Where is sheafification in the definition of exact sequence of sheaves?

Question

I am reading Andreas Gathmann's notes on Algebraic geometry,http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf

Def 7.1.14(iv)says the following

As usual, a sequence of sheaves and morphisms $$ \cdots \rightarrow F_{i−1} \rightarrow F_i \rightarrow F_{i+1} \rightarrow \cdots $$ is called exact if ker$( F_i \rightarrow F_{i+1}) = $im$( F_{i−1}\rightarrow F_i ) $ for all $i$.

However，even every arrow is a morphism of sheaves, the image of a sheaf will not necessarily be a sheaf, so what does this definition really mean? I have two guesses.

One is that if one of the image is not a sheaf, then the sequnce is never exact. The other is that the notation im$( F_{i−1}\rightarrow F_i ) $ stands for the sheafification of the presheaf image.

Which one is true? (or both are wrong?)

Answer 1

Sheaves of abelian groups form an abelian category, and the definition given is the usual definition of the exactness of a sequence in an abelian category. Image here means the categorical image in an abelian category, which is equivalently either the kernel of the cokernel or the cokernel of the kernel; either way, to compute it you have to compute a cokernel, and that cokernel can be computed as a presheaf cokernel and then sheafified.

Answer 2

In any Abelian category, the image of a morphism is the kernel of the cokernel.

We can use the relationship bewteen sheaves and presheaves to compute the image, however: if $i,a$ are the inclusion and sheafification functors, then if $f$ is any morphism:

$$ \begin{align}\mathop{\mathrm{im}} f &= \ker \mathop{\mathrm{coker}} f \\&\cong \ker ai(\mathop{\mathrm{coker}} f) \\&\cong a(\ker \mathop{\mathrm{coker}}if) \\&\cong a(\mathop{\mathrm{im}} (if)) \end{align}$$

using the fact that $ai \cong 1$, $a$ is left exact, and $i$ is right exact. Thus, the image of a sheaf morphism is indeed the sheafification of its image when viewed as a presheaf morphism.

Note, for example, $\mathop{\mathrm{im}} (f)$ means the image as a sheaf morphism, and $\mathop{\mathrm{im}} (if)$ means the image as a presheaf morphism. Allow me to emphasize that the result of this calculation literally is the image of a morphism in the category of sheaves of abelian groups.