}) be the image of α_{1} under the composition

Ch_{0} (Y′ ×_{L(X)} Spec F (X)(Y)) → Ch_{0}(Y′_{F (X)(Y)}) → Ch_{0}(N′_{F (X)(Y)}) ↪ Ch_{i}(X_{F (X)(Y)}),

where the first map is the push-forward with respect to the closed imbedding

Y′ ×_{L(X)} Spec F (X)(Y) ↪ Y′_{F (X)(Y)} := Y′ ×_{F (X)} Spec F (X)(Y).

Since τ_{L(X)} ⊂ τ, the variety X has an F (Y)-point and therefore the field extension F (X)(Y)/F (Y) is purely transcendental. Consequently, the element α_{2} is F (Y)-rational and lifts to an element α_{3} ∈ Ch_{dim Y +i}(Y × X). We mean here a lifting with respect to the composition

Ch_{dim Y +i}(Y × X) →→ Ch_{i}(X_{F (Y)}) ⟶^{res_{F (X)(Y )/F (Y )}} Ch_{i}(X_{F (X)(Y)})

where the first map is the epimorphism given by the pull-back with respect to the morphism X_{F (Y)} → Y × X induced by the generic point of the variety Y.

We define the morphism α as the composition

M(Y)(i) ⟶^{α_{3}} M(X) ⟶ M

where the second map is the projection of M(X) onto its summand M.

We proceed by constructing β. Let β_{1} ∈ Ch_{dim Y′}(Y′ ×_{F (X)} Y_{F (X)}) be the class of the closure of the graph of a rational map of L(X)-varieties Y′ −−→ Y_{F (X)} (which exists because τ ⊂ τ_{L(X)} ∪ τ′). Note that this graph is a subset of Y′ ×_{L(X)} Y_{F (X)}, which we consider as a subset of Y′ ×_{F (X)} Y_{F (X)} via the closed imbedding Y′ ×_{L(X)} Y_{F (X)} ↪ Y′ ×_{F (X)} Y_{F (X)}. Let β_{2} be the image of β_{1} under the composition

Ch^{dim Y} (Y′ ×_{F (X)} Y_{F (X)}) =

Ch^{dim Y} (M(Y′) ⊗ M(Y_{F (X)})) → Ch^{dim Y} (N′ ⊗ M(Y_{F (X)})) →

Ch^{dim Y +i} (M(X_{F (X)}) ⊗ M(Y_{F (X)})) = Ch^{dim Y +i} ((X × Y)_{F (X)})

where the first arrow is induced by the projection M(Y′) → N′ and the second arrow is induced by the imbedding N′(i) → M(X_{F (X)}). The element β_{2} lifts to an element

β_{3} ∈ Ch^{dim Y +i}(X × X × Y).

We mean here a lifting with respect to the epimorphism

Ch^{dim Y +i}(X × X × Y) →→ Ch^{dim Y +i} ((X × Y)_{F (X)})

given by the pull-back with respect to the morphism X × X × Y → (X × Y)_{F (X)} induced by the generic point of the second factor in this triple direct product.

Let π ∈ Ch_{dim X}(X × X) be the projector defining the summand M of M(X). Considering β_{3} as a correspondence from X to X × Y, we define

β_{4} ∈ Ch^{dim Y +i}(X × X × Y)

as the composition β_{3} ∘ π. We get

β_{5} ∈ Ch^{dim Y +i}(X × Y) = Ch_{dim X−i}(X × Y)

as the image of β_{4} under the pull-back with respect to the diagonal of X. Finally, we define the morphism β as the composition

M ⟶ M(X) ⟶^{β_{5}} M(Y)(i).

The verification of the relation mult(β ∘ α) = 1, finishing the proof, is similar to that of [5, proof of Theorem 3.5]. □

Remark 4.1. Theorem 1.1 can be also proved using a weaker result in place of [1, Theorem 4.2], namely, [2, Theorem 7.5].

References

[1] Brosnan, P. On motivic decompositions arising from the method of Białynicki-Birula. Invent. Math. 161, 1 (2005), 91–111.

[2] Chernousov, V., Gille, S., and Merkurjev, A. Motivic decomposition of isotropic projective homogeneous varieties. Duke Math. J. 126, 1 (2005), 137–159.

[3] Chernousov, V., and Merkurjev, A. Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem. Transform. Groups 11, 3 (2006), 371–386.

[4] Elman, R., Karpenko, N., and Merkurjev, A. The algebraic and geometric theory of quadratic forms, vol. 56 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2008.

[5] Karpenko, N. A. Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties. Linear Algebraic Groups and Related Structures (preprint server) 333 (2009, Apr 3, revised: 2009, Apr 24), 18 pages.

[6] Tits, J. Classification of algebraic semisimple groups. In Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965). Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62.

UPMC Univ Paris 06, Institut de Mathématiques de Jussieu, F-75252 Paris, FRANCE

Web page: www.math.jussieu.fr/~karpenko
E-mail address: karpenko at math.jussieu.fr