Voevodsky reformulated the Milnor-Bloch-Kato conjecture as a change-of-topology morphism from the Zariski to the étale topology of a *field* with torsion coefficients being an isomorphism. The Beilinson-Lichtenbaum conjecture is more generally about such an isomorphism for *varieties* and integer coefficients. How does the former imply the latter (sketch/reference)?



Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture
To go from finite coefficients to integral coefficients, one notes that
rationally, Zariski and etale cohomology agree (this boils down to the fact that higher
Galois cohomology is torsion), and compares the two long exact sequences associated to the short exact sequence of coefficients $\mathbb Z(n) \to \mathbb Q(n) \to \mathbb Q/\mathbb Z(n)$.
One can go up to degree $n+1$ because of (the analog of Hilbert's theorem 90) 
that the degree $n+1$ etale cohomology vanishes. 


To go from fields to smooth varieties over a field, one compares the local-to-global
spectral sequences
$$ \bigoplus\_{x\in X^{(s)}} H^{t-s}(k(x),\mathbb Z/m(n-s)) \Rightarrow H^{s+t}(X,\mathbb Z/m(n))$$
for both theories, 
where $x$ runs through the points $x$ of codimension $s$ with residue field $k(x)$.


