Does Milnor K-Theory arise from Waldhausen K-Theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:38:01Z http://mathoverflow.net/feeds/question/9321 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9321/does-milnor-k-theory-arise-from-waldhausen-k-theory Does Milnor K-Theory arise from Waldhausen K-Theory philip314 2009-12-18T22:09:44Z 2011-05-22T00:02:50Z <p>Quillens higher K-groups of rings can be realized as &pi;<sub>n</sub>K(C) - the Waldhausen K-Theory of a suitable Waldhausen category C. Is this also true for Milnor K-Theory of Rings? Is there a functor F from rings to waldhausen categories s.t. $K^M_n(R)\cong \pi_n(K(F(R))$?</p> http://mathoverflow.net/questions/9321/does-milnor-k-theory-arise-from-waldhausen-k-theory/9545#9545 Answer by Evgeny Shinder for Does Milnor K-Theory arise from Waldhausen K-Theory Evgeny Shinder 2009-12-22T16:22:02Z 2009-12-22T16:22:02Z <p>I don't know if there any evidence for this to be true. Note that Quillen K-groups <em>are defined</em> as homotopy groups of some space (+-construction, Q-construction, Waldhausen construction etc), whereas Milnor K-groups were defined in terms of generators and relations, which generalize generators and relations for classical K_2.</p> <p>More invariantly Milnor K-groups can be constructed using homology of GL_n (paper of Suslin and Nesterenko) or as certain motivic cohomology groups of a field (Suslin-Voevodsky). However, these constructions are unrelated to any homotopy groups.</p> <p>Also, I'm not sure how you define Milnor K-theory for a general ring R? (I was interpreting your question with "ring R" replaced by "field F".)</p> http://mathoverflow.net/questions/9321/does-milnor-k-theory-arise-from-waldhausen-k-theory/40640#40640 Answer by John Rognes for Does Milnor K-Theory arise from Waldhausen K-Theory John Rognes 2010-09-30T16:23:35Z 2010-10-03T13:25:34Z <p>Bob Thomason proved that there is no Milnor K-theory functor for schemes, with a reasonable map to Quillen K-theory, in:</p> <blockquote> <p>Le principe de scindage et l'inexistence d'une $K$-theorie de Milnor globale. [The splitting principle and the nonexistence of a global Milnor $K$-theory] Topology 31 (1992), no. 3, 571--588.</p> </blockquote> <ul> <li>John</li> </ul> http://mathoverflow.net/questions/9321/does-milnor-k-theory-arise-from-waldhausen-k-theory/65685#65685 Answer by Dan Grayson for Does Milnor K-Theory arise from Waldhausen K-Theory Dan Grayson 2011-05-22T00:02:50Z 2011-05-22T00:02:50Z <p>The original question seems not to have been answered yet. One answer might be that it would be unnatural to expect all the Milnor K-groups of a field R to arise as the homotopy groups of a single space $K(F(R))$, because the natural way they currently arise is as homotopy groups of separate spaces, or better, of separate spectra. The spectra are the Eilenberg-MacLane spectra $\mathbb Z(n)$ associated to the chain complexes that compute motivic cohomology of $R$, namely, $K_n^M R = \pi_{-n} \mathbb Z(n)$.</p>