Local modules for multiplicative subsets

Definition 8.4.3. Let \(S \subseteq \pi _*(R)\) be a multiplicative subset of homogeneous elements. An \(R\)-module \(N\) is \(S\)-local if multiplication by every \(s \in S\) is an isomorphism \[ s\colon N[\abs {s}] \iso N. \] Equivalently, multiplication by every \(s \in S\) is an isomorphism on the graded group \(\pi _*(N)\). We write \(\Mod _R^{\Loc (S)} \subseteq \Mod _R\) for the full subcategory of \(S\)-local modules.