Dummit Foote 10.5.1(d) commutative diagram of exact sequences.

Question

I solved other problems, except (d): if $\beta$ is injective, $\alpha$ and $\gamma$ are surjective, then $\gamma$ is injective.

Unlike others, I don't know where to start.

Answer 1

As the comments mention, this exercise is false as stated. Here's a counterexample: let $A$ and $B$ be groups, with usual inclusion and projection homomorphisms $$\iota_A(a) = (a,1),$$ $$\iota_B(b) = (1,b)$$ and $$\pi_B(a,b) = b.$$

Then the following diagram meets the stated requirements, except $\pi_B$ is not injective.

$$\require{AMScd} \begin{CD} A @>{\iota_A}>> A\times B @>{\iota_B \circ \pi_B}>> A\times B\\ @V{\operatorname{Id}}VV @V{\operatorname{Id}}VV @V{\pi_B}VV\\ A @>{\iota_A}>> A \times B @>{\pi_B}>> B \end{CD}$$

This indeed exploits the fact that $\varphi$ is not required to be surjective.