On the kernels in Milnor’s K-theory under function field extensions

A. Vishik

March 30, 1998

Contents

0 Introduction 1

1 Universal symbol 2

2 Pure part of the kernel 4

0 Introduction

In this paper we will consider the following question:

Question 1.

Let k be a field of characteristic different from 2, and Q/k be some quadric.

Is it true, that the ideal Ker_{Q} := Ker(K_{∗}^{M}(k)/2 → K_{∗}^{M}(k(Q))/2) in K_{∗}^{M}(k)/2 is generated by pure symbols ?

The answer to our question is known to be positive in the case Q - Pfister neigbour (i.e.: a subquadric of dimension more or equal than half in a big Pfister quadric) by [6], and for 2-dimensional non-Pfister quadric by [8].

The positive answer to our question would imply the following:

1) Conjecture of B.Kahn, M.Rost and R.J.Sujatha (see Conjecture 1 of the original version of [3], where authors asked basically the same question about the low degree part of the kernel).

2) The fact that any conservative quadric (i.e. such quadric, for which the kernel above is nontrivial) is a subquadric in a big Pfister quadric.

3) The fact, that if some quadric Q is a normvariety for some h ∈ K_{m}^{M}(k)/2, then h is a pure symbol. From this, on it’s part, follows nice motivic description of excellent quadrics.

We will show, that our question is equivalent to the fact that Ker_{Q}, considered as a cyclic module of M.Rost (see [7]) is generated by one precisely described pure symbol. So, in the case of positive answer to the question we would have explicit description of the Ker_{Q}; and we have such a description for the pure part of the kernel unconditionally.

I would like to thank Yuri Ivanovich Manin for helpful remarks to the original version of the paper. Also I’m grateful to Max-Planck Institut für Mathematik for supporting me during the writing of this work.

1 Universal symbol

In this section we will show, that for arbitrary quadric Q/k there exists some purely transcendental extension k(A_{Q})/k, and some pure symbol α_{Q} ∈ K_{∗}^{M}(k(A_{Q}))/2, s.t. for arbitrary field extension E/k, our quadric Q has Epoint if and only if α_{Q}|_{E(A_{Q})} = 0.

Statement 2.

Suppose we have complete flag F of quadrics Q_{0} ⊂ Q_{1} ⊂ ⋯ ⊂ Q_{n} = Q, where n = dim(Q).

Then there exists purely transcendental field extension k(A_{F})/k, and a pure symbol {y_{0}, …, y_{n}} ∈ K_{n+1}^{M}(k(A_{F}))/2, s.t. for arbitrary field extension E/k, Q_{i} has E-point if and only if {y_{0}, …, y_{i}}|_{E(A_{F})} = 0.

Proof

Induction on the dimension of Q.

1) q = ⟨1, −a_{0}⟩. Evidently, α_{k\sqrt{a_{0}}} = \{a_{0}\} ∈ K_{1}^{M}(k)/2 will have desired property.

2) Let now q_{i} = ⟨1, −a_{0}, …, −a_{i}⟩, and F′ is a flag: Q_{0} ⊂ ⋯ ⊂ Q_{n−1}. By inductive assumption: we have α_{F′} = {y_{0}, …, y_{n−1}} ∈ K_{n}^{M}(k(A_{F′}))/2, which has desired property with respect to F′.

In particular α_{F′}|_{k(Q_{n−1})(A_{F′})} = 0 (Since over k(Q_{n−1}) Q_{n−1} has a point). By the “Third representation theorem” (see [5], IX, Theorem 2.8) this means, that q_{n−1}|_{k(A_{F′})} is similar to a subform of ⟨⟨y_{0}, …, y_{n−1}⟩⟩, and since q_{n−1} represents 1, it is a subform itself. We have: q_{n−1}|_{k(A_{F′})} ⊥ q′′ = ⟨⟨y_{0}, …, y_{n−1}⟩⟩, where q′′ is some form, defined over k(A_{F′}).

