Is the derived category of coherent sheaves idempotent closed?

Question

I am wondering whether the bounded derived category $D^b Coh(X)$ is idempotent complete, i.e. whether every idempotent morphism splits. This is true for $Perf(X)$ since perfect complexes are the compact objects in the category of quasi-coherent sheaves and compactness is preserved under taking retracts.

For example, if X is smooth, then $D^bCoh$ and $Perf$ agree and so $D^b Coh$ is also idempotent closed.

Question : Are there any general conditions on a scheme X that imply that $D^b Coh$ is also idempotent closed?

Answer 1

I think it's true if $X$ is Noetherian.

By Prop. 36.11.2 of the Stacks Project, the natural functor $$D^b\left(\text{Coh}(\mathcal{O}_X)\right)\to D^b_{\text{Coh}}(\mathcal{O}_X)$$ is then an equivalence, and $D^b_{\text{Coh}}(\mathcal{O}_X)$ is idempotent closed, since the category $\text{Coh}(\mathcal{O}_X)$ of coherent sheaves is abelian and therefore idempotent closed.