### A short note about short exact sequences.

We will discuss short exact sequences here. They are of the form $A\xrightarrow{f} B\xrightarrow{g} C$, where im $f$= ker $g$.

Example: $\Bbb{Z}\xrightarrow{f}\Bbb{Q}\xrightarrow{g}\Bbb{R}$, such that $f(a)=a$ and $g(b)=b+\Bbb{Z}$.

What is the point of such a construction? Is there any way to visualize this? Whatever the image of $f$, you want to completely obliterate it. Leave no trace of it. Or maybe you want to check how similar the elements of the co-domain are to the image.

What if you have a mapping of the sort $0\to A\xrightarrow{\alpha} B\xrightarrow{\beta} C\to 0$? Assume that $0$ maps only to $0$. Also, if $A,B$ and $C$ are modules, $0$ would be the additive identity. The whole of $C$ maps to $0$. Hence, the kernel is the whole of $C$. Which means the im $\beta$ is the whole of $C$; i.e. $\beta$ is surjective. Also, $\alpha$ is injective, as its kernel is precisely $0$.

What if we had $0\to S\xrightarrow{\alpha} A\xrightarrow{\beta} B\xrightarrow{\gamma} C\to 0$? Here $\gamma$ is surjective, and $\alpha$ is injective. Can’t say much about $\beta$.