From this: if for some field extension E/k, {y_{0}, …, y_{n−1}} ≠ 0 (as an element of K_{n}^{M}(E(A_{F′}))/2), then the following four conditions are equivalent:

a) q|_{E} - is isotropic.

b) q|_{E(A_{F′})} - is isotropic.

c) (⟨⟨y_{0}, …, y_{n−1}⟩⟩ ⊥ −(q′′ ⊥ ⟨a_{n}⟩))|_{E(A_{F′})} is dim(q′′) + 1-times isotropic.

d) (q′′ ⊥ ⟨a_{n}⟩)|_{E(A_{F′})} - is a subform in ⟨⟨y_{0}, …, y_{n−1}⟩⟩|_{E(A_{Q′})}.

By the “Third representation theorem” (see [5], IX, Theorem 2.8) the last condition is equivalent to the fact: ⟨⟨y_{0}, …, y_{n−1}⟩⟩|_{E(A_{F′})(V_{q′′⊥⟨an⟩})} represents the “generic value” (q′′ ⊥ ⟨a_{n}⟩)(𝔳), where V_{q′′⊥⟨a_{n}⟩} is the underlying k(A_{F′})- vector space of quadratic form q′′ ⊥ ⟨a_{n}⟩ (defined over that field), and 𝔳 - coordinates in V_{q′′⊥⟨a_{n}⟩}. And this is equivalent to: {y_{0}, …, y_{n−1}, (q′′ ⊥ ⟨a_{n}⟩)(𝔳)} = 0 ∈ K_{n}^{M}(E(A_{F′})(V_{q′′⊥⟨a_{n}⟩}))/2.

Put: α_{F} = {y_{0}, …, y_{n−1}, (q′′ ⊥ ⟨a_{n}⟩)(𝔳)}, and k(A_{F}) = k(A_{F′})(V_{q′′⊥⟨a_{n}⟩}). If {y_{0}, …, y_{n−1}}|_{E(A_{F′})} ≠ 0, then we already proved that Q_{E} has a point

if and only if α_{F} |_{E(A_{F})} = 0.

If {y_{0}, …, y_{n−1}}|_{E(A_{F′})} = 0, then Q_{n−1} ⊂ Q has a rational point over E, and, on the other hand, trivially, α_{F} |_{E(A_{F})} = 0.

Corollary 3. All anisotropic quadrics are stably embedable, i.e., for each such quadric Q/k there exists purely transcendental extension k(A_{Q}), s.t. Q|_{k(A_{Q})} ⊂ Q_{α}, where Q_{α}/k(A_{Q}) is Pfister quadric.

Proof

In the previous statement take: E = k(Q), k(A_{Q}) = k(A_{F}) and α = α_{F} for some full flag F of subquadrics in Q. Then since Q|_{k(Q)} is isotropic, we have: α_{Q}|_{k(A_{Q})(Q)} = 0. So, α ∈ K_{n+1}^{M}(k(A_{Q})) is a nontrivial representative of the kernel. And Q|_{k(A_{Q})} is a subquadric in Q_{α_{Q}}.

Corollary 4.

For anisotropic quadric Q, and some quadric R we have, that Q|_{k(R)} is isotropic if and only if R|_{k(A_{Q})} is a subquadric in a Pfister quadric Q_{α_{Q}}.

Remark By the result of D.W.Hoffmann, Q|_{k(R)} is anisotropic if dim(R) ≥ 2^{k} − 1 > dim(Q) (for some k) (see [2], Theorem 1).

This shows, that Q_{α_{Q}} does not contain subquadrics of big dimension defined over k.

Corollary 5.

Let Q, R be quadrics over k, and dim(Q) = dim(R) = n. Let Q^{i} (resp. R^{i}) be variety of i-dimensional projective planes on Q (resp R).

Then the following is equivalent:

(1) M(Q) = M(R), where M(X) is a Chow motive of X.

(2) ∀0 ≤ i ≤ [n/2], ∀E/k - field extension, Ker(K_{∗}^{M}(E)/2 → K_{∗}^{M}(E(Q^{i}))/2) = Ker(K_{∗}^{M}(E)/2 → K_{∗}^{M}(E(R^{i}))/2).

