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. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 42.12: Proper pushforward

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F92. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 42.12.3 as pdf