I know that a finite sequence of module homomorphisms $f_i: A_i\to A_{i+1}$, is exact if and only if $$Image(f_i)= Kernel(f_{i+1})\quad i\in [0,\ldots ,n]$$
But if we have an exact sequence of the form $0\to A_0\to \ldots, A_{n+1}\to0$
Does that necessarily mean that the first half of $f_i$ are injective, while the second half is surjective?
In other words, does that necessarily mean $f_i$ is injective $\forall i\in[0,\ldots,\frac{n}{2}]$ and $f_i$ is surjective for $\forall i\in[\frac{n}{2},\ldots,n]$?
If so, then is this always the case with all exact sequences?
Anything you can say would be greatly appreciated! Thank you :)
No, for instance consider the following exact sequence of $A$-modules and homomorphisms $$ 0\rightarrow A\stackrel{f}\rightarrow A\times A\stackrel{g}\rightarrow A\times A \stackrel{h}\rightarrow A\times A\stackrel{k}\rightarrow A\rightarrow 0, $$ where $f(a)=(a,0)$, $g(a,b)=(0,b)$, $h(a,b)=(a,0)$ and $k(a,b)=b$.
(The sequence can be made arbitrarily long)