THE MOTIVE OF A PFISTER FORM

MARKUS ROST

Contents

Introduction 1 1. Correspondences 1 1.1. Algebra of correspondences 1 1.2. Motives 2 1.3. The Tate motive 3 1.4. Splitting off a point 3 1.5. Chow groups of fibrations 4 2. Isotropic and split quadrics 5 2.1. Motives of quadrics and Witt equivalence 5 2.2. Zero cycles on a quadric 7 2.3. The Chow ring of a split quadric 7 3. Endomorphisms of quadrics 8 4. Construction of the motive 11 References 13

Introduction

This is a draft which deserves some reorganization and supplements. However it contains all details concerning the construction and properties of the motive.

Unfortunately only very late I noticed that Proposition 1 and Proposition 9 with Corollaries should have been formulated for motives instead only for varieties. But this may be complemented without much difficulty.

1. Correspondences

In this section we recall some basic facts about correspondences and Grothendieck motives. The basic reference is [1], in particular [1, Example 16.1.12].

Let k be a fixed ground field.

1.1. Algebra of correspondences. By a variety X we understand a separated scheme of finite type over k. We denote by CH_{p}(X) the Chow group of p-dimen-sional cycles on X. If X is smooth, we denote by CH^{p}(X) the Chow group of pcodimensional cycles on X. In case X is irreducible, one has CH^{p}(X) = CH_{d−p}(X) where d = dim X.

Date: November 12, 1998.

Let \overline{V} be the following additive category: The objects of \overline{V} are the smooth proper varieties over k. The morphism groups in \overline{V} are

Hom\overline{V}(X, Y) = CH_{∗}(X × Y) = ⊕_{p} CH_{p}(X × Y).

The composition law in \overline{V} is given by the composition of correspondences:

CH_{∗}(X × Y) × CH_{∗}(Y × Z) → CH_{∗}(X × Z),
(f, g) ↦ g ∘ f = π_{∗}(Δ^{∗}(f × g)),

with

Δ: X × Y × Z → X × Y × Y × Z,  Δ(x, y, z) = (x, y, y, z),
π: X × Y × Z → X × Z,  π(x, y, z) = (x, z).

The transpose of f ∈ Hom\overline{V}(X, Y) is f^{t} = τ_{∗}(f) ∈ Hom\overline{V}(Y, X) where τ: X × Y → Y × X, τ(x, y) = (y, x). The category \overline{V} is endowed with the duality ^{t}: \overline{V} → \overline{V}^{op} given by X^{t} = X on objects and by the transpose on morphisms.

One has Hom\overline{V}(Spec k, X) = CH_{∗}(X). This way X ↦ CH_{∗}(X) defines a functor on \overline{V} (by composition of morphisms).

Let V = V(k) be the following subcategory of \overline{V}: The objects of V are the objects of \overline{V}. The morphism groups in V are

Hom(X, Y) = ⊕_{i} CH_{dim X_{i}}(X_{i} × Y) = ⊕_{j} CH^{dim Y_{j}}(X × Y_{j}) ⊂ Hom\overline{V}(X, Y).

Here X_{i} resp. Y_{j} are the connected components of X resp. Y. The composition law in V is induced from the composition law in \overline{V}.

We write

End(X) = Hom(X, X)

for the endomorphism ring of X. If X is irreducible, then

End(X) = CH_{d}(X × X) = CH^{d}(X × X), d = dim X.

The assignments X ↦ CH_{p}(X) resp. X ↦ CH^{p}(X) are covariant resp. contravariant functors on V. For f ∈ Hom(X, Y) we denote the associated maps by f_{∗}: CH_{p}(X) → CH_{p}(Y) resp. f^{∗}: CH^{p}(Y) → CH^{p}(X).

For a morphism f: X → Y of varietes over k we denote by the same letter the class of its graph:

f = [Graph(f)] ∈ Hom(X, Y).

In this case the maps f_{∗} resp. f^{∗} are the standard push forward resp. pull back maps.

1.2. Motives. By a motive we understand a pair (X, p) with X an object in V and p ∈ End(X) a projector: p ∘ p = p. The Chow groups of (X, p) are defined by

CH_{r}((X, p)) = p_{∗}(CH_{r}(X)),
CH^{r}((X, p)) = p^{∗}(CH^{r}(X)).

The category M is defined as follows: Its objects are the pairs (X, p). Its morphism groups are

Hom((X, p), (Y, q)) = q ∘ Hom(X, Y) ∘ p ⊂ Hom(X, Y).

The composition of morphisms in M is induced from the composition law in V.

