Motivic Splitting Lemma

A. Vishik, K. Zainoulline^{1}

Received: September 7, 2007 Revised: March 25, 2008

Communicated by Ulf Rehmann

Abstract. Let M be a Chow motive over a field F. Let X be a smooth projective variety over F and N be a direct summand of the motive of X. Assume that over the generic point of X the motives M and N become isomorphic to a direct sum of twisted Tate motives. The main result of the paper says that if a morphism f : M → N splits over the generic point of X then it splits over F, i.e., N is a direct summand of M. We apply this result to various examples of motives of projective homogeneous varieties.

2000 Mathematics Subject Classification:

Keywords and Phrases: Chow motive, homogeneous variety

1 Introduction

By a variety X over a field F we always mean a reduced and irreducible scheme of finite type over F. By F (X) we denote the function field of X.

Definition 1.1. Let M be a Chow motive over F. We say M is split over F if it is a direct sum of twisted Tate motives over F. We say a motive M is generically split if there exists a smooth projective variety X over F and an integer l such that M is a direct summand of the twisted motive M(X){l} of X and M is split over F (X). In particular, a smooth projective variety X is called generically split if its Chow motive M(X) is split over F (X).

The classical examples of such varieties are Severi-Brauer varieties, Pfister quadrics and maximal orthogonal Grassmannians. In the present paper we provide useful technical tool to study motivic decompositions of generically split varieties (motives). Namely, we prove the following Theorem 1.2. Let M be a Chow motive over a field F. Let X be a smooth projective variety over F and N be a direct summand of the motive of X. Assume that M and N are split over F (X). Then a morphism M → N splits, i.e. N is a direct summand of M, if it splits over F (X).

1Partially supported by SFB 701, INTAS 05-1000008-8118 and DFG GI 706/1-1.

Documenta Mathematica 13 (2008) 81–96

The paper is organized as follows. In section 2 we introduce the category of Chow motives over a relative base. In section 3 we provide the version of the Rost nilpotence theorem for generically split varieties. In section 4 we prove the main result of this paper (see the above theorem). The last section is devoted to various applications and examples.

2 Chow motives over a relative base

Let X be a variety over a field F. We say X is essentially smooth over F if it is an inverse limit of smooth varieties X_{i} over F taken with respect to open embeddings. Let CH^{m}(X; Λ) = CH^{m}(X) ⊗_{ℤ} Λ denote the Chow group of codimension m cycles on X with coefficients in a commutative ring Λ. If X is essentially smooth, then CH^{m}(X; Λ) = \underrightarrow{lim} CH^{m}(X_{i}; Λ), where the limit is taken with respect to the pull-backs induced by open embeddings. In the present section we introduce the category of Chow motives over an essentially smooth variety X with Λ-coefficients. Our arguments follow the paper [9].

I. First, we define the category of correspondences C(X; Λ). The objects of C(X; Λ) are smooth projective maps Y → X. The morphisms are given by

Hom([Y → X], [Z → X]) = ⊕_{i} CH^{dim(Z_{i}/X)}(Y ×_{X} Z_{i}; Λ),

where the sum is taken over all irreducible components Z_{i} of Z of relative dimensions dim(Z_{i}/X). The composition of two morphisms is given by the usual correspondence product

ψ ∘ ϕ = (p_{Y,T})_{∗}((p_{Y,Z})^{∗}(ϕ) · (p_{Z,T})^{∗}(ψ)),

where ϕ ∈ Hom([Y → X], [Z → X]), ψ ∈ Hom([Z → X], [T → X]) and p_{Y,T}, p_{Y,Z}, p_{Z,T} are projections Y ×_{X} Z ×_{X} T → Y ×_{X} T, Y ×_{X} Z, Z ×_{X} T. The category C(X; Λ) is a tensor additive category, where the direct sum is given by [Y → X] ⊕ [Z → X] := [Y ∐ Z → X] and the tensor product by [Y → X] ⊗ [Z → X] := [Y ×_{X} Z → X] (cf. [9, §2-4]). As usual we denote by ϕ^{t} ∈ CH(Z ×_{X} Y) the transposition of a cycle ϕ ∈ CH(Y ×_{X} Z). The category of effective Chow motives Chow^{eff}(X; Λ) can be defined as the pseudo-abelian completion of C(X; Λ). Namely, the objects are pairs (U, ρ), where U is an object of C(X; Λ) and ρ ∈ End_{C(X;Λ)}(U) is a projector, i.e. ρ ∘ ρ = ρ. The morphisms between (U_{1}, ρ_{1}) and (U_{2}, ρ_{2}) are given by the group ρ_{2} ∘ Hom_{C(X;Λ)}(U_{1}, U_{2}) ∘ ρ_{1}. The composition of morphisms is induced by the correspondence product. In the case X = Spec(F) and Λ = ℤ we obtain the usual category Chow^{eff}(F) of effective Chow motives over F with integral coefficients (cf. [9, §5]). Consider the projective line ℙ^{1} over F. The projector ρ = [Spec(F) × ℙ^{1}] ∈ CH^{1}(ℙ^{1} × ℙ^{1}) defines an object (ℙ^{1}, ρ) in Chow^{eff}(F) called the Tate motive over F and denoted by ℤ{1} (cf. [9, §6]).

II. We have two types of restriction functors.

1) For any morphism f: X_{1} → X_{2} of essentially smooth varieties we have a tensor additive functor

res_{X_{2}/X_{1}} : C(X_{2}; Λ) → C(X_{1}; Λ)

given on the objects by [Y_{2} → X_{2}] ↦ [Y_{2} ×_{X_{2}} X_{1} → X_{1}] and on the morphisms by ϕ ↦ (id × f)^{∗}(ϕ), where id × f : (Y_{2} ×_{X_{2}} Z_{2}) ×_{X_{2}} X_{1} → Y_{2} ×_{X_{2}} Z_{2} is the natural map. It induces a functor on pseudo-abelian completions

res_{X_{2}/X_{1}} : Chow^{eff}(X_{2}; Λ) → Chow^{eff}(X_{1}; Λ).

