Lemma 17.6.3. Let $i : Z \to X$ be the inclusion of a closed subset into the topological space $X$. The functor $\textit{Ab}(X) \to \textit{Ab}(Z)$, $\mathcal{F} \mapsto \mathcal{H}_ Z(\mathcal{F})$ of Remark 17.6.2 is a right adjoint to $i_* : \textit{Ab}(Z) \to \textit{Ab}(X)$. In particular $i_*$ commutes with arbitrary colimits.

**Proof.**
We have to show that for any abelian sheaf $\mathcal{F}$ on $X$ and any abelian sheaf $\mathcal{G}$ on $Z$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}(Z)}(\mathcal{G}, \mathcal{H}_ Z(\mathcal{F})) \]

This is clear because after all any section of $i_*\mathcal{G}$ has support in $Z$. Details omitted. $\square$

## 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Your email address will not be published.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.

## Comments (4)

Comment #1854 by Keenan Kidwell on

Comment #1893 by Johan on

Comment #6035 by Yuzhou Gu on

Comment #6179 by Johan on