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.

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)

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.

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*.

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.

Last revised on September 1, 2022 at 02:38:56. See the history of this page for a list of all contributions to it.