nLab numerical motive

Idea

A pure motive defined by applying numerical equivalence of algebraic cycles.

Properties

Semi-simple abelian category

The category of numerical motives is a semisimple abelian category.

In fact, numerical equivalence is precisely the only adequate equivalence relation relation that gives the category of pure motives this property.

(Jannsen 92)

Related concepts

• motivic Galois group

References

A standard account is in

• U. Jannsen, S. Kleiman and J.-P. Serre, Motives, Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held at the University of Washington, Seattle, Washington, July 20–August 2, 1991. Proceedings of Symposia in Pure Mathematics 55, part 1 and 2. American Mathematical Society, Providence, RI, 1994.

The fact that the category of numerical motives is semisimple abelian is in

• Uwe Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. math. 107 (1992) (pdf)

A brief review of numerical noncommutative motives is in section 4 of

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