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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