2) For any homomorphism of commutative rings h: Λ → Λ′ we have a tensor additive functor

res_{Λ′/Λ} : C(X, Λ) → C(X; Λ′)

which is identical on objects and is given by id ⊗ h: CH(Y ×_{X} Z; Λ) → CH(Y ×_{X} Z; Λ′) on morphisms. Again, it induces a functor on pseudo-abelian completions

res_{Λ′/Λ} : Chow^{eff}(X; Λ) → Chow^{eff}(X; Λ′).

Observe that the functor res_{Λ′/Λ} commutes with res_{X_{2}/X_{1}}. We denote by res_{X_{2}/X_{1},Λ′/Λ} the composite res_{X_{2}/X_{1}} ∘ res_{Λ′/Λ}. To simplify the notation we omit X_{2} (resp. Λ), if X_{2} = Spec F (resp. Λ = ℤ). Let f: X → Spec F and h: ℤ → Λ be the structure maps. Then res_{X,Λ} : Chow^{eff}(F) → Chow^{eff}(X; Λ). Given a motive N over F we denote by N_{X,Λ} its image res_{X,Λ}(N) in Chow^{eff}(X; Λ). The image ℤ{1}_{X,Λ} of the Tate motive is denoted by T and is called the Tate motive over X. Let M be a motive from Chow^{eff}(X; Λ) and l ≥ 0 be an integer. The tensor product M ⊗ T^{⊗l} is denoted by M{l} and is called the twist of M. The trivial Tate motive T^{⊗0} will be denoted Λ (thus, T^{⊗l} = Λ{l}). The same arguments as in the proof of [9, Lemma of §8] show that for any motives U and V from Chow^{eff}(X; Λ) and l ≥ 0 the natural map

Hom_{Chow^{eff} (X;Λ)}(U, V) → Hom_{Chow^{eff} (X;Λ)}(U{l}, V {l})  (1)

given by ϕ ↦ ϕ ⊗ id_{T} is an isomorphism.

III. We define the category Chow(X; Λ) of Chow motives over X with Λcoefficients as follows. The objects are pairs (U, l), where U is an object of Chow^{eff}(X; Λ) and l is an integer. The morphisms are given by

Hom((U, l), (V, m)) := lim_{N→+∞} Hom_{Chow^{eff} (X;A)}(U{N + l}, V {N + m}).

This is again a tensor additive category, where the sum and the product are given by

(U, l) ⊕ (V, m) := (U{l − n} ⊕ V {m − n}, n), where n = min(l, m),
(U, l) ⊗ (V, m) := (U ⊗ V, l + m).

Observe that the Tate motive T is isomorphic to ([id : X → X], 1) and, hence, it is invertible in (Chow(X; Λ), ⊗). Moreover, we can say that Chow(X; Λ) is obtained from Chow^{eff}(X; Λ) by inverting T (cf. [9, §8]). According to (1) the natural functor Chow^{eff}(X; Λ) → Chow(X; Λ) given by U ↦ (U, 0) is fully faithful and the restriction res_{X,Λ} descend to the respective functor res_{X,Λ} : Chow(F) → Chow(X; Λ). For a smooth projective morphism Y → X we denote by M(Y → X) its effective motive ([Y → X], id) considered as an object of Chow(X; Λ). If X = Spec F and Λ = ℤ, then we denote the motive M(Y → X) simply by M(Y). By definition there is a natural identification

Hom_{Chow(X;Λ)}(M(Y → X){i}, M(Z →X){j}) = CH^{dim(Z/X)+j−i}(Y ×_{X} Z; Λ).

IV. Let M be an object of Chow(X; Λ). We define the Chow group with low index CH_{m}(M; Λ) of M as

CH_{m}(M; Λ) := Hom_{Chow(X;Λ)}(Λ{m}, M)

and the Chow group with upper index CH^{m}(M; Λ) as

CH^{m}(M; Λ) := Hom_{Chow(X;Λ)}(M, Λ{m}).

Observe that if M = M(Y → X), then we obtain the usual Chow groups CH^{dim(Y/X)−m}(Y; Λ) and CH^{m}(Y; Λ) of a variety Y. A composite with a morphism f : M → N induces a homomorphism between the Chow groups R_{m}(f): CH_{m}(M) → CH_{m}(N) and R^{m}(f): CH^{m}(N) → CH^{m}(M) called the realization map.

3 The Rost nilpotence

We will extensively use the following version of the Rost nilpotence (cf. [14, Proposition 9])

Proposition 3.1. Let N be a generically split motive over a field F. Then for any field extension E/F and any coefficient ring Λ the kernel of the restriction

res_{E/F} : End_{F} (N) → End_{E}(N_{E})

consists of nilpotents.

To simplify the notation we denote by End_{X}(M) the endomorphism group Hom_{Chow(X;Λ)}(M, M), where M is a motive over a variety X.

