A collection of sheaf morphisms $(\phi^k \colon \mathcal{F}^k \to \mathcal{F}^{k+1})_{k \in \mathbb{Z}}$ is called
- exact if $\operatorname{im} \phi^k = \ker \phi^{k+1}$, and
- a complex if $\phi^{k+1} \circ \phi^k = 0$.
My question is: does an exact sequence of sheaf morphism always define a complex of sheaves? This implication holds in other categories (e.g. for $R$-modules). But I guess, something could go wrong with global sections.
There is nothing special about sheaves. The statement holds in every abelian category. Clearly, $f \circ g = 0$ is equivalent to $\mathrm{im}(g) \subseteq \mathrm{ker}(f)$, so $\mathrm{im}(g) = \mathrm{ker}(f)$ is a stronger statement.