The Stacks project

Lemma 42.12.3. 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

\[ \xymatrix{ Z_ k(X_1 \cap X_2) \ar[r] & Z_ k(X_1) \oplus Z_ k(X_2) \ar[r] & Z_ k(X) \ar[r] & 0 } \]

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. First assume $X$ is quasi-compact. Then $Z_ k(X)$ is a free $\mathbf{Z}$-module with basis given by the elements $[Z]$ where $Z \subset X$ is integral closed of $\delta $-dimension $k$. The groups $Z_ k(X_1)$, $Z_ k(X_2)$, $Z_ k(X_1 \cap X_2)$ are free on the subset of these $Z$ such that $Z \subset X_1$, $Z \subset X_2$, $Z \subset X_1 \cap X_2$. This immediately proves the lemma in this case. The general case is similar and the proof is omitted. $\square$

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