Associating to X the motive (X, id_{X}) identifies V as a full subcategory of M. We simply write X = (X, id_{X}). The functors M ↦ CH_{p}(M) resp. M ↦ CH^{p}(M) extend the corresponding functors on V.

The sum and the product in M (and in V and \overline{V}) are defined by disjoint union and product:

(X, p) ⊕ (Y, q) = (X ∪ Y, p + q),
(X, p) ⊗ (Y, q) = (X × Y, p × q).

1.3. The Tate motive. The endomorphism ring of ℙ^{1} is

End(ℙ^{1}) = ℤ ⊕ ℤ,

generated by the cycle classes p_{0} = [ℙ^{1}×P] and p_{1} = [P ×ℙ^{1}] with P a k-point on ℙ^{1}. The endomorphisms p_{0}, p_{1} are orthogonal projectors. One has (ℙ^{1}, p_{0}) = Spec k.

The Tate motive is the motive L = (ℙ^{1}, p_{1}). For i ≥ 0 we denote by L^{⊗i} the i-fold product L ⊗ ⋯ ⊗ L with L^{⊗0} = Spec k. One has

CH_{p}(M ⊗ L^{⊗i}) = CH_{p−i}(M)

If X is irreducible, then

Hom(X ⊗ L^{⊗i}, Y ⊗ L^{⊗j}) = CH_{(dim X)+i−j}(X × Y)

For projective spaces there is the following decomposition in M:

ℙ^{n} = ⊕_{i=0}^{n}L^{⊗i}.

1.4. Splitting off a point. Let X be equidimensional of dimension d. We denote by [X] ∈ CH_{d}(X) the fundamental class of X.

Let

P ∈ ∐_{x∈X_{(0)}} ℤ

be a zero cycle. The classes

[P] ∈ CH_{0}(X) = Hom(L^{⊗0}, X) = Hom(X, L^{⊗d}),
[X] ∈ CH_{d}(X) = Hom(L^{⊗d}, X) = Hom(X, L^{⊗0}),

define morphisms

L^{⊗0} \overrightarrow{⟵}_{g_{X}}^{f_{P}} X, L^{⊗d} \overrightarrow{⟵}_{g_{P}}^{f_{X}} X.

If P has degree 1, one has g_{X} ∘ f_{P} = id_{L^{⊗0}} and g_{P} ∘ f_{X} = id_{L^{⊗d}}. Then f_{P} ∘ g_{X}, f_{X} ∘ g_{P} ∈ End(X) are projectors. If additionally d > 0, these projectors are orthogonal. They identify L^{⊗0} ⊕ L^{⊗d} as a direct summand of X and we have a decomposition

X = L^{⊗0} ⊕ (X, π_{P}) ⊕ L^{⊗d} (1)

with π_{P} = id_{X} − f_{P} ∘ g_{X} − f_{X} ∘ g_{P}.

Suppose that the degree map

deg: CH_{0}(X) → ℤ

is bijective. Then the correspondences f_{P}, g_{P}, and π_{P} do not depend on the choice of P. Moreover we have a canonical decomposition

End(X) = End(L^{⊗0}) ⊕ End((X, π_{P})) ⊕ End(L^{⊗d}).

1.5. Chow groups of fibrations. Let X, B be smooth and proper varieties over k and let π: B × X → B be the projection. For b ∈ B we write X_{b} = Spec κ(b) × X for the fibre over b. Moreover, for f ∈ End(X) we denote by f_{b} ∈ End_{V(κ(b))}(X_{b}) the element obtained by base change k → κ(b).

We will need the following observation:

Proposition 1. Let f ∈ End(X) and suppose that

(f_{b})_{∗}(CH_{p}(X_{b})) = 0

for all b ∈ B and 0 ≤ p ≤ dim B. Then

f^{(1+dim B)} ∘ Hom(B, X) = 0.

(Here the power of f is taken in the ring End(X).)

Proof. We use the setting of [3]. For simplicity we assume that X and B are irreducible of dimensions d = dim X and e = dim B.

First note that the map Hom(B, X) → Hom(B, X), g ↦ f ∘ g is given by the composite of the maps

(2)  CH_{p}(B × X) ⟶^{×f} CH_{p+d}(B × X × X × X),
(3)  CH_{p+d}(B × X × X × X) ⟶^{Δ^{∗}} CH_{p}(B × X × X),
(4)  CH_{p}(B × X × X) ⟶^{g_{∗}} CH_{p}(B × X),

with Δ(b, x, y) = (b, x, x, y) and g(b, x, y) = (b, y).

Similarly, after replacing B by Spec κ(b) and f by f_{b}, the composition of these maps yield the action of f_{b} on CH_{p}(X_{b}).

