Determine if the following short exact sequence is split.

Question

Do the following short exact sequences split?

$$0\longrightarrow A\longrightarrow B\longrightarrow \mathbb{Z}^2 \longrightarrow 0$$

$$0\longrightarrow\mathbb{Z}\longrightarrow A\longrightarrow B\longrightarrow 0$$

This is a question on a Ph.D Topology exam. I know what it means to be a split short exact sequence. In order for the short exact sequence $0\longrightarrow A\longrightarrow B\longrightarrow C \longrightarrow 0$ split you need one of the following:

1. there exists map $B\longrightarrow A$ such that $A\longrightarrow B\longrightarrow A$ is the identity on $A$.

2. there exists map $C\longrightarrow B$ such that $C\longrightarrow B\longrightarrow C$ is the identity on $C$.

3. $B$ is isomorphic to the direct sum of $A$ and $C$.

I have tried to find examples and nonexamples online of split short exact sequences and I've tried to figure out how to answer the above question, but I am struggling hard. I've tried to use the fact that these sequences are exact so we have the fact that the $Im(f_i)=Ker(f_{i+1})$. If someone could please give me an explanation to this question I would be very grateful.

Thanks.

Answer 1

I assume the groups involved here are commutative. The first sequence splits because $Z^2$ is a free abelian group. if $f:B\rightarrow Z^2$, $f(u)=e_1,f(v)=e_2,$ where $e_1,e_2$ are generators of $Z^2$, write $g(e_1)=u, g(e_2)=v$.

Answer 2

I assume the sequences are of abelian groups. The first one splits since the right term is a free abelian group. The second one does not necessarily split, since one can take $A=\mathbb Q$ and $B$ the quotient. Since the middle term is torsion free, the sequence cannot split.

Answer 3

For the first short exact sequence, note that $\mathbb{Z}^2$ is a free $\mathbb{Z}$-module, it is thus projective and the short exact sequence thus splits. In fact, a characterizing property for projective modules is the following.

$M$ is a projective $R$-module if and only if any short exact sequence $$0\longrightarrow K\longrightarrow L\longrightarrow M\longrightarrow 0$$of $R$-modules splits.

The above proposition already implies the second short exact sequence does not necessarily split. An explicit example can be the following

$$0\longrightarrow \mathbb{Z}\longrightarrow \mathbb{Z}\longrightarrow \mathbb{Z}/2\mathbb{Z}\longrightarrow 0,$$ where the first map is the multiply by $2$ map while the second may is the natural projection(or modulo $2$ map). Note this short exact sequence does not split since $\mathbb{Z}$ is torsion free.

Hope the above helps.