Let
$$
A\_1\twoheadrightarrow
A\_2\twoheadrightarrow
A\_3\twoheadrightarrow
A\_4\twoheadrightarrow
\cdots
$$
be an inductive sequence of **countable** abelian groups, the connecting homomorphisms of which are surjective and **split**, that is, we have embeddings $A\_{n+1}\rightarrowtail A\_n$ such that the composition $A\_{n+1}\rightarrowtail A\_n\twoheadrightarrow A\_{n+1}$ is the identity for every $n$. This means that $A\_{n+1}$ is a direct summand of $A\_n$.


Let $\varinjlim A\_n$ denote the inductive limit of the system
$$
A\_1\twoheadrightarrow
A\_2\twoheadrightarrow
A\_3\twoheadrightarrow
A\_4\twoheadrightarrow
\cdots
$$
and let $\varprojlim A\_n$ denote the projective limit of the system
$$
A\_1\leftarrowtail
A\_2\leftarrowtail
A\_3\leftarrowtail
A\_4\leftarrowtail
\cdots.
$$
We get an induced map
$$
\varprojlim A\_n\to\varinjlim A\_n.
$$
As [Zhen Lin](https://math.stackexchange.com/users/5191/zhen-lin) has shown in [over here](https://math.stackexchange.com/a/254702/367), this map need not be surjective. Here is a weaker question:


**Question:**
If we have $\varinjlim A\_n=0$, then can we conclude that $\varprojlim A\_n=0$?


This would, of course, follow if the map $\varprojlim A\_n\to\varinjlim A\_n$ was always injective. Is there any reason to expect this?


[Earlier versions of this question were posted [here](https://math.stackexchange.com/questions/254649) and [here](https://math.stackexchange.com/questions/254822) on MSE.]



Inductive vs projective limit of sequence of split surjections
I think it's true that $\varprojlim A\_n\to\varinjlim A\_n$ is always injective.


We may as well assume that 


$A\_1\leftarrowtail
A\_2\leftarrowtail
A\_3\leftarrowtail
A\_4\leftarrowtail
\cdots$


is a sequence of inclusions of nested subgroups, so $\varprojlim A\_n$ is just the intersection. An element of the kernel of $\varprojlim A\_n\to\varinjlim A\_n$ is just an element $a$ of $\bigcap A\_n$ that is in the kernel of the map $A\_1\twoheadrightarrow A\_k$ for some $k$. But this implies $a=0$ since this map is a splitting of the inclusion $A\_k\rightarrowtail A\_1$.


This doesn't use countability.