Let Z = X, X × X, or X × X × X. The projections π: B × Z → B induce spectral sequences

E_{p,q}^{2} = A_{p}(B, A_{q}[π, M]) ⟹ A_{p+q}(B × Z, M), (5)

see [3, Sect. 8]. Here M can be any cycle module in the sense of [3]; for our purpose we may restrict to the case M = K_{∗}^{M} given by Milnor’s K-theory.

The maps (2), (3), and (4) extend to the Chow groups A_{p}(B × Z, M) and, similarly, to the groups A_{p}(Z_{b}, M). Moreover they are compatibel with the spectral sequences (5). For the product map ×f this is fairly obvious from the definitions, see [3, Sect. 14]. For the pull back map Δ^{∗} see [3, Theorem 12.7]. For the push forward map g_{∗} see [3, Proposition 8.5.1].

The filtration on the Chow groups of B × X corresponding to the spectral sequence (5) for Z = X is

CH_{p,0}(π) ⊂ CH_{p,1}(π) ⊂ ⋯ ⊂ CH_{p}(B × X)

where CH_{p,r}(π) ⊂ CH_{p}(B × X) is the subgroup generated by the classes of cycles V ⊂ Z with dim π(V) ≤ r.

Consider the quotients

\overline{CH}_{p,r}(π) = CH_{p,r}(π)/ CH_{p,r−1}(π)

The direct summands Ẽ_{r,p−r}^{2} of the groups E_{r,p−r}^{2} corresponding to the quotients \overline{CH}_{p,r}(π) are the cokernels of the divisor maps

∐_{b∈B_{(r+1)}} A_{p−r}(X_{b}, K_{∗}^{M}, 1 − p + r) ⟶^{d} ∐_{b∈B_{(r)}} CH_{p−r}(X_{b}).

The group \overline{CH}_{p,r}(π) is a quotient of Ẽ_{r,p−r}^{2}. Therefore f acts trivially on \overline{CH}_{p,r}(π) as long as f_{b} acts trivially on CH_{p−r}(X_{b}) for all b ∈ B.

Since the filtration on CH_{p}(B × X) has length 1 + e and since f acts trivially on the filtration quotients, it follows that the action of f on CH_{p}(B × X) is nilpotent of order 1 + e.

2. Isotropic and split quadrics

For the remaining parts of this article, the characteristic of the ground field k is different from 2. For generalities on quadratic forms and in particular Pfister forms, see [2, 4].

For a quadratic form φ we denote by X_{φ} the associated projective quadric. If not mentioned otherwise, we assume φ to be regular so that X_{φ} is smooth. One has dim X_{φ} = dim φ − 2.

2.1. Motives of quadrics and Witt equivalence. We call a quadric X_{φ} isotropic if φ is isotropic. A quadric X is isotropic if and only if X has a k-rational point.

The following proposition shows that the motive of a quadric X_{φ} depends essentially (up to elementary operations involving the Tate motive) on the class of φ in the Witt ring of k.

Proposition 2. Let φ = ℍ ⊥ ψ where ℍ is a hyperbolic plane and let X = X_{φ} and Y = X_{ψ}. Then

X = L^{⊗0} ⊕ Y ⊗ L ⊕ L^{⊗d}

where d = dim X.

The case d = 0 is easily verified and therefore we may assume d > 0. We choose coordinates (u, v, y) such that φ(u, v, y) = uv + ψ(y).

The proof of Proposition 2 is based on the stratification of X given by the following data:

Z = { u = 0 } ⊂ X,
Z′ = Z \ P, P = [0, 1, 0],
r: Z′ → Y, r([0, v, y]) = [y].

Then X \ Z ≃ 𝔸^{d} with d = dim X, the point P is the singular point of Z, and r is a 1-dimensional vector bundle.

Suppose for a moment that we work in a category of motives which include the motives of arbitrary (possibly non smooth and non proper) varieties. Then the data above immediately give the desired decomposition of the motive of X: The decomposition X = Z ∪ 𝔸^{d} indicates that X = Z ⊕ L^{⊗d}. Similarly one would have Z = L^{⊗0} ⊕ Z′. Furthermore, since r is a 1-dimensional vector bundle, homotopy invariance gives Z′ = Y ⊗ L.

Constructions of such a category of motives have been announced by Morel and Voevodsky, and (in a different manner by extending the framework of [3] to a bivariant theory of cycles) by the author. However, for the sake of completeness, we continue here to work in the category M. The arguments are of a completely formal nature starting from the described data.

Let

S ⊂ Y × X

be the closure of the image of the morphism

Z′ ⟶^{(r,i)} Y × X

where i: Z′ → X is the inclusion. The cycle class f = [S] ∈ CH_{d−1}(Y × X) defines a morphism

Y ⊗ L ⟶^{f} X.