Proof. Recall that (see Definition 1.1) a motive N over F is generically split if there exists a smooth projective variety X and l ∈ ℤ such that N is a direct summand of M(X){l} and N_{K} = res_{K/F} (N) is split, where K = F (X) denotes the function field of X. We may assume that N is a direct summand of M(X) (that is, l = 0). Since for a split motive M and a field extension E/L, the map End_{L}(M_{L}) → End_{E}(M_{E}) is an isomorphism, we may assume that E = K. Consider the composite of ring homomorphisms

res_{K/F} : End_{F} (N) ⟶^{res_{X/F}} End_{X}(N_{X}) ⟶^{res_{K/X}} End_{K}(N_{K}),

where the last map is induced by passing to the generic point Spec K → X. Observe that End_{K}(N_{K}) = \underrightarrow{lim} End_{U}(N_{U}), where the limit is taken over all open subvarieties U ⊂ X. Then ker(res_{K/X}) = ∪_{U} ker(res_{U/X}) and by Lemma 3.2 the kernel of res_{K/X} consists of nilpotents. On the other hand, the map res_{X/F} is injective. Indeed, since N is a direct summand of M(X), End_{F} (N) is a subring of End_{F} (M(X)) and End_{X}(N_{X}) is a subring of End_{X}(M(X)_{X}). So, it is sufficient to prove the injectivity for the case N = M(X). The restriction res_{X/F} : End_{F} (M(X)) → End_{X}(M(X)_{X}) coincides with the pull-back π_{1,2}^{∗}: CH(X ×X; Λ) → CH(X ×X ×X; Λ) induced by the projection on the first two coordinates. And π_{1,2}^{∗} splits by (id_{X} × Δ_{X})^{∗}: CH(X × X × X; Λ) → CH(X × X; Λ), where Δ_{X}: X → X × X is the diagonal. The proposition is proven.

Lemma 3.2. Let X be a smooth projective variety over F and Λ be a commutative ring. Let U ⊂ X be an open embedding. Then for any motive M from Chow(X; Λ) the kernel of the restriction map

res_{U/X} : End_{X}(M) → End_{U}(M_{U})

consists of nilpotents.

Proof. If M is a direct summand of [Y → X]{i}, then End_{X}(M) is a subring of End_{X}(M(Y → X)) and it is sufficient to study the case M = M(Y → X). Recall that End_{X}(M(Y → X)) = CH^{dim(Y)−dim(X)}(Y ×_{X} Y; Λ). Let ϕ be an element from the kernel of res_{U/X}. Let j: Z → X be the reduced closed complement to U in X. Then by the localization sequence for Chow groups the cycle ϕ belongs to the image of the induced push-forward

(id_{(Y ×_{X}Y)} × j)_{∗}: CH((Y ×_{X} Y) ×_{X} Z; Λ) → CH(Y ×_{X} Y; Λ).

Let codim(Z) be the minimum of codimensions of irreducible components of Z, and d := [\frac{dim(X)}{codim(Z)}]+1. We claim that the d-th power ϕ^{∘d} of ϕ taken with respect to the correspondence product is trivial. Indeed, ϕ^{∘d} = (π_{1,d+1})_{∗}(ϕ_{1}·ϕ_{2}·. . .·ϕ_{d}), where ϕ_{i} = π_{i,i+1}^{∗}(ϕ) and the map π_{i,i′}: Y^{×(d+1)} → Y ×_{X} Y is the projection on the i-th and i′-th components. Since π_{i,i′}^{∗} ∘ (id_{(Y ×_{X}Y)} × j)_{∗} coincides with (id_{Y^{×(d+1)}} × j)_{∗} ∘ (π_{i,i′} × id_{Z})^{∗}, all cycles ϕ_{i} belong to the image of the pushforward

(id_{Y^{×(d+1)}} × j)_{∗}: CH(Y^{×(d+1)} ×_{X} Z) → CH(Y^{×(d+1)}).

By Proposition 6.1 applied to the projection Y^{×(d+1)} → X and the closed embedding j: Z ↪ X we obtain that the product

ϕ_{1} · . . . · ϕ_{d} ∈ ((id_{Y^{×(d+1)}} × j)_{∗} CH(Y^{×(d+1)} ×_{X} Z))^{d}

is trivial. Therefore, ϕ^{∘d} is trivial as well.

We finish this section with the following

Definition 3.3. Given motive M over a field F and a field extension L/F we say a cycle in CH(M_{L}) is rational if it is in the image of the restriction map res_{L/F}.

Observe that the rationality of cycles is preserved by push-forward and pullback maps. It also respects addition, intersection and correspondence product of cycles.

4 Motivic splitting lemma

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

Theorem 4.1. Let M be a Chow motive over a field F. Let X be a smooth projective variety over F and N be a direct summand of the motive of X. Assume that M and N are split over the function field K = F (X). Then a morphism f: M → N splits, i.e. N is a direct summand of M, if it splits over K.

Proof. To construct a section of f we apply recursively the following procedure starting from g = 0 and such m that CH^{i}(N_{K}) = 0, for i < m.

For a morphism g : N → M such that the realization morphism R^{i}(f_{K} ∘ g_{K}) is the identity on CH^{i}(N_{K}) for i < m, we construct a new morphism g′ : N → M such that R^{i}(f_{K} ∘ g′_{K}) is the identity on CH^{i}(N_{K}) for i ≤ m.

Since the motive N_{K} splits, for the corresponding projector ρ_{N} over K we may write (ρ_{N})_{K} = ∑_{l} ω_{l} × ω_{l}^{∨} for certain ω_{l} ∈ CH^{∗}(X_{K}) and ω_{l}^{∨} ∈ CH_{∗}(X_{K}) such that deg(ω_{l} · ω_{m}^{∨}) = δ_{l,m}. Elements ω_{l} form a basis of CH^{∗}(N_{K}) = (ρ_{N})_{K} ∘ CH^{∗}(X_{K}) ⊂ CH^{∗}(X_{K}). Consider the surjection CH^{m}(X × X) ↠ CH^{m}(K ×_{F} X) = CH^{m}(X_{K}). Let Ω_{l} be a preimage of an element ω_{l} of CH^{m}(X_{K}). Consider the difference id − f ∘ g and denote it by h. Assume that over K it sends a basis element ω_{j} to a cycle α_{j}. Since R^{i}(h_{K}) is trivial for all i < m, the cycle h_{K} = h_{K} ∘ (ρ_{N})_{K} can be written as

h_{K} = ∑_{codim α_{l}=m} α_{l} × ω_{l}^{∨} + ∑_{codim α_{j}>m} α_{j} × ω_{j}^{∨} ∈ CH^{dim X}(X_{K} × X_{K}). (2)

