Lemma 62.6.8 (0H57)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 62: Relative Cycles
Section 62.6: Relative cycles
Lemma 62.6.8 (cite)

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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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