The Stacks project

17.7 A canonical exact sequence

We give this exact sequence its own section.

Lemma 17.7.1. Let $X$ be a topological space. Let $U \subset X$ be an open subset with complement $Z \subset X$. Denote $j : U \to X$ the open immersion and $i : Z \to X$ the closed immersion. For any sheaf of abelian groups $\mathcal{F}$ on $X$ the adjunction mappings $j_{!}j^{-1}\mathcal{F} \to \mathcal{F}$ and $\mathcal{F} \to i_*i^{-1}\mathcal{F}$ give a short exact sequence

\[ 0 \to j_{!}j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0 \]

of sheaves of abelian groups. For any morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of abelian sheaves on $X$ we obtain a morphism of short exact sequences

\[ \xymatrix{ 0 \ar[r] & j_{!}j^{-1}\mathcal{F} \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & i_*i^{-1}\mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & j_{!}j^{-1}\mathcal{G} \ar[r] & \mathcal{G} \ar[r] & i_*i^{-1}\mathcal{G} \ar[r] & 0 } \]

Proof. The functoriality of the short exact sequence is immediate from the naturality of the adjunction mappings. We may check exactness on stalks (Lemma 17.3.1). For a description of the stalks in question see Sheaves, Lemmas 6.31.6 and 6.32.1. $\square$

Comments (2)

Comment #7843 by weng yi xiang on

in lemma 02UT,upper * should be upper -1

Comment #8066 by on

Thanks and fixed here. If you wanted to be listed as a contributor can you (re)confirm your name?

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Your email address will not be published.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 02US. Beware of the difference between the letter 'O' and the digit '0'.