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

$\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X)}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U)}(\mathcal{F}, \mathcal{G}|_ U)$

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 $U$. $\square$

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