nLab Chow motive

Definition

The category of effective pure Chow motives , $Mot_{rat}^{eff}(k)$, is the idempotent completion of the category $Corr_{rat}(k)$ whose objects are smooth projective varieties over some field $k$, and whose hom-sets are Chow groups in the product of two varietes (see for instance Vishik09, def. 2.1).

$$Hom_{Corr_{rat}(k)}(Y,X) \coloneqq CH^{dim X}(X \times Y) \,.$$

Hence a morphism $X \to Y$ in $Mor_{rat}^{eff}(k)$ is an equivalence class of linear combinations of correspondences/spans of the form

$$X \leftarrow \Sigma \rightarrow Y \,.$$

If one furthermore inverts the Lefschetz motive $\mathbf{L}$ then one obtains the category of pure Chow motives

$$Mot_{rat}(k) \coloneqq Mot_{rat}(k)[\mathbf{L}^{-1}] \,.$$

This was introduced by Grothendieck. See e.g. (Vishik09, p. 6), (Mazza-Voevodsky-Weibel, p. 181).

This is a special case of the more general notion of pure motives. See there for more.

Properties

Relation to Voevodsky motives

There is a full and faithful functor from the category of Chow motives into that of Voevodsky motives:

$$Chow^{eff} \hookrightarrow DM^{eff} \,.$$

(e.g Mazza-Voevodsky-Weibel, prop. 20.1)

Relation to noncommutative Chow motives

The relation between Chow motives and noncommutative Chow motives is recalled as theorem 4.6 in (Tabuada 11).

This relation is best understood via K-motives, see there.

Relation to KK-theory

The definition of Chow motives in algebraic geometry is somewhat analogous to the construction of KK-theory in noncommutative topology. See at KK-theory – Relation to motives.

Related concepts

• pure motive

  • Chow motive

  • numerical motive

• motivic cohomology

• category of spans

References

A quick and complete statement of the definition is in

• Alexander Vishik, Chow groups and motives, lecture notes 2009 (pdf)

See also

• Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lectures in motivic cohomology (web pdf)

A review of noncommutative Chow motives is in section 4 of

• Goncalo Tabuada, A guided tour through the garden of noncommutative motives, (arxiv1108.3787);

Discussion of how the derived category of a scheme determines its commutative and noncommutative Chow motive is in

• Adeel Khan, On derived categories and noncommutative motives of varieties, arXiv.