Kernel sheaf
= Kernel sheaf
{title2=$\ker\phi$}
For a <morphism of sheaves> of abelian groups or modules, the kernel sheaf has sections $\ker(\mathcal F(U)\to\mathcal G(U))$. These already satisfy the <sheaf gluing axiom>. Its <stalk> at $x$ is the kernel of $\mathcal F_x\to\mathcal G_x$, since filtered colimits of modules preserve exactness. A sheaf morphism is injective exactly when its kernel sheaf is zero.