Hi.
I am trying to prove that the top row is an exact sequence. I have already proved exactness at ker $\beta$. Do I need to prove that $\bar g$ and $\bar h$ are injective and surjective respectively?
Note: The assumption is, the middle and bottom rows are exact. The top row was not part of the diagram originally. I added it to the diagram since that is what I need to work on.
Thanks in advance.
From what you’ve stated, if you’ve shown exactness at the top middle then the top sequence is exact by definition. Showing that $\tilde g$ is injective would be exactness at the top left if there was a $0$ object with a map into $ker\space\alpha$. And similarly for surjectivity at the top right. It is true however that $g$ injective would imply $\tilde g$ is.
By the way, providing context helps. It looks like you are trying to prove the Snake Lemma which states that your top row is exact assuming your starting diagram (2nd and 3rd rows) commutes and has each row exact. It actually concludes that there is a 6 term exact sequence (three cokernels to the right of your top row).