Definition 62.6.1. Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $r \geq 0$ be an integer. A relative $r$-cycle on $X/S$ is a family $\alpha $ of $r$-cycles on fibres of $X/S$ such that for every morphism $g : S' \to S$ where $S'$ is the spectrum of a discrete valuation ring we have
\[ sp_{X'/S'}(\alpha _\eta ) = \alpha _0 \]
where $sp_{X'/S'}$ is as in Section 62.4 and $\alpha _\eta $ (resp. $\alpha _0$) is the value of the base change $g^*\alpha $ of $\alpha $ at the generic (resp. closed) point of $S'$. The group of all relative $r$-cycles on $X/S$ is denoted $z(X/S, r)$.
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