Motivic Cohomology

Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry. It had a basis in category theory for drawing consequences from those conjectures; Grothendieck and Enrico Bombieri showed the depth of this approach by deriving a conditional proof of the Weil conjectures by this route. The standard conjectures, however, resisted proof.

This left the motive (motif in French) theory as having heuristic status. Serre, for example, preferred to work more concretely with a compatible system of ℓ-adic representations, which at least conjecturally should be as good as having a motive, but instead listed the data obtainable from a motive by means of its 'realisations' in the étale cohomology theories with l-adic coefficients, as l varied over prime numbers. From the Grothendieck point of view, motives should further contain the information provided by algebraic de Rham cohomology, and crystalline cohomology. In some sense motivic cohomology would be the mother of all cohomology theories in algebraic geometry; the other cohomology theories would be specializations.

Grothendieck gave a solution for Weil cohomology theories over a field in 1967. This involved extending the category of smooth projective varieties to the category of Chow motives. This is an additive category, but not an abelian category unless one takes rational coefficients and passes to numerical equivalence. The study of motives for arbitrary varieties (mixed motives) began in the early 1970s with Deligne's notion of 1-motives. The hope is that there is something like an abelian category of mixed motives, containing all varieties over the field, and a universal cohomology theory on mixed motives in the sense of Homological Algebra. However, progress towards realizing this picture was slow; Deligne's absolute Hodge cycles provided one technical fix. Beilinson's absolute Hodge cohomology provided a universal cohomology theory with rational coefficients (and without any category of motives) using algebraic K-theory.