Some new results on the Chowgroups of quadrics

Markus Rost Regensburg, January 1990

§ 1 Introduction

The computation of some K-cohomology groups of certain normvarieties plays an essential role in the investigation of the bijectivity of the Galoissymbol

K_{n}^{M}F/p → H^{n}(F; μ_{p}^{⊗n}).

In the case p = 2 these normvarieties are quadrics associated with Pfisterforms.

In this note we describe some new results and conjectures about such quadrics. In short we have the following results:

It turns out that for a n-fold Pfisterform φ the Chow-motive of its associated quadric X_{φ} can be described as

X_{φ} ≃ M_{φ} × ℙ^{d_{φ}}, d_{φ} = 2^{n−1} − 1,

where M_{φ} is a certain Chow-motive associated with φ. We give a complete description of the Chowgroups of M_{φ} and of the generators of H^{p}(M_{φ}; K_{p+1}); moreover we have a precise conjecture about (the Milnor-K-theory version of) H^{p}(M_{φ}; K_{p+r}) for r ≤ 2.

This note contains no proofs. Detailed proofs will be prepared as soon as possible.

§ 2 Motific decomposition of certain quadrics

We work in the category of Chow-motives over a field F (Char F ≠ 2) (see e.g. [ Fulton; Intersection Theory, § 16 ]). For a motive M = (X, p) let CH_{k}(M) = p_{∗}(CH_{k}(X)), where CH_{k}(X) is the group of k-dimensional cycles modulo rational equivalence. We denote by L = (ℙ^{1}, p) the Tate motive and by L^{i} its i-th power. For a quadratic form φ over F we denote by X_{φ} the corresponding projective quadric (dim X_{φ} = dim φ − 2) and by \overline{X}_{φ} its associated Chow-motive.

One first observation is Proposition 1 Let φ = ψ⊥h, where h is a hyperbolic plane and dim ψ ≥ 0. Then

\overline{X}_{φ} = \overline{X}_{ψ} ⊗ L ⊕ L^{0} ⊕ L^{d}; d = dim ψ.

Hence the motive of the quadric X_{φ} depends essentially only on the class of φ in the Wittring of F. Moreover the proposition shows that the motive of a trivial quadric X_{φ} (i.e. φ = Σ(−1)^{i}x_{i}^{2}) is just a sum of certain powers of the Tate motive.

Proposition 2 Let K|F be any field extension. Then α ^{∈} End(\overline{X}_{φ}) is invertible if and only if α_{K} ^{∈} End((\overline{X}_{φ})_{K}) is invertible.

This is basic to all what follows. To proof proposition 2 one considers the filtration on End(\overline{X}_{φ}) = CH_{d}(X_{φ}×X_{φ}) (d = dim X_{φ}) given by the dimension of cycles after projection to one factor.

For a n-fold Pfisterform φ we call n_{φ} = n the degree of φ; moreover we put d_{φ} = 2^{n−1}−1. Note that dim X_{φ} = 2d_{φ}.

Theorem 3 For a Pfisterform φ there exists a unique motive M = M_{φ} such that for extensions K|E|F there exists

i) an isomorphism

M_{E} = L_{E}^{0} ⊕ L_{E}^{d_{φ}}

if φ_{E} is isotropic.

ii) a commutative diagram of isomorphisms

ℤ
i_{E} ↙  ↘ i_{K}
CH_{d_{φ}}(M_{E}) ⟶^{res_{K|E}} CH_{d_{φ}}(M_{K})

iii) a commutative diagram of injective homomorphisms

CH_{0}(M_{E}) ⟶^{res_{K|E}} CH_{0}(M_{K})
↘ N_{E}  ↙ N_{K}
ℤ

with

Im N_{E} = \{ \begin{matrix}2 ℤ & if φ_{E} is anisotropic \\ ℤ & if φ_{E} is isotropic. \end{matrix}

