Relative cycle

For the algebraic topology concept, see Relative homology § Definition.

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