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

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

2012-01-31T19:02:49Z

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)?

Answer by Timo Keller for Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture

2012-02-01T18:14:21Z

From the Handbook of K-Theory (mentioned by Mikhail Bondarko in the above comments):

p. 202, Thomas Geisser:

"In [91], Suslin and Voevodsky show that, assuming resolution of singularities, the Bloch–Kato conjecture (1.10) implies the Beilinson–Lichtenbaum conjecture (1.11) with mod m-coefficients; in [34] the hypothesis on resolution of singularities is removed."

[91] A. Suslin, V. Voevodsky, Bloch–Kato conjecture and motivic cohomology with finite coefficients. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., 548 (2000), 117–189.

[34] T. Geisser, M. Levine, The Bloch–Kato conjecture and a theorem of Suslin– Voevodsky. J. Reine Angew. Math. 530 (2001), 55–103.

Answer by Thomas Geisser for Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture

2012-09-28T08:35:22Z

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 $k(x)$.