# The Stacks Project

## Tag 00A3

Lemma 6.31.4. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.

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 $$\mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(X)}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(U)}(\mathcal{F}, \mathcal{G}|_U)$$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
3. Let $\mathcal{F}$ be a sheaf of sets on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows $$j_{!}\mathcal{F}_x = \left\{ \begin{matrix} \emptyset & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right.$$
4. On the category of presheaves of $U$ we have $j_pj_{p!} = \text{id}$.
5. On the category of sheaves of $U$ we have $j^{-1}j_! = \text{id}$.

Proof. To map $j_{p!}\mathcal{F}$ into $\mathcal{G}$ it is enough to map $\mathcal{F}(V) \to \mathcal{G}(V)$ whenever $V \subset U$ compatibly with restriction mappings. And by Lemma 6.31.1 the same description holds for maps $\mathcal{F} \to \mathcal{G}|_U$. The adjointness of $j_!$ and restriction follows from this and the properties of sheafification. The identification of stalks is obvious from the definition of the extension by the empty set and the definition of a stalk. Statements (4) and (5) follow by computing the value of the sheaf on any open of $U$. $\square$

The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4476–4508 (see updates for more information).

\begin{lemma}
\label{lemma-j-shriek}
Let $X$ be a topological space.
Let $j : U \to X$ be the inclusion of an open subset.
\begin{enumerate}
\item The functor $j_{p!}$ is a left adjoint to the
restriction functor $j_p$ (see Lemma \ref{lemma-j-pullback}).
\item The functor $j_!$ is a left adjoint to restriction,
in a formula
$$\Mor_{\Sh(X)}(j_!\mathcal{F}, \mathcal{G}) = \Mor_{\Sh(U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \Mor_{\Sh(U)}(\mathcal{F}, \mathcal{G}|_U)$$
bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
\item Let $\mathcal{F}$ be a sheaf of sets on $U$.
The stalks of the sheaf $j_!\mathcal{F}$ are described
as follows
$$j_{!}\mathcal{F}_x = \left\{ \begin{matrix} \emptyset & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right.$$
\item On the category of presheaves of $U$ we have $j_pj_{p!} = \text{id}$.
\item On the category of sheaves of $U$ we have $j^{-1}j_! = \text{id}$.
\end{enumerate}
\end{lemma}

\begin{proof}
To map $j_{p!}\mathcal{F}$ into $\mathcal{G}$
it is enough to map $\mathcal{F}(V) \to \mathcal{G}(V)$
whenever $V \subset U$ compatibly with restriction
mappings. And by Lemma \ref{lemma-j-pullback}
the same description holds for maps
$\mathcal{F} \to \mathcal{G}|_U$.
The adjointness of $j_!$ and restriction follows
from this and the properties of sheafification.
The identification of stalks is obvious from the
definition of the extension by the empty set
and the definition of a stalk.
Statements (4) and (5) follow by computing the
value of the sheaf on any open of $U$.
\end{proof}

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