1 On Theorem 1.5

The proof of this theorem will appear in a joint paper with D. Orlov and A. Vishik. Here is a sketch. In what follows I denote the operations Q_{i}^{alg} by d^{i}.

Lemma 1 There are exact sequences:

0 → H^{i+n+1}(X\underline{a}, Z_{(2)}(i + n − 1)) → K_{i+n}^{M}(k)/2 → K_{i+n}^{M}(k(Q\underline{a}))/2.
⊕_{x∈Q\underline{a}}K_{i}^{M}(k_{x}) → K_{i}^{M}(k) → H^{2^{n}−1+i}(X\underline{a}, Z_{(2)}(2^{n−1} − 1 + i)) → 0

Proof: The first one was constructed in [2, Lemma 6.4]. To get the second one note that by [2, Theorem 4.5] we have an exact sequence

H^{2^{n}−2+i}(M\underline{a}, Z_{(2)}(2^{n−1} − 1 + i)) → H^{i}(X\underline{a}, Z_{(2)}(i)) ⟶^{∧μ\underline{a}}

→ H^{2^{n}−1+i}(X\underline{a}, Z_{(2)}(2^{n−1} − 1 + i)) → H^{2^{n}−1+i}(M\underline{a}, Z_{(2)}(2^{n−1} − 1 + i))

The last group is zero since M\underline{a} is a direct summand of the motive of a variety of dimension 2^{n−1} − 1. The group H^{i}(X\underline{a}, Z_{(2)}(i)) is isomorphic to K_{i}^{M}(k) ⊗ Z_{(2)} (a corollary of [1, Theorem 1.2]). The first group may be replaced without changing its image by the corresponding motivic cohomology group of Q\underline{a} and simple reasoning shows that its image in K_{i}^{M}(k) is exactly the image of ⊕_{x∈Q\underline{a}}K_{i}^{M}(k_{x}).

I claim now that there is a commutative diagram of the form

\begin{aligned}K_{i}^{M}(k) & → & H^{2^{n}−1+i}(X\underline{a}, Z_{(2)}(2^{n−1} − 1 + i)) \\ −∧\underline{a} ↓ & ↘ & ↑ϕ \\ K_{i+n}^{M}(k) & ← & H^{i+n+1}(X\underline{a}, Z_{(2)}(i + n − 1)) \end{aligned}

where ϕ is a monomorphism. Together with the following theorem (due to A. Vishik) it would imply [1, Theorem 1.5].

Theorem 2 Let X be a smooth projective quadric of then the image of ⊕_{x∈X}K_{n}^{M}(k_{x}) in K_{n}^{M}(k) is the same as the image of the corresponding sum taken over points of degree 2 over k.

Let γ be the element of H^{n+1}(X\underline{a}, Z_{(2)}(n − 1)) which corresponds by the first exact sequence to the symbol \underline{a}. We define the morphism

K_{i}^{M}(k) → H^{i+n+1}(X\underline{a}, Z_{(2)}(i + n − 1))

by the rule x ↦ x ∧ γ. One verifies easily that the lower/left triange commutes. We define now ϕ to be the composition d^{n−2} ∘ … ∘ d^{1}. To verify that the upper/right triangle commutes we need the following lemma which follows easily from [1, Theorem 2.3].

Lemma 3 μ\underline{a} = ϕ(γ).

Multiplicative properties of d^{i}’s imply now the commutativity of the second triangle. It remains to show that ϕ is a monomorphism. Since the motivic cohomology groups we are looking at have degree greater than weight we may replace in our diagram X by X̃. Our result follows then from [1, Theorem 2.3] since H^{i}(X̃\underline{a}, Z_{(2)}(i)) = 0 for all i (a corollary of the Bloch-Kato-... conjecture).

Using similar arguments one can get a complete computation of motivic cohomology of X̃\underline{a}. Denote by Θ\underline{a} ; i the cokernel

coker( ⨁_{x∈Q\underline{a}} K_{i}^{M}(k_{x}) →^{⊕tr} K_{i}^{M}(k))

The answer looks as follows.

Theorem 4

H^{a+b}(X̃\underline{a}, Z_{(2)}(a)) = \{ \begin{matrix}Θ\underline{a} ; c(a,b) & if b > 0 and b is even \\ 0 & otherwise \end{matrix}

where c(a, b) = a − b − n + 3 + [b/2^{n−1}] + ϵ(b) and ϵ(b) is the number of units in the first n − 1 “bits” of the binary decomposition of b.

References

[1] The previous message.

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