Recall the decomposition (1). One has g_{X} ∘ f = 0 and g_{P} ∘ f = 0 by dimension reasons. Therefore f defines actually a morphism

Y ⊗ L ⟶^{f̂} (X, π_{P}).

By Manin’s identity principle [1, Example 16.1.12], f̂ is an isomorphism, if for any B in V and p ≥ 1 the map

(id_{B} × f̂)_{∗}: CH_{p−1}(B ⊗ Y) → CH_{p}(B ⊗ (X, π_{P}))

is bijective. This follows from the following two Lemmata.

Lemma 3. Let X′ = X \ P and let i: Z′ → X′, j: X′ → X be the inclusions. The following diagram is commutative:

CH_{p−1}(B ⊗ Y) ⟶^{f_{∗}} CH_{p}(B ⊗ X)
r^{∗}↓  ↓^{j^{∗}}
CH_{p}(B ⊗ Z′) ⟶^{i_{∗}} CH_{p}(B ⊗ X′)

The pull back map r^{∗} is bijective.

Proof. The bijectivity of r^{∗} follows from the fact that r is a 1-dimensional vector bundle.

Let S′ = S ∩ (Y × X′). The scheme S′ is just the image of the closed immersion Z′ → Y × X′.

The map j^{∗}∘f_{∗} is given by taking cross product with X′, intersecting with B×S′ and taking push forward along Y. Now S′ ⊂ Y × Z′ and Z′ ⊂ X′ is a smooth hyperplane. Therefore we can also take first cross product with Z′, intersect with B × S′ inside B × Y × Z′, take push forward along Y (this yields the pull back map r^{∗}, since S′ is the transpose of the graph of r) and finally take the direct image with i_{∗}.

Lemma 4. There is an isomorphism ρ: CH_{p}(B⊗(X, π_{P})) → CH_{p}(B⊗Z′) making the following diagram commutative:

CH_{p}(B ⊗ (X, π_{P})) ⟶^{inclusion} CH_{p}(B ⊗ X)
ρ↓^{≃}  ↓^{j^{∗}}
CH_{p}(B ⊗ Z′)  ⟶^{i_{∗}} CH_{p}(B ⊗ X′)

The map i_{∗} is injective.

Proof. The map j^{∗} fits into an exact sequence

CH_{p}(B × P) ⟶^{(i_{P})_{∗}} CH_{p}(B × X) ⟶^{j^{∗}} CH_{p}(B × X′) → 0

The push forward map (i_{P})_{∗} is injective, since the projection λ_{X}: X → Spec k gives a left inverse.

The map i_{∗} fits into a long exact sequence (see [3, Sect. 5])

⋯ → A_{p+1}(B × X′, K_{∗}^{M}, −p) ⟶^{ℓ^{∗}} A_{p+1}(B × 𝔸^{d}, K_{∗}^{M}, −p) ⟶^{∂}

⟶^{∂} CH_{p}(B × Z′) ⟶^{i_{∗}} CH_{p}(B × X′) ⟶^{ℓ^{∗}} CH_{p}(B × 𝔸^{d}) → 0

Here the maps ℓ^{∗} are induced from the inclusion ℓ: 𝔸^{d} → X′.

Let λ: 𝔸^{d} → Spec k be the projection. The pullback maps λ^{∗}: A_{p−d}(B) → A_{p}(B × 𝔸^{d}) are isomorphims. The maps λ_{X}^{∗} ∘ (λ^{∗})^{−1} are left inverses to ∂. Hence the long exact sequence splits up into short exact sequences.

Putting things together yields a decomposition

CH_{p}(B × X) = CH_{p}(B × P) ⊕ CH_{p}(B × Z′) ⊕ CH_{p}(B × 𝔸^{d})

It is easy to see that this decomposition coincides with the decomposition given by (1).

The proof of Proposition 2 is now complete.

One may check that the inverse of the correspondence f̂ is given by the cycle T = σ(S) ⊂ X × Y where

σ: Y × X → X × Y,
σ([y′], [u, v, y]) = ([v, u, y], [y′]).

(T is the transpose of S reflected by u ↔ v.)

2.2. Zero cycles on a quadric. We recall the computation of the group of zero cycles of a quadric (see [5]).

Lemma 5. For a quadric X the degree map deg: CH_{0}(X) → ℤ is injective. Its image is

deg(CH_{0}(X) → ℤ) = \begin{cases}ℤ & if X is isotropic, \\ 2ℤ & if X is anisotropic. \end{cases}

2.3. The Chow ring of a split quadric. The following material is well known. For n ≥ 2 let τ_{n} be the following n-dimensional quadratic form:

τ_{n} = \begin{cases}∑_{i=1}^{m} x_{i}y_{i} + z^{2} & if n = 2m + 1, \\ ∑_{i=1}^{m} x_{i}y_{i} & if n = 2m. \end{cases}

We call a quadratic form φ split if φ is similar to τ_{n} for some n.

