Hi all

I thought that the following was true but was unable to think of a proof and after browsing the internet now I am not so sure. My understanding of graded rings and modules is not great so I'm thinking this may be obvious to some people but not me.

In the category of coherent sheaves on a projective variety $X$, is a locally free sheaf a projective (in the categorical sense)? I'm mainly interested when $X$ is a nonsingular curve but would be curious as to the general answer.

Thanks.