Proof

By [8], Proposition 5.1 we have: (1)⇔ Universal splitting towers of M.Knebusch for Q and R are equivalent (see [4], or [8], Definition 2.4.19).

That is: M(Q) = M(R) ⇔ ∀0 ≤ i ≤ [n/2], Q^{i} has a rational point over k(R^{i}), and R^{i} has a rational point over k(Q^{i}).

So, evidently, (1)⇒(2). And to prove the opposite conclusion, it is enough to show, that if Ker(K_{∗}^{M}(E)/2 → K_{∗}^{M}(E(Q))/2) = Ker(K_{∗}^{M}(E)/2 → K_{∗}^{M}(E(R))/2), ∀E/k, then Q has a rational point over k(R) and R has a rational point over k(Q) (the case of arbitrary 0 ≤ i ≤ [n/2] can be easily obtained from this).

Take E = k(Q)(A_{R}) (notations as above). Then Ker(K_{∗}^{M}(E)/2 → K_{∗}^{M}(E(Q))/2) = 0. Hence the second kernel is zero as well, and so, α_{R}|_{k(Q)(A_{R})} = 0 ∈ K_{∗}^{M}(E)/2. By Statement 2, R has a rational point over k(Q). Similarly, Q has a rational point over k(R).

Remark If M(Q) or M(R) is undecomposable, then in (2) it is enough to consider only i = 0 (see [8], Corollary 3.15).

2 Pure part of the kernel

In this section we will show that for arbitrary quadric Q/k the set of pure symbols in Ker(K_{∗}^{M}(k)/2 → K_{∗}^{M}(k(Q))/2) is generated (under the action of the semigroup of pure symbols from K_{∗}^{M}(k)/2) by the specializations of one element α_{Q} ∈ K_{n+1}^{M}(k(A_{Q})), where α_{Q} can be taken as α_{F} for some flag F (and k(A_{Q}) as k(A_{F})).

Let me remind (see [7]), that the cyclic module over k is a collection of ℤ-graded abelian groups M(E), where E runs through all finitely generated field extensions E/k, which are connected by the following operations:

(1) For any φ : Spec(E) → Spec(F) over k, the homomorphism φ^{∗} : M(F) → M(E) of degree 0 (“pull-back”).

(2) For any finite morphism φ : Spec(E) → Spec(F) over k, the homomorphism φ_{∗} : M(E) → M(F) of degree 0 (“push-forward”).

(3) For any F/k, M(F) has a structure of K_{∗}^{M}(F)-module, which respects grading on both objects.

(4) For a discrete valuation 𝔳 on F/k, there is the homomorphism ∂_{𝔳} : M(F) → M(k(𝔳)) of degree −1 (“derivation”).

All this data should satisfy natural compatibility rules (see [7], notice, that our notations are a bit different). Typical example is M(E) = K_{∗}^{M}(E).

It is not difficult to see, that M(E) = Ker_{Q|_{E}} will also be one (really, any submodule of cyclic module, closed under operations is a cyclic module; and for a cyclic module M, M′ := M/r, where r is some fixed integer, is also a cyclic module).

Together with a derivation we can consider “specialization” map. Namely, if π is uniformizing parameter for 𝔳, then we have: S_{𝔳}^{π} : M(F) → M(k(𝔳)) of degree 0, where S_{𝔳}^{π}(x) := ∂_{𝔳}({−π} · x). If ∂_{𝔳}(x) = 0, then S_{𝔳}^{π}(x) does not depend on the choice of π. If X is a smooth variety, and Spec(E) is a point on it, given by some ℤ^{r} - valuation, then the composition S_{𝔳_{r}}^{π_{r}} ∘ ⋯ ∘ S_{𝔳_{1}}^{π_{1}} defines a specialization map from M_{0,E}(k(X)) to M(E), where M_{0,E}(k(X)) := Ker(M(k(X)) →^{∂_{D}} ⊕_{Spec(E)∈D−div.ofX}M(k(D))). This map does not depend on the choice of ℤ^{r} - valuation, or on the uniformizers (see [7]). Notice, that the specialization map is a composition of maps of type (3) and (4).

