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 $i_* : \textit{Ab}(Z) \to \textit{Ab}(Z)$. 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. $\square$

Comment #1854 by Keenan Kidwell on

The second sentence says "...the support...is supported on..." Probably change the second part into "is contained in" or something along those lines.

Comment #6035 by Yuzhou Gu on

In the statement, it should be $i_*: Ab(Z)\to Ab(X)$ instead of $i_*: Ab(Z)\to Ab(Z)$.

(It looks like math formulas in preview and in actual comment work differently.)

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).