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



Stable vector bundle Projective or Injective?
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 $\geq 1$. 