Suppose we have a cyclic module M, some (finitely generated) field extension E/k, and some element x ∈ M(E). Then it is natural to say, that cyclic submodule generated by x is the minimal cyclic submodule M_{x} of M, containing x. Since we know, that cyclic submodules are precisely submodules, closed under operations, we have M_{x}(F) consists of those y ∈ M(F), that y can be obtained from x using operations (1), (2), (3), (4). Similarly, we say, that submodule weakly generated by x is the minimal submodule (not nesessarily cyclic) M_{x}^{w} of M, closed under (1), (3) and (4).

Denote by PKer_{Q} the ideal in K_{∗}^{M}(k)/2, generated by pure symbols from Ker_{Q}. Also, we will denote the same way the corresponding submodule in K_{∗}^{M} /2: PKer_{Q}(E) := PKer_{Q|_{E}}.

Proposition 6.

PKer_{Q} coinsides with a submodule of K_{∗}^{M} /2 weakly generated by α_{Q}. Proof

As we know from Statement 2, α_{Q} ∈ PKer_{Q}(k(A_{Q})). Also, it is clear, that operations (1) and (3) preserve PKer_{Q}.

Lemma 7. Operation (4) (acting on K_{∗}^{M}) maps pure symbols to pure ones. Proof Really, for a valuation 𝔳 on a filed F, any pure symbol in F can be written

as β = {π^{k_{1}} · b_{1}, …, π^{k_{n}} · b_{n}}, where π is uniformizing parameter, and b_{j}’s are units for 𝔳.

Applying the fact, that {π^{k} · a, π^{l} · b} = {π^{k} · a, −π^{l−k} · b/a}, we get, that β = {π^{d} ·c_{1}, c_{2}, …, c_{n}}, where d = g.c.d.(k_{j}; j = 1, …, n), and c_{m}’s are units.

Then ∂_{𝔳}(β) = {c_{2}^{d}, c_{3}, …, c_{n}} is pure.

So, we have, that (K_{∗}^{M} /2)_{α_{Q}}^{w} ⊂ PKer_{Q}. Before we prove the opposite inclusion, let us generalize a bit the con-

struction of Universal symbol.

Suppose we have two quadratic forms p′ ⊂ p′′, full flag G of forms “between them”: p′ = p_{0} ⊂ p_{1} ⊂ ⋯ ⊂ p_{s} = p′′, where dim(p_{t+1}) − dim(p_{t}) = 1, and an inclusion of p′ into some Pfister form ⟨⟨γ⟩⟩. Let p_{j} = p_{0} ⊥ ⟨−c_{1}, …, −c_{j}⟩. For any 0 ≤ j ≤ t ≤ s, we will denote as G^{[j,t]} the subflag p_{j} ⊂ ⋯ ⊂ p_{t}.

Then we can construct some purely transcendental field extension k(A_{G,γ}), and some pure symbol α_{G,γ} in the following way.

Let ⟨⟨γ⟩⟩ = p_{0} ⊥ q_{G^{[0,0]},γ}, then take: k(A_{G^{[0,1]},γ}) := k(V_{q_{G^{[0,0]},γ}⊥⟨c_{1}⟩}), and take α_{G^{[0,1]},γ} := γ · {q_{G^{[0,0]},γ} ⊥ ⟨c_{1}⟩(𝔳)}, where q_{G^{[0,0]},γ} ⊥ ⟨c_{1}⟩(𝔳) is the generic value of q_{G^{[0,0]},γ} ⊥ ⟨c_{1}⟩. We have: p_{1}|_{k(A_{G^{[0,1]},γ})} ⊂ ⟨⟨α_{G^{[0,1]},γ}⟩⟩.

Now, by definition, k(A_{G^{[0,i+1]},γ}) := k(A_{G^{[0,i]},γ})(A_{G^{[i,i+1]},α_{G^{[0,i]},γ}}), and α_{G^{[0,i+1]},γ} := α_{G^{[i,i+1]},α_{G^{[0,i]},γ}}. Also, by definition: ⟨⟨α_{G^{[0,l]},γ}⟩⟩ = p_{l}|_{k(A_{G^{[0,l]},γ})} ⊥ q_{G^{[0,l]},γ}.