Lemma 6. A quadratic form φ (dim φ ≥ 2) is split if and only if X_{φ} is a direct sum of powers of the Tate motive.

In this case one has with d = dim X_{φ}

X_{φ} = \begin{cases}⊕_{i=0}^{d}L^{⊗i} & if d is odd, \\ ⊕_{i=0}^{d}L^{⊗i} ⊕ L^{⊗(d/2)} & if d is even. \end{cases}

Proof. Using Proposition 2 one reduces to the case when φ is anisotropic. But then X_{φ} has no rational point and cannot be a sum of powers of the Tate motive (cf. Lemma 5).

A quadric is called split if it satisfies the properties of Lemma 6.

Lemma 7. Let X be a split quadric. Then

End(X) = ⊕_{p} End_{ℤ}(CH_{p}(X))

Proof. This holds in fact for any sum of powers of the Tate motive.

Lemma 8. Let X = X_{τ_{n}} and d = dim X = n − 2. Then (i) For p < d/2 one has CH^{p}(X) = ℤ generated by the class of the plane section

{ x_{1} = ⋯ = x_{p} = 0 }.

(ii) If d = 2m − 2, then CH^{d/2}(X) = ℤ ⊕ ℤ generated by the classes of the two maximal linear subspaces

{ x_{1} = ⋯ = x_{m−1} = x_{m} = 0 },
{ x_{1} = ⋯ = x_{m−1} = y_{m} = 0 }.

(iii) For p > d/2 one has CH^{p}(X) = ℤ generated by the class of the linear subspace

{ x_{1} = ⋯ = x_{m} = z = y_{1} = ⋯ = y_{p−m} = 0 }  if d = 2m − 1,
{ x_{1} = ⋯ = x_{m} = y_{1} = ⋯ = y_{p+1−m} = 0 }  if d = 2m − 2.

Proof. This well known fact follows e.g. by analyzing the proof of Proposition 2. In fact, it easy to check that the maps

f_{∗}: CH_{p−1}(Y) → CH_{p}(X)

sends plane sections to plane sections and linear subspaces to linear subspaces.

Let h ∈ CH^{1}(X) and u ∈ CH^{(d+1)/2}(X), v, w ∈ CH^{d/2}(X) the generators described in Lemma 8. The Chow ring of X_{τ_{n}} has as a ring over ℤ the following presentation (d > 0):

⟨ h, u | h^{(d+1)/2} = 2u, h^{d+1} = 0, u^{2} = 0 ⟩  if d ≡ 1 mod 2,
⟨ h, u | h^{d/2} = v + w, hv = hw, vw = 0, v^{2} = w^{2} = h^{d}, h^{d+1} = 0 ⟩ if d ≡ 0 mod 4,
⟨ h, u | h^{d/2} = v + w, hv = hw, vw = h^{d}, v^{2} = w^{2} = 0, h^{d+1} = 0 ⟩ if d ≡ 2 mod 4.

3. Endomorphisms of quadrics

We are now ready to prove the following proposition, which provides an important tool to construct correspondences between quadrics.

Proposition 9. For d ≥ 0 there exist a number N(d) with the following property:

Let X be a smooth projective quadric of dimension d, let f ∈ End(X) and let F/k be some field extension. If f_{F} = 0 ∈ End_{V(F)}(X_{F}), then f^{N(d)} = 0.

Proof. We argue by induction on d. Let φ be a quadratic form defining X.

Suppose first that X has a k-rational point P. Then φ = ℍ⊥ψ as in Proposition 2 and

End(X) = [X × P]ℤ ⊕ End(X_{ψ}) ⊕ [P × X]ℤ.

If f_{F} = 0, then necessarily f ∈ End(X_{ψ}), since the other summands do not change under field extensions. By induction we have f^{N(d−2)} = 0.

In the general case we apply Proposition 1 with B = X. Since all the quadrics X_{b} have a rational point, we may apply the previous step and see that f_{b}^{N(d−2)} = 0 for all b ∈ B. It suffices to take N(d) = (1 + d)N(d − 2).

Corollary 10. Let X be a smooth projective quadric, let f ∈ End(X) and let F/k be some field extension.

(i) If f_{F} is nilpotent, then f is nilpotent. (ii) If f_{F} is an isomorphism, then f is an isomorphism.

Proof. (i) is clear from Proposition 9.

