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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.

## References

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 × ℙ^{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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Last updated on Friday November 30, 2007 at 08:33:13 PST (GMT -0800)

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