Lemma 42.19.3. Let (S, \delta ) be as in Situation 42.7.1. Let X be a scheme locally of finite type over S. Let U \subset X be an open subscheme, and denote i : Y = X \setminus U \to X as a reduced closed subscheme of X. Let k \in \mathbf{Z}. Suppose \alpha , \beta \in Z_ k(X). If \alpha |_ U \sim _{rat} \beta |_ U then there exist a cycle \gamma \in Z_ k(Y) such that
\alpha \sim _{rat} \beta + i_*\gamma .
In other words, the sequence
\xymatrix{ \mathop{\mathrm{CH}}\nolimits _ k(Y) \ar[r]^{i_*} & \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r]^{j^*} & \mathop{\mathrm{CH}}\nolimits _ k(U) \ar[r] & 0 }
is an exact complex of abelian groups.
Proof.
Let \{ W_ j\} _{j \in J} be a locally finite collection of integral closed subschemes of U of \delta -dimension k + 1, and let f_ j \in R(W_ j)^* be elements such that (\alpha - \beta )|_ U = \sum (i_ j)_*\text{div}(f_ j) as in the definition. Set W_ j' \subset X equal to the closure of W_ j. Suppose that V \subset X is a quasi-compact open. Then also V \cap U is quasi-compact open in U as V is Noetherian. Hence the set \{ j \in J \mid W_ j \cap V \not= \emptyset \} = \{ j \in J \mid W'_ j \cap V \not= \emptyset \} is finite since \{ W_ j\} is locally finite. In other words we see that \{ W'_ j\} is also locally finite. Since R(W_ j) = R(W'_ j) we see that
\alpha - \beta - \sum (i'_ j)_*\text{div}(f_ j)
is a cycle supported on Y and the lemma follows (see Lemma 42.14.2).
\square
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