Suppose that f_{F} is an isomorphism. We may assume that X_{F} is a split quadric. Then, by Lemma 7, f_{F} is completely determined by its action on the groups CH_{p}(X_{F}) ≃ ℤ, ℤ ⊕ ℤ. Thus f_{F} satisfies an equation (t^{2} − 1)(t^{2} + mt ± 1) = 0 with m ∈ ℤ. It follows that the inverse of f_{F} is an integral polynomial in f_{F} and therefore defined over k.

Hence we may assume that f_{F} = (id_{X})_{F}. But then (ii) is immediate from (i): If f = 1 + g and g is nilpotent, then f is invertible.

Corollary 11. Let X, Z be quadrics and f ∈ Hom(X, Z). If F/k is a field extension such that X_{F} and Z_{F} are split and if

(f_{F})_{∗}: CH_{p}(X_{F}) → CH_{p}(Z_{F})

is an isomorphism for all p, then f is an isomorphism.

Proof. We must have dim X = dim Z. Let g = f^{t}. Then (g_{F})_{∗}: CH_{p}(Z_{F}) → CH_{p}(X_{F}) is also an isomorphism for all p. By Lemma 7, the endomorphism (f ∘g)_{F} is an isomorphism. By Corollary 10 (ii) f ∘g is an isomorphism. Similarly it follows that g ∘ f is an isomorphism.

Definition 12. Let X be a quadric of dimension d. We call a projector p ∈ End(X) special, if there exists a quadric Z of dimension d − 2 such that

(X, id_{X} − p) ≃ Z ⊗ L.

Example: (cf. (1) and Proposition 2) If X is isotropic, then π_{P} is a special projector on X.

Proposition 13. On a quadric X there exists at most one special projector.

Proof. First suppose that X is isotropic. Then there is an isomorphism

(X, p) ⊕ Z ⊗ L ⟶^{∼} (X, π_{P}) ⊕ Y ⊗ L

with Y as in Proposition 2. Since

Hom(Z ⊗ L, (X, π_{P})) = 0

by dimension reasons, the induced morphism Z ⊗ L → Y ⊗ L has a left inverse. Hence Z is a direct summand of Y. The morphism Z → Y must be an isomorphism by Corollary 11. It follows that (X, π_{P}) ≃ (X, p). Hence

Hom((X, p), Y ⊗ L) = 0

similar as above and the decomposition is unique.

In the general case let p, p′ be two special projectors on X with quadrics Z resp. Z′ and isomorphisms

(X, p) ⊕ Z ⊗ L ⟶^{∼} X ⟶^{∼} (X, p′) ⊕ Z′ ⊗ L.

Let F = k(Z). Note that X_{F} is isotropic. Namely Z_{F} is isotropic and any F -rational point on Z_{F} corresponds to an element u ∈ CH_{1}(X_{F}) which gives in the split case a generator. Intersecting u with a hyperplane section gives an F -rational point on X_{F}. By the previous step we have

Hom(Z_{F} ⊗ L, (X, p)_{F} ) = 0.

By a filtration argument one sees that Hom(Z ⊗ L, (X, p′)) is trivial. Similarly for Hom((X, p), Z′ ⊗ L) = 0. Hence the decomposition is unique.

Lemma 14. Let p be a special projector. Then p^{t} = p.

Proof. If f: (X, id_{X} −p) → Z⊗L is an isomorphism, then f^{t}: Z⊗L → (X, id_{X} −p^{t}) is an isomorphism. Hence p^{t} is special und must be equal to p.

Lemma 15. Let X, Z be quadrics with dim Z = dim X − 2 and let F be a field extension of k such that X, Z are split. Suppose that there exists a morphism

f: Z ⊗ L → X

such that

(f_{F})_{∗}: CH_{(p−1)}(Z_{F}) → CH_{p}(X_{F})

is an isomorphism for all 1 < p < d. Then there exists a special projector p on X and f induces an isomorphism

Z ⊗ L ≃ (X, p).

Proof. The morphism f^{t} ∘ f is seen to be an isomorphism, by similar arguments as above. Then id_{X} − f ∘ (f^{t} ∘ f)^{−1} ∘ f^{t} is a special projector.

Let p ∈ End(X) be a special projector and M = (X, p). We compute the endomorphism ring of M. Let F be a splitting field of X and let (d = dim X)

ω: End(M) → End(M_{F}) = End(L^{⊗0} ⊕ L^{⊗d}) = ℤ ⊕ ℤ

be the natural map. Let

Γ = 2ℤ ⊕ 2ℤ + (1, 1)ℤ ⊂ ℤ ⊕ ℤ.

