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Let $X$ be scheme. Consider an extension $$ 0 \to \mathcal{O}_X^n \to E \to \mathcal{O}_X^m \to 0 $$ with $E$ a locally free $\mathcal{O}_X$-module.

My question is : does $E$ have to be free ? I think it's false but I couldn't find a counter example.

This is related to the following question : is a locally free $\mathcal{O}_X$-module projective in the category of quasi-coherent sheaves over $X$ ?

This is true (I think) if $X$ is affine since in this case a locally free sheaf is just a locally free module over a ring which is projective. But I don't think it's true over a general scheme although I don't know a counter example.

The link with my question is that if locally free sheaves are projective it would mean that the above exact sequence has a section and thus $E$ is free.

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  • $\begingroup$ Since $\mathrm{Ext^1}(\mathcal{O}, \mathcal{O}) = 0$ any such exact sequence splits and $E = \mathcal{O}^{n+m}$ $\endgroup$ – qwenty May 19 '18 at 19:47
  • $\begingroup$ Thanks ! Is it really true though that $Ext^1(\mathcal{O},\mathcal{O}) = 0$ if you are not affine ? $\endgroup$ – ayokointhesand May 19 '18 at 19:48
  • $\begingroup$ Oops it's not true for example for elliptic curves.. $\endgroup$ – qwenty May 19 '18 at 19:51
  • $\begingroup$ Ah so I guess this will provide a counter example ? $\endgroup$ – ayokointhesand May 19 '18 at 19:52
  • $\begingroup$ I'am not sure that in general it is not possible that $E = \mathcal{O}^2$ for non-trivial extension. But that's true for an elliptic curve: maths.tcd.ie/pub/ims/bull60/R6005.pdf example 4.4 but may be there is much more easier proof $\endgroup$ – qwenty May 19 '18 at 20:02
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No. As in the comments, the issue is precisely that $\text{Ext}^1(\mathcal{O}, \mathcal{O}) \cong H^1(X, \mathcal{O})$ need not vanish. For example, if $X$ is a smooth projective curve of genus $g$ then we have

$$\dim H^1(X, \mathcal{O}) = g$$

and so we have nontrivial extensions of $\mathcal{O}$ by $\mathcal{O}$ whenever $g \ge 1$, e.g. if $X$ is an elliptic curve as also mentioned in the comments.

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