Motivic Splitting Lemma

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 the motives M and N split over the generic point of X as direct sums of shifted copies of a Tate motive. 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. We say a motive M is split if it is isomorphic to a direct sum of shifted copies of a Tate motive. We say a motive M is generically split if there exists a smooth projective variety X and an integer l such that M is split over the generic point of X and M is a direct summand of the shifted motive M(X)(l)[2l] of X. In particular, a variety X is called generically split if its Chow motive M(X) is split over the generic point of 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. 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 the motives M and N are split over the field of fractions K of X. To prove the theorem we use the following auxiliary facts 1

Theorem 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 the motives M and N are split over the field of fractions K of X.Then a morphism f : M → N splits, i.e., N is a direct summand of M, if it splits over K.
To prove the theorem we use the following auxiliary facts Chow motives over a relative base.For a smooth variety X over F we introduce the category of Chow motives over X following [5].First, we define the category of correspondences C(X) whose objects are smooth projective morphisms Y → X and (for connected Y ) with natural composition law.Now the category of effective Chow motives Chow ef f (X) can be defined as the Karoubian envelope of C(X).One has the restriction functor For a motive N we denote N X := res X/F (N).In particular, the image of the Tate motive Z(1) [2] gives us the Tate motive Z X (1) [2] in Chow ef f (X).The category Chow ef f (X) has natural tensor structure Finally, Chow (X) is obtained from Chow ef f (X) by inverting Z X (1) [2].We use the following standard notation which agrees with [5].For a motive M ∈ Ob(Chow(X)) we denote by Hence, for shifts we have We identify the Chow group with low index CH m (M) of a motive M with Hom X (Z X (m)[2m], M) and the Chow group with upper index CH m (M) with Hom X (M, Z X (m)[2m]).For a smooth projective variety X over a field F we denote by M(X) the motive [X → Spec F ] of Chow (F ).As usual we denote by For a given motive N over F and a field extension L/F we say a cycle in CH(N L ) is rational if it is in the image of the restriction map res L/F .Rost Nilpotence Theorem for generically split motives.Assume a motive N is generically split, i.e., there is a smooth projective variety X and l ∈ Z such that N is a direct summand of M(X)(l)[2l], and N K is split, where K is the field of fractions of X.We will extensively use the following version of the Rost Nilpotence Theorem (cf.[2]) Proposition 1.Let N be a generically split motive over F .Then Rost Nilpotence Theorem holds for N.In other words, for any field extension E/F , the kernel of the map res E/F : End F (N) → End E (N E ) consists of nilpotents.
Proof.We may assume that N is a direct summand of M(X) (that is, l = 0).Since for a split motive M and for arbitrary field extension E/L, the map End L (M L ) → End E (M E ) is an isomorphism, we may assume that E = K is the field of fractions of X.We have ring homomorphisms where the last one is induced by the generic point Spec K → X.
By Lemma 1 the kernel of res K consists of nilpotents.On the other hand, the map res X/F is split injective with the section induced by the composite Let f be an element from the kernel.Let j :

