An exact sequence for K_{∗}^{M}/2 with applications to quadratic forms

D. Orlov^{1}, A. Vishik^{2} and V. Voevodsky^{3}

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 An exact sequence for K_{∗}^{M}/2 . . . . . . . . . . . . . . . . . . . 2 3 Reduction to points of degree 2 . . . . . . . . . . . . . . . . . 8 4 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.1 Milnor’s Conjecture on quadratic forms. . . . . . . . . 12 4.2 Kahn-Rost-Sujatha Conjecture . . . . . . . . . . . . . 13 4.3 J– filtration conjecture. . . . . . . . . . . . . . . . . . . 13

1 Introduction

Let k be a field of characteristics zero. For a sequence \underline{a} = (a_{1}, …, a_{n}) of invertible elements of k consider the homomorphism

K_{∗}^{M}(k)/2 → K_{∗+n}^{M}(k)/2

in Milnor’s K-theory modulo elements divisible by 2 defined by the multiplication with the symbol corresponding to \underline{a}. The goal of this paper is to construct a four-term exact sequence (18) which provides information about the kernel and cokernel of this homomorphism.

The proof of our main theorem (Theorem 3.2) consists of two independent parts. Let Q\underline{a} be the norm quadric defined by the sequence \underline{a} (see below). First, we use the techniques of [12] to establish a four term exact sequence (1) relating the kernel and cokernel of the multiplication by \underline{a} with Milnor’s K-theory of the closed and the generic points of Q\underline{a} respectively. This is done in the first section. Then, using elementary geometric arguments, we show that the sequence can be rewritten in its final form (18) which involves only the generic point and the closed points with residue fields of degree 2.

1supported by the NSF grant DMS 97-29992 and the RFFI–99–01–01144

2supported by the NSF grant DMS 97-29992

3Supported by the NSF grants DMS-97-29992 and DMS-9901219 and The Ambrose Monell Foundation

As an application we establish, for fields of characteristics zero, the validity of three conjectures in the theory of quadratic forms - the Milnor conjecture on the structure of the Witt ring, the Khan-Rost-Sujatha conjecture and the J-filtration conjecture. All these results require only the first form of our exact sequence. Using the final form of the sequence we also show that the kernel of multiplication by \underline{a} is generated, as a K_{∗}^{M}(k)-module, by its components of degree ≤ 1.

This paper is a natural extension of [12] and we feel free to refer to the results of [12] without reproducing them here. Most of the mathematics used in this paper was developed in the spring of 1996 when all three authors were at Harvard. In its present form the paper was written while the authors were members of the Institute for Advanced Study in Princeton. We would like to thank both institutions for their support.

2 An exact sequence for K_{∗}^{M}/2

Let \underline{a} = (a_{1}, ..., a_{n}) be a sequence of elements of k^{∗}. Recall that the n–fold Pfister form ⟨⟨a_{1}, ..., a_{n}⟩⟩ is defined as the tensor product

⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩

where ⟨1, −a_{i}⟩ is the norm form in the quadratic extension k(\sqrt{a_{i}}). Denote by Q\underline{a} the projective quadric of dimension 2^{n−1} − 1 defined by the form q\underline{a} = ⟨⟨a_{1}, ..., a_{n−1}⟩⟩ − ⟨a_{n}⟩. This quadric is called the small Pfister quadric or the norm quadric associated with the symbol \underline{a}. Denote by k(Q\underline{a}) the function field of Q\underline{a} and by (Q\underline{a})_{0} the set of closed points of Q\underline{a}. The following result is the main theorem of the paper.

Theorem 2.1 Let k be a field of characteristic zero. Then for any sequence of invertible elements (a_{1}, …, a_{n}) the following sequence of abelian groups is exact

∐_{x∈(Q\underline{a})_{(0)}} K_{∗}^{M}(k(x))/2 →^{Tr_{k(x)/k}} K_{∗}^{M}(k)/2 →^{·\underline{a}} K_{∗+n}^{M}(k)/2 → K_{∗+n}^{M}(k(Q\underline{a}))/2 (1)

The proof goes as follows. We first construct two exact sequences of the form

0 → K → K_{∗+n}^{M}(k)/2 → K_{∗+n}^{M}(k(Q\underline{a}))/2  (2)

and

∐_{x∈(Q\underline{a})_{(0)}} K_{∗}^{M}(k(x))/2 →^{Tr_{k(x)/k}} K_{∗}^{M}(k)/2 → I → 0  (3)

and then construct an isomorphism I → K such that the composition

K_{∗}^{M}(k)/2 → I → K → K_{∗+n}^{M}(k)/2

is the multiplication by \underline{a}.

Our construction of the sequence (2) makes sense for any smooth scheme X and we shall do it in this generality. Recall that we denote by Č(X) the simplicial scheme such that Č(X)_{n} = X^{n+1} and faces and degeneracy morphisms are given by partial projections and diagonal embeddings respectively. We will use repeatedly the following lemma which is an immediate corollary of [12, Proposition 2.7] and [12, Corollary 2.13].