From (2) we immediately see that

α_{l} = pr_{1}_{∗}(Ω_{l,K} · h_{K}) ∈ CH^{m}(X_{K}) is rational.  (3)

Also, α_{l} ∘ (ρ_{N})_{K} = α_{l}. The realization R^{m}(f_{K}) is a ℤ-linear map CH^{m}(N_{K}) → CH^{m}(M_{K}). Let C = (c_{ij}) be the respective matrix of coefficients, i.e.,

R^{m}(f_{K}) : ω_{i} ↦ ∑_{j} c_{ji}θ_{j},

where {θ_{i}} is a ℤ-basis of CH^{m}(M_{K}). Let s : N_{K} → M_{K} be a section of f_{K}. The realization map R^{m}(s) is a left inverse to R^{m}(f_{K}). Hence, for the respective matrix of coefficients D = (d_{ij}) we have

R^{m}(s) : θ_{i} ↦ ∑_{j} d_{ji}ω_{j}

and D · C = id, i.e., ∑_{j} d_{ij}c_{jk} = δ_{ik}. For each α_{l} define the morphism u_{l} : N → M as

u_{l} = ∑_{i} d_{li}Θ_{i}^{∨} ∘ (pr_{1}^{∗}(α_{l}) · Δ_{X}) ∘ p_{N},

where Θ_{i}^{∨} is a preimage of an element θ_{i}^{∨} of CH_{m}(M_{K}) by means of the canonical surjection Hom_{F} (M(X)(m)[2m], M) → CH_{m}(M_{K}) and p_{N} : N → M(X) be the morphism presenting N as a direct summand of M(X). By definition, u_{l} is a rational morphism and the realization R^{m}(u_{l}) is given by

θ_{i} ↦ d_{li}α_{l}

Hence, the composite R^{m}(f ∘ u_{l}) = R^{m}(u_{l}) ∘ R^{m}(f) maps ω_{i} to δ_{il}α_{l}. Set g̃ = g + ∑_{l} u_{l}. By construction, the realization R(f ∘ g̃) is the identity on CH^{i}(N_{K}) for i ≤ m. Consider the endomorphism id − f ∘ g̃ of N. Over K its realization R^{i}(id − f ∘ g̃) is trivial for each i ≤ m. Recursion step is proven and we obtain map g′ : N → M such that (f ∘ g′)_{K} = id_{N_{K}}. Let q = id − f ∘ g′. By the Proposition 3.1, q^{r} = 0, for some r. Set g = g′ ∘ (id + q + q^{∘2} + … + q^{∘(r−1)}). Then f ∘ g = id_{N} and N is a direct summand of M.

5 Examples and Applications

Geometric construction of a generalized Rost motive. Let p be a prime and F be a field of characteristic different from p. Let n be a positive integer. To each nonzero cyclic subgroup ⟨α⟩ in K_{n}^{M}(F)/p consisting of pure symbols one can assign some motive M_{α} in the category Chow(F; ℤ/pℤ), which satisfies the following property For an arbitrary field extension E/F

α|_{E} ≠ 0 ⟺ (M_{α})_{E} = res_{E/F} (M_{α}) is indecomposable;
α|_{E} = 0 ⟺ (M_{α})_{E} is split.

It follows from the results of V. Voevodsky and M. Rost that for a given subgroup such motive always exists and is unique (see [17, § 5] and [15, Prop. 5.9]). Moreover, when split it is isomorphic to

⨁_{i=0}^{p−1} ℤ/pℤ\{i · \frac{p^{n−1}−1}{p−1}\}.

Such a motive is called a generalized Rost motive with ℤ/pℤ-coefficients.

Definition 5.1. A motive with integral coefficients which specializes modulo p into a generalized Rost motive and splits modulo q for every prime q different from p will be called an integral generalized Rost motive and denoted by R_{n,p}.

Integral generalized Rost motives, hypothetically, should be parameterized not by the pure cyclic subgroups of K_{n}^{M}(F)/p, but by the pure symbols of K_{n}^{M}(F)/p up to a sign. The existence of integral generalized Rost motives is known for n = 2 and arbitrary p, for p = 2 and arbitrary n, and for the pair n = 3, p = 3. All these examples are essentially due to M. Rost. As the first application of Theorem 4.1 we obtain the construction of the classical integral Rost motive corresponding to a Pfister form.

Corollary 5.2. (cf. [14, Theorem 17.(9) and Proposition 19]) Let X be a hyperplane section of a n-fold Pfister quadric Y over a field F. Then M(Y) ≃ M(X){1} ⊕ R_{n,2}, where R_{n,2} is an integral Rost motive.