Now we are ready to prove the main result of the paper
Proof of Theorem.To construct a section of f we apply recursively the following procedure starting from g = 0 and m = 0.
For a morphism g : N → M such that the realization morphism Since the motive N K splits, for the corresponding projector ρ N over K we may write (ρ 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 can be written as (1) From ( 1) we immediately see that 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), 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 is the identity on CH i (N K ) for i ≤ m and v is rational.Let g be a morphism defined over the base field such that gK = v.Consider the endomorphism id Recursion step is proven and we obtain map g : Geometric construction of a generalized Rost motive.
Let p be a prime and n be a positive integer.To each nonzero cyclic subgroup α in K M n (F )/p consisting of pure symbols one can assign some motive M α in the category Chow (F ) with Z/pZ-coefficients, which has the property that for arbitrary field extension E/F , (M α ) E is either indecomposable, which happens if and only if α| E = 0, or (M α ) E is split, which happens if and only if α| E = 0.It follows from the results of V. Voevodsky and M. Rost that for a given subgroup such motive always exists and is unique (see [10]).Moreover, when split it is isomorphic to Such a motive is called a generalized Rost motive (with Z/pZ-coefficients).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 .The integral generalized Rost motives, hypothetically, should be parameterized not by the pure cyclic subgroups of K M n (F )/p, but by the pure symbols of K M n (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.
Proof.It is known that the variety X (a maximal Pfister neighbor of Y ) and Y become cellular over the generic point of X, i.e., the motives M(X) and M(Y ) are split over K. Let Γ i be the graph of the closed embedding i : X ֒→ Y .Its realization R(Γ i ) over K coincides with the induced pull-back i * K which maps an additive generator of CH(Y K ) to an additive generator of CH(X K ) and, hence, splits.The latter means that correspondences Γ i and the transposed Γ t i split over K. Take f : M(Y ) → M(X)(1) [2] to be the morphism induced by Γ t i and apply the theorem.Corollary 2. Let X be a hyperplane section of a twisted form Y of a Cayley plane which splits by a cubic field extension.Then M(Y ) ≃ M(X) (1)[2] ⊕ R 3,3 , where R 3,3 is an integral generalized Rost motive.
Proof.Consider the closed embedding X ֒→ Y , where X = F 4 (J)/P 4 and Y = OP 2 (J) are the twisted forms of F 4 /P 4 and the Cayley plane OP 2 = E 6 /P 6 corresponding to a Jordan algebra J defined by means of the first Tits construction (i.e., which splits by a cubic field extension) and proceed as in the previous proof.Finally, observe that the specialization of R 3,3 with Z/3Z-coefficients is a generalized Rost motive corresponding to a symbol given by the Rost-Serre invariant g 3 .
Remark 1. Observe that in view of the main result of [7] we obtain So from the motivic point of view the variety OP 2 (J) is a 3-analog of a Pfister quadric and F 4 (J)/P 4 is a 3-analog of a maximal Pfister neighbor.
Twisted forms of Grassmannians.
PGL n : Consider a Grassmannian G(d, n) of d-dimensional planes in a ndimensional affine space.Its twisted form is called a generalized Severi-Brauer variety and denoted by SB d (A), where A is the respective c.s.a. of degree n.The next corollary relates the motive of a generalized Severi-Brauer variety with the motive of usual Severi-Brauer variety (cf.[11]).Proof.It is known that X is a maximal Pfister neighbor of a 3-fold Pfister quadric, both X and Y are cellular over the generic points of each other and split by a quadratic field extension L/F .Since X is a maximal Pfister neighbor, it splits as a direct sum of shifted copies of a Rost motive R 3,2 .To construct a motivic isomorphism f : M(X) of the motivic decomposition of X.Then we prove that all these f i satisfy the conditions of the theorem and, hence, split.Finally, we define f to be the direct sum f = ⊕ i f i .
To construct such f i we proceed as follows.Consider the twisted form Z of the variety of complete flags G 2 /B.Observe that the variety Z has dimension 6.Let α ∈ Pic(Z L ) be an element of the Picard group of Z. Since Tits algebras for G 2 are trivial, Pic(Z) ≃ Pic(Z L ), i.e., the cycle α is defined over the base field.Set α ′ ∈ CH 5 (X × Y ) to be the image of α by means of the push-forward (pr X * , pr Y * ) : Z → X × Y induced by the canonical quotient maps Z → X and Z → Y .Define f i to be the composite So the problem reduces to finding a cycle α ∈ Pic(Z L ) such that the map is surjective.Since the Chow groups of X L and Y L have only one additive generator in each codimension, it is enough to show that for each i the map The latter can be done easily by writing down the surjectivity conditions in terms of Z-bases of the respective Chow groups and, then, solving arising system of Z-linear equations.
Observe that the map induced by α ′ never splits.So it is not possible to construct a map from M(X) → M(Y ) in this way without decomposing M(X).Proof.The proof is the same as in the case of G 2 and uses the motivic decomposition of the variety X provided in [7].
Hence, it remains to identify N with the motive R 3,3 .To do this recall 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 M 3 (F )/3.This implies that there is a morphism f : N → R 3,3 which becomes an isomorphism over the separable closure of F .Since N is split over the generic point of the twisted form of F 4 /P 4 , by Thm. 1 we conclude that R 3,3 is a direct summand of N. Hence, N ≃ R 3,3 ⊕ S, where S is a phantom motive, i.e., the one which becomes trivial over the separable closure of F .

Lemma 1 .
are the morphisms defining N as a direct summand of M(X).Proposition is proven.For any M ∈ Ob(Chow (X)), the kernel of the map induced by an open embedding U → X res

Corollary 3 .
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 [6, II.10.A].By the result of Karpenko [4, 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)).Then the condition that f splits over the generic point of SB(A) is equivalent to the fact that for each m = 0, . . ., n − 1 g.c.d.Schubert varieties generating CH m (G(d, n)).The latter can be easily computed using explicit formulas for degrees of Schubert varieties provided for instance in [3, Ch. 14, Ex. 14.7.11.(ii)].Finally, observe that the motives M(SB(A)) and M(SB(A op )) are isomorphic.So replacing A by A op doesn't change anything.

G 2 :
Let G 2 /P 1 and G 2 /P 2 denote projective homogeneous varieties of a split group of type G 2 and maximal parabolic subgroups P i corresponding to the respective vertices i = 1, 2 of the Dynkin diagram.These are non-isomorphic varieties of dimension 5.The following corollary provides a shortened proof of Bonnet result[1].Corollary 4. Let X and Y be twisted forms (by means of the same cocycle) of projective homogeneous varieties G 2 /P 1 and G 2 /P 2 respectively.Then M(X) ≃ M(Y ).

F 4 :
Let F 4 /P 4 and F 4 /P 3 denote projective homogeneous varieties of a split group of type F 4 and maximal parabolic subgroups corresponding to the 4-th and the 3-rd vertices of the Dynkin diagram.The first variety has dimension 15 and the second -21.Corollary 5. Let X and Y be twisted forms of varieties F 4 /P 4 and F 4 /P 3 by means of the (same) cocycle which splits by a cubic field extension.Then the motive M(X) is a direct summand of the motive M(Y ).