Lemma 62.6.6. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ locally of finite type. Let $r \geq 0$. Let $\alpha $ and $\beta $ be relative $r$-cycles on $X/S$. The following are equivalent
$\alpha = \beta $, and
$\alpha _\eta = \beta _\eta $ for any generic point $\eta \in S$ of an irreducible component of $S$.
Proof. The implication (1) $\Rightarrow $ (2) is immediate. Assume (2). For every $s \in S$ we can find an $\eta $ as in (2) which specializes to $s$. By Properties, Lemma 28.5.10 we can find a morphism $g : S' \to S$ from the spectrum $S'$ of a discrete valuation ring which maps the generic point $\eta '$ to $\eta $ and maps the closed point $0$ to $s$. Then $\alpha _ s$ and $\beta _ s$ are elements of $Z_ r(X_ s)$ which base change to the same element of $Z_ r(X_{0'})$, namely $sp_{X_{S'}/S'}(\alpha _{\eta '})$ where $\alpha _{\eta '}$ is the base change of $\alpha _\eta $. Since the base change map $Z_ r(X_ s) \to Z_ r(X_{0'})$ is injective as discussed in Section 62.3 we conclude $\alpha _ s = \beta _ s$. $\square$
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