Definition 6.31.5. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.

1. Let $\mathcal{F}$ be an abelian presheaf on $U$. We define the extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $0$ to be the abelian presheaf on $X$ defined by the rule

$j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} 0 & \text{if} & V \not\subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right.$

with obvious restriction mappings.

2. Let $\mathcal{F}$ be an abelian sheaf on $U$. We define the extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $0$ to be the sheafification of the abelian presheaf $j_{p!}\mathcal{F}$.

3. Let $\mathcal{C}$ be a category having an initial object $e$. Let $\mathcal{F}$ be a presheaf on $U$ with values in $\mathcal{C}$. We define the extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $e$ to be the presheaf on $X$ with values in $\mathcal{C}$ defined by the rule

$j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} e & \text{if} & V \not\subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right.$

with obvious restriction mappings.

4. Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. Let $\mathcal{F}$ be a sheaf of algebraic structures on $U$ (of the give type). We define the extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $e$ to be the sheafification of the presheaf $j_{p!}\mathcal{F}$ defined above.

5. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}|_ U$-modules. In this case we define the extension by $0$ to be the presheaf of $\mathcal{O}$-modules which is equal to $j_{p!}\mathcal{F}$ as an abelian presheaf endowed with the multiplication map $\mathcal{O} \times j_{p!}\mathcal{F} \to j_{p!}\mathcal{F}$.

6. Let $\mathcal{O}$ be a sheaf of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}|_ U$-modules. In this case we define the extension by $0$ to be the $\mathcal{O}$-module which is equal to $j_!\mathcal{F}$ as an abelian sheaf endowed with the multiplication map $\mathcal{O} \times j_!\mathcal{F} \to j_!\mathcal{F}$.

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