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' which is rationally equivalent to Z on X, such that Y and Z' intersect properly. The lemma is one of the key ingredients in developing intersection theory and the Chow ring, 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 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
• Roberts, Joel (1972). "Chow's moving lemma. Appendix 2 to: "Motives" by Steven L. Kleiman.". Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.). Groningen, Wolters-Noordhoff. pp. 89–96. ISBN 9001670806. MR 0382269. OCLC 579160.