A motive is an object designed to encode the common algebraic part of the different cohomology theories of algebraic varieties. Pure motives are constructed from smooth projective varieties using algebraic correspondences with rational coefficients modulo an appropriate equivalence relation, formally splitting idempotent correspondences and adjoining Tate twists. This construction permits direct sums, tensor products, and duals. Singular, de Rham, and étale cohomology give realizations when the equivalence relation is compatible with the chosen cohomology theory. Rational equivalence has this compatibility; for numerical equivalence it requires the corresponding standard conjecture identifying numerical and homological equivalence (Milne 2012, pp. 8-9).
The theory of mixed motives extends this framework to varieties that are not smooth and projective, but its most general expected category remains conjectural.
