The Bloch--Kato Conjecture

The Bloch--Kato conjecture, now a theorem due to the combined work of Voevodsky and Rost, is one of the deepest results in arithmetic and algebraic geometry. It identifies Milnor K-theory modulo a prime with Galois cohomology, providing the definitive link between algebraic K-theory and etale cohomology.

Statement

Theorem4.2Bloch--Kato / Norm Residue Isomorphism Theorem

Let $F$ be a field and $\ell$ a prime with $\operatorname{char}(F) \neq \ell$ . The norm residue homomorphism

$h_\ell^n: K_n^M(F) / \ell \xrightarrow{\;\sim\;} H^n_{\text{et}}(\operatorname{Spec} F, \mu_\ell^{\otimes n})$

is an isomorphism for all $n \geq 0$ .

More generally, for any $m \geq 1$ with $\operatorname{char}(F) \nmid m$ :

$K_n^M(F) / m \xrightarrow{\;\sim\;} H^n_{\text{et}}(\operatorname{Spec} F, \mu_m^{\otimes n}).$

The proof strategy (Voevodsky)

RemarkOverview of the proof

The proof of the Bloch--Kato conjecture for $\ell = 2$ (the Milnor conjecture) and then for odd primes follows a sophisticated strategy:

Step 1: Reduction to symbol length 1. Using the norm maps and transfer arguments, the problem reduces to showing that the map $h_\ell^n$ is an isomorphism for fields $F$ such that $K_n^M(F)/\ell$ is generated by a single symbol $\{a_1, \ldots, a_n\}$ .

Step 2: Construction of splitting varieties. For a symbol $\alpha = \{a_1, \ldots, a_n\} \in K_n^M(F)/\ell$ , construct a smooth projective variety $X_\alpha$ (a norm variety or splitting variety) such that:

$\alpha$ vanishes in $K_n^M(F(X_\alpha))/\ell$ (the symbol "splits" over the function field of $X_\alpha$ ).
$\dim X_\alpha = \ell^{n-1} - 1$ .
$X_\alpha$ has specific properties related to the Rost degree formula.

Step 3: Motivic cohomology computations. Using the motivic Steenrod algebra, show that the map from motivic cohomology $H^{n,n}_{\text{mot}}(F, \mathbb{Z}/\ell)$ to etale cohomology $H^n_{\text{et}}(F, \mu_\ell^{\otimes n})$ is an isomorphism. This uses:

The identification $K_n^M(F)/\ell \cong H^{n,n}_{\text{mot}}(F, \mathbb{Z}/\ell)$ (Nesterenko--Suslin, Totaro).
The motivic-to-etale spectral sequence.
Vanishing results for motivic cohomology.

Step 4: The Rost--Voevodsky machinery. The crucial technical ingredient is the construction of motivic cohomology operations (analogues of the Steenrod squares/powers) and their use to show vanishing of certain obstruction groups.

Key ingredients

Definition4.7Motivic cohomology

For a smooth variety $X$ over a field $k$ , the motivic cohomology groups $H^{p,q}_{\text{mot}}(X, \mathbb{Z})$ are defined (following Voevodsky) as:

$H^{p,q}_{\text{mot}}(X, \mathbb{Z}) = H^p_{\text{Zar}}(X, \mathbb{Z}(q))$

where $\mathbb{Z}(q)$ is the motivic complex of weight $q$ . Key properties:

$H^{n,n}_{\text{mot}}(\operatorname{Spec} F, \mathbb{Z}) \cong K_n^M(F)$ (Nesterenko--Suslin--Totaro).
$H^{2q,q}_{\text{mot}}(X, \mathbb{Z}) \cong CH^q(X)$ (Chow groups).
There is a comparison map $H^{p,q}_{\text{mot}}(X, \mathbb{Z}/\ell) \to H^p_{\text{et}}(X, \mu_\ell^{\otimes q})$ .

The Bloch--Kato conjecture for the diagonal bidegree $(n, n)$ at $\operatorname{Spec} F$ is exactly the statement that the comparison map is an isomorphism.

ExampleMotivic Steenrod operations

Voevodsky constructed motivic Steenrod operations:

For $\ell = 2$ : $Sq^{2i}: H^{p,q}_{\text{mot}}(X, \mathbb{Z}/2) \to H^{p+2i, q+i}_{\text{mot}}(X, \mathbb{Z}/2)$

For $\ell$ odd: $P^i: H^{p,q}_{\text{mot}}(X, \mathbb{Z}/\ell) \to H^{p+2i(\ell-1), q+i(\ell-1)}_{\text{mot}}(X, \mathbb{Z}/\ell)$

These satisfy analogues of the Adem relations and the Cartan formula. The key property used in the proof: on $H^{n,n}_{\text{mot}}$ , the top Steenrod operation detects whether a symbol is zero, allowing inductive arguments on the weight.

Consequences

ExampleGeneration of the Brauer group

The Bloch--Kato conjecture for $n = 2$ (Merkurjev--Suslin theorem) implies:

For a field $F$ containing $\mu_\ell$ : every $\ell$ -torsion element in $\operatorname{Br}(F)$ is a sum of cyclic algebras. More precisely, $\operatorname{Br}(F)[\ell] \cong K_2^M(F)/\ell$ , and every class is represented by $\sum \{a_i, b_i\}$ , corresponding to $\sum [(a_i, b_i)_\ell]$ in the Brauer group.

This resolved a major open question in the theory of central simple algebras and motivated much of the subsequent development.

RemarkBeyond Bloch--Kato: the motivic Bloch--Kato conjecture

The full Beilinson--Lichtenbaum conjecture extends Bloch--Kato to all bidegrees: the comparison map

$H^{p,q}_{\text{mot}}(X, \mathbb{Z}/\ell) \to H^p_{\text{et}}(X, \mu_\ell^{\otimes q})$

is an isomorphism for $p \leq q$ (and all smooth $X$ over a field). This is now proved as a consequence of the Bloch--Kato conjecture via the motivic-to-etale spectral sequence:

$E_2^{p,q} = H^p_{\text{Zar}}(X, H^q_{\text{et}}(\mu_\ell^{\otimes n})) \implies H^{p+q}_{\text{et}}(X, \mu_\ell^{\otimes n}).$

The Bloch--Kato conjecture computes the $E_2$ -terms, and the spectral sequence degenerates for $p + q \leq n$ , giving the Beilinson--Lichtenbaum conjecture.