Lemma 62.6.4. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r, e \geq 0$ be integers. Let $\alpha $ be a family of $r$-cycles on fibres of $X/S$. Let $\{ f_ i : X_ i \to X\} $ be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension $e$. Then $\alpha $ is a relative $r$-cycle if and only if each flat pullback $f_ i^*\alpha $ is a relative $r$-cycle.
Proof. If $\alpha $ is a relative $r$-cycle, then each pull back $f_ i^*\alpha $ is a relative $r$-cycle by Lemma 62.6.2. Assume each $f_ i^*\alpha $ is a relative $r$-cycle. Let $g : S' \to S$ be a morphism where $S'$ is the spectrum of a discrete valuation ring. After replacing $S$ by $S'$, $X$ by $X' = X \times _ S S'$, and $\alpha $ by $\alpha ' = g^*\alpha $ we reduce to the problem studied in the next paragraph.
Assume $S$ is the spectrum of a discrete valuation ring with closed point $0$ and generic point $\eta $. We have to show that $sp_{X/S}(\alpha _\eta ) = \alpha _0$. Denote $f_{i, 0} : X_{i, 0} \to X_0$ the base change of $f_ i$ to the closed point of $S$. Similarly for $f_{i, \eta }$. Observe that
\[ f_{i, 0}^*sp_{X/S}(\alpha _\eta ) = sp_{X_ i/S}(f_{i, \eta }^*\alpha _\eta ) = f_{i, 0}^*\alpha _0 \]
Namely, the first equality holds by Lemma 62.4.4 and the second by assumption. Since the family of maps $f_{i, 0}^* : Z_ r(X_0) \to Z_ r(X_{i, 0})$ is jointly injective (due to the fact that $f_{i, 0}$ is jointly surjective), we conclude what we want. $\square$
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