LIFTING OF COEFFICIENTS FOR CHOW MOTIVES OF QUADRICS

OLIVIER HAUTION

Abstract. We prove that the natural functor from the category of Chow motives of smooth projective quadrics with integral coefficients to the category with coefficients modulo 2 induces a bijection on the isomorphism classes of objects.

Contents

1. Introduction  1
2. Chow groups of quadrics  2
3. Lifting of coefficients  3
4. Surjectivity in main Theorem  4
5. Injectivity in main Theorem  5
References  8

1. Introduction

A. Vishik has given a description of the Chow motives of quadrics with integral coefficients in [4]. It uses much subtler methods than the ones used to give a similar description with coefficients in ℤ/2, found for example in [1], but the description obtained is the same. The result presented here allows to recover Vishik’s results from the modulo 2 description.

In order to state the main result, we first define the categories involved. Let Λ be a commutative ring. We write Q_{F} for the class of smooth projective quadrics over a field F. We consider the additive category C(Q_{F} , Λ), where objects are quadrics in Q_{F} and if X, Y are two such quadrics, Hom(X, Y) is the group of correspondences of degree 0, namely CH_{dim X}(X × Y, Λ). We write CM(Q_{F} , Λ) for the idempotent completion of C(Q_{F} , Λ). This is the category of graded Chow motives of smooth projective quadrics with coefficients in Λ. If (X, ρ), (Y, σ) are two such motives then we have :

Hom((X, ρ), (Y, σ)) = σ ∘ CH_{dim X}(X × Y, Λ) ∘ ρ.

We will prove the following :

Theorem 1. The functor CM(Q_{F} , ℤ) → CM(Q_{F} , ℤ/2) induces a bijection on the isomorphism classes of objects.

Date: October 2, 2007.

2000 Mathematics Subject Classification. 11E04, 14C25 .

Key words and phrases. Quadratric forms, Chow motives, Correpondences.

The proof mostly relies on the low rank of the homogeneous components of the Chow groups of quadrics when passing to a splitting field. These components are almost always indecomposable if we take into account the Galois action. The only exception is the component of rank 2 when the discriminant is trivial but in this case the Galois action on the Chow group is trivial which allows the proof to go through. It seems that this method doesn’t generalize to other projective homogeneous varieties, at least to other quadratic grassmannians since in this case the ranks of the homogeneous components of the Chow groups are not bounded.

I am very grateful to Nikita Karpenko for his useful suggestions.

2. Chow groups of quadrics

We first recall some facts and fix the notations that we will use.

Let F be a field and φ be a non-degenerate quadratic form on a F -vector space V of dimension D + 2. The associated projective quadric X is smooth of dimension D = 2d or 2d + 1. Let L/F be a splitting extension for X, i.e. an extension such that V_{L} has a totally isotropic space of dimension d+1. We write h^{i}, l_{i} for the usual basis of CH(X_{L}), where 0 ≤ i ≤ d. The class h is the pull-back of the hyperplane class of the projective space of V_{L}, the class l_{i} is the class of a totally isotropic subspace of V_{L} of dimension i + 1.

If D is even, then CH_{d}(X_{L}) is freely generated by h^{d} and l_{d}. There are exactly two classes of maximal totally isotropic spaces, l_{d} and l_{d}′. They correspond to spaces exchanged by a reflection and verify the relation l_{d} + l_{d}′ = h^{d}.

The group Aut(L/F) acts on CH(X_{L}). It acts trivially on the i-th homogeneous components of CH(X_{L}), as long as 2i ≠ D.

Another scheme that may be associated to a quadratic form φ is G(φ), the scheme of maximal totally isotropic subspaces of V. We can view it as a closed subscheme of the Grassmannian variety of V.

When D is even, if L is a splitting field for φ, then G(φ)_{L} has two connected components exchanged by any reflection. We orient X and G(φ) accordingly: a totally isotropic space whose class is l_{d} lies in the component G of G(φ)_{L}, we call G′ the other component. See [1] for proofs of all these facts.

If Z is a scheme over a field F and L/F an extension we say that a cycle in CH(Z_{L}) is rational if it belongs to the image of the pull-back homomorphism CH(Z) → CH(Z_{L}).

