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I have a very easy question, which I couldn't get in the literature. Please forgive me if it is so easy!!! Question: Is a stable vector bundle over a curve $C$ is projective (as a $\mathcal O_C$-module) ? is it injective? Could you give me a reference?

One may ask a further question: in which cases a locally free sheaf is projective (resp. injective)?

Thanks

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A vector bundle $E\neq 0$ on $C$ is neither projective nor injective. Let $L$ be a line bundle; then $\mathrm{Ext}_C^1(E,L)\cong H^1(C,E^*\otimes L)$, which is nonzero by Serre duality when $\deg(L)\ll 0$. Similarly $\mathrm{Ext}_C^1(L,E)\cong H^1(C,E\otimes L^{-1})$ is nonzero for $\deg(L)\gg 0$.

An argument along the same lines gives the same property for any nonzero coherent sheaf on a projective variety of dimension $\geq 1$.

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  • $\begingroup$ I'm confused. You should take the Ext sheaves, not the Ext groups, or I'm mistaken? As far as I remember, a finitely generated module is locally free if and only it is projective. So a coherent, locally free $\mathcal{O}_X$-module should be projective. I'm missing something? $\endgroup$
    – Francesco Polizzi
    Commented Feb 9, 2015 at 15:33
  • $\begingroup$ The Ext sheaves correspond to a property we may call "internal projectivity" (exactness of ext-sheaves in the first variable), which is indeed a property that holds for vector bundles on schemes. $\endgroup$
    – Leo Alonso
    Commented Feb 9, 2015 at 15:39
  • $\begingroup$ @Francesco : just take the standard definition of injective or projective (the usual lifting property) in the category of $\mathscr{O}$-modules; this is equivalent to the vanishing of global Ext. $\endgroup$
    – abx
    Commented Feb 9, 2015 at 16:18
  • $\begingroup$ @abx: right, thanks. I was confused because I was thinking about the Serre-Swan theorem, but it actually holds for affine varieties. $\endgroup$
    – Francesco Polizzi
    Commented Feb 9, 2015 at 16:32

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