Lemma 2.2 For any smooth scheme X over k and any p ≤ q the homomorphism

H^{p,q}(Spec(k), Z/2) → H^{p,q}(Č(X), Z/2)

defined by the canonical morphism Č(X) → Spec(k), is an isomorphism.

Proposition 2.3 For any n ≥ 0 there is an exact sequence of the form

0 → H^{n,n−1}(Č(X), Z/2) → K_{n}^{M}(k)/2 → K_{n}^{M}(k(X))/2  (4)

Proof: The computation of motivic cohomology of weight 1 shows that

Hom(Z/2, Z/2(1)) ≅ H^{0,1}(Spec(k), Z/2) ≅ Z/2

The nontrivial element τ : Z/2 → Z/2(1) together with the multiplication morphism Z(n − 1) ⊗ Z/2(1) →^{∼} Z/2(n) defines a morphism τ : Z/2(n − 1) → Z/2(n). The Beilinson-Lichtenbaum conjecture implies immediately the following result.

Lemma 2.4 The morphism τ extends to a distinguished triangle in DM_{−}^{eff} of the form

Z/2(n − 1) →^{·τ} Z/2(n) → \underline{H}^{n,n}(Z/2)[−n],  (5)

where \underline{H}^{n,n}(Z/2(n)) is the n–th cohomology sheaf of the complex Z/2(n).

Consider the long sequence of morphisms in the triangulated category of motives from the motive of Č(X) to the distinguished triangle (5). It starts as

0 → H^{n,n−1}(Č(X), Z/2) → H^{n,n}(Č(X), Z/2) → H^{0}(Č(X), \underline{H}^{n,n}(Z/2))

By Lemma 2.2 there are isomorphisms

H^{n,n}(Č(X), Z/2) = H^{n,n}(Spec(k), Z/2) = K_{n}^{M}(k)/2

On the other hand, since \underline{H}^{n,n}(Z/2) is a homotopy invariant sheaf with transfers, we have an embedding

H^{0}(Č(X), \underline{H}^{n,n}(Z/2)) ↪ \underline{H}^{n,n}(Z/2)(Spec(k(X)))

The right hand side is isomorphic to H^{n,n}(Spec(k(X)), Z/2) = K_{n}^{M}(k(X))/2. This completes the proof of the proposition.

Let us now construct the exact sequence (3). Denote the standard simplicial scheme Č(Q\underline{a}) by X\underline{a}. Recall that we have a distinguished triangle of the form

M(X\underline{a})(2^{n−1} − 1)[2^{n} − 2] →^{φ} M\underline{a} →^{ψ} M(X\underline{a}) →^{μ′} M(X\underline{a})(2^{n−1} − 1)[2^{n} − 1] (6)

where M\underline{a} is a direct summand of the motive of the quadric Q\underline{a}. Denote the composition

M(X\underline{a}) →^{μ′} M(X\underline{a})(2^{n−1} − 1)[2^{n} − 1] →^{pr} Z/2(2^{n−1} − 1)[2^{n} − 1]  (7)

by μ ∈ H^{2^{n}−1,2^{n−1}−1}(X\underline{a}, Z/2). By Lemma 2.2 we have

H^{i,i}(X\underline{a}, Z/2) = H^{i,i}(Spec(k), Z/2) = K_{i}M(k)/2

Therefore, multiplication with μ defines a homomorphism

K_{i}^{M}(k)/2 →^{·μ} H^{i+2^{n}−1,i+2^{n−1}−1}(X\underline{a}, Z/2).

Proposition 2.5 The sequence

∐_{x∈(Q\underline{a})_{(0)}} K_{i}^{M}(k(x))/2 →^{Tr_{k(x)/k}} K_{i}^{M}(k)/2 →^{·μ} H^{i+2^{n}−1,i+2^{n−1}−1}(X\underline{a}, Z/2) → 0 (8)

is exact.

Proof: Let us consider morphisms in the triangulated category of motives from the distinguished triangle (6) to the object Z/2(i+2^{n−1}−1)[i+2^{n}−1]. By definition, M\underline{a} is a direct summand of the motive of the smooth projective variety Q\underline{a} of dimension 2^{n−1}−1, therefore, the group H^{i+2^{n}−1,i+2^{n−1}−1}(M\underline{a}, Z/2) is trivial by [12, Corollary 2.3]. Using this fact, we obtain the following exact sequence:

H^{i+2^{n}−2,i+2^{n−1}−1}(M\underline{a}, Z/2) →^{φ^{∗}} H^{i,i}(X\underline{a}, Z/2) →^{μ′^{∗}}  (9)
→ H^{i+2^{n}−1,i+2^{n−1}−1}(X\underline{a}, Z/2)→0

