Abstract
During the early and mid-eighties, Beilinson [2] and Deligne [24] independently described a conjectural abelian tensor category of mixed motives over a given base field k, ℳℳ k , which, in analogy to the category of mixed Hodge structures, should contain Grothendieck’s category of pure (homological) motives as the full subcategory of semi-simple objects, but should have a rich enough structure of extensions to allow one to recover the weight-graded pieces of algebraic K-theory. More specifically, one should have, for each smooth scheme X of finite type over a given field k, an object h(X) in the derived category D b(ℳℳ k ), as well as Tate twists h(X)(n), and natural isomorphisms
,
where K p (X)(n) is the weight n eigenspace for the Adams operations. The abelian groups
should form the universal Bloch–Ogus cohomology theory on smooth k-schemes of finite type; as this theory should arise from mixed motives, it is called motivic cohomology.
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Levine, M. (2005). Mixed Motives. In: Friedlander, E., Grayson, D. (eds) Handbook of K-Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27855-9_10
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