Lemma 17.6.3. Let $i : Z \to X$ be the inclusion of a closed subset into the topological space $X$. Denote1 $i^! : \textit{Ab}(X) \to \textit{Ab}(Z)$ the functor $\mathcal{F} \mapsto i^{-1}\mathcal{H}_ Z(\mathcal{F})$. Then $i^!$ is a right adjoint to $i_*$, in a formula

$\mathop{Mor}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Ab}(Z)}(\mathcal{G}, i^!\mathcal{F}).$

In particular $i_*$ commutes with arbitrary colimits.

Proof. Note that $i_*i^!\mathcal{F} = \mathcal{H}_ Z(\mathcal{F})$. Since $i_*$ is fully faithful we are reduced to showing that

$\mathop{Mor}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{H}_ Z(\mathcal{F})).$

This follows since the support of the image via any homomorphism of a section of $i_*\mathcal{G}$ is contained in $Z$, see Lemma 17.5.2. $\square$

[1] This is likely nonstandard notation.

## Comments (2)

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.

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