1 Results.

Theorem 1.1 For any field k of characteristic ≠ 2 the canonical homomorphisms

\begin{matrix}1. & K_{n}^{M}(k)/2 → H_{et}^{n}(k, Z_{2}) \\ 2. & K_{n}^{M}(k)/2 → I^{n}/I^{n+1} \end{matrix}

are isomorphisms.

Theorem 1.2 For any field k of characteristic ≠ 2 which admits resolution of singularities one has

H_{M}^{n}(Spec(k), Z/2(m)) = \{ \begin{matrix}0 & for n > m \\ H_{et}^{n}(k, Z/2) & for n ≤ m \end{matrix}

Theorem 1.3 For any field k of characteristic ≠ 2 which admits resolution of singularities there is a spectral sequence which starts with

E_{2}^{p,q} = \{ \begin{matrix}H_{et}^{p−q}(k, Z/2) & for p, q ≤ 0 \\ 0 & otherwise \end{matrix}

and converges to K_{−p−q}/2(k) (K-theory with Z/2-coefficients).

Corollary 1.4 For any scheme of finite type X over a field k satisfying the assumptions of Theorem 1.3 the canonical homomorphism

K_{n}/2(X) → K_{n}^{et}/2(X)

is an isomorphism for n ≥ dim(X/k) + cd_{2}(k) − 1 and a monomorohism for n = dim(X/k) + cd_{2}(k) − 2.

For a_{1}, …, a_{n} ∈ k^{∗} denote by Q\underline{a} the projective quadric (of dimension 2^{n−1} − 1) given by the equation

<< a_{1}, …, a_{n−1} >>= a_{n}t^{2}.

Theorem 1.5 For any field k of characteristic ≠ 2 which admits resolution of singularities and any sequence of invertible elements a_{1}, …, a_{n} ∈ k^{∗} the following sequence is exact

⨁_{x∈Q\underline{a}} H_{et}^{m}(k_{x}) ⟶^{⊕tr_{kx/k}} H_{et}^{m}(k) ⟶^{∧(a_{1},...,a_{n})} H_{et}^{m+n}(k) → H_{et}^{m+n}(k(Q\underline{a})).

where the direct sum is taken over the closed points x of Q\underline{a} such that [k_{x} : k] = 2 and cohomology are with Z/2-coefficients.

Corollary 1.6 For any k and a_{1}, …, a_{n} as above the ideal

ker(⊕_{i≥0}H_{et}^{i}(k, Z/2) ⟶^{∧(a_{1},...,a_{n})} ⊕_{i≥0}H_{et}^{i}(k, Z/2))

is generated by the first component (i.e. by ker ∩ H_{et}^{1}(k, Z/2)).

2 The way it’s done.

I’ll outline here the proof of the first isomorphism in Theorem 1.1. The rest is done by similar technique. The fact that Theorem 1.2 follows from 1.1(1) was proven in [3]. Theorem 1.3 follows from Theorem 1.2 and Bloch-Lichtenbaum spectral sequence from motivic cohomology to algebraic Ktheory. The second isomorphism in 1.1 follows by a well known trick from the first one and 1.5.

Note also that one can prove easily that provided Theorem 1.1 holds for all fields of charactersictic zero it holds for all fields of characteristic ≠ 2. This allows us to drop the resolution of singularities assumption. In what follows I assume that k admits resolution of singularities.

Denote by X\underline{a} the simplcial scheme with terms being Qn ; \underline{a} and faces and degeneraces given by projections and diagonals. Morally speaking, X\underline{a} is a simplex with the “scheme of vertices” being Q\underline{a}. We will say that BK(n, 2) holds over k if for any field K over k and any m ≤ n the homomorphism K_{m}^{M}(K)/2 → H^{m}(K, Z/2) is an isomorphism. The following theorem is a weaker analog of [4, Proposition 4.15].

Theorem 2.1 Assume that BK(n, 2) holds over k and for any K/k and a_{1}, …, a_{n} ∈ K^{∗} one has

H^{n+1}(X\underline{a}, Z_{(2)}(n)) = 0.

Then BK(n, 2) holds over k.

Finally the following result is a corollary of two theorems of M. Rost ([1], [2]). The proof is given in [4, Corollary 4.17].

Theorem 2.2 For \underline{a} = (a_{1}, …, a_{n}) one has:

H^{2^{n}−1}(X\underline{a}, Z_{(2)}(2^{n−1})) = 0.

