I have recently come across the category of algebraic vector bundles over a scheme $X$. In short, it is the category of locally free $\mathcal{O}_X$-modules of finite rank. An additive category $\mathcal{A}$ is idempotent complete if for any idempotent endomorphism $p: A \rightarrow A$, $ker(p) \in \mathcal{A}$. So the category of finitely generated projective module $P(R)$ is idempotent complete. Since $ker(p) \oplus im(p) \cong P$; for $p : P \rightarrow P$ in $P(R).$
Then should that make the category of algebraic vector bundles $VB(X)$ over any scheme $X$ an idempotent complete category? If $X$ is quasi-compact will it be idempotent complete?
I would be grateful for any hints or suggestions.