I have just asked this question on math.stackexchange, and would like to repost it here:

The norm residue isomorphism theorem establishes that the norm residue map between Milnor K-theory of a field $k$ mod $\ell$ and étale cohomology

$\partial^n: \ K_n^M(k)/\ell \rightarrow H^n_{ét}(k, \mu^{\otimes n}_\ell)$

is an isomorphism. Just how much information about the non-reduced K-theory of a field can be derived from étale cohomology?

  • $\begingroup$ You should delete the MSE posting (or maybe refer readings of that posting to this one): it is considered impolite to simultaneously post in both; and this is the more sensible setting. $\endgroup$ Commented Jan 14, 2015 at 7:06
  • $\begingroup$ OK, just deleted MSE copy. $\endgroup$
    – user6419
    Commented Jan 14, 2015 at 7:12
  • $\begingroup$ The rational K-theory of algebraically closed fields is still not understood (except for $\overline{\mathbb{Q}}$ and $\overline{\mathbb{F}_q(T)}$), and there is no information from etale cohomology that could help with that. On the other hand, rational K-theory has Galois descent. $\endgroup$ Commented Jan 14, 2015 at 9:19
  • $\begingroup$ @Matthias Thank you; however, the purpose of the question was not just to find out how to understand rational K-theory via étale cohomology; I wanted to know what sort of connections (conjectural or not) can be established in the rational case, motivated by the reduced case. $\endgroup$
    – user6419
    Commented Jan 14, 2015 at 16:24


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.