Nilpotent-local decomposition

Proof sketch. By Lemma 8.4.15, the \(\infty \)-category \(\LMod _R^{S\dnil }\) is the localizing subcategory generated by the set of objects \(\{ (R/Rs)[n] \mid s \in S, n \in \Z \}\). It is therefore presentable by Proposition 22.2.7(2). Its inclusion into \(\LMod _R\) preserves colimits by Observation 8.4.10, so Theorem 22.2.5 gives a right adjoint \(G\). For any module \(M\), the counit of this adjunction gives a map \(G(M) \to M\). We define \(M' := G(M)\) and we let \(M''\) be the cofiber of this map. It remains to show that \(M''\) is \(S\)-local, for which we use criterion (3) from Proposition 8.4.16. Let \(N\) be an \(S\)-nilpotent left \(R\)-module. Applying \(\hom _{\LMod _R}(N,-)\) provides an exact sequence of spectra \[ \hom _{\LMod _R}(N,M') \to \hom _{\LMod _R}(N,M) \to \hom _{\LMod _R}(N,M''). \] Both functors in the adjunction are exact, so the adjunction is enriched in spectra. The first map is therefore an isomorphism, and it follows that \(\hom _{\LMod _R}(N,M'') = 0\), as desired. □