Beilinson--Lichtenbaum Conjecture

The Beilinson--Lichtenbaum conjecture identifies motivic cohomology with finite coefficients and etale cohomology in a precise range. Now a theorem (following from the Bloch--Kato conjecture), it provides the definitive comparison between algebraic and arithmetic cohomology theories.

Statement

Theorem6.4Beilinson--Lichtenbaum Conjecture (Theorem)

Let $X$ be a smooth variety over a field $k$ and $\ell$ a prime with $\operatorname{char}(k) \neq \ell$ . The natural comparison map

$\alpha: H^n_{\text{mot}}(X, \mathbb{Z}/\ell(p)) \to H^n_{\text{et}}(X, \mu_\ell^{\otimes p})$

from motivic cohomology (Nisnevich) to etale cohomology satisfies:

$\alpha$ is an isomorphism for $n \leq p$ .
$\alpha$ is an injection for $n = p + 1$ .
$\alpha$ may fail to be surjective for $n \geq p + 1$ .

Equivalently: the motivic-to-etale change-of-topology map $\tau: \mathbb{Z}/\ell(p)_{\text{Nis}} \to R\tau_* \mu_\ell^{\otimes p}$ induces isomorphisms on cohomology sheaves $\mathcal{H}^n$ for $n \leq p$ .

Relation to Bloch--Kato

RemarkDerivation from Bloch--Kato

The Beilinson--Lichtenbaum conjecture is a formal consequence of the Bloch--Kato conjecture (norm residue isomorphism). The key steps:

Step 1: The Bloch--Kato conjecture states $K_n^M(F)/\ell \cong H^n_{\text{et}}(F, \mu_\ell^{\otimes n})$ for all fields $F$ . This is the case $X = \operatorname{Spec} F$ of Beilinson--Lichtenbaum (with $n = p$ ).

Step 2: For general $X$ , use the Gersten resolution on both sides. On the motivic side:

$0 \to H^n_{\text{mot}}(X, \mathbb{Z}/\ell(p)) \to \bigoplus_{x \in X^{(0)}} H^n(k(x), \mathbb{Z}/\ell(p)) \to \bigoplus_{x \in X^{(1)}} H^{n-1}(k(x), \mathbb{Z}/\ell(p-1)) \to \cdots$

On the etale side, the analogous Gersten-type spectral sequence (Bloch--Ogus):

$E_1^{a,b} = \bigoplus_{x \in X^{(a)}} H^{b-a}_{\text{et}}(k(x), \mu_\ell^{\otimes (p-a)}) \implies H^{a+b}_{\text{et}}(X, \mu_\ell^{\otimes p}).$

Step 3: The Bloch--Kato conjecture identifies the $E_1$ -terms of both sequences for $b - a \leq p - a$ , i.e., $b \leq p$ . A spectral sequence comparison then gives the desired isomorphism for $n \leq p$ .

Consequences for K-theory

ExampleQuillen--Lichtenbaum conjecture

The Beilinson--Lichtenbaum conjecture implies the Quillen--Lichtenbaum conjecture: for a smooth variety $X$ over a field of characteristic not $\ell$ , the natural map

$K_n(X; \mathbb{Z}/\ell) \to K_n^{\text{et}}(X; \mathbb{Z}/\ell)$

from algebraic K-theory to etale K-theory (defined via the etale sheafification of the K-theory presheaf) is an isomorphism for $n \geq \dim X$ (and more generally, for $n$ sufficiently large relative to the etale cohomological dimension).

For $X = \operatorname{Spec} \mathcal{O}_F[1/\ell]$ (the ring of $S$ -integers of a number field, $S = \{\ell\}$ ):

$K_n(\mathcal{O}_F[1/\ell]; \mathbb{Z}/\ell) \cong K_n^{\text{et}}(\mathcal{O}_F[1/\ell]; \mathbb{Z}/\ell) \quad \text{for } n \geq 2$

since $\operatorname{cd}_\ell(\mathcal{O}_F[1/\ell]) \leq 2$ .

