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Let $X$ be a locally ringed space, in the category of $\mathscr O_X$- modules, is a locally free sheaf of finite rank a projective object?

Born to be proud
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    What is true (almost trivially) is that it's "internally projective": if $\mathscr{E}$ is locally free and $\mathscr{F} \to \mathscr{F}''$ is epimorphic, then $\mathscr{Hom}(\mathscr{E}, \mathscr{F}) \to \mathscr{Hom}(\mathscr{E}, \mathscr{F}''$ is also epimorphic. – Daniel Schepler May 18 '23 at 00:28

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Let $X = \mathbb{P}^1$. Following this question, it is easy to show that the structure sheaf $\mathscr{O}$ is not projective. Indeed, Serre duality implies $\operatorname{Ext}^1(\mathscr{O}, \omega_{\mathbb{P}^1}) \cong H^0(\mathbb{P}^1, \mathscr{O})^\vee \neq 0$.

You can avoid Serre duality with a longer argument given here.

Daniel
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