# Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture

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)?

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The Beilinson-Lichtenbaum conjecture is about torsion coefficients.:) You could have a look at math.uiuc.edu/K-theory/handbook/1-351-428.pdf –  Mikhail Bondarko Jan 31 '12 at 19:27
Also note: when you consider a morphism of topologies, it suffices to verify an isomorphisms of complexes of sheaves at (the corresponding) points. If you have rigidity (or purity), Nisnevich points reduce to fields. –  Mikhail Bondarko Jan 31 '12 at 19:47
Thank you. I will have a look at it. –  Timo Keller Feb 1 '12 at 13:49

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 $k(x)$.

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