Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any idempotent $\alpha: X\rightarrow X$ has a splitting in $\mathcal{C}$, i.e. there exists an object $Y$ together with morphisms $i: Y\rightarrow X$ and $p: X\rightarrow Y$ such that $i\circ p=\alpha$ and $p\circ i=id\_Y$.


A famous result by in Bokstedt&Neeman in "Homotopy limits in triangulated categories" is
that if $\mathcal{C}$ is a triangulated category with (possibly infinite) direct sums, then it is idempotent complete. See Proposition 3.2 of that paper.


It is clear from their result that the derived category of complexes sheaves of $\mathcal{O}\_X$-modules $D(X)$ is idempotent complete. However, the derived category of perfect complexes, $D\_{\text{perf}}(X)$ does not allow infinite direct sum. $\textbf{My question}$ is: is $D\_{\text{perf}}(X)$ still idempotent complete?


By the way, Bokstedt and Neeman has proved a very similar result, which claims that the derived category of finite complexes of finitely generated projective modules of a ring is idempotent complete. This illustrate that the infinite direct sums condition is not always necessary.



Is the derived category of perfect complexes idempotent complete?
The derived category of perfect complexes is idempotent complete, because it is the sub category of compact objects in the derived category of quasi coherent sheaves (which is idempotent complete by the result you mention) and compact objects are stable under retracts.