To explain the key point of the proof I need a notion of cohomological operation in motivic cohomology. For simplicity I assume below that our field is of characteristic zero and there is an embedding k ⊂ C. By definition an operation in motivic cohomology with coefficients in A = Z/l, Z, Z_{(l)} etc. of degree (i, j) is a family of (additive) natural transformations

H^{∗}(−, A(∗)) → H^{∗+i}(−, A(∗ + j))

which commute with two suspension isomorphisms. For more details see [5] and the enclosed preprint. We denote the bigraded algebra of cohomological operations by U^{∗,∗}(k, A). Clearly, for any embedding k ⊂ C we get a homomorphism from U^{∗,∗}(k, A) to the corresponding algebra of cohomological operations in ordinary cohomology with coefficients in the abelian group A which forgets the second filtration.

It is not hard to reproduce the classical construction of Steenrod squares (or more generally reduced power operations) and obtain elements Sq_{alg}^{i} of degree (i, [i/2]) (where [i/2] is the integral part of i/2) in U^{∗,∗}(k, Z/2) which have properties very similar to the properties of the usual Steenrod squares and map to them for any embeding k ⊂ C. Define Q_{i}^{alg} inductively setting

Q_{0}^{alg} = Sq_{alg}^{1} = β
Q_{i+1}^{alg} = [Sq_{alg}^{2^{i+1}}, Q_{i}^{alg}]

(where [−, −] is the commutant). These operations map to the Milnor elements Q_{i} in the usual Steenrod algebra. They also have the following two properties:

1. Q_{i}^{alg} maps the image of H ∗ (−, Z_{2}(∗)) in H ∗ (−, Z/2(∗)) to itself.

2. Q_{i}^{alg}(xy) = Q_{i}^{alg}(x)y + xQ_{i}^{alg}(y) + {−1}Q_{i−1}^{alg}(x)Q_{i−1}^{alg}(y) in particular if k contains a square root of −1 Q_{i}^{alg} is a derivation.

3. (Q_{i}^{alg})^{2} = 0.

The last property shows that for any (pointed) simplcial scheme X we may consider complexes

...→H^{n−2i+1+1}(X ,Z/2(m−2^{i}+1)) →^{Q{alg \atop i}}H^{n}(X ,Z/2(m)) →^{Q{alg \atop i}}H^{n+2i+1−1}(X ,Z/2(m+2^{i}−1))→...

Denote the cohomology group of this complex by Ψ_{i}^{n,m}(X). Let X̃\underline{a} be the homotopy fiber of the natural morphism X\underline{a} → Spec(k) in any reasonable stable category (such that there is the corresponding long exact sequence of motivic cohomology). The key point in the proof is the following

Theorem 2.3 Let \underline{a} = (a_{1}, …, a_{n}). Then for any i ≤ n − 1 one has

Ψ_{i}^{∗,∗}(X̃\underline{a}) = 0.

Remark: In fact in Theorem 2.3 one can replace X̃\underline{a} by the simplicial gadget constructed in the same way from any quadric of dimension ≥ 2^{i−1} − 1.

I’ll outline the proof in a few moments, but first let’s see how it implies all we need. Note that X̃\underline{a} = 0 in the etale topology. In particular if we know BK(n − 1, 2) the main theorem of [3] implies that all motivic cohomology of X̃\underline{a} of the form H^{i}(−, Z/2(j)) with i ≤ j ≤ n − 1 are zero. Note also that for i > j motivic cohomology of X̃\underline{a} and X\underline{a} coincide. In particular it is sufficient to show that H^{n+1}(X̃\underline{a}, Z_{(2)}(n)) = 0. Trivial transfer argument shows that it is a group of exponent 2. Thus it is sufficient to show that the map

H^{n+1}(X̃\underline{a}, Z_{(2)}(n)) → H^{n+1}(X̃\underline{a}, Z/2(n))

is zero. Let α ∈ H^{n+1}(X̃\underline{a}, Z/2(n)) be an element which lifts to cohomology with Z_{(2)}-coefficients. Then Q_{n−2}^{alg}Q_{n−3}^{alg} … Q_{1}^{alg}(α) belongs to the group H^{2^{n}−1}(X̃\underline{a}, Z/2(2^{n−1})) and by the first property of Q_{i}^{alg}’s and Theorem 2.2 is zero. Now Theorem 2.3 with the fact that H^{i}(X̃\underline{a}, Z/2(j)) = 0 for i ≤ j implies that α = 0.

Playing around with this technique one may produce a complete computation of motivic cohomology of X̃\underline{a} and then using [4, Theorem 4.5] motivic cohomology of Rost’s motive M\underline{a}. This computation agrees with the predictions made in [2].