By definition (see [12, p.43]) the morphism φ is given by the composition

M(X\underline{a})(2^{n−1} − 1)[2^{n} − 2] →^{pr} Z(2^{n−1} − 1)[2^{n} − 2] → M\underline{a}  (10)

and the composition of the second arrow with the canonical embedding M\underline{a} → M(Q\underline{a}) is the fundamental cycle map

Z(2^{n−1} − 1)[2^{n} − 2] → M(Q\underline{a})

which corresponds to the fundamental cycle on Q\underline{a} under the isomorphism

Hom(Z(2^{n−1} − 1)[2^{n} − 2], M(Q\underline{a})) = CH_{2^{n−1}−1}(Q\underline{a}) ≅ Z

(see [12, Theorem 4.5]). On the other hand by Lemma 2.2 the homomorphism

H^{i,i}(Spec(k), Z/2) → H^{i,i}(X\underline{a}, Z/2)

defined by the first arrow in (10) is an isomorphism. This implies immediately that the exact sequence (9) defines an exact sequence of the form

H^{i+2^{n}−2,i+2^{n−1}−1}(Q\underline{a}, Z/2) →^{φ^{∗}} H^{i,i}(Spec(k), Z/2) →^{μ′^{∗}}  (11)
→ H^{i+2^{n}−1,i+2^{n−1}−1}(X\underline{a}, Z/2)→0

By [12, Corollary 2.4] there is an isomorphism

H^{i+2^{n}−2,i+2^{n−1}−1}(Q\underline{a}, Z/2) ≅ H^{2^{n−1}−1}(Q\underline{a}, \underline{K}_{i+2^{n−1}−1}^{M}/2)

The Gersten resolution for the sheaf \underline{K}_{m}^{M}/2 (see, for example, [10]) shows that the group H^{2^{n−1}−1}(Q\underline{a}, \underline{K}_{i+2^{n−1}−1}^{M}/2) can be identified with the cokernel of the map:

∐_{y∈(Q\underline{a})_{(1)}} K_{i+1}^{M}(k(y))/2 →^{∂} ∐_{x∈(Q\underline{a})_{(0)}} K_{i}^{M}(k(x))/2,

and the map H^{i+2^{n}−2,i+2^{n−1}−1}(Q\underline{a}, Z/2)→H^{i,i}(Spec(k), Z/2) defined by the fundamental cycle corresponds in this description to the map

∐_{x∈(Q\underline{a})_{(0)}} K_{i}^{M}(k(x))/2 →^{Tr_{k(x)/k}} K_{i}^{M}(k)/2 = H^{i,i}(Spec(k), Z/2)

This finishes the proof of Proposition 2.5.

We are going to show now that the map K_{∗}^{M}(k)/2 →^{\underline{α}} K_{∗+n}^{M}(k)/2 glues the exact sequences (4) and (8) in one. Denote by ℍ^{i}(X\underline{a}) the direct sum ⊕_{m}H^{m+i,m}(X\underline{a}, Z/2). It has a natural structure of a graded module over the ring K_{∗}^{M}(k)/2 and one can easily see that the sequences (4) and (8) define sequences of K_{∗}^{M}(k)/2–modules of the form

0 → ℍ^{1}(X\underline{a}) → K_{∗}^{M}(k)/2 → K_{∗}^{M}(k(Q\underline{a}))/2  (12)
∐_{x∈(Q\underline{a})_{(0)}} K_{∗}^{M}(k(x))/2 →^{Tr_{k(x)/k}} K_{∗}^{M}(k)/2 →^{·μ} ℍ^{2^{n−1}}(X\underline{a}) → 0  (13)

Consider cohomological operations

Q_{i} : H^{•,∗}(−, Z/2) → H^{•+2^{i+1}−1,∗+2^{i}−1}(−, Z/2)

introduced in [12, p.32]. The composition Q_{n−2} ⋯ Q_{0} defines a homomorphism of graded abelian groups d : ℍ^{1}(X\underline{a}) → ℍ^{2^{n−1}}(X\underline{a}) and [12, Theorem 3.17(2)] together with the fact that H^{p,q}(Spec(k), Z/2) = 0 for p > q implies that d is a homomorphism of K_{∗}^{M}(k)/2-modules. We are going to show that d is an isomorphism and that the composition

K_{∗}^{M}(k)/2 →^{·μ} ℍ^{2^{n−1}}(X\underline{a}) →^{d^{−1}} ℍ^{1}(X\underline{a}) → K_{∗}^{M}(k)/2  (14)

is the multiplication with \underline{a}.

Lemma 2.6 The homomorphism d is injective.

