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Let $\mathcal{F},\mathcal{G}$ are sheaves of abelian groups on a topological space $X$ and $\phi:\mathcal{F}\mapsto\mathcal{G}$ be a sheaf homomorphism. Then $coker\phi$ turns out to be a pre-sheaf. Does it become a sheaf under some additional natural conditions imposed on the morphism or on the sheaves?(e.g. flasque sheaves, injective morphism etc.)

Sam
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  • This is almost never the case and sheafification is usually required. – Pol van Hoften Nov 06 '17 at 22:08
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    If $\mathcal{F}$ is a flasque sheaf and $\phi$ is injective, then using the acyclicity of $\mathcal{F}$, one can see that $(\mathcal{G}/\mathcal{F})(U) = \mathcal{F}(U)/\mathcal{G}(U)$ and hence the cokernel presheaf is already a sheaf. You maynot assume that $\phi$ is injective but then one can show that $Im(\phi)$ is a flasque sheaf and hence use the previous method. For more details look at https://math.stackexchange.com/a/209535/172843 – random123 Nov 07 '17 at 12:04

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