nLab Voevodsky motive

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.

Related concepts

See also

• motivic homotopy theory

• motive

  • pure motive

    • Chow motive, numerical motive

  • mixed motive

  • noncommutative motive

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]