# Non-trivial $2$-extension of graded $k[x,y]$-modules

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?

• The algebra $A$ has global dimension two, so every submodule of a free module has projective dimension at most one. – Andrew Hubery Sep 13 '19 at 12:43
• @AndrewHubery I'm not so sure whether I understand how this answers my second question. I suppose there cannot be such a non-split sequence. If I write down a projective resolution $0\to Q\to P\to N\to 0$, I get maps $P\to M$ and $Q\to L$. How can I use these to see that the 2-extension must split? – Bubaya Sep 16 '19 at 10:04
• @AndrewHubery I see; if $N$ has projective dimension $1$, then $\operatorname{Ext}^2(N, M)=0$, so there cannot be a non-trivial $2$-extension. Feel free to post this as an answer. – Bubaya Sep 16 '19 at 16:25