Generators for nilpotent modules

Lemma 8.4.15 ([Lurie (2017), Lemma 7.2.3.13]). Let \(R\) be an associative ring spectrum and let \(S \subseteq \pi _*(R)\) be a multiplicative subset satisfying the left Ore condition. Then every \(S\)-nilpotent left \(R\)-module \(M\) can be written as the colimit of a sequence of left \(R\)-modules \[ 0 = M_0 \to M_1 \to M_2 \to \cdots \] in which the fiber of each map \(M_i \to M_{i+1}\) is a coproduct of modules of the form \((R/Rs_{\alpha })[n_{\alpha }]\) with \(s_{\alpha } \in S\) and \(n_{\alpha } \in \Z \). In particular, \(\LMod _R^{S\dnil }\) is generated under colimits by the set of objects \[ \{\, (R/Rs)[n] \mid s \in S, \, n \in \Z \,\}. \]