Proof: We have to show that the composition of operations

Q_{n−2} … Q_{0} : H^{∗+n,∗+n−1}(X\underline{a}, Z/2)→H^{∗+2n−1,∗+2^{n−1}−1}(X\underline{a}, Z/2)

is injective. Let X̃\underline{a} be the simplicial cone of the morphism X\underline{a} → Spec(k) which we consider as a pointed simplicial scheme. The long exact sequence of cohomology defined by the cofibration sequence

(X\underline{a})_{+} → Spec(k)_{+} → X̃\underline{a} → Σ_{s}^{1}((X\underline{a})_{+})  (15)

together with the fact that H^{p,q}(Spec(k), Z/2) = 0 for p > q shows that for p > q + 1 we have a natural isomorphism H^{p,q}(X̃\underline{a}, Z/2) = H^{p−1,q}(X\underline{a}, Z/2) compatible with the actions of cohomological operations. Therefore, it is sufficient to prove injectivity of the composition Q_{2} … Q_{0} on motivic cohomology groups of the form H^{∗+n+1,∗+n−1}(X̃\underline{a}, Z/2). To show that Q_{n−2} ⋯ Q_{0} is a monomorphism it is sufficient to check that the operation Q_{i} acts monomorphically on the group

H^{∗+n−i+2^{i+1}−1),∗+n−i−2^{i}−2}(X̃\underline{a}, Z/2)

for all i = 0, ..., n − 2. For any i ≤ n − 1 we have ker(Q_{i}) = Im(Q_{i}) by [12, Theorem 3.25] and [12, Lemma 4.11]. Therefore, the kernel of Q_{i} on our group is the image of H^{∗+n−i),∗+n−i−1}(X̃\underline{a}, Z/2). On the other hand, the cofibration sequence (15) together with Lemma 2.2 implies that for p ≤ q + 1 we have H^{p,q}(X̃\underline{a}, Z/2) = 0 which proves the lemma.

Denote by γ the element of H^{n,n−1}(X\underline{a}, Z/2) which corresponds to the symbol \underline{a} under the embedding into K_{n}^{M}(k)/2 (sequence (4)). To prove that d is surjective and that the composition (14) is multiplication with \underline{a} we use the following lemma.

Lemma 2.7 The composition K_{∗}^{M}(k)/2 →^{·γ} ℍ^{1}(X\underline{a}) →^{d} ℍ^{2^{n−1}}(X\underline{a}) coincides with the multiplication by μ.

Proof: Since our maps are homomorphisms of K_{∗}^{M}(k)-modules it is sufficient to verify that the cohomological operation d sends γ ∈ H^{n,n−1}(X\underline{a}, Z/2) to μ ∈ H^{2^{n}−1,2^{n−1}−1}(X\underline{a}, Z/2). By Lemma 2.6 d is injective. Therefore, the element d(γ) iz nonzero. On the other hand, sequence (8) shows that

H^{2^{n}−1,2^{n−1}−1}(X\underline{a}, Z/2) ≅ K_{0}^{M}(k)/2 ≅ Z/2

and μ is a generator of this group. Therefore, d(γ) = μ.

Lemma 2.8 The homomorphism d is surjective.

Proof: Follows immediately from Lemma 2.7 and surjectivity of multiplication by μ (Proposition 2.5).

Lemma 2.9 The composition (14) is the multiplication with \underline{a}.

Proof: Since Q\underline{a} has a rational point over the field k(Q\underline{a}) the symbol \underline{a} is divisible by 2 in K_{n}^{M}(k(Q\underline{a})) and therefore the morphism of the multiplication by \underline{a} factors through ℍ^{1}(X\underline{a}). To verify that (14) is the multiplication with \underline{a} it remains to check that d of the image of the generator of K_{∗}^{M}/2(k) in ℍ^{1}(X\underline{a}) is its image in ℍ^{2^{n−1}}(X\underline{a}). This is exactly the statement of Lemma 2.7.

This finishes the proof of Theorem 2.1.

The following statement, which is easily deduced from the exact sequence (1), is the key to many applications. Let E/k be a field. For any element h ∈ K_{n}^{M}(k) denote by h|_{E}, as usually, the restriction of h on E, i.e. image of h under the natural morphism K_{n}^{M}(k) → K_{n}^{M}(E).

Theorem 2.10 For any field k and any nonzero h ∈ K_{n}^{M}(k)/2 there exist a field E/k and a pure symbol α = {a_{1}, …, a_{n}} ∈ K_{n}^{M}(k)/2 such that h|_{E} = α|_{E} is a nonzero pure symbol of K_{n}^{M}(E).

Proof: Let h = α_{1} + ⋯ + α_{l}, where α_{i} are pure symbols corresponding to sequences \underline{a}_{i} = (a_{1i}, …, a_{ni}). Let Q_{\underline{a}_{i}} be the norm quadric corresponding to the symbol α_{i}. For any 0 < i ≤ l denote by E_{i} the field k(Q_{\underline{a}_{1}} ×⋯×Q_{\underline{a}_{i}}). It is clear that h|_{E_{l}} = 0. Let us fix i such that h|_{E_{i+1}} = 0 and h|_{E_{i}} is a nonzero element. Then h|_{E_{i}} belongs to

ker(K_{n}^{M}(E_{i})/2 → K_{n}^{M}(E_{i+1})/2)

By Theorem 2.1, the kernel is covered by K_{0}^{M}(E_{i}) ≅ Z/2 and is generated by α_{i+1}|_{E_{i}}. Thus, we have α_{i+1}|_{E_{i}} = h|_{E_{i}} ≠ 0.

