\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*}

The Stacks project

Lemma 6.31.6. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. Consider the functors of restriction and extension by $0$ for abelian (pre)sheaves.

  1. The functor $j_{p!}$ is a left adjoint to the restriction functor $j_ p$ (see Lemma 6.31.1).

  2. The functor $j_!$ is a left adjoint to restriction, in a formula

    \[ \mathop{Mor}\nolimits _{\textit{Ab}(X)}(j_!\mathcal{F}, \mathcal{G}) = \mathop{Mor}\nolimits _{\textit{Ab}(U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{Mor}\nolimits _{\textit{Ab}(U)}(\mathcal{F}, \mathcal{G}|_ U) \]

    bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.

  3. Let $\mathcal{F}$ be an abelian sheaf on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows

    \[ j_{!}\mathcal{F}_ x = \left\{ \begin{matrix} 0 & \text{if} & x \not\in U \\ \mathcal{F}_ x & \text{if} & x \in U \end{matrix} \right. \]
  4. On the category of abelian presheaves of $U$ we have $j_ pj_{p!} = \text{id}$.

  5. On the category of abelian sheaves of $U$ we have $j^{-1}j_! = \text{id}$.

Proof. Omitted. $\square$


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