The Stacks project

Lemma 62.6.8. In the situation of Example 62.5.4 assume $S$ is locally Noetherian and $Z$ is flat over $S$ in dimensions $\geq r$. Then $[Z/X/S]_ r$ is a relative $r$-cycle on $X/S$.

Proof. The assumption means that $\mathcal{O}_ Z$ is flat over $S$ in dimensions $\geq r$. Thus applying Lemma 62.6.7 with $\mathcal{F} = (Z \to X)_*\mathcal{O}_ Z$ we conclude. $\square$

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