nLab
Chow motive

Contents

Definition

The category of effective pure Chow motives , Mot rat eff(k)Mot_{rat}^{eff}(k), is the idempotent completion of the category Corr rat(k)Corr_{rat}(k) whose objects are smooth projective varieties over some field kk, 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)CH dimX(X×Y). Hom_{Corr_{rat}(k)}(Y,X) \coloneqq CH^{dim X}(X \times Y) \,.

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

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

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

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

This was introduced by Grothendieck. See e.g. (Vishik09, p. 6), (Mazza-Voevosky-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 effDM eff. 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.

References

A quick and complete statement of the definition is in

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

See also

A review of noncommutative Chow motives is in section 4 of

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 July 3, 2014 at 09:52:58. See the history of this page for a list of all contributions to it.