kernels, cokernels, etc. on sheaves of modules

In this subsection we restrict ourselves to deal with sheaves of modules.

To shorten our statements, we call a presheaf which satisfies (only) the sheaf axiom (1) (locality) a “(1)-presheaf”.

LEMMA 07.34   Let $\varphi: \mathcal F\to \mathcal G$ be a homomorphism between sheaves of modules. Then we have

1. The presheaf kernel of $\varphi$ is a sheaf. We call it the sheaf kernel $\operatorname{Ker}(\varphi)$ of $\varphi$.
2. The presheaf image of $\varphi$ is not necessarily a sheaf, but it is a (1)-presheaf. We call the sheafication of the presheaf image as the sheaf image $\operatorname{Image}(\varphi)$ of $\varphi$.
3. The presheaf cokernel of $\varphi$ is not necessarily a sheaf. We call the sheafication of the cokernel as the sheaf cokernel $\operatorname{Coker}(\varphi)$ of $\varphi$.

DEFINITION 07.35   A sequence of homomorphisms of sheaves of modules

$$\mathcal F_1 \overset{f_1}{\to} \mathcal F_2 \overset{f_2}{\to} \mathcal F_3$$

is said to be exact if $\operatorname{Image}(f_1)=\operatorname{Ker}(f_2)$ holds.

LEMMA 07.36   A sequence of homomorphisms of sheaves of modules

$$\mathcal F_1 \overset{f_1}{\to} \mathcal F_2 \overset{f_2}{\to} \mathcal F_3$$

is exact if and only if it is exact stalk wise, that means, if and only if the sequence

$$(\mathcal F_1)_P \overset{f_1}{\to} (\mathcal F_2)_P \overset{f_2}{\to} (\mathcal F_3)_P$$

is exact for all point $P$.