# Contents

## Idea

A. Suslin and V. Voevodsky have realized a triangulated category ${D}^{b}\left({\mathrm{ℳℳ}}_{k}\right)$ 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.