Lemma 16. The map ω is injective. One has

Im ω = \begin{cases}ℤ ⊕ ℤ & if X is isotropic, \\ Γ & if X is anisotropic. \end{cases}

Proof. We have (1, 1) = ω(id_{M}) ∈ Im ω. Moreover 2ℤ ⊕ 2ℤ ⊂ Im ω. Namely X becomes isotropic over a quadratic extension F/k and then

ω ∘ cor_{F/k}(End(M_{F})) = 2ℤ ⊕ 2ℤ.

If (1, 0) ∈ Im ω, there exists a zero cycle of degree 1 on X and X is isotropic. It remains to show that ω is injective.

Let B = X and let π: B × X → B be the projection to the second factor and let

E_{s,t}^{2} = A_{s}(B, A_{t}[π, K_{∗}^{M}]) ⟹ A_{s+t}(B × X, K_{∗}^{M}) (6) be the associated spectral sequence, see [3, Sect. 8]. This spectral sequence commutes with the projector p acting on X. Hence we have a spectral sequence

E_{s,t}^{2} = A_{s}(B, p_{∗}(A_{t}[π, K_{∗}^{M}])) ⟹ p_{∗}(A_{s+t}(B × X, K_{∗}^{M})) (7)

Since all fibers X_{b} are isotropic, we have

p_{∗}(A_{t}[π, K_{∗}^{M}]) = \begin{cases}0 & if 0 < t < d, \\ K_{∗}^{M} & if t = 0, d. \end{cases}

It follows that there is a short exact sequence

CH_{0}(B) → CH_{d}(B ⊗ M) → CH_{d}(B) → 0.

Hence CH_{d}(B⊗M) is free of rank ≤ 2 and the same is true for the subgroup End(M).

4. Construction of the motive

Let a_{n} ∈ k^{∗}, n ≥ 1 be a sequence of elements. We denote by

φ_{n} = ⟨⟨a_{1}, …, a_{n}⟩⟩ = ⊗_{i=1}^{n}⟨1, −a_{i}⟩

the Pfister form corresponding to a_{1}, …, a_{n}. Moreover φ′_{n} denotes the prime subform of φ_{n} defined by φ_{n} = ⟨1⟩ ⊥ φ′_{n}. Furthermore we put ψ_{n} = φ_{n−1} ⊥ ⟨−a_{n}⟩. Note that

ψ_{n} = ⟨1, −a_{n}⟩ ⊥ φ′_{n}.

We write X_{n} = X_{ψ_{n}} and Z_{n} = X_{φ′_{n}}. Let d_{n} = dim X_{n} = 2^{n−1} − 1.

Theorem 17. On the quadric X_{n} there exist a special projector p_{n}.

Let M_{n} = (X_{n}, p_{n}). Then

(8)  (X_{n}, id_{X_{n}} − p_{n}) ≃ Z_{n−1} ⊗ L

and

(9)  Z_{n} ≃ M_{n} ⊗ ⨁_{i=0}^{d_{n}−1} L^{⊗i}

Proof. For n = 1, 2 one takes p_{n} = id_{X_{n}}.

Step 1: We first assume that the theorem holds for n and construct p_{n+1} with property (8).

We may assume that X_{n} is anisotropic.

We first choose a certain element

Θ ∈ Hom(M_{n} ⊗ L^{⊗d_{n}}, X_{n+1}) = CH_{2d_{n}}(M_{n} ⊗ X_{n+1}).

Let F = k(X_{n}) and consider the sequence

CH_{2d_{n}}(M_{n} ⊗ X_{n+1}) \overrightarrow{⟵}_{p_{n}}^{incl.} CH_{2d_{n}}(X_{n} × X_{n+1}) ⟶^{j} CH_{d_{n}}((X_{n+1})_{F} )

Here j is the projection map. j is surjective (by taking closure of cycles). Since (X_{n+1})_{F} is split, the rightmost term is ≃ ℤ generated by the class u of a maximal linear subspace.

We claim that j(ker p_{n}) = 0. To check this, one may pass to a splitting field in which case the claim is easy to verify. Let Θ be an element which maps to the generator u.

Let F be a splitting field of φ_{n} and let

ρ: Hom(M_{n} ⊗ L^{⊗d_{n}}, X_{n+1}) → Hom((M_{n})_{F} ⊗ L^{⊗d_{n}}, (X_{n+1})_{F} ) =
= Hom(L^{⊗d_{n}} ⊗ L^{⊗2d_{n}}, (X_{n+1})_{F} ) = ℤ ⊕ ℤ

be the restriction map. Note that

ρ(Θ) ∈ (1, ℤ).

Lemma 18. If X_{n} is anisotropic, then 2ℤ ⊕ 2ℤ ⊂ Im ρ ⊂ Γ.