Let us return now to the proof of Proposition 6. Suppose we have some pure symbol β in Ker(K_{∗}^{M}(k)/2 → K_{∗}^{M}(k(Q))/2). Let F be full flag of quadrics: Q_{0} ⊂ Q_{1} ⊂ ⋯ ⊂ Q_{n} = Q, q_{i} =

⟨1, −a_{0}, …, −a_{i}⟩, which we used in the construction of α_{Q} = α_{F}. Notice, that q_{0} = ⟨⟨a_{0}⟩⟩. Then it is easy to see, that k(A_{F}) = k(A_{F,{a_{0}}}), and α_{F} = α_{F,{a_{0}}}.

We have q_{n} = q ⊂ ⟨⟨β⟩⟩ for some β ∈ K_{∗}^{M}(k)/2 (since q represents 1). Take β_{Q_{0}} := {a_{0}}. We have: q_{0} ⊂ ⟨⟨β_{Q_{0}}⟩⟩ ⊂ ⟨⟨β⟩⟩.

Suppose q_{i} ⊂ ⟨⟨β_{Q_{i}}⟩⟩ ⊂ ⟨⟨β⟩⟩, deg(β_{Q_{i}}) = j, then deg(α_{F^{[i,n]},β_{Q_{i}}}) = j+n−i.

We have: α_{F^{[i,n]},β_{Q_{i}}} = β_{Q_{i}} · {y_{i+1}, …, y_{n}}, where y_{l} is a pull-back of (q_{F^{[i,l−1]},β_{Q_{i}}} ⊥ ⟨a_{l}⟩)(𝔳_{l}) to k(A_{F^{[i,n]},β_{Q_{i}}}).

Let q_{i} ⊂ ⟨⟨β_{Q_{i}}⟩⟩, and ⟨⟨β_{Q_{i}}⟩⟩ = q_{i} ⊥ q_{F^{[i,i]},β_{Q_{i}}}. We have two possibilities: 1)q_{F^{[i,i]},β_{Q_{i}}} ⊥ ⟨a_{i+1}⟩ is isotropic, i.e. there are t_{0}, z_{0}, that q_{F^{[i,i]},β_{Q_{i}}}(t_{0}) + a_{i+1}z_{0}^{2} = 0. 2)q_{F^{[i,i]},β_{Q_{i}}} ⊥ ⟨a_{i+1}⟩ is anisotropic.

1) In this case we have: divisor y_{i+1} = 0 on A_{F^{[i,i+1]},β_{Q_{i}}} = V_{q_{F^{[i,i]},β_{Q_{i}}} ⊥⟨a_{i+1}⟩} has rational point (t_{0}, z_{0}), and q_{i+1} ⊂ ⟨⟨β_{Q_{i}}⟩⟩. So, we can construct k(A_{F^{[i+1,n]},β_{Q_{i}}}), and α_{F^{[i+1,n]},β_{Q_{i}}}. Let Y_{i+1} = 0 be the preimage of the divisor y_{i+1} = 0 under the natural projection A_{F^{[i,n]},β_{Q_{i}}} → V_{q_{F^{[i,i]},β_{Q_{i}}} ⊥⟨a_{i+1}⟩}.

Let p′_{l} be the specialization of q_{F^{[i,l]},β_{Q_{i}}} over k(A_{F^{[i,l]},β_{Q_{i}}}) at the generic point E_{l} of the fiber of A_{F^{[i,l]},β_{Q_{i}}} over the rational point (t_{0}, z_{0}) of V_{q_{F^{[i,i]},β_{Q_{i}}} ⊥⟨a_{i+1}⟩}.

Then E_{l} is naturally a purely transcendental extension of k(A_{F^{[i+1,l]},β_{Q_{i}}}), and if j : Spec(E_{l}) → Spec(k(A_{F^{[i+1,l]},β_{Q_{i}}})) is corresponding morphism, then we have: j^{∗}(q_{F^{[i+1,l]},β_{Q_{i}}}) is a nondegenerate part of p′_{l} (it can be easily proved by induction on l).

