Relation between motivic cohomology and Quillen K-theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:29:51Z http://mathoverflow.net/feeds/question/30917 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30917/relation-between-motivic-cohomology-and-quillen-k-theory Relation between motivic cohomology and Quillen K-theory norondion 2010-07-07T16:53:30Z 2010-07-07T21:54:31Z <p>What's the relation between Voevodsky's motivic cohomology and Quillen K-theory of a scheme?</p> http://mathoverflow.net/questions/30917/relation-between-motivic-cohomology-and-quillen-k-theory/30937#30937 Answer by norondion for Relation between motivic cohomology and Quillen K-theory norondion 2010-07-07T19:00:29Z 2010-07-07T19:00:29Z <p>I found that for a smooth quasi-projective variety $k$: $CH^q(X,p)_\mathbf{Q} = K_p(X)^{(q)}_\mathbf{Q}$, where $CH^q(X,p) = H^{2q-p,q}(X)$ are Bloch's higher Chow groups.</p> http://mathoverflow.net/questions/30917/relation-between-motivic-cohomology-and-quillen-k-theory/30955#30955 Answer by Dan Ramras for Relation between motivic cohomology and Quillen K-theory Dan Ramras 2010-07-07T21:07:46Z 2010-07-07T21:54:31Z <p>You should look at Marc Levine's preprint "K-theory and motivic cohomology of schemes, I". The version on the UIUC K-theory server seems to be older than the version on his <a href="http://www.uni-due.de/~bm0032/publ/Publ.html" rel="nofollow"> website </a>.</p> <p>Roughly speaking, the motivic spectral sequence starts from motivic cohomology and converges to algebraic K-theory. This spectral sequence was conjectured by Beilinson and first written down for nice fields by Bloch and Lichtenbaum (pre-print on the UIUC server), and was extended to more general schemes by Friedlander and Suslin (Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 6, 773--875). I think these papers were written before the bulk of Voevodsky's work, and the E_2 term was described in terms of Bloch's higher Chow groups. The correspondence between motivic cohomology and higher Chow groups is Theorem 1.2 in Levine's paper, and is due to Voevodsky ("Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic," Int. Math. Res. Not. 2002, no. 7, 351--355).</p>