Suppose $A$ is an algebra. It's a theorem that the Karoubi envelope (aka idempotent completion) of the category of free $A$-modules is equivalent to the category of projective $A$-modules.
Is there a proof of this theorem? I've come up empty handed. Thanks.
Edit: Is this the idea? If $P$ is a projective module, it is a direct summand of a free module $F$, so there is a surjection $\pi\colon F\to P$. Being projective, there is a section $s\colon P\to F$ such that $\pi s=1_P$. It follows that $(s\pi)^2=s\pi s\pi=s\pi$ is idempotent, and $s\pi(F)=s(P)\cong P$.
So the equivalence is given by the functor $(F,e)\mapsto e(F)$. This functor is essentially surjective by the above. Also, $e(F)$ is projective since $F\cong e(F)\oplus (1-e)(F)$, so it's a direct summand of $F$.