Contents

# 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.

## 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.

Last revised on May 7, 2021 at 19:41:33. See the history of this page for a list of all contributions to it.