Proof. In the proof we use several auxiliary facts concerning quadrics and their motives which can be found in [5]. Let ϕ_{X} and ϕ_{Y} be the quadratic forms which define X and Y. By definition ϕ_{X} is a subform of codimension 1 of the Pfister form ϕ_{Y}. According to [5, Def.5.1.2 and Thm.5.3.4.(a)] Y becomes isotropic over K = F (X). This fact together with [5, Prop.4.2.1] implies that both ϕ_{X} and ϕ_{Y} become totally split (hyperbolic) over K. Then by [5, E.10.8] the motives M(X)_{K} and M(Y)_{K} are split over K. Let Γ_{e} be the graph of the closed embedding e: X ↪ Y. The respective correspondence cycle [Γ_{e}] ∈ CH_{dim X}(X × Y) induces the realization map R^{∗}(Γ_{e}) which coincides with the pull-back e^{∗}: CH^{∗}(Y) → CH^{∗}(X) (see §2.IV). It is known that the Chow ring of a hyperbolic quadric is generated by two elements ⟨h, l⟩, where h is the class of a hyperplane section and l is the class of a maximal totally isotropic subspace. In this notation the pull-back e_{K}^{∗} maps h_{Y} ↦ h_{X} and l_{Y} ↦ l_{X}, i.e. maps the ring CH(Y_{K}) onto the ring CH(X_{K}). The latter means that R^{∗}(Γ_{e}) and, therefore, the transposed correspondence cycle [Γ_{e}]^{t} ∈ CH_{dim Y −1}(Y × X) have a section over K. Take f = [Γ_{e}]^{t}: M(Y) → M(X){1} and apply Theorem 1.2. We obtain the decomposition M(Y) = M(X){1} ⊕ N, where N is such that

N_{K} = ℤ ⊕ ℤ{2^{n−1} − 1}.

Let E/F be a field extension. The Pfister quadric Y corresponds to some pure symbol α ∈ K_{n}^{M}(F)/2 (see [5, §9.4]) with the property that α|_{E} = 0 if and only if Y_{E} has a rational point. Consider the specialization N_{E,ℤ/2} with ℤ/2-coefficients. We have the following chain of equivalences: N_{E,ℤ/2} is decomposable ⇔ N_{E,ℤ/2} contains ℤ/2 as a direct summand ⇔ M(Y; ℤ/2)_{E} contains ℤ/2 as a direct summand ⇔ (see [14, §1.4]) Y_{E} has a zero-cycle of odd degree ⇔ (Springer Theorem) Y_{E} has a rational point. At the same time, the specialization N_{ℤ/p} is split for any odd prime p, since M(Y; ℤ/p) is split. Hence, N is an integral generalized Rost motive corresponding to the symbol α.

To provide the next application we use several auxiliary facts concerning Albert algebras and Cayley planes which can be found in [4], [8], [11], [12]. We use the notation of [12, §3]. Consider an Albert algebra J defined by means of the first Tits construction. Let F_{4}(J) and E_{6}(J) denote the respective simple groups of types F_{4} and E_{6}. Let X be the variety of maximal parabolic subgroups of F_{4}(J) of type P_{4}. Let Y be the variety of maximal parabolic subgroups of E_{6}(J) of type P_{1} Here P_{i} corresponds to a standard parabolic subgroup generated by the Borel subgroup and all unipotent subgroups corresponding to linear spans of all simple roots with no i-th terms (our enumeration of roots follows Bourbaki). The variety Y is called a (twisted) Cayley plane. Observe that there is a closed embedding e: X ↪ Y such that over the splitting field K of J the class [X_{K}] ∈ Pic Y_{K} generates the Picard group of Y_{K}. In other words, X_{K} is a hyperplane section of Y_{K} (see [8, 6.3]).

Corollary 5.3. Let X and Y be as above. Then M(Y) ≃ M(X){1} ⊕ R_{3,3}, where R_{3,3} is an integral generalized Rost motive corresponding to the Serre-Rost invariant g_{3}(J) in K_{3}^{M}(F)/3.

Proof. We follow the previous proof step by step. Let K denote the function field of X. Analyzing the Tits indices of F_{4}(J) we conclude that J becomes reduced over K. Moreover, since J is defined by means of the first Tits construction, J becomes split over K. By definition it implies that both groups and varieties become split over K.

Consider now the graph Γ_{e} of the closed embedding e: X ↪ Y. As before, the respective correspondence cycle [Γ_{e}] induces the realization map R^{∗}(Γ_{e}) which coincides with the pull-back e^{∗}. The Chow rings CH(X_{K}) and CH(Y_{K}) are generated by ⟨h, g_{1}^{4}⟩ (see [10, 4.10]) and ⟨H, σ′_{4}, σ_{8}⟩ (see [4, 5.1]). By the Lefschetz hyperplane theorem the pull-back e^{∗} has to be an isomorphisms on all graded components of codimensions ≤ 7. This immediately implies that e^{∗} maps H ↦ h and σ′_{4} ↦ g_{1}^{4}, i.e. maps the ring CH(Y_{K}) onto the ring CH(X_{K}). So R^{∗}(Γ_{e}) and, therefore, the transposed cycle [Γ_{e}]^{t} have a section over K. Take f = [Γ_{e}]^{t}: M(Y) → M(X){1} and apply Theorem 1.2. We obtain the decomposition M(Y) = M(X){1} ⊕ N, where the motive N is such that

N_{K} = ℤ ⊕ ℤ{4} ⊕ ℤ{8}.

Let E/F be a field extension. Let α = g_{3}(J) ∈ K_{3}^{M}(F)/3 be the Serre-Rost invariant of the Jordan algebra J (see [13]). Analyzing the Tits indices of E_{6}(J) we see that α|_{E} = 0 if and only if Y_{E} has a zero-cycle of degree coprime to 3. Consider the specialization N_{E,ℤ/3} with ℤ/3-coefficients. Similar to the quadric case there is a chain of equivalences which says that N_{E,ℤ/3} is decomposable ⇔ Y_{E} has a zero-cycle of degree coprime to 3. At the same time, the specialization N_{ℤ/p} is split for any prime p ≠ 3, since M(Y; ℤ/p) is split. Therefore, N is an integral generalized Rost motive corresponding to the symbol α.

Remark 1. Observe that in view of the main result of [10] we obtain the following decomposition

M(Y) ≃ ⨁_{i=0}^{8} R_{3,3}{i}.

So from the motivic point of view the variety Y is a 3-analog of a Pfister quadric.

