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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?

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    $\begingroup$ The algebra $A$ has global dimension two, so every submodule of a free module has projective dimension at most one. $\endgroup$ – Andrew Hubery Sep 13 '19 at 12:43
  • $\begingroup$ @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? $\endgroup$ – Bubaya Sep 16 '19 at 10:04
  • $\begingroup$ @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. $\endgroup$ – Bubaya Sep 16 '19 at 16:25

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