Lemma 8.1.8 (Perfect modules and compact objects, [Lurie (2017), Proposition 7.2.4.2]). Let \(R\) be an associative ring spectrum.
- (1)
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The regular left \(R\)-module \(R\) is compact.
- (2)
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The \(\infty \)-category \(\LMod _R\) is compactly generated by \(R\).
- (3)
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A left \(R\)-module is perfect if and only if it is compact.
Proof. The functor \(\Hom _{\LMod _R}(R,-)\) is naturally isomorphic to the composite \[ \LMod _R \xrightarrow {\fgt } \Sp \xrightarrow {\Omega ^\infty } \An , \] which preserves filtered colimits. Thus \(R\) is compact. Its shifts jointly detect isomorphisms, since \[ \pi _n(M) \cong \pi _0\Hom _{\LMod _R}(R[n],M). \] Hence \(R\) is a compact generator of \(\LMod _R\). The characterization of compact objects in a compactly generated stable \(\infty \)-category from Proposition 22.3.4 now identifies the compact objects with the thick subcategory generated by \(R\), which is \(\Perf (R)\). □
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