The Stacks project

Remark 62.8.9. Let $f : X \to S$ be a morphism of schemes. Let $r \geq 0$. Let $Z \subset X$ be a closed subscheme. Assume

1. $S$ is Noetherian and geometrically unibranch,

2. $f$ is of finite type, and

3. $Z \to S$ has relative dimension $\leq r$.

Then for all sufficiently divisible integers $n \geq 1$ there exists a unique effective relative $r$-cycle $\alpha$ on $X/S$ such that $\alpha _\eta = n[Z_\eta ]_ r$ for every generic point $\eta$ of $S$. This is a reformulation of [Theorem 3.4.2, SV]. If we ever need this result, we will precisely state and prove it here.