Relative cycle

In algebraic geometry, a relative cycle is a type of algebraic cycle on a scheme. In particular, let ${\displaystyle X}$ be a scheme of finite type over a Noetherian scheme ${\displaystyle S}$, so that ${\displaystyle X\rightarrow S}$. Then a relative cycle is a cycle on ${\displaystyle X}$ which lies over the generic points of ${\displaystyle S}$, such that the cycle has a well-defined specialization to any fiber of the projection ${\displaystyle X\rightarrow S}$.(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 (2012-12-18). "Triangulated categories of mixed motives". arXiv:0912.2110 [math.AG].
• 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.CS1 maint: ref=harv (link)
• Appendix 1A of Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology, Clay Mathematics Monographs, 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284