The Stacks project

Lemma 42.19.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $X_1, X_2 \subset X$ be closed subschemes such that $X = X_1 \cup X_2$ set theoretically. For every $k \in \mathbf{Z}$ the sequence of abelian groups

\[ \xymatrix{ \mathop{\mathrm{CH}}\nolimits _ k(X_1 \cap X_2) \ar[r] & \mathop{\mathrm{CH}}\nolimits _ k(X_1) \oplus \mathop{\mathrm{CH}}\nolimits _ k(X_2) \ar[r] & \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r] & 0 } \]

is exact. Here $X_1 \cap X_2$ is the scheme theoretic intersection and the maps are the pushforward maps with one multiplied by $-1$.

Proof. By Lemma 42.12.3 the arrow $\mathop{\mathrm{CH}}\nolimits _ k(X_1) \oplus \mathop{\mathrm{CH}}\nolimits _ k(X_2) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$ is surjective. Suppose that $(\alpha _1, \alpha _2)$ maps to zero under this map. Write $\alpha _1 = \sum n_{1, i}[W_{1, i}]$ and $\alpha _2 = \sum n_{2, i}[W_{2, i}]$. Then we obtain a locally finite collection $\{ W_ j\} _{j \in J}$ of integral closed subschemes of $X$ of $\delta $-dimension $k + 1$ and $f_ j \in R(W_ j)^*$ such that

\[ \sum n_{1, i}[W_{1, i}] + \sum n_{2, i}[W_{2, i}] = \sum (i_ j)_*\text{div}(f_ j) \]

as cycles on $X$ where $i_ j : W_ j \to X$ is the inclusion morphism. Choose a disjoint union decomposition $J = J_1 \amalg J_2$ such that $W_ j \subset X_1$ if $j \in J_1$ and $W_ j \subset X_2$ if $j \in J_2$. (This is possible because the $W_ j$ are integral.) Then we can write the equation above as

\[ \sum n_{1, i}[W_{1, i}] - \sum \nolimits _{j \in J_1} (i_ j)_*\text{div}(f_ j) = - \sum n_{2, i}[W_{2, i}] + \sum \nolimits _{j \in J_2} (i_ j)_*\text{div}(f_ j) \]

Hence this expression is a cycle (!) on $X_1 \cap X_2$. In other words the element $(\alpha _1, \alpha _2)$ is in the image of the first arrow and the proof is complete. $\square$

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