3 How one proves Theorem 2.3.

We will need some basic techniques from the “stable homotopy theory of algebraic varieties” see [6]. First of all we need to know that there is a tensor triangulated category SHot(k) together with a “suspension spectrum” functor Ssp(−) from (pointed) simplicial schemes to it. This category has to “shift” functors - the triangulated shift X ↦ X[1] and the Tate twist X ↦ X(1) such that X(1)[2] = X ⊗ Ssp(P^{1}, ∞). It also has an object Z which represents motivic cohomology i.e.

Hom_{SHot(k)}(Ssp(X), Z(j)[i]) = H^{i}(X , Z(j))

and Hom_{SHot(k)}(Z, Z(j)[i]) = U^{i,j}(k, Z). The same of course true for finite coefficients. In particular we have a morphism

Z/2 → Z/2(2^{i} − 1)[2^{i+1} − 1]

which corresponds to Q_{i}^{alg}. Denote by Φ_{i} its “homotopy fiber” i.e. an object such that there is a distinguished triangle

Z/2(2^{i} − 1)[2^{i+1} − 2] → Φ_{i} → Z/2 → Z/2(2^{i} − 1)[2^{i+1} − 1].

Denote by MU the algebraic Thom spectrum (its “terms” are the Thom spaces over BU(n)’s considered as simplicial algebrac varieties). As in the topological situation it is a ring spectrum in the sense that there are morphisms

e : 1 → MU
MU ⊗ MU → MU

which make it into a commutative, associative monoid in (SHot(k), ⊗). An orientation for a theory represented by an object E of SHot(k) is a module structure on E over MU i.e. a morphism ϕ : MU ⊗E → E satisfying obvious axioms.

The following theorem is proven in [6].

Theorem 3.1 Let (E, ϕ) be an oriented theory. Then it has direct image homomorphisms for smooth proper morphisms. In particular if X is a smooth projective variety of dimension d over k there and X is a pointed simplcial scheme is a canonical (with respect to (E, ϕ) and X) morphism

E^{∗,∗}(X ∧ X) → E^{∗−2d,∗−d}(X).

The Eilenberg-Maclane spectrum Z representing motivic cohomology has a canonical orientation. The properties of Q_{i}^{alg} given above imply easily that the theory Φ_{i} also has an orientation (in fact it has many). Note that for any X of dimension 2^{i−1} − 1, any orientation of Φ_{i} and any X we have a square

Φ_{i}^{∗,∗}(X)  →  Φ_{i}^{∗,∗}(X ∧ X)
↓  ↓
H^{∗}(X , Z/2(∗)) → Φ_{i}^{∗+2−2^{i},∗+1−2^{i−1}}(X)

where the upper horizontal arrow is given by the projection X → Spec(k), the right vertical one comes from orientation by Theorem 3.1 and the other two come from the definition of Φ_{i}. One can show easily that for any X this diagram commute up to multiplication by some n_{i}(X) ∈ Z/2 which is idependend on X.

Proposition 3.2 For a quadric Q of dimension 2^{i−1} −1 we have n_{i}(Q) = 1.

To prove this fact we consider an embedding k ⊂ C and the corresponding frunctor from SHot(k) to the usual stable homotopy category SHot (if k is not of charactersitic zero one can use etale homotopy types instead but it is more tricky). All the constructions above “commute” with this functor and Proposition 3.2 follows from the topological fact that for any orientation of the topological analog of Φ_{i} the number n_{i}(Q) is one (which is a corollary of the fact that Q represents the generator v_{i} in BP_{∗}...).

Note now that the condition Ψ_{i}^{∗,∗}(X) = 0 is equivalent to the condition that the composition

Φ_{i}^{∗,∗}(X) → H^{∗}(X , Z/2(∗)) → Φ_{i}^{∗+2−2^{i},∗+1−2^{i−1}}(X)

corresponding to one of the compositions in the diagram given above is zero. In the case of X̃\underline{a} it follows from Proposition 3.2 and the trivial fact that

X̃\underline{a} ∧ Q\underline{a} = 0

since the morphism X\underline{a} → Spec(k) becomes a homotopy equivalence after multiplication with Q\underline{a}.

References

[1] M. Rost. On the spinornorm and A_{0}(X, K_{1}) for quadrics. Preprint, 1988.

[2] M. Rost. Some new results on the Chowgroups of quadrics. Preprint, 1990.

[3] A. Suslin and V. Voevodsky. Motivic cohomology with finite coefficients and Bloch-Kato conjecture. Preprint, Spring 1995.

[4] V. Voevodsky. Algebraic Morava K-theories and the Bloch-Kato conjecture for Z/2 coefficients. Preprint, June 1995.

[5] V. Voevodsky. Cohomological operations in motivic cohomology. In preparation, 1996.

[6] V. Voevodsky. Generalized cohomology theories on the category of algebraic varieties. In preparation, 1996.