Nilpotent and locally nilpotent modules

Definition 8.4.9. Let \(R\) be an associative ring spectrum and let \(S \subseteq \pi _*(R)\) be a multiplicative subset of homogeneous elements.

(1)

A left \(R\)-module \(M\) is called \(S\)-nilpotent if for every element \(x \in \pi _n(M)\), there exists an element \(s \in S\) such that \(s \cdot x = 0\) in \(\pi _*(M)\).

(2)

A left \(R\)-module \(N\) is called \(S\)-local if, for every element \(s \in S\) of degree \(d\) and every \(k\in \Z \), left multiplication by \(s\) induces an isomorphism \[ s\cdot -\colon \pi _k(N)\iso \pi _{k+d}(N). \]

We denote by \(\LMod _R^{S\dnil }\) and \(\LMod _R^{\Loc (S)}\) the full subcategories of \(\LMod _R\) spanned by the \(S\)-nilpotent and \(S\)-local modules, respectively.