Lemma 62.6.14. Let $S$ be a locally Noetherian scheme. Let $i : X \to X'$ be a thickening of schemes locally of finite type over $S$. Let $r \geq 0$. Then $i_* : z(X/S, r) \to z(X'/S, r)$ is a bijection.
Proof. Since $i_ s : X_ s \to X'_ s$ is a thickening it is clear that $i_*$ induces a bijection between families of $r$-cycles on the fibres of $X/S$ and families of $r$-cycles on the fibres of $X'/S$. Also, given a family $\alpha $ of $r$-cycles on the fibres of $X/S$ $\alpha \in z(X/S, r) \Leftrightarrow i_*\alpha \in z(X'/S, r)$ by Lemma 62.6.5. The lemma follows. $\square$
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