Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:14:51Z http://mathoverflow.net/feeds/question/87162 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87162/milnor-bloch-kato-conjecture-implies-the-beilinson-lichtenbaum-conjecture Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture Timo Keller 2012-01-31T19:02:49Z 2012-09-28T08:35:22Z <p>Voevodsky reformulated the Milnor-Bloch-Kato conjecture as a change-of-topology morphism from the Zariski to the étale topology of a <em>field</em> with torsion coefficients being an isomorphism. The Beilinson-Lichtenbaum conjecture is more generally about such an isomorphism for <em>varieties</em> and integer coefficients. How does the former imply the latter (sketch/reference)?</p> http://mathoverflow.net/questions/87162/milnor-bloch-kato-conjecture-implies-the-beilinson-lichtenbaum-conjecture/87260#87260 Answer by Timo Keller for Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture Timo Keller 2012-02-01T18:14:21Z 2012-02-01T18:14:21Z <p>From the Handbook of K-Theory (mentioned by Mikhail Bondarko in the above comments):</p> <p>p. 202, Thomas Geisser:</p> <p>"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."</p> <p>[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.</p> <p>[34] T. Geisser, M. Levine, The Bloch–Kato conjecture and a theorem of Suslin– Voevodsky. J. Reine Angew. Math. 530 (2001), 55–103.</p> http://mathoverflow.net/questions/87162/milnor-bloch-kato-conjecture-implies-the-beilinson-lichtenbaum-conjecture/108325#108325 Answer by Thomas Geisser for Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture Thomas Geisser 2012-09-28T08:35:22Z 2012-09-28T08:35:22Z <p>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. </p> <p>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 $k(x)$.</p>