# The Stacks Project

## Tag 00A7

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.

1. The functor $j_{p!}$ is a left adjoint to restriction, in a formula $$\mathop{\rm Mor}\nolimits_{\textit{PMod}(\mathcal{O})}(j_{p!}\mathcal{F}, \mathcal{G}) = \mathop{\rm Mor}\nolimits_{\textit{PMod}(\mathcal{O}|_U)}(\mathcal{F}, \mathcal{G}|_U)$$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
2. The functor $j_!$ is a left adjoint to restriction, in a formula $$\mathop{\rm Mor}\nolimits_{\textit{Mod}(\mathcal{O})}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\rm Mor}\nolimits_{\textit{Mod}(\mathcal{O}|_U)}(\mathcal{F}, \mathcal{G}|_U)$$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
3. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows $$j_{!}\mathcal{F}_x = \left\{ \begin{matrix} 0 & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right.$$
4. On the category of sheaves of $\mathcal{O}|_U$-modules on $U$ we have $j^{-1}j_! = \text{id}$.

Proof. Omitted. $\square$

The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4698–4737 (see updates for more information).

\begin{lemma}
\label{lemma-j-shriek-modules}
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.
\begin{enumerate}
\item The functor $j_{p!}$ is a left adjoint to restriction,
in a formula
$$\Mor_{\textit{PMod}(\mathcal{O})}(j_{p!}\mathcal{F}, \mathcal{G}) = \Mor_{\textit{PMod}(\mathcal{O}|_U)}(\mathcal{F}, \mathcal{G}|_U)$$
bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
\item The functor $j_!$ is a left adjoint to restriction,
in a formula
$$\Mor_{\textit{Mod}(\mathcal{O})}(j_!\mathcal{F}, \mathcal{G}) = \Mor_{\textit{Mod}(\mathcal{O}|_U)}(\mathcal{F}, \mathcal{G}|_U)$$
bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
\item Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $U$.
The stalks of the sheaf $j_!\mathcal{F}$ are described
as follows
$$j_{!}\mathcal{F}_x = \left\{ \begin{matrix} 0 & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right.$$
\item On the category of sheaves of $\mathcal{O}|_U$-modules on $U$
we have $j^{-1}j_! = \text{id}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

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