Lemma 62.6.15. Let $S$ be a locally Noetherian scheme. Let $X$ be a scheme locally of finite type over $S$. Let $r \geq 0$. Let $U \subset X$ be an open such that $X \setminus U$ has relative dimension $< r$ over $S$, i.e., $\dim (X_ s \setminus U_ s) < r$ for all $s \in S$. Then restriction defines a bijection $z(X/S, r) \to z(U/S, r)$.
Proof. Since $Z_ r(X_ s) \to Z_ r(U_ s)$ is a bijection by the dimension assumption, we see that restriction induces a bijection between families of $r$-cycles on the fibres of $X/S$ and families of $r$-cycles on the fibres of $U/S$. These restriction maps $Z_ r(X_ s) \to Z_ r(U_ s)$ are compatible with base change and with specializations, see Lemma 62.5.1 and 62.4.4. The lemma follows easily from this; details omitted. $\square$
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