If $0\rightarrow F\rightarrow G\rightarrow H \rightarrow 0$ is an extension of $\mathcal{O}$-modules with $F$ and $G$ locally free (each of constant finite rank, i.e. vector bundles), then is $H$ locally free? The same question can be asked with the roles of $F$, $G$, and $H$ interchanged, too.
In other words, is a quotient of a locally free sheaf by a subsheaf locally free?
Best Answer
No. A locally free sheaf of finite rank on a noetherian affine scheme is a finitely-generated projective module, but $$0 \longrightarrow 2 \mathbb{Z} \longrightarrow \mathbb{Z} \longrightarrow \mathbb{Z} / 2 \mathbb{Z} \longrightarrow 0$$ is an exact sequence of $\mathbb{Z}$-modules and $\mathbb{Z} / 2 \mathbb{Z}$ is not projective.