In the next proposition, X is a smooth projective quadric of dimension D = 2d associated to a quadratic space (V, φ), L/F is a splitting extension for X, and disc X is the discriminant algebra of φ.

Proposition 2. Under the natural Aut(L/F)-actions, we can identify the pair {l_{d}, l_{d}′} and the connected components of Spec(disc X ⊗ L).

Proof. Consider the closed subscheme E of V × G(φ) consisting of pairs (u, U) such that u ∈ U. It is vector bundle of rank d over G(φ). The projective bundle P(E) can be seen as a closed subscheme of X × G(φ). There are unique elements e_{k} ∈ CH^{k}(G(φ)_{L}), e′_{0} ∈ CH^{0}(G(φ)_{L}) such that :

[P(E)] = l_{d} × [G] + l_{d}′ × [G′] + ∑_{k=1}^{d} h^{d−k} × e_{k}

It is proven in [1] that the products p = [G]e_{1}...e_{d} and p′ = [G′]e_{1}...e_{d} are classes of rational points over L, and that they form a basis of CH_{0}(G(φ)_{L}). For k = 1, .., d, the cycle e_{k} is invariant under Aut(L/F). Therefore the group Aut(L/F) acts on {p, p′} the way it acts on the connected components {G, G′}.

Taking pull-backs, we see that [P(E)]^{∗}(p) = l_{d} and [P(E)]^{∗}(p′) = l_{d}′. Moreover the correspondence [P(E)] is rational, therefore is equivariant under the action of Aut(L/F). Hence, as an Aut(L/F)-set, the pair {l_{d}, l_{d}′} is isomorphic to {p, p′} ≃ {G, G′}. But the scheme G(φ) has a morphism to Spec(disc X) (see [1]) and the classes of G and G′ in CH(G(φ)_{L}) are the pull-backs of the classes of the connected components of Spec(disc X ⊗ L). The statement follows.

Here is another proof. Let γ be in Aut(L/F), W be a maximal totally isotropic space of V_{L}. Choose an isometry f sending W to ^{γ}W. We may assume that the group of isometries of φ_{L} is generated by reflections : The only case where the Cartan-Dieudonné theorem does not apply is when L = ℤ/2ℤ (and D = 2). In this case the quadric is already split over F and there is nothing to prove.

In any other case f can be written as a product of m reflections. The integer m does depend on f and on its decomposition, but its parity only depends on γ : If W lies in a connected component C of G(φ)_{L}, then ^{γ}W lies in the component ^{γ}C. Since these components are exchanged by any reflection, ^{γ}C = C if and only if m is even.

As already noticed these components are in natural correspondence with those of Spec(disc X ⊗ L). On the other hand [W] = [f(W)] ∈ CH(X_{L}) if and only if m even, which completes the proof. □

3. Lifting of coefficients

We now give a useful caracterization of rational cycles (Proposition 7). The proof will rely on the following theorem ([3], Proposition 9):

Theorem 3 (Rost’s nilpotence for quadrics). Let X be a smooth projective quadric over F, and let α ∈ End_{CM(Q_{F} ,Λ)}(X, id). If α_{L} ∈ CH(X_{L}^{2}) vanishes for some field extension L/F, then α is nilpotent.

Proofs for the following corollaries can be found in [1] :

Corollary 4. Let X be a smooth projective quadric over F and L/F a field extension. Let π a projector in End_{CM(Q_{L},Λ)}(X_{L}) that is the restriction of some element in End_{CM(Q_{F} ,Λ)}(X). Then there exist a projector φ in End_{CM(Q_{F} ,Λ)}(X) such that φ_{L} = π.

Corollary 5. Let f : (X, ρ) → (Y, σ) be a morphism in CM(Q_{F} , Λ). If f_{L} is an isomorphism for some field extension L/F then f is an isomorphism

Here we use the ideas found in [6] and [2].

Proposition 6. For any n ≥ 1, the functor CM(Q_{F} , ℤ/2^{n}) → CM(Q_{F} , ℤ/2) is bijective on the isomorphism classes of objects.

