The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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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