Relative cycle explained

In algebraic geometry, a relative cycle is a type of algebraic cycle on a scheme. In particular, let

X be a scheme of finite type over a Noetherian scheme S , so that X → S . Then a relative cycle is a cycle on X which lies over the generic points of S , such that the cycle has a well-defined specialization to any fiber of the projection X → S .

The notion was introduced by Andrei Suslin and Vladimir Voevodsky in 2000; the authors were motivated to overcome some of the deficiencies of sheaves with transfers.

References

• Book: 0912.2110. Denis-Charles. Cisinski. Frédéric. Déglise. Triangulated Categories of Mixed Motives. Springer Monographs in Mathematics. 2019. 10.1007/978-3-030-33242-6. 978-3-030-33241-9. 115163824.
• Book: Cycles, Transfers and Motivic Homology Theories. Voevodsky. Vladimir. Suslin. Andrei. Princeton University Press. 2000. 9780691048147. Annals of Mathematics Studies, vol. 143. 10 - 86. Relative cycles and Chow sheaves. 43895658.
• Appendix 1A of