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 $\beta$.