Proof. This is just reformulating Corollary 2.7 of [2]. □

Any smooth projective quadric admits a (non-canonical) finite Galois splitting extension, of degree a power of 2. This will be used together with the following proposition in order to prove rationality of cycles.

Proposition 7. Let X, Y ∈ Q_{F} and L/F be a splitting Galois extension of degree m for X and Y. A correspondence in CH((X × Y)_{L}, ℤ) is rational if and only if it is invariant under the group Gal(L/F) and its image in CH((X × Y)_{L}, ℤ/m) is rational.

Proof. We write Z for X ×Y. We first prove that if d is a Gal(L/F)-invariant cycle in CH(Z_{L}) then [L : F]d is rational.

Let τ : L → F̄ be a separable closure so that we have a F̄-isomorphism L ⊗ F̄ → F̄ × ... × F̄ given by u ⊗ 1 ↦ (τ ∘ γ(u))_{γ∈Gal(L/F)}.

We have a fibrered square :

\xymatrix{Z_{L} \ar[d] & Z_{L⊗F̄} \ar[l] \ar[d] \\ Z & ZF̄ \ar[l]}

It follows that we have a commutative diagram of pull-backs and push-forwards:

\xymatrix{CH(Z_{L}) \ar[d] \ar[r] & CH(Z_{L⊗F̄}) \ar[d] \\ CH(Z) \ar[r] & CH(ZF̄)}

The top map followed by the map on the right is:

x ↦ ∑_{γ∈Gal(L/F)} t^{∗}(^{γ}x)

where t : ZF̄ → Z_{L} is the map induced by τ. Using the commutativity of the diagram and the injectivity of t^{∗}, we see that the composite CH(Z_{L}) → CH(Z) → CH(Z_{L}) maps x to ∑^{γ}x, where γ runs in Gal(L/F). The claim follows.

Now suppose that u is a cycle in CH(Z_{L}, ℤ) invariant under Gal(L/F), and that its image in CH(Z_{L}, ℤ/m) is rational. We can find a rational cycle v in CH(Z_{L}, ℤ) and a cycle δ in CH(Z_{L}, ℤ) such that mδ = v − u. Since CH(Z_{L}, ℤ) is torsion-free, δ is invariant under Gal(L/F) . The first claim ensures that v − u is rational, hence u is rational. □

Let us remark that if X ∈ Q_{F}, L/F is a splitting extension, and 2i < dim X then 2l_{i} = h^{dim X−i} ∈ CH(X_{L}) is always rational. It follows that 2 CH_{i}(X_{L}) consists of rational cycles when 2i ≠ dim X.

4. Surjectivity in main Theorem

Proposition 8. The functor CM(Q_{F} , ℤ) → CM(Q_{F} , ℤ/2) is surjective on the isomorphism classes of objects.

Proof. Let (X, π) ∈ CM(Q_{F} , ℤ/2) and L/F a finite splitting Galois extension for X of degree 2^{n}. By Proposition 6, we can lift (X, π) to some (X, τ) ∈ CM(Q_{F} , ℤ/2^{n}). It will be enough to build a projector ρ ∈ CH(X × X, ℤ) which reduces modulo 2^{n} to τ. We will find a Gal(L/F)-invariant projector in CH_{dim X}(X ×X)_{L} which gives modulo 2^{n} the projector τ_{L}. Then Proposition 7 will give a cycle in CH(X × X, ℤ) lifting τ. This cycle may be chosen to be a projector by Corollary 4.

For any commutative ring Λ, projectors in CH((X ×X)_{L}, Λ) are in bijective correspondence with ordered pairs of subgroups of CH(X_{L}, Λ) which form a direct sum decomposition. This bijection is compatible with the natural Gal(L/F)-actions. A projector is of degree 0 if and only if the two summands in the associated decomposition are graded subgroups of CH(X_{L}, Λ).

When dim X is odd or when disc X is a field, each homogeneous component of CH(X_{L}, Λ) is Gal(L/F)-indecomposable, hence a projector of degree 0 of CH(X_{L}, Λ) is just a subset of {0, ..., dim X}. It follows that we can lift any such projector and that every projector is invariant under the action of Gal(L/F).

