Recent Progress
In the mid-1990s, several people proposed candidates for the derived category of the conjectural category of motives. The most successful has been Vladimir Voevodsky's construction. By applying techniques from homotopy theory and K-theory to algebraic geometry, Voevodsky constructed a bigraded motivic cohomology theory
for algebraic varieties. It is not known whether these groups vanish for negative ; this property 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 homotopy theory for algebraic varieties, in the form of a model category, and
- a triangulated category of motives.
If the vanishing conjecture holds, there is an abelian category of motives, and is its derived category.
Read more about this topic: Motivic Cohomology
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