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 etale 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. Progress towards realizing this picture was slow; Deligne's absolute Hodge cycles provided one technical fix.

Recent progress

By applying techniques from homotopy theory and K-theory to algebraic geometry, Voevodsky has constructed a bigraded motivic cohomology theory $H^{p,q}(X)$ for algebraic varieties. It is not known whether these groups vanish for negative $p$; this is known as the Vanishing Conjecture. Otherwise, this theory is known to satisfy all of the properties suggested by Grothendieck. Voevodsky provided two constructions of motivic cohomology for algebraic varieties, via: (a) a homotopy theory for algebraic varieties, in the form of a model category, and (b) a triangulated category $DM$ of motives. If the Vanishing Conjecture holds, there is an abelian category of motives, and $DM$ is its derived category.

References

• Kleiman, Steven L. (1972), "Motives", in Oort, F. (ed.), Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970), Groningen: Wolters-Noordhoff, pp. 53–82, MR0382267
• Uwe Jannsen ... eds. (1994), Serre, Jean-Pierre; Jannsen, Uwe; Kleiman, Steven L. (eds.), Motives, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1637-0 {{citation}}: |author= has generic name (help)
• Carlo Mazza, Vladimir Voevodsky, Charles Weibel. (2006), Lectures in Motivic Cohomology, American Mathematical Society, ISBN 978-0-8218-3847-1 {{citation}}: Cite has empty unknown parameter: |unused_data= (help); External link in |title= (help); Text "Clay Math Monographs vol.2" ignored (help)CS1 maint: multiple names: authors list (link)

See also

• Motive (algebraic geometry)