When dim X is even and disc X is trivial, CH_{i}(X_{L}, Λ) is indecomposable if 2i ≠ 2d_{X} = dim X. The group Gal(L/F) acts trivially on CH_{d_{X}} (X_{L}, Λ). If the rank of the restriction of (τ_{L})_{∗} to CH_{d_{X}} (X_{L}, ℤ/2^{n}) is 0 or 2, the projector τ_{L} clearly lifts to a Gal(L/F)-invariant projector in CH_{dim X}(X × X)_{L}.

The last case is when the rank is 1. Once we fix a decomposition of the group CH_{d_{X}} (X_{L}, ℤ/2^{n}) as rank 1 summands, any other such decomposition is given by some element of SL_{2}(ℤ/2^{n}). The next lemma ensures that we can lift such an element to SL_{2}(ℤ), thus that τ_{L} lifts to a projector with integral coefficients. It remains to notice that Gal(L/F) acts trivially on CH_{dim X}(X × X)_{L} since disc X is trivial, to conclude the proof. □

Lemma 9. For any positive integers k and p, the reduction homomorphism SL_{k}(ℤ) → SL_{k}(ℤ/pℤ) is surjective.

Proof. Any matrix in SL_{k}(ℤ/pℤ) can be written as a product of elementary matrices, which correspond to elementary transformations. But elementary matrices lie in the image of SL_{k}(ℤ) → SL_{k}(ℤ/pℤ). □

5. Injectivity in main Theorem

In order to prove injectivity in Theorem 1, we may assume that we are given two motives (X, ρ), (Y, σ) in CM(Q_{F} , ℤ) and an isomorphism between their images in CM(Q_{F} , ℤ/2). We will build an isomorphism with integral coefficients between the two motives.

We fix a finite Galois splitting extension L/F for X and Y of degree 2^{n}. Using Proposition 6 we may assume that there exists an isomorphism α between (X, ρ) and (Y, σ) in CM(Q_{F} , ℤ/2^{n}). By Proposition 7 and Corollary 5, it is enough to build an isomorphism (X_{L}, ρ_{L}) → (Y_{L}, σ_{L}) which reduces to a rational correspondence modulo 2^{n} and which is equivariant under Gal(L/F).

Let d_{X} be such that dim X = 2d_{X} or 2d_{X} + 1 and d_{Y} defined similarly for Y. Let r(X, ρ) be the rank of CH_{d_{X}} (X_{L}) ∩ im (ρ_{L})_{∗} if dim X is even and r(X, ρ) = 0 if dim X is odd. We define r(Y, σ) in a similar fashion. We will distinguish cases using these integers.

A basis of CH(X_{L}) ∩ im (ρ_{L})_{∗} gives an isomorphism of (X_{L}, ρ_{L}) with twists of Tate motives, thus choosing bases for the groups CH(X_{L}) ∩ im (ρ_{L})_{∗} and CH(Y_{L}) ∩ im (σ_{L})_{∗}, we can see morphisms between the two motives as matrices.

We fix a basis (e_{i}) of CH(X_{L}) as follows : we choose e_{i} ∈ CH_{i}(X_{L}) among the cycles h^{dim X−i}, l_{i} for 2i ≠ dim X. We are done when r(X, ρ) = 0.

If r(X, ρ) = 2 we complete the basis with e_{d_{X}} = l_{d_{X}} , e′_{d_{X}} = l′_{d_{X}} ∈ CH_{d_{X}} (X_{L}). If r(X, ρ) = 1 we choose a generator e_{d_{X}} of CH_{d_{X}} (X_{L}) ∩ im (ρ_{L})_{∗} to complete

the basis.

We choose a basis (f_{i}) for CH(Y_{L}) ∩ im (σ_{L})_{∗} in a similar way.

If we write ρ̃ and σ̃ for the reduction modulo 2^{n} of ρ and σ, these bases reduce to bases (ẽ_{i}) of CH(X_{L}, ℤ/2^{n}) ∩ im(ρ̃_{L})_{∗} and (f̃_{i}) of CH(Y_{L}, ℤ/2^{n}) ∩ im(σ̃_{L})_{∗}. In these homogeneous bases the matrix of a correspondence of degree 0 is diagonal by blocks. The sizes of the blocks are the ranks of the homogeneous components of im (ρ_{L})_{∗}.

