Motivic splitting lemma

A. Vishik, K. Zainoulline

June 9, 2006

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 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. 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)

Hom_{X}([Y → X], [Z → X]) = CH_{dim(Y)}(Y ×_{X} Z)

with natural composition law. Now the category of effective Chow motives Chow^{eff}(X) can be defined as the Karoubian envelope of C(X). One has the restriction functor

res_{X/F} : Chow^{eff}(F) → Chow^{eff}(X)

sending [Y] to [Y ×_{F} X → X]. 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^{eff}(X). The category Chow^{eff}(X) has natural tensor structure

[Y → X] ⊗ [Z → X] := [Y ×_{X} Z → X].

Finally, Chow(X) is obtained from Chow^{eff}(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 M(i)[2i] its shift M ⊗ ℤ_{X}(1)[2]^{⊗i}. Hence, for shifts we have

Hom_{X}([Y → X](i)[2i], [Z → X](j)[2j]) = CH_{dim(Y)+i−j}(Y ×_{X} Z).

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 c^{t} ∈ CH(Z ×_{X} Y) the transposition of a cycle c ∈ CH(Y ×_{X} Z). For a morphism f : M → N the composition operation induces realization maps R^{m}(f) : CH^{m}(N) → CH^{m}(M).

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

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

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

N ⊗N ⟶^{id⊗ϕ_{N}} N ⊗M(X) ⟶^{id⊗Δ_{X}} N ⊗M(X)⊗M(X) ⟶^{id⊗p_{N} ⊗id} N ⊗N ⊗M(X),

where N ⟶^{ϕ_{N}} M(X) ⟶^{p_{N}} N are the morphisms defining N as a direct summand of M(X). Proposition is proven.

Lemma 1. For any M ∈ Ob(Chow(X)), the kernel of the map induced by an open embedding U → X

res_{U} : End_{X}(M) → End_{U}(M_{U})

consists of nilpotents.

Proof. If M is a direct summand of [Y → X](i)[2i], then End_{X}(M) is a direct summand of End_{X}([Y → X]), and it is sufficient to study the case M = [Y → X].

Let f be an element from the kernel. Let j : Z → X be the reduced closed complement to U in X. Under the identification End_{X}([Y → X]) = CH_{dim(Y)}(Y ×_{X} Y), f belongs to the image of the induced push-forward j_{∗} : CH_{dim(Y)}((Y ×_{X} Y) ×_{X} Z) → CH_{dim(Y)}(Y ×_{X} Y). Then f^{∘(dim(X)+1)} must be zero, since [Z]^{dim(X)+1} is zero in CH(X). Lemma is proven.

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 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} can be written as

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

(1)

From (1) we immediately see that

α_{l} = pr_{1}_{∗}(Ω_{l,K} · h_{K}) ∈ CH^{m}(X_{K}) is rational.  (2)

Also, (ρ_{N})_{K} ∘ α_{l} = 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} = (pr_{1}^{∗}(α_{l}) · ∑_{i} d_{li}Θ_{i,K}^{∨}) ∘ (p_{N})_{K},

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

θ_{i} ↦ d_{li}α_{l}

Hence, the composite R^{m}(f_{K} ∘ u_{l}) = R^{m}(u_{l}) ∘ R^{m}(f_{K}) maps ω_{i} to δ_{il}α_{l}.

Set v = g_{K} + ∑_{l} u_{l}. By construction, the realization R(f_{K} ∘ v) is the identity on CH^{i}(N_{K}) for i ≤ m and v is rational. Let g̃ be a morphism defined over the base field such that g̃_{K} = v. Consider the endomorphism id−f ∘g̃ of N. Over K its realization R^{i}(id−f_{K} ∘v) is trivial for each i ≤ m.

Recursion step is proven and we obtain map g̃ : N → M such that (f ∘ g̃)_{K} = id_{N_{K}}. Let q = id − f ∘ g̃. By the Proposition 1, q^{r} = 0, for some r. Set g = g̃ ∘ (id + q + q^{∘2} + … + q^{∘(r−1)}). Then f ∘ g = id_{N} and N is a direct summand of M.

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_{n}^{M}(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

⨁_{i=0}^{p−1} ℤ/pℤ(i · \frac{p^{n−1}−1}{p−1})[2i · \frac{p^{n−1}−1}{p−1}]

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_{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.

Corollary 1. Let X be a hyperplane section of a n-fold Pfister quadric Y. Then M(Y) ≃ M(X)(1)[2] ⊕ R_{n,2}, where R_{n,2} is an integral Rost motive (cf. [8]).

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_{i}^{t} split over K. Take f : M(Y) → M(X)(1)[2] to be the morphism induced by Γ_{i}^{t} 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

M(𝕆ℙ^{2}(J)) ≃ ⨁_{i=0}^{8} R_{3,3}(i)[2i].

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]).

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

where c_{i}^{(m)} are degrees of 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).

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) ⟶^{≃} M(Y) we construct certain morphisms f_{i} : R_{3,2}(i)[2i] → M(Y) on each component R_{3,2}(i)[2i] 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

f_{i} : R_{3,2}(i)[2i] → M(X) ⟶^{α′} M(Y).

So the problem reduces to finding a cycle α ∈ Pic(Z_{L}) such that the map

R(f)_{L} = ⊕_{i}R(f_{i})_{L} : CH(Y_{L}) → ⨁_{i} CH(R_{3,2}(i)[2i]_{L})

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 R(f_{i})_{L} : CH(Y_{L}) → CH(R_{3,2}(i)[2i]_{L}) is surjective.

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).

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).

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].

Compactifications of a Merkurjev-Suslin variety

Here we follow definitions and notation of [9]. 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 twisted form X = SB_{3}(M_{2}(A)) of a smooth hyperplane section of Grassmannian 𝔾(3, 6). Using Thm. 1 we can simplify the proof of the following result proven by N. Semenov in [9]

Corollary 6. Let D be a twisted form (corresponding to a cubic division algebra) of a smooth hyperplane section of 𝔾(2, 6). Then

M(D) ≃ ⨁_{i=1}^{5} M(SB(A))(i) ⊕ N,

where N is ismorphic to R_{3,3} up to a ‘phantom’ motive in the category Chow(F). In other words from the motivic point of view the variety D can be viewed as a 3-analog of a Norm quadric (cf. [9]).

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} corresponding to the third vertex of the respective Dynkin diagram. According to the Tits diagrams for the group PGL_{M_{2}(A)} the parabolic subgroup P_{3} is defined over F and X is isotropic. By [2, Thm. 7.5] the motive of X splits as

M(X) = ℤ ⊕ Q(1)[2] ⊕ Q(4)[8] ⊕ ℤ(9)[18],

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

M(X) = ℤ ⊕ ⨁_{i=1}^{6} M(SB(A))(i)[2i] ⊕ ℤ(9)[18].  (3)

Now 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)[2i] of (3). Observe that the motive M(D) splits over the generic point of SB(A). The direct computations (using multiplication tables provided in [9]) show that f splits over F (SB(A)). By Thm. 1 we conclude that M(D) ≃ ⨁_{i=1}^{5} M(SB(A))(i)[2i] ⊕ N for some motive N which splits over F (SB(A)) as a direct sum ℤ ⊕ ℤ(4)[8] ⊕ ℤ(8)[16].

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_{3}^{M}(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.

Acknowledgments We would like to thank Ivan Panin for very stimulating discussions. We are grateful to Ulf Rehmann, Markus Rost and SFB701 for creating a nice working atmosphere and opportunity for us to work together.

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http://www.mathematik.uni-bielefeld.de/~rost/

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