A short note about short exact sequences.

We will discuss short exact sequences here. They are of the form , where im = ker .

Example: , such that and .

What is the point of such a construction? Is there any way to visualize this? Whatever the image of , 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 ? Assume that maps only to . Also, if and are modules, would be the additive identity. The whole of maps to . Hence, the kernel is the whole of . Which means the im is the whole of ; i.e. is surjective. Also, is injective, as its kernel is precisely .

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 $\beta$.