Lemma 10. If r(X, ρ) = 1 then disc X is trivial.

Proof. Assume disc X is not trivial. The correspondence ρ induces a projection of CH_{d_{X}} (X_{L}) which is equivariant under the action of Gal(L/F). But CH_{d_{X}} (X_{L}) is indecomposable as a Gal(L/F)-module. It follows that (ρ_{L})_{∗} is either the identity or 0 when restricted to CH_{d_{X}} (X_{L}) , hence r(X, ρ) ≠ 1. □

Corollary 11. If r(X, ρ) ≠ 2 then Gal(L/F) acts trivially on im (ρ_{L})_{∗}.

Lemma 12. If r(X, ρ) = 2 then r(Y, ρ) = 2, dim Y = dim X and disc Y = disc X.

Proof. Since the isomorphism (α_{L})_{∗} is graded, the d_{X}-th homogeneous component of im(α_{L})_{∗} has rank 2. This image is a subgroup of the Chow group with coefficients in ℤ/2^{n} of a split quadric, thus the only possibility is that dim Y is even, d_{X} = d_{Y} and r(Y, σ) = 2.

The isomorphism (α_{L})_{∗} is equivariant under the action of Gal(L/F). Therefore the pair {l_{d_{X}} , l_{d_{X}} ′} ⊂ CH(X_{L}, ℤ/2^{n}) is Gal(L/F)-isomorphic to {l_{d_{Y}} , l_{d_{Y}} ′} ⊂ CH(Y_{L}, ℤ/2^{n}). Obviously, the same holds with integral coefficients. By proposition 2, we have a Gal(L/F)-isomorphism between the split étale algebras disc X ⊗ L and disc Y ⊗ L. Hence disc X and disc Y correspond to the same cocycle in H^{1}(Gal(L/F), ℤ/2), thus are isomorphic. □

We now proceed with the proof of the injectivity.

Let us first assume that r(X, ρ) ≠ 2. Then r(Y, σ) ≠ 2 by the preceding lemma. By Corollary 11 the group Gal(L/F) acts trivially on im (ρ_{L})_{∗} and on im (σ_{L})_{∗}, therefore any morphism (X_{L}, ρ_{L}) → (Y_{L}, σ_{L}) is defined by a cycle invariant under Gal(L/F).

Since the isomorphism α_{L} is of degree 0, its matrix in our graded bases of the modulo 2^{n} Chow groups is diagonal. Let λ_{i} ∈ (ℤ/2^{n})^{×} be the coefficients in the diagonal so that we have (α_{L})_{∗}(ẽ_{i}) = λ_{i}f̃_{i} for all i such that CH_{i}(X_{L})∩im (ρ_{L})_{∗} ≠ ∅.

If r(X, ρ) = 1 then λ_{d_{X}} is defined and we consider the cycle β = (λ_{d_{X}})^{−1}α_{L}. If r(X, ρ) = 0, we just put β = α_{L}.

Now we take k_{i} ∈ ℤ/2^{n} such that λ_{i}^{−1} = 2k_{i} + 1. Let Δ ∈ End(X_{L}, ρ̃_{L}) be the graph of the diagonal. We consider the rational cycle

γ = Δ + 2 ∑ k_{i}ẽ_{i} × ẽ_{dim X−i}

where the sum is taken over all i such that CH_{i}(X_{L}) ⊂ im (ρ_{L})_{∗} (which implies in case r(X, ρ) = 1 that we do not take i = d_{X}). The composite β ∘ γ is rational, and its matrix in our bases is the identity matrix. This correspondence lifts to an isomorphism with integral coefficients (X_{L}, ρ_{L}) → (Y_{L}, σ_{L}).

Next assume that r(X, ρ) = 2. Then we have dim X = dim Y, r(Y, σ) = 2 and disc X = disc Y. The matrix of (α_{L})_{∗} is diagonal by blocks :

(\begin{matrix}ν_{i_{1}} \\ & . & . & . \\ & & & & ν_{i_{r}} \\ & & & & & B \\ & & & & & & ν_{i_{r+1}} \\ & & & & & & & . & . & . \\ & & & & & & & & & & ν_{i_{p}} \end{matrix} )

where ν_{i} ∈ (ℤ/2^{n})^{×} and B ∈ GL_{2}(ℤ/2^{n}).