Projective homogeneous varieties of type F_{4}. As before let J be an Albert algebra defined by means of the first Tits construction. Let F_{4}(J) be the respective group of type F_{4}. Let X be the same as before, i.e. the variety of maximal parabolic subgroups of type P_{4} of F_{4}(J). Let Y be the variety of maximal parabolic subgroups of type P_{3} of F_{4}(J). Observe that Y has dimension 20.

Corollary 5.4. Let X and Y be as above. Then the motive M(X; ℤ) is isomorphic to a direct summand of the motive M(Y; ℤ).

Proof. Since the Albert algebra J splits over the function field K of X, the motives M(X) and M(Y) become split over K as well. By the main result of [10] M(X) splits as

M(X) ≃ ⨁_{i=0}^{7} R_{3,3}{i},

where R_{3,3} is the integral generalized Rost motive corresponding to g_{3}(J). Let Z be the variety of parabolic subgroups of type P_{3,4} of F_{4}(J). Observe that Z has dimension 21 and there is a map pr_{XY} = (pr_{X}, pr_{X}): Z → X ×Y, where pr_{X}, pr_{Y} are the quotient maps. For each i = 0 … 7 consider the composite

f_{α_{i}} : M(Y) ⟶^{pr_{XY ∗}(α_{i})} M(X) → R_{3,3}{i}, where α_{i} ∈ Pic Z.

Set f = ⨁_{i=0}^{7} f_{α_{i}}: M(Y) → M(X). Assume that we can choose α_{i} ∈ Pic Z in such a way that the realization map R^{∗}(f) becomes split injective over K. Then by Theorem 1.2 applied f, the motive M(X) is isomorphic to a direct summand of M(Y). So to prove the corollary it is enough to find α_{i} ∈ Pic Z, i = 0 … 7, such that R^{∗}(f) is split injective over K. Observe that the restriction map res_{K/F} : Pic Z → Pic Z_{K} is an isomorphism (see [10, Lemma 4.3]). Therefore, we may assume that α_{i} ∈ Pic Z_{K}. Observe also that the ring structures of CH(X_{K}), CH(Y_{K}) and CH(Z_{K}) are known. We have R^{∗}(f_{α_{i}})_{K} = R^{∗}(α_{i})_{K} ∘ R^{∗}(ρ_{i})_{K}, where ρ_{i} is an idempotent defining R_{3,3}{i}. Both realizations R^{∗}(ρ_{i})_{K} and R^{∗}(α_{i})_{K} can be described explicitly on generators. Indeed, the realization R^{∗}(α_{i})_{K} is given by the composite

CH(X_{K}) ⟶^{pr_{X}^{∗}} CH(Z_{K}) ⟶^{·α_{i}} CH(Z_{K}) ⟶^{pr_{Y ∗}} CH(Y_{K}), where the maps pr_{X}^{∗} and pr_{Y ∗} can be described using [10, §3]. The explicit description of the cycles (ρ_{i})_{K} is provided in [10, 5.5]. Let {α_{i} = c_{1i}g_{1}+c_{2i}g_{2}}_{i=0...7}, c_{1i}, c_{2i} ∈ ℤ, be the presentation of the cycles α_{i} in terms of a fixed ℤ-basis ⟨g_{1}, g_{2}⟩ of Pic Z_{K}. Since all realization maps R^{∗}(α_{i})_{K}, R^{∗}(ρ_{i})_{K} are ℤ-linear, the question of split injectivity of R^{∗}(f)_{K} translates into the problem of solving certain system of ℤ-linear equations in 16 variables {c_{1i}, c_{2i}}_{i=0...7}. Direct computations show that this system has a solution. This finishes the proof of the corollary.

Twisted forms of Grassmannians. Consider a Grassmannian 𝔾(d, n) of d-dimensional planes in a n-dimensional affine space. Its twisted form is called a generalized Severi-Brauer variety and denoted by SB_{d}(A), where A is the respective central simple algebra of degree n (see [7, §1.C]). The next corollary relates the motive of a generalized Severi-Brauer variety with the motive of usual Severi-Brauer variety.

Corollary 5.5. Let A and B be two central division algebras of degree n with [A] = ±d[B] in the Brauer group Br(F), where d and n are coprime. Then the motive of the Severi-Brauer variety SB(A) is a direct summand in the motive of the generalized Severi-Brauer variety SB_{d}(B).

Proof. We construct the morphism f : M(SB_{d}(B)) → M(SB(A)) as follows. Consider the Plücker embedding pl : SB_{d}(B) → SB(Λ^{d}B). It induces the morphism M(SB_{d}(B)) → M(SB(Λ^{d}B)), where Λ^{d}B is the d-th lambda power of B (see [7, II.10.A]). By [6, Cor. 1.3.2] the motive M(SB(Λ^{d}B)) splits as a direct sum of shifted copies of M(SB(A)), where [A] = d[B] in Br(F). Take f to be the composite of the Plücker embedding and the projection M(SB(Λ^{d}B)) → M(SB(A)). We claim that f has a section (splits) over the generic point of SB(A). Indeed, it is equivalent to the fact that for each m = 0, …, n − 1

g.c.d._{i}(c_{i}^{(m)}) = 1

where c_{i}^{(m)} are degrees of the Schubert varieties generating CH^{m}(𝔾(d, n)). The latter can be computed using explicit formulas for degrees of Schubert varieties provided for instance in [3, Ch. 14, Ex. 14.7.11.(ii)]. Then by Theorem 1.2 the motive M(SB(A)) is a direct summand in M(SB_{d}(B)). Observe that the motives M(SB(A)) and M(SB(A^{op})) are isomorphic. So replacing [A] by [A^{op}] = −[A] doesn’t change anything.

