The Stacks project

Lemma 82.15.2. In Situation 82.2.1 let $X/B$ be good. Let $U \subset X$ be an open subspace. Let $Y$ be the reduced closed subspace of $X$ with $|Y| = |X| \setminus |U|$ and denote $i : Y \to X$ the inclusion morphism. 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 subspaces 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. Let $W_ j' \subset X$ be the corresponding integral closed subspace of $X$, i.e., having the same generic point as $W_ j$. 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 on $X$ whose restriction to $U$ is zero. The lemma follows by applying Lemma 82.10.2. $\square$

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