We say a category $\mathcal{N}$ is exact if it is additive and is endowed with an exact structure. In brief, it is an additive category with a predetermined class of short exact sequences in its ambient abelian category $\mathcal{A}$. It is idempotent complete if for every idempotent endomorphism $p$ ($p^2 = p$) in $\mathcal{N}$; $ker(p) \in \mathcal{N}$ (up to isomorphism).
As an example, we can take $P(R)$ the category of finitely generated projective $R$-modules ($R$ is a commutative ring with unity). Since for any idempotent endomorphism $p : P \rightarrow P$, the exact sequence $0\rightarrow ker(p) \rightarrow P \rightarrow Im(p) \rightarrow 0$ splits.
My question is if I take the category $P(X)$; which we define as the category of locally free $\mathcal{O}_X$-modules of finite rank, where $X$ is a quasi-compact scheme. Then will $P(X)$ be idempotent complete? I understand that if I take $X = Spec(R)$ the category $P(X)$ coincides with $P(R)$, but is that sufficient to conclude that $P(X)$ is idempotent complete?
