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Consider sheaves of abelian groups $\mathcal{F}$ and $\mathcal{G}$ and a morphism of sheaves $\varphi: \mathcal{F} \to \mathcal{G}$. So for each open $U$ we have a grouphomomorphism $\varphi_U : \mathcal{F}(U) \to \mathcal{G}(U)$. Consider for each open $U$ the association $\mathcal{ker}(\varphi)(U) := \ker(\varphi_u)$.

This is a sheaf. So it is also a presheaf. However I am wondering what the restriction map $\mathrm{res}^{U}_V: \mathrm{ker}(\varphi)(U) \to \mathrm{ker}(\varphi)(V)$ is.

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It is the usual restriction map. Let $s \mapsto s_{|V}$ denotes the restriction map for $F,G$ and $s \mapsto res^U_V(s)$ the restriction map for the kernel sheaf.

If $s \in ker(\phi)(U)$ that means that $s \in F(U)$ and $\phi_U(s) = 0$. We define $res^U_V(s) = s_{|V}$. Since $\phi_V(s_{|V}) = \phi_U(s)_{|V} = 0_{|V} = 0$ this is well defined. (I used that $\phi$ is a morphism of sheaves, i.e commutes with restriction map).

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