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This is Exercise 2.5.A. in Vakil's notes of Algebraic Geometry. This section is about sheaves of abelian groups on a topological space $X$.

Show that the stalk of the kernel is the kernel of the stalks: for all $p\in X$, there is a natural isomorphism $$(\ker(\mathscr{F}\rightarrow\mathscr{G}))_p\cong \ker(\mathscr{F}_p\rightarrow \mathscr{G}_p).$$

My attempt:

We consider the following diagram:

enter image description here

where the downward arrow is the induced morphism on the stalks, as defined in Exercise 2.4.D. Since the three are all sheaves, the downward arrows are all injective. We want to show that $(\ker\phi)_p$ is the kernel of $\phi_p$ for all $p$. Since the second row is induced from the first row, we have $\phi_p\circ i_p=0$ for all $p$. Since all kernels are monomorphisms, by Exercise 2.4.N., the map $i_p$ is also a monomorphism. Hence it is a kernel of its cokernel.

My question: I am not sure how to show that $i_p$ is the kernel of $\phi_p$. I tried the universal property, but cannot connect it with the upper arrows, so cannot find a unique map.

Thank you for your help!

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    $\begingroup$ In general one use the fact that filtrant directed limits are exact in the category of abelian groups, so they commute with kernels. Note that the functor stalk at $p$ is indeed a limit over a directed system. (You cannot simply use the universal property, you will have to use this property of the category of abelian group) $\endgroup$ – Roland Feb 6 '17 at 10:21
  • $\begingroup$ @Roland: I see! Thank you very much! $\endgroup$ – KittyL Feb 6 '17 at 10:30

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