Consider a topological space $X$ and a direct limit of sheaves and morphisms $\{ \cal{F}_i, f_{ij}\}$. Define the direct limit presheaf by $U \to \varinjlim \cal{F}_i $. In general this is just a presheaf. If $X$ is noetherian it is a sheaf, and this is the result I want to prove.
My attemp:
I tried to prove the first axiom of sheaves (I suspect that the second will need similar arguments). Let be an open set $U$ and an open cover $\{ V_i \}_i$ of $U$. Since $X$ is noetherian it can be can supposed that the cover is finite. Let $s \in \varinjlim \cal{F_i(U)}$ such that $s|_{V_i}=0 \quad \forall i$. The goal is to show that then $s=0$. Being $s \in \varinjlim \cal{F_i(U)}$, we know that there exists some $j / \quad s=f_j(U)(t_j)$ for some $t_j \in \cal{F}_j(U)$ ($f_i(U)$ are the morphisms of the direct limit of the abealian groups $\cal{F}_i(U)$ ). I can consider the germs ${t_j}_{V_i}$. If I could show that $s|_{V_i}=0 \Rightarrow {t_j}_{V_i}=0$ then the problem would be done, because the $\cal{F}_i$ are sheaves, in particular they verify the first axiom of sheaves, so $t_j=0$ and then $s=f_j(t_j)=0$.