nLab
Voevodsky motive

A. Suslin and V. Voevodsky have realized a triangulated category $D^b({\mathrm{ℳℳ}}_k)$ which is supposed to be the bounded derived category of the hypothetical abelian category of mixed motive?s over $k$, predicted by Grothendieck–Beilinson–Deligne. There are variants developed by Hanamura and M. Levine. There is a different “derived” approach to mixed motives, namely the $A^1$-homotopy theory of F. Morel and V. Voevodsky. Voevodsky used the derived category of mixed motives to solve Milnor's conjecture in algebraic K-theory.

See also

• pure motive, motive

• wikipedia article on motives

• A. Beilinson, V. Vologodsky, A DG guide to Voevodsky’s motives, math.AG/0604004

• M.V. Bondarko, Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura, math.AG/0601713

• M.V. Bondarko, Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences: a survey, arxiv:0903.0091

• V. Voevodsky, Motives over simplicial schemes, Journal of K-Theory, Volume 5, Issue 01 , pp 1 - 38, (preliminary version in K-theory preprint archive: here.