Lemma 6.31.1. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset $U$ into $X$.

Let $\mathcal{G}$ be a presheaf of sets on $X$. The presheaf $j_ p\mathcal{G}$ (see Section 6.21) is given by the rule $V \mapsto \mathcal{G}(V)$ for $V \subset U$ open.

Let $\mathcal{G}$ be a sheaf of sets on $X$. The sheaf $j^{-1}\mathcal{G}$ is given by the rule $V \mapsto \mathcal{G}(V)$ for $V \subset U$ open.

For any point $u \in U$ and any sheaf $\mathcal{G}$ on $X$ we have a canonical identification of stalks

\[ j^{-1}\mathcal{G}_ u = (\mathcal{G}|_ U)_ u = \mathcal{G}_ u. \]On the category of presheaves of $U$ we have $j_ pj_* = \text{id}$.

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

The same description holds for (pre)sheaves of abelian groups, (pre)sheaves of algebraic structures, and (pre)sheaves of modules.

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