Definitions

# Equivalence relations on algebraic cycles

In mathematics, equivalence relations of algebraic cycles are used in order to obtain a well-working theory of algebraic cycles, including well-defined intersection products. They also form an integral part of the category of pure motives.

Possible (and useful) adequate equivalence relations include the rational, algebraic, homological and numerical equivalence. "Adequate" means that the relations behave well with respect to functoriality, i.e. push-forward and pull-back of cycles.

definition remarks
rational equivalence Z ∼rat Z' if there is a cycle V on X × 1, such that V ∩ X × {0} = Z and V ∩ X × {∞} = Z' .

the finest adequate equivalence relation. "∩" denotes intersection in the cycle-theoretic sense (i.e. with multiplicities)
algebraic equivalence Z ∼alg Z' if there is a curve C and a cycle V on X × C, such that V ∩ X × {c} = Z and V ∩ X × {d} = Z' for two points c and d on the curve.

homological equivalence for a given Weil cohomology H, Z ∼hom Z' if the image of the cycles under the cycle class map agrees

depends a priori of the choice of H, but does not assuming the standard conjecture D
numerical equivalence Z ∼num Z' if Z ∩ T = Z' ∩ T, where T is any cycle such that dim T = codim Z (so that the intersection is a linear combination of points)

the coarsest equivalence relation

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