3 Reduction to points of degree 2

In this section we prove the following result.

Theorem 3.1 Let k be a field such that char(k) ≠ 2 and Q be a smooth quadric over k. Let Q_{(0)} be the set of closed points of Q and Q_{(0,≤2)} the subset in Q_{(0)} of points x such that [k_{x} : k] ≤ 2. Then, for any n ≥ 0, the image of the map

⊕tr_{k_{x}/k} : ⊕_{x∈Q_{(0)}}K_{n}^{M}(k_{x}) → K_{n}^{M}(k)  (16)

coincides with the image of the map

⊕tr_{k_{x}/k} : ⊕_{x∈Q_{(0,≤2)}}K_{n}^{M}(k_{x}) → K_{n}^{M}(k)  (17)

Combining Theorem 2.1 with Theorem 3.1 we get the following result.

Theorem 3.2 Let k be a field of characteristic zero and \underline{a} = (a_{1}, …, a_{n}) a sequence of invertible elements of k. Then the sequence

⊕_{x∈(Q\underline{a})_{(0,≤2)}}K_{i}^{M}(k_{x})/2 → K_{i}^{M}(k)/2 →^{\underline{a}} K_{i+n}^{M}(k)/2 → K_{i+n}^{M}(k(Q\underline{a}))/2 (18)

is exact.

Theorem 3.2 together with the well known result of Bass and Tate (see [1, Corollary 5.3]) implies the following.

Theorem 3.3 Let k be a field of characteristic zero and \underline{a} = (a_{1}, …, a_{n}) a sequence of invertible elements of k such that the corresponding elements of K_{n}^{M}(k)/2 is not zero. Then the kernel of the homomorphism K_{∗}^{M}(k)/2 →^{\underline{a}} K_{∗+n}^{M}(k)/2 is generated, as a module over K_{∗}^{M}(k), by the kernel of the homomorphism K_{1}^{M}(k)/2 → K_{1+n}^{M}(k)/2.

Let us start the proof of Theorem 3.1 with the following two lemmas.

Lemma 3.4 Let E be an extension of k of degree n and V a k-linear subspace in E such that 2dim(V) > n. Then, for any n > 0, K_{n}^{M}(E) is generated, as an abelian group, by elements of the form (x_{1}, …, x_{n}) where all x_{i}’s are in V.

Proof: It is sufficient to prove the statement for n = 1. Let x be an invertible element of E. Since 2dim(V) > dim_{k}E we have V ∩ xV ≠ 0. Therefore x is a quotient of two elements of V ∩ E^{∗}.

Lemma 3.5 Let k be an infinite field and p a closed point in P_{k}^{n}, n ≥ 2 of degree m. Then there exists a rational curve C of degree m − 1 such that p ∈ C and C has at most one singular point.

Proof: We may assume that p lies in A^{n} ⊂ P^{n}. Then there exists a linear function x_{1} on A^{n} such that the map of the residue fields k_{x_{1}(p)} → k_{p} is an isomorphism. Let (x_{1}, …, x_{n}) be a coordinate system starting with x_{1}. Since the restriction of x_{1} to p is an isomorphism the inverse gives a collection of regular functions x̄_{2}, …, x̄_{n} on x_{1}(p) ⊂ A^{1}. Each of this functions has a representative f_{i} in k[x_{1}] of degree at most m − 1. The projective closure of the affine curve given by the equations x_{i} = f_{i}(x_{1}), i = 2, …, n satisfies the conditions of the lemma.

Let Q be any quadric over k. If Q has a rational point (or even a point of odd degree, which is the same by Springer’s theorem, [3, VII,Theorem 2.3]), then Theorem 3.1 for Q holds for obvious reasons. Therefore we may assume that Q has no points of odd degree. It is well known (see e.g. [11, Th.2.3.8, p.39]) that any smooth quadric of dimension > 0 over a finite field of odd characteristic has a rational point. Since the statement of the theorem is obvious for dim(Q) = 0 we may assume that k is infinite.

Let e be a point on Q with the residue field E. We have to show that the image of the transfer map K_{n}^{M}(E) → K_{n}^{M}(k) lies in the image of the map (17). We proceed by induction on d where 2d = [E : k]. If d = 1 there is nothing to prove. Assume by induction that for any closed point f of Q such that [k_{f} : k] < 2d the image of the transfer map K_{n}^{M}(k_{f}) → K_{n}^{M}(k) lies in the image of (17).