ExampleK-theory of finite fields revisited

For $X = \operatorname{Spec} \mathbb{F}_q$ with $\ell | q - 1$ (so $\mu_\ell \subset \mathbb{F}_q$ ):

$H^n_{\text{et}}(\mathbb{F}_q, \mu_\ell^{\otimes p}) = \begin{cases} \mathbb{Z}/\ell & n = 0, 1 \\ 0 & n \geq 2 \end{cases}$

since $\operatorname{cd}(\mathbb{F}_q) = 1$ . The Beilinson--Lichtenbaum conjecture gives $H^n_{\text{mot}}(\mathbb{F}_q, \mathbb{Z}/\ell(p)) = H^n_{\text{et}}(\mathbb{F}_q, \mu_\ell^{\otimes p})$ for $n \leq p$ , confirming the motivic spectral sequence computation of Quillen's $K$ -groups for finite fields with finite coefficients.

Proof strategy

Proof

We outline how the Beilinson--Lichtenbaum conjecture follows from the Bloch--Kato conjecture.

Step 1: Reformulation via motivic complexes. By Voevodsky's work, motivic cohomology with $\mathbb{Z}/\ell$ coefficients is computed by:

$H^n_{\text{mot}}(X, \mathbb{Z}/\ell(p)) = H^n_{\text{Nis}}(X, \mathbb{Z}/\ell(p))$

where $\mathbb{Z}/\ell(p)$ is the motivic complex reduced mod $\ell$ .

Step 2: The change-of-topology map. There is a natural map of sites $\varepsilon: X_{\text{et}} \to X_{\text{Nis}}$ and a map of complexes $\mathbb{Z}/\ell(p)_{\text{Nis}} \to R\varepsilon_* \varepsilon^* \mathbb{Z}/\ell(p)$ . The conjecture asserts $\mathcal{H}^n(\text{cone}) = 0$ for $n \leq p$ .

Step 3: Reduction to fields. The cone is a Nisnevich sheaf, so it suffices to show vanishing on stalks, i.e., on essentially smooth local rings and their fraction fields. By the localization sequence, it suffices to check for fields.

Step 4: Application of Bloch--Kato. For a field $F$ , the motivic complex $\mathbb{Z}/\ell(p)$ on $\operatorname{Spec} F$ is concentrated in degrees $\leq p$ (by the Beilinson--Soule conjecture for fields, which is known). The comparison map on $\mathcal{H}^p$ is exactly the Bloch--Kato map:

$K_p^M(F)/\ell = H^p_{\text{mot}}(F, \mathbb{Z}/\ell(p)) \to H^p_{\text{et}}(F, \mu_\ell^{\otimes p})$

which is an isomorphism by hypothesis.

Step 5: Induction on weight. Use the exact triangle $\mathbb{Z}/\ell(p-1) \to \mathbb{Z}/\ell(p) \to C(p) \to$ and induction, combined with the Bloch--Kato isomorphism at the top weight, to deduce the full statement. $\square$

■

Beyond Beilinson--Lichtenbaum

RemarkIntegral versions and p-adic comparisons

The Beilinson--Lichtenbaum conjecture has integral and $p$ -adic refinements:

Integral version: The map $H^n_{\text{mot}}(X, \mathbb{Z}(p)) \to H^n_{\text{et}}(X, \mathbb{Z}_\ell(p))$ (with $\ell$ -adic coefficients) gives, after taking the inverse limit, information about the $\ell$ -adic K-groups $K_n(X) \hat{\otimes} \mathbb{Z}_\ell$ .
$p$ -adic version ( $p = \operatorname{char}(k)$ ): The comparison uses logarithmic de Rham--Witt cohomology instead of etale cohomology: $H^n_{\text{mot}}(X, \mathbb{Z}/p(p)) \cong H^n(X, \nu_r(p))$ where $\nu_r(p) = W_r\Omega^p_{X, \log}$ is the logarithmic part of the de Rham--Witt complex. This was established by Bloch--Kato--Gabber.
Syntomic cohomology: For schemes over $p$ -adic fields, the comparison goes to syntomic cohomology, mediating between motivic and etale via crystalline theory.