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What's the relation between Voevodsky's motivic cohomology and Quillen K-theory of a scheme?

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up vote 7 down vote accepted

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 website .

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).

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The spectral sequence exists for all schemes (say, over a field) if one uses higher Chow groups and K'-theory. It then gives the desired result for smooth schemes because higher Chow groups are isomorphic to motivic cohomology and K'-theory to K-theory. Also, Marc Levine has several papers where he gives a proof of the spectral sequence without referring to the Bloch-Lichtenbaum paper which never appeared. –  Thomas Geisser Sep 25 '12 at 8:19

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.

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