Compactifications of a Merkurjev-Suslin variety. Here we follow definitions and notation of [16]. Let A be a cubic division algebra over F. Recall that a smooth compactification D of a Merkurjev-Suslin variety MS(A, c) can be identified with the smooth hyperplane section of the twisted form X = SB_{3}(M_{2}(A)) of Grassmannian 𝔾(3, 6). Using Theorem 1.2 one obtains a shortened proof of the main result of [16]

Corollary 5.6. Let D be the smooth projective variety introduced above. Then

M(D) ≃ ⨁_{i=1}^{5} M(SB(A)){i} ⊕ R_{3,3},

where R_{3,3} is an integral generalized Rost motive. In other words, from the motivic point of view the variety D can be viewed as a 3-analog of a Norm quadric.

Proof. Let i : D ↪ X denote the closed embedding. It induces the map Γ_{i}: M(D) → M(X). The variety X is a projective homogeneous PGL_{6}-variety corresponding to a maximal parabolic subgroup of type P_{3}. According to the Tits indices for the group PGL_{M_{2}(A)} the parabolic subgroup P_{3} is defined over F and, hence, X is isotropic. By [2, Thm. 7.5] the motive of X splits as

M(X) = ℤ ⊕ Q{1} ⊕ Q{4} ⊕ ℤ{9},

where Q = M(SB(A) × SB(A^{op})) = ⨁_{i=0}^{2} M(SB(A)){i} by the projective bundle theorem. Hence, we obtain

M(X) = ℤ ⊕ ⨁_{i=1}^{6} M(SB(A)){i} ⊕ ℤ{9}.  (4)

We define f to be the composite of Γ_{i} and the canonical projection from M(X) to the direct summand ⨁_{i=1}^{5} M(SB(A)){i} of (4). Observe that the motive M(D) splits over the generic point of SB(A). The direct computations (using multiplication tables provided in [16]) show that f has a section over F (SB(A)). By Theorem 1.2 we conclude that

M(D) ≃ ⨁_{i=1}^{5} M(SB(A)){i} ⊕ N

for some motive N which splits over F (SB(A)) as ℤ ⊕ ℤ{4} ⊕ ℤ{8}. Note that both D and the twisted form of F_{4}/P_{4} (given by the first Tits construction) split the same symbol 𝔤_{3} in K_{3}^{M}(F)/3. This implies that there is a morphism f : N_{ℤ/3} → R_{3,3} of motives with ℤ/3-coefficients which becomes an isomorphism over the separable closure of F, where R_{3,3} is a generalized Rost motive corresponding to 𝔤_{3}. Since N is split over the generic point of the twisted form of F_{4}/P_{4}, R_{3,3} is a direct summand of N_{ℤ/3} which implies that R_{3,3} ≃ N_{ℤ/3}. Finally observe that N_{ℤ/p} splits if p ≠ 3.

6 Appendix

Proposition 6.1. Let X be a smooth quasi-projective variety, π: Y → X a smooth morphism and i: Z ↪ X a closed embedding. Consider the Cartesian square

\xymatrix{Y_{Z} \ar[r]^{i′} \ar[d]_{π_{Z}} & Y \ar[d]^{π} \\ Z \ar[r]^{i} & X}

Then (im i′_{∗})^{d} = 0 for d = [\frac{dim(X)}{codim_{X}(Z)}] + 1, where codim_{X}(Z) is the minimum of codimensions of irreducible components of Z.

It is sufficient to prove the following:

Lemma 6.2. Let π: Y → X be a smooth morphism, with X smooth quasiprojective, and i_{1} : Z_{1} ↪ X, i_{2} : Z_{2} ↪ X closed embeddings. Then there exists a closed embedding i_{3} : Z_{3} ↪ X such that

codim(Z_{3}) ≥ codim(Z_{1}) + codim(Z_{2}) and im(i′_{1})_{∗} · im(i′_{2})_{∗} ⊂ im(i′_{3})_{∗},

where i′_{l}: Y_{Z_{l}} ↪ Y, l = 1, 2, 3 is obtained from the respective Cartesian square.

Proof. We have (i′_{1})_{∗}(a) · (i′_{2})_{∗}(b) = Δ_{X}^{∗}((i′_{1} × i′_{2})_{∗}(a × b)). The diagonal map

Δ_{Y} : Y → Y × Y is the composition Y ⟶^{ϕ} Y ×_{X} Y ⟶^{f_{W}} Y × Y, where ϕ is the relative diagonal and the second map is the natural embedding. By Lemma 6.3 applied to B = X × X, V = X, f = Δ_{X}, T = Z_{1} × Z_{2} and W = Y × Y we obtain a closed embedding h: Z ↪ X such that

codim(Z) ≥ codim(Z_{1}) + codim(Z_{2}) and im(f_{W}^{∗} ∘ (i′_{1} × i′_{2})_{∗}) ⊂ im(h_{W ∗}).

Consider the Cartesian square

\xymatrix{Y \ar[r]^{ϕ} & Y ×_{X} Y \\ Y_{Z} \ar[r]^{ϕ_{Z}} \ar[u]^{h′} & (Y ×_{X} Y)_{Z} \ar[u]_{h_{W}}}

By [3, Theorem 6.2(a)], ϕ^{∗} ∘h_{W ∗} = h′_{∗} ∘ϕ^{!}. Hence, im(Δ_{X}^{∗} ∘(i_{1} ×i_{2})_{∗}) ⊂ im(h′_{∗}) and the lemma is proven.

