Motive (algebraic Geometry) - Mixed Motives

For a fixed base field k, the category of mixed motives is a conjectural abelian tensor category MM(k), together with a contravariant functor

Var(k) → MM(X)

taking values on all varieties (not just smooth projective ones as it was the case with pure motives). This should be such that motivic cohomology defined by

Ext*_MM(1, ?)

coincides with the one predicted by algebraic K-theory, and contains the category of Chow motives in a suitable sense (and other properties). The existence of such a category was conjectured by Beilinson. This category is yet to be constructed.

Instead of constructing such a category, it was proposed by Deligne to first construct a category DM having the properties one expects for the derived category

Db(MM(k)).

Getting MM back from DM would then be accomplished by a (conjectural) motivic t-structure.

The current state of the theory is that we do have a suitable category DM. Already this category is useful in applications. Voevodsky's Fields medal-winning proof of the Milnor conjecture uses these motives as a key ingredient.

There are different definitions due to Hanamura, Levine and Voevodsky. They are known to be equivalent in most cases and we will give Voevodsky's definition below. The category contains Chow motives as a full subcategory and gives the "right" motivic cohomology. However, Voevodsky also shows that (with integral coefficients) it does not admit a motivic t-structure.

• Start with the category Sm of smooth varieties over a perfect field. Similarly to the construction of pure motives above, instead of usual morphisms smooth correspondences are allowed. Compared to the (quite general) cycles used above, the definition of these correspondences is more restrictive; in particular they always intersect properly, so no moving of cycles and hence no equivalence relation is needed to get a well-defined composition of correspondences. This category is denoted SmCor, it is additive.
• As a technical intermediate step, take the bounded homotopy category Kb(SmCor) of complexes of smooth schemes and correspondences.
• Apply localization of categories to force any variety X to be isomorphic to X × A1 and also, that a Mayer-Vietoris-sequence holds, i.e. X = U ∪ V (union of two open subvarieties) shall be isomorphic to U ∩ V → U ⊔ V.
• Finally, as above, take the pseudo-abelian envelope.

The resulting category is called the category of effective geometric motives. Again, formally inverting the Tate object, one gets the category DM of geometric motives.