nLab
Voevodsky motive
Contents
Context
Motivic cohomology
Contents
Idea
A. Suslin and V. Voevodsky have realized a triangulated category $D^b(\mathcal{MM}_k)$ which is supposed to be the bounded derived category of the hypothetical abelian category of mixed motives 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 .

See at motive the section Contructions of the derived category of mixed motives .

Applications
Voevodsky used the derived category of mixed motives to solve Milnor's conjecture in algebraic K-theory .

See also

References
Wikipedia article on motives and references therein.

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 .

Models for Voevodsky motives:

Peter Bonart, Triangulated Categories of Big Motives via Enriched Functors (2023) [arXiv:2310.17349 ]
Last revised on October 28, 2023 at 06:06:41.
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