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 $X\to S$. 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 $X\to S$.(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

• Appendix 1A of