Lemma 62.6.5. Let $S$ be a locally Noetherian scheme. Let $i : X \to Y$ be a closed immersion of schemes locally of finite type over $S$. Let $r \geq 0$. Let $\alpha $ be a family of $r$-cycles on fibres of $X/S$. Then $\alpha $ is a relative $r$-cycle on $X/S$ if and only if $i_*\alpha $ is a relative $r$-cycle on $Y/S$.
Proof. Since base change commutes with $i_*$ (Lemma 62.5.1) it suffices to prove the following: if $S$ is the spectrum of a discrete valuation ring with generic point $\eta $ and closed point $0$, then $sp_{X/S}(\alpha _\eta ) = \alpha _0$ if and only if $sp_{Y/S}(i_{\eta , *}\alpha _\eta ) = i_{0, *}\alpha _0$. This is true because $i_{0, *} : Z_ r(X_0) \to Z_ r(Y_0)$ is injective and because $i_{0, *}sp_{X/S}(\alpha _\eta ) = sp_{Y/S}(i_{\eta , *}\alpha _\eta )$ by Lemma 62.4.5. $\square$
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