If $\{X\_i, i\in I\}$ is a directed system of abelian groups such that we have
$$\varinjlim\_{i\in I}X\_i = 0$$
is it true that for every $i$ and large enough $j\ge i$ the transition map $f\_{i,j} : X\_i\to X\_j$ is the zero map?


If the transition maps satisfy this, then of course the direct limit is zero. If the $X\_i$ are finitely generated abelian groups then the converse is also clearly true. The question is for general abelian groups $X\_i$ (I'm happy to assume the $X\_i$ are all countably generated and that $I=\mathbb{N}$).



Transition maps in trivial direct limit
No. Take each $G\_n$ free abelian with basis $e\_k$ with $k\in \mathbb N$. Here $n$ runs over the natural number. Let the map from $G\_n$ to $G\_{n+1}$ kill $e\_0$ and send $e\_k$ to $e\_{k-1}$ for $k>0$. Then no map is zero but each element in each group eventually maps to $0$ far enough down. So the direct limit is 0.


