Lemma 6.31.8. Let $(X, \mathcal{O})$ be a ringed space. Let $j : (U, \mathcal{O}|_ U) \to (X, \mathcal{O})$ be an open subspace. Consider the functors of restriction and extension by $0$ for (pre)sheaves of modules defined above.

The functor $j_{p!}$ is a left adjoint to restriction, in a formula

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}(j_{p!}\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O}|_ U)}(\mathcal{F}, \mathcal{G}|_ U) \]bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.

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

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O}|_ U)}(\mathcal{F}, \mathcal{G}|_ U) \]bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.

Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows

\[ j_{!}\mathcal{F}_ x = \left\{ \begin{matrix} 0 & \text{if} & x \not\in U \\ \mathcal{F}_ x & \text{if} & x \in U \end{matrix} \right. \]On the category of sheaves of $\mathcal{O}|_ U$-modules on $U$ we have $j^{-1}j_! = \text{id}$.

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