Now if disc X = disc Y is a field, there is an element in Gal(L/F) that simultaneously exchanges the cycles in the bases {l_{d_{X}} , l_{d_{X}} ′} of CH_{d_{X}} (X_{L}, ℤ/2^{n}) and {l_{d_{Y}} , l_{d_{Y}} ′} of CH_{d_{Y}} (Y_{L}, ℤ/2^{n}). It follows that we may write B as :

(\begin{matrix}a & b \\ b & a \end{matrix})

for some a and b in ℤ/2^{n}. The determinant of B is (a + b)(a − b) ∈ (ℤ/2^{n})^{×}, hence (a − b) ∈ (ℤ/2^{n})^{×}. Thus we may replace α_{L} by (a − b)^{−1}α_{L} and assume that b = a + 1.

As before we may write ν_{i}^{−1} = 2k_{i} + 1 and replace α_{L} by the rational cycle :

α_{L} ∘ (Δ + 2 ∑ k_{i}ẽ_{i} × ẽ_{dim X−i})

the sum being taken over all i such that CH_{i}(X_{L}) ∩ im (ρ_{L})_{∗} ≠ ∅ and i ≠ d_{X}. Therefore we may assume that ν_{i} = 1 for all i and that we have a matrix of the shape :

(\begin{matrix}I_{s} & & & 0 \\ & a+1 & a \\ & a & a+1 \\ 0 & & & I_{t} \end{matrix})

The matrix of the rational cycle h^{d_{X}} × h^{d_{Y}} ∈ CH((X × Y)_{L}, ℤ/2^{n}) is :

(\begin{matrix}0 & & & 0 \\ & 1 & 1 \\ & 1 & 1 \\ 0 & & & 0 \end{matrix})

Now α_{L} − a(h^{d_{X}} × h^{d_{Y}}) is rational and its matrix is the identity. This cycle is invariant under Gal(L/F) and lifts to an isomorphism (X_{L}, ρ_{L}) → (Y_{L}, σ_{L}).

It remains to treat the case when disc X is trivial. In this case the group Gal(L/F) acts trivially on CH(X_{L}) (and on CH(Y_{L}) since disc Y is also trivial).

As before, composing with a rational cycle, we may assume that ν_{i} = 1 for all i. We write det B^{−1} = 2k + 1.

The cycle Δ + k(h^{d_{X}} × h^{d_{X}}) ∈ End(X_{L}, ρ̃_{L}) is rational and its matrix in our basis is :

(\begin{matrix}I_{p} & & & 0 \\ & 1+k & k \\ & k & 1+k \\ 0 & & & I_{r} \end{matrix})

We see that the determinant of this matrix is 1 + 2k. Therefore the composite α_{L} ∘ (Δ + k(h^{d_{X}} × h^{d_{X}})) has determinant 1. We use Lemma 9 to conclude, which completes the proof of Theorem 1.

References

[1] Elman R., Karpenko N., Merkurjev A., Algebraic and Geometric Theory of Quadratic Forms

   (2007), http://www.institut.math.jussieu.fr/∼karpenko.

[2] Petrov V., Semenov N., Zainoulline K., J-invariant of linear algebraic groups preprint (2006),

   http://www.mathematik.uni-bielefeld.de/LAG/ .

[3] Rost.  M,  The  motive  of  a  Pfister  form  preprint  (1998)

   http://www.math.uni-bielefeld.de/∼rost/motive.html

[4] Vishik A., Motives of Quadrics with Applications to the Theory of Quadratic Forms (2004) in Geometric methods in the algebraic theory of quadratic forms, Lect. Notes in Math. 1835, Springer, 2004, 25-101.

[5] Vishik A., On the Chow groups of quadratic grassmannians (2005) , Doc. Math. 10, 111-130 (electronic).

[6] Vishik A., Yagita N., Algebraic Cobordisms of a Pfister quadric preprint (2005),

   http://www.math.uiuc.edu/K-theory/0739/.

Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie - Paris 6, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE.

   E-mail address: olivier.haution@gmail.com