Relative cycle

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 $X\rightarrow S$ . 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 $X\rightarrow S$ .(Voevodsky & Suslin 2000)

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

Cisinski, Denis-Charles; Déglise, Frédéric (2019). Triangulated Categories of Mixed Motives. Springer Monographs in Mathematics. arXiv:0912.2110. doi:10.1007/978-3-030-33242-6. ISBN 978-3-030-33241-9. S2CID 115163824.
Voevodsky, Vladimir; Suslin, Andrei (2000). "Relative cycles and Chow sheaves". Cycles, Transfers and Motivic Homology Theories. Annals of Mathematics Studies, vol. 143. Princeton University Press. pp. 10–86. ISBN 9780691048147. OCLC 43895658.
Appendix 1A of Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284

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