Corollary 20.2.5 (Perfect complexes). The object \(R[0]\) is a compact generator of \(\D (R)\). If \[ \Perf (R):=\D (R)^{\omega } \] denotes the full subcategory of compact objects, then \(\Perf (R)\) is the smallest thick subcategory of \(\D (R)\) containing \(R[0]\). Its objects are precisely the retracts of bounded complexes of finitely generated free \(R\)-modules, and there is an equivalence of symmetric monoidal \(\infty \)-categories \[ \D (R)\simeq \Ind (\Perf (R)). \] Here the symmetric monoidal structure on the right is the one supplied by Theorem 22.5.4.
Proof. Since \(R\) is projective as an \(R\)-module, Corollary 6.4.5 gives \[ \pi _n\hom _{\D (R)}(R[0],X)\cong H_n(X) \] for every \(X\in \D (R)\). The canonical t-structure on \(\D (R)\) is compatible with filtered colimits by [Lurie (2017), Proposition 1.3.5.21], so each homology functor preserves filtered colimits. Since homotopy groups of spectra preserve filtered colimits by Lemma 4.4.28, the displayed formula shows that \(R[0]\) is compact. Its shifts jointly detect zero objects, since a complex is zero in \(\D (R)\) precisely when all its homology groups vanish. It follows from Proposition 22.3.4 that \(R[0]\) is a compact generator.
By Proposition 22.3.4, the compact objects form the smallest thick subcategory containing \(R[0]\); in particular, \(\Perf (R)\) is small. Bounded complexes of finitely generated free modules are obtained from shifts of \(R[0]\) by finite cofiber sequences, using their finite filtrations by degree. Conversely, their retracts form a thick subcategory. This gives the asserted explicit description of \(\Perf (R)\).
The underlying equivalence \(\D (R)\simeq \Ind (\Perf (R))\) now follows from Proposition 22.3.7. The derived tensor product restricts to \(\Perf (R)\): this follows from its exactness separately and the description of \(\Perf (R)\) as the thick subcategory generated by the unit. The uniqueness assertion in Theorem 22.5.4 upgrades the equivalence to a symmetric monoidal equivalence. □
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