Chow's moving lemma

In algebraic geometry, Chow's moving lemma, proved by Wei-Liang Chow (1956), states: given algebraic cycles Y, Z on a nonsingular quasi-projective variety X, there is another algebraic cycle Z' on X such that Z' is rationally equivalent to Z and Y and Z' intersect properly. The lemma is one of key ingredients in developing the intersection theory, as it is used to show the uniqueness of the theory.

Even if Z is an effective cycle, it is not, in general, possible to choose the cycle Z' to be effective.

References[]

Chow, Wei-Liang (1956), "On equivalence classes of cycles in an algebraic variety", Annals of Mathematics, 64 (3): 450–479, doi:10.2307/1969596, ISSN 0003-486X, JSTOR 1969596, MR 0082173
Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

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