Let $A = \mathbb Z$ and $B = \mathbb Q.$ Then Ext$(A, B)$ gives the set of all equivalent extensions of $A$ by $B.$ I have few questions.
- Is this sequence $0\rightarrow \mathbb Z \stackrel{\alpha}{\longrightarrow} \mathbb Z \oplus \mathbb Q \stackrel{\beta} {\longrightarrow} \mathbb Q \rightarrow 0$ split?
My answer is yes if $\beta$ is the canonical projection on second components. Since we can get splitting map $\delta: \mathbb Q \rightarrow \mathbb Z \oplus \mathbb Q $ defined by $\delta(a) = (0,a).$ This satisfies $\beta\circ\delta$ identity on $\mathbb Q.$
Suppose $\beta$ is not a projection. Is the sequence in (1) split or not? I do not know exactly if there exists $\beta$ other than projection so that the sequence (1) is short exact.
Suppose the above sequence in (1) is split and suppose we have another short exact sequence $0\rightarrow \mathbb Z \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} \mathbb Q \rightarrow 0,$ which is equivalent with the sequence in $(1).$ Can we say $G\cong \mathbb Z\oplus\mathbb Q$? Thanks