Mixed Motives

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 $$ \text{Hom}_{D^{b}(\mathcal{MM}_{k})}(1,h(X)(n)[m]) \otimes \mathbb{Q} \cong K_{2n-m}(X)^{(n)} $$ , where K p (X)(n) is the weight n eigenspace for the Adams operations. The abelian groups $$ H^{p}_{\mathcal{M}}(X,\mathbb{Z}(q)) := \text{Hom}_{D^{b}(\mathcal{MM}_{k})}(1,h(X)(q)[p]) $$ 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.