ScalingStacks

Proof. First note that in characteristic zero, $BGL_{n}$ is clearly perfect: the compact and dualizable objects are both finite dimensional representations which generate.

If $G$ is an affine algebraic group, we can embed $G\hookrightarrow GL_{n}$ as a subgroup of $GL_{n}$ for some $n$. Thus we obtain a morphism $BG\to BGL_{n}$ with fiber $GL_{n}/G$. By a theorem of Chevalley [Ch], $GL_{n}/G$ is a quasi-projective variety, and so by Proposition 3.21, $BG$ itself is perfect.

Finally, for a quasi-projective derived scheme $X$ with a linear action of $G$, the morphism $X/G\to BG$ is quasi-projective, so applying Proposition 3.21 again, we conclude that $X/G$ is perfect. ∎