Hence, α_{F^{[i+1,n]},β_{Q_{i}}} ∈ K_{j+n−i−1}^{M}(k(A_{F^{[i+1,n]},β_{Q_{i}}}))/2 is a specialization of the derivative ∂_{Y_{i+1}=0}(α_{F^{[i,n]},β_{Q_{i}}}).

We can take β_{Q_{i+1}} = β_{Q_{i}}. Then q_{i+1} ⊂ ⟨⟨β_{Q_{i+1}}⟩⟩ ⊂ ⟨⟨β⟩⟩.

2) We have: q_{i} ⊂ ⟨⟨β_{Q_{i}}⟩⟩ ⊂ ⟨⟨β⟩⟩, and q_{i} ⊂ q_{i+1} ⊂ ⟨⟨β⟩⟩. Since q_{F^{[i,i]},β_{Q_{i}}} ⊥ ⟨a_{i+1}⟩ is anisotropic, there exist some subform p of ⟨⟨β⟩⟩ of dimension = dim(⟨⟨β_{Q_{i}}⟩⟩) + 1, which contains both ⟨⟨β_{Q_{i}}⟩⟩ and q_{i+1}. That means, that there exist (t_{0}, z_{0}), s.t. p = ⟨⟨β_{Q_{i}}⟩⟩ ⊥ ⟨q_{F^{[i,i]},β_{Q_{i}}}(t_{0}) + a_{i+1}z_{0}^{2}⟩.

Take: β_{Q_{i+1}} := β_{Q_{i}} · {q_{F^{[i,i]},β_{Q_{i}}}(t_{0}) + a_{i+1}z_{0}^{2}} ∈ K_{j+1}^{M}(k)/2. Then q_{i+1} ⊂ ⟨⟨β_{Q_{i+1}}⟩⟩ ⊂ ⟨⟨β⟩⟩.

Evidently, β_{Q_{i+1}} is a specialization of α_{F^{[i,i+1]},β_{Q_{i}}} ∈ K_{j+1}^{M}(k(V_{q_{F^{[i,i]},β_{Q_{i}}} ⊥⟨a_{i+1}⟩}))/2 at the rational point (t_{0}, z_{0}).

Also, we have natural identification: k(A_{F^{[i+1,l]},β_{Q_{i+1}}}) with E_{l}, where E_{l} is defined precisely as in the case (1). Under this identification, q_{F^{[i+1,l]},β_{Q_{i+1}}} is isomorphic to the specialization of q_{F^{[i,l]},β_{Q_{i}}} at the generic point (E_{l}) of the fiber of the natural projection A_{F^{[i,l]},β_{Q_{i}}} → V_{q_{F^{[i,i]},β_{Q_{i}}} ⊥⟨a_{i+1}⟩} over the point (t_{0}, z_{0}).

Hence, α_{F^{[i+1,n]},β_{Q_{i+1}}} ∈ K_{j+n−i}^{M}(k(A_{F^{[i+1,n]},β_{Q_{i+1}}}))/2 is a specialization of α_{F^{[i,n]},β_{Q_{i}}} ∈ K_{j+n−i}^{M}(k(A_{F^{[i,n]},β_{Q_{i}}}))/2.

Continuing this way, we get: q = q_{n} ⊂ ⟨⟨β_{Q_{n}}⟩⟩ ⊂ ⟨⟨β⟩⟩, and for each 0 ≤ i < n, α_{F^{[i+1,n]},β_{Q_{i+1}}} ∈ K_{j+n−i}^{M}(k(A_{F^{[i+1,n]},β_{Q_{i+1}}}))/2 is obtained from α_{F^{[i,n]},β_{Q_{i}}} ∈ K_{j+n−i}^{M}(k(A_{F^{[i,n]},β_{Q_{i}}}))/2 using derivations and specializations.

