Relative cycle

In algebraic geometry, a relative cycle is a type of algebraic cycle on a scheme. In particular, let [math]\displaystyle{ X }[/math] be a scheme of finite type over a Noetherian scheme [math]\displaystyle{ S }[/math], so that [math]\displaystyle{ X \rightarrow S }[/math]. Then a relative cycle is a cycle on [math]\displaystyle{ X }[/math] which lies over the generic points of [math]\displaystyle{ S }[/math], such that the cycle has a well-defined specialization to any fiber of the projection [math]\displaystyle{ X \rightarrow S }[/math].(Voevodsky Suslin) 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. doi:10.1007/978-3-030-33242-6. ISBN 978-3-030-33241-9. 
• 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, 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1