The motive M_{φ} can be described more precisely as follows: Let φ = μ⊗⟨⟨a⟩⟩ where μ is a Pfisterform of degree n_{μ} = n_{φ} − 1 and put ρ = μ⊥⟨−a⟩. Then M_{φ} = (X_{ρ}, p) for a certain projector p ^{∈} End(\overline{X}_{ρ}) such that p_{∗} is the identity on CH_{d_{φ}}(X_{ρ}) and CH_{0}(X_{ρ}) (note that d_{φ} = dim X_{ρ}). Hence CH_{d_{φ}}(M_{φ}) = ℤ with the cycle X_{ρ} as canonical generator; we denote this generator by [ M_{φ}]. Moreover CH_{0}(M_{φ}) = CH_{0}(X_{ρ}); the homomorphisms N in iii) are given by the degree of a zero-cycle. The motive of X_{ρ} decomposes as

\overline{X}_{ρ} = M_{φ} ⊕ \overline{X}_{μ′} ⊗ L,

where μ′ is the pure subform of μ, i.e. μ = μ′⊥⟨1⟩. More generally, one has

Proposition 4 Let φ be a Pfisterform and let ρ be a subform of φ with dim ρ = \frac{1}{2} dim φ + k, k > 0. Suppose that

ρ_{F (X_{φ})} ≃ η_{F (X_{φ})}⊥h,

where η is a form over F and h is hyperbolic of dimension 2k. Then

\overline{X}_{ρ} = ⨁_{i=0}^{k−1} M_{φ} ⊗ L^{i} ⊕ \overline{X}_{η} ⊗ L^{k}.

Hence, if ρ is an excellent form, i.e. the class of φ in the Wittring of F is the alternating sum of a sequence φ_{0}, …, φ_{n} of Pfisterforms φ_{i} with φ_{i} a subform of φ_{i+1} , one has a decomposition of the motive X_{ρ} in terms of the motives M_{φ_{i}}.

§ 3 On the K-cohomology of M_{φ}

For a variety X over F we denote by A_{p}(X, K_{n}^{M}) the homology of the complex

⨁_{v^{∈} X_{(p+1)}} K_{n+p+1}^{M}κ(v) ⟶^{d} ⨁_{v^{∈} X_{(p)}} K_{n+p}^{M}κ(v) ⟶^{d} ⨁_{v^{∈} X_{(p−1)}} K_{n+p−1}^{M}κ(v),

where K_{n}^{M} denotes the n-th Milnor K-group, X_{(p)} denotes the set of all points of X of dimension p and d is given by the tame symbol.

Using the deformation to the normal cone (see Fulton) one can define intersection theory for the groups A_{p}(X; K_{n}^{M}). Hence the functors A_{p}( , K_{n}^{M}) are defined on the category of Chow-motives.

Note that A_{p}(X; K_{−p}^{M}) = CH_{p}(X) and that for a smooth variety X of dimension d there is a natural homomorphism

A_{p}(X; K_{n}^{M}) → H^{d−p}(X; K_{n+d})

induced by the canonical map from Milnor-K-theory to Quillen’s K-groups (which is an isomorphism for n + p ≤ 2).

For a nonsingular quadratic form φ let D_{0}(φ) ⊂ K_{0}F = ℤ be the subgroup

D_{0}(φ) = \{ \begin{matrix}K_{0}F & if φ is isotropic \\ 2K_{0}F & if φ is nonisotropic \end{matrix}

and for n ≥ 1 let D_{n}(φ) ⊂ K_{n}^{M}F be the subgroup generated by symbols in which one entry is represented by φ.

One can show that D_{n}(φ) is exactly the image of the normmap

N : A_{0}(X_{φ}K_{n}^{M}) → K_{n}^{M}F.

Theorem 5 For a Pfisterform φ of degree n_{φ} ≥ 2 one has

A_{p}(M_{φ}; K_{−p}) = CH_{p}(M_{φ}) = \{ \begin{matrix}K_{0}F & for p = d_{φ} = 2^{n_{φ}−1} − 1 \\ K_{0}F/D_{0}(φ) & for p = 2^{k} − 1; k = 1, …, n_{φ} − 2 \\ D_{0}(φ) & for p = 0 \\ 0 & else \end{matrix}

The generators of CH_{p}(M_{φ}) can be described as follows.

For a Pfistersubform ψ of φ there is a natural morphism

i_{ψ,φ} : M_{ψ} → M_{φ} ,

compatible with the norm maps CH_{0} → ℤ_{0} . It is induced by inclusion Xρ̃ → X_{ρ} for appropriate choices of representations M_{φ} = (X_{ρ}, p), M_{ψ} = (Xρ̃, p̃) as described above.

