Present state of the Bloch-Kato conjecture

compiled from notes of the talk delivered by A. Suslin

Ever since Quillen introduced higher K-theory, attempts have been made to understand its relationships with other cohomology theories, such as étale cohomology, chow rings and also Milnor K-theory, which in the case of fields is given by an explicit presentation

K_{∗}^{M}(F) = ∐_{n≥0}(F^{∗})^{⊗n}/⟨a ⊗ (1 − a)⟩.

In the beginning of the eighties, Beilinson produced very explicit conjectures predicting what one should expect in general. For practical purposes, let k be a field of characteristic zero (since resolution of singularities is needed in some proofs). Let Sm/F be the category of smooth schemes over F.

Beilinson conjecture. There exist a complexes Z(n), n ≥ 0 of abelian sheaves on the big Zariski site (Sm/F)_{Zar} satisfying

(i) Z(0) = Z, Z(1) = O^{∗}[−1]; (ii) (The Beilinson-Soulé vanishing conjecture.) For n > 0, Z(n) is acyclic outside [1, n] and H^{n}(Z(n)) = K_{n}^{M}, the sheaf of Milnor K-groups; (iii) There exists a strongly convergent spectral sequence

E_{2}^{p,q} = H^{p}(X, Z(q)) ⇒ K_{2q−p}(X);

(iv) (The Beilinson-Lichtenbaum conjecture.) For any prime p (different from char k), there is an isomorphism

Z/p(n) ≅ τ_{≤n}Rπ_{∗}(μ_{p}^{⊗n}),

where π : (Sm/F)_{ét} → (Sm/F)_{Zar}; in particular

H^{i}(F, Z/p(n)) = \{ \begin{matrix}Hi_{ét}(F, μ_{p}^{⊗n}), & i ≤ n, \\ 0, & i > n. \end{matrix}

One special case of the Beilinson-Lichtenbaum conjectures (for i = n) is known as the:

Bloch-Kato conjecture. For a field F, there is a natural isomorphism

K_{n}^{M}(F)/p ≅ Hn_{ét}(F, μ_{p}^{⊗n}).

The first successful attempt to define the motivic cohomology theory (though not the complexes Z(n)) was the work of S. Bloch [B]. The motivic complexes Z(n) were defined by Voevodsky [V], using the theory of sheaves with transfers.

The sheaves Z(n) of Voevodsky satisfy the Beilinson property (i) and part of property (ii) for trivial reasons. However, the other part – acyclicity in degrees ≤ 0 is still open. The spectral sequence relating algebraic K-theory to motivic cohomology was constructed in the case of fields by S. Bloch and S. Lichtenbaum [B-L] and generalized to the case of arbitrary smooth schemes by E. Friedlander and A. Suslin [F-S].

A. Suslin and V. Voevodsky [S-V] have shown that the Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture. Finally, V. Voevodsky has proved in his famous paper on the Milnor conjecture [V2] that the Bloch-Kato conjecture holds in the case p = 2. More generally, Voevodsky proved in [V2] the following theorem, which may be used also for primes p ≠ 2.

Theorem (Voevodsky). Assume the Beilinson-Lichtenbaum conjecture holds in weights < n. Assume further that for any field extension F′/F and any n-tuple (a_{1}, ..., a_{n}) ∈ (F′)^{∗}, there exists a universal splitting variety X = X_{a} for the symbol a = {a_{1}, ..., a_{n}} with the following properties:

(0) X is smooth and projective over F′;

(1) X has dimension d = p^{n−1} − 1;

(2) S_{d}(X)/p is prime to p (here, S_{d}(X) is an appropriate characteristic number of X, which is known to be divisible by p);

(3) The norm map N : A_{0}(X_{a}, K_{1}) → (F′)^{∗} is injective; then the Bloch-Kato conjecture and hence also the Beilinson-Lichtenbaum conjecture holds also in weight n.

For p = 2, one can take X = X_{a} to be the Pfister (neighbour) quadric, in which case (0),(1) and (2) are trivial and (3) was proved previously by M. Rost. In the general case, M. Rost [R] has proved that one can always find X = X_{a} satisfying (0),(1) and (2), and moreover proved half of (3).

References

[B] S. Bloch, Algebraic cycles and higher K-theory, Adv. Math. 61, 267-304

[B-L] S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, preprint 1994.

[F-L] E. Friedlander and A. Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology, preprint 1999.

[R] M. Rost, Chain lemma for splitting fields of symbols, preprint

http://www.physik.uni-regensburg.de/ rom03516/papers.html

[S-V] A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients , K-theory preprint 341, 1999

http://www.mathematik.uni-osnabrueck.de/K-theory/

[V1] V. Voevodsky, Triangulated categories of motives over a field, in: Cycles, Transfers and Motivic Homology Theories, Annals of Math. Studies, 1999.

[V2] V. Voevoedsky, The Milnor conjecture, MPI-preprint 1998-8,

http://www.mpim-bonn.mpg.de/cgi-bin/preprint/

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