Property of short exact sequences.

Question

If $0 \to L \to M \to N \to 0$ is a short exact sequence of $A$-modules then the following statements are equivalent:

1. There is an isomorphism $M \cong L \oplus N$ under which $L \to M$ is given by $m \mapsto (m, 0)$ and $M \to N$ by $(m, n) \mapsto n$.
2. There exists a section of $M \to N$.

I understand that the condition "there is an isomorphism $M \cong L \oplus N$" is much weaker than the condition 1. above, so

I wonder if there are some simple $M, L$ and $N$ such that $M \cong L \oplus N$, but there is no section of $M \to N$.

Answer 1

That's a great question!

Usually counterexamples are already easy to find for $\mathbb{Z}$-modules (also known as abelian groups), but sadly, in this case, if you look only at finitely generated abelian groups, then any short exact sequence $$0 \to A' \to A \to A'' \to 0$$ with $A \cong A'\oplus A''$ is actually split. (It is possible to see it by revising the classification of finitely generated abelian groups and calculations of the Yoneda $\operatorname{Ext}^1_\mathbb{Z} (A'',A')$.)

So we need to look for counterexamples among abelian groups that are not finitely generated. Maybe the easiest is the following. Take an infinite direct sum of copies of $\mathbb{Z}/n\mathbb{Z}$ and consider the map

\begin{align*} p\colon \mathbb{Z} \oplus \bigoplus_{i \ge 0} \mathbb{Z}/n\mathbb{Z} & \twoheadrightarrow \bigoplus_{i \ge 0} \mathbb{Z}/n\mathbb{Z},\\ (x,y_0,y_1,y_2,\ldots) & \mapsto (x \mod{n}, y_0, y_1, y_2, \ldots). \end{align*}

This is not the projection, but this is a surjective homomorphism, and we have a legitimate short exact sequence $$0 \to \mathbb{Z} \xrightarrow{x \mapsto (n x, ~ 0,0,0,\ldots)} \mathbb{Z} \oplus \bigoplus_{i \ge 0} \mathbb{Z}/n\mathbb{Z} \xrightarrow{p} \bigoplus_{i \ge 0} \mathbb{Z}/n\mathbb{Z} \to 0$$

But you can see that $p$ doesn't have a section.

Answer 2

How about taking

$$ 0 \to \Bbb Z/2 \Bbb Z \to \Bbb Z/2 \Bbb Z $$

and

$$ \Bbb Z \xrightarrow{\cdot 2} \Bbb Z \to \Bbb Z / 2 \Bbb Z. $$

Both are short exact. But if you take the countable direct sum of both then the middle term is the direct sum of the both other terms, but there is no section.