But, evidently, k(A_{F^{[n,n]},β_{Qn}}) = k, and α_{F^{[n,n]},β_{Qn}} = β_{Q_{n}}.

On the other hand, as we saw, α_{F^{[0,n]},β_{Q_{0}}} = α_{F}.

Hence, β is divisible by some pure symbol, obtained from α_{F} using derivations and specializations.

So, PKer_{Q} ⊂ (K_{∗}^{M} /2)_{α_{Q}}^{w}.

Proposition is proven.

Remark 1 In particular we see, that for any pure symbol β ∈ Ker(K_{∗}^{M}(k)/2 → K_{∗}^{M}(k(Q))/2), β is divisible by a pure symbol from the kernel of degree ≤ dim(Q) + 1. This result was known to M.Knebusch, as E.R.Gentile and D.B.Shapiro point out (see [1], Remark after Corollary 8).

Remark 2 Actually, we could make our universal symbol α_{Q} a bit more “universal”. Namely, we can consider F (ℙ^{n+1}) - variety of full flags in ℙ^{n+1} ⊃ Q.

Then taking the generic point k(F (ℙ^{n+1})) of later variety with generic flag F_{gen}, we would get corresponding universal symbol α_{gen,Q} := α_{F_{gen}} ∈ K_{n+1}^{M}(k(F (ℙ^{n+1}))(A_{F_{gen}}))/2.

This way, α_{gen,Q} does not depend on anything but Q itself, and our usual α_{F_{0}} is just a specialization of α_{gen,Q} at the generic point of the fiber of A_{F} over k-rational point F_{0} of F (ℙ^{n+1}) (notice, that we can consider A_{F} as a fibration over F (ℙ^{n+1}) with rational fibers).

From Proposition 6 it follows that (K_{∗}^{M} /2)_{α_{F_{0}}}^{w} = (K_{∗}^{M} /2)_{α_{gen,Q}}^{w}. Really, the first module is a submodule of the later (since α_{F_{0}} is a specialization of α_{gen,Q}), and the later module is, evidently, a submodule in PKer_{Q}.

Since the same is true for any extension E/k, we get: (K_{∗}^{M} /2)_{α_{F_{0}}} = (K_{∗}^{M} /2)_{α_{gen,Q}} as well (see [7]).

In the light of Proposition 6, our Question 1 is equivalent to the following: Question 8. Is the following true?

a) Ker_{Q} = (K_{∗}^{M} /2)_{α_{Q}}.

b) PKer_{Q} is stable under transfers.

Notice, that Question 8)b) is trivial for dim(Q) ≤ 2; to the contrary - Question 8)a) for dim(Q) = 0 is more or less equivalent to Milnor’s conjecture (on e’tale cohomology). Hopefully, due to V.Voevodsky’s methods we know that the answer is positive for Q - Pfister quadric (see [6]). Natural strategy for Question 8)a) would be to reduce the case of arbitrary quadric Q/k to that of Pfister one Q_{α_{Q}}/k(A_{Q}), using the universality of the symbol α_{Q}.

References

[1] E.R.Gentile, D.B.Shapiro, Conservative Quadratic Forms., Math. Zeit., 163 (1978), 15-23.

[2] D.W.Hoffmann, Isotropy of quadratic forms over the function field of a quadric., Math. Zeit., 220 (1995), 461-476.

[3] B.Kahn, M.Rost, R.J.Sujatha, Unramified cohomology of quadrics., to appear in Amer.J.Math.

[4] M.Knebusch, Generic splitting of quadratic forms, I., Proc. London Math. Soc. 33 (1976), 65-93.

[5] T.Y.Lam, Algebraic theory of quadratic forms., Massachusets, Benjamin 1973.

[6] D.Orlov, A.Vishik, V.Voevodsky, Motivic cohomology of Pfister quadrics and Milnor’s conjecture on quadratic forms. Preprint.

[7] M.Rost, Chow groups with coefficients., Doc. Math. 1 (1996), 319-393. MR 98a:14006, Zbl 864.14002.

[8] A.Vishik, Integral Motives of Quadrics, MPIM-preprint, 1998 (13).