A short note about short exact sequences.

by ayushkhaitan3437

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 $\beta$.