I'm interested in the $\infty$-categorical version of this question. More precisely, let $R$ be a ring spectrum and consider the presentable stable $\infty$-category $R$-Mod of left $R$-modules. Let $\operatorname{Perf} R$ denote the subcategory of $R$-Mod consisting of compact objects. Is it true that $\operatorname{Perf} R$ are generated by $R$ under finite colimits and then retractions?
From my superficial understanding of $\infty$-categories, Ahkil's answer should work line by line. In short, "$R$ is a compact object" plus "compact objects are preserved under finite colimit and restriction" implies one direction. The other direction is implied by the fact "any object can be written as a filtered colimit of compact objects" by Grothendieck construction plus retraction again. Are these claims correct?
There's probably a decent chance that he answers can be found in Lurie's Higher Topos theory and Higher Algebra. However, I would be appreciate if someone can point out which sections I should look for or if there's some other thinner reference. I would be more appreciate if someone can give a "model free" argument since the models sometimes confuse me about what is actually happen in the proof.