Lemma 17.6.3 (01AZ)—The Stacks project

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. $\square$

Comments (4)

Comment #1854 by Keenan Kidwell on March 06, 2016 at 00:45

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

Comment #1893 by Johan on April 01, 2016 at 23:33

Thanks, fixed here.

Comment #6035 by Yuzhou Gu on April 05, 2021 at 11:35

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.)

Comment #6179 by Johan on May 01, 2021 at 23:08

Thanks and fixed here.

Comments (4)

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