If dim(Q) = 0 our statement is obvious. Consider the case of a conic dim(Q) = 1. Let D be any effective divisor on Q of degree 2d−2. Denote by h^{0}(D) the linear space H^{0}(Q, O(D)) which can be identified with the space of rational functions f such that D + (f) is effective. Evaluating elements of h^{0}(D) on e we get a homomorphism h^{0}(D) → E which is injective since deg(D) < 2d. By the Riemann-Roch theorem we have dim(h^{0}(D)) = 2d − 1 and therefore, by Lemma 3.4, K_{n}^{M}(E) is generated by elements of the form {f_{1}(e), …, f_{n}(e)} where f_{i} ∈ h^{0}(D). Let now D′ be an effective divisor on Q of degree 2 (it exists since Q is a conic). Using again the Riemann-Roch theorem we see that dim(|e−D′|) > 0 i.e. that there exists a rational function f with a simple pole in e and a zero in D′. In particular, the degrees of all the points where f has singularities other than e is strictly less than 2d. Consider the symbol {f_{1}, …, f_{n}, f} ∈ K_{n+1}^{M}(k(Q)). Let

∂ : K_{n+1}^{M}(Q) → ⊕_{x∈Q_{(0)}}K_{n}^{M}(k_{x})

be the residue homomorphism. By [10, Prop. 2.2(RC) p.338] its composition with (16) is zero. On the other hand we have

∂({f_{1}, …, f_{n}, f}) = {f_{1}(e), …, f_{n}(e)} + u

where u is a sum of symbols concentrated in the singular points of f_{1}, …, f_{n} and singular points of f other than e. Therefore, by our construction u belongs to ⊕_{x∈Q_{(0),<2d}}K_{n}^{M}(k_{x}) and we conclude that tr_{E/k}{f_{1}(e), …, f_{n}(e)} lies in the image of (17) by induction.

Let now Q be a quadric in P^{n} where n ≥ 3. Let c be a rational point of P^{n} outside Q and π : Q → P^{n−1} be the projection with the center in c. The ramification locus of π is a quadric on P^{n−1} which has no rational points.

Assume first that there exists c such that the degree of π(e) is d. Then, by Lemma 3.5, we can find a (singular) rational curve C′ in P^{n−1} of degree d−1 which contains π(e). Consider the curve C = π^{−1}(C′) ⊂ Q. Let C̃, tildeC′ be the normalizations of C and C′ and π̃ : C̃ → C̃′ the morphism corresponding to π. Since deg(e) = 2d and deg(π(e)) = d the point e does not belong to the ramification locus of π : Q → P^{n−1}. This implies that e lifts to a point ẽ of C̃ of degree 2d and that ẽ = p̃^{−1}(π̃(ẽ)). Since the ramification locus of π has no rational points the singular point of C′ is unramified. This implies that π̃ is ramified in ≤ 2(d − 1) points and, therefore, C̃ is a hyperelliptic curve of genus less or equal to d − 2. Let D be an effective divisor on C̃ of degree 2d − 2. By the Riemann-Roch theorem we have dim(h^{0}(D)) ≥ d + 1. On the other hand, since deg(D) < 2d, the homomorphism h^{0}(D) → E defined by evaluation at ẽ is injective. Therefore, by Lemma 3.4 any element in K_{n}^{M}(E) is of the form {f_{1}(ẽ), …, f_{n}(ẽ)} for f_{i} ∈ h^{0}(D). Let D′ be an effective divisor on C̃ of degree 2. By the Riemann-Roch theorem we have dim(h^{0}(ẽ−D′)) ≥ d+1 > 0. Therefore, there exists a rational function f with simple pole in ẽ and such that all its other singularities are located in points of degree < 2d. We can conclude now that tr_{E/k}{f_{1}(ẽ), …, f_{n}(ẽ)} belongs to the image of (17) in the same way as we did in the case of dim(Q) = 1.

Consider now the general case - we still assume that n ≥ 3 but not that we can find a center of projection c such that deg(π(e)) = d. Taking a general c we may assume that deg(π(e)) = 2d and that e does not belong to the ramification locus of π. By Lemma 3.5 we can find a rational curve C′ in P^{n−1} of degree 2d − 1 which contains π(e). Consider the curve C = π^{−1}(C′) ⊂ Q. Let C̃, tildeC′ be the normalizations of C and C′ and π̃ : C̃ → C̃′ the morphism corresponding to π. Since the point e does not belong to the ramification locus of π it lifts to a point ẽ of C̃ of degree 2d. Since the ramification locus of π does not have rational points and the only singular point of C′ is rational, π̃ is ramified in no more than 2(2d − 1) points and, therefore, C̃ is a hyperelliptic curve of genus ≤ 2d − 2. Let D be an effective divisor on C̃′ = P^{1} of degree d. We have dim(h^{0}(D)) = d + 1 and since π̃(ẽ) has degree 2d the evaluation at π̃(ẽ) gives an injective homomorphism h^{0}(D) → E. By Lemma 3.4, we conclude that any element of K_{n}^{M}(E) is a linear combination of elements of the form {f_{1}(π̃(ẽ)), …, f_{n}(π̃(ẽ))} where f_{i} are in h^{0}(D). Let D′ be an effective divisor of degree 2 on C̃. By the Riemann-Roch theorem we have dim(h^{0}(ẽ − D′)) ≥ 1 i.e. there exists a rational function f with a simple pole in ẽ and a zero in D′. If d > 1 then all the singular points of f, but ẽ, are of degree < 2d and by the same reasoning as in the previous two cases we conclude that tr_{E/k}({f_{1}(π̃(ẽ)), …, f_{n}(π̃(ẽ))}) is a linear combination of the form