The generator of CH_{p}(M_{φ}), p = 2^{k} − 1 for some k ^{∈} {1, …, n_{φ} − 1}, is then given by the image (i_{ψ,φ})_{∗}([ M_{ψ}]) of the fundamental cycle [ M_{ψ}] ^{∈} CH_{dψ}(M_{ψ}), where ψ is any Pfistersubform of φ of degree n_{ψ} = k + 1.

Theorem 6 Let φ be a Pfisterform of degree n_{φ} ≥ 2.

i) If p = 2^{k} + 2^{ℓ} − 1, 0 < ℓ < k < n_{φ} − 1, then A_{p}(M_{φ}; K_{1−p}) is cyclic of order at most 2.

ii) If p = 0, then the normmap A_{0}(M_{φ}; K_{1}) → K_{1}F is injective with image D_{1}(φ).

iii) For all other values of p the multiplication map

CH_{p}(M_{φ}) ⊗ K_{1}F → A_{p}(M_{φ}; K_{1−p})

is surjective.

To give some information about the generators for the groups in i) we describe now a general conjecture about certain elements in the groups A_{p}(M_{φ}; K_{n}).

Let ℓ : ℕ → ℕ be the function with the property

p + 1 = ∑_{i=0}^{ℓ(p)} 2^{k_{i}} for some 0 ≤ k_{0} < k_{1} < ⋯ < k_{ℓ(p)}

Conjecture 7 For Pfisterforms φ of degree n_{φ} ≥ 2 there exist unique classes

γ_{p}(φ) ^{∈} A_{p}(M_{φ}; K_{ℓ(p)−p}^{M}) for 0 < p ≤ d_{φ}, p odd,

such that

I) γ_{d_{φ}}(φ) ^{∈} CH_{d_{φ}}(M_{φ}) is the generator [ M_{φ}].

II) For a Pfister-subform ψ < φ one has

(i_{ψ,φ})_{∗}(γ_{p}(ψ)) = γ_{p}(φ) for 0 < p ≤ d_{ψ}, p odd.

III) For a Pfister-subform ψ < φ with φ = ψ ⊗ ⟨⟨a⟩⟩ one has

(i_{ψ,φ})^{∗}(γ_{p}(φ)) = {a} · γ_{p−d_{φ}+d_{ψ}}(ψ) for d_{ψ} < p < d_{φ}, p odd.

For III) note that ℓ(p − d_{φ} + d_{ψ}) = ℓ(p) − 1. For ℓ(p) = 0 the classes γ_{p}(φ) are exactly the generators of CH_{p}(M_{φ}) as described above (this follows from I) and II)). I have constructed the classes γ_{p}(φ) for ℓ(p) ≤ 1 (which are the generators for A_{p}(M_{φ}; K_{1−p})) and it is probable that I can do this for ℓ(p) ≤ 2. Moreover our methods should lead to a proof of

Conjecture 8 For a Pfisterform φ of degree n_{φ} ≥ 2 one has for n ≤ 2:

A_{p}(M_{φ}; K_{n−p}^{M}) = \{ \begin{matrix}K_{n}^{M}F & for p = d_{φ} \\ [ K_{n−ℓ(p)}^{M}F/D_{n−ℓ(p)}(φ) ] & for 0 < p < d_{φ}, p odd, \\ 0 & for 0 < p < d_{φ}, p even \\ D_{n}(φ) & p = 0 \end{matrix}

Here we understand K_{r}^{M}F = 0 for r < 0. For 0 < p ≤ d_{φ}, p odd, the isomorphism is given by multiplication with γ_{p}(φ) and for p = 0 by the normmap to K_{n}^{M}F.

Conjecture 8 is definitely false for n ≥ 3 (e.g. for n_{φ} = 2 the motive M_{φ} is the conic

corresponding to φ and K_{3}^{M}F ⟶^{[ M_{φ} ]} A_{1}(M_{φ}; K_{2}^{M}) is neither surjective nor injective in general). Nevertheless the classes γ_{p}(φ) should form a (part of a) fundamental set of generators of the groups A_{p}(M_{φ}; K_{n}^{M}).