Lemma 6.3. Let V ⟶^{f} B ←^{g} T be closed embeddings with regular f, and smooth quasi-projective B. Let ε: W → B be a smooth morphism. Consider two Cartesian diagrams:

\xymatrix{W_{V} \ar[r]^{f_{W}} \ar[d]_{ε_{V}} & W \ar[d]^{ε} & W_{T} \ar[l]_{g_{W}} \ar[d]^{ε_{T}} & and & T \ar[r]^{g} & B \\ V \ar[r]^{f} & B & T \ar[l]_{g} &  & T̃ \ar[r]^{g̃} \ar[u]^{f̃} & V \ar[u]_{f}}

There exists a closed embedding h: Z ↪ V such that codim(h) ≥ codim(g) and im(f_{W}^{∗} ∘ g_{W ∗}) ⊂ im(h_{W ∗}).

Proof. Consider the Cartesian square

\xymatrix{W_{T} \ar[r]^{g_{W}} & W \\ WT̃ \ar[r]^{g̃_{W}} \ar[u]^{f̃_{W}} & W_{V} \ar[u]_{f_{W}}}

By [3, Theorem 6.2(a)], f_{W}^{∗} ∘g_{W ∗} = g̃_{W ∗}∘f_{W}^{!} . The morphism f_{W}^{!} : CH^{∗}(W_{T}) → CH^{∗}(WT̃) is given by the composition:

CH^{∗}(W_{T}) ⟶^{σ} CH^{∗}(C_{W}) ⟶^{ρ_{W ∗}} CH^{∗}(N_{W}) ⟶^{s^{∗}} CH^{∗}(WT̃),

where σ is the specialization map from [3, §5.2], C_{W} = C_{W_{T}} (WT̃) = C_{T} (T̃) ×_{B} W is the normal cone of the morphism f̃_{W} and N_{W} = WT̃ ×_{W_{V}} N_{f_{W}} = (T̃ ×_{V} N_{f}) ×_{B} W is the total space of the vector bundle g̃_{W}^{∗} (N_{f_{W}}) = (εT̃ ∘ g̃)^{∗}(N_{f}) over WT̃, ρ_{W} : C_{W} ↪ N_{W} is the closed embedding and s: WT̃ → N_{W} is the zero section. Consider the Cartesian square of projective completions of C_{W} and N_{W}

\xymatrix{ℙ(C_{W} ⊕ O) \ar[r]^{^{\overline{ρ}^{W}}} & ℙ(N_{W} ⊕ O) \\ C_{W} \ar[r]^{ρ_{W}} \ar[u]^{e_{C}} & N_{W} \ar[u]_{e_{N}}}

Documenta Mathematica 13 (2008) 81–96

By [3, Proposition 3.3] the morphism s^{∗} ∘ ρ_{W_{∗}}: CH^{∗}(C_{W}) → CH^{∗}(WT̃) is given by s^{∗} ∘ ρ_{W_{∗}}(x) = π_{W_{∗}}(c_{d}(g̃_{W}^{∗} N_{f_{W}} ⊗ O(1)) · \overline{ρ}_{W_{∗}}(y)), where e_{C}^{∗}(y) = x, π_{W} : ℙ(N_{W} ⊕ O) → WT̃ is the projection and d = codim(f_{W}) = codim(f). By Lemma 6.4, we can choose a cycle α representing c_{d}(g̃^{∗}N_{f} ⊗ O(1)) in such a way that |α| ∩ ℙ(C ⊕ O) has codimension d in ℙ(C ⊕ O). Consider Z := π(|α| ∩ ℙ(C ⊕ O)) and the closed embedding j: Z ↪ T̃. Then for arbitrary x ∈ CH^{∗}(ℙ(C_{W} ⊕ O)) we have |π_{W_{∗}}(ε∗ ; T̃(α) · \overline{ρ}_{W_{∗}}(x))| ⊂ ε^{−1}(Z). This implies that im(f_{W}^{!} ) ⊂ im(j_{W_{∗}}) and im(g̃_{W_{∗}} ∘ f_{W}^{!} ) ⊂ im(h_{W_{∗}}), where h = g̃ ∘ j. At the same time, codim(h) ≥ codim(g), and the lemma is proven.

Lemma 6.4. Let X be a quasi-projective variety, and Z_{l}, l = 1, …, n be closed irreducible subvarieties of dimensions d_{l}. Let V be a vector bundle over X. Then there exists a representative α_{d} of c_{d}(V) such that |α_{d}|∩Z_{l} has dimension ≤ d_{l} − d.

Proof. The total Chern class c_{•}(V) is the inverse of the total Segre class s_{•}(V), and s_{i}(V) = π_{∗}(c_{1}(O(1))^{n−1+i}), where π : ℙ_{X}(V) → X is the projection, and n = dim(V). Thus, the general case of our statement follows by the inductive application of the one with d = 1, and V - linear bundle. Indeed, since c_{d}([X]) = − ∑_{j=1}^{d} π_{∗}(c_{1}(O(1))^{n−1+j}(π^{−1}(c_{d−j}([X])))), and α_{d−j} can be chosen with the needed property, it is sufficient to apply the above particular case to the set of irreducible components of π^{−1}(Z_{l} ∩ |α_{d−j}|), l = 1, …, n; j = 1, …, d inside ℙ_{X}(V). Finally, the case d = 1 and linear V follows from the presentation V = L_{1} ⊗ L_{2}^{−1}, where L_{i} have ”sufficiently many sections”, which is possible, since X is quasi-projective.

Acknowledgments We would like to thank Ivan Panin for very stimulating discussions on the subject of this paper. We are grateful to Ulf Rehmann and Markus Rost for creating a nice working atmosphere and opportunity for us to work together. We are also grateful to the referee for numerous comments which helped to improve the presentation of the paper.

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A. Vishik School of Mathematical Sciences University of Nottingham University Park, Nottingham, NG7 2RD UK alexander.vishik@nottingham.ac.uk K. Zainoulline Mathematisches Institut der LMU München Theresienstr. 39 80333 München Germany kirill@mathematik.uni-muenchen.de