Let $A=k[x,y]$ as bigraded $k$-algebra. I am looking for a non-trivial exact sequence $0\to K\to L\to M\to N\to 0$ of finitely generated $A$-modules; i.e., an exact sequence that does not split neither at $L$ nor at $M$. I struggle to find an example for that, although I think that such a sequence must exists.
Plus, I want that $K$ is a quotient and $N$ is a submodule of a free module. Is this even possible? If not, why?