Proof. The first inclusion is easily seen by passing to a quadratic splitting field and taking norms.

We replace k by k(X_{n+1}). Then X_{n+1} is isotropic, but X_{n} is still anisotropic. One finds using the induction hypothesis:

Hom(M_{n} ⊗ L^{⊗d_{n}}, X_{n+1}) = Hom(M_{n} ⊗ L^{⊗d_{n}}, L^{⊗0} ⊕ L^{⊗d_{n+1}} ⊕ M_{n} ⊗ ⨁_{i=1}^{d_{n}} L^{⊗i})

= Hom(M_{n} ⊗ L^{⊗d_{n}}, M_{n} ⊗ ⨁_{i=1}^{d_{n}} L^{⊗i})

The map ρ is then given by projection onto the summand for i = d_{n} followed by passing to the split case. Hence it factors through the map ω: End(M_{n}) → ℤ ⊕ ℤ and Lemma 16 yields the claim.

Since ρ(Θ) ∉ 2ℤ ⊕ 2ℤ we must have Γ ⊂ Im ρ and we may assume that

ρ(Θ) = (1, 1).

Let h ∈ CH^{1}(X_{n+1}) be a hyperplane section and put

f = Θ · (1 + h + h^{2} + ⋯ + h^{d_{n}−1}) ∈

Hom(M_{n} ⊗ ⨁_{i=1}^{d_{n}} L^{⊗i}, X_{n+1}) = Hom(Z_{n} ⊗ L, X_{n+1}).

One easily checks that f satisfies the assumptions in Lemma 15. Hence Lemma 15 yields the projector p_{n+1} together with property (8).

Step 2: We now assume the existence of p_{n} with property (8) and show (9). We may again assume that X_{n} is anisotropic. This time we look for an element

Θ ∈ Hom(M_{n} ⊗ L^{⊗d_{n}−1}, Z_{n}) = CH_{2d_{n}−1}(M_{n} ⊗ Z_{n}).

Similar as above we have sequence (with F = k(X_{n}))

CH_{2d_{n}−1}(M_{n} ⊗ Z_{n}) \overrightarrow{⟵}_{p_{n}}^{incl.} CH_{2d_{n}−1}(X_{n} × Z_{n}) ⟶^{j} CH_{d_{n}−1}((Z_{n})_{F} ) = uℤ

and find Θ mapping to u.

By a spectral sequence argument (with base Z_{n} and using the fact that M_{n} ≃ L^{⊗0} ⊕ L^{⊗d_{n}} over any residue class field of Z_{n}) on gets an exact sequence

CH_{d_{n}−1}(Z_{n}) ⟶^{M_{n}×} CH_{2d_{n}−1}(M_{n} ⊗ Z_{n}) ⟶^{ℓ} CH_{2d_{n}−1}(Z_{n}) → 0.

Suppose that ℓ(Θ) ∈ CH_{2d_{n}−1}(Z_{n}) = [Z_{n}]ℤ is even. Then we can arrange ℓ(Θ) = 0 still keeping j(Θ) = u. But then Θ would come from CH_{d_{n}−1}(Z_{n}) and the maximal linear subspace u would be defined over k. This contradicts the fact that X_{n} is anisotropic. We my therefore find Θ with

ℓ(Θ) = [Z_{n}], j(Θ) = u.

Let h ∈ CH^{1}(Z_{n}) be a hyperplane section and put

f = Θ · (1 + h + h^{2} + ⋯ + h^{d_{n}−1}) ∈ Hom(M_{n} ⊗ ⨁_{i=0}^{d_{n}−1} L^{⊗i}, Z_{n}).

Since p_{n}^{t} = p_{n}, the transpose f^{t} is well defined. It is easy to check that f ∘ f^{t} satisfies the assumption of Corollary 11 and hence

f ∘ f^{t} is an isomorphism.

Similarly one proves that f^{t} ∘ f is an isomorphism (by extending Corollary 11 to motives of the form M_{n} ⊗ ℙ^{m}).

Proposition 19. One has

X_{φ_{n}} ≃ M_{n} ⊗ ℙ^{d_{n}}.

Proof. This follows as for (9).

References

[1] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984.

[2] T. Y. Lam, The algebraic theory of quadratic forms, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1980, Revised second printing, Mathematics Lecture Note Series.

[3] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393 (electronic).

[4] W. Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985.

[5] R. G. Swan, Zero cycles on quadric hypersurfaces, Proc. Amer. Math. Soc. 107 (1989), no. 1, 43–46.

NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Germany

E-mail address: markus.rost@mathematik.uni-regensburg.de
URL: http://www.physik.uni-regensburg.de/~rom03516