∑_{x∈C̃_{(0),<2d}} tr_{k_{x}/k}(u_{x}) + ∑_{i,x∈(f_{i}∘p̃)} tr_{k_{y}/k}(v_{i,y})

Summands of the first type are in the image of (17) by the inductive assumption. The fact that summands of the second type are in the image of (17) follows from the case deg(π(e)) = d considered above.

4 Some applications

4.1 Milnor’s Conjecture on quadratic forms.

As the first corollary of Theorem 2.10 we get Milnor’s Conjecture on quadratic forms.

As usually, we denote by W (k) the Witt ring of quadratic forms over k, and by I ⊂ W (k) the ideal of even-dimensional forms. The filtration W (k) ⊃ I ⊃ I^{2} ⊃ ⋯ ⊃ I^{n} ⊃ … by the powers of I is called the I-filtration on W. We denote the associated graded ring by Gr_{I^{·}}^{∗}(W (k)). Consider the map

K_{1}^{M}(k)/2 = k^{∗}/(k^{∗})^{2} →^{φ_{1}} Gr_{I^{·}}^{1}(W (k))

which sends {a} to ⟨1, −a⟩. Since (⟨1, −a⟩ + ⟨1, −b⟩ − ⟨1, −ab⟩) ∈ I^{2} is is a group-homomorphism and one can easily see that it is an isomorphism. For any a ∈ k^{∗}\1, the form ⟨⟨a, 1 − a⟩⟩ is hyperbolic and, therefore, the isomorphism φ_{1} can be extended to a ring homomorphism φ : K_{∗}^{M}(k)/2 → Gr_{I^{·}}^{∗}(W (k)). Since Gr_{I^{·}}^{∗}(W (k)) is generated by the first-degree component φ is surjective. The Milnor Conjecture on quadratic forms states that φ is an isomorphism i.e. that it is injective. It was proven in degree 2 by J.Milnor [6, Th. 4.1], in degree 3 by M.Rost [8] and A.Merkurjev-A.Suslin [7], and in degree 4 by M.Rost. Moreover, R.Elman and T.Y.Lam [2] proved that the map φ is injective on pure symbols.

Theorem 4.1 Let k be a field of characteristic zero. Then, the natural map φ : K_{∗}^{M}(k)/2 → Gr_{I^{·}}^{∗}(W (k)) is an isomorphism.

Proof: We already know that φ is surjective. Let h ≠ 0 be an element of K_{n}^{M}(k)/2. By Theorem 2.10 there exists a field extension E/k such that h|_{E} is a nonzero pure symbol. By a result of R.Elman and T.Y.Lam ([2]) the map φ is injective on pure symbols. Hence φ(h|_{E}) is a nonzero element of Gr_{I^{·}}^{n}(W (E)). Since the morphism φ is commutes with field extensions, the element φ(h) ∈ Gr_{I^{·}}^{n}(W (k)) is also nonzero. Therefore, φ is injective.

4.2 Kahn-Rost-Sujatha Conjecture

In [5] B.Kahn, M.Rost and R.Sujatha proved that for any quadric Q of dimension m the ker(K_{i}^{M}(k)/2 → K_{i}^{M}(k(Q))) is trivial for any i < log_{2}(m+2), if i ≤ 4 (actually, in [5] authors worked with H_{et}^{i}(k, Z/2) instead of K_{i}^{M}(k)/2, but because of [12] we can use K_{i}^{M}(k)/2 here). The authors also conjectured 4 (among other things) that the same is true without the restriction i ≤ 4. The following result proves this conjecture.

Theorem 4.2 Let Q be an m-dimensional quadric over a field k of characteristic zero. Then ker(K_{i}^{M}(k)/2 → K_{i}^{M}(k(Q))/2) is trivial for any i < log_{2}(m + 2).

Proof: Denote by q a quadratic form which defines the quadric Q. Assume that h is a nonzero element of ker(K_{i}^{M}(k)/2 → K_{i}^{M}(k(Q))/2). Using Theorem 2.10 we can find an extension E/k such that h|_{E} is a nonzero pure symbol of the form \underline{a} = \{a_{1}, ..., a_{n}\}. Then, since h|_{E(Q)} = 0, the corresponding Pfister quadric Q\underline{a}/E becomes hyperbolic over E(Q). Since Q\underline{a}|_{E(Q)} is hyperbolic the form t · q|_{E} is isomorphic to a subform of the Pfister form ⟨⟨a_{1}, ..., a_{n}⟩⟩ for some coefficient t ∈ E^{∗} by [11, ch.4, Theorem 5.4]. In particular, m+2 = dim(Q)+2 = dim(q) ≤ 2^{i}. Therefore, we have i ≥ log_{2}(m+2).

4.3 J– filtration conjecture.

Together with the I– filtration on W (k) we can consider the following socalled J– filtration. Let x ∈ W (k) be an element, q an anisotropic quadratic form which represents x and Q the corresponding projective quadric. Since Q has a point over the field k(Q), we have a decomposition of the form

q|_{k(Q)} = q_{1} ⊥ \underbrace{H ⊥ ... ⊥ H}_{i_{1}(q)},

4only in the original version of the paper

where q_{1} is an anisotropic form over k(Q), and H is the elementary hyperbolic form. The number i_{1}(q) is called the first higher Witt index of q. In the same way we can decompose q_{1}|_{k(Q)(Q_{1})} etc. obtaining a sequence of quadratic forms q, q_{1}, ..., q_{s−1}, where each q_{i} is an anisotropic form defined over k(Q)...(Q_{i−1}), and

q_{s−1}|_{k(Q)...(Q_{s−1})} = \underbrace{H ⊥ ... ⊥ H}_{i_{s}(q)}

is a hyperbolic form. By [4, Theorem 5.8] (see also [11, ch 4.,Theorem 5.4]), any quadratic form q′ over a field E, such that q′|_{E(Q′)} is hyperbolic, is proportional to some Pfister form. This implies that the form q_{s−1} is proportional to an n–fold Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩, where {a_{1}, ..., a_{n}} ∈ K_{n}^{M}(k(Q)...(Q_{s−2})). This procedure defines for any element x ∈ W (k) a natural number n which we will call the degree of x.

Let us define J_{n}(W (k)) as the subset of W (k) consisting of all elements of degree ≥ n. It can be easily checked that I^{n} ⊆ J_{n}. It was conjectured in [4, Question 6.7] and in [11] that the J_{·} coincides with the I^{·}. The following theorem proves this conjecture.

Theorem 4.3 J_{n} = I^{n}.

Proof: Let x be an element of J_{n}(W (k)) which is represented by a quadratic form q. As above we have a sequence of quadrics Q, Q_{1}, ..., Q_{s−1} such that q|_{k(Q)(Q_{1})...(Q_{s−1})} is hyperbolic. This means that x goes to 0 under the natural map from W (k) to W (k(Q)(Q_{1})...(Q_{s−1})).

All quadrics Q, Q_{1}, …, Q_{s−1} have dimensions ≥ 2^{n} − 2 > 2^{n−1} − 2. By Theorem 4.2, for any 0 ≤ i ≤ n − 1, the kernel ker(K_{i}^{M}(k)/2 → K_{i}^{M}(k(Q)...(Q_{s−1}))) is trivial. Therefore, applying the Milnor conjecture (Theorem 4.1), we conclude that the map

Gr_{I^{·}}^{i}(W (k)) → Gr_{I^{·}}^{i}(W (k(Q)...(Q_{s−1})))

is a monomorphism for all 0 ≤ i ≤ n − 1. Therefore the map

W (k)/I^{n}(W (k)) → W (k(Q)...(Q_{s−1}))/I^{n}(W (k(Q)...(Q_{s−1})))

is a monomorphism as well. Therefore, x belongs to I^{n}(W (k)).

References

[1] H. Bass and J. Tate, The Milnor ring of a global field, Lecture Notes in Math. 342 (1973), 349-446

[2] Richard Elman, T.Y.Lam, Pfister forms and K-theory of fields, J.Algebra 23 (1972), 181-213.

[3] T.Y.Lam, Algebraic theory of quadratic forms, Benjamin, Readings, Mass., 1973.

[4] M.Knebusch, Generic splitting of quadratic forms, I., Proc. London Math. Soc. 33 (1976), 67-93.

[5] B.Kahn, M.Rost, and R.J.Sujatha, Unramified cohomology of quadrics, I., Amer.J.Math. 120 (1998), no.4, 841-891.

[6] J.Milnor, Algebraic K - theory and quadratic forms, Invent. Math. 9 (1969/70), 318-344.

[7] A.S.Merkurjev, A.A.Suslin, The norm-residue homomorphism of degree 3 (in Russian), Izv. Akad. Nauk SSSR 54 (1990), 339-356. English translation: Math. USSR Izv. 36 (1991), 349-368.

[8] M.Rost, Hilbert theorem 90 for K_{3}^{M} for degree-two extensions, Preprint, Regensburg, 1986.

[9] M.Rost, Some new results on the Chow-groups of quadrics, Preprint, Regensburg, 1990.

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

[11] W.Scharlau, Quadratic and Hermitian Forms, Springer, Berlin, Heidelberg, New-York, Tokyo, 1985.

[12] V.Voevodsky, The Milnor conjecture, Preprint MPI/97-8 (see also www.math.uiuc.edu/K-theory/0170/).