# Is the derived category of perfect complexes idempotent complete?

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.

## 1 Answer

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.

• The result is clear with your reasoning. Nevertheless, do you know some reference for this result? Mar 27, 2015 at 3:56
• I'd go look somewhere in Thomason-Trobaugh. I don't know though. Mar 27, 2015 at 3:59
• One good reference is Proposition 2.1.1 here: arxiv.org/abs/math/0204218.
– AAK
Mar 27, 2015 at 5:51
• In the language of infinity-categories, the relevant result is Proposition 5.5.7.8 in Higher Topos Theory. See also Section 2.4 of arxiv.org/pdf/1001.2282.pdf, especially the text right before Lemma 2.20, for a discussion adapted to the stable setting.
– AAK
Mar 27, 2015 at 6:02
• The result is true even without the assumption that perfect objects are compact (this is not true for any scheme). A reference is stacks.math.columbia.edu/tag/08GA Oct 23, 2018 at 8:12