Motive

Summary

A motive is an object of a category built from algebraic varieties and algebraic cycles on their products. Grothendieck proposed this category as a universal cohomology theory for varieties. Every good cohomology theory, such as ℓ-adic cohomology, algebraic de Rham cohomology or singular cohomology, should factor through the category of motives. Pure motives over a field are well defined, with morphisms given by correspondences modulo an equivalence relation such as rational equivalence. Many of their expected properties depend on the standard conjectures.