The Stacks project

Lemma 62.5.10. Let $g : S' \to S$ be a bijective morphism of schemes which induces isomorphisms of residue fields. Let $f : X \to S$ be locally of finite type. Set $X' = S' \times _ S X$. Let $r \geq 0$. Then base change by $g$ determines a bijection between the group of families of $r$-cycles on fibres of $X/S$ and the group of families of $r$-cycles on fibres of $X'/S'$.

Proof. Omitted. $\square$

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