Motivic cohomology

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Motivic cohomology is a homological theory in mathematics, the existence of which was first conjectured by 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 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 system of compatible 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 etale cohomology theories with l-adic coefficients, as l varied over prime numbers. In 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. Progress towards realizing this picture was slow; Deligne's absolute Hodge cycles provided one technical fix.

[edit] Recent progress

By applying techniques from homotopy theory and K-theory to algebraic geometry, Voevodsky has recently demonstrated a motivic version of homotopy theory for algebraic varieties in the form of a model category. The resulting category can then be used to produce a motivic cohomology theory